Robust and Optimal Control
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The document is a book titled "Robust and Optimal Control" that provides an overview of key concepts in robust and optimal control theory. It covers topics such as linear algebra, linear dynamical systems, performance specifications, stability and performance of feedback systems, performance limitations, model uncertainty and robustness, and linear quadratic optimization. The book is intended to serve as both a textbook and reference for control engineers and mathematicians working in robust and optimal control.
![Preface
This book is inspired by the recent development in the robust and H1 control theory,
particularly the state-space H1 control theory developed in the paper by Doyle, Glover,
Khargonekar, and Francis 1989] (known as DGKF paper). We give a fairly comprehensive and step-by-step treatment of the state-space H1 control theory in the style
of DGKF. We also treat the robust control problems with unstructured and structured
uncertainties. The linear fractional transformation (LFT) and the structured singular
value (known as ) are introduced as the uni ed tools for robust stability and performance analysis and synthesis. Chapter 1 contains a more detailed chapter-by-chapter
review of the topics and results presented in this book.
We would like to thank Professor Bruce A. Francis at University of Toronto for his
helpful comments and suggestions on early versions of the manuscript. As a matter
of fact, this manuscript was inspired by his lectures given at Caltech in 1987 and his
masterpiece { A Course in H1 Control Theory. We are grateful to Professor Andre Tits
at University of Maryland who has made numerous helpful comments and suggestions
that have greatly improved the quality of the manuscript. Professor Jakob Stoustrup,
Professor Hans Henrik Niemann, and their students at The Technical University of
Denmark have read various versions of this manuscript and have made many helpful
comments and suggestions. We are grateful to their help. Special thanks go to Professor
Andrew Packard at University of California-Berkeley for his help during the preparation
of the early versions of this manuscript. We are also grateful to Professor Jie Chen at
University of California-Riverside for providing material used in Chapter 6. We would
also like to thank Professor Kang-Zhi Liu at Chiba University (Japan) and Professor
Tongwen Chen at University of Calgary for their valuable comments and suggestions.
In addition, we would like to thank G. Balas, C. Beck, D. S. Bernstein, G. Gu, W.
Lu, J. Morris, M. Newlin, L. Qiu, H. P. Rotstein and many other people for their
comments and suggestions. The rst author is especially grateful to Professor Pramod
P. Khargonekar at The University of Michigan for introducing him to robust and H1
control and to Professor Tryphon Georgiou at University of Minnesota for encouraging
him to complete this work.
Kemin Zhou
xiii](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-10-320.jpg)
![NOTATION AND SYMBOLS
In
aij ]
xvii
n n identity matrix
a matrix with aij as its i-th row and j -th column element
diag(a1 : : : an ) an n n diagonal matrix with ai as its i-th diagonal element
AT
transpose
A
adjoint operator of A or complex conjugate transpose of A
A;1
inverse of A
A+
pseudo inverse of A
A;
shorthand for (A;1 )
det(A)
determinant of A
Trace(A)
trace of A
(A)
eigenvalue of A
(A)
spectral radius of A
(A)
the set of spectrum of A
(A)
largest singular value of A
(A)
smallest singular value of A
i-th singular value of A
i (A)
(A)
condition number of A
kAk
spectral norm of A: kAk = (A)
Im(A), R(A)
image (or range) space of A
Ker(A), N(A)
kernel (or null) space of A
X; (A)
stable invariant subspace of A
X+ (A)
antistable invariant subspace of A
Ric(H )
g f
hi
x?y
D?
S?
L2 (;1 1)
L2 0 1)
L2 (;1 0]
L2+
L2;
l2+
l2;
L2 (j R)
L2 (@ D )
H2 (j R)
the stabilizing solution of an ARE
convolution of g and f
Kronecker product
direct sum or Kronecker sum
angle
inner product
orthogonal, hx yi = 0
orthogonal complement of D, i.e., D D?
D
or D is unitary
?
?
orthogonal complement of subspace S , e.g., H2
time domain Lebesgue space
subspace of L2 (;1 1)
subspace of L2 (;1 1)
shorthand for L2 0 1)
shorthand for L2 (;1 0]
shorthand for l2 0 1)
shorthand for l2 (;1 0)
square integrable functions on C 0 including at 1
square integrable functions on @ D
subspace of L2 (j R) with analytic extension to the rhp](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-14-320.jpg)
![NOTATION AND SYMBOLS
xviii
H2 (@ D )
?
H2 (j R)
?
H2 (@ D )
L 1 ( j R)
L1 (@ D )
H1 (j R)
H1 (@ D )
;
H1 (j R)
; (@ D )
H1
subspace of L2 (@ D ) with analytic extension to the inside of @ D
subspace of L2 (j R) with analytic extension to the lhp
subspace of L2 (@ D ) with analytic extension to the outside of @ D
functions bounded on Re(s) = 0 including at 1
functions bounded on @ D
the set of L1 (j R) functions analytic in Re(s) > 0
the set of L1 (@ D ) functions analytic in jz j < 1
the set of L1 (j R) functions analytic in Re(s) < 0
the set of L1 (@ D ) functions analytic in jz j > 1
pre x B or B closed unit ball, e.g. BH1 and B
pre x Bo
open unit ball
pre x R
real rational, e.g., RH1 and RH2 , etc
R s]
Rp (s)
polynomial ring
rational proper transfer matrices
G (s)
G (z )
A B
C D
shorthand for GT (;s) (continuous time)
shorthand for GT (z ;1 ) (discrete time)
shorthand for state space realization C (sI ; A);1 B + D
or C (zI ; A);1 B + D
F` (M Q)
F u ( M Q)
S (M N )
lower LFT
upper LFT
star product](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-15-320.jpg)
![1
Introduction
1.1 Historical Perspective
This book gives a comprehensive treatment of optimal H2 and H1 control theory and
an introduction to the more general subject of robust control. Since the central subject
of this book is state-space H1 optimal control, in contrast to the approach adopted
in the famous book by Francis 1987]: A Course in H1 Control Theory, it may be
helpful to provide some historical perspective of the state-space H1 control theory to
be presented in this book. This section is not intended as a review of the literature
in H1 theory or robust control, but rather only an attempt to outline some of the
work that most closely touches on our approach to state-space H1 . Hopefully our lack
of written historical material will be somewhat made up for by the pictorial history of
control shown in Figure 1.1. Here we see how the practical but classical methods yielded
to the more sophisticated modern theory. Robust control sought to blend the best of
both worlds. The strange creature that resulted is the main topic of this book.
The H1 optimal control theory was originally formulated by Zames 1981] in an
input-output setting. Most solution techniques available at that time involved analytic
functions (Nevanlinna-Pick interpolation) or operator-theoretic methods Sarason, 1967
Adamjan et al., 1978 Ball and Helton, 1983]. Indeed, H1 theory seemed to many to
signal the beginning of the end for the state-space methods which had dominated control
for the previous 20 years. Unfortunately, the standard frequency-domain approaches to
H1 started running into signi cant obstacles in dealing with multi-input multi-output
(MIMO) systems, both mathematically and computationally, much as the H2 (or LQG)
theory of the 1950's had.
1](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-18-320.jpg)
![INTRODUCTION
2
Figure 1.1: A picture history of control
Not surprisingly, the rst solution to a general rational MIMO H1 optimal control
problem, presented in Doyle 1984], relied heavily on state-space methods, although
more as a computational tool than in any essential way. The steps in this solution were as
follows: parameterize all internally-stabilizing controllers via Youla et al., 1976] obtain
realizations of the closed-loop transfer matrix convert the resulting model-matching
problem into the equivalent 2 2-block general distance or best approximation problem
involving mixed Hankel-Toeplitz operators reduce to the Nehari problem (Hankel only)
solve the Nehari problem by the procedure of Glover 1984]. Both Francis, 1987] and
Francis and Doyle, 1987] give expositions of this approach, which will be referred to as
the 1984" approach.
In a mathematical sense, the 1984 procedure solved" the general rational H1 optimal control problem and much of the subsequent work in H1 control theory focused
on the 2 2-block problems, either in the model-matching or general distance forms.
Unfortunately, the associated complexity of computation was substantial, involving several Riccati equations of increasing dimension, and formulae for the resulting controllers
tended to be very complicated and have high state dimension. Encouragement came](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-19-320.jpg)
![1.1. Historical Perspective
3
from Limebeer and Hung 1987] and Limebeer and Halikias 1988] who showed, for
problems transformable to 2 1-block problems, that a subsequent minimal realization
of the controller has state dimension no greater than that of the generalized plant G.
This suggested the likely existence of similarly low dimension optimal controllers in the
general 2 2 case.
Additional progress on the 2 2-block problems came from Ball and Cohen 1987],
who gave a state-space solution involving 3 Riccati equations. Jonckheere and Juang
1987] showed a connection between the 2 1-block problem and previous work by
Jonckheere and Silverman 1978] on linear-quadratic control. Foias and Tannenbaum
1988] developed an interesting class of operators called skew Toeplitz to study the
2 2-block problem. Other approaches have been derived by Hung 1989] using an
interpolation theory approach, Kwakernaak 1986] using a polynomial approach, and
Kimura 1988] using a method based on conjugation.
The simple state space H1 controller formulae to be presented in this book were rst
derived in Glover and Doyle 1988] with the 1984 approach, but using a new 2 2-block
solution, together with a cumbersome back substitution. The very simplicity of the new
formulae and their similarity with the H2 ones suggested a more direct approach.
Independent encouragement for a simpler approach to the H1 problem came from
papers by Petersen 1987], Khargonekar, Petersen, and Zhou 1990], Zhou and Khargonekar 1988], and Khargonekar, Petersen, and Rotea 1988]. They showed that for
the state-feedback H1 problem one can choose a constant gain as a (sub)optimal controller. In addition, a formula for the state-feedback gain matrix was given in terms
of an algebraic Riccati equation. Also, these papers established connections between
H1 -optimal control, quadratic stabilization, and linear-quadratic di erential games.
The landmark breakthrough came in the DGKF paper (Doyle, Glover, Khargonekar,
and Francis 1989]). In addition to providing controller formulae that are simple and
expressed in terms of plant data as in Glover and Doyle 1988], the methods in that
paper are a fundamental departure from the 1984 approach. In particular, the Youla
parameterization and the resulting 2 2-block model-matching problem of the 1984
solution are avoided entirely replaced by a more purely state-space approach involving
observer-based compensators, a pair of 2 1 block problems, and a separation argument.
The operator theory still plays a central role (as does Redhe er's work Redhe er, 1960]
on linear fractional transformations), but its use is more straightforward. The key
to this was a return to simple and familiar state-space tools, in the style of Willems
1971], such as completing the square, and the connection between frequency domain
inequalities (e.g kGk1 < 1), Riccati equations, and spectral factorizations. This book
in some sense can be regarded as an expansion of the DGKF paper.
The state-space theory of H1 can be carried much further, by generalizing timeinvariant to time-varying, in nite horizon to nite horizon, and nite dimensional to
in nite dimensional. A ourish of activity has begun on these problems since the publication of the DGKF paper and numerous results have been published in the literature,
not surprising, many results in DGKF paper generalize mutatis mutandis, to these cases,
which are beyond the scope of this book.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-20-320.jpg)
![2.1. Linear Subspaces
19
to see that rank(A) = rank(AT ) = rank(PA) if T and P are nonsingular matrices with
appropriate dimensions.
A square matrix U 2 F n n whose columns form an orthonormal basis for Fn is called
an unitary matrix (or orthogonal matrix if F = R), and it satis es U U = I = UU .
The following lemma is useful.
Lemma 2.1 Let D = d1 : : : dk 2 Fn k (n > k) be such that D D = I , so
di i = 1 2 : : : k are orthonormal. Then there exists a matrix D? 2 Fn (n;k) such that
D D? is a unitary matrix. Furthermore, the columns of D?, di i = k + 1 : : : n,
form an orthonormal completion of fd1 d2 : : : dk g.
The following results are standard:
Lemma 2.2 Consider the linear equation
where A 2 Fn l and
B
AX = B
2 Fn m are given matrices. Then the following statements are
equivalent:
(i) there exists a solution X 2 Fl m .
(ii) the columns of B 2 ImA.
(iii) rank A B =rank(A).
(iv) Ker(A ) Ker(B ).
Furthermore, the solution, if it exists, is unique if and only if A has full column rank.
The following lemma concerns the rank of the product of two matrices.
Lemma 2.3 (Sylvester's inequality) Let A 2 Fm n and B 2 Fn k . Then
rank (A) + rank(B ) ; n rank(AB ) minfrank (A) rank(B )g:
For simplicity, a matrix M with mij as its i-th row and j -th column's element will
sometimes be denoted as M = mij ] in this book. We will mostly use I as above to
denote an identity matrix with compatible dimensions, but from time to time, we will
use In to emphasis that it is an n n identity matrix.
Now let A = aij ] 2 C n n , then the trace of A is de ned as
Trace(A) :=
n
X
i=1
aii :
Trace has the following properties:
Trace( A) = Trace(A) 8 2 C A 2 C n n
Trace(A + B ) = Trace(A) + Trace(B ) 8A B 2 C n n
Trace(AB ) = Trace(BA) 8A 2 C n m B 2 C m n :](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-36-320.jpg)
![LINEAR ALGEBRA
24
2.4 Matrix Calculus
Let X = xij ] 2 C m n be a real or complex matrix and F (X ) 2 C be a scalar real or
complex function of X then the derivative of F (X ) with respect to X is de ned as
@
@
@X F (X ) := @xij F (X ) :
Let A and B be constant complex matrices with compatible dimensions. Then the
following is a list of formulas for the derivatives2:
@
@X TracefAXB g
@
T
@X TracefAX B g
@
@X TracefAXBX g
@
T
@X TracefAXBX g
@
k
@X TracefX g
= AT B T
@
k
@X TracefAX g
=
@
;1
@X TracefAX B g
@
@X log det X
@
T
@X det X
@
k
@X detfX g
= ;(X ;1 BAX ;1)T
= BA
= AT X T B T + B T X T AT
= AT XB T + AXB
= k(X k;1 )T
Pk;1
i=0
X i AX k;i;1
T
= (X T );1
@
= @X det X = (detX )(X T );1
= k(detX k )(X T );1 :
And nally, the derivative of a matrix A( ) 2 C m n with respect to a scalar 2 C is
de ned as
dA := daij
d
d
so that all the rules applicable to a scalar function also apply here. In particular, we
have
d(AB ) = dA B + A dB
d
d
d
dA;1 = ;A;1 dA A;1 :
d
d
2 Note that transpose rather than complex conjugate transpose should be used in the list even if the
involved matrices are complex matrices.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-41-320.jpg)
![LINEAR ALGEBRA
28
2.7 Vector Norms and Matrix Norms
In this section, we will de ne vector and matrix norms. Let X be a vector space, a realvalued function k k de ned on X is said to be a norm on X if it satis es the following
properties:
(i) kxk 0 (positivity)
(ii) kxk = 0 if and only if x = 0 (positive de niteness)
(iii) k xk = j j kxk, for any scalar (homogeneity)
(iv) kx + yk kxk + kyk (triangle inequality)
for any x 2 X and y 2 X . A function is said to be a semi-norm if it satis es (i), (iii),
and (iv) but not necessarily (ii).
Let x 2 C n . Then we de ne the vector p-norm of x as
kxkp :=
n
X
i=1
jxi jp
!1=p
for 1 p 1:
In particular, when p = 1 2 1 we have
kxk1 :=
kxk2 :=
n
X
i=1
jxi j
v
u n
uX
t
i=1
jxi j2
kxk1 := 1maxn jxi j:
i
Clearly, norm is an abstraction and extension of our usual concept of length in 3dimensional Euclidean space. So a norm of a vector is a measure of the vector length",
for example kxk2 is the Euclidean distance of the vector x from the origin. Similarly,
we can introduce some kind of measure for a matrix.
Let A = aij ] 2 C m n , then the matrix norm induced by a vector p-norm is de ned
as
kAxk
kAkp := sup kxk p :
x6=0
p
In particular, for p = 1 2 1, the corresponding induced matrix norm can be computed
as
m
X
kAk1 = 1maxn jaij j (column sum)
j
i=1](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-45-320.jpg)
![LINEAR ALGEBRA
30
1. (A) kAk (This is also true for F norm and any induced matrix norm).
2. kAB k kAk kB k. In particular, this gives A;1
kAk;1 if A is invertible.
(This is also true for any induced matrix norm.)
3. kUAV k = kAk, and kUAV kF = kAkF , for any appropriately dimensioned unitary
matrices U and V .
4. kAB kF kAk kB kF and kAB kF kB k kAkF .
Note that although pre-multiplication or post-multiplication of a unitary matrix on
a matrix does not change its induced 2-norm and F -norm, it does change its eigenvalues.
For example, let
A= 1 0 :
1 0
Then 1 (A) = 1
2 (A) = 0.
Now let
U=
"
1
p2
1
; p2
then U is a unitary matrix and
#
1
p2
1
p2
p
2 0
0 0
UA =
p
with 1 (UA) = 2, 2 (UA) = 0. This property is useful in some matrix perturbation
problems, particularly, in the computation of bounds for structured singular values
which will be studied in Chapter 10.
Lemma 2.10 Let A be a block partitioned matrix with
2A A
6 A11 A12
A=6 .
6 .21 ..22
4 . .
A1q
A2q
Am1 Am2
.
.
.
Amq
3
7
7 =: Aij ]
7
5
and let each Aij be an appropriately dimensioned matrix. Then for any induced matrix
p-norm
2
3
kA11 k
kAkp
kA12 k
6 kA21 kp kA22 kp
p
6 . p
6 .
.
.
4 .
.
kAm1 kp kAm2 kp
kA1q kp
kA2q kp 7
7
.
.
.
kAmq kp
7 :
5
Further, the inequality becomes an equality if the F -norm is used.
p
(2:2)](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-47-320.jpg)
![2.7. Vector Norms and Matrix Norms
31
Proof. It is obvious that if the F -norm is used, then the right hand side of inequality (2.2) equals the left hand side. Hence only the induced p-norm cases, 1 p 1,
will be shown. Let a vector x be partitioned consistently with A as
2x
6 x1
x = 6 ..
6 2
4 .
xq
and note that
kxkp =
Then
k Aij ]kp :=
=
=
2 kx1k
6 kx2kp
6 .p
6 .
4 .
kxq kp
3
7
7 :
7
5
p
2 Pq A1j xj 3
6 Pjq=1 A2j xj 7
7
sup k Aij ] xkp = sup 6 j=1..
6
7
4P .
5
kxkp =1
kxkp=1
q A x
mj j
p
2 Pq
3 j=1
2 Pq kA1j k kxj k
6 Pjq=1 A1j xj p 7
6
6 j=1 A2j xj p 7
7
6 Pjq=1 kA2j kp kxj kp
7
sup 6
6
7 kxsup 6 j=1 .. p p
6
..
7
kxkp =1 6
kp =1 4 P
.
6 P .
7
q kA k kx k
4 q
5
j =1 mj p j p
j =1 Amj xj p p
2 kA11 k kA12k
kA1q kp 3 2 kx1 kp 3
p
p
6 kA21 kp kA22kp
kA2q kp 7 6 kx2 kp 7
76
7
sup 6 ..
6
..
.. 7 6 .. 7
4 .
54 . 5
kxkp =1
.
.
kxq kp p
kAm1 kp kAm2 kp
kA1q kp
h
i
sup
kxkp =1
=
3
7
7
7
5
h
kAij kp
kAij kp
i
p
p
3
7
7
7
5
p
kxkp
:
2](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-48-320.jpg)
![LINEAR ALGEBRA
32
2.8 Singular Value Decomposition
A very useful tool in matrix analysis is Singular Value Decomposition (SVD). It will be
seen that singular values of a matrix are good measures of the size" of the matrix and
that the corresponding singular vectors are good indications of strong/weak input or
output directions.
Theorem 2.11 Let A 2 Fm n . There exist unitary matrices
U = u1 u2 : : : um] 2 Fm m
V = v1 v2 : : : vn ] 2 Fn n
such that
A=U V
= 01 0
0
2
6
6
1=6
4
where
and
1
0
1
0
.
.
.
0
2
.
.
.
0
p
2
0
0
... .
.
.
p
3
7
7
7
5
0 p = minfm ng:
Proof. Let = kAk and without loss of generality assume m n. Then from the
de nition of kAk, there exists a z 2 Fn such that
kAz k = kz k :
~
~ ~
By Lemma 2.8, there is a matrix U 2 F m n such that U U = I and
~
Az = Uz:
Now let
We have Ax = y. Let
and
z
x = kz k 2 Fn
~
y = Uz 2 Fm :
~
Uz
V = x V1 2 Fn n
U = y U1 2 Fm m
be unitary.4 Consequently, U AV has the following structure:
A1 := U AV = 0 w
B
4 Recall that it is always possible to extend an orthonormal set of vectors to an orthonormal basis
for the whole space.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-49-320.jpg)
![2.8. Singular Value Decomposition
35
2
Some useful properties of SVD are collected in the following lemma.
Lemma 2.13 Let A 2 Fm n and
1
2
r > r+1 = = 0 r minfm ng:
Then
1. rank(A) = r
2. KerA = spanfvr+1 : : : vn g and (KerA)? = spanfv1 : : : vr g
3. ImA = spanfu1 : : : ur g and (ImA)? = spanfur+1 : : : um g
4. A 2 Fm n has a dyadic expansion:
A=
5.
6.
7.
8.
r
X
i=1
i ui vi = Ur r Vr
where Ur = u1 : : : ur ], Vr = v1 : : : vr ], and r = diag ( 1 : : : r )
2
2
2
kAk2 = 1 + 2 + + r
F
kAk = 1
i (U0 AV0 ) = i (A) i = 1 : : : p for any appropriately dimensioned unitary matrices U0 and V0
P
Let k < r = rank(A) and Ak := k=1 i ui vi , then
i
min kA ; B k = kA ; Ak k = k+1 :
rank(B) k
Proof. We shall only give a proof for part 8. It is easy to see that rank(Ak ) k and
kA ; Ak k = k+1 . Hence, we only need to show that min kA ; B k k+1 . Let
rank(B) k
B be any matrix such that rank(B ) k. Then
kA ; B k = kU V ; B k = k ; U BV k
+1
Ik+1 0 ( ; U BV ) Ik0
=
^
k+1 ; B
+1
^
^
where B = Ik+1 0 U BV Ik0 2 F(k+1) (k+1) and rank(B ) k. Let x 2 Fk+1
^
be such that Bx = 0 and kxk = 1. Then
^
^
kA ; B k
( k+1 ; B )x = k k+1 xk k+1 :
k+1 ; B
Since B is arbitrary, the conclusion follows.
2](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-52-320.jpg)
![44
LINEAR ALGEBRA
2.12 Notes and References
A very extensive treatment of most topics in this chapter can be found in Brogan
1991], Horn and Johnson 1990,1991] and Lancaster and Tismenetsky 1985]. Golub
and Van Loan's book 1983] contains many numerical algorithms for solving most of the
problems in this chapter. The matrix dilation theory can be found in Davis, Kahan,
and Weinberger 1982].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-61-320.jpg)
![3.2. Controllability and Observability
47
The input/output relationship (i.e., with zero initial state: x0 = 0) can be described by
the convolution equation
y(t) = (g u)(t) :=
Z1
;1
g(t ; )u( )d =
Zt
;1
g(t ; )u( )d :
3.2 Controllability and Observability
We now turn to some very important concepts in linear system theory.
De nition 3.1 The dynamical system described by the equation (3.1) or the pair
(A B ) is said to be controllable if, for any initial state x(0) = x0 , t1 > 0 and nal
state x1 , there exists a (piecewise continuous) input u( ) such that the solution of (3.1)
satis es x(t1 ) = x1 . Otherwise, the system or the pair (A B ) is said to be uncontrollable.
The controllability (or observability introduced next) of a system can be veri ed through
some algebraic or geometric criteria.
Theorem 3.1 The following are equivalent:
(i) (A B ) is controllable.
(ii) The matrix
Wc (t) :=
Zt
0
eA BB eA d
is positive de nite for any t > 0.
(iii) The controllability matrix
C = B AB A2 B : : : An;1 B
P
has full row rank or, in other words, hA jImB i := n=1 Im(Ai;1 B ) = Rn .
i
(iv) The matrix A ; I B ] has full row rank for all in C .
and x be any eigenvalue and any corresponding left eigenvector of A, i.e.,
x A = x , then x B 6= 0.
(v) Let
(vi) The eigenvalues of A + BF can be freely assigned (with the restriction that complex
eigenvalues are in conjugate pairs) by a suitable choice of F .](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-64-320.jpg)
![LINEAR DYNAMICAL SYSTEMS
48
Proof.
(i) , (ii): Suppose Wc (t1 ) > 0 for some t1 > 0, and let the input be de ned as
u( ) = ;B eA (t1 ; )Wc (t1 );1 (eAt1 x0 ; x1 ):
Then it is easy to verify using the formula in (3.3) that x(t1 ) = x1 . Since x1 is
arbitrary, the pair (A B ) is controllable.
To show that the controllability of (A B ) implies that Wc (t) > 0 for any t > 0,
assume that (A B ) is controllable but Wc (t1 ) is singular for some t1 > 0. Since
eAtBB eA t 0 for all t, there exists a real vector 0 6= v 2 Rn such that
v eAtB = 0 t 2 0 t1 ]:
Now let x(t1 ) = x1 = 0, and then from the solution (3.3), we have
0 = eAt1 x(0) +
Zt
1
0
eA(t1 ; )Bu( )d :
Pre-multiply the above equation by v to get
0 = v eAt1 x(0):
If we chose the initial state x(0) = e;At1 v, then v = 0, and this is a contradiction.
Hence, Wc (t) can not be singular for any t > 0.
(ii) , (iii): Suppose Wc (t) > 0 for all t > 0 (in fact, it can be shown that Wc (t) > 0
for all t > 0 i , for some t1 , Wc (t1 ) > 0) but the controllability matrix C does not
have full row rank. Then there exists a v 2 Rn such that
v Ai B = 0
for all 0 i n ; 1. In fact, this equality holds for all i 0 by the CayleyHamilton Theorem. Hence,
v eAt B = 0
for all t or, equivalently, v Wc (t) = 0 for all t this is a contradiction, and hence,
the controllability matrix C must be full row rank. Conversely, suppose C has full
row rank but Wc (t) is singular for some t1 . Then there exists a 0 6= v 2 Rn such
that v eAt B = 0 for all t 2 0 t1 ]. Therefore, set t = 0, and we have
v B = 0:
Next, evaluate the i-th derivative of v eAt B = 0 at t = 0 to get
v Ai B = 0 i > 0:
Hence, we have
v B AB A2 B : : : An;1 B = 0
or, in other words, the controllability matrix C does not have full row rank. This
is again a contradiction.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-65-320.jpg)
![LINEAR DYNAMICAL SYSTEMS
50
De nition 3.3 The dynamical system (3.1), or the pair (A B), is said to be stabilizable
if there exists a state feedback u = Fx such that the system is stable, i.e., A + BF is
stable.
Therefore, it is more appropriate to call this stabilizability the state feedback stabilizability to di erentiate it from the output feedback stabilizability de ned later.
The following theorem is a consequence of Theorem 3.1.
Theorem 3.2 The following are equivalent:
(i) (A B ) is stabilizable.
(ii) The matrix A ; I B ] has full row rank for all Re
(iii) For all and x such that x A = x
0.
0, x B 6= 0.
and Re
(iv) There exists a matrix F such that A + BF is Hurwitz.
We now consider the dual notions of observability and detectability of the system
described by equations (3.1) and (3.2).
De nition 3.4 The dynamical system described by the equations (3.1) and (3.2) or
by the pair (C A) is said to be observable if, for any t1 > 0, the initial state x(0) = x0
can be determined from the time history of the input u(t) and the output y(t) in the
interval of 0 t1]. Otherwise, the system, or (C A), is said to be unobservable.
Theorem 3.3 The following are equivalent:
(i) (C A) is observable.
(ii) The matrix
Wo (t) :=
is positive de nite for any t > 0.
(iii) The observability matrix
Zt
0
eA C CeA d
2 C 3
6 CA 7
6
7
O = 6 CA2 7
6 . 7
6 .. 7
4
5
CAn;1
T
has full column rank or n=1 Ker(CAi;1 ) = 0.
i
(iv) The matrix
A; I
C
has full column rank for all in C .](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-67-320.jpg)
![3.2. Controllability and Observability
51
(v) Let and y be any eigenvalue and any corresponding right eigenvector of A, i.e.,
Ay = y, then Cy 6= 0.
(vi) The eigenvalues of A + LC can be freely assigned (with the restriction that complex
eigenvalues are in conjugate pairs) by a suitable choice of L.
(vii) (A C ) is controllable.
Proof. First, we will show the equivalence between conditions (i) and (iii). Once this
is done, the rest will follow by the duality or condition (vii).
(i) ( (iii): Note that given the input u(t) and the initial condition x0 , the output in
the time interval 0 t1] is given by
y(t) = CeAt x(0) +
Zt
0
CeA(t; ) Bu( )d + Du(t):
Since y(t) and u(t) are known, there is no loss of generality in assuming u(t) =
0 8t. Hence,
y(t) = CeAt x(0) t 2 0 t1 ]:
From this equation, we have
3 2 C 3
CA 7
7 6 CA2 7
7=6
6
7 6 . 7 x(0)
7
5 6 .. 7
4
5
(n;1) (0)
y
2 y(0)
6 y_ (0)
6 .
6 ..
4
CAn;1
where y(i) stands for the i-th derivative of y. Since the observability matrix O
has full column rank, there is a unique solution x(0) in the above equation. This
completes the proof.
(i) ) (iii): This will be proven by contradiction. Assume that (C A) is observable but
that the observability matrix does not have full column rank, i.e., there is a vector
x0 such that Ox0 = 0 or equivalently CAi x0 = 0 8i 0 by the Cayley-Hamilton
Theorem. Now suppose the initial state x(0) = x0 , then y(t) = CeAt x(0) = 0.
This implies that the system is not observable since x(0) cannot be determined
from y(t) 0.
2
De nition 3.5 The system, or the pair (C A), is detectable if A + LC is stable for
some L.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-68-320.jpg)
![LINEAR DYNAMICAL SYSTEMS
62
Proof. Without loss of generality, assume that A is in the Jordan canonical form and
that the matrix B is partitioned accordingly:
2J
6 1 J2
A=6
6
4
where Ji is in the form of
2
6
6
6
Ji = 6
6
6
4
...
i
1
i
Jk
3
2B
7
6 B1
7 B = 6 .2
7
6 ..
5
4
Bk
...
... ...
i
1
i
3
7
7
7
5
3
7
7
7
7
7
7
5
and i 6= j if i 6= j . By PBH test, the pair (A B ) is controllable if and only if, for each
i = 1 : : : k, the last row of Bi is not zero. Let bi 2 Rm be the last row of Bi , and then
we only need to show that, for almost all v 2 Rm , bi v 6= 0 for each i = 1 : : : k which is
clear since for each i, the set v 2 Rm such that bi v = 0 has measure zero in Rm since
bi 6= 0.
2
The cyclicity assumption in this theorem is essential. Without this assumption, the
theorem does not hold. For example, the pair
A= 1 0
B= 1 0
0 1
0 1
is controllable but there is no v 2 R2 such that (A Bv) is controllable since A is not
cyclic.
Since a matrix A with distinct eigenvalues is cyclic, by de nition we have the following lemma.
Lemma 3.13 If (A B) is controllable, then for almost any K 2 Rm n , all the eigenvalues of A + BK are distinct and, consequently, A + BK is cyclic.
A proof can be found in Brasch and Pearson 1970], Davison 1968], and Heymann
1968].
Now it is evident that given a multiple input controllable pair (A B ), there is a
matrix K 2 Rm n and a vector v 2 Rm such that A + BK is cyclic and (A + BK Bv)
is controllable. Moreover, from the pole placement results for the single input system,
there is a matrix f 2 Rn so that the eigenvalues of (A + BK )+(Bv)f can be arbitrarily
assigned. Hence, the eigenvalues of A + BF can be arbitrarily assigned by choosing a
state feedback in the form of
u = Fx = (K + vf )x:](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-79-320.jpg)
![3.5. Observers and Observer-Based Controllers
63
A dual procedure can be applied for the output injection A + LC .
The canonical form for single input or single output system can also be generalized to
multiple input and multiple output systems at the expense of notation. The interested
reader may consult Kailath 1980] or Chen 1984].
If a system is not completely controllable, then the Kalman controllable decomposition can be applied rst and the above procedure can be used to assign the system
poles corresponding to the controllable subspace.
3.5 Observers and Observer-Based Controllers
We have shown in the last section that if a system is controllable and, furthermore, if
the system states are available for feedback, then the system closed loop poles can be
assigned arbitrarily through a constant feedback. However, in most practical applications, the system states are not completely accessible and all the designer knows are the
output y and input u. Hence, the estimation of the system states from the given output
information y and input u is often necessary to realize some speci c design objectives.
In this section, we consider such an estimation problem and the application of this state
estimation in feedback control.
Consider a plant modeled by equations (3.1) and (3.2). An observer is a dynamical
system with input of (u y) and output of, say x, which asymptotically estimates the
^
state x. More precisely, a (linear) observer is a system such as
q_ = Mq + Nu + Hy
x = Qq + Ru + Sy
^
so that x(t) ; x(t) ! 0 as t ! 1 for all initial states x(0), q(0) and for every input u( ).
^
Theorem 3.14 An observer exists i (C A) is detectable. Further, if (C A) is detectable, then a full order Luenberger observer is given by
q_ = Aq + Bu + L(Cq + Du ; y)
x = q
^
where L is any matrix such that A + LC is stable.
(3.5)
(3.6)
Proof. We rst show that the detectability of (C A) is su cient for the existence
of an observer. To that end, we only need to show that the so-called Luenberger
observer de ned in the theorem is indeed an observer. Note that equation (3.5) for q
is a simulation of the equation for x, with an additional forcing term L(Cq + Du ; y),
which is a gain times the output error. Equivalent equations are
q_ = (A + LC )q + Bu + LDu ; Ly
x = q:
^](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-80-320.jpg)
![64
LINEAR DYNAMICAL SYSTEMS
These equations have the form allowed in the de nition of an observer. De ne the error,
e := x ; x, and then simple algebra gives
^
e = (A + LC )e
_
therefore e(t) ! 0 as t ! 1 for every x(0), q(0), and u(:).
To show the converse, assume that (C A) is not detectable. Take the initial state
x(0) in the undetectable subspace and consider a candidate observer:
q_ = Mq + Nu + Hy
x = Qq + Ru + Sy:
^
Take q(0) = 0 and u(t) 0. Then the equations for x and the candidate observer are
x = Ax
_
q_ = Mq + HCx
x = Qq + SCx:
^
Since an unobservable subspace is an A-invariant subspace containing x(0), it follows
that x(t) is in the unobservable subspace for all t 0. Hence, Cx(t) = 0 for all t 0,
and, consequently, q(t) 0 and x(t) 0. However, for some x(0) in the undetectable
^
subspace, x(t) does not converge to zero. Thus the candidate observer does not have
the required property, and therefore, no observer exists.
2
The above Luenberger observer has dimension n, which is the dimension of the state
x. It's possible to get an observer of lower dimension. The idea is this: since we can
measure y ; Du = Cx, we already know x modulo Ker C , so we only need to generate
the part of x in Ker C . If C has full row rank and p := dim y, then the dimension of
Ker C equals n ; p, so we might suspect that we can get an observer of order n ; p.
This is true. Such an observer is called a minimal order observer". We will not pursue
this issue further here. The interested reader may consult Chen 1984].
Recall that, for a dynamical system described by the equations (3.1) and (3.2), if
(A B ) is controllable and state x is available for feedback, then there is a state feedback
u = Fx such that the closed-loop poles of the system can be arbitrarily assigned.
Similarly, if (C A) is observable, then the system observer poles can be arbitrarily
placed so that the state estimator x can be made to approach x arbitrarily fast. Now
^
let us consider what will happen if the system states are not available for feedback
so that the estimated state has to be used. Hence, the controller has the following
dynamics:
_
x = (A + LC )^ + Bu + LDu ; Ly
^
x
u = F x:
^](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-81-320.jpg)
![3.6. Operations on Systems
67
De nition 3.7 The transpose of a transfer matrix G(s) or the dual system is de ned
as
G 7;! GT (s) = B (sI ; A );1 C + D
or equivalently
A C
A B
C D 7;! B D :
De nition 3.8 The conjugate system of G(s) is de ned as
G 7;! G (s) := GT (;s) = B (;sI ; A );1 C + D
or equivalently
A B
;A ;C
C D 7;! B
D
In particular, we have G (j!) := G(j!)] = G (j!).
:
^
De nition 3.9 A real rational matrix G(s) is called a right (left) inverse of a transfer
^
^
^
matrix G(s) if G(s)G(s) = I ( G(s)G(s) = I ). Moreover, if G(s) is both a right inverse
and a left inverse of G(s), then it is simply called the inverse of G(s).
Lemma 3.15 Let Dy denote a right (left) inverse of D if D has full row (column) rank.
Then
y
y
Gy = A ; BD C ;BD
yC
y
D
D
is a right (left) inverse of G.
Proof. The right inverse case will be proven and the left inverse case follows by duality.
Suppose DDy = I . Then
GGy =
=
2 A BDy C
4 0 A ; BDy C
yC
2 C DDy C
A BD
4 0 A ; BDy C
C
3
BDy
;BDy 5
DDy
3
BDy
;BDy 5 :
I
C
Performing similarity transformation T = I I on the above system yields
0 I
3
2A
0
0
GGy = 4 0 A ; BDy C ;BDy 5
C
0
I
= I:
2](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-84-320.jpg)
![80
LINEAR DYNAMICAL SYSTEMS
3.11 Multivariable System Poles and Zeros
Let R s] denote the polynomial ring with real coe cients. A matrix is called a polynomial matrix if every element of the matrix is in R s]. A square polynomial matrix is
called a unimodular matrix if its determinant is a nonzero constant (not a polynomial
of s). It is a fact that a square polynomial matrix is unimodular if and only if it is
invertible in R s], i.e., its inverse is also a polynomial matrix.
De nition 3.11 Let Q(s) 2 R s] be a (p m) polynomial matrix. Then the normal
rank of Q(s), denoted normalrank (Q(s)), is the rank in R s] or, equivalently, is the
maximum dimension of a square submatrix of Q(s) with nonzero determinant in R s].
In short, sometimes we say that a polynomial matrix Q(s) has rank(Q(s)) in R s] when
we refer to the normal rank of Q(s).
To show the di erence between the normal rank of a polynomial matrix and the
rank of the polynomial matrix evaluated at certain point, consider
1
Q(s) = ss2 1 :
Then Q(s) has normal rank 2 since det Q(s) = s ; s2 6 0. However, Q(0) has rank 1.
It is a fact in linear algebra that any polynomial matrix can be reduced to a socalled Smith form through some pre- and post- unimodular operations. cf. Kailath,
1984, pp.391].
Lemma 3.25 (Smith form) Let P (s) 2 R s] be any polynomial matrix, then there
exist unimodular matrices U (s) V (s) 2 R s] such that
2 (s) 0
3
0 0
1
6 0 2 (s)
0 07
6 .
6 .
. ..
.
. 7
.
. 7
U (s)P (s)V (s) = S (s) := 6 .
. .
.
.
. 7
6 0
7
4
0
(s) 0 5
r
0
0
0 0
and i (s) divides i+1 (s).
S (s) is called the Smith form of P (s). It is also clear that r is the normal rank of P (s).
We shall illustrate the procedure of obtaining a Smith form by an example. Let
2 s + 1 (s + 1)(2s + 1) s(s + 1) 3
P (s) = 4 s + 2 (s + 2)(s2 + 5s + 3) s(s + 2) 5 :
1
2s + 1
s
The polynomial matrix P (s) has normal rank 2 since
+
+
det(P (s)) 0 det s + 1 (s (s 2)(1)(2s 5s 1) 3) = (s + 1)2 (s + 2)2 6 0:
s + 2 + s2 + +](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-97-320.jpg)
![3.11. Multivariable System Poles and Zeros
81
First interchange the rst row and the third row and use row elementary operation to
zero out the s+1 and s+2 elements of P (s). This process can be done by pre-multiplying
a unimodular matrix U to P (s):
20 0 1 3
U = 4 0 1 ;(s + 2) 5 :
1 0 ;(s + 1)
Then
21
3
2s + 1
s
P1 (s) := U (s)P (s) = 4 0 (s + 1)(s + 2)2 0 5 :
0
0
0
Next use column operation to zero out the 2s + 1 and s terms in P1 . This process can
be done by post-multiplying a unimodular matrix V to P1 (s):
2 1 ;(2s + 1) ;s 3
1
0 5
V (s) = 4 0
0
0
1
and
21
3
0
0
P1 (s)V (s) = 4 0 (s + 1)(s + 2)2 0 5 :
0
0
0
Then we have
21
3
0
0
S (s) = U (s)P (s)V (s) = 4 0 (s + 1)(s + 2)2 0 5 :
0
0
0
Similarly, let Rp (s) denote the set of rational proper transfer matrices.1 Then any
real rational transfer matrix can be reduced to a so-called McMillan form through some
pre- and post- unimodular operations.
Lemma 3.26 (McMillan form) Let G(s) 2 Rp(s) be any proper real rational transfer
matrix, then there exist unimodular matrices U (s) V (s) 2 R s] such that
3
2 1 (s) 0
0 0
1 (s)
6 0 2((ss))
0 07
7
6
7
6 . 2. .
U (s)G(s)V (s) = M (s) := 6 .
.
.. . . 7
. . 7
6 .
.
. . 7
6
r (s)
5
4 0
0
r (s) 0
0
0
0 0
and i (s) divides i+1 (s) and i+1 (s) divides i (s).
1 It is obvious that a similar notion of normal rank can be extended to a matrix in R ( ). In
particular, let ( ) 2 R ( ) be a rational matrix and write ( ) = ( ) ( ) such that ( ) is a
p s
G s
p s
G s
N s =d s
d s
polynomial and ( ) is a polynomial matrix, then ( ) is said to have normal rank if ( ) has
normal rank .
N s
r
G s
r
N s](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-98-320.jpg)
![LINEAR DYNAMICAL SYSTEMS
84
Proof. We shall outline a proof of this lemma. We shall rst show that there is a
vector u0 2 C m such that G(z0 )u0 = 0 if z0 2 C is a transmission zero. Without loss of
generality, assume
2
6
6
6
G(s) = U1 (s) 6
6
6
4
1
1
(s)
(s)
0
0
..
.
0
0
2
2
(s)
(s)
..
.
0
0
...
0
0
..
.
m (s)
m (s)
0
3
7
7
7 V (s)
7 1
7
7
5
for some unimodular matrices U1 (s) and V1 (s) and suppose z0 is a zero of 1 (s), i.e.,
1 (z0 ) = 0. Let
u0 = V1;1 (z0 )e1 6= 0
where e1 = 1 0 0 : : :] 2 Rm . Then it is easy to verify that G(z0 )u0 = 0. On the other
hand, suppose there is a u0 2 C m such that G(z0 )u0 = 0. Then
2
6
6
6
U1 (z0 ) 6
6
6
4
De ne
Then
1
1
(z0 )
(z0 )
0
0
..
.
0
0
2
2
(z0 )
(z0 )
..
.
0
0
...
0
0
..
.
m (z0 )
m (z0 )
0
3
7
7
7 V (z )u = 0:
7 1 0 0
7
7
5
2u 3
6 u1 7
6 .2 7 = V1 (z0)u0 6= 0:
6 .. 7
4 5
um
2
6
6
6
4
1 (z0 )u1
2 (z0 )u2
..
.
m (z0 )um
3
7
7 = 0:
7
5
This implies that z0 must be a root of one of polynomials i (s) i = 1 : : : m.
2
Note that the lemma may not be true if G(s) does not have full column normal rank.
This can be seen from the following example. Consider
1
G(s) = s + 1 1 1
1 1
1
u0 = ;1 :](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-101-320.jpg)
![3.12. Notes and References
89
The following lemma characterizes the relationship between zeros of a transfer function and poles of its inverse.
A B
Lemma 3.38 Suppose that G = C D is a square transfer matrix with D nonsingular, and suppose z0 is not an eigenvalue of A (note that the realization is not
necessarily minimal). Then there exists x0 such that
(A ; BD;1 C )x0 = z0 x0 Cx0 6= 0
i there exists u0 6= 0 such that
G(z0 )u0 = 0:
Proof. (() G(z0)u0 = 0 implies that
;1
;1
G;1 (s) = A ; BD C ;BD
D;1 C
D;1
has a pole at z0 which is observable. Then, by de nition, there exists x0 such that
(A ; BD;1 C )x0 = z0 x0
and
Cx0 6= 0:
()) Set u0 = ;D;1 Cx0 6= 0. Then
(z0 I ; A)x0 = ;BD;1Cx0 = Bu0 :
Using this equality, one gets
G(z0 )u0 = C (z0 I ; A);1 Bu0 + Du0 = Cx0 ; Cx0 = 0:
2
The above lemma implies that z0 is a zero of an invertible G(s) if and only if it is a
pole of G;1 (s).
3.12 Notes and References
Readers are referred to Brogan 1991], Chen 1984], Kailath 1980], and Wonham 1985]
for the extensive treatment of the standard linear system theory. The balanced realization was rst introduced by Mullis and Roberts 1976] to study the roundo noise
in digital lters. Moore 1981] proposed the balanced truncation method for model
reduction which will be considered in Chapter 7.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-106-320.jpg)
![4
Performance Speci cations
The most important objective of a control system is to achieve certain performance
speci cations in addition to providing the internal stability. One way to describe the
performance speci cations of a control system is in terms of the size of certain signals
of interest. For example, the performance of a tracking system could be measured
by the size of the tracking error signal. In this chapter, we look at several ways of
de ning a signal's size, i.e., at several norms for signals. Of course, which norm is most
appropriate depends on the situation at hand. For that purpose, we shall rst introduce
some normed spaces and some basic notions of linear operator theory, in particular, the
Hardy spaces H2 and H1 are introduced. We then consider the performance of a system
under various input signals and derive the worst possible outputs with the class of input
signals under consideration. We show that H2 and H1 norms come out naturally as
measures of the worst possible performance for many classes of input signals. Some
state space methods of computing real rational H2 and H1 transfer matrix norms are
also presented.
4.1 Normed Spaces
Let V be a vector space over C (or R) and let k k be a norm de ned on V . Then V
is a normed space. For example, the vector space C n with any vector p-norm, k kp ,
for 1 p 1, is a normed space. As another example, consider the linear vector
space C a b] of all bounded continuous functions on the real interval a b]. Then C a b]
91](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-108-320.jpg)
![PERFORMANCE SPECIFICATIONS
92
becomes a normed space if a supremum norm is de ned on the space:
kf k1 := sup jf (t)j:
t2 a b]
A sequence fxn g in a normed space V is called a Cauchy sequence if kxn ; xm k ! 0
as n m ! 1. A sequence fxn g is said to converge to x 2 V , written xn ! x, if
kxn ; xk ! 0. A normed space V is said to be complete if every Cauchy sequence in V
converges in V . A complete normed space is called a Banach space. For example, Rn
and C n with the usual spatial p-norm, k kp for 1 p 1, are Banach spaces. (One
should not be confused with the notation k kp used here and the same notation used
below for function spaces because usually the context will make the meaning clear.)
The following are some more examples of Banach spaces:
lp 0 1) spaces for 1 p < 1:
For each 1 p < 1, lp 0 1) consists of all sequences x = (x0 x1 : : :) such that
1
X p
jxi j < 1. The associated norm is de ned as
i=0
kxkp :=
1
X
i=0
jxi jp
!1=p
:
l1 0 1) space:
l1 0 1) consists of all bounded sequences x = (x0 x1 : : :), and the l1 norm is
de ned as
kxk1 := sup jxi j:
i
Lp (I ) spaces for 1 p 1:
For each 1 p < 1, Lp (I ) consists of all Lebesgue measurable functions x(t)
de ned on an interval I
kxkp :=
and
R
Z
I
such that
jx(t)jp dt
1=p
< 1 for 1 p < 1
kx(t)k1 := ess sup jx(t)j:
t2I
Some of these spaces, for example, L2 (;1 0], L2 0 1) and L2 (;1 1), will be
discussed in more detail later on.
C a b] space:
C a b] consists of all continuous functions on the real interval a b] with the norm
de ned as
kxk1 := sup jx(t)j:
t2 a b]](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-109-320.jpg)
![PERFORMANCE SPECIFICATIONS
94
Note that many important metric notions and geometrical properties such as length,
distance, angle, and the energy of physical systems can be deduced from this inner
product. For instance, the length of a vector x 2 C n is de ned as
p
kxk := hx xi
and the angle between two vectors x y 2 C n can be computed from
yi
cos (x y) = khx kyk (x y) 2 0 ]:
xk
The two vectors are said to be orthogonal if (x y) = 2 .
We now consider a natural generalization of the inner product on C n to more general
(possibly in nite dimensional) vector spaces.
De nition 4.2 Let V be a vector space over C . An inner product on V is a complex
valued function,
h i : V V 7;! C
such that for any x y z 2 V and
2C
(i) 1 hx y + z i = hx yi + hx z i
(ii) hx yi = hy xi
(iii) hx xi > 0 if x 6= 0.
A vector space V with an inner product is called an inner product space.
p
It is clear that the inner product de ned above induces a norm kxk := hx xi, so
that the norm conditions in Chapter 2 are satis ed. In particular, the distance between
vectors x and y is d(x y) = kx ; yk.
Two vectors x and y in an inner product space V are said to be orthogonal if
hx yi = 0, denoted x ? y. More generally, a vector x is said to be orthogonal to a set
S V , denoted by x ? S , if x ? y for all y 2 S .
The inner product and the inner-product induced norm have the following familiar
properties.
Theorem 4.1 Let V be an inner product space and let x y 2 V . Then
(i) jhx yij kxk kyk (Cauchy-Schwarz inequality). Moreover, the equality holds if
and only if x = y for some constant or y = 0.
(ii) kx + yk2 + kx ; yk2 = 2 kxk2 + 2 kyk2 (Parallelogram law) .
(iii) kx + yk2 = kxk2 + kyk2 if x ? y.
1 This is the other way round to the usual mathematical convention since we want to have h
rather than
for
2 C n.
y x
x y
x y
i=
x y](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-111-320.jpg)
![4.2. Hilbert Spaces
95
A Hilbert space is a complete inner product space with the norm induced by its
inner product. Obviously, a Hilbert space is also a Banach space. For example, C n
with the usual inner product is a ( nite dimensional) Hilbert space. More generally, it
is straightforward to verify that C n m with the inner product de ned as
hA B i := Trace A B =
n m
XX
i=1 j =1
aij bij 8A B 2 C n m
is also a ( nite dimensional) Hilbert space.
Here are some examples of in nite dimensional Hilbert spaces:
l2 (;1 1):
l2 (;1 1) consists of the set of all real or complex square summable sequences
x = (: : : x;2 x;1 x0 x1 x2 : : :)
i.e,
1
X
i=;1
jxi j2 < 1
with the inner product de ned as
hx yi :=
1
X
i=;1
xi yi
for x y 2 l2 (;1 1). The subspaces l2 (;1 0) and l2 0 1) of l2 (;1 1) are
de ned similarly and consist of sequences of the form x = (: : : x;2 x;1 ) and
x = (x0 x1 x2 : : :), respectively.
Note that we can also de ne a corresponding Hilbert space even if each component
xi is a vector or a matrix in fact, the following inner product will su ce:
hx yi :=
1
X
i=;1
Trace(xi yi ):
L2 (I ) for I R:
L2 (I ) consists of all square integrable and Lebesgue measurable functions de ned
on an interval I
R
with the inner product de ned as
Z
hf gi := f (t) g(t)dt
I
for f g 2 L2 (I ). Similarly, if the function is vector or matrix valued, the inner
product is de ned correspondingly as
Z
hf gi := Trace f (t) g(t)] dt:
I](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-112-320.jpg)
![PERFORMANCE SPECIFICATIONS
96
Some very often used spaces in this book are L2 0 1) L2 (;1 0] L2 (;1 1).
More precisely, they are de ned as
L2 = L2 (;1 1): Hilbert space of matrix-valued functions on R, with inner
product
hf gi :=
Z1
;1
Trace f (t) g(t)] dt:
L2+ = L2 0 1): subspace of L2 (;1 1) with functions zero for t < 0.
L2; = L2 (;1 0]: subspace of L2 (;1 1) with functions zero for t > 0.
Let H be a Hilbert space and M H a subset. Then the orthogonal complement of
M , denoted by H M or M ?, is de ned as
M ? = fx : hx yi = 0 8y 2 M x 2 Hg :
It can be shown that M ? is closed (hence M ? is also a Hilbert space). For example,
let M = L2+ L2 , then M ? = L2; is a Hilbert space.
Let M and N be subspaces of a vector space V . V is said to be the direct sum of
M and N , written V = M N , if M N = f0g, and every element v 2 V can be
expressed as v = x + y with x 2 M and y 2 N . If V is an inner product space and
M and N are orthogonal, then V is said to be the orthogonal direct sum of M and N .
As an example, it is easy to see that L2 is the orthogonal direct sum of L2; and L2+ .
Similarly, l2 (;1 1) is the orthogonal direct sum of l2 (;1 0) and l2 0 1).
The following is a version of the so-called orthogonal projection theorem:
Theorem 4.2 Let H be a Hilbert space, and let M be a closed subspace of H. Then for
each vector v 2 H, there exist unique vectors x 2 M and y 2 M ? such that v = x + y,
i.e., H = M M ?. Moreover, x 2 M is the unique vector such that d(v M ) = kv ; xk.
Let H1 and H2 be two Hilbert spaces, and let A be a bounded linear operator from
H1 into H2 . Then there exists a unique linear operator A : H2 7;! H1 such that for
all x 2 H1 and y 2 H2
hAx yi = hx A yi:
A is called the adjoint of A. Furthermore, A is called self-adjoint if A = A .
Let H be a Hilbert space and M H be a closed subspace. A bounded operator P
mapping from H into itself is called the orthogonal projection onto M if
P (x + y) = x 8 x 2 M and y 2 M ? :](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-113-320.jpg)
![4.3. Hardy Spaces H2 and H1
97
4.3 Hardy Spaces H2 and H1
Let S
C
be an open set, and let f (s) be a complex valued function de ned on S :
f (s) : S 7;! C :
Then f (s) is said to be analytic at a point z0 in S if it is di erentiable at z0 and also
at each point in some neighborhood of z0 . It is a fact that if f (s) is analytic at z0 then
f has continuous derivatives of all orders at z0 . Hence, a function analytic at z0 has
a power series representation at z0 . The converse is also true, i.e., if a function has a
power series at z0 , then it is analytic at z0 . A function f (s) is said to be analytic in S if
it has a derivative or is analytic at each point of S . A matrix valued function is analytic
in S if every element of the matrix is analytic in S . For example, all real rational stable
transfer matrices are analytic in the right-half plane and e;s is analytic everywhere.
A well known property of the analytic functions is the so-called Maximum Modulus
Theorem.
Theorem 4.3 If f (s) is de ned and continuous on a closed-bounded set S and analytic
on the interior of S , then the maximum of jf (s)j on S is attained on the boundary of
S , i.e.,
max jf (s)j = s2@S jf (s)j
max
s2S
where @S denotes the boundary of S .
Next we consider some frequently used complex (matrix) function spaces.
L2 (j R) Space
L2 (j R) or simply L2 is a Hilbert space of matrix-valued (or scalar-valued) func-
tions on j R and consists of all complex matrix functions F such that the integral
below is bounded, i.e.,
Z1
;1
Trace F (j!)F (j!)] d! < 1:
The inner product for this Hilbert space is de ned as
1 Z 1 Trace F (j!)G(j!)] d!
hF Gi := 2
;1
for F G 2 L2 , and the inner product induced norm is given by
p
kF k2 := hF F i:
For example, all real rational strictly proper transfer matrices with no poles on the
imaginary axis form a subspace (not closed) of L2 (j R) which is denoted by RL2 (j R) or
simply RL2 .](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-114-320.jpg)
![PERFORMANCE SPECIFICATIONS
98
H2 Space2
H2 is a (closed) subspace of L2 (j R) with matrix functions F (s) analytic in Re(s) >
0 (open right-half plane). The corresponding norm is de ned as
Z1
kF k2 := sup 21
Trace F ( + j!)F ( + j!)] d! :
2
>0
It can be shown3 that
kF k2 = 21
2
;1
Z1
;1
Trace F (j!)F (j!)] d!:
Hence, we can compute the norm for H2 just as we do for L2 . The real rational
subspace of H2 , which consists of all strictly proper and real rational stable transfer
matrices, is denoted by RH2 .
?
H2 Space
?
H2 is the orthogonal complement of H2 in L2 , i.e., the (closed) subspace of functions in L2 that are analytic in the open left-half plane. The real rational subspace
?
of H2 , which consists of all strictly proper rational transfer matrices with all poles
in the open right half plane, will be denoted by RH? . It is easy to see that if G
2
is a strictly proper, stable, and real rational transfer matrix, then G 2 H2 and
?
G 2 H2 . Most of our study in this book will be focused on the real rational
case.
The L2 spaces de ned above in the frequency domain can be related to the L2 spaces
de ned in the time domain. Recall the fact that a function in L2 space in the time
domain admits a bilateral Laplace (or Fourier) transform. In fact, it can be shown that
this bilateral Laplace (or Fourier) transform yields an isometric isomorphism between
the L2 spaces in the time domain and the L2 spaces in the frequency domain (this is
what is called Parseval's relations or Plancherel Theorem in complex analysis):
L2 (;1 1) = L2 (j R)
L2 0 1) = H2
?
L2 (;1 0] = H2 :
As a result, if g(t) 2 L2 (;1 1) and if its Fourier (or bilateral Laplace) transform is
G(j!) 2 L2 (j R), then
kGk2 = kgk2 :
2 The H space and H space de ned below together with the H spaces,
1, which will not be
1
p
2
introduced in this book, are usually called Hardy spaces named after the mathematician G. H. Hardy
(hence the notation of H).
3 See Francis 1987].
p](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-115-320.jpg)
![4.3. Hardy Spaces H2 and H1
99
Hence, whenever there is no confusion, the notation of functions in the time domain
and in the frequency domain will be used interchangeably.
De ne an orthogonal projection
P+ : L2 (;1 1) 7;! L2 0 1)
such that, for any function f (t) 2 L2 (;1 1), we have g(t) = P+ f (t) with
g(t) := f (t) for t < 0:
0
for t 0
In this book, P+ will also be used to denote the projection from L2 (j R) onto H2 .
Similarly, de ne P; as another orthogonal projection from L2 (;1 1) onto L2 (;1 0]
?
(or L2 (j R) onto H2 ). Then the relationships between L2 spaces and H2 spaces can be
shown as in Figure 4.1.
Laplace Transform L2 0 1)
H2
Inverse Transform
P+
6
L2 (;1 1)
P;
?
L2 (;1 0]
6
P+
Laplace Transform
Inverse Transform
-
L2 (j R)
P;
Laplace Transform
Inverse Transform
-
?
?
H2
Figure 4.1: Relationships among function spaces
Other classes of important complex matrix functions used in this book are those
bounded on the imaginary axis.
L1 (j R) Space
L1 (j R) or simply L1 is a Banach space of matrix-valued (or scalar-valued) functions that are (essentially) bounded on j R, with norm
kF k1 := ess sup F (j!)] :
! 2R](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-116-320.jpg)
![PERFORMANCE SPECIFICATIONS
100
The rational subspace of L1 , denoted by RL1 (j R) or simply RL1 , consists of
all rational proper transfer matrices with no poles on the imaginary axis.
H1 Space
H1 is a (closed) subspace in L1 with functions that are analytic in the open
right-half plane and bounded on the imaginary axis. The H1 norm is de ned as
kF k1 := sup F (s)] = ess sup F (j!)] :
Re(s)>0
! 2R
The second equality can be regarded as a generalization of the maximum modulus
theorem for matrix functions. The real rational subspace of H1 is denoted by
RH1 which consists of all proper and real rational stable transfer matrices.
;
H1 Space
;
H1 is a (closed) subspace in L1 with functions that are analytic in the open
;
left-half plane and bounded on the imaginary axis. The H1 norm is de ned as
kF k1 := sup F (s)] = ess sup F (j!)] :
Re(s)<0
! 2R
;
The real rational subspace of H1 is denoted by RH; which consists of all proper
1
rational transfer matrices with all poles in the open right half plane.
;
De nition 4.3 A transfer matrix G(s) 2 H1 is usually said to be antistable or anti-
causal.
Some facts about L1 and H1 functions are worth mentioning:
(i) if G(s) 2 L1 , then G(s)L2 := fG(s)f (s) : f (s) 2 L2 g L2 .
(ii) if G(s) 2 H1 , then G(s)H2 := fG(s)f (s) : f (s) 2 H2 g H2 .
;
?
?
?
(ii) if G(s) 2 H1 , then G(s)H2 := fG(s)f (s) : f (s) 2 H2 g H2 .
Remark 4.1 The notation for L1 is somewhat unfortunate it should be clear to the
reader that the L1 space in the time domain and in the frequency domain denote
completely di erent spaces. The L1 space in the time domain is usually used to denote
signals, while the L1 space in the frequency domain is usually used to denote transfer
functions and operators.
~
Let G(s) 2 L1 be a p q transfer matrix. Then a multiplication operator is de ned
as
MG : L2 7;! L2
MG f := Gf:](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-117-320.jpg)
![4.3. Hardy Spaces H2 and H1
101
In writing the above mapping, we have assumed that f has a compatible dimension. A
more accurate description of the above operator should be
MG : Lq 7;! Lp
2
2
i.e., f is a q-dimensional vector function with each component in L2 . However, we shall
suppress all dimensions in this book and assume all objects have compatible dimensions.
It is easy to verify that the adjoint operator MG = MG .
A useful fact about the multiplication operator is that the norm of a matrix G in
L1 equals the norm of the corresponding multiplication operator.
Theorem 4.4 Let G 2 L1 be a p q transfer matrix. Then kMGk = kGk1 .
Remark 4.2 It is also true that this operator norm equals the norm of the operator
?
restricted to H2 (or H2 ), i.e.,
kMGk = kMGjH2 k := sup fkGf k2 : f 2 H2 kf k2 1g :
This will be clear in the proof where an f 2 H2 is constructed.
~
Proof. By de nition, we have
kMGk = sup fkGf k2 : f 2 L2 kf k2 1g :
First we see that kGk1 is an upper bound for the operator norm:
Z1
f (j!)G (j!)G(j!)f (j!) d!
kGf k2 = 21
2
;1
Z1
kGk2 21
kf (j!)k2 d!
1
;1
= kGk2 kf k2:
1 2
To show that kGk1 is the least upper bound, rst choose a frequency !o where G(j!)]
is maximum, i.e.,
G(j!0 )] = kGk1
and denote the singular value decomposition of G(j!0 ) by
G(j!0 ) = u1 (j!0 )v1 (j!0 ) +
r
X
i=2
i ui (j!0 )vi (j!0 )
where r is the rank of G(j!0 ) and ui vi have unit length.
If !0 < 1, write v1 (j!0 ) as
2 ej 3
6 1 ej 7
v1 (j!0 ) = 6 .. 7
6 2. 7
4
5
1
2
q ej q](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-118-320.jpg)
![PERFORMANCE SPECIFICATIONS
102
where i 2 R is such that i 2 ; 0) and q is the column dimension of G. Now let
i 0 be such that
i ; j!0
i=
+ j!
i
and let f be given by
2
6
6
f (s) = 6
6
4
0
;s
1 1 +s
1
;s
2 2 +s
2
..
.
;s
q q +s
q
3
7
7 f^(s)
7
7
5
where a scalar function f^ is chosen so that
jf^(j!)j = c if j! ; !o j < or j! + !oj <
0 otherwise
where is a small positive number and c is chosen so that f has unit 2-norm, i.e.,
p =2 . This in turn implies that f has unit 2-norm. Then^
c=
kGf k2
2
=
1
2
h
G(;j!o )]2 + G(j!o )]2
G(j!o )]2 = kGk2 :
1
i
If !0 = 1, then G(j!0 ) is real and v1 is real. In this case, the conclusion follows by
letting f (s) = v1 f^(s).
2
4.4 Power and Spectral Signals
In this section, we introduce two additional classes of signals that have been widely used
in engineering. These classes of signals have some nice statistical and frequency domain
representations. Let u(t) be a function of time. De ne its autocorrelation matrix as
Ruu ( ) := Tlim 21
!1 T
ZT
;T
u(t + )u (t)dt
if the limit exists and is nite for all . It is easy to see from the de nition that Ruu ( ) =
Ruu (; ). Assume further that the Fourier transform of the signal's autocorrelation
matrix function exists (and may contain impulses). This Fourier transform is called the
spectral density of u, denoted Suu (j!):
Suu (j!) :=
Z1
;1
Ruu ( )e;j! d :](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-119-320.jpg)
![4.4. Power and Spectral Signals
103
Thus Ruu ( ) can be obtained from Suu (j!) by performing an inverse Fourier transform:
Z1
Ruu ( ) := 21
Suu (j!)ej! d!:
;1
We will call signal u(t) a power signal if the autocorrelation matrix Ruu ( ) exists and
is nite for all , and moreover, if the power spectral density function Suu (j!) exists
(note that Suu (j!) need not be bounded and may include impulses).
The power of the signal is de ned as
s
1 Z T ku(t)k2 dt = pTrace R (0)]
kukP = Tlim 2T
uu
!1
;T
where k k is the usual Euclidean norm and the capital script P is used to di erentiate
this power semi-norm from the usual Lebesgue Lp norm. The set of all signals having
nite power will be denoted by P .
The power semi-norm of a signal can also be computed from its spectral density
function
Z1
2 = 1
Trace Suu (j!)]d!:
kukP 2
;1
This expression implies that, if u 2 P , Suu is strictly proper in the sense that Suu (1) =
0. We note that if u 2 P and ku(t)k1 := supt ku(t)k < 1, then kukP kuk1.
However, not every signal having nite 1-norm is a power signal since the limit in the
de nition may not exist. For example, let
2k
2k+1 for k = 0 1 2 : : :
u(t) = 1 2 < t <:2
0 otherwise
R
T
Then limT !1 21 ;T u2 dt does not exist. Note also that power signals are persistent
T
signals in time such as sines or cosines clearly, a time-limited signal has zero power, as
does an L2 signal. Thus k kP is only a semi-norm, not a norm.
Now let G be a linear system transfer matrix with convolution kernel g(t), input u(t), and output z (t). Then Rzz ( ) = g( ) Ruu ( ) g (; ) and Szz (j!) =
G(j!)Suu (j!)G (j!). These properties are useful in establishing some input and output relationships in the next section.
A signal u(t) is said to have bounded power spectral density if kSuu (j!)k1 < 1.
The set of signals having bounded spectral density is denoted as
S := fu(t) 2 Rm : kSuu (j!)k1 < 1g:
p
The quantity kuks := kSuu (j!)k is called the spectral density norm of u(t). The set
S can be used to model signals with xed spectral characteristics by passing white noise
signals through a weighting lter. Similarly, P could be used to model signals whose
spectrum is not known but which are nite in power.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-120-320.jpg)
![4.5. Induced System Gains
105
Input u(t)
Output z (t)
kz k2 = kGu0 k2 = kgu0k2
u(t) = u0 (t) u0 2 Rq
kz k1 = kgu0k1
kz kP = 0
kz k2 = 1
u(t) = u0 sin !0 t u0 2 Rq
lim sup max jzi (t)j = kG(j!0 )u0 k1 = max jGi (j!0 )u0 j
i
i
t!1
lim sup kz (t)k = kG(j!0 )u0 k
t!1
1
kz kP = p kG(j!0 )u0 k
2
Table 4.1: Output norms with xed inputs
Proof.
u = u0 (t): If u(t) = u0 (t), then z (t) = g(t)u0, so kz k2 = kgu0k2 and kz k1 =
kgu0k1 . But by Parseval's relation, kgu0k2 = kGu0k2 . On the other hand,
1 Z T u g (t)g(t)u dt
0
P = Tlim 2T ;T 0
!1
Z1
Tracefg (t)g(t)g dt ku0 k2
lim 1
T !1 2T ;1
= Tlim 21 kGk2 ku0k2 = 0:
2
!1 T
kz k2
u = u0 sin !0 t: With the input u(t) = u0 sin(!0 t), the i-th output as t ! 1 is
zi (t) = jGi (j!0 )u0 j sin !0 t + argfGi (j!0 )u0 g]:
(4:1)](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-122-320.jpg)
![4.5. Induced System Gains
107
Input u(t) Output z (t)
Signal Norms
kuk2 =
2
L2
L2
S
S
S
P
kuk2 = 21
P
P
P
kuk2 = 21
P
L1
Z1
0
Induced Norms
kuk2 dt
kGk1
kuk2 = kSuu k1
S
Z1
;1
Z1
;1
kGk1
TracefSuu (j!)gd!
kGk2
TracefSuu (j!)gd!
kGk1
kuk1 = sup max jui (t)j
i
max kgi k1
i
t
L1
kuk1 = sup ku(t)k
Z1
t
Table 4.2: Induced System Gains
Proof.
S 7;! S : If u 2 S , then
Szz (j!) = G(j!)Suu (j!)G (j!):
Now suppose
G(j!0 )] = kGk1
and take a signal u such that Suu (j!0 ) = I . Then
kSzz (j!)k1 = kGk2 :
1
S 7;! P : By de nition, we have
1 Z 1 TracefG(j!)S (j!)G (j!)g d!:
uu
P = 2 ;1
kz k2
0
kg(t)k dt](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-124-320.jpg)
![PERFORMANCE SPECIFICATIONS
108
Now let u be white with unit spectral density, i.e., Suu = I . Then
Z1
2 = 1
kz kP 2
TracefG(j!)G (j!)g d! = kGk2 :
2
;1
P 7;! P :
If u is a power signal, then
Z1
2 = 1
kz kP 2
TracefG(j!)Suu (j!)G (j!)g d!
;1
and immediately, we get that
kz kP kGk1 kukP :
To achieve the equality, assume that !0 is such that
G(j!0 )] = kGk1
and denote the singular value decomposition of G(j!0 ) by
G(j!0 ) = u1 (j!0 )v1 (j!0 ) +
r
X
i=2
i ui (j!0 )vi (j!0 )
where r is the rank of G(j!0 ) and ui vi have unit length.
If !0 < 1, write v1 (j!0 ) as
2 ej 3
6 1ej 7
v1 (j!) = 6 .. 7
6 2. 7
4
5
1
2
q ej q
where i 2 R is such that i 2 ; 0) and q is the column dimension of G. Now
let i 0 be such that
i ; j!0
i=
+ j!
i
0
and let the input u be generated from passing u through a lter:
^
2
6
6
u(t) = 6
6
4
;s
1
1 1 +s
2 ;s
2 2 +s
..
.
;s
q q +s
q
3
7
7 u(t)
7^
7
5](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-125-320.jpg)
![4.5. Induced System Gains
109
where
p
u(t) = 2 sin(!o t):
^
Then Ruu ( ) = cos(!o ), so
^^
kukP = Ruu (0) = 1:
^
^^
Also,
Then
(! ; !o ) + (! + !o)] :
Suu (j!) =
^^
2
6
6
Suu (j!) = 6
6
4
and it is easy to show
;j!
1
1 1 +j!
;j!
2 2 +j!
2
..
.
q ;j!
q q +j!
3
2
7
7 S (j!) 6
7 uu 6
7 ^^ 6
6
5
4
;j!
1 1 +j!
1
;j!
2 2 +j!
2
..
.
;j!
q q +j!
q
3
7
7
7
7
5
kukP = 1:
Finally,
1
kz k2 = 2 G(j!o )]2 + 1 G(;j!o )]2
P
2
2
= G(;j!o)]
= kGk2 :
1
Similarly, if !0 = 1, then G(j!) is real. The conclusion follows by letting u =
v1 u(t) and !o ! 1.
^
L1 7;! L1 :
1. First of all, maxi kgik1 is an upper bound on the 1-norm/1-norm system
gain:
jzi (t)j =
=
Zt
0
gi ( )u(t ; ) d
Z tX
q
jgij ( )j d kuk1
0 j =1
kgi k1 kuk1
Zt
0
jgi ( )u(t ; )j d
Z 1X
q
0 j =1
and
kz k1 := max sup jzi (t)j
i t
max kgi k1 kuk1 :
i
jgij ( )j d kuk1](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-126-320.jpg)
![4.6. Computing L2 and H2 Norms
113
with A stable. Then we have
kGk2 = trace(B Lo B ) = trace(CLc C )
2
(4:2)
where Lo and Lc are observability and controllability Gramians which can be obtained
from the following Lyapunov equations
ALc + LcA + BB = 0 A Lo + LoA + C C = 0:
Proof. Since G is stable, we have
At
g(t) = L;1 (G) = Ce B t < 0
0
t 0
and
kGk2
2
=
=
Z1
Z01
0
Tracefg (t)g(t)g dt =
TracefB
eA t C
Z1
0
Tracefg(t)g(t) g dt
CeAt B g dt =
Z1
0
TracefCeAt BB eA t C g dt:
The lemma follows from the fact that the controllability Gramian of (A B ) and the
observability Gramian of (C A) can be represented as
Lo =
Z1
0
eA t C CeAt dt Lc =
which can also be obtained from
ALc + LcA + BB = 0
Z1
0
eAt BB eA t dt
A Lo + LoA + C C = 0:
2
To compute the L2 norm of a rational transfer function, G(s) 2 L2 , using state space
approach. Let G(s) = G(s)]+ + G(s)]; with G+ 2 RH2 and G; 2 RH? . Then
2
2 = k G(s)] k2 + k G(s)] k2
kGk2
+ 2
; 2
where k G(s)]+ k2 and k G(s)]; k2 = k G(;s)]+ k2 can be computed using the above
lemma.
Still another useful characterization of the H2 norm of G is in terms of hypothetical
input-output experiments. Let ei denote the ith standard basis vector of Rm where m
is the input dimension of the system. Apply the impulsive input (t)ei ( (t) is the unit
impulse) and denote the output by zi (t)(= g(t)ei ). Assume D = 0, and then zi 2 L2+
and
m
X
kGk2 = kzi k2 :
2
2
i=1
Note that this characterization of the H2 norm can be appropriately generalized for
nonlinear time varying systems, see Chen and Francis 1992] for an application of this
norm in sampled-data control.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-130-320.jpg)
![4.7. Computing L1 and H1 Norms
115
Thus the conclusion follows if the above realization has neither uncontrollable modes
nor unobservable modes on the imaginary axis. Assume that j!0 is an eigenvalue
of H but not a pole of ;1 (s). Then j!0 must be either an unobservable mode of
BR;1
( R;1 D C R;1 B H ) or an uncontrollable mode of (H ;C DR;1 ). Now
suppose j!0 is an unobservable mode of ( R;1 D C R;1 B H ). Then there exists
an x0 = x1 6= 0 such that
x
2
Hx0 = j!0 x0
R;1 D C R;1 B
x0 = 0:
These equations can be simpli ed to
(j!0 I ; A)x1 = 0
(j!0 I + A )x2 = ;C Cx1
D Cx1 + B x2 = 0:
Since A has no imaginary axis eigenvalues, we have x1 = 0 and x2 = 0. This contradicts
our assumption, and hence the realization has no unobservable modes on the imaginary
axis.
Similarly, a contradiction will also be arrived if j!0 is assumed to be an uncontrolBR;1
lable mode of (H ;C DR;1 ).
2
Bisection Algorithm
Lemma 4.7 suggests the following bisection algorithm to compute RL1 norm:
(a) select an upper bound u and a lower bound l such that l kGk1 u
(b) if ( u ; l )= l speci ed level, stop kGk ( u + l )=2. Otherwise go to next
step
(c) set = ( l + u )=2
(d) test if kGk1 < by calculating the eigenvalues of H for the given
(e) if H has an eigenvalue on j R set l = otherwise set u = go back to step (b).
Of course, the above algorithm applies to H1 norm computation as well. Thus L1
norm computation requires a search, over either or !, in contrast to L2 (H2 ) norm
computation, which does not. A somewhat analogous situation occurs for constant
matrices with the norms kM k2 = trace(M M ) and kM k1 = M ]. In principle, kM k2
2
2
can be computed exactly with a nite number of operations, as can the test for whether
(M ) < (e.g. 2 I ; M M > 0), but the value of (M ) cannot. To compute (M ),
we must use some type of iterative algorithm.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-132-320.jpg)
![116
PERFORMANCE SPECIFICATIONS
Remark 4.4 It is clear that kGk1 < i
;1 G
1 < 1. Hence, there is no loss of
generality to assume = 1. This assumption will often be made in the remainder of
this book. It is also noted that there are other fast algorithms to carry out the above
norm computation nevertheless, this bisection algorithm is the simplest.
~
The H1 norm of a stable transfer function can also be estimated experimentally
using the fact that the H1 norm of a stable transfer function is the maximum magnitude
of the steady-state response to all possible unit amplitude sinusoidal input signals.
4.8 Notes and References
The basic concept of function spaces presented in this chapter can be found in any
standard functional analysis textbook, for instance, Naylor and Sell 1982] and Gohberg
and Goldberg 1981]. The system theoretical interpretations of the norms and function
spaces can be found in Desoer and Vidyasagar 1975]. The bisection H1 norm computational algorithm is rst developed in Boyd, Balakrishnan, and Kabamba 1989]. A more
e cient L1 norm computational algorithm is presented in Bruinsma and Steinbuch
1990].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-133-320.jpg)
![126
STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS
5.4 Coprime Factorization over RH1
Recall that two polynomials m(s) and n(s), with, for example, real coe cients, are said
to be coprime if their greatest common divisor is 1 (equivalent, they have no common
zeros). It follows from Euclid's algorithm1 that two polynomials m and n are coprime
i there exist polynomials x(s) and y(s) such that xm + yn = 1 such an equation is
called a Bezout identity. Similarly, two transfer functions m(s) and n(s) in RH1 are
said to be coprime over RH1 if there exists x y 2 RH1 such that
xm + yn = 1:
The more primitive, but equivalent, de nition is that m and n are coprime if every
common divisor of m and n is invertible in RH1 , i.e.,
h mh;1 nh;1 2 RH1 =) h;1 2 RH1 :
More generally, we have
De nition 5.3 Two matrices M and N in RH1 are right coprime over RH1 if they
have the same number of columns and if there exist matrices Xr and Yr in RH1 such
that
Xr Yr M = Xr M + Yr N = I:
N
~
~
Similarly, two matrices M and N in RH1 are left coprime over RH1 if they have the
same number of rows and if there exist matrices Xl and Yl in RH1 such that
~ ~
M N
Xl = MX + NY = I:
~ l ~ l
Yl
Note that these de nitions are equivalent to saying that the matrix M is leftN
~ ~
invertible in RH1 and the matrix M N is right-invertible in RH1 . These two
equations are often called Bezout identities.
Now let P be a proper real-rational matrix. A right-coprime factorization (rcf)
of P is a factorization P = NM ;1 where N and M are right-coprime over RH1 .
~ ~
~
~
Similarly, a left-coprime factorization (lcf) has the form P = M ;1N where N and M
are left-coprime over RH1 . A matrix P (s) 2 Rp (s) is said to have double coprime
factorization if there exist a right coprime factorization P = NM ;1 , a left coprime
~ ~
factorization P = M ;1N , and Xr Yr Xl Yl 2 RH1 such that
Xr Yr
~ ~
;N M
M ;Yl
N Xl
= I:
(5:14)
~
Of course implicit in these de nitions is the requirement that both M and M be square
and nonsingular.
1
See, e.g., Kailath, 1980, pp. 140-141].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-143-320.jpg)
![5.5. Feedback Properties
133
high frequency range, which in turn implies that the loop gain, (Lo ), should be small
at those frequencies.
Still another tradeo is with the sensor noise error reduction. The con ict between
the disturbance rejection and the sensor noise reduction is evident in equation (5.24).
Large (Lo(j!)) values over a large frequency range make errors due to d small. However, they also make errors due to n large because this noise is passed through" over
the same frequency range, i.e.,
y = To (r ; n) + So Pdi + So d (r ; n):
Note that n is typically signi cant in the high frequency range. Worst still, large loop
gains outside of the bandwidth of P , i.e., (Lo (j!)) 1 or (Li (j!)) 1 while
(P (j!)) 1, can make the control activity (u) quite unacceptable, which may cause
the saturation of actuators. This follows from
u = KSo(r ; n ; d) ; Ti di = Si K (r ; n ; d) ; Ti di P ;1 (r ; n ; d) ; di :
Here, we have assumed P to be square and invertible for convenience. The resulting
equation shows that disturbances and sensor noise are actually ampli ed at u whenever
the frequency range signi cantly exceeds the bandwidth of P , since for ! such that
(P (j!)) 1, we have
1
P ;1 (j!)] = P (j!)] 1:
Similarly, the controller gain, (K ), should also be kept not too large in the frequency
range where the loop gain is small in order to not saturate the actuators. This is because
for small loop gain (Lo (j!)) 1 or (Li (j!)) 1
u = KSo (r ; n ; d) ; Ti di K (r ; n ; d):
Therefore, it is desirable to keep (K ) not too large when the loop gain is small.
To summarize the above discussion, we note that good performance requires in some
frequency range, typically some low frequency range (0 !l ):
(PK )
1
(KP )
1
(K )
1
and good robustness and good sensor noise rejection require in some frequency range,
typically some high frequency range (!h 1)
(PK )
1
(KP )
1
(K ) M
where M is not too large. These design requirements are shown graphically in Figure 5.4.
The speci c frequencies !l and !h depend on the speci c applications and the knowledge
one has on the disturbance characteristics, the modeling uncertainties, and the sensor
noise levels.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-150-320.jpg)
![5.7. Weighted H2 and H1 Performance
137
It is easy to show that for large , the weighting functions are as shown in Figure 5.5, and thus the above loop shaping technique cannot be applied. However,
it is also easy to show that the system with controller
K = I2
gives
1
1
Ws S = s+2 0
1
0 s+2
Wt T = s+10 0
1
0 s+10
and, therefore, the robust performance criterion is satis ed.
Even if all of the above problems can be avoided, it may still be di cult to nd a
matrix function Lo so that K = P ;1 Lo is stabilizing. This becomes much harder
if P is non-minimum phase and/or unstable.
Hence some new methodologies have to be introduced to solve complicated problems.
The so-called LQG/LTR (Linear Quadratic Gaussian/Loop Transfer Recovery) procedure developed rst by Doyle and Stein 1981] and extended later by various authors
can solve some of the problems, but it is essentially limited to nominally minimum
phase and output multiplicative uncertain systems. For these reasons, it will not be
introduced here. This motivates us to consider the closed-loop performance directly in
terms of the closed-loop transfer functions instead of open loop transfer functions. The
following section considers some simple closed-loop performance problem formulations.
5.7 Weighted H2 and H1 Performance
In this section, we consider how to formulate some performance objectives into mathematically tractable problems. As shown in section 5.5, the performance objectives of
a feedback system can usually be speci ed in terms of requirements on the sensitivity
functions and/or complementary sensitivity functions or in terms of some other closedloop transfer functions. For instance, the performance criteria for a scalar system may
be speci ed as requiring
js(j!)j < 1 8! !0
js(j!)j > 1 8! > !0
1
where s(j!) = 1+p(j!)k(j!) . However, it is much more convenient to re ect the system
performance objectives by choosing appropriate weighting functions. For example, the
above performance objective can be written as
jws (j!)s(j!)j 1 8!
with
jws (j!)j =
;1
;1
8! !0
8! > !0 :](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-154-320.jpg)
![5.8. Notes and References
141
This problem can also be regarded as minimizing the maximum power of the error
subject to all bounded power disturbances: let
e
e := u
^
~
then
W
W
See = W e So Wd Sd~d~ W e So Wd
^^
KSoWd
u
u KSo Wd
and
sup kek2 = sup
^P
~
~
kdkP
1
kdkP
1
1 Z 1 Trace S (j!) d! =
ee
~~
2 ;1
We SoWd
Wu KSo Wd
2
1
:
Alternatively, if the system robust stability margin is the major concern, the weighted
complementary sensitivity has to be limited. Thus the whole cost function may be
We So Wd
W1 ToW2
1
where W1 and W2 are the frequency dependent uncertainty scaling matrices. These
design problems are usually called H1 mixed sensitivity problems. For a scalar system,
an H1 norm minimization problem can also be viewed as minimizing the maximum
magnitude of the system's steady-state response with respect to the worst case sinusoidal
inputs.
5.8 Notes and References
The presentation of this chapter is based primarily on Doyle 1984]. The discussion
of internal stability and coprime factorization can also be found in Francis 1987] and
Vidyasagar 1985]. The loop shaping design is well known for SISO systems in the
classical control theory. The idea was extended to MIMO systems by Doyle and Stein
1981] using LQG design technique. The limitations of the loop shaping design are
discussed in detail in Stein and Doyle 1991]. Chapter 18 presents another loop shaping
method using H1 control theory which has the potential to overcome the limitations
of the LQG/LTR method.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-158-320.jpg)
![PERFORMANCE LIMITATIONS
144
and complementary sensitivity functions. A well-known theorem due to Bode states
that for single-input single-output open-loop stable systems with more than one polezero excess, the integral of the logarithmic magnitude of the sensitivity function over
all frequencies must equal to zero. This integral relation therefore suggests that in the
presence of bandwidth constraints, the desirable property of sensitivity reduction in
one frequency range must be traded o against the undesirable property of sensitivity
increase at other frequencies. A result by Freudenberg and Looze 1985] further extends Bode's theorem to open-loop unstable systems, which shows that the presence
of open-loop unstable poles makes the sensitivity tradeo a more di cult task. In the
same reference, the limitations imposed by the open loop non-minimum phase zeros
upon the feedback properties were also quanti ed using the Poisson integral. The results presented here are some multivariable extensions of the above mentioned integral
relations.
We shall now consider a linear time-invariant feedback system with an n n loop
transfer matrix L. Let S (s) and T (s) be the sensitivity function and the complementary
sensitivity function, respectively
S (s) = (I + L(s));1 T (s) = L(s)(I + L(s));1 :
(6:1)
Before presenting the multivariable integral relations, recall that a point z 2 C is a transmission zero of L(s), which has full normal rank and a minimal state space realization
(A B C D), if there exist vectors and such that the relation
A ; zI B
C D
=0
holds, where
= 1, and is called the input zero direction associated with z . Analogously, a transmission zero z of L(s) satis es the relation
x
w
A ; zI B = 0
C D
where x and w are some vectors with w satisfying the condition w w = 1. The vector
w is called the output zero direction associated with z . Note also that p 2 C is a pole of
L(s) if and only if it is a zero of L;1(s). By a slight abuse of terminology, we shall call
the input and output zero directions of L;1 (s) the input and output pole directions of
L(s), respectively.
In the sequel we shall preclude the possibility that z is both a zero and pole of L(s).
Then, by Lemma 3.27 and 3.28, z is a zero of L(s) if and only if L(z ) = 0 for some
vector ,
= 1, or w L(z ) = 0 for some vector w, w w = 1. Similarly, p is a pole of
L(s) if and only if L;1(p) = 0 for some vector ,
= 1, or w L;1(p) = 0 for some
vector w, w w = 1.
It is well-known that a non-minimum phase transfer function admits a factorization
that consists of a minimum phase part and an all-pass factor. Let zi 2 C + , i = 1
k,
be the non-minimum phase zeros of L(s) and let i i i = 1 be the input directions
generated from the following iterative procedure](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-161-320.jpg)
![PERFORMANCE LIMITATIONS
148
p
+ 1 (Re(p1 ; p2 ))2 + 4Rep1 Rep2 cos2 (
2
1 2 ):
2
The proof is now completed.
The utility of this corollary is clear. This result fully characterizes the limitation
imposed by a pair of open-loop unstable poles on the sensitivity reduction properties.
This limitation depends not only on the relative distances of the poles to the imaginary
axis, but also on the principal angle between the two pole directions.
Next, we investigate the design constraints imposed by open-loop non-minimum
phase zeros upon sensitivity properties. The results below may be considered to be a
matrix extension of the Poisson integral relation.
Theorem 6.4 Let S (s) 2 H1 be factorized as in (6.3) and assume that
lim
max
R!1 2 ; =2 =2]
ln (S (Rej ))
R
= 0:
(6:8)
Then, for any non-minimum phase zero z = x0 + jy0 2 C + of L(s) with output direction
w, w w = 1,
Z1
Z1
x
ln (S (j!)) x2 + (!0; y )2 d!
0
;1
0
x
ln (S (j!)) x2 + (!0; y )2 d!
0
;1
0
ln (Sm (z ))
;
ln w Bk 1 (z )
ln (S (z )):
;
B1 1 ( z ) :
(6:9)
(6:10)
Note that the condition (6.8) is satis ed if L(s) is a proper rational transfer matrix.
Furthermore, for single-input single-output systems (n = 1),
w
and
Z1
k
Y
i=1
Bi;1 (z ) =
k
Y z + pi
i=1
z ; pi
k
Y z + pi
x
ln jS (j!)j x2 + (!0; y )2 d! = ln
:
0
;1
0
i=1 z ; pi
The more interesting result, however, is the inequality (6.10). This result again suggests
that the multivariable sensitivity properties are closely related to the pole and zero
directions. This result implies that the sensitivity reduction ability of the system may
be severely limited by the open-loop unstable poles and non-minimum phase zeros,
especially when these poles and zeros are close to each other and the angles between
their directions are small.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-165-320.jpg)
![6.3. Design Limitations and Sensitivity Bounds
149
6.3 Design Limitations and Sensitivity Bounds
The integral relations derived in the preceding section are now used to analyze the design tradeo s and the limitations imposed by the bandwidth constraint and right-half
plane poles and zeros upon sensitivity reduction properties. Similar to their counterparts for single-input single-output systems, these integral relations show that there will
necessarily exist frequencies at which the sensitivity function exceeds one if it is to be
kept below one over other frequencies, hence exhibiting a tradeo between the reduction
of the sensitivity over one frequency range against its increase over another frequency
range.
Suppose that the feedback system is designed such that the level of sensitivity reduction is given by
(S (j!)) ML < 1 8! 2 ;!L !L ]
(6:11)
where ML > 0 is a given constant. Let z = x0 + jy0 2 C + be an open right-half plane
zero of L(s) with output direction w. De ne also
(z ) :=
Z !L
x0
2 + (! ; y0 )2 d!:
;!L x0
The following lower bound on the maximum sensitivity displays a limitation due to the
open right-half plane zeros upon the sensitivity reduction properties.
Corollary 6.5 Let the assumption in Theorem 6.4 holds. In addition, suppose that the
condition (6.11) is satis ed. Then, for each open right-half plane zero z 2 C + of L(s)
with output direction w,
kS (s)k1
1
ML
and
kS (s)k1
(z )
; (z)
1
ML
;
w Bk 1 (z )
(z )
; (z )
;
B1 1 (z )
( (S (z )))
; (z)
; (z)
:
(6:12)
(6:13)
Proof. Note that
Z1
x
ln (S (j!)) x2 + (!0; y )2 d! ( ; (z )) ln kS (j!)k1 + (z ) ln(ML):
0
;1
0
Then the inequality (6.12) follows by applying inequality (6.10) and inequality (6.13)
follows by applying inequality (6.9)
2
The interpretation of Corollary 6.5 is similar to that in single-input single-output
systems. Roughly stated, this result shows that for a non-minimum phase system, its](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-166-320.jpg)
![PERFORMANCE LIMITATIONS
150
sensitivity must increase beyond one at certain frequencies if the sensitivity reduction is
to be achieved at other frequencies. Of particular importance here is that the sensitivity
function will in general exhibit a larger peak in multivariable systems than in singleinput single-output systems, due to the fact that (S (z )) 1.
The design tradeo s and limitations on the sensitivity reduction which arise from
bandwidth constraints as well as open-loop unstable poles can be studied using the
extended Bode integral relations. However, these integral relations by themselves do
not mandate a meaningful tradeo between the sensitivity reduction and the sensitivity
increase, since the sensitivity function can be allowed to exceed one by an arbitrarily
small amount over an arbitrarily large frequency range so as not to violate the Bode
integral relations. However, bandwidth constraints in feedback design typically require
that the open-loop transfer function be small above a speci ed frequency, and that it roll
o at a rate of more than one pole-zero excess above that frequency. These constraints
are commonly needed to ensure stability robustness despite the presence of modeling
uncertainty in the plant model, particularly at high frequencies. One way of quantifying
such bandwidth constraints is by requiring the open-loop transfer function to satisfy
MH
< 1 8! 2 !H 1)
(6:14)
!1+k
where !H > !L , and MH > 0, k > 0 are some given constants. With the bandwidth
(L(j!))
constraint given as such, the following result again shows that the sensitivity reduction
speci ed by (6.11) can be achieved only at the expense of increasing the sensitivity at
certain frequencies.
Corollary 6.6 Suppose the Assumptions 6.1-6.2 hold. In addition, suppose that the
conditions (6.11) and (6.14) are satis ed for some !H and !L such that !H > !L .
Then
!L
!
1 !H ;!L (1 ; ) k(!HH !L)
;
(6:15)
max (S (j!)) e
ML
!2 !L !H ]
where
=
max
Pk
i=1 (Repi ) i i
!H ; !L
:
Proof. Note rst that for ! !H ,
(S (j!)) = (I + 1 (j!))
L
and
Z1
1
1 ; (L(j!))
1
MH d! = X Z 1 1 MH i d!
; ln 1 ; !1+k
1+k
!H
i=1 !H i !
1
MH
1 ; !1+k](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-167-320.jpg)
![6.4. Bode's Gain and Phase Relation
151
!i
1
X1
!H
MH
=
1+
i i(1 + k) ; 1 !H k
i=1
!i
!
1
!H X 1 MH = ; !H ln 1 ; MH
1+
1+
k i=1 i !H k
k
!H k
; !kH ln(1 ; ):
Then
Z1
0
ln (S (j!))d! =
Z !L
0
ln (S (j!))d! +
Z !H
!L
ln (S (j!))d! +
Z1
Z1
!H
ln (S (j!))d!
MH
ln 1 ; !1+k d!
L H
!H
!H ln(1 ; )
!L ln ML + (!H ; !L ) !2max ] ln (S (j!)) ; k
!L !H
and the result follows from applying (6.4).
!L ln ML + (!H ; !L ) !2max ] ln (S (j!)) ;
! !
2
The above lower bound shows that the sensitivity can be very signi cant in the
transition band.
6.4 Bode's Gain and Phase Relation
In the classical feedback theory, the Bode's gain-phase integral relation (see Bode 1945])
has been used as an important tool to express design constraints in scalar systems. The
following is an extended version of the Bode's gain and phase relationship for an openloop stable scalar system with possible right-half plane zeros, see Freudenberg and Looze
1988] and Doyle, Francis, and Tannenbaum 1992]:
k
1 Z 1 d ln jLj ln coth j j d + Y j!0 + zi
L(j!0 ) =
2
;1 d
i=1 j!0 ; zi
where zi 's are assumed to be the right-half plane zeros of L(s) and := ln(!=!0 ). Note
L
that d ln jd (j!)j is the slope of the Bode plot which is almost always negative. It follows
that L(j!0 ) will be large if the gain L attenuates slowly and small if it attenuates
rapidly. The behavior of L(j!) is particularly important near the crossover frequency
!c where jL(j!c )j = 1 since + L(j!c) is the phase margin of the feedback system,
and further the return di erence is given by
L
j1 + L(j!c )j = j1 + L;1(j!c )j = 2 sin + 2 (j!c)](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-168-320.jpg)
![PERFORMANCE LIMITATIONS
152
which must not be too small for good stability robustness. If + L(j!c ) is forced to
be very small by rapid gain attenuation, the feedback system will amplify disturbances
and exhibit little uncertainty tolerance at and near !c . Since j!0 + z i 0 for each i,
j!0 ; zi
a non-minimum phase zero contributes an additional phase lag and imposes limitations
upon the rollo rate of the open-loop gain. The con ict between attenuation rate
and loop quality near crossover is thus clearly evident. A thorough discussion of the
limitations these relations impose upon feedback control design is given by Bode 1945],
Horowitz 1963], and Freudenberg and Looze 1988]. See also Freudenberg and Looze
1988] for some multivariable generalizations.
In the classical feedback theory, it has been common to express design goals in
terms of the shape" of the open-loop transfer function. A typical design requires that
the open-loop transfer function have a high gain at low frequencies and a low gain at
high frequencies while the transition should be well-behaviored. The same conclusion
applies to multivariable system where the singular value plots should be well-behaviored
between the transition band.
6.5 Notes and References
The results presented in this chapter are based on Chen 1992a, 1992b, 1995]. Some
related results can be found in Boyd and Desoer 1985] and Freudenberg and Looze
1988]. The related results for scalar systems can be found in Bode 1945], Horowitz
1963], Doyle, Francis, and Tannenbaum 1992], and Freudenberg and Looze 1988].
The study of analytic functions, harmonic functions1 , and various integral relations
in the scalar case can be found in Garnett 1981] and Ho man 1962].
1 A function : C 7;! R is said to be a harmonic function (subharmonic function) if 52 ( ) = 0
(52 ( ) 0) where the symbol 52 ( ) with = + denotes the Laplacian of ( ) and is de ned
by
2
2
52 ( ) := (2 ) + (2 )
f
f s
f s
f s
s
f s
x
jy
f s
@ f s
@ f s
@x
@y
:](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-169-320.jpg)
![7.1. Model Reduction by Balanced Truncation
157
where i has multiplicity si i = 1 2 : : : N and s1 + s2 + + sN = n. Then the
truncated system
Gr (s) = A11 B1
C1 D
is balanced and asymptotically stable. Furthermore
kG(s) ; Gr (s)k1 2( r+1 + r+2 + + N )
and the bound is achieved if r = N ; 1, i.e.,
kG(s) ; GN ;1 (s)k1 = 2 N :
Proof. The stability of Gr follows from Theorem 7.1. We shall now prove the error
bound for the case si = 1 for all i. The case where the multiplicity of i is not equal to
one is more complicated and an alternative proof is given in the next chapter. Hence,
we assume si = 1 and N = n.
Let
(s) := (sI ; A11 );1
(s) := sI ; A22 ; A21 (s)A12
~
B (s) := A21 (s)B1 + B2
~
C (s) := C1 (s)A12 + C2
then using the partitioned matrix results of section 2.3,
G(s) ; Gr (s) = C (sI ; A);1 B ; C1 (s)B1
sI ; A11 ;A12 ;1 B1 ; C (s)B
1
1
;A21 sI ; A22
B2
~
~
= C (s) ;1 (s)B (s)
=
C1 C2
computing this quantity on the imaginary axis to get
h
i
~
~
~
~
G(j!) ; Gr (j!)] = 1=2 ;1 (j!)B (j!)B (j!) ; (j!)C (j!)C (j!) : (7:13)
max
~
~
~
~
Expressions for B (j!)B (j!) and C (j!)C (j!) are obtained by using the partitioned
form of the internally balanced Gramian equations (7.3){(7.8).
~
~
An expression for B (j!)B (j!) is obtained by using the de nition of B (s), substituting for B1 B1 , B1 B2 and B2 B2 from the partitioned form of the Gramian equations
(7.3){(7.5), we get
~
~
B (j!)B (j!) = (j!) 2 + 2 (j!):
~
~
The expression for C (j!)C (j!) is obtained analogously and is given by
~
~
C (j!)C (j!) = 2 (j!) + (j!) 2 :](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-174-320.jpg)
![158
MODEL REDUCTION BY BALANCED TRUNCATION
~
~
~
~
These expressions for B (j!)B (j!) and C (j!)C (j!) are then substituted into (7.13)
to obtain
G(j!) ; Gr (j!)] = 1=2
max
2+
;1 (j!) 2 (j!)
2+
Now consider one-step order reduction, i.e., r = n ; 1, then
; (j!) 2 (j!) :
2= n
and
G(j!) ; Gr (j!)] = n 1=2 1 + ;1(j!) 1 + (j!)]
(7:14)
max
where := ; (j!) (j!) = ; is an all pass" scalar function. (This is the only
place we need the assumption of si = 1) Hence j (j!)j = 1.
Using triangle inequality we get
G(j!) ; Gr (j!)]
n 1 + j (j! )j] = 2 n :
(7:15)
This completes the bound for r = n ; 1.
The remainder of the proof is achieved by using the order reduction by one step
results and by noting that Gk (s) = A11 B1 obtained by the k-th" order partiC1 D
tioning is internally balanced with balanced Gramian given by
1 = diag( 1 Is1
2 Is2
: : : k Isk ):
Let Ek (s) = Gk+1 (s) ; Gk (s) for k = 1 2 : : : N ; 1 and let GN (s) = G(s). Then
Ek (j!)] 2 k+1
since Gk (s) is a reduced order model obtained from the internally balanced realization
of Gk+1 (s) and the bound for one-step order reduction, (7.15) holds.
Noting that
G(s) ; Gr (s) =
by the de nition of Ek (s), we have
G(j!) ; Gr (j!)]
N ;1
X
k =r
N ;1
X
k=r
Ek (s)
Ek (j!)] 2
N ;1
X
k=r
k+1 :
This is the desired upper bound.
To see that the bound is actually achieved when r = N ; 1, we note that (0) = I .
Then the right hand side of (7.14) is 2 N at ! = 0.
2
The singular values i are called the Hankel singular values. A useful consequence
of the above theorem is the following corollary.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-175-320.jpg)
![7.3. Relative and Multiplicative Model Reductions
163
Proof. Since the input weighting matrix Wi = I , it is obvious that the input weighted
Gramian is given by P . Now the output weighted transfer matrix is given by
G;1 (G ; D) =
2 A
3 "
#
0
B
;1 C A ; BD;1 C 0 5 =: A B :
4 ;BD
C 0
;1
;1
D C
It is easy to verify that
D C
0
Q := Q Q
Q Q
satis es the following Lyapunov equation
QA + A Q + C C = 0:
Hence Q is the output weighted Gramian.
The proof for part (b) is more involved and needs much more work. We refer readers
to the references at the end of the chapter for details.
2
In the above theorem, we have assumed that the system is square, we shall now
extend the results to include non-square case. Let G(s) be a minimum phase transfer
matrix with a minimal realization
A B
G(s) = C D
and assume that D has full row rank. Without loss of generality, we shall also assume
that D is normalized such that DD = I . Let D? be a matrix with full row rank such
D
that D is square and unitary.
?
Lemma 7.6 A complex number z 2 C is a zero of G(s) if and only if z is an uncontrollable mode of (A ; BD C BD? ).
A B
Proof. Since D has full row rank and G(s) = C D is a minimal realization, z
is a transmission zero of G(s) if and only if
A ; zI B
C D
does not have full row rank. Now note that
A ; zI B
C D
I
0
0 D D? ]
2 I 0
4 ;C I
0
0
0
0 I
3
5](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-180-320.jpg)
![7.4. Notes and References
167
It can be shown that the balanced stochastic realization and the self-weighted balanced
p
realization in Theorem 7.5 are the same and i = i = 1 + i2 if G(s) is minimum phase.
To illustrate, consider a third order stable and minimum phase transfer function
s3 + 2s2 + 3s + 4
G(s) = s3 + 3s2 + 4s + 4 :
It is easy to show that the Hankel singular values of the phase function is given by
1 = 0:55705372196966
2 = 0:53088390857035
3 = 0:03715882438832
and a rst order BST approximation is given by
^ s + 0 00375717470515
G = s + 0::00106883052786:
The relative approximation error is 2:51522024045904 and the error bound is
3
Y 1+
i
; 1 = 2:51522024046226:
i=2 1 ; i
7.4 Notes and References
The balanced model reduction method was rst introduced by Moore 1981]. The stability properties of the reduced order model were shown by Pernebo and Silverman 1982].
The error bound for the balanced model reduction was shown by Enns 1984] and Glover
1984] subsequently gave an independent proof. The frequency weighted balanced model
reduction method was also introduced by Enns 1984] from a somewhat di erent perspective. The error bounds for the relative and multiplicative approximations using the
self-weighted balanced realization were shown by Zhou 1993]. The Balanced Stochastic
Truncation (BST) method was proposed by Desai and Pal 1984] and generalized by
Green 1988a,1988b] and many other people. The relative error bound for the Balanced
Stochastic Truncation method was obtained by Green 1988a] and the multiplicative
error bound for the BST was obtained by Wang and Safonov 1992]. It was also shown
in Zhou 1993] that the frequency self-weighted method and the BST method are the
same if the transfer matrix is minimum phase. Improved error bounds for the BST
reduction were reported in Wang and Safonov 1990,1992] where it was claimed that
the following error bounds hold
n
X
i
^
G;1 (G ; G) 1 2
1; i
i=r+1
^
^
G;1 (G ; G)
1
2
n
X
i
1; i:
i=r+1](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-184-320.jpg)
![168
MODEL REDUCTION BY BALANCED TRUNCATION
However, the example in the last section gives
2
3
X
i
1 ; i = 2:34052280324021
i=2
which is smaller than the actual error. Other weighted model reduction methods can
be found in Al-Saggaf and Franklin 1988], Glover 1986,1989], Glover, Limebeer and
Hung 1992], Hung and Glover 1986] and references therein. Discrete time balance
model reduction results can be found in Al-Saggaf and Franklin 1987], Hinrichsen and
Pritchard 1990], and references therein.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-185-320.jpg)
![HANKEL NORM APPROXIMATION
170
The above bounds are then used to show that the k-th order optimal Hankel norm
^
approximation, G(s), together with some constant matrix D0 satis es
^
G(s) ; G(s) ; D0 1 ( k+1 + : : : + n ):
We shall also provide an alternative proof for the error bounds derived in the last chapter
for the truncated balanced realizations using the results obtained in this chapter.
Finally we consider the Hankel operator in discrete time and o er some alternative
proof of the well-known Nehari's theorem.
8.1 Hankel Operator
Let G(s) 2 L1 be a matrix function. The Hankel operator associated with G will be
denoted by ;G and is de ned as
?
;G : H2 7;! H2
?
;G f := (P+ MG)f = P+ (Gf ) for f 2 H2
i.e., ;G = P+ MGjH? . This is shown in the following diagram:
2
MG
?
H2
;G
>
L2
P+
-
?
H2
There is a corresponding Hankel operator in the time domain. Let g(t) denote the
inverse (bilateral) Laplace transform of G(s). Then the time domain Hankel operator
is
;g : L2 (;1 0] 7;! L2 0 1)
;g f := P+ (g f ) for f 2 L2 (;1 0]:
Thus
R 0 g(t ; )f ( )d t 0
(;g f )(t) = ;1
0
t < 0:
Because of the isometric isomorphism property between the L2 spaces in the time domain and in the frequency domain, we have
k;g k = k;G k :](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-187-320.jpg)
![HANKEL NORM APPROXIMATION
172
Γ
g
u(t)
0
y(t)
t
0
t
Figure 8.1: System theoretical interpretation of Hankel operators
from the initial state to the future output. These two operators will be called the controllability operator, c , and the observability operator, o, respectively, and are de ned
as
c : L2 (;1 0] 7;! C n
c u :=
and
Z0
;1
e;A Bu( )d
o : C n 7;! L2 0 1)
o x0 := CeAt x0 t 0:
(If all the data are real, then the two operators become c : L2 (;1 0] 7;! Rn and
o : Rn 7;! L2 0 1).) Clearly, x0 = c u(t) for u(t) 2 L2 (;1 0] is the system state
at t = 0 due to the past input and y(t) = o x0 t 0, is the future output due to the
initial state x0 with the input set to zero.
It is easy to verify that
;g = o c :
c
L2 (;1 0]
>
Cn
;g
ZZ
Z
o
ZZ
~
Z
- L2 0 1 )
The adjoint operators of c and o can also be obtained easily from their de nitions
as follows: let u(t) 2 L2 (;1 0], x0 2 C n , and y(t) 2 L2 0 1), then
h cu
x0 iC n
=
Z0
u ( )B e;A x0 d = hu B e;A x0 iL2 (;1 0] = hu c x0 iL2 (;1 0]
;1](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-189-320.jpg)
![8.1. Hankel Operator
and
h o x0 yiL2 0 1) =
Z1
0
x
173
(t)eA t C y(t)dt = hx0
Z1
0
eA t C y(t)dtiC n = hx0 o yiC n
where h iX denotes the inner product in the Hilbert space X . Therefore, we have
n
c : C 7;! L2 (;1 0]
;A x0
0
c x0 = B e
and
n
o : L2 0 1) 7;! C
o y (t) =
Z1
0
eA t C y(t)dt:
This also gives the adjoint of ;g :
;g = ( o c) = c o : L2 0 1) 7;! L2 (;1 0]
;g y =
Z1
0
B eA (t; )C y(t)dt
0:
Let Lc and Lo be the controllability and observability Gramians of the system, i.e.,
Lc =
Lo =
Then we have
Z1
Z0
0
1
eAt BB eA t dt
eA t C CeAt dt:
c c x0 =
o o x0 =
Lcx0
Lox0
for every x0 2 C n . Thus Lc and Lo are the matrix representations of the operators
c c and o o .
Theorem 8.1 The operator ;g ;g (or ;G;G) and the matrix LcLo have the same
p
nonzero eigenvalues. In particular k;g k = (Lc Lo).
Proof. Let
6= 0 be an eigenvalue of ;g ;g , and let 0 6= u 2 L2 (;1 0] be a corresponding eigenvector. Then by de nition
;g ;g u = c o o cu = 2 u:
(8:1)
Pre-multiply (8.1) by c and de ne x = cu 2 C n to get
LcLo x = 2 x:
(8:2)
2](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-190-320.jpg)
![HANKEL NORM APPROXIMATION
174
Note that x = cu 6= 0 since otherwise 2 u = 0 from (8.1) which is impossible. So 2
is an eigenvalue of LcLo .
On the other hand, suppose 2 6= 0 and x 6= 0 are an eigenvalue and a corresponding
eigenvector of LcLo . Pre-multiply (8.2) by c Lo and de ne u = c Lox to get (8.1). It
is easy to see that u 6= 0 since c u = c c Lox = LcLo x = 2 x 6= 0. Therefore 2 is
an eigenvalue of ;g ;g .
Finally, since G(s) is rational, ;g ;g is compact and self-adjoint and has only discrete
spectrum. Hence k;g k2 = ;g ;g = (Lc Lo).
2
Remark 8.1 Let
6= 0 be an eigenvalue of ;g ;g and 0 6= u 2 L2 (;1 0] be a
corresponding eigenvector. De ne
v := 1 ; u 2 L 0 1):
2
g
2
Then (u v) satisfy
;g u =
;g v =
v
u:
This pair of vectors (u v) are called a Schmidt pair of ;g . The proof given above suggests
a way to construct this pair: nd the eigenvalues and eigenvectors of LcLo , i.e., i2 and
xi such that
Lc Loxi = i2 xi :
Then the pairs (ui vi ) given below are the corresponding Schmidt pairs:
u = ( 1 L x ) 2 L (;1 0] v = x 2 L 0 1):
i
c i o i
i
2
o i
2
~
Remark 8.2 As seen in various literature, there are some alternative ways to write a
Hankel operator. For comparison, let us examine some of the alternatives below:
(i) Let v(t) = u(;t) for u(t) 2 L2 (;1 0], and then v(t) 2 L2 0 1). Hence, the
Hankel operator can be written as
;g : L2 0 1) 7;! L2 0 1) or ;G : H2 7;! H2
(;g v)(t) =
=
R 1 g(t + )v( )d
0
t 0
t<0
1
CeA(t+ )Bv( )d for t 0:
0
Z
0](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-191-320.jpg)
![8.2. All-pass Dilations
175
(ii) In some applications, it is more convenient to work with an anticausal operator G
and view the Hankel operator associated with G as the mapping from the future
input to the past output. It will be clear in later chapters that this operator is
closely related to the problem of approximating an anticausal function by a causal
function, which is the problem at the heart of the H1 Control Theory.
A
Let G(s) = C B be an antistable transfer matrix, i.e., all the eigenvalues of
0
A have positive real parts. Then the Hankel operator associated with G(s) can
be written as
^
;g : L2 0 1) 7;! L2 (;1 0]
^
(;g v)(t) =
=
R 1 g(t ; )v( )d
0
t 0
t>0
1
CeA(t; )Bv( )d for t 0
0
Z
0
or in the frequency domain
?
^
;G = P; MGjH2 : H2 7;! H2
^
;Gv = P; (Gv) for v 2 H2 :
?
Now for any v 2 H2 and u 2 H2 , we have
hP; (Gv) ui = hGv ui = hv G ui = hv P+ (G u)i:
^
Hence, ;G = ;G .
~
8.2 All-pass Dilations
This section considers the dilation of a given transfer function to an all-pass transfer
function. This transfer function dilation is the key to the optimal Hankel norm approximation in the next section. But rst we need some preliminary results and some state
space characterizations of all-pass functions.
De nition 8.1 The inertia of a general complex, square matrix A denoted In(A) is the
triple ( (A) (A) (A)) where
(A) = number of eigenvalues of A in the open right half-plane.
(A) = number of eigenvalues of A in the open left half-plane.
(A) = number of eigenvalues of A on the imaginary axis.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-192-320.jpg)
![HANKEL NORM APPROXIMATION
196
Furthermore, let H2 (@ D ) be the (closed) subspace of L2 (@ D ) with matrix functions F ( )
analytic in D , i.e.,
H2 (@ D ) = F ( ) 2 L2 (@ D ) : 21
Z
2
0
F (ej )ejn d = 0 for all n > 0
?
and let H2 (@ D ) be the (closed) subspace of L2 (@ D ) with matrix functions F ( ) analytic
in C nD .
It can be shown that the Fourier transform (or bilateral Z transform) gives the
following isometric isomorphism:
L2 (@ D ) = l2 (;1 1)
H2 (@ D ) = l2 0 1)
?
H2 (@ D ) = l2 (;1 0):
Remark 8.3 It should be kept in mind that, in contrast to the variable z in the standard Z -transform, here denotes = z ;1 .
~
Analogous to the space in the half plane, L1 (@ D ) is used to denote the Banach
space of essentially bounded matrix functions with norm
kF k1 = ess sup F (ej ) :
202 ]
The Hardy space H1 (@ D ) is the closed subspace of L1 (@ D ) with functions analytic in
D and is de ned as
H1 (@ D ) = F ( ) 2 L1 (@ D ) :
The H1 norm is de ned as
Z
2
0
F (ej )ejn d = 0 for all n > 0 :
kF k1 := sup F ( )] = ess sup F (ej ) :
2D
202 ]
It is easy to see that L1 (@ D ) L2 (@ D ) and H1 (@ D ) H2 (@ D ). (However, it should
be pointed out that these inclusions are not true for functions in the half planes or for
continuous time functions.)
Example 8.4 Let F ( ) 2 L2 (@ D ), and let F ( ) have a power series representation as
follows:
F( ) =
1
X
i=;1
Fi i :
?
Then F ( ) 2 H2 (@ D ) if and only if Fi = 0 for all i < 0 and F ( ) 2 H2 (@ D ) if and only
if Fi = 0 for all i 0.
3](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-213-320.jpg)
![8.8. Nehari's Theorem*
203
Corollary 8.20 Suppose G 2 L1, then
^
inf kG ; Qk1 = ;G
Q2H1
and the in mum is achieved.
Proof. This follows from the fact that
^
inf kG ; Qk1 = inf ; kG ; Q k1 = k;G k = ;G :
Q2H1
Q 2H1
2
Let G 2 RH1 and g be the impulse response of G and let 1 be the largest Hankel
singular value of G, i.e., k;G k = 1 . Suppose u 2 L2 (;1 0] (or l2 (;1 0)) and
v 2 L2 0 1) (or l2 0 1)) are the corresponding Schmidt pairs:
;g u = 1 v
;g v = 1 u:
Now denote the Laplace transform (or Z -transform) of u and v as U 2 RH? and
2
V 2 RH2 .
Lemma 8.21 Let G 2 RH1 and 1 be the largest Hankel singular value of G. Then
inf; kG ; Qk1 = 1
Q2H1
and
(G ; Q)U = 1 V:
Moreover, if G is a scalar function, then Q = G ; 1 V=U is the unique solution and
G ; Q is all-pass.
Proof. Let H := (G ; Q)U and note that ;GU 2 RH2 and P+H = P+(GU ) = ;GU .
Then
0
=
=
=
=
=
=
kH ; ;GU k2
2
2 + k; U k2 ; hH ; U i ; h; U H i
kH k2
G 2
G
G
2 + k; U k2 ; hP H ; U i ; h; U P H i
kH k2
G 2
+
G
G
+
2 ; h; U ; U i
kH k2 G G
kH k2 ; hU ;G ;GU i
2
2
kH k2 ; 1 hU U i
2
2
kH k2 ; 1 kU k2
2
2
2
kG ; Qk2 kU k2 ; 1 kU k2
1
2
2
= 0:](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-220-320.jpg)
![HANKEL NORM APPROXIMATION
204
Hence, H = ;G U , i.e., (G ; Q)U = ;GU = 1 V .
Now if G is a scalar function, then Q is uniquely determined and Q = G ; 1 V=U .
We shall prove through explicit construction below that V=U is an all-pass.
2
Formulas for Continuous Time
Let G(s) 2 RH1 and let
A
G(s) = C B :
0
Let Lc and Lo be the corresponding controllability and observability Gramians:
And let
2
1,
ALc + LcA + BB = 0
A Lo + Lo A + C C = 0:
be the largest eigenvalue and the corresponding eigenvector of Lc Lo:
Lc Lo =
De ne
2
1
:
:= 1 Lo :
1
Then
u =
v =
;A 2 L2 (;1 0]
c =B e
At 2 L2 0 1)
o = Ce
are the Schmidt pair and
U (s) =
V (s) =
;A
?
;B 0 2 RH2
A
C 0 2 RH2 :
It is easy to show that if G is a scalar, then U V are scalar transfer functions and
U U = V V . Hence, V=U is an all-pass. The details are left as an exercise for the
reader.
Formulas for Discrete Time
Similarly, let G be a discrete time transfer matrix and let
A
G( ) = C ( ;1 I ; A);1 B = C B :
0](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-221-320.jpg)
![HANKEL NORM APPROXIMATION
206
Suppose H has a singular value decomposition"
H = U V
= diagf 1
U = u1 u2
V = v1 v2
2
with U U = UU = I and V V = V V = I . Then
2
u11
6 u12
u = u1 = 4 .
..
3
7
5
g
2
v11
6 v12
v = v1 = 4 .
..
3
7
5
where u and v are partitioned such that u1i 2 Rp and v1i 2 Rq . Finally, U ( ) and V ( )
can be obtained as
U( ) =
V( ) =
1
X
i=1
1
X
i=1
u1i ;i 2 RH?
2
v1i i;1 2 RH2 :
In particular, if G( ) is an n-th order matrix polynomial, then matrix H has only a
nite number of nonzero elements and
H = Hn 0
0 0
with
2
6
6
=6
6
6
4
G1 G2
G2 G3
G3 G4
Gn;1 Gn
Gn 0
3
7
7
7
7
7
5
0
0 :
..
..
..
.
.
.
Gn 0
0
0
Hence, u and v can be obtained from a standard SVD of Hn .
Hn
..
.
8.9 Notes and References
The study of the Hankel operators and of the optimal Hankel norm approximation
theory can be found in Adamjan, Arov, and Krein 1978], Bettayeb, Silverman, and
Safonov 1980], Kung and Lin 1981], Power 1982], Francis 1987], Kavranoglu and
Bettayeb 1994] and references therein. The presentation of this chapter is based on
Francis 1987] and Glover 1984,1989].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-223-320.jpg)
![9.1. Model Uncertainty
209
A simple example of this involves using frequency-domain bounds on transfer functions
to describe a model set. To use such sets to describe physical systems, the bounds
must roughly grow with frequency. In particular, at su ciently high frequencies, phase
is completely unknown, i.e., 180o uncertainties. This is a consequence of dynamic
properties which inevitably occur in physical systems. This gives a less structured
representation of uncertainty.
Examples of less-structured representations of uncertainty are direct set membership
statements for the transfer function matrix of the model. For instance, the statement
(j!)] < 1 8! 0
(9:1)
P (s) = P (s) + W1 (s) (s)W2 (s)
where W1 and W2 are stable transfer matrices that characterize the spatial and frequency
structure of the uncertainty, con nes the matrix P to a neighborhood of the nominal
model P . In particular, if W1 = I and W2 = w(s)I where w(s) is a scalar function,
then P describes a disk centered at P with radius w(j!) at each frequency as shown
in Figure 9.1. The statement does not imply a mechanism or structure which gives rise
to . The uncertainty may be caused by parameter changes, as mentioned above or by
neglected dynamics or by a host of other unspeci ed e ects. An alternative statement
to (9.1) is the so-called multiplicative form:
P (s) = (I + W1 (s) (s)W2 (s))P (s):
(9:2)
This statement con nes P to a normalized neighborhood of the nominal model P . An
advantage of (9.2) over (9.1) is that in (9.2) compensated transfer functions have the
same uncertainty representation as the raw model (i.e., the weighting functions apply to
PK as well as P ). Some other alternative set membership statements will be discussed
later.
P(j ω )
w(jω )
Figure 9.1: Nyquist Diagram of an Uncertain Model
The best choice of uncertainty representation for a speci c FDLTI model depends, of
course, on the errors the model makes. In practice, it is generally possible to represent](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-226-320.jpg)
![210
MODEL UNCERTAINTY AND ROBUSTNESS
some of these errors in a highly-structured parameterized form. These are usually
the low frequency error components. There are always remaining higher frequency
errors, however, which cannot be covered this way. These are caused by such e ects as
in nite-dimensional electro-mechanical resonance, time delays, di usion processes, etc.
Fortunately, the less-structured representations, e.g., (9.1) or (9.2), are well suited to
represent this latter class of errors. Consequently, (9.1) and (9.2) have become widely
used generic" uncertainty representations for FDLTI models. An important point is
that the construction of the weighting matrices W1 and W2 for multivariable systems is
not trivial.
Motivated from these observations, we will focus for the moment on the multiplicative description of uncertainty. We will assume that P in (9.2) remains a strictly proper
FDLTI system for all . More general perturbations (e.g., time varying, in nite dimensional, nonlinear) can also be covered by this set provided they are given appropriate
conic sector" interpretations via Parseval's theorem. This connection is developed in
Safonov, 1980] and Zames, 1966] and will not be pursued here.
When used to represent the various high frequency mechanisms mentioned above,
the weighting functions in (9.2) commonly have the properties illustrated in Figure 9.2.
They are small ( 1) at low frequencies and increase to unity and above at higher
frequencies. The growth with frequency inevitably occurs because phase uncertainties
eventually exceed 180 degrees and magnitude deviations eventually exceed the nominal transfer function magnitudes. Readers who are skeptical about this reality are
encouraged to try a few experiments with physical devices.
nominal model
log ω
actual model
Figure 9.2: Typical Behavior of Multiplicative Uncertainty: p (s) = 1 + w(s) (s)]p(s)
Also note that the representation of uncertainty in (9.2) can be used to include perturbation e ects that are in fact certain. A nonlinear element, may be quite accurately](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-227-320.jpg)
![9.2. Small Gain Theorem
211
modeled, but because our design techniques cannot e ectively deal with the nonlinearity, it is treated as a conic linearity1 . As another example, we may deliberately choose
to ignore various known dynamic characteristics in order to achieve a simple nominal
design model. This is the model reduction process discussed in the previous chapters.
The following terminologies are used in this book.
De nition 9.1 Given the description of an uncertainty model set and a set of performance objectives, suppose P 2 is the nominal design model and K is the resulting
controller. Then the closed-loop feedback system is said to have
Nominal Stability (NS): if K internally stabilizes the nominal model P .
Robust Stability (RS): if K internally stabilizes every plant belong to .
Nominal Performance (NP): if the performance objectives are satis ed for the
nominal plant P .
Robust Performance (RP): if the performance objectives are satis ed for every
plant belong to .
The nominal stability and performance can be easily checked using various standard
techniques. The conditions for which the robust stability and robust performance are
satis ed under various assumptions on the uncertainty set will be considered in the
following sections.
9.2 Small Gain Theorem
This section and the next section consider the stability test of a nominally stable system
under unstructured perturbations. The basis for the robust stability criteria derived in
the sequel is the so-called small gain theorem.
Consider the interconnected system shown in Figure 9.3 with M (s) a stable p q
transfer matrix.
w1
- e e1
+ 6
+
M
+
e2 e + w2
?
Figure 9.3: Small Gain Theorem
1
See, for example, Safonov 1980] and Zames 1966].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-228-320.jpg)
![9.2. Small Gain Theorem
213
(2) 0 < !0 < 1: write u1 and v1 in the following form:
2
6
u1 = 6
6
4
u11 ej 1
u12 ej 2
..
.
u1p ej p
3
2
7
7
7
5
v1 = 6
6
4
6
v11 ej
v12 ej
..
.
v1q ej
3
1
2 7
7
7
5
q
where u1i 2 R and v1j 2 R are chosen so that i j 2 ; 0) for all i j .
Choose i 0 and j 0 so that
i ; j!0 =
i
i + j!0
for i = 1 2 : : : p and j = 1 2 : : : q. Let
(s) = 1
1
Then k k1 = 1=
1
2
6
4
v11 1 ;s
1 +s
..
.
v1q q ;s
q +s
j ; j!0 =
j
j + j!0
3
7
5
h
u11 1 ;s
1 +s
1 and (j!0 ) = 11 v1 u1 .
s i
u1p p ;s 2 RH1 :
p+
2
The theorem still holds even if and M are in nite dimensional. This is summarized
as the following corollary.
Corollary 9.2 The following statements are equivalent:
(i) The system is well-posed and internally stable for all 2 H1 with k k1 < 1=
(ii) The system is well-posed and internally stable for all 2 RH1 with k k1 < 1=
(iii) The system is well-posed and internally stable for all 2 C q p with k k < 1=
(iv) kM k1 .
Remark 9.1 It can be shown that the small gain condition is su cient to guarantee
internal stability even if is a nonlinear and time varying stable" operator with an
appropriately de ned stability notion, see Desoer and Vidyasagar 1975].
~
The following lemma shows that if kM k1 > , there exists a destabilizing with
k k1 < 1= such that the closed-loop system has poles in the open right half plane.
(This is stronger than what is given in the proof of Theorem 9.1.)
Lemma 9.3 Suppose M 2 RH1 and kM k1 > . Then there exists a 0 > 0 such
that for any given 2 0 0 ] there exists a 2 RH1 with k k1 < 1= such that
det(I ; M (s) (s)) has a zero on the axis Re(s) = .](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-230-320.jpg)
![MODEL UNCERTAINTY AND ROBUSTNESS
214
Proof. Without loss of generality, assume = 1. Since M 2 RH1 and kM k1 > 1,
there is a su ciently small 0 > 0 such that kM ( 0 + s)k1 := sup kM ( 0 + s)k > 1.
Re(s)>0
Now let 2 0 0 ]. Then there exists a 0 < !0 < 1 such that kM ( + j!0)k > 1.
Let M ( + j!0 ) = U V be a singular value decomposition with
up
U = u1 u2
V = v1 v2
2
=6
4
Write u1 and v1 in the following form:
2
6
u1 = 6
6
4
u11 ej 1
u12 ej 2
..
.
u1p ej p
1
vq
3
2
...
7
5
3
2
7
7
7
5
v1 = 6
6
4
6
:
v11 ej
v12 ej
..
.
v1q ej
3
1
2 7
7
7
5
q
where u1i 2 R and v1j 2 R are chosen so that i j 2 ; 0) for all i j .
Choose i 0 and j 0 so that
i ; ; j!0 =
i
i + + j!0
for i = 1 2 : : : p and j = 1 2 : : : q. Let
2
(s) = 1
1
6
4
1 ;s
v11 1 +s
..
.
s
v1q q ;s
q+
j ; ; j!0 =
j
j + + j!0
3
7
5
h
;s
u11 1 +s
1
i
u1p p;s 2 RH1 :
p +s
Then k k1 = 1= 1 < 1 and det (I ; M ( + j!0 ) ( + j!0 )) = 0. Hence s = + j!0
is a zero for the transfer function det (I ; M (s) (s)).
2
The above lemma plays a key role in the necessity proofs of many robust stability
tests in the sequel.
9.3 Stability under Stable Unstructured Uncertainties
The small gain theorem in the last section will be used here to derive robust stability
tests under various assumptions of model uncertainties. The modeling error will again](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-231-320.jpg)
![9.3. Stability under Stable Unstructured Uncertainties
W1 2 RH1 W2 2 RH1
Perturbed Model Sets
2 RH1 k k1 < 1
(I + W1 W2 );1 P
Representative Types of
Uncertainty Characterized
output (sensor) errors
neglected HF dynamics
changing # of rhp zeros
input (actuators) errors
neglected HF dynamics
changing # of rhp zeros
LF parameter errors
changing # of rhp poles
P (I + W1 W2 );1
LF parameter errors
changing # of rhp poles
(I + W1 W2 )P
P (I + W1 W2 )
P (I + W1 W2 P );1
additive plant errors
neglected HF dynamics
uncertain rhp zeros
LF parameter errors
uncertain rhp poles
~
~
(M + ~ M );1 (N + ~ N )
~ ~
P = M ;1 N
= ~N ~M
(N + N )(M + M );1
P = NM ;1
N
=
LF parameter errors
neglected HF dynamics
uncertain rhp poles & zeros
LF parameter errors
neglected HF dynamics
uncertain rhp poles & zeros
P + W1 W2
M
221
Robust Stability Tests
kW2 ToW1 k1 1
kW2 Ti W1 k1 1
kW2 So W1 k1 1
kW2 Si W1 k1 1
kW2 KSoW1 k1 1
kW2 So PW1 k1 1
K
I
~
So M ;1
M ;1 Si K I ]
1
1
1
1
Table 9.1: Unstructured Robust Stability Tests (# stands for `the number'.)](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-238-320.jpg)
![9.4. Unstructured Robust Performance
225
Theorem 9.7 Suppose P 2 f(I + W1 W2)P :
2 RH1 k k1 < 1g and K internally stabilizes P . Then the system robust performance is guaranteed if either one of
the following conditions is satis ed
(i) for each frequency !
(Wd ) (We So) + (W1 ) (W2 To ) 1
(9:7)
(W1;1 Wd ) (We So Wd ) + (W2 To W1 ) 1
(9:8)
(ii) for each frequency !
where W1 and Wd are assumed to be invertible and (W1;1 Wd ) is the condition
number.
Proof. It is obvious that both condition (9.7) and condition (9.8) guarantee that
kW2 To W1 k1 1. So it is su cient to show that Ted~ 1 1 8 2 RH1 k k1 < 1.
Now for any frequency !, it is easy to see that
(Ted~)
(We So ) (I + W1 W2 To);1 ] (Wd )
S (W
= (I(WeWo ) W d ) )
+ 1 2 To
(We So) (Wd )
1 ; (W1 W2 To )
(We So) (Wd )
1 ; (W1 ) (W2 To ) ( ) :
Hence condition (9.7) guarantees (Ted ) 1 for all 2 RH1 with k k1 < 1 at all
~
frequencies.
Similarly, suppose W1 and Wd are invertible write
Ted~ = We So Wd (W1;1 Wd );1 (I + W2 ToW1 );1 (W1;1 Wd )
and then
(Ted~)
Hence by condition (9.8), (Ted )
~
at all frequencies.
(We So Wd ) (W1;1 Wd ) :
1 ; (W2 ToW1 ) ( )
1 is guaranteed for all 2 RH1 with k k1 < 1
2
Remark 9.2 It is not hard to show that either one of the conditions in the theorem is
also necessary for scalar valued systems.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-242-320.jpg)
![228
MODEL UNCERTAINTY AND ROBUSTNESS
Remark 9.7 It is important to note that in this case, the robust stability condition is
given in terms of Li = KP while the nominal performance condition is given in terms
of Lo = PK . These classes of problems are called skewed problems or problems with
skewed speci cations.2 Since, in general, PK 6= KP , the robust stability margin or
tolerances for uncertainties at the plant input and output are generally not the same.
~
Remark 9.8 It is also noted that the robust performance condition is related to the
condition number of the weighted nominal model. So in general if the weighted nominal
model is ill-conditioned at the range of critical frequencies, then the robust performance
condition may be far more restrictive than the robust stability condition and the nominal
performance condition together. For simplicity, assume W1 = I , Wd = I and W2 = wt I
where wt 2 RH1 is a scalar function. Further, P is assumed to be invertible. Then
robust performance condition (9.10) can be written as
(We So ) + (P ) (wt To ) 1 8!:
Comparing these conditions with those obtained for non-skewed problems shows that
the condition related to robust stability is scaled by the condition number of the plant3 .
Since (P ) 1, it is clear that the skewed speci cations are much harder to satisfy
if the plant is not well conditioned. This problem will be discussed in more detail in
section 11.3.3 of Chapter 11.
~
~
Remark 9.9 Suppose K is invertible, then T ~ can be written as
ed
~
Ted~ = We K ;1 (I + Ti W1 W2 );1 Si KWd:
Assume further that We = I Wd = ws I W2 = I where ws 2 RH1 is a scalar function.
Then a su cient condition for robust performance is given by
(K ) (Si ws ) + (Ti W1 ) 1 8!
with (K ) := (K ) (K ;1 ). This is equivalent to treating the input multiplicative plant
uncertainty as the output multiplicative controller uncertainty.
~
These skewed speci cations also create problems for MIMO loop shaping design
which has been discussed brie y in Chapter 5. The idea of loop shaping design is
based on the fact that robust performance is guaranteed by designing a controller with
a su cient nominal performance margin and a su cient robust stability margin. For
example, if (W1;1 Wd ) 1, the output multiplicative perturbed robust performance is
guaranteed by designing a controller with twice the required nominal performance and
robust stability margins.
See Stein and Doyle 1991].
Alternative condition can be derived so that the condition related to nominal performance is scaled
by the condition number.
2
3](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-245-320.jpg)
![9.5. Gain Margin and Phase Margin
229
The fact that the condition number appeared in the robust performance test for
skewed problems can be given another interpretation by considering two sets of plants
1 and 2 as shown below.
1
2
- wt -
:= fP (I + wt ) :
:= f(I + wt )P :
~
2 RH1 k k1 < 1g
2 RH1 k k1 < 1g :
- wt ~
?
-e - P
-
- P
?
-e -
Figure 9.10: Converting Input Uncertainty to Output Uncertainty
Assume that P is invertible, then
2
1
if jwt j jwt j (P ) 8!
~
since P (I + wt ) = (I + wt P P ;1 )P .
The condition number of a transfer matrix can be very high at high frequency which
may signi cantly limit the achievable performance. The example below, taken from
the textbook Franklin, Powell, and Workman, 1990], shows that the condition number
shown in Figure 9.11 may increase with the frequency:
2
;0:2 0:1 1 0 1 3
6 ;0:05
0 0 0 0:7 7
6
7
s + 1)(s + 0 07)
6
0 ;1 1 0 7 = 1 ;0s:05 0:(7(s + 1)(s + :0:13)
P (s) = 6 0
7
a(s)
4
1
0 0 0 0 5
0
1
0 0 0
where a(s) = (s + 1)(s + 0:1707)(s + 0:02929).
9.5 Gain Margin and Phase Margin
In this section, we show that the gain margin and phase margin de ned in classical
control theory may not be su cient indicators of a system's robustness. Let L(s) be a
scalar transfer function and consider a unit feedback system such as the one shown in
the following diagram:](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-246-320.jpg)
![9.7. Notes and References
237
δ 22
0.707
0.707
δ 11
Figure 9.18: Stability Region for a = 1
The stability region is drawn in the Figure 9.18. This clearly shows that the analysis of
an MIMO system using SISO methods can be misleading and can even give erroneous
results. Hence an MIMO method has to be used.
9.7 Notes and References
The small gain theorem was rst presented by Zames 1966]. The book by Desoer and
Vidyasagar 1975] contains a quite extensive treatment and applications of this theorem
in various forms. Robust stability conditions under various uncertainty assumptions are
discussed in Doyle, Wall, and Stein 1982]. It was rst shown in Kishore and Pearson
1992] that the small gain condition may not be necessary for robust stability with
closed-ball perturbed uncertainties. In the same reference, some extensions of stability
and performance criteria under various structured/unstructured uncertainties are given.
Some further extensions are also presented in Tits and Fan 1994].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-254-320.jpg)
![10.2. Examples of LFTs
where
and
251
2
A 0 B1 0 B2
6 0 0
0 I 0
6
M (s) = 6 C1 0 D11 0 D12
6
4 0 I
0 0 0
C2 0 D21 0 D22
3
7
7
7
7
5
B
F = AK DK :
CK K
Note that F is a constant matrix, not a system matrix. Hence if the controller structure
is xed (or constrained), then the corresponding control problem becomes a constant
(constrained) output feedback problem.
Parametric Uncertainty: A Mass/Spring/Damper System
One natural type of uncertainty is unknown coe cients in a state space model. To motivate this type of uncertainty description, we will begin with a familiar mass/spring/damper
system, shown below in Figure 10.1.
6
F
m
X
X
k XX
X
X
X
c
Figure 10.1: Mass/Spring/Damper System
The dynamical equation of the system motion can be described by
F
c_ k
x + mx + mx = m:
Suppose that the 3 physical parameters m c, and k are not known exactly, but are
believed to lie in known intervals. In particular, the actual mass m is within 10% of
a nominal mass, m, the actual damping value c is within 20% of a nominal value of
c, and the spring sti ness is within 30% of its nominal value of k. Now introducing
perturbations m , c, and k , which are assumed to be unknown but lie in the interval
;1 1], the block diagram for the dynamical system can be shown in Figure 10.2.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-268-320.jpg)
![LINEAR FRACTIONAL TRANSFORMATION
254
^
^
^
^
Here A Ai 2 Rn n B Bi 2 Rn nu C Ci 2 Rny n , and D Di 2 Rny nu .
The various terms in these state equations are interpreted as follows: the nominal
system description G(s), given by known matrices A B C and D, is (A B C D) and
the parametric uncertainty in the nominal system is re ected by the k scalar uncertain
parameters 1 : : : k , and we can specify them, say by i 2 ;1 1]. The structural
^ ^ ^
^
knowledge about the uncertainty is contained in the matrices Ai Bi Ci , and Di . They
re ect how the i'th uncertainty, i , a ects the state space model.
Now, we consider the problem of describing the perturbed system via the LFT
formulae so that all the uncertainty can be represented as a nominal system with the
unknown parameters entering it as the feedback gains. This is shown in Figure 10.4.
Since G (s) = Fu (M 1 I ) where
s
2
M
6
6
=6
6
4
A+
C+
k
X
i=1
k
X
i=1
^
i Ai B +
^
i Ci
D+
k
X
i=1
k
X
i=1
3
^
i Bi
7
7
7
7
5
^
i Di
we need to nd an LFT representation for the matrix M with respect to
p = diag f 1 I 2 I
: : : kIg :
To achieved this with the smallest possibly size of repeated blocks, let qi denote the
rank of the matrix
"
#
^ ^
Ai Bi 2 R( n+ny ) (n+nu )
Pi = ^ ^
Ci Di
for each i. Then Pi can be written as
"
Pi = Li
Wi
#"
Ri
Zi
#
where Li 2 Rn qi , Wi 2 Rny qi , Ri 2 Rn qi and Zi 2 Rnu qi . Hence, we have
"
Li
i Pi =
Wi
#
"
Ri
i Iqi ]
Zi
#
and M can be written as
z"
M =
M11
}|
A B
C D
#{
z"
+
M12
L1
W1
}|
Lk
Wk
z
#{ 2
6
6
4
p
1 Iq1
}|
{2
3z
...
76
76
54
k Iqk
M21
}|
R1 Z1
..
.
..
.
Rk Zk
{
3
7
7
5](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-271-320.jpg)
![256
LINEAR FRACTIONAL TRANSFORMATION
10.3 Basic Principle
We have studied several simple examples of the use of LFTs and, in particular, their
role in modeling uncertainty. The basic principle at work here in writing a matrix
LFT is often referred to as pulling out the s". We will try to illustrate this with
another picture (inspired by Boyd and Barratt 1991]). Consider a structure with four
substructures interconnected in some known way as shown in Figure 10.5.
. . . . . . . . . . . . . . . . . . . .
.......... . . . . ....................................................................
.... .
............................ .. .. .. .. .... . ..
1
.....
. .
. . ..
. . . . . . . . . ....................................
2 .... . .
. . . .... ...... ............ ..... ... .......
. .
.... ...... ............ ..... ... ... ....... . . . . .........
..... . .
.................................................................
.... .
. .. . . . . .. . .
... ...
.....
3 ... ............. ........
... .. K ..........
.
... .
.
.... .. .. ... .. ....................................
. . .
........................................ . . . . . ...........
. . . . . . . . . . . . . . . . . . . .
Figure 10.5: Multiple Source of Uncertain Structure
This diagram can be redrawn as a standard one via pulling out the s" in Figure 10.6. Now the matrix M " of the LFT can be obtained by computing the corresponding transfer matrix in the shadowed box.
We shall illustrate the above principle with an example. Consider an input/output
relation
a+b c 2
z = 1 + d 2 + + 1 2 w =: Gw
e2
1 2
1
where a b c d and e are given constants or transfer functions. We would like to write
G as an LFT in terms of 1 and 2 . We shall do this in three steps:
1. Draw a block diagram for the input/output relation with each separated as
shown in Figure 10.7.
2. Mark the inputs and outputs of the 's as y's and u's, respectively. (This is
essentially pulling out the s).
3. Write z and y's in terms of w and u's with all 's taken out. (This step is equivalent
to computing the transformation in the shadowed box in Figure 10.6.)
2
6
6
6
6
6
6
4
y1
y2
y3
y4
z
3
2
u1
7
6
7
6 u2
7
6
7 = M 6 u3
7
6
7
6
5
4 u4
w
3
7
7
7
7
7
7
5](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-273-320.jpg)
![LINEAR FRACTIONAL TRANSFORMATION
260
C = S
D = S
"
"
C1
C2
D11
D21
D12
D22
D12
D22
# "
#
DK 11
DK 21
"
DK 11
DK 21
#!
CK 1
CK 2
DK 12
DK 22
#!
:
10.5 Notes and References
This chapter is based on the lecture notes by Packard 1991] and the paper by Doyle,
Packard, and Zhou 1991].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-277-320.jpg)
![STRUCTURED SINGULAR VALUE
264
d2
d1 e1
?e-e - K -e 2 P
;6
e
- W2 3
-
d3
- W1
?
-e
Figure 11.3: Multiplicatively Perturbed Systems
uncertainty. As we have shown in the last chapter, if a system is built from components
which are themselves uncertain, then, in general, the uncertainty in the system level is
structured involving typically a large number of real parameters. The stability analysis
involving real parameters is much more involved and is beyond the scope of this book
and that, instead, we shall simply cover the real parametric uncertainty with norm
bounded dynamical uncertainty. Moreover, the interconnection model M can always be
chosen so that (s) is block diagonal, and, by absorbing any weights, k k1 < 1. Thus
we shall assume that (s) takes the form of
(s) = fdiag
: : : s IrS 1 : : : F ] : i (s) 2 RH1 j 2 RH1 g
with k i k1 < 1 and k j k1 < 1. Then the system is robustly stable i the intercon1 Ir1
nected system in Figure 11.4 is stable.
-
F (s)
..
.
- 1 (s)I
..
.
M11 (s)
..
.
Figure 11.4: Robust Stability Analysis Framework
The results of Table 11.1 can be applied to the analyses of the system's robust
stability in two ways:](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-281-320.jpg)
![11.2. Structured Singular Value
267
Lemma 11.1 Let be a structured set and M (s) 2 RH1. Then
sup
s2C +
(M (s)) = sup
(M (s)) = sup
(M (j!)):
(M (s)) = sup
(M (s)) sup
(M (j!)):
s2C +
!
Proof. It is clear that
sup
s2C +
s2C +
!
Now suppose sups2C + (M (s)) > 1= , then by the de nition of , there is an so 2
f1g and a complex structured such that ( ) < and det(I ; M (so ) ) = 0.
This implies that there is a 0 ! 1 and 0 <
^
1 such that det(I ; M (j !) ) =
^
0. This in turn implies that (M (j !)) > 1= since ( ) < . In other words,
^
sups2C + (M (s)) sup! (M (j!)). The proof is complete.
2
C+
The formal de nition and characterization of the structured singular value of a
constant matrix will be given below.
11.2.2 De nitions of
This section is devoted to de ning the structured singular value, a matrix function
denoted by ( ). We consider matrices M 2 C n n . In the de nition of (M ), there is an
underlying structure (a prescribed set of block diagonal matrices) on which everything
in the sequel depends. For each problem, this structure is, in general, di erent it
depends on the uncertainty and performance objectives of the problem. De ning the
structure involves specifying three things: the type of each block, the total number of
blocks, and their dimensions.
There are two types of blocks: repeated scalar and full blocks. Two nonnegative
integers, S and F , represent the number of repeated scalar blocks and the number of
full blocks, respectively. To bookkeep their dimensions, we introduce positive integers
r1 : : : rS m1 : : : mF . The i'th repeated scalar block is ri ri , while the j 'th full block
is mj mj . With those integers given, we de ne
C n n as
= diag 1 Ir1 : : : s IrS 1 : : : F ] : i 2 C j 2 C mj mj :
(11:3)
For consistency among all the dimensions, we must have
S
X
i=1
ri +
F
X
j =1
mj = n:
Often, we will need norm bounded subsets of , and we introduce the following notation:
B = f 2 : ( ) 1g
(11:4)
Bo = f 2 : ( ) < 1g
(11:5)](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-284-320.jpg)
![STRUCTURED SINGULAR VALUE
268
where the superscript o" symbolizes the open ball. To keep the notation as simple as
possible in (11.3), we place all of the repeated scalar blocks rst in actuality, they can
come in any order. Also, the full blocks do not have to be square, but restricting them
as such saves a great deal in terms of notation.
De nition 11.1 For M 2 C n
n
(M ) is de ned as
(M ) := min f ( ) :
2
1
det (I ; M ) = 0g
(11:6)
2 makes I ; M singular, in which case (M ) := 0.
Remark 11.1 Without a loss in generality, the full blocks in the minimal norm can
unless no
each be chosen to be dyads (rank = 1). To see this, assume S = 0, i.e., all blocks are
full blocks. Suppose that I ; M is singular for some 2 . Then there is an x 2 C n
such that M x = x. Now partition x conformably with :
2
6
6
4
3
x1
x2
.
7
7
7
7
5
x = 6 ..
6
xF
xi 2 C mi i = 1 : : : F
and de ne
D = diag(d1 Im1 : : : dF ImF )
where di = 1 if xi = 0 and di = 1= kxi k if xi 6= 0. Next de ne
2
6
6
x1
~
x2
~
x = 6 ..
~ 6
4
and
.
xF
~
2
6
6
y1
y2
4
.
y = 6 ..
6
3
2
7
7
7
7
5
:= Dx = 6
6
3
2
7
7
7
7
5
:= x = 6
~ 6
6
6
4
6
6
d1 x1
d2 x2
..
.
dm xF
3
7
7
7
7
5
3
~
1 x1
~ 7
2 x2 7
7
.. 7 :
. 5
yF
xF
~
It follows that kxi k = 1 if xi 6= 0, kxi k = 0 if xi = 0 , and y 6= 0. Hence, de ne a new
~
~
perturbation ~ 2 C n n as
~ := diag y1 x1 : : : yF xF ] :
~
~
4](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-285-320.jpg)
![11.2. Structured Singular Value
271
Remark 11.2 Note that the scaling set D in Theorem 11.3 and in the inequality (11.14)
does not necessarily be restricted to Hermitian. In fact, they can be replaced by any
set of nonsingular matrices that satisfy (11.12). However, enlarging the set of scaling
matrices does not improve the upper bound in inequality (11.14). This can be shown
as follows: Let D be any nonsingular matrix such that D = D. Then there exist a
Hermitian matrix 0 < R = R 2 D and a unitary matrix U such that D = UR and
inf
D
;
DMD;1 = inf
D
;
;
URMR;1U = R2D RMR;1 :
inf
Therefore, there is no loss of generality in assuming D to be Hermitian.
~
11.2.3 Bounds
In this section we will concentrate on the bounds
max (UM )
U 2U
(M )
inf
D2D
;
DMD;1 :
The lower bound is always an equality Doyle, 1982].
Theorem 11.4 max (MU ) = (M ).
U 2U
Unfortunately, the quantity (UM ) can have multiple local maxima which are not
global. Thus local search cannot be guaranteed to obtain , but can only yield a lower
bound. For computation purposes one can derive a slightly di erent formulation of
the lower bound as a power algorithm which is reminiscent of power algorithms for
eigenvalues and singular values Packard, Fan, and Doyle, 1988]). While there are open
questions about convergence, the algorithm usually works quite well and has proven to
be an e ective method to compute .
The upper bound can be reformulated as a convex optimization problem, so the
global minimum can, in principle, be found. Unfortunately, the upper bound is not
always equal to . For block structures satisfying 2S + F 3, the upper bound is
always equal to (M ), and for block structures with 2S + F > 3, there exist matrices
for which is less than the in mum. This can be summarized in the following diagram,
which shows for which cases the upper bound is guaranteed to be equal to .
Theorem 11.5 (M ) = D2D (DMD;1) if 2S + F 3
inf
S=
0
1
2
F=
0
yes
no
1
2
3
4
yes yes yes no
yes no no no
no no no no](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-288-320.jpg)
![STRUCTURED SINGULAR VALUE
272
Several of the boxes have connections with standard results.
S = 0, F = 1 :
(M ) = (M ).
;
(M ) = (M ) = D2D DMD;1 . This is a standard result in
inf
linear algebra. In fact, without a loss in generality, the matrix M can be assumed
in Jordan Canonical form. Now let
S = 1, F = 0 :
2
J1 =
6
6
6
6
6
6
6
4
1
3
1
... ...
1
2
7
7
7
7
7
7
7
5
6
6
6
=6
6
6
6
4
D1
1
3
k
...
kn1 ;2
kn1 ;1
7
7
7
7
7
7
7
5
2 C n1
n1 :
;
;
Then inf1 n1 (D1 J1 D1 1 ) = klim (D1 J1 D1 1 ) = j j. (Note that by Re!1
D1 2C n
mark 11.2, the scaling matrix does not need to be Hermitian.) The conclusion
follows by applying this result to each Jordan block.
It is also equivalent to the fact that Lyapunov asymptotic stability and exponential stability are equivalent for discrete time systems. This is because (M ) < 1
(exponential stability of a discrete time system matrix M ) implies for some nonsingular D 2 C n n
(DMD;1 ) < 1 or (D;1 ) M D DMD;1 ; I < 0
which in turn is equivalent to the existence of a P = D D > 0 such that
M PM ; P < 0
(Lyapunov asymptotic stability).
S = 0, F = 2 : This case was studied by Redhe er 1959].
S = 1, F = 1 : This is equivalent to a state space characterization of the H1
norm of a discrete time transfer function, see Chapter 21.
S = 2, F = 0 : This is equivalent to the fact that for multidimensional systems
(2-d, in fact), exponential stability is not equivalent to Lyapunov stability.
S = 0, F
4 : For this case, the upper bound is not always equal to . This
is important, as these are the cases that arise most frequently in applications.
Fortunately, the bound seems to be close to . The worst known example has a
ratio of over the bound of about :85, and most systems are close to 1.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-289-320.jpg)
![11.2. Structured Singular Value
273
The above bounds are much more than just computational schemes. They are also
theoretically rich and can unify a number of apparently quite di erent results in linear
systems theory. There are several connections with Lyapunov asymptotic stability, two
of which were hinted at above, but there are further connections between the upper
bound scalings and solutions to Lyapunov and Riccati equations. Indeed, many major
theorems in linear systems theory follow from the upper bounds and from some results
of Linear Fractional Transformations. The lower bound can be viewed as a natural
generalization of the maximum modulus theorem.
Of course one of the most important uses of the upper bound is as a computational
scheme when combined with the lower bound. For reliable use of the theory, it is
essential to have upper and lower bounds. Another important feature of the upper bound
is that it can be combined with H1 controller synthesis methods to yield an ad-hoc
-synthesis method. Note that the upper bound when applied to transfer functions is
simply a scaled H1 norm. This is exploited in the D ; K iteration procedure to perform
approximate -synthesis (Doyle 1982]), which will be brie y introduced in section 11.4.
11.2.4 Well Posedness and Performance for Constant LFTs
Let M be a complex matrix partitioned as
"
M = M11 M12
M21 M22
#
(11:15)
and suppose there are two de ned block structures, 1 and 2 , which are compatible
in size with M11 and M22 , respectively. De ne a third structure as
("
=
1
0
0
2
#
:
1
2
1
22
)
2
:
(11:16)
Now, we may compute with respect to three structures. The notations we use to keep
track of these computations are as follows: 1 ( ) is with respect to 1 , 2 ( ) is with
respect to 2 , and ( ) is with respect to . In view of these notations, 1 (M11 ),
(M ) all make sense, though, for instance, 1 (M ) does not.
2 (M22 ) and
This section is interested in following constant matrix problems:
determine whether the LFT Fl (M 2 ) is well posed for all 2 2 2 with ( 2 )
(< ), and,
if so, then determine how large" Fl (M 2 ) can get for this norm-bounded set
of perturbations.
Let 2 2 2 . Recall that Fl (M 2 ) is well posed if I ; M22 2 is invertible. The
rst theorem is nothing more than a restatement of the de nition of .
Theorem 11.6 The linear fractional transformation Fl (M
2)
is well posed](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-290-320.jpg)
![STRUCTURED SINGULAR VALUE
274
(a) for all 2 2 B 2 if and only if 2 (M22) < 1.
(b) for all 2 2 Bo 2 if and only if 2 (M22 ) 1.
As the perturbation" 2 deviates from zero, the matrix Fl (M 2 ) deviates from
M11 . The range of values that 1 (Fl (M 2 )) takes on is intimately related to (M ),
as shown in the following theorem:
Theorem 11.7 (MAIN LOOP THEOREM) The following are equivalent:
()
(M ) < 1
()
(M ) 1
8
>
>
<
>
>
:
8
>
>
<
>
>
:
2 (M22 ) < 1
max
22
B
1 (Fl (M
2
2 (M22 )
max
o
22
B
and
2 )) < 1:
1 and
2
1 ( Fl ( M
1:
2 ))
Proof. We shall only prove the rst part of the equivalence. The proof for the second
part is similar.
( Let i 2 i be given, with ( i ) 1, and de ne = diag 1 2 ]. Obviously
2 . Now
"
#
I ; M11 1 ;M12 2 :
det (I ; M ) = det
(11:17)
;M
I ;M
By hypothesis I ; M22
21 1
2
is invertible, and hence, det (I ; M ) becomes
det (I ; M22 2 ) det I ; M11
Collecting the
22 2
1 ; M12 2 (I ; M22 2 )
;1 M
21 1
:
terms leaves
det (I ; M ) = det (I ; M22 2 ) det (I ; Fl (M 2 ) 1 ) :
(11:18)
But, 1 (Fl (M 2 )) < 1 and 1 2 B 1 , so I ; Fl (M 2 ) 1 must be nonsingular.
Therefore, I ; M is nonsingular and, by de nition, (M ) < 1.
) Basically, the argument above is reversed. Again let 1 2 B 1 and 2 2
B 2 be given, and de ne = diag 1 2]. Then 2 B and, by hypothesis,
det (I ; M ) 6= 0. It is easy to verify from the de nition of that (always)
(M ) max f 1 (M11 ) 2 (M22 )g :
We can see that 2 (M22 ) < 1, which gives that I ;M22 2 is also nonsingular. Therefore,
the expression in (11.18) is valid, giving
det (I ; M22 2 ) det (I ; Fl (M 2 ) 1 ) = det (I ; M ) 6= 0:
1](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-291-320.jpg)
![11.3. Structured Robust Stability and Performance
w1
- e e1
+ 6
+
277
G(s)
+
e2 ?+ w2
e
Proof. ((=) By Lemma 11.1, sups2C + (G(s)) = sup!2R (G(j!)) . Hence
det(I ; G(s) (s)) 6= 0 for all s 2 C + f1g whenever k k1 < 1= , i.e., the system is
robustly stable.
(=)) Suppose sup!2R (G(j!)) > . Then there is a 0 < !o < 1 such that
(G(j!o )) > . By Remark 11.1, there is a complex c 2 that each full block
has rank 1 and ( c ) < 1= such that I ; G(j!o ) c is singular. Next, using the same
construction used in the proof of the small gain theorem (Theorem 9.1), one can nd a
rational (s) such that k (s)k1 = ( c ) < 1= , (j!o ) = c , and (s) destabilizes
the system.
2
Hence, the peak value on the plot of the frequency response determines the size
of perturbations that the loop is robustly stable against.
Remark 11.4 The internal stability with closed ball of uncertainties is more complicated. The following example is shown in Tits and Fan 1994]. Consider
"
#
0 ;1
1
G(s) = s + 1
1 0
and = (s)I2 . Then
sup (G(j!)) = sup jj! 1 1j = (G(j 0)) = 1:
+
! 2R
! 2R
On the other hand, (G(s)) < 1 for all s 6= 0 s 2 C + , and the only matrices in the
form of ; = I2 with j j 1 for which
det(I ; G(0);) = 0
are the complex matrices jI2 . Thus, clearly, (I ; G(s) (s));1 2 RH1 for all real
rational (s) = (s)I2 with k k1 1 since (0) must be real. This shows that
sup!2R (G(j!)) < 1 is not necessary for (I ; G(s) (s));1 2 RH1 with the closed
ball of structured uncertainty k k1 1. Similar examples with no repeated blocks are
1
generated by setting G(s) = s+1 M where M is any real matrix with (M ) = 1 for
which there is no real 2 with ( ) = 1 such that det(I ; M ) = 0. For example,
let
82
9
2
3
3
"
#
0
>
>
1
<
=
6
7 ;
6
7
M =4
= >4
5
5
i 2 C>
2
0 ;
:
;
3](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-294-320.jpg)
![STRUCTURED SINGULAR VALUE
278
with 2 = 1 and 2 + 2 2 = 1. Then it is shown in Packard and Doyle 1993] that
2
(M ) = 1 and all 2 with ( ) = 1 that satisfy det(I ; M ) = 0 must be
complex.
~
Remark 11.5 Let 2 RH1 be a structured uncertainty and
"
#
G(s) = G11 (s) G12 (s) 2 RH1
G21 (s) G22 (s)
then Fu (G ) 2 RH1 does not necessarily imply (I ; G11 );1 2 RH1 whether
in an open ball or is in a closed ball. For example, consider
2
1
s+1
G(s) = 6 0
4
1
"
and
=
admissible
0
10
s+1
0
1
0
0
is
3
7
5
#
1
with k k1 < 1. Then Fu (G ) =
1 2 RH1 for all
1 ; 1 s+1
2
(k k1 < 1) but (I ; G11 );1 2 RH1 is true only for k k1 < 0:1. ~
1
11.3.2 Robust Performance
Often, stability is not the only property of a closed-loop system that must be robust to
perturbations. Typically, there are exogenous disturbances acting on the system (wind
gusts, sensor noise) which result in tracking and regulation errors. Under perturbation,
the e ect that these disturbances have on error signals can greatly increase. In most
cases, long before the onset of instability, the closed-loop performance will degrade to
the point of unacceptability, hence the need for a robust performance" test. Such a
test will indicate the worst-case level of performance degradation associated with a given
level of perturbations.
Assume Gp is a stable, real-rational, proper transfer function with q1 + q2 inputs
and p1 + p2 outputs. Partition Gp in the obvious manner
"
#
Gp (s) = G11 G12
G21 G22
so that G11 has q1 inputs and p1 outputs, and so on. Let
C q1
structure, as in equation (11.3). De ne an augmented block structure
("
0
#
f 2 C q2 p2
)
p1 be a block
: 2
:
0 f
The setup is to theoretically address the robust performance questions about the
loop shown below
P :=](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-295-320.jpg)
![STRUCTURED SINGULAR VALUE
280
with k k1 < 1 and that the interconnection model G is given by
"
#
G(s) = G11 (s) G12 (s) 2 RH1 :
G21 (s) G22 (s)
(G(j!)) 1.
Then the closed-loop system is well-posed and internally stable i sup!
Let
"
#
d! I
D! =
d! 2 R+
I
then
#
"
;
D! G(j!)D! 1 = 1G11 (j!) d! G12 (j!) :
d! G21 (j! ) G22 (j! )
Hence by Theorem 11.5, at each frequency !
"
(G(j!)) = d inf
2R
!
G11 (j!) d! G12 (j!)
1 G21 (j! ) G22 (j! )
d!
+
#!
:
(11:22)
Since the minimization is convex in log d! see, Doyle, 1982], the optimal d! can be found
by a search however, two approximations to d! can be obtained easily by approximating
the right hand side of (11.22):
(1) From Lemma 2.10, we have
(G(j!)) d inf
! 2R+
s
inf
d! 2R+
=
q
"
kG11 (j!)k d! kG12 (j!)k
1 kG21 (j! )k kG22 (j! )k
d!
#!
kG11 (j!)k2 + d2 kG12 (j!)k2 + d12 kG21 (j!)k2 + kG22 (j!)k2
!
!
kG11 (j!)k2 + kG22 (j!)k2 + 2 kG12 (j!)k kG21 (j!)k
with the minimizing d! given by
^
d! =
8 q
kG21 (j!)k
>
>
<
kG12 (j!)k
>
>
:
0
1
if G12 6= 0 & G21 6= 0
if G21 = 0
if G12 = 0:
(11:23)
(2) Alternative approximation can be obtained by using the Frobenius norm
(G(j!)) d inf
! 2R+
"
G11 (j!) d! G12 (j!)
1 G21 (j! ) G22 (j! )
d!
#
F](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-297-320.jpg)
![11.3. Structured Robust Stability and Performance
285
Using Lemma 11.10, it is easy to show that the maximum is achieved at either or
and that optimal d is given by
j t
d2 = jww"j j
s
so the structured singular value is
s
(G(j!)) = jws "j2 + jwt j2 + jws "jjwt j (P ) + (1 ) ]:
P
(11:29)
Note that if jws "j and jwt j are not too large, which are guaranteed if the nominal
performance and robust stability conditions are satis ed, then the structured singular
value is proportional to the square root of the plant condition number:
(G(j!))
p
jws "jjwt j (P ) :
(11:30)
3
This con rms our intuition that an ill-conditioned plant with skewed speci cations is
hard to control.
11.3.4 Approximation of Multiple Full Block
The approximations given in the last subsection can be generalized to the multiple block
problem by assuming that M is partitioned consistently with the structure of
= diag(
so that
2
M=
and
6
6
6
6
4
1
2
::: F)
M11 M12
M21 M22
M1F
M2F
MF 1 MF 2
MFF
..
.
..
.
..
.
3
7
7
7
7
5
D = diag(d1 I : : : dF ;1 I I ):
Now
d
DMD;1 = Mij di
j
And hence
(M )
dF := 1:
inf (DMD;1 ) = D2D
inf
D2D
d
Mij d i
j](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-302-320.jpg)
![11.4. Overview on Synthesis
289
D
D;1
G
- K
Figure 11.7: -Synthesis via Scaling
(i) Fix an initial estimate of the scaling matrix D! 2 D pointwise across frequency.
(ii) Find scalar transfer functions di (s) d;1 (s) 2 RH1 for i = 1 2 : : : (F ; 1) such
i
that jdi (j!)j d! . This step can be done using the interpolation theory Youla
i
and Saito, 1967] however, this will usually result in very high order transfer
functions, which explains why this process is currently done mostly by graphical
matching using lower order transfer functions.
(iii) Let
D(s) = diag(d1 (s)I : : : dF ;1 (s)I I ):
Construct a state space model for system
"
^
G(s) = D(s)
#
I
"
;1
G(s) D (s)
#
I
:
(iv) Solve an H1 -optimization problem to minimize
^
F` (G K )
1
over all stabilizing K 's. Note that this optimization problem uses the scaled
^
version of G. Let its minimizing controller be denoted by K .
^ ;
(v) Minimize D! F` (G K )D! 1 ] over D! , pointwise across frequency.4 Note that
^
this evaluation uses the minimizing K from the last step, but that G is unscaled.
The minimization itself produces a new scaling function. Let this new function
^
be denoted by D! .
^
(vi) Compare D! with the previous estimate D! . Stop if they are close, but, otherwise,
^
replace D! with D! and return to step (ii).
With either K or D xed, the global optimum in the other variable may be found using
the and H1 solutions. Although the joint optimization of D and K is not convex and
4
The approximate solutions given in the last section may be used.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-306-320.jpg)
![290
STRUCTURED SINGULAR VALUE
the global convergence is not guaranteed, many designs have shown that this approach
works very well see e.g. Balas, 1990]. In fact, this is probably the most e ective design
methodology available today for dealing with such complicated problems. The detailed
treatment of analysis is given in Packard and Doyle 1991]. The rest of this book will
focus on the H1 optimization which is a fundamental tool for synthesis.
11.5 Notes and References
This chapter is partially based on the lecture notes given by Doyle 1984] in Honeywell
and partially based on the lecture notes by Packard 1991] and the paper by Doyle,
Packard, and Zhou 1991]. Parts of section 11.3.3 come from the paper by Stein and
Doyle 1991]. The small theorem for systems with non-rational plants and uncertainties is proven in Tits 1994]. Other results on can be found in Fan and Tits 1986],
Fan, Tits, and Doyle 1991], Packard and Doyle 1993], Packard and Pandey 1993],
Young 1993], and references therein.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-307-320.jpg)
![12
Parameterization of
Stabilizing Controllers
The basic con guration of the feedback systems considered in this chapter is an LFT
as shown in Figure 12.1, where G is the generalized plant with two sets of inputs: the
exogenous inputs w, which include disturbances and commands, and control inputs u.
The plant G also has two sets of outputs: the measured (or sensor) outputs y and the
regulated outputs z . K is the controller to be designed. A control problem in this setup
is either to analyze some speci c properties, e.g., stability or performance, of the closedloop or to design the feedback control K such that the closed-loop system is stable
in some appropriate sense and the error signal z is speci ed, i.e., some performance
condition is satis ed. In this chapter we are only concerned with the basic internal
stabilization problems. We will see again that this setup is very convenient for other
general control synthesis problems in the coming chapters.
Suppose that a given feedback system is feedback stabilizable. In this chapter, the
problem we are mostly interested in is parameterizing all controllers that stabilize the
system. The parameterization of all internally stabilizing controllers was rst introduced
by Youla et al 1976] in their parameterization, the coprime factorization technique is
used. All of the existing results are mainly in the frequency domain although they can
also be transformed to state-space descriptions. In this chapter, we consider this issue
in the general setting and directly in state space without adopting coprime factorization
technique. The construction of the controller parameterization is done via considering
a sequence of special problems, which are so-called full information (FI) problems, disturbance feedforward (DF) problems, full control (FC) problems and output estimation
291](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-308-320.jpg)
![12.2. Duality and Special Problems
295
12.2 Duality and Special Problems
In this section, we will discuss four problems from which the output feedback solutions
are constructed via a separation argument. These special problems are fundamental
to the approach taken for synthesis in this book, and, as we shall see, they are also of
importance in their own right.
12.2.1 Algebraic Duality and Special Problems
Before we get into the study of the algebraic structure of control systems, we now introduce the concept of algebraic duality which will play an important role. It is well known
that the concepts of controllability (stabilizability) and observability (detectability) of
a system (C A B ) are dual because of the duality between (C A B ) and (B T AT C T ).
So, to deal with the issues related to a system's controllability and/or observability, we
only need to examine the issues related to the observability and/or controllability of its
dual system, respectively. The notion of duality can be generalized to a general setting.
Consider a standard system block diagram
z
y
G
- K
w
u
where the plant G and controller K are assumed to be linear time invariant. Now
consider another system shown below
z
~
w
~
GT
y
~
- KT
u
~
whose plant and controller are obtained by transposing G and K . We can check easily
T
that Tzw = F`(G K )]T = F`(GT K T ) = Tzw . It is not di cult to see that K internally
~~
stabilizes G i K T internally stabilizes GT . And we say that these two control structures
are algebraically dual, especially, GT and K T which are dual objects of G and K ,
respectively. So as far as stabilization or other synthesis problems are concerned, we
can obtain the results for GT from the results for its dual object G if they are available.
Now, we consider some special problems which are related to the general OF problems stated in the last section and which are important in constructing the results for
OF problems. The special problems considered here all pertain to the standard block
diagram, but to di erent structures than G. The problems are labeled as](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-312-320.jpg)
![12.3. Parameterization of All Stabilizing Controllers
303
given stabilizing controller K . This approach, however, does not give much insight into
the controller construction and thus can not be generalized to other synthesis problems.
The conventional Youla approach to this problem is via coprime factorization Youla
et al, 1976, Vidyasagar, 1985, Desoer et al, 1982], which will be adopted in the later
part of this chapter as an alternative approach.
In the following sections, we will present a novel approach to this problem without
adopting coprime factorizations. The idea of this approach is to reduce the output
feedback problem into some simpler problems, such as FI and OE or FC and DF which
admit simple solutions, and then to solve the output feedback problem by the separation
argument. The advantages of this approach are that it is simple and that many other
synthesis problems, such as H2 and H1 optimal control problems in Chapters 14 and 16,
can be solved by using the same machinery.
Readers should bear in mind that our objective here is to nd all admissible controllers for the OF problem. So at rst, we will try to build up enough tools for this
objective by considering the special problems. We will see that it is not necessary to parameterize all stabilizing controllers for these special problems to get the required tools.
Instead, we only parameterize some equivalent classes of controllers which generate the
same control action.
^
De nition 12.2 Two controllers K and K are of equivalent control actions if their
^
corresponding closed loop transfer matrices are identical, i.e. Fl (G K ) = Fl (G K ),
^.
written as K = K
Algebraically, the controller equivalence is an equivalence relation. We will see that
for di erent special problems we have di erent re ned versions of this relation. We
will also see that the characterizations of equivalent classes of stabilizing controllers for
special problems are good enough to construct the parameterization of all stabilizing
controllers for the OF problem. In the next two subsections, we mainly consider the
stabilizing controller characterizations for special problems. Also, we use the solutions
to these special problems and the approach provided in the last section to characterize
all stabilizing controllers of OF problems.
12.3.2 Stabilizing Controllers for FI and FC Problems
In this subsection, we rst examine the FI structure
z
yFI
GFI
- KFI
w
u
where the transfer matrix GFI is given in section 12.2. The purpose of this subsection
is to characterize the equivalent classes of stabilizing controllers KFI that stabilize](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-320-320.jpg)
![304
PARAMETERIZATION OF STABILIZING CONTROLLERS
internally GFI and to build up enough tools to be used later. For this problem, we
^
say two controllers KFI and KFI are equivalent if they produce the same closed-loop
transfer function from w to u. Obviously, this also guarantees that Fl (GFI KFI ) =
^
Fl (GFI KFI ). The characterization of equivalent classes of controllers can be suitably
called the control parameterization in contrast with controller parameterization. Note
that the same situation will occur in FC problems by duality.
Since we have full information for feedback, our controller will have the following
general form:
h
i
KFI = K1 (s) K2(s)
with K1 (s) stabilizing internally (sI ; A);1 B2 and arbitrary K2 (s) 2 RH1 . Note that
the stability of K2 is required to guarantee the internal stability of the whole system
since w is fed directly through K2.
Lemma 12.9 Let F be a constant matrix such that A + B2F is stable. Then all stabilizing controllers, in the sense of generating all admissible closed-loop transfer functions,
for FI can be parameterized as
h
i
KFI = F Q
with any Q 2 RH1 .
Note that for the parameter matrix Q 2 RH1 , it is reasonable to assume that the
realization of Q(s) is stabilizable and detectable.
Proof. It is easy to see that the controller given in the above formula stabilizes the
system GFI . Hence we only need to show that the given set of controllers parameterizes
all stabilizing control action, u, i.e., there is a choice of Q 2 RH1 such that the transfer
h
i
functions from w to u for any stabilizing controller KFI = K1 (s) K2 (s) and for
h
i
0
KFI = F Q are the same. To show this, make a change of control variable as
v = u ; Fx, where x denotes the state of the system GFI then the system with the
controller KFI will be as shown in the following diagram:
z
yFI
with
2
A + B2 F
6
6 C1 + D12 F
~
GFI = 6 " #
6
I
4
0
w
~
GFI
v
- KFI
~
B1
D11#
"
0
I
B2
D12#
"
0
0
3
7
7
7
7
5
~
KFI = KFI ; F 0]:](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-321-320.jpg)
![318
PARAMETERIZATION OF STABILIZING CONTROLLERS
2
3
A + LC2 ;(B2 + LD22) L
6
=4 F
I
0 7
5
C2
;D22
I
where F and L are chosen such that A + B2 F and A + LC2 are both stable.
Then Jy can be computed as
"
~
~
V0 ;U0
~ ~
;N M
#
2
3
A + B2 F + LC2 + LD22F ;L B2 + LD22
6
7
Jy = 4
F
0
I
5:
;(C2 + D22 F )
I
;D22
Proof. This follows from some tedious algebra.
2
Remark 12.11 Note that Jy is exactly the same as the J in Theorem 12.8 and that
;1
K0 := U0V0 is an observer-based stabilizing controller with
"
#
K0 := A + B2 F + LC2 + LD22 F ;L :
F
0
~
12.7 Notes and References
The special problems FI, DF, FC, and OE were rst introduced in Doyle, Glover, Khargonekar, and Francis 1989] for solving the H1 problem, and they have been since used
in many other papers for di erent problems. The new derivation of all stabilizing controllers was reported in Lu, Zhou, and Doyle 1991]. The paper by Moore et al 1990]
contains some other related interesting results. The conventional Youla parameterization can be found in Youla et al 1976], Desoer et al 1980], Doyle 1984], Vidyasagar
1985], and Francis 1987]. The parameterization of all two-degree-of-freedom stabilizing
controllers is given in Youla and Bongiorno 1985] and Vidyasagar 1985].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-335-320.jpg)
![ALGEBRAIC RICCATI EQUATIONS
322
Remark 13.1 It is now clear that to obtain solutions to the Riccati equation, it is
necessary to be able to construct bases for those invariant subspaces of H . One way of
constructing those invariant subspaces is to use eigenvectors and generalized eigenvectors
of H . Suppose i is an eigenvalue of H with multiplicity k (then i+j = i for all
j = 1 : : : k ; 1), and let vi be a corresponding eigenvector and vi+1 : : : vi+k;1 be the
corresponding generalized eigenvectors associated with vi and i . Then vj are related
by
( H ; i I )v i
= 0
(H ; i I )vi+1
= vi
..
.
(H ; i I )vi+k;1 = vi+k;2
and the spanfvj j = i : : : i + k ; 1g is an invariant subspace of H .
~
Example 13.1 Let
"
#
"
#
"
#
A = ;3 2 R = 0 0
Q= 0 0 :
;2 1
0 ;1
0 0
The eigenvalues of H are 1 1 ;1 ;1, and the corresponding eigenvector and generalized
eigenvector of 1 are
2
3
2
3
2
7
7
7
7
5
6
6
=6
6
4
3
1
;1 7
6
7
6
6 2 7
6 ;3=2 7
7
6
7:
v1 = 6
6 2 7 v2 = 6
17
4
5
4
5
;2
0
The corresponding eigenvector and generalized eigenvector of ;1 are
2
v3 =
6
6
6
6
4
1
1
0
0
v4
3
1
7
3=2 7
7:
0 7
5
0
All solutions of the Riccati equation under various combinations are given below:
"
#
"
X1 = v v ], then X = X X ;1 =
spanfv1 v2 g is H -invariant: let
1 2
2 1
X2
is a solution and (A + RX ) = f1 1g
"
#
"
X1 = v v ], then X = X X ;1 =
spanfv1 v3 g is H -invariant: let
1 3
2 1
X2
is also a solution and (A + RX ) = f1 ;1g
;10 6
6 ;4
;2 2
2 ;2
#
#](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-339-320.jpg)
![13.1. All Solutions of A Riccati Equation
"
323
#
X1 = v v ], then X = 0 is a solution and
spanfv3 v4 g is H -invariant: let
3 4
X2
(A + RX ) = f;1 ;1g
spanfv1 v4 g spanfv2 v3 g, and spanfv2 v4 g are not H -invariant subspaces. Read-
ers can verify that the matrices constructed from those vectors are not solutions
to the Riccati equation (13.1).
3
Up to this point, we have said nothing about the structure of the solutions given by
Theorem 13.1 and 13.2. The following theorem gives a su cient condition for a Riccati
solution to be hermitian (not necessarily real symmetric).
Theorem 13.3 Let V be an n-dimensional H -invariant subspace and let X1 X2 2
Cn
n be such that
"
#
V = Im X1 :
X2
Then i + j 6= 0 for all i j = 1 : : : n, i j 2 (H jV ) implies that X1 X2 is hermi;
tian, i.e., X1 X2 = (X1 X2 ) . Furthermore, if X1 is nonsingular, then X = X2 X1 1 is
hermitian.
Proof. Since V is an invariant subspace of H , there is a matrix representation for
H jV such that ( ) = (H jV ) and
"
#
"
H X1 = X1
X2
X2
"
X1
Pre-multiply this equation by
X2
"
X1
X2
#
#
"
#
:
J to get
#
"
JH X1 = X1
X2
X2
#
"
J X1
X2
#
:
(13:4)
Note that JH is hermitian (actually symmetric since H is real) therefore, the left-hand
side of (13.4) is hermitian as well as the right-hand side:
"
i.e.,
X1
X2
#
"
J X1
X2
#
"
=
X1
X2
#
J
"
#
X1 = ;
X2
"
X1
X2
(;X1 X2 + X2 X1 ) = ; (;X1 X2 + X2 X1 ):
#
"
J X1
X2
#](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-340-320.jpg)
![13.2. Stabilizing Solution and Riccati Operator
327
(ii) Start with the equation
"
#
"
#
H X1 = X1 H;
X2
X2
;
and post-multiply by X1 1 to get
"
#
"
#
;
H I = I X1 H; X1 1:
X
X
Now pre-multiply by X ; I ]:
"
(13:8)
#
X ; I ]H I = 0:
X
This is precisely the Riccati equation.
(iii) Pre-multiply (13.8) by I 0] to get
;
A + RX = X1 H; X1 1:
2
Thus A + RX is stable because H; is.
Now, we are going to state one of the main theorems of this section which gives the
necessary and su cient conditions for the existence of a unique stabilizing solution of
(13.1) under certain restrictions on the matrix R.
Theorem 13.6 Suppose H has no imaginary eigenvalues and R is either positive semide nite or negative semi-de nite. Then H 2 dom(Ric) if and only if (A R) is stabilizable.
Proof. (() To prove that H 2 dom(Ric), we must show that
X; (H ) Im
"
0
#
I
are complementary. This requires a preliminary step. As in the proof of Theorem 13.5
de ne X1 X2 H; so that
"
#
X; (H ) = Im X1
"
#
"
X2
#
H X1 = X1 H; :
X2
X2
(13:9)](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-344-320.jpg)
![ALGEBRAIC RICCATI EQUATIONS
328
We want to show that X1 is nonsingular, i.e., Ker X1 = 0. First, it is claimed that Ker
X1 is H; -invariant. To prove this, let x 2 Ker X1 . Pre-multiply (13.9) by I 0] to get
AX1 + RX2 = X1 H; :
(13:10)
Pre-multiply by x X2 , post-multiply by x, and use the fact that X2 X1 is symmetric
(see (13.6)) to get
x X2 RX2 x = 0:
Since R is semide nite, this implies that RX2 x = 0. Now post-multiply (13.10) by x to
get X1 H; x = 0, i.e. H; x 2 Ker X1 . This proves the claim.
Now to prove that X1 is nonsingular, suppose, on the contrary, that Ker X1 6= 0.
Then H; jKer X1 has an eigenvalue, , and a corresponding eigenvector, x:
H; x = x
(13:11)
Re < 0 0 6= x 2 Ker X1 :
Pre-multiply (13.9) by 0 I ]:
; QX1 ; A X2 = X2 H; :
(13:12)
Post-multiply the above equation by x and use (13.11):
(A + I )X2 x = 0:
Recall that RX2 x = 0, we have
x X2 A + I R] = 0:
Then stabilizability of (A R) implies X2 x = 0. But if both X1 x = 0 and X2 x = 0, then
"
#
x = 0 since X1 has full column rank, which is a contradiction.
X2
()) This is obvious since H 2 dom(Ric) implies that X is a stabilizing solution and
that A + RX is asymptotically stable. It also implies that (A R) must be stabilizable.
2
Theorem 13.7 Suppose H has the form
"
#
A ;BB :
H=
;C C ;A
Then H 2 dom(Ric) i (A B ) is stabilizable and (C A) has no unobservable modes on
the imaginary axis. Furthermore, X = Ric(H ) 0 if H 2 dom(Ric), and Ker(X ) = 0
if and only if (C A) has no stable unobservable modes.
Note that Ker(X ) Ker(C ), so that the equation XM = C always has a solution
for M , and a minimum F -norm solution is given by X yC .](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-345-320.jpg)
![13.2. Stabilizing Solution and Riccati Operator
329
Proof. It is clear from Theorem 13.6 that the stabilizability of (A B) is necessary,
and it is also su cient if H has no eigenvalues on the imaginary axis. So we only need
to show that, assuming (A B ) is stabilizable, H has no imaginary eigenvalues i (C A)
has no unobservable modes on the imaginary axis. Suppose that j! is an eigenvalue
"
#
x is a corresponding eigenvector. Then
and 0 6=
z
Ax ; BB z = j!x
;C Cx ; A z = j!z:
Re-arrange:
(A ; j!I )x = BB z
; (A ; j!I ) z = C Cx:
Thus
(13:13)
(13:14)
hz (A ; j!I )xi = hz BB z i = kB z k2
;hx (A ; j!I ) z i = hx C Cxi = kCxk2
so hx (A ; j!I ) z i is real and
;kCxk2 = h(A ; j!I )x z i = hz (A ; j!I )xi = kB z k2:
Therefore B z = 0 and Cx = 0. So from (13.13) and (13.14)
(A ; j!I )x = 0
(A ; j!I ) z = 0:
Combine the last four equations to get
z A ; j!I B ] = 0
"
#
A ; j!I x = 0:
C
The stabilizability of (A B ) gives z = 0. Now it is clear that j! is an eigenvalue of H
i j! is an unobservable mode of (C A).
Next, set X := Ric(H ). We'll show that X 0. The Riccati equation is
A X + XA ; XBB X + C C = 0
or equivalently
(A ; BB X ) X + X (A ; BB X ) + XBB X + C C = 0:
(13:15)](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-346-320.jpg)
![ALGEBRAIC RICCATI EQUATIONS
334
Example 13.4 Let
2
A=6
4
1 0 0
0 2 0
0 0 ;3
3
2
7
5
B=6
4
"
1
1
1
A ;BB
The eigenvalues of matrix H =
;C C ;A
3
h
7
5
i
C= 0 0 0 :
#
are
= 1 2 = ;1 3 = 2 4 = ;2 5 = ;3 6 = 3
and their corresponding eigenvectors are
3
2
3
2
3
2
3
2
3
2
3
2
;3 7
0
4
0
3
1
6
7
6
7
6
7
6
7
6
7
6
6 ;6 7
6 0 7
6 3 7
6 1 7
6 2 7
6 0 7
7
6
7
6
7
6
7
6
7
6
7
6
7
6
7
6
7
6
6 0 7
6 ;3 7
6 0 7
6 ;1 7
6 1 7
6 ;12 7
7
6
7
6
7
6
v1 = 6 7 v2 = 6
7:
7 v6 = 6
7 v5 = 6
7 v4 = 6
7 v3 = 6
6 0 7
6 0 7
6 0 7
6 0 7
6 6 7
6 0 7
7
6
7
6
7
6
7
6
7
6
7
6
7
6
7
6
7
6
7
6
6 0 7
6 0 7
4 0 5
4 0 5
4 12 5
4 0 5
5
4
5
4
6
0
0
0
0
0
There are four distinct nonnegative de nite symmetric solutions depending on the chosen invariant subspaces:
2
3
"
#
0 0 0
X1 = v v v ] Y = X X ;1 = 6
7
4 0 0 0 5
1 3 5 1
2 1
X2
0 0 0
1
"
"
#
2
X1 = v v v ] Y = X X ;1 = 6
4
2 3 5 2
2 1
X2
#
2
X1 = v v v ] Y = X X ;1 = 6
4
1 4 5 3
2 1
X2
2
2 0 0
0 0 0
0 0 0
3
0 0 0
0 4 0
0 0 0
3
7
5
7
5
3
18 ;24 0
X1 = v v v ] Y = X X ;1 = 6
36 0 7.
4 ;24
5
2 4 5 4
2 1
X2
0
0 0
These solutions can be ordered as Y4 Yi Y1 i = 2 3. Of course, this is only a partial
ordering since Y2 and Y3 are not comparable. Note also that only Y4 is a stabilizing
solution, i.e., A ; BB Y4 is stable. Furthermore, Y4 and Y1 are the maximal" and
minimal" solutions, respectively.
3
"
#](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-351-320.jpg)
![13.4. Spectral Factorizations
345
Then (j!) > 0 8!. Thus from part (i), there is an X = X
0 such that
A ; BB X is stable and
A X + X A ; X BB X ; 2 C C = 0:
It is easy to see from Theorem 13.14 that X is monotone-decreasing with , i.e.,
X 1 X 2 if 1 2 . To show that lim !1 X exists, we need to show that X is
bounded below for all 0 < < 1.
In the following, it will be assumed that (A B ) is controllable. The controllability
assumption will be removed later.
Let Y be the destabilizing solution to the following Riccati equation:
A Y + Y A ; Y BB Y = 0
with (A ; BB Y ) C + (note that the existence of such a solution is guaranteed by
the controllability assumption). Then it is easy to verify that
(A ; BB Y ) (X ; Y ) + (X ; Y )(A ; BB Y ) ; (X ; Y )BB (X ; Y ) ; 2C C = 0:
This implies that
X ;Y =
1
Z
0
e;(A;BB Y ) t (X ; Y )BB (X ; Y ) + 2 C C ]e;(A;BB Y )t dt 0:
Thus X is bounded below with Y as the lower bound, and lim !1 X exists. Let
X := lim !1 X then from continuity argument, X satis es the Riccati equation
A X + XA ; XBB X + P = 0
and (A ; BB X ) C ; .
Now suppose (A B ) is not controllable, and then assume without loss of generality
that
"
#
"
#
h
i
A11 A12
B1
A=
B=
C = C1 C2
0 A22
0
so that (A11 B1 ) is controllable and A11 and A22 are stable. Then the Riccati equation
for
#
"
X11 X12
X =
(X12 ) X22
can be written as three equations
A11 X11 + X11 A11 ; X11 B1 B1 X11 ; 2 C1 C1 = 0
(A11 ; B1 B1 X11 ) X12 + X12 A22 + X11 A12 ; 2 C1 C2 = 0
A22 X22 + X22 A22 + (X12 ) A12 + A12 X12 ; (X12 ) B1 B1 X12 ; 2 C2 C2 = 0](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-362-320.jpg)
![13.4. Spectral Factorizations
Proof. Note that
"
#
"
P S = I SR;1=2
S R
0 R1=2
347
#"
^
P 0
0 I
#"
I
#
0
R;1=2 S
R 1= 2
:
Hence the function (s) can be written as
h
(s) = B (;sI ; A );1 ' (s)
i
"
^
P 0
0 I
#"
(sI ; A);1 B
'(s)
#
with '(s) = R1=2 + R;1=2 S (sI ; A);1 B . It is easy to verify that
^ (s) = ';1 (s)] (s)';1 (s):
2
Hence the conclusion follows.
Now we are ready to state and prove one of the main results of this section. The
following theorem and corollary characterize the relations among spectral factorizations,
Riccati equations, and decomposition of Hamiltonians.
Theorem 13.19 Let A B P S R be matrices of compatible dimensions such that P =
P , R = R > 0, with (A B ) stabilizable. Suppose either one of the following assump-
tions is satis ed:
(A1) A has no eigenvalues on j!-axis
(A2) P is sign de nite, i.e., P 0 or P 0 and (P A) has no unobservable modes on
the j!-axis.
Then
(I) The following statements are equivalent:
(a) The parahermitian rational matrix
h
(s) = B (;sI ; A );1 I
i
"
P S
S R
#"
(sI ; A);1 B
#
I
satis es
(j!) > 0 for all 0 ! 1:
(b) There exists a unique real X = X such that
(A ; BR;1 S ) X + X (A ; BR;1 S ) ; XBR;1B X + P ; SR;1S = 0
and that A ; BR;1 S ; BR;1 B X is stable.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-364-320.jpg)
![ALGEBRAIC RICCATI EQUATIONS
358
This lemma immediately leads to one characterization of inner matrices in terms
of their state space representations. Simply add the condition that D D = I to
Lemma 13.29 to get N N = I .
Corollary 13.30 Suppose N =
"
A B
C D
#
is stable and minimal, and X is the ob-
servability grammian. Then N is inner if and only if
(a) D C + B X = 0
(b) D D = I .
A transfer matrix N? is called a complementary inner factor (CIF) of N if N N? ]
is square and inner. The dual notion of the complementary co-inner factor is de ned in
the obvious way. Given an inner N , the following lemma gives a construction of its CIF.
The proof of this lemma follows from straightforward calculation and from the fact that
CX y X = C since Im(I ; X + X ) Ker(X ) Ker(C ).
"
#
Lemma 13.31 Let N = A B be an inner and X be the observability grammian.
C D
Then a CIF N? is given by
"
y
N? = A ;X C D?
C
D?
#
where D? is an orthogonal complement of D such that D D? ] is square and orthogonal.
13.7 Inner-Outer Factorizations
In this section, some special form of coprime factorizations will be developed. In particular, explicit realizations are given for coprime factorizations G = NM ;1 with inner
numerator N and inner denominator M , respectively. The former factorization in the
case of G 2 RH1 will give an inner-outer factorization1. The results will be proven for
the right coprime factorizations, while the results for left coprime factorizations follow
by duality.
Let G 2 Rp be a p m transfer matrix and denote R1=2 R1=2 = R. For a given
full column rank i
matrix D, let D? denote for any orthogonal complement of D so that
h
;1=2 D? (with R = D D > 0) is square and orthogonal. To obtain an rcf of
DR
G with N inner, we note that if NM ;1 is an rcf, then (NZr )(MZr );1 is also an rcf for
any nonsingular real matrix Zr . We simply need to use the formulas in Theorem 5.9 to
solve for F and Zr .
1 A p m (p
m) transfer matrix Go 2 RH1 is called an outer if Go (s) has full row rank in the
open right half plane, i.e., Go (s) has full row normal rank and has no open right half plane zeros.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-375-320.jpg)
![ALGEBRAIC RICCATI EQUATIONS
364
"
I X
Now let T =
0 I
"
#
then
#
(A + BF )
0
=T
;(A + BF )
;BR;1 B
"
#
~
~
A ; BD R;1 C
;C R;1 C
;1 B
~
;BR
;(A ; BD R;1 C )
T ;1:
This shows that the stable invariant subspaces for these two Hamiltonian matrices are
related by
X;
or
"
(A + BF )
;BR;1B
"
#
#
"
~
~
0
A ; BD R;1 C
;C R;1 C
= T X;
;1 B
~
;(A + BF )
;BR
;(A ; BD R;1 C )
"
#
"
I = Im I + XY
= T Im
Y
Y
Hence we have P = Y (I + XY );1 .
I
Im
P
#
"
#
#
I
:
= Im
Y (I + XY );1
2
13.9 Notes and References
The general solutions of a Riccati equation are given by Martensson 1971]. The iterative procedure for solving ARE was rst introduced by Kleinman 1968] for a special
case and was further developed by Wonham 1968]. It was used by Coppel 1974], Ran
and Vreugdenhil 1988], and many others for the proof of the existence of maximal
and minimal solutions. The comparative results were obtained in Ran and Vreugdenhil
1988]. The paper by Wimmer 1985] also contains comparative results for some special
cases. The paper by Willems 1971] contains a comprehensive treatment of ARE and
the related optimization problems. Some matrix factorization results are given in Doyle
1984]. Numerical methods for solving ARE can be found in Arnold and Laub 1984],
Dooren 1981], and references therein. The state space spectral factorization for functions singular at 1 or on imaginary axis is considered in Clements and Glover 1989]
and Clements 1993].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-381-320.jpg)
![14
H2 Optimal Control
In this chapter we treat the optimal control of linear time-invariant systems with a
quadratic performance criterion. The material in this chapter is standard, but the
treatment is somewhat novel and lays the foundation for the subsequent chapters on
H1 -optimal control.
14.1 Introduction to Regulator Problem
Consider the following dynamical system:
x = Ax + B2 u x(t0 ) = x0
_
(14:1)
where x0 is given but arbitrary. Our objective is to nd a control function u(t) de ned
on t0 T ] which can be a function of the state x(t) such that the state x(t) is driven to a
(small) neighborhood of origin at time T . This is the so-called Regulator Problem. One
might suggest that this regulator problem can be trivially solved for any T > t0 if the
system is controllable. This is indeed the case if the controller can provide arbitrarily
large amounts of energy since, by the de nition of controllability, one can immediately
construct a control function that will drive the state to zero in an arbitrarily short
time. However, this is not practical since any physical system has the energy limitation,
i.e., the actuator will eventually saturate. Furthermore, large control action can easily
drive the system out of the region where the given linear model is valid. Hence certain
limitations have to be imposed on the control in practical engineering implementation.
365](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-382-320.jpg)
![H2 OPTIMAL CONTROL
366
The constraints on control u may be measured in many di erent ways for example,
Z
T
t0
T
Z
kuk dt
t0
kuk2 dt
sup kuk
t2 t0 T ]
i.e., in terms of L1 -norm, L2 -norm, and L1 -norm, or more generally, weighted L1 -norm,
L2 -norm, and L1 -norm
T
Z
t0
kWu uk dt
T
Z
t0
kWu uk2 dt
sup kWu uk
t2 t0 T ]
for some constant weighting matrix Wu .
Similarly, one might also want to impose some constraints on the transient response
x(t) in a similar fashion
Z
T
t0
kWx xk dt
T
Z
t0
kWx xk2 dt
sup kWx xk
t2 t0 T ]
for some weighting matrix Wx . Hence the regulator problem can be posed as an optimal
control problem with certain combined performance index on u and x, as given above.
In this chapter, we shall be concerned exclusively with the L2 performance problem or
quadratic performance problem. Moreover, we will focus on the in nite time regulator
problem, i.e., T ! 1, and, without loss of generality, we shall assume t0 = 0. In this
case, our problem is as follows: nd a control u(t) de ned on 0 1) such that the state
x(t) is driven to the origin at t ! 1 and the following performance index is minimized:
min
u
1 " x(t)
Z
0
# "
u(t)
Q S
S R
#"
#
x(t) dt
u(t)
(14:2)
for some Q = Q , S , and R = R > 0. This problem is traditionally called a Linear
Quadratic Regulator problem or simply an LQR problem. Here we have assumed R > 0
to emphasis that the control energy has to be nite, i.e., u(t) 2 L2 0 1). So this is the
space over which the integral is minimized. Moreover, it is also generally assumed that
"
Q S
S R
#
0:
(14:3)
Since R is positive de nite, it has a square root, R1=2 , which is also positive-de nite.
By the substitution
R1=2 u
we may as well assume at the start that R = I . In fact, we can even assume S = 0 by
using a pre-state feedback u = ;S x + v provided some care is exercised however, this
u](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-383-320.jpg)
![H2 OPTIMAL CONTROL
386
Proof. Items 1 and 2 follow immediately from the proof of the state feedback results
because the argument that Tvw = 0 did not depend on the restriction to state feedback
only. Thus we only need to prove item 3. Let K be an admissible controller such that
kTzw k2 < . As in the SF proof, de ne a new control variable v = u ; F2 x then the
closed-loop system is as shown below
z
y
with
2
AF2
6
6 C1F2
~
GFI = 6 " #
6
I
4
~
GFI
"
0
0
v
-
~
K
#
B2
D12#
"
B1
w
0
0
3
7
7
7
7
5
~
K = K ; F2 0]:
I
0
Denote by Q the transfer matrix from w to v it belongs to RH2 by internal stability
and the fact that D12 has full column rank and Tzw with zi= C1F2 x + D12 Qw hasi nite
h
h
H2 norm. Then u = F2 x + v = F2 x + Qw = F2 Q y so K = F2 Q , and
kTzw k2 = kGc B1 + UQk2 = kGc B1 k2 + kQk2 hence,
2
2
2
2
kQk2 = kTzw k2 ; kGcB1 k2 < 2 ; kGc B1 k2 :
2
2
2
2
Likewise, one can show that every controller of the form given in item no.3 is admissible
and suboptimal.
2
The results for DF, OI, FC, and OE follow from the parallel development of Chapter 12.
Disturbance Feedforward:
2
A B1 B2
6
GDF (s) = 4 C1 0 D12
C2 I 0
3
7
5
This problem inherits the same assumptions (i)-(iii) as in the state feedback problem,
in addition to the stability condition of A ; B1 C2 .
1. min kTzw k2 = kGc B1 k2
"
A + B2 F2 ; B1 C2 B1
2. K (s) =
F2
0
#](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-403-320.jpg)
![392
14.11 Notes and References
H2 OPTIMAL CONTROL
The detailed treatment of H2 related theory, LQ optimal control, Kalman ltering, etc.,
can be found in Anderson and Moore 1990] or Kwakernaak and Sivan 1972].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-409-320.jpg)
![LINEAR QUADRATIC OPTIMIZATION
394
Although this result is well-known, we will include a time-domain proof similar in technique to the proofs of the optimal H2 and H1 control results.
Lemma 15.1 w2L2; kwk2 x(0) = x0 = x0 y0 where y0 solves Lcy0 = x0.
inf
2
Proof. Assume (A B) is controllable otherwise, factor out the uncontrollable subspace. Then Lc is invertible and y0 = L;1x0 . Moreover, w 2 L2; can be used to
c
produce any x(0) = x0 given x(;1) = 0. We need to show
inf kwk2 x(0) = x0 = x0 L;1 x0 :
(15:2)
2
c
w2L2;
To show this, we can di erentiate x(t) L;1 x(t) along the solution of (15.1) for any given
c
input w as follows:
d (x L;1 x) = x L;1 x + x L;1 x = x (A L;1 + L;1 A)x + 2hw B L;1 xi:
_ c
c
c _
c
c
c
dt
Using Lc equation to substitute for A L;1 + L;1 A and completion of the squares gives
c
c
d
;1
2
;1 2
dt (x Lc x) = kwk ; kw ; B Lc xk :
Integration from t = ;1 to t = 0 with x(;1) = 0 and x(0) = x0 gives
x0 L;1x0 = kwk2 ; kw ; B L;1 xk2 kwk2 :
c
2
c
2
2
c
If w(t) = B e;A t L;1x0 = B L;1e(A+BB L;1)t x0 on (;1 0], then w 2 L2; , w =
c
;1 x and equality is achieved, cthus proving (15.2).
B Lc
2
Lemma 15.2
supw2BL2; kP+z k2 = supw2BH? kP+ MG wk2 = (LoLc):
2
2
2
Proof. Given x(0) = x0 and w = 0, for t 0 the norm of z(t) = CeAtx0 can be found
from
kP+z k2 =
2
1
Z
0
x0 eA t C CeAt x0 dt = x0 Lo x0 :
Combine this result with Lemma 15.1 to give
+ z k2
sup kP+ z k2 = sup kPwk2 2 = max x0 Lo1x0 = (LoLc ):
2
x0 6=0 x0 L; x0
w2BL2;
06=w2L2; k 2
c
2](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-411-320.jpg)
![LINEAR QUADRATIC OPTIMIZATION
396
Proof. We can di erentiate x(t) Xx(t) as in the last section, use the Riccati equation
(15.3) to substitute for A X + XA, and complete the squares to get
d (x Xx) = ;kz k2 + kwk2 ; kR;1=2 Rw ; (B X + D C )x]k2 :
dt
If w 2 L2+ , then x 2 L2+ , so integrating from t = 0 to t = 1 gives
kz k2 ; kwk2 = x0 Xx0 ; kR;1=2 Rw ; (B X + D C )x]k2 x0 Xx0 :
(15:4)
2
2
2
;1 (B X +D C )]t
If we let w = R;1 (B X + D C )x = R;1 (B X + D C )e A+BR
x0 , then
w 2 L2+ because A + BR;1 (B X + D C ) is stable. Thus the inequality in (15.4) can
be made an equality and the proof is complete. Note that the sup is achieved for a w
which is a linear function of the state.
2
As a direct consequence of this lemma, we have the following
Corollary 15.4 Let x(0) = x0 and X0 = X0 > 0.
(i) if kGk1 < and X = Ric(H ). Then
sup
kz k2 ; 2 (kwk2 + x0 X0 x0 )
2
2
n
06=(x0 w)2R L2 0 1)
= sup
8
>
<
>
:
06=x0 2Rn
<0
=0
= +1
x0 Xx0 ; 2 x0 X0 x0
max (X ; 2 X0 ) < 0
max (X ; 2 X0 ) = 0
max (X ; 2 X0 ) > 0:
kP+z k2 < 2 if and only if (D) < , H 2 dom(Ric),
sup
2
n L2 0 1) kw k2 + x0 X0 x0
06=(x0 w)2R
and max (X ; 2 X0 ) < 0.
Remark 15.2 The matrix X0 has the interpretation of the con dence on the initial
condition x0 . So if (X0 ) is small, then the initial condition is probably not known
very well. In that case min will be large where min denotes the smallest such
that max (X ; 2 X0 ) 0. On the other hand, a large (X0 ) implies that the initial
condition is known very well and that min is determined essentially by the condition
H 2 dom(Ric).
~
A dual version of Lemma 15.3 can also be obtained and is useful in characterizing
the so-called 2 2 block mixed Hankel-Toeplitz operator. To do that, we rst note that1
(ii)
2
GT (s) =
"
A
B
C
D
#
1 Note that since the system matrices are real, AT = A B T = B , etc. The conjugate transpose is
used here for the transpose for the sake of consistency in notation.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-413-320.jpg)
![15.3. Mixed Hankel-Toeplitz Operators
397
and kGk1 < 1 i kGT k1 < 1. Let J denote the following Hamiltonian matrix
"
J= A
;BB
#
"
C
+
;A
;BD
0
#
h
~
R;1 DB
C
i
~
~
where R := I ; DD . Then J 2 dom(Ric), Y = Ric(J ) 0, A + (Y C + BD )R;1 C
is stable and
~
AY + Y A + (Y C + BD )R;1 (CY + DB ) + BB = 0
(15:5)
if kGk1 < 1. For simplicity, we shall assume that (A B ) is controllable hence, Y > 0.
The case in which Y is singular can be obtained by restricting x0 2 Im(Y ) and replacing
Y ;1 by Y + .
Lemma 15.5 Suppose kGk1 < 1 and (A B) is controllable. Then
sup kP;z k2 ; kwk2 x(0) = x0 = ;x0 Y ;1 x0
2
2
w2L2;
and the sup is achieved.
Proof. Analogous to the proof of Lemma 15.3, we can di erentiate x(t) Y ;1x(t), use
the Riccati equation (15.5) to substitute for AY + Y A , and complete the squares to
get
d
;1
2
2
;1=2
;1
2
dt (x Y x) = ;kz k + kwk ; kR Rw ; (B Y + D C )x]k
where R = I ; D D > 0. If w 2 L2; , then x 2 L2; and x(;1) = 0 so integrating
from t = ;1 to t = 0 gives
kz k2 ; kwk2 = ;x0 Y ;1 x0 ; kR;1=2 Rw ; (B Y ;1 + D C )x]k2 ;x0 Y ;1 x0 : (15:6)
2
2
2
If we let w = R;1 (B Y ;1 + D C )x = R;1 (B Y ;1 + D C )e A+BR;1 (B Y ;1 +D C )]tx0 ,
~
then w 2 L2; because A + BR;1(B Y ;1 + D C ) = ;Y fA +(Y C + BD )R;1 C g Y ;1
~ ;1 C is stable. Thus the inequality can be made an equality
and A + (Y C + BD )R
and the proof is complete.
2
15.3 Mixed Hankel-Toeplitz Operators
h
i
h
i
Now suppose that the input is partitioned so that B = B1 B2 , D = D1 D2 ,
h
i
G(s) = G1 (s) G2 (s) 2 RH1](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-414-320.jpg)
![LINEAR QUADRATIC OPTIMIZATION
398
and w is partitioned conformably. Then kG2 k1 < 1 i
HW :=
"
A
0
;C C ;A
#
"
+
#
(D2 ) < 1, and
h
B2
;
R2 1 D2 C B2
;C D2
i
is in dom(Ric) where R2 := I ; D2 D2 > 0. For HW 2 dom(Ric), de ne W = Ric(HW ).
Let
("
#
)
w1 w 2 H ? w 2 L :
? L2 =
w 2 W := H2
(15:7)
1
2
2 2
w2
We are interested in a test for supw2BW kP+ z k2 < 1, or, equivalently,
sup k;wk2 < 1
w2BW
(15:8)
where ; = P+ MG1 MG2 ] : W ! H2 is a mixed Hankel-Toeplitz operator:
"
w
; 1
w2
#
= P+
= P+
h
h
"
#
w1
?
w1 2 H2 w2 2 L2
G1 G2
w2
i
G1 G2 P; w + P+ G2 P+ w2 :
i
?
?
Thus ; is the sum of the Hankel operator P+ MG : H2 H2 ! H2 and the Toeplitz
operator P+ MG2 : H2 ! H2 . The following lemma generalizes Corollary 13.24 (B1 =
0 D1 = 0) and Lemma 15.2 (B2 = 0 D2 = 0).
Lemma 15.6 supw2BW k;wk2 < 1 i the following two conditions hold:
(i) (D2 ) < 1 and HW 2 dom(Ric)
(ii) (WLc) < 1.
Proof. By Corollary 13.24, condition (i) is necessary for (15.8), so we will prove that
given condition (i), (15.8) holds i condition (ii) holds. We will do this by showing,
equivalently, that (WLc) 1 i supw2BW k;wk2 1. By de nition of W , if w 2 W
then
kP+ z k2 ; kwk2 = kP+ z k2 ; kP+ w2 k2 ; kP; wk2 :
2
2
2
2
2
2 through x(0). Thus if Lc is invertible,
Note that the last term only contributes to kP+ z k2
then Lemma 15.1 and 15.3 yield
sup kP+ z k2 ; kwk2 x(0) = x0 = x0 Wx0 ; x0 L;1 x0
2
2
c
w2W
(15:9)
and the supremum is achieved for some w 2 W that can be constructed from the
previous lemmas. Since (WLc ) 1 i 9 x0 6= 0 such that the right-hand side of](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-415-320.jpg)
![LINEAR QUADRATIC OPTIMIZATION
402
"
(i)'
A ; j!I B2
C1
D12
#
has full column rank for all ! 2 R, and
~
(iii)' D12 has full column rank.
It is clear from the assumptions that H2 2 dom(Ric) and X2 = Ric(H2 ) > 0, where
"
#
A
;B2 B2 :
H2 =
;C1 C1 ;A
Let F2 = ;B2 X2 and D? be such that D12 D? ] is an orthogonal matrix. De ne
"
Gc (s) := AF2 I
C1F2 0
and
"
#
"
U (s) := AF2 B2
C1F2 D12
#
#
;1
(15:13)
U? = AF2 ;X2 C1 D?
C1F2
D?
where AF2 = A + B2 F2 and C1F2 = C1 + D12 F2 . The following is easily proven using
Lemma 13.29 by obtaining a state-space realization and by eliminating uncontrollable
states using a little algebra involving the Riccati equation for X2 .
h
i
Lemma 15.8 U U?] is square and inner and a realization for Gc U U? is
"
#
;1
Gc U U? = AF2 ;B2 X2 C1 D? 2 RH2 :
X2 0
0
h
i
(15:14)
This implies that U and U? are each inner and that both U? Gc and U Gc are
in RH? . To answer our earlier question, de ne a Hamiltonian matrix H1 and the
2
associated Riccati equation as
"
A
B1 B1 ; B2 B2
H1 :=
;C1 C1
;A
#
A X1 + X1 A + X1 B1 B1 X1 ; X1 B2 B2 X1 + C1 C1 = 0:
Theorem 15.9 sup umin kzk2 < 1 if and only if H1 2 dom(Ric) and X1 =
2L2+
w2BL2+
Ric(H1 ) > 0. Furthermore, if the condition is satis ed, then u = F1 x with F1 :=
;B2 X1 is an optimal control.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-419-320.jpg)
![15.5. Linear Quadratic Max-Min Problem
403
Proof. ()) As in Chapter 14, de ne := u ; F2x to get
z = Gc B1 w + U :
Then the hypothesis implies that
sup min kz k2 < 1:
(15:15)
w2BH2 2H2
Since by Lemma 15.8 U U? ] is square and inner, kz k2 = k U U?] z k2, and
h
U U?
i
"
z = U G c B1 w +
U? Gc B1 w
Thus
sup min kz k2 = sup min
w2BH2 2H2
#
"
w2BH2 2H2
"
#
P (U Gc B1 w) + P+ (U Gc B1 w + ) :
= ;
U? Gc B1 w
P; (U Gc B1 w) + P+ (U Gc B1 w + )
U? Gc B1 w
#
2
and the right hand of the above equation is minimized by = ;P+ (U Gc B1 w) we
have
"
P; (U Gc B1 w)
sup min kz k2 = sup
U? Gc B1 w
w2BH2 2H2
w2BH2
=: sup k; wk2 < 1
#
2
w2BH2
where ; : L2+ ! W is de ned as
"
#
"
P (U Gc B1 w) = P; U
; w= ;
U? Gc B1 w
U?
with
W :=
("
q1
q2
#
#
Gc B1 w
)
?
q1 2 H2 q2 2 L2 :
Note that from equation (15.10) the adjoint operator ; : W ! H2 is given by
"
#
h
q
; 1 = P+ (B1 Gc (Uq1 + U? q2 )) = P+ B1 Gc U U?
q2
where Gc U U? ] 2 RH2 has the realization in (15.14). So we have
sup k;qk2 < 1:
q2BW
i
"
q1
q2
#](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-420-320.jpg)
![404
LINEAR QUADRATIC OPTIMIZATION
This is just the condition (15.8), so from Lemma 15.6 and equation (15.14) we have
that
"
;
; #
AF2
X2 1C1 C1 X2 1 2 dom(Ric)
HW :=
;X2B1 B1 X2
;AF2
and W = Ric(HW ) 0. Note that the observability of (C1 A) implies X2 > 0.
;
Furthermore, the controllability Gramian for the system (15.14) is X2 1 since
;
;
;
;
AF2 X2 1 + X2 1 AF2 + B2 B2 + X2 1 C1 C1 X2 1 = 0:
;
Lemma 15.6 also implies (WX2 1) < 1 or, equivalently, X2 # W . Using the Ric>
"
;I X2;1 provides a similarity
;X2 0
cati equation for X2 , one can verify that T :=
transformation between H1 and HW , i.e., H1 = THW T ;1. Then
"
#
"
#
;1
X; (H1 ) = T X; (HW ) = T Im I = Im I ; X2 W
W
X2
so H1 2 dom(Ric) and X1 = X2 (X2 ; W );1 X2 > 0.
(() If H1 2 dom(Ric) and X1 = Ric(H1 ) > 0, A + B1 B1 X1 ; B2 B2 X1 is
stable. De ne
AF1 := A + B2 F1
C1F1 := C1 + D12 F1 :
Then the Riccati equation can be written as
AF1 X1 + X1 AF1 + C1F1 C1F1 + X1 B1 B1 X1 = 0:
We conclude from Lyapunov theory that AF1 is stable since (AF1 B1 X1 ) is detectable
and X1 > 0. Now with the given control law u = F1 x, the dynamical system becomes
x = AF1 x + B1 w
_
z = C1F1 x:
So by Corollary 13.24, kTzw k1 < 1, i.e., supw2L2+ kz k2 < 1 for the given control law.
2
Theorem 15.9 will be used in the next chapter to solve the FI H1 control problem.
15.6 Notes and References
Di erential game is a well-studied topic, see e.g., Bryson and Ho 1975]. The paper
by Mageirou and Ho 1977] is one of the early papers that are relevant to the topics
covered in this chapter. The current setup and proof are taken from Doyle, Glover,
Khargonekar, and Francis 1989]. The application of game theoretic results to H1
problems can be found in Basar and Bernhard 1991], Khargonekar, Petersen, and Zhou
1990], Limebeer, Anderson, Khargonekar, and Green 1992], and references therein.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-421-320.jpg)
![H1 CONTROL: SIMPLE CASE
406
where the plant G and controller K are assumed to be real-rational and proper. It will
be assumed that state space models of G and K are available and that their realizations
are assumed to be stabilizable and detectable. Recall again that a controller is said to
be admissible if it internally stabilizes the system. Clearly, stability is the most basic
requirement for a practical system to work. Hence any sensible controller has to be
admissible.
Optimal H1 Control: nd all admissible controllers K (s) such that kTzw k1
is minimized.
It should be noted that the optimal H1 controllers as de ned above are generally not
unique for MIMO systems. Furthermore, nding an optimal H1 controller is often both
numerically and theoretically complicated, as shown in Glover and Doyle 1989]. This
is certainly in contrast with the standard H2 theory, in which the optimal controller
is unique and can be obtained by solving two Riccati equations without iterations.
Knowing the achievable optimal (minimum) H1 norm may be useful theoretically since
it sets a limit on what we can achieve. However, in practice it is often not necessary
and sometimes even undesirable to design an optimal controller, and it is usually much
cheaper to obtain controllers that are very close in the norm sense to the optimal ones,
which will be called suboptimal controllers. A suboptimal controller may also have other
nice properties over optimal ones, e.g., lower bandwidth.
Suboptimal H1 Control: Given > 0, nd all admissible controllers
K (s) if there is any such that kTzw k1 < .
For the reasons mentioned above, we focus our attention in this book on suboptimal
control. When appropriate, we brie y discuss what will happen when approaches the
optimal value.
16.2 Output Feedback H1 Control
16.2.1 Internal Stability and Input/output Stability
Now suppose K is a stabilizing controller for the system G. Then the internal stability
guarantees Tzw = F` (G K ) 2 RH1 , but the latter does not necessarily imply the
internal stability. The following lemma provides the additional (mild) conditions to the
equivalence of Tzw = F`(G K ) 2 RH1 and internal stability of the closed-loop system.
To state the lemma, we shall assume that G and K have the following stabilizable and
detectable realizations, respectively:
2
A B1 B2
6
G(s) = 4 C1 D11 D12
C2 D21 D22
3
7
5
"
#
^ ^
K (s) = A B :
^ D
C ^](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-423-320.jpg)
![H1 CONTROL: SIMPLE CASE
408
16.2.2 Contraction and Stability
One of the keys to the entire development of H1 theory is the fact that the contraction
and internal stability is preserved under an inner linear fractional transformation.
Theorem 16.2 Consider the following feedback system:
z
r
P
-
Q
w
v
"
#
P = P11 P12 2 RH1
P21 P22
;
Suppose that P P = I , P211 2 RH1 and that Q is a proper rational matrix. Then
the following are equivalent:
(a) The system is internally stable, well-posed, and kTzw k1 < 1.
(b) Q 2 RH1 and kQk1 < 1.
Proof. (b) ) (a). Note that since P Q 2 RH1, the system internal stability is guaranteed if (I ; P22 Q);1 2 RH1 . Therefore, internal stability and well-posedness follow
easily from kP22 k1 1, kQk1 < 1, and a small gain argument. (Note that kP22 k1 1
follows from the fact that P22 is a compression of P .)
To show that kTzw k1 < 1, consider the closed-loop system at any frequency s = j!
with the signals xed as complex constant vectors. Let kQk1 =: < 1 and note that
;
Twr = P211 (I ; P22 Q) 2 RH1 . Also let := kTwr k1 . Then kwk krk, and P inner
implies that kz k2 + krk2 = kwk2 + kvk2. Therefore,
kz k2 kwk2 + ( 2 ; 1)krk2 1 ; (1 ; 2 ) ;2 ]kwk2
which implies kTzw k1 < 1 .
(a) ) (b). To show that kQk1 < 1, suppose there exist a (real or in nite) frequency
! and a constant nonzero vector r such that at s = j! , kQrk krk. Then setting
;
w = P211 (I ; P22 Q)r, v = Qr gives v = Tvw w. But as above, P inner implies that
2 + krk2 = kwk2 + kv k2 and, hence, that kz k2 kwk2 , which is impossible since
kz k
kTzw k1 < 1. It follows that max (Q(j!)) < 1 for all !, i.e., kQk1 < 1 since Q is
rational.
To show Q 2 RH1 , let Q = NM ;1 with N M 2 RH1 be a right coprime factorization, i.e., there exist X Y 2 RH1 such that XN + Y M = I . We shall show that
M ;1 2 RH1 . By internal stability we have
Q(I ; P22 Q);1 = N (M ; P22 N );1 2 RH1
and
(I ; P22 Q);1 = M (M ; P22 N );1 2 RH1 :](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-425-320.jpg)
![16.2. Output Feedback H1 Control
409
Thus
XQ(I ; P22 Q);1 + Y (I ; P22 Q);1 = (M ; P22 N );1 2 RH1 :
This implies that the winding number of det(M ; P22 N ), as s traverses the Nyquist
contour, equals zero. Now note the fact that, for all s = j!, det M ;1 6= 0, det(I ;
P22 Q) 6= 0 for all in 0,1] (this uses the fact that kP22 k1 1 and kQk1 < 1). Also,
det(I ; P22 Q) = det(M ; P22 N ) det M ;1, and we have det(M ; P22 N ) 6= 0 for all
in 0,1] and all s = j!. We conclude that the winding number of det M also equals
zero. Therefore, Q 2 RH1 , and the proof is complete.
2
16.2.3 Simplifying Assumptions
In this chapter, we discuss a simpli ed version of H1 theory. The general case will be
considered in the next chapter. The main reason for doing so is that the general case
has its unique features but is much more involved algebraically. Involved algebra may
distract attention from the essential ideas of the theory and therefore lose insight into
the problem. Nevertheless, the problem considered below contains the essential features
of the H1 theory.
The realization of the transfer matrix G is taken to be of the form
2
3
A B1 B2
6
G(s) = 4 C1 0 D12 7 :
5
C2 D21 0
The following assumptions are made:
(i) (A B1 ) is stabilizable and (C1 A) is detectable
(ii) (A B2 ) is stabilizable and (C2 A) is detectable
h
i
h
"
#
(iii) D12 C1 D12 = 0 I
"
(iv)
#
i
B1 D = 0 .
D21 21
I
Assumption (i) is made for a technical reason: together with (ii) it guarantees that
the two Hamiltonian matrices (H2 and J2 in Chapter 14) in the H2 problem belong
to dom(Ric). This assumption simpli es the theorem statements and proofs, but if
it is relaxed, the theorems and proofs can be modi ed so that the given formulae are
still correct, as will be seen in the next chapter. An important simpli cation that is
a consequence of the assumption (i) is that internal stability is essentially equivalent
to input-output stability (Tzw 2 RH1 ). This equivalence enables us to focus only on
input/output stability and is captured in Corollary 16.3 below. Of course, assumption
(ii) is necessary and su cient for G to be internally stabilizable, but is not needed](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-426-320.jpg)
![16.8. Separation Theory
427
16.8.1 H1 Controller Structure
The H1 controller formulae from Theorem 16.4 are
"
#
^
A1 ;Z1 L1
Ksub (s) :=
F1
0
^
A1 := A + ;2B1 B1 X1 + B2 F1 + Z1 L1 C2
F1 := ;B2 X1 L1 := ;Y1 C2 Z1 := (I ; ;2 Y1 X1 );1
where X1 := Ric(H1 ) and Y1 := Ric(J1 ). The necessary and su cient conditions
for the existence of an admissible controller such that kTzw k1 < are
(i) H1 2 dom(Ric) and X1 := Ric(H1 ) 0
(ii) J1 2 dom(Ric) and Y1 := Ric(J1 0
(iii) (X1 Y1 ) < 2 .
We have seen that condition (i) corresponds to the Full Information condition and
that (ii) corresponds to the Full Control condition. It is easily shown that, given the FI
and FC results, these conditions are necessary for the output feedback case as well.
Lemma 16.14 Suppose there exists an admissible controller making kTzw k1 < .
Then conditions (i) and (ii) hold.
Proof. Let K be an admissible controller for which kTzw k1 < . The controller
K C2 D21 ] solves the FI problem hence from Theorem 16.9, H1 2 dom(Ric) and
X1 := Ric(H1 ) 0. Condition (ii) follows by the dual argument.
2
We would also expect some condition beyond these two, and that is provided by
(iii), which is an elegant combination of elements from FI and FC. Note that all the
conditions of Theorem 16.4 are symmetric in H1 , J1 , X1 , and Y1 , but the formula
for the controller is not. Needless to say, there is a dual form that can be obtained by
inspection from the above formula. For a symmetric formula, the state equations above
;
can be multiplied through by Z11 and put in descriptor form. A simple substitution
from the Riccati equation for X1 will then yield a symmetric, though more complicated,
formula:
_
(I ; ;2 Y1 X1 )x = As x ; L1 y
^
^
(16.10)
u = F1 x
^
(16.11)
where As := A + B2 F1 + L1 C2 + ;2Y1 A X1 + ;2 B1 B1 X1 + ;2Y1 C1 C1 .
To emphasize its relationship to the H2 controller formulae, the H1 controller can
be written as
_
x = Ax + B1 wworst + B2 u + Z1 L1 (C2 x ; y)
^
^
^
^](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-444-320.jpg)
![H1 CONTROL: SIMPLE CASE
436
16.10 Controller Interpretations
This section considers some additional connections with the minimum entropy solution
and the work of Whittle and will be of interest primarily to readers already familiar
with them. The connection with the Q-parameterization approach will be considered
in the next chapter for the general case.
Section 16.10.2 gives another separation interpretation of the central H1 controller
of Theorem 16.4 in the spirit of Whittle (1981). It has been shown in Glover and Doyle
1988] that the central controller corresponds exactly to the steady state version of the
optimal risk sensitive controller derived by Whittle, 1981], who also derives a separation
result and a certainty equivalence principle (see also Whittle, 1986]).
16.10.1 Minimum Entropy Controller
Let T be a transfer matrix with kT k1 < . Then the entropy of T (s) is de ned by
I (T ) = ; 2
2Z 1
;1
;
ln det I ; ;2T (j!)T (j!) d!:
It is easy to see that
I (T ) = ; 2
2Z 1X
;1 i
ln 1 ; ;2 i2 (T (j!)) d!
and I (T )
show that
0, where i (T (j!)) is the ith singular value of T (j!). It is also easy to
Z 1 X
2
2
lim I (T ) = 21
i (T (j! )) d! = kT k2 :
!1
;1 i
Thus the entropy I (T ) is in fact a performance index measuring the tradeo between
the H1 optimality ( ! kT k1 ) and the H2 optimality ( ! 1).
It has been shown in Glover and Mustafa 1989] that the central controller given in
Theorem 16.4 is actually the controller that satis es the norm condition kTzw k1 <
and minimizes the following entropy:
;2
2Z 1
;
ln det I ; ;2 Tzw (j!)Tzw (j!) d!:
;1
Therefore, the central controller is also called the minimum entropy controller (maxi~
mum entropy controller if the entropy is de ned as I (T ) = ;I (T )).
16.10.2 Relations with Separation in Risk Sensitive Control
Although Whittle, 1981] treats a nite horizon, discrete time, stochastic control problem, his separation result has a clear interpretation for the present in nite horizon,](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-453-320.jpg)
![16.12. Notes and References
439
Remark 16.4 It is simple;to show that if X11 and Y11 are nonsingular and if X1 =
;1
1
X12 X11 and Y1 = Y12 Y11 , then condition (iii) in the above theorem is equivalent to
X1 0, Y1 0, and (Y1 X1 ) 2 . So in this case, the conditions for the existence
of an optimal controller can be obtained from taking the limit" of the corresponding
conditions in Theorem 16.4. Moreover, the controller given above is reduced to the
descriptor form given in equations (16.12) and (16.13).
~
16.12 Notes and References
This chapter is based on Doyle, Glover, Khargonekar, Francis 1989], and Zhou 1992].
The minimum entropy controller is studied in detail in Glover and Mustafa 1989] and
Mustafa and Glover 1990]. The risk sensitivity problem is treated in Whittle 1981,
1986]. The connections between the risk sensitivity controller and the central H1
controller are explored in Doyle and Glover 1988]. The complete characterization of
optimal controllers for the general setup can be found in Glover, Limebeer, Doyle,
Kasenally, and Safonov 1991].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-456-320.jpg)
![H1 CONTROL: GENERAL CASE
442
17.1 General H1 Solutions
Consider the system described by the block diagram
z
y
G
-
K
w
u
where, as usual, G and K are assumed to be real rational and proper with K constrained
to provide internal stability. The controller is said to be admissible if it is real-rational,
proper, and stabilizing. Although we are taking everything to be real, the results
presented here are still true for the complex case with some obvious modi cations. We
will again only be interested in characterizing all suboptimal H1 controllers.
The realization of the transfer matrix G is taken to be of the form
2
A B1 B2
G(s) = 6 C1 D11 D12
4
C2 D21 0
3
7
5
"
A B
=
C D
#
which is compatible with the dimensions z (t) 2 Rp1 , y(t) 2 Rp2 , w(t) 2 Rm1 , u(t) 2
Rm2 , and the state x(t) 2 Rn . The following assumptions are made:
(A1) (A B2 ) is stabilizable and (C2 A) is detectable
(A2) D12 =
"
0
I
#
h
and D21 = 0 I
"
A ; j!I B2
C1
D12
#
"
A ; j!I B1
C2
D21
#
(A3)
(A4)
i
has full column rank for all !
has full row rank for all !.
Assumption (A1) is necessary for the existence of stabilizing controllers. The assumptions in (A2) mean that the penalty on z = C1 x + D12 u includes a nonsingular,
normalized penalty on the control u, that the exogenous signal w includes both plant
disturbance and sensor noise, and that the sensor noise weighting is normalized and
nonsingular. Relaxation of (A2) leads to singular control problems see Stroorvogel
1990]. For those problems that have D12 full column rank and D21 full row rank but
do not satisfy assumption (A2), a normalizing procedure is given in the next section so
that an equivalent new system will satisfy this assumption.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-459-320.jpg)
![17.1. General H1 Solutions
443
Assumptions (A3) and (A4) are made for a technical reason: together with (A1)
they guarantee that the two Hamiltonian matrices in the corresponding H2 problem
belong to dom(Ric), as we have seen in Chapter 14. It is tempting to suggest that
(A3) and (A4) can be dropped, but they are, in some sense, necessary for the methods
presented in the last chapter to be applicable. A further discussion of the assumptions
and their possible relaxation will be discussed in Section 17.3.
The main result is now stated in terms of the solutions of the X1 and Y1 Riccati
equations together with the state feedback" and output injection" matrices F and L.
R := D1 D1 ;
"
"
2 Im
0
2 Ip
1
#
0
0
#
0
0
where D1 := D11 D12 ]
#
"
D11
~
where D 1 :=
R := D 1 D 1 ;
D21
0
"
# "
#
h
i
A
0
B
H1 :=
;
R;1 D1 C1 B
;C1 C1 ;A
;C1 D1
"
# "
#
h
i
A
0
C
~
J1 :=
;
R;1 D 1 B1 C
;B1 B1 ;A
;B1D 1
1
X1 := Ric(H1)
F :=
L :=
"
h
#
F11 := ;R;1 D C + B X ]
1
1 1
F21
i
~
L11 L21 := ; B1 D 1 + Y1 C ]R;1
Partition D, F11 , and L11 are as follows:
2
"
Y1 := Ric(J1 )
3
F111 F121 F21
0 # 6 L
7
F = 6 111 D1111 D1112 0 7 :
7
6
6
0 D
L
I 7
5
4 L121 D1121 D1122
0
I
0
L21
Remark 17.1 In the above matrix partitioning, some matrices may not exist depending on whether D12 or D21 is square. This issue will be discussed further later. For the
time being, we shall assume all matrices in the partition exist.
~
Theorem 17.1 Suppose G satis es the assumptions (A1){(A4).
(a) There exists an admissible controller K (s) such that jjF` (G K )jj1 < (i.e.
kTzw k1 < ) if and only if](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-460-320.jpg)
![H1 CONTROL: GENERAL CASE
444
(i) > max( D1111 D1112 ] D1111 D1121 ])
(ii) H1 2 dom(Ric) with X1 = Ric(H1 ) 0
(iii) J1 2 dom(Ric) with Y1 = Ric(J1 ) 0
(iv) (X1 Y1 ) < 2 .
(b) Given that the conditions of part (a) are satis ed, then all rational internally
stabilizing controllers K (s) satisfying jjF` (G K )jj1 < are given by
where
K = F`(M1 Q)
for arbitrary Q 2 RH1
2
such that kQk1 <
3
^ ^
^
A B1 B2 7
6
^ ^
^ 5
M1 = 6 C1 D11 D12 7
4
^ ^
C2 D21 0
^
D11 = ;D1121 D1111 ( 2 I ; D1111 D1111 );1 D1112 ; D1122
^ 12 2 Rm2 m2 and D21 2 Rp2 p2 are any matrices (e.g. Cholesky factors) satisfying
^
D
^ ^
D12 D12 = I ; D1121 ( 2 I ; D1111 D1111 );1 D1121
^ ^
D21 D21 = I ; D1112 ( 2 I ; D1111 D1111 );1 D1112
and
where
^
^
B2 = Z1 (B2 + L121 )D12
^
^
C2 = ;D21 (C2 + F121 )
^
^ ^; ^
B1 = ;Z1L21 + B2 D121 D11
^
^ ^; ^
C1 = F21 + D11 D211 C2
^
^ ^; ^
A = A + BF + B1 D211 C2
Z1 = (I ; ;2 Y1 X1 );1 :
(Note that if D11 = 0 then the formulae are considerably simpli ed.)
Some Special Cases:
Case 1: D12 = I
In this case
1. in part (a), (i) becomes > (D1121 ).
2. in part (b)
^
D11 = ;D1122
^ ^
D12 D12 = I ; ;2 D1121 D1121
^ ^
D21 D21 = I:](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-461-320.jpg)
![450
H1 CONTROL: GENERAL CASE
Violating A1 and either or both of A3 and A4
Sensible control problems can be posed which violate A1 and either or both of A3 and
A4. For example, cases when A has modes at s = 0 which are unstabilizable through B2
and/or undetectable through C2 arise when an integrator is included in a weight on a
disturbance input or an error term. In these cases, either A3 or A4 are also violated, or
the closed-loop H1 norm cannot be made nite. In many applications, such problems
can be reformulated so that the integrator occurs inside the loop (essentially using the
internal model principle) and is hence detectable and stabilizable. We will show this
process in the next section.
An alternative approach to such problems which could potentially avoid the problem
reformulation would be to pursue the techniques in the last chapter but to relax internal
stability to the requirement that all closed-loop modes be in the closed left half plane.
Clearly, to have nite H1 norms, these closed-loop modes could not appear as poles
in Tzw . The formulae given in this chapter will often yield controllers compatible with
these assumptions. The user would then have to decide whether closed-loop poles on
the imaginary axis were due to weights and hence acceptable or due to the problem
being poorly posed, as in the above example.
A third alternative is to again introduce perturbations so that A1, A3, and A4 are
satis ed. Roughly speaking, this would produce sensible answers for sensible problems,
but the behavior as ! 0 could be problematic.
Relaxing A2
In the cases that either D12 is not full column rank or that D21 is not full row rank,
improper controllers can give a bounded H1 -norm for Tzw , although the controllers will
not be admissible by our de nition. Such singular ltering and control problems have
been well-studied in H2 theory and many of the same techniques go over to the H1 -case
(e.g. Willems 1981], Willems, Kitapci, and Silverman 1986], and Hautus and Silverman
1983]). In particular, the structure algorithm of Silverman 1969] could be used to make
the terms D12 and D21 full rank by the introduction of suitable di erentiators in the
controller. A complete solution to the singular problem can be found in Stroorvogel,
1990].
17.4 H2 and H1 Integral Control
It is interesting to note that the H2 and H1 design frameworks do not in general
produce integral control. In this section we show how to introduce integral control into
the H2 and H1 design framework through a simple disturbance rejection problem. We
consider a feedback system shown in Figure 17.1. We shall assume that the frequency
contents of the disturbance w are e ectively modeled by the weighting Wd 2 RH1 and
the constraints on control signal are limited by an appropriate choice of Wu 2 RH1 . In
order to let the output y track the reference signal r, we require K contain an integral,](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-467-320.jpg)
![H1 CONTROL: GENERAL CASE
454
which is independent of as expected. Similarly, we can solve an optimal H2 controller
+ 1)(s +
+
; 0 069)
^
K2 = ;s43:487((s 2 + 30:94s )(s411100)(ss; 7::964)
( + )2 s
+ :81)(
and
+ 100)(
^
K2 (s) = ;K2(s)M (s) = s43:2487(s :+ 1)(s 411:81)(ss; 0::069) :
(s + 30 94s +
; 7 964)
An approximate integral control can also be achieved without going through the
above process by letting
1
~
We = We = s + M (s) = 1
for a su ciently small > 0. For example, a controller for = 0:001 is given by
s+
7:85(s + 1)(s + 100)(s ; 0:1545)
K1 = (s316880(s + 1)(32)(s100)(s ; 0:1545)
+ 0:001)(s +
+ 40370)(s ; 20)
s(s + 32)(s ; 20)
which gives the closed-loop H1 norm of 7:85. Similarly, an approximate H2 integral
controller is obtained as
+ 1)(s + 100)(s 0679)
K2 = (s + 043:47(s 2 + 30:93s + 411; 0:s ; 7:9718) :
:001)(s
:7)(
17.5 H1 Filtering
In this section we show how the ltering problem can be solved using the H1 theory
developed early. Suppose a dynamic system is described by the following equations
x = Ax + B1 w(t) x(0) = 0
_
(17.1)
y = C2 x + D21 w(t)
(17.2)
z = C1 x + D11 w(t)
(17.3)
The ltering problem is to nd an estimate z of z in some sense using the measurement
^
of y. The restriction on the ltering problem is that the lter has to be causal so
that it can be realized, i.e., z has to be generated by a causal system acting on the
^
measurements. We will further restrict our lter to be unbiased, i.e., given T > 0 the
estimate z(t) = 0 8t 2 0 T ] if y(t) = 0 8t 2 0 T ]. Now we can state our H1 ltering
^
problem.
H1 Filtering: Given a > 0, nd a causal lter F (s) 2 RH1 if it exists
such that
^2
J := sup kz ; z2k2 < 2
with z = F (s)y.
^
w2L2 0 1)
kwk2](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-471-320.jpg)
![17.6. Youla Parameterization Approach*
=
459
kR11 + Qk2 +
2
"
0
R21
R12
R22
#
2
2
Furthermore,
:
kR11 + Qk2 = kR11 ; D12 D11 D21 k2 + kQ + D12 D11 D21 k2
2
2
2
?
since (R11 ; D12 D11 D21 ) 2 H2 and (Q + D12 D11 D21 ) 2 H1 . In fact,
(Q + D12 D11 D21 ) has to be in H2 to guarantee that the 2-norm be nite.
Hence the unique optimal solution is given by a projection to H1 :
Qopt = ;R11 ]+ = ;D12 D11 D21 :
The optimal controller is given by
Kopt = F` (M2 Qopt):
In particular, if D11 = 0 then Kopt = F` (M2 0).
H1 case: Recall the de nition of the 1-norm:
"
"
#
#!
R11 + Q R12 (j!):
= sup
R21 R22
!
1
Consider the minimization problem with respect to Q frequency by frequency
R11 + Q R12
R21 R22
it becomes a constant matrix dilation problem if no causality restrictions are
imposed on Q. The H1 optimal control follows this idea. However, it should
be noted that since Q is restricted to be an RH1 matrix, the term R11 + Q
cannot be made into an arbitrary matrix. Therefore, the problem will be
signi cantly di erent from the constant matrix case, and, generically,
"
min
Q2H1
where
(
0 := max
h
R11 + Q R12
R21 R22
R21 R22
#
1
"
i
1
>
R12
R22
0
#
)
1
:
It is convenient to use the characterization in Corollary 2.23. Recall that
"
i
R11 + Q R12
R21 R22
#
(< ) for >
0
(I ; WW );1=2 (Q + R11 + WR22 Z )(I ; Z Z );1=2
1
1
(< )](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-476-320.jpg)
![H1 CONTROL: GENERAL CASE
462
formulae in Theorems 16.4 and 16.5. This approach is exactly the procedure used in
Doyle 1984] and Francis 1987] with Lemma 15.7 used to solve the general distance
problem. Although this approach is conceptually straightforward and was, in fact, used
to obtain the rst proof of the current state space results in this chapter, it seems
unnecessarily cumbersome and indirect.
17.8 State Feedback and Di erential Game
It has been shown in Chapters 15 and 16 that a (central) suboptimal full information
H1 control law is actually a pure state feedback if D11 = 0. However, this is not true
in general if D11 6= 0, as will be shown below. Nevertheless, the state feedback H1
control is itself a very interesting problem and deserves special treatment. This section
and the section to follow are devoted to the study of this state feedback H1 control
problem and its connections with full information control and di erential game.
Consider a dynamical system
x = Ax + B1 w + B2 u
_
z = C1 x + D11 w + D12 u
(17.7)
(17.8)
where z (t) 2 Rp1 , y(t) 2 Rp2 , w(t) 2 Rm1 , u(t) 2 Rm2 , and x(t) 2 Rn . The following
assumptions are made:
(AS1) (A B2 ) is stabilizable
h
i
(AS2) There is a matrix D? such that D12 D? is unitary
"
(AS3)
A ; j!I B2
C1
D12
#
has full column rank for all !.
We are interested in the following two related quadratic min-max problems: given > 0,
check if
sup
min kz k2 <
and
w2B L2 0 1) u2L2 0 1)
min
sup
u2L2 0 1) w2B L2 0 1)
kz k2 < :
The rst problem can be regarded as a full information control problem since the control
signal u can be a function of the disturbance w and the system state x. On the other
hand, the optimal control signal in the second problem cannot depend on the disturbance
w (the worst disturbance w can be a function of u and x). In fact, it will be shown that
the control signal in the latter problem depends only on the system state hence, it is
equivalent to a state feedback control.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-479-320.jpg)
![17.8. State Feedback and Di erential Game
463
Theorem 17.6 Let > 0 be given and de ne
R := D1 D1 ;
"
2 Im
0
"
A
0
H1 :=
;C1 C1 ;A
h
0
0
1
#
i
#
where D1 := D11 D12 ]
"
B
;
;C1 D1
#
h
R;1 D1 C1 B
i
where B := B1 B2 .
(a)
sup
min kz k2 < if and only if
w2B L2 0 1) u2L2 0 1)
(D? D11 ) <
H1 2 dom(Ric) X1 = Ric(H1 ) 0:
Moreover, an optimal u is given by
h
i
u = ;D12D11 w + D12 D11 I Fx
and a worst w worst is given by
w
where
(b)
min
"
worst
= F11 x
#
F := F11 := ;R;1 D1 C1 + B X1 ] :
F21
sup
u2L2 0 1) w2B L2 0 1)
kz k2 < if and only if
(D11 ) <
H1 2 dom(Ric) X1 = Ric(H1 0:
Moreover, an optimal u is given by
u = F21 x = ;D12D11 w
worst +
h
i
D12 D11 I Fx
and the worst wsfworst is given by
wsfworst = ( 2 I ; D11 D11 );1 f(D11 C1 + B1 X1 )x + D11 D12 ug :
Proof. The condition for part (a) can be shown in the same way as in Chapter 15
and is, in fact, the solution to the FI problem for the general case. We now prove the
condition for part (b). It is not hard to see that (D11 ) < is necessary since control
u cannot feed back w directly, so the D11 term cannot be (partially) eliminated as in](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-480-320.jpg)
![468
H1 CONTROL: GENERAL CASE
17.10 Notes and References
The detailed derivation of the H1 solution for the general case is treated in Glover
and Doyle 1988, 1989]. The loop-shifting procedures are given in Safonov, Limebeer,
and Chiang 1989]. The idea is also used in Zhou and Khargonekar 1988] for state
feedback problems. A fairly complete solution to the singular H1 problem is obtained
in Stoorvogel 1992]. The H1 ltering and smoothing problems are considered in detail
in Nagpal and Khargonekar 1991]. The Youla parameterization approach is treated
very extensively in Doyle 1984] and Francis 1987] and in the references therein. The
presentation of the state feedback H1 control in this chapter is based on Zhou 1992].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-485-320.jpg)
![18.4. Notes and References
487
2
This theorem shows that all closed-loop objectives are guaranteed to have bounded
magnitude and the bounds depend only on , W1 W2 , and G.
18.4 Notes and References
The H1 loop shaping using normalized coprime factorization was developed by McFarlane and Glover 1990, 1992]. In the same references, some design examples were
also shown. The method has been applied to the design of scheduled controllers for
a VSTOL aircraft in Hyde and Glover 1993]. The robust stabilization of normalized
coprime factors is closely related to the robustness in the gap metric and graph topology, see El-Sakkary 1985], Georgiou and Smith 1990], Glover and McFarlane 1989],
McFarlane, Glover, and Vidyasagar 1990], Qiu and Davison 1992a, 1992b] Vinnicombe
1993], Vidyasagar 1984, 1985], Zhu 1989], and references therein.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-504-320.jpg)
![19.2. H1 Controller Reductions
499
Proof. We shall assume N21 has full row rank. The case when N12 has full column
rank can be shown in the same way.
To show that kTzw k1 < 1, consider the closed-loop system at any frequency s = j!
with the signals xed as complex constant vectors. Let kQk1 =: < 1 and note that
+
+
Twy = N21 (I ; N22 Q) where N21 is a right inverse of N21 . Also let := kTwy k1 . Then
kwk2 kyk2, and kN k1 1 implies that kz k2 + kyk2 kwk2 + kuk2. Therefore,
2
2
2
2
2 kwk2 + ( 2 ; 1)kyk2 1 ; (1 ; 2 ) ;2 ]kwk2
kz k2
2
2
2
which implies kTzw k1 < 1 .
2
19.2.1 Additive Reduction
Consider the class of (reduced order) controllers that can be represented in the form
^
K = K0 + W2 W1
where K0 may be interpreted as a nominal, higher order controller, is a stable perturbation, with stable, minimum phase, and invertible weighting functions W1 and W2 .
Suppose that kF` (G K0 )k1 < . A natural question is whether it is possible to obtain
^
a reduced order controller K in this class such that the H1 performance bound remains
^ is in place of K0 . Note that this is somewhat a special case of the above
valid when K
^
^
general problem the speci c form of K restricts that K and K0 must possess the same
right half plane poles, thus to a certain degree limiting the set of attainable reduced
order controllers.
^
Suppose K is a suboptimal H1 controller, i.e., there is a Q 2 RH1 with kQk1 <
^ = F` (M1 Q). It follows from simple algebra that
such that K
; ^
Q = F`(Ka 1 K )
where
"
#
"
#
0 I
;
M11 0 I :
a
I 0
I 0
Furthermore, it follows from straightforward manipulations that
; ^
kQk1 < () F` (Ka 1 K ) 1 <
;
() F` (Ka 1 K0 + W2 W1 ) 1 <
~
() F` (R ) < 1
where
K ;1 :=
~
R=
"
;1=2 I
0
0
W1
#"
1
R11 R12
R21 R22
#"
;1=2 I
0
0
W2
#](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-516-320.jpg)
![CONTROLLER ORDER REDUCTION
500
and R is given by the star product
"
#
R11 R12 = S (K ;1
a
R21 R22
"
#
Ko I ):
I 0
~
~
It is easy to see that R12 and R21 are both minimum phase and invertible, and
hence have full column and full row rank, respectively for all ! 2 R 1. Consequently,
~
by invoking Lemma 19.3, we conclude that if R is a contraction and k k1 < 1 then
~
F` (R ) 1 < 1. This guarantees the existence of a Q such that kQk1 < , or
^
^
equivalently, the existence of a K such that F` (G K ) < . This observation leads
1
to the following theorem.
Theorem 19.4 Suppose W1 and W2 are stable, minimum phase and invertible transfer
~
matrices such that R is a contraction. Let K0 be a stabilizing controller such that
^
^
kF`(G K0 )k1 < . Then K is also a stabilizing controller such that F` (G K ) 1 <
if
^
k k1 = W2;1 (K ; K0 )W1;1 1 < 1:
~
Since R can always be made contractive for su ciently small W1 and W2 , there are
in nite many W1 and W2 that satisfy the conditions in the theorem. It is obvious that
^
^
to make W2;1 (K ; K0 )W1;1 < 1 for some K , one would like to select the largest"
1
W1 and W2 .
Lemma 19.5 Assume kR22 k1 < and de ne
2
0
"
#
6 ;R ;R11 R0 R012
6
L = L1 L2 = F`(6 11 0
6 0 R21 021 ;R22
L2 L3
4
R12
;R22
0
~
Then R is a contraction if W1 and W2 satisfy
"
(W1 W1 );1
0
0
(W2 W2 );1
# "
Proof. See Goddard and Glover 1993].
0
3
7
7
7
7
5
;1 I ):
#
L1 L2 :
L2 L 3
2
An algorithm that maximizes det(W1 W1 ) det(W2 W2 ) has been developed by Goddard and Glover 1993]. The procedure below, devised directly from the above theorem,
can be used to generate a required reduced order controller which will preserve the
^
closed-loop H1 performance bound F`(G K ) < .
1](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-517-320.jpg)
![CONTROLLER ORDER REDUCTION
504
(i) Let K0 = 12 ;1(= ~ ;1 ~ 21) be a suboptimal H1 central controller (Q = 0) such
22
22
that kTzw k1 < .
^ ;1 ^
^ ^^
~ ~
(ii) Find a reduced order controller K = U V ;1 (or V U ) such that the following
frequency weighted H1 error
" ;1 #
" # " ^ #!
p
I 0 ;1
12 ; U
< 1= 2
^
0 I
V
22
or
"
# "^
~ 21
~
; U
^
~ 22
~
V
#!
~ ;1
"
;1 I 0
#
I
0
1
p
1
< 1= 2:
^
(iii) The closed-loop system with the reduced order controller K is stable and the
performance is maintained with the reduced order controller, i.e.,
^
kTzw k1 = F`(G K )
1
< :
19.3 Frequency-Weighted L1 Norm Approximations
We have shown in the previous sections that controller reduction problems are equivalent
to frequency weighted model reduction problems. To that end, the frequency weighted
balanced model reduction approach in Chapter 7 can be applied. In this section, we
propose another method based on the frequency weighted Hankel norm approximation
method.
Theorem 19.9 Let W1(s) 2 RH; and W2(s) 2 RH; with minimal state space real1
1
izations
"
W1 (s) = A1w B1w
C1w D1w
#
"
W2 (s) = A2w B2w
C2w D2w
"
#
#
^ ^
A B
^
and let G(s) 2 RH1 . Suppose that G1 (s) = ^1 ^ 1 2 RH1 is an r-th order
C1 D1
optimal Hankel norm approximation of W1 GW2 ]+ , i.e.,
^
G1 = arg degQ r W1 GW2 ]+ ; Q H
inf
and assume
"
A1w ; I B1w
C1w
D1w
# "
A2w ; I B2w
C2w
D2w
#](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-521-320.jpg)
![19.3. Frequency-Weighted L1 Norm Approximations
505
^
have respectively full row rank and full column rank for all = i (A1 ) i = 1 : : : r.
Then there exist matrices X Y Q, and Z such that
^
A1w X ; X A1 + B1w Y = 0
(19.12)
^1
C1w X + D1w Y = C
(19.13)
^1 Q + ZC2w = 0
QA2w ; A
(19.14)
^
QB2w + ZD2w = B1 :
(19.15)
#
" ^
A1 Z is the frequency weighted optimal Hankel norm approxFurthermore, Gr :=
Y 0
imation, i.e.,
^
inf W1 (G ; G)W2 = kW1 (G ; Gr )W2 kH = r+1 ( W1 GW2 ]+ ) :
^
H
degG r
Proof. We shall assume W2 = I for simplicity. The general case can be proven
^
similarly. Assume without loss of generality that A1 has a diagonal form
^
A1 = diag 1 2 : : : r ]:
^
(The proof can be easily modi ed if A1 has a general Jordan canonical form). Partition
^
X , Y , and C1 as
^
^ ^
^
X = X1 X2 : : : Xr ] Y = Y1 Y2 : : : Yr ] C1 = C11 C12 : : : C1r ]:
Then the equations (19.12) and (19.13) can be rewritten as
"
A1w ; i I B1w
C1w
D1w
By the assumption, the matrix
"
#"
# "
Xi = 0
^
Yi
C1i
A1w ; i I B1w
C1w
D1w
#
i = 1 2 : : : r:
#
has full row rank for all i and thus the existence of X and Y is guaranteed. Let
"
#
^
A1w ;X B1 2 RH; :
^
W1 =
1
^
C1w ;D1
Then using equations (19.12) and (19.13), we get
2 A B Y 0 3 2 A A X ; X A + B Y ;X B
^
^1
1
1
1
6 0w Aw1 B1 7 = 6 0w 1w A1 1 1w
^
^ 5 4
^
^1
W1 Gr = 4
B
C1w D1w Y 0
C1w
C1w X + D1w Y
0
3
7
5](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-522-320.jpg)
![CONTROLLER ORDER REDUCTION
506
=
2 A 0 ;X B 3
^
1
6 0w A1 B1 1 7 = W1 + G1:
^
^ 5 ^ ^
4
^
C1w C1
0
Using this expression, we have
kW1 (G ; Gr )kH =
=
^ ^
^
W1 G]+ + W1 G]; ; W1 ; G1 H = W1 G]+ ; G1 H
^
inf W1 (G ; G)
kW1 (G ; Gr )kH :
r+1 ( W1 G]+ )
^
H
degG r
2
^
^
Note that Y = C1 if W1 = I , Z = B1 if W2 = I , and the rank conditions in the
^
above theorem are actually equivalent to the statements that the poles of G1 are not
zeros of W1 and W2 . These conditions will of course be satis ed automatically if W1 (s)
and W2 (s) have all zeros in the right half plane. Numerical experience shows that if
the weighted Hankel approximation is used to obtain a L1 norm approximation, then
choosing W1 (s) and W2 (s) to have all poles and zeros in the right half plane may reduce
the L1 norm approximation error signi cantly.
Corollary 19.10 Let W1 (s) 2 RH; W2 (s) 2 RH; and G(s) 2 RH1. Then
1
1
^
^
inf W1 (G ; G)W2
inf W (G ; G)W2 = r+1 ( W1 GW2 ]+ ) :
^
^
1 degG r 1
H
degG r
The lower bound in the above corollary is not necessarily achievable. To make the 1norm approximation error as small as possible, a suitable constant matrix Dr should
^
be chosen so that W1 (G ; G ; Dr )W2 is made as small as possible. This Dr can
1
usually be obtained using any standard convex optimization algorithm. To further
reduce the approximation error, the following optimization is suggested.
Weighted L1 Model Reduction Procedures
Let W1 and W2 be any antistable transfer matrices with all zeros in the right half
plane.
(i) Let
#
" ^
A1 Z
^
G1 =
Y 0
be a weighted optimal Hankel norm approximation of G.
^
(ii) Let the reduced order model G be parameterized as
" ^
#
" ^
#
^ ( ) = A1 B
^ ( ) = A1 Z :
G
or G
Y D
C
D](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-523-320.jpg)
![19.4. An Example
509
"
~ ;1
~
;C R;1 C
Y = Ric ;(A ; BD ~R 1 C )
~
;B (I ; D R; D)B (A ; BD R;1 C )
#
0
~
with R := DD > 0. Then we have the following spectral factorization
WW = W W
where W W ;1 2 RH; and
1
#
"
;1
~;1=2
:
W = A + BB X (I + Y X ) (BD=2; Y C )R
~
R1
C + DB X
19.4 An Example
We consider a four-disk control system studied by Enns 1984]. We shall set up the
dynamical system in the standard linear fractional transformation form
x = Ax + B1 w#+ B2" #
_
u
"p
q1 H x + 0 u
z =
0
I
h
i
y = C2 x + 0 I w
where q1 = 1 10;6 q2 = 1 and
2 ;0:161 ;6:004 ;0:58215 ;9:9835 ;0:40727 ;3:982
6 1
0
0
0
0
0
6
6 0
6
1
0
0
0
0
6
6
0
1
0
0
0
A=6 0
6 0
0
0
1
0
0
6
6 0
0
0
0
1
0
6
6 0
4
0
0
0
0
1
0
h
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
3
7
7
7
7
7
7
7
7
7
7
7
7
5
0
0
0
0
0
h
i
h
B1 = pq2 B2 0
H = 0 0 0 0 0:55 11 1:32 18
213
607
6 7
607
6 7
6 7
6 7
B2 = 6 0 7
607
6 7
607
6 7
607
4 5
i
0
i
C2 = 0 0 6:4432 10;3 2:3196 10;3 7:1252 10;2 1:0002 0:10455 0:99551 :
The optimal H1 norm for Tzw is opt = 1:1272. We choose = 1:2 to compute an
8th order suboptimal controller Ko . The coprime factorizations of Ko are obtained
using Lemma 19.6 as Ko = 12 ;1 = ~ ;1 ~ 21 . The controller is reduced using several
22
22
methods and the results are listed in Table 19.2 where the following abbreviations are
made:](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-526-320.jpg)
![19.5. Notes and References
513
reduction criterion may have little relevance to the closed-loop H1 performance. For
example, Table 19.4 shows that the 7-th order weighted approximation error s using
the Hankel/Optimization method is very small, however, the H1 performance is very
far away from the desired level.
Although the unweighted right coprime factor reduction method gives very good
results for this example, one should not be led to conclude that the unweighted right
coprime factor method will work well in general. If this is true, then one can easily
conclude that the unweighted left coprime factor method will do equally well by considering a dual problem. Of course, Table 19.2 shows that this is not true because the
unweighted left coprime factor method does not give good results. The only reason for
the good performance of the right coprime factor method for this example is the special
data structure of this example, i.e., the relationship between the B1 matrix and B2
matrix:
h
i
B1 = pq2 B2 0 :
The interested reader may want to explore this special structure further.
^
Order of K
Lower bounds
r
7
6
5
4
3
2
1
0
r+1 0.295 0.303 0.385 0.405 0.635 0.668 0.687 0.702
B 1.009 0.626 4.645 0.750 71.8 6.59 127.2 2.029
H/O 0.295 0.323 0.389 0.658 0.960 1
1
1
Table 19.3: PWRCF: and lower bounds
19.5 Notes and References
The stability oriented controller reduction criterion is rst proposed by Enns 1984]. The
weighted and unweighted coprime factor controller reduction methods are proposed by
Liu and Anderson 1986, 1990], Liu, Anderson, and Ly 1990], Anderson and Liu 1989],
and Anderson 1993]. For normalized H1 controller, Mustafa and Glover 1991] have
proposed a controller reduction method with a prior performance bounds. Normalized
coprime factors have been used in McFarlane and Glover 1990] for controller order
reductions. Lenz, Khargonekar and Doyle 1987] have also proposed another H1 controller reduction method with guaranteed performance for a class of H1 problems. The
main results presented in this chapter are based on the work of Goddard and Glover
1993,1994].
We should note that a satisfactory solution to the general frequency weighted L1
norm model reduction problem remains unavailable and this problem has a crucial
implication toward controller reduction with preserving closed-loop H1 performance
as its objective. The frequency weighted Hankel norm approximation is considered in
Latham and Anderson 1986], Hung and Glover 1986], and Zhou 1993]. The L1 model
reduction procedures discussed in this chapter are due to Zhou 1993].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-530-320.jpg)
![20.1. Lagrange Multiplier Method
517
Theorem 20.2 Suppose that x0 2 Rn is a local minimum of f (x) subject to the constraints H (x) = 0. Then there exists
"
0
#
2 Rm+1
such that 0 rf (x0 ) + rH (x0 ) = 0.
Remark 20.1 Although some second order necessary and su cient conditions for local
minimality can be given, they are usually not very useful in our applications for the
reason that they are very hard to verify except in some very special cases. Furthermore,
even if some su cient conditions can be veri ed, it is still not clear whether the minima
is global. It is a common practice in many applications that the rst-order necessary
conditions are used to derive some necessary conditions for the existence of a minima.
Then nd a solution (or solutions) from these necessary conditions and check if the
solution(s) satis es our objectives regardless of the solution(s) being a global minima
or not.
~
In most of the control applications, constraints are given by a symmetric matrix
function, and in this case, we have the following lemma.
Lemma 20.3 Let T (x) = T (x) 2 Rl l be a symmetric matrix function and let x0 2 Rn
be such that T (x0 ) = 0. Then x0 is a regular point of the constraints T (x) = T (x) = 0
if, for P = P 2 Rl l , r Trace(T (x0 )P ) = 0 has the unique solution P = 0.
Proof. Since T (x) = tij (x)] is a symmetric matrix, tij = tji and the e ective constraints for T (x) = 0 are given by the l(l + 1)=2 equations, tij (x) = 0 for i = 1 2 : : : l
and i j l. By de nition, x0 is a regular point for the e ective constraints (tij (x) = 0
for i = 1 2 : : : l and i j l) if the following equation has the unique solution pij = 0
for i = 1 2 : : : l and i j l:
(x0 ) :=
l
X
i=1
rtii pii +
l l
XX
i=1 j =i+1
2rtij pji = 0:
Now the result follows by de ning pij := pji and by noting that (x0 ) can be written
as
0
1
l l
XX
(x0 ) = r @
with P = P .
i=1 j =1
tij pji A = r Trace(T (x0 )P )
2
Corollary 20.4 Suppose that x0 2 Rn is a local minimum of f (x) subject to the constraints T (x) = 0 where T (x) = T (x) 2 Rl l and suppose further that x0 is a regular](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-534-320.jpg)
![20.3. Notes and References
525
Remark 20.5 This method can also be used to derive the H1 results presented in the
previous chapters. The interested reader should consult the references for details. It
should be pointed out that this method su ers a severe de ciency: global results are
hard to nd. This is due to (a) only rst order necessary conditions can be relatively
easily derived (b) the controller order must be xed hence even if a xed-order optimal
controller can be found, it may not be optimal over all stabilizing controllers.
~
20.3 Notes and References
Optimization using the Lagrange multiplier can be found in any standard optimization
textbook. In particular, the book by Hestenes 1975] contains the nite dimensional
case, and the one by Luenberger 1969] contains both nite and in nite dimensional
cases. The Lagrange multiplier method has been used extensively by Hyland and Bernstein 1984], Bernstein and Haddard 1989], and Skelton 1988] in control applications.
Theorem 20.6 was originally shown in Hyland and Bernstein 1984].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-542-320.jpg)
![DISCRETE TIME CONTROL
530
vectors corresponding to a generalized eigenvalue
fx1 : : : xk g satisfying
"
"
#
A 0 ;
;Q I
"
#
are referred to the set of vectors
"
A 0 x =
;Q I 1
#!
I G
xi =
0 A
#
I G x
1
0 A
#
I G x
i;1 i = 2 : : : k:
0 A
"
~
See Dooren 1981] and Arnold and Laub 1984] for details.
"
Proof. (a): Since S 2 dom(Ric), 9 T = T1
T2
#
such that
"
X; (S ) = Im T1
T2
and T1 is invertible. Let X := T2 T1;1, then
"
#
"
#
#
"
#
X; (S ) = Im T1 = Im I T1 = Im I :
T2
X
X
Obviously, X is unique since
"
#
"
I = Im I
Im
X1
X2
#
i X1 = X2 . Now let us show that X is symmetric. Since
XT1 = T2
pre-multiply by T1 to get
T1 XT1 = T1 T2:
(21:3)
We only need to show that T1 T2 is symmetric.
Since X; (S ) is a stable invariant subspace, there is a stable n n matrix S; such
that
ST = TS;:
(21:4)
Pre-multiply by S; T J and note that S JS = J we get
(S; T J )TS; = (S; T J )ST = (S; T )JST = T S JST = T JT
S; (T JT )S; ; T JT = 0:](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-547-320.jpg)
![21.2. Discrete Riccati Equations
535
Theorem 21.7 Let G and Q be positive semi-de nite matrices and
"
#
A + G(A );1 Q ;G(A );1 :
S=
;(A );1 Q
(A );1
Then S 2 dom(Ric) i (A G) is stabilizable and (Q A) has no unobservable modes on
the unit circle. Furthermore, X = Ric(S ) 0 if S 2 dom(Ric) and X > 0 if and only
if (Q A) has no unobservable stable modes.
Proof. Let Q = C C for some matrix C . The rst half of the theorem follows from
Lemmas 21.5 and 21.6. Now rewrite the discrete Riccati equation as
A (I + XG);1 X (I + GX );1 A ; X + A X (I + GX );2 GXA + C C = 0 (21:15)
and note that by de nition (I + GX );1 A is stable and A X (I + GX );2 GXA + C C 0.
Thus X 0 by Lyapunov theorem. To show that the kernel of X has the refereed
property, suppose x 2 KerX , pre-multiply (21.15) by x , and post-multiply by x to get
XAx = 0 Cx = 0:
(21:16)
This implies that KerX is an A-invariant subspace. If KerX 6= 0, then there is an
0 6= x 2 KerX , so Cx = 0, such that Ax = x. But for x 2 KerX , (I + GX );1 Ax =
(I + GX );1 x = x, so j j < 1 since (I + GX );1A is stable. Thus is an unobservable
stable mode of (Q A).
On the other hand, suppose that j j < 1 is a stable unobservable mode of (Q A).
Then there exists a x 2 C n such that Ax = x and Cx = 0 do the same pre- and postmultiplications on (21.15) as before to get
j j2 x (XG + I );1 Xx ; x Xx = 0:
This can be rewritten as
x X 1=2 j j2 (I + X 1=2 GX 1=2 );1 ; I ]X 1=2 x = 0:
Now X 0 G 0, and j j < 1 imply that j j2 (I + X 1=2GX 1=2 );1 ; I < 0. Hence
Xx = 0, i.e., X is singular.
2
Lemma 21.8 Suppose that D has full column rank and let R = D D > 0 then the
following statements are equivalent:
"
#
A ; ej I B has full column rank for all 2 0 2 ].
(i)
C
D
;(I ; DR;1D )C A ; BR;1D C has no unobservable modes on the unit circle,
(ii)
or, equivalently, (D? C A ; BR;1 D C ) has no unobservable modes on the unit
circle.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-552-320.jpg)
![DISCRETE TIME CONTROL
536
;
Proof. Suppose that ej is an unobservable mode of (I ; DR;1D )C A ; BR;1D C
then there is an x = 0 such that
6
(A ; BR;1 D C )x = ej x (I ; DR;1 D )Cx = 0
i.e.,
"
But this implies that
A ; ej I B
C
D
#"
"
I
0
#" #
x = 0:
;R;1D C I
A ; ej I B
C
D
0
#
(21:17)
does not have full column rank. Conversely, suppose that (21.17) does not have full
" #
column rank for some
then there exists
"
A ; ej I B
C
D
Now let
" # "
#" #
" # "
Then
u 6= 0 such that
v
and
u = 0:
v
u =
I
0
;1D C I
v
;R
x =
I
0
;1D C I
y
R
#" #
x :
y
#" #
u 6= 0
v
(A ; BR;1 D C ; ej I )x + By = 0
(I ; DR;1D )Cx + Dy = 0:
Pre-multiply (21.19) by D to get y = 0. Then we have
(A ; BR;1 D C )x = ej x (I ; DR;1 D )Cx = 0
;
i.e., ej is an unobservable mode of (I ; DR;1 D )C A ; BR;1 D C .
(21:18)
(21:19)
2
Corollary 21.9 Suppose that D has full column rank and denote R = D D > 0. Let
S have the form
"
#
;1
;1
S = E + G(E;) Q ;G(E ;)1
1Q
;(E )
(E )
where E = A ; BR;1 D C , G = BR;1 B , Q = C (I ; DR;1D )C , and E is assumed
"
#
A ; ej I B has
to be invertible. Then S 2 dom(Ric) i (A B ) is stabilizable and
C
D
full column rank for all 2 0 2 ]. Furthermore, X = Ric(S ) 0.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-553-320.jpg)
![21.3. Bounded Real Functions
537
Note that the Riccati equation corresponding to the simplectic matrix in Corollary 21.9 is
E XE ; X ; E XG(I + XG);1 XE + Q = 0:
This equation can also be written as
A XA ; X ; (B XA + D C ) (D D + B XB );1 (B XA + D C ) + C C = 0:
21.3 Bounded Real Functions
Let a real rational transfer matrix be given by
"
M (z ) = A B
C D
#
where again A is assumed to be nonsingular and the realization is assumed to have
no uncontrollable and no unobservable modes on the unit circle. Note again that all
results hold for A singular case with the same modi cation as in the last section. De ne
M (z ) := M T (z ;1 ). Then
"
;1
;(A );1 C
M (z ) = (A ) ;1
B (A ) D ; B (A );1 C
"
#
:
#
Lemma 21.10 Let M (z) = A B 2 RL1 and let S be a simplectic matrix de ned
C 0
by
"
#
A ;1
);1
S := A ; BB (;1 ) C C BB (A;1
:
;(A ) C C
(A )
Then the following statements are equivalent:
(i) kM (z )k1 < 1
(ii) S has no eigenvalues on the unit circle and kC (I ; A);1 B k < 1.
Proof. It is easy to compute that
2 A ; BB (A );1C C ;BB (A );1 B 3
I ; M (z )M (z )];1 = 6
(A );1 C C
(A );1
0 7:
4
5
;1 C C ;B (A );1
I
;B (A )
It is claimed that I ; M (z )M (z )];1 has no uncontrollable and/or unobservable modes
on the unit circle.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-554-320.jpg)
![538
DISCRETE TIME CONTROL
To show " #
that, suppose that = ej is an uncontrollable mode of I ; M (z )M (z )];1 .
q
Then 9 q = 1 2 C 2n such that
q2
"
#
" #
A ; BB (A );1 C C ;BB (A );1 = ej q q B = 0:
q
(A );1 C C
(A );1
0
Hence q1 B = 0 and
h
i
h
i
q1 A + q2 (A );1 C C q2 (A );1 = ej q1 q2 :
There are two possibilities:
1. q2 6= 0. Then we have q2 (A );1 = ej q2 , i.e., Aq2 = e;j q2 . This implies e;j is
an eigenvalue of A. This is a contradiction since M (z ) 2 RL1 .
2. q2 = 0. Then q1 A = ej q1 , which again implies that M (z ) has a mode on the unit
circle if q1 6= 0, again a contradiction.
Similar proof can be done for observability, hence the claim is true.
Now note that
"
#"
#"
#
;I
A ; BB (A );1 C C ;BB (A );1
;I
S=
:
I
(A );1 C C
(A );1
I
Hence S does not have eigenvalues on the unit circle. It is clear that we have already
proven that S has no eigenvalues on the unit circle i (I ; M M );1 2 RL1 . So it is
su cient to show that
kM (z )k1 < 1 , (I ; M M );1 2 RL1 andkM (1)k < 1:
It is obvious that the right hand side is necessary. To show that it is also su cient, suppose kM (z )k1 1, then max (M (ej )) = 1 for some 2 0 2 ], since max (M (1)) < 1
and M (ej ) is continuous in . This implies that 1 is an eigenvalue of M (e;j )M (ej ),
so I ; M (e;j )M (ej ) is singular. This contradicts to (I ; M M );1 2 RL1 .
2
In the above Lemma, we have assumed that the transfer matrix is strictly proper.
We shall now see how to handle non-strictly proper case. For that purpose we shall
focus our attention on the stable system, and we shall give an example below to show
why this restriction is sometimes necessary for the technique to work.
We rst note that H1 -norm of a stable system is de ned as
kM (z )k1 = sup (M (z )) :
jzj 1
Then it is clear that kM (z )k1
(M (1)) = kDk. Thus in particular if kM (z )k1 < 1
then I ; D D > 0.
On the other hand, if a function is only known to be in RL1 , the above condition
may not be true if the system is not stable.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-555-320.jpg)
![21.3. Bounded Real Functions
541
The last equality is obtained from substituting in Riccati equation. Now pre-multiply
the above equation by B (z ;1 I ; A);1 and post-multiply by (zI ; A);1 B to get
I ; M (z ;1 )M (z ) = W (z ;1 )W (z )
where
#
"
A
B
W (z ) =
;1=2 B XA ;(I ; B XB )1=2 :
(I ; B XB )
Suppose W (ej )v = 0 for some and v then ej is a zero of W (z ) if v 6= 0. However,
all the zeros of W (z ) are given by the eigenvalues of
A + B (I ; B XB );1 B XA = (I ; BB X );1 A
that are all inside of the unit circle. Hence ej cannot be a zero of W .
Therefore, we get I ; M (e;j )M (ej ) > 0 for all 2 0 2 ], i.e., kM k1 < 1. 2
The following more general results can be proven easily following the same procedure
as in the proof of (c))(a).
"
#
Corollary 21.13 Let M (z) = A B 2 RL1 and suppose 9 X = X such that
C 0
A XA ; X + A XB (I ; B XB );1B XA + C C = 0:
Then
where
I ; M (z ;1)M (z ) = W (z ;1 )(I ; B XB )W (z )
"
#
A
B
W (z ) =
;1 B XA ;I :
(I ; B XB )
Moreover, the following statements hold:
(1) if I ; B XB > 0 (< 0), then kM (z )k1 1 ( 1)
(2) if I ; B XB > 0 (< 0) and j i f(I ; BB X );1 Agj 6= 1, then kM (z )k1 < 1 (> 1).
Remark 21.4 As in the continuous time case, the equivalence between (a) and (b) in
Theorem 21.12 can be used to compute the H1 norm of a discrete time transfer matrix.
~](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-558-320.jpg)
![21.4. Matrix Factorizations
545
Some coprime factorizations are particularly interesting, for example, the coprime
factorization with inner numerator. This factorization in the case of G(z ) 2 RH1
yields an inner-outer factorization.
"
#
j
B
Theorem 21.19 Assume that (A B) is stabilizable, A ;Ce I D has full column
rank for all 2 0 2 ], and D has full column rank. Then there exists a right coprime
factorization G = NM ;1 such that N is inner. Furthermore, a particular realization is
given by
"
# 2 A + BF
M := 6
4 F
N
C + DF
where
BR;1=2
R;1=2
DR;1=2
3
7
5
R = D D + B XB
F = ;R;1 (B XA + D C )
and X = X
0 is the unique stabilizing solution
AD X (I + B (D D);1 B X );1AD ; X + C D? D? C = 0
where AD := A ; B (D D);1 D C .
Using Lemma 21.17, the complementary inner factor of N in Theorem 21.19 can be
obtained as follows
"
#
A + BF Y
N? =
C + DF Z
where Y and Z satisfy
A XY + C Z = 0
B XY + D Z = 0
Z Z + Y XY = I:
Note that Y and Z are only related to F implicitly through X .
Remark 21.5 If G(z) 2 RH1 , then the denominator matrix M in Theorem 21.19 is
an outer. Hence, the factorization G = N (M ;1) is an inner-outer factorization.
~
Suppose that the system G(z ) is not stable then a coprime factorization with inner
denominator can also be obtained by solving a special Riccati equation.](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-562-320.jpg)
![21.5. Discrete Time H2 Control
549
The realization of the transfer matrix G is taken to be of the form
2 A B B 3 "
#
1
2
6 C1 D11 D12 7 =: A B :
G(z ) = 4
5
C D
C2 D21
0
The following assumptions are made:
(A1) (A B2 ) is stabilizable and (C2 A) is detectable
h
i
(A2) D12 is full column rank with D12 D? unitary and D21 is full row rank with
"
#
D21 unitary
~
D?
(A3)
(A4)
"
"
A ; ej I B2
C1
D12
A ; ej I B1
C2
D21
#
#
has full column rank for all 2 0 2 ]
has full row rank for all 2 0 2 ].
The problem in this section is to nd an admissible controller K which minimizes
kTzw k2 .
Denote
Let X2 0 and Y2
Ax := A ; B2 D12 C1
Ay := A ; B1 D21 C2 :
0 be the stabilizing solutions to the following Riccati equations:
Ax (I + X2 B2 B2 );1 X2 Ax ; X2 + C1 D? D? C1 = 0
and
~ ~
Ay (I + Y2 C2 C2 );1 Y2 Ay ; Y2 + B1 D? D? B1 = 0:
Note that the stabilizing solutions exist by the assumptions (A3) and (A4). Note also
that if Ax and Ay are nonsingular, the solutions can be obtained through the following
two simplectic matrices:
"
;1
( ;1
H2 := Ax + B2 B2 ;Ax ) C1 D?D? C1 ;B2 B2 (A1x )
;(Ax ) 1 C1 D? D? C1
(Ax );
"
#
;1
C ;1 ~ ~
J2 := Ay + C2;12 Ay ~ B1~D? D?B1 ;C2 C21Ay :
;Ay B1 D? D? B1
A;
y
#](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-566-320.jpg)
![21.7. Model Reduction Using Coprime Factors
Suppose that a dynamic system is given by
"
G= A B
C D
559
#
and assume that the realization is stabilizable and detectable. Recall from Theorem 21.21 that there exists a normalized right coprime factorization G = NM ;1 such
" #
M is inner.
that
N
Lemma 21.29 Let
p (Y X ).
i
i =
given by
i=
Proof. This is obvious since
Then the Hankel singular values of
"
M
N
#
are
i
p1 + 2 < 1:
i
i (Y X )
2
i = i (PQ) = 1 + i (Y X )
and i (Y X ) 0.
2
It is known that there exists a transformation such that X and Y are balanced:
"
#
1 0
X=Y = =
0
2
with 1 = diag 1 Is1 : : : r Isr ] > 0.
Now partitioning the system G and matrix F accordingly,
2A A B 3
6 A11 A12 B1 7 F = h F1 F2 i :
G = 4 21 22 2 5
C1
C2 D
Then the reduced coprime factors
"
# 2 A11 + B1F1 B1Z ;1=2 3
^
M := 6
Z ;1=2 7 2 RH1
4 F1
5
^
N
DZ ;1=2
C1 + DF1
satisfy the following error bound.
" # " ^ #
Lemma 21.30 M ; M
^
N
N
1
2
X
i r+1
i =2
X
i r+1
i
p1 + 2 :
i](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-576-320.jpg)
![21.8. Notes and References
561
Proof. Pre-multiply equation (21.35) by
U=
"h
I 0
i
#
0
I
0
and post-multiply equation (21.35) by U to get
"
or
"
#
A11 1 A11 ; 1 + C1 C1 + A21 2 A21 A11 1 B1 + C1 D + A21 2 B2 = 0
B1 1 A11 + D C1 + B2 2 A21
B1 1 B1 + D D ; I + B2 2 B2
#
"
A11 1 A11 ; 1 + C1 C1 A11 1 B1 + C1 D = ; A21
B1 1 A11 + D C1
B1 1 B1 + D D ; I
B2
This gives
"
A11 B1
C1 D
#"
Now let T = 1=2 , and then
1
1 0
0 I
"
#"
A11 B1
C1 D
TA11 T ;1 TB1
C1 T ;1 D
"
# "
#
A21
i.e., kNr k1 1. It is clear that Nr is inner if
B2
may not be true).
;
1 0
0 I
# "
A
2 B21
2
#
#
:
0:
1
#
2 = 0 (although the converse
2
21.8 Notes and References
The results for the discrete Riccati equation are based mostly on the work of Kucera
1972] and Molinari 1975]. Numerical algorithms for solving discrete ARE with singular A matrix can be found in Arnold and Laub 1984], Dooren 1981], and references
therein. The matrix factorizations are obtained by Chu 1988]. The normalized coprime
factorizations are obtained by Meyer 1990] and Walker 1990]. The detailed treatment
of discrete time H1 control can be found in Stoorvogel 1990], Limebeer, Green, and
Walker 1989], and Iglesias and Glover 1991].](https://image.slidesharecdn.com/robustandoptimalcontrol-140223075418-phpapp01/85/Robust-and-Optimal-Control-578-320.jpg)
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Robust and Optimal Control
- 2. ROBUST AND OPTIMAL CONTROL KEMIN ZHOU with JOHN C. DOYLE and KEITH GLOVER PRENTICE HALL, Englewood Cli s, New Jersey 07632
- 4. Contents Preface xiii Notation and Symbols xvi List of Acronyms xx 1 Introduction 1 1.1 1.2 1.3 Historical Perspective : : : : : : : : : : : : : : : : : : : : : : : : : : How to Use This Book : : : : : : : : : : : : : : : : : : : : : : : : : Highlights of The Book : : : : : : : : : : : : : : : : : : : : : : : : : 2 Linear Algebra 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 Linear Subspaces : : : : : : : : : : : : : Eigenvalues and Eigenvectors : : : : : : Matrix Inversion Formulas : : : : : : : : Matrix Calculus : : : : : : : : : : : : : : Kronecker Product and Kronecker Sum : Invariant Subspaces : : : : : : : : : : : : Vector Norms and Matrix Norms : : : : Singular Value Decomposition : : : : : : Generalized Inverses : : : : : : : : : : : Semide nite Matrices : : : : : : : : : : : Matrix Dilation Problems* : : : : : : : : Notes and References : : : : : : : : : : : 3 Linear Dynamical Systems 3.1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Descriptions of Linear Dynamical Systems : : : : : : : : : : : : : : vii 1 4 6 17 17 20 22 24 25 26 28 32 36 36 38 44 45 45
- 5. CONTENTS viii 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 Controllability and Observability : : : : : : : : Kalman Canonical Decomposition : : : : : : : : Pole Placement and Canonical Forms : : : : : : Observers and Observer-Based Controllers : : : Operations on Systems : : : : : : : : : : : : : : State Space Realizations for Transfer Matrices : Lyapunov Equations : : : : : : : : : : : : : : : Balanced Realizations : : : : : : : : : : : : : : Hidden Modes and Pole-Zero Cancelation : : : Multivariable System Poles and Zeros : : : : : : Notes and References : : : : : : : : : : : : : : : 4 Performance Speci cations 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Normed Spaces : : : : : : : : : Hilbert Spaces : : : : : : : : : : Hardy Spaces H2 and H1 : : : Power and Spectral Signals : : Induced System Gains : : : : : Computing L2 and H2 Norms : Computing L1 and H1 Norms Notes and References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 Stability and Performance of Feedback Systems 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Feedback Structure : : : : : : : : : Well-Posedness of Feedback Loop : Internal Stability : : : : : : : : : : Coprime Factorization over RH1 : Feedback Properties : : : : : : : : The Concept of Loop Shaping : : : Weighted H2 and H1 Performance Notes and References : : : : : : : : 6 Performance Limitations 6.1 6.2 6.3 6.4 6.5 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Introduction : : : : : : : : : : : : : : : : : : Integral Relations : : : : : : : : : : : : : : : Design Limitations and Sensitivity Bounds : Bode's Gain and Phase Relation : : : : : : Notes and References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 47 53 58 63 65 68 71 72 78 80 89 91 91 93 97 102 104 112 114 116 117 117 119 121 126 130 134 137 141 143 143 145 149 151 152
- 6. CONTENTS ix 7 Model Reduction by Balanced Truncation 7.1 7.2 7.3 7.4 Model Reduction by Balanced Truncation : : : Frequency-Weighted Balanced Model Reduction Relative and Multiplicative Model Reductions : Notes and References : : : : : : : : : : : : : : : 8 Hankel Norm Approximation 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 Hankel Operator : : : : : : : : : : : : : : : : All-pass Dilations : : : : : : : : : : : : : : : : Optimal Hankel Norm Approximation : : : : L1 Bounds for Hankel Norm Approximation Bounds for Balanced Truncation : : : : : : : Toeplitz Operators : : : : : : : : : : : : : : : Hankel and Toeplitz Operators on the Disk* : Nehari's Theorem* : : : : : : : : : : : : : : : Notes and References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 Model Uncertainty and Robustness 9.1 9.2 9.3 9.4 9.5 9.6 9.7 Model Uncertainty : : : : : : : : : : : : : : : : : : Small Gain Theorem : : : : : : : : : : : : : : : : : Stability under Stable Unstructured Uncertainties : Unstructured Robust Performance : : : : : : : : : Gain Margin and Phase Margin : : : : : : : : : : : De ciency of Classical Control for MIMO Systems Notes and References : : : : : : : : : : : : : : : : : 10 Linear Fractional Transformation 10.1 10.2 10.3 10.4 10.5 Linear Fractional Transformations : Examples of LFTs : : : : : : : : : Basic Principle : : : : : : : : : : : Redhe er Star-Products : : : : : : Notes and References : : : : : : : : 11 Structured Singular Value 11.1 11.2 11.3 11.4 11.5 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : General Framework for System Robustness : : Structured Singular Value : : : : : : : : : : : Structured Robust Stability and Performance Overview on Synthesis : : : : : : : : : : : : Notes and References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 153 154 160 161 167 169 170 175 184 188 192 194 195 200 206 207 207 211 214 222 229 232 237 239 239 246 256 257 260 261 262 266 276 287 290
- 7. CONTENTS x 12 Parameterization of Stabilizing Controllers 12.1 12.2 12.3 12.4 12.5 12.6 12.7 Existence of Stabilizing Controllers : : : : : : : : : : Duality and Special Problems : : : : : : : : : : : : : Parameterization of All Stabilizing Controllers : : : : Structure of Controller Parameterization : : : : : : : Closed-Loop Transfer Matrix : : : : : : : : : : : : : Youla Parameterization via Coprime Factorization* : Notes and References : : : : : : : : : : : : : : : : : : 13 Algebraic Riccati Equations 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 All Solutions of A Riccati Equation : : : : Stabilizing Solution and Riccati Operator Extreme Solutions and Matrix Inequalities Spectral Factorizations : : : : : : : : : : : Positive Real Functions : : : : : : : : : : : Inner Functions : : : : : : : : : : : : : : : Inner-Outer Factorizations : : : : : : : : : Normalized Coprime Factorizations : : : : Notes and References : : : : : : : : : : : : 14 H2 Optimal Control 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Introduction to Regulator Problem : : : : : : : : : Standard LQR Problem : : : : : : : : : : : : : : : Extended LQR Problem : : : : : : : : : : : : : : : Guaranteed Stability Margins of LQR : : : : : : : Standard H2 Problem : : : : : : : : : : : : : : : : Optimal Controlled System : : : : : : : : : : : : : H2 Control with Direct Disturbance Feedforward* Special Problems : : : : : : : : : : : : : : : : : : : Separation Theory : : : : : : : : : : : : : : : : : : Stability Margins of H2 Controllers : : : : : : : : : Notes and References : : : : : : : : : : : : : : : : : 15 Linear Quadratic Optimization 15.1 15.2 15.3 15.4 15.5 15.6 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Hankel Operators : : : : : : : : : : : : : : : : : : : : : Toeplitz Operators : : : : : : : : : : : : : : : : : : : : Mixed Hankel-Toeplitz Operators : : : : : : : : : : : : Mixed Hankel-Toeplitz Operators: The General Case* Linear Quadratic Max-Min Problem : : : : : : : : : : Notes and References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 291 292 295 301 312 314 315 318 319 320 325 333 342 353 357 358 362 364 365 365 367 372 373 375 378 379 381 388 390 392 393 393 395 397 399 401 404
- 8. CONTENTS 16 H1 Control: Simple Case 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 16.12 Problem Formulation : : : : : : : : Output Feedback H1 Control : : : Motivation for Special Problems : : Full Information Control : : : : : : Full Control : : : : : : : : : : : : : Disturbance Feedforward : : : : : Output Estimation : : : : : : : : : Separation Theory : : : : : : : : : Optimality and Limiting Behavior Controller Interpretations : : : : : An Optimal Controller : : : : : : : Notes and References : : : : : : : : xi : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 H1 Control: General Case 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 17.10 General H1 Solutions : : : : : : : : : : : : : : : : : Loop Shifting : : : : : : : : : : : : : : : : : : : : : : Relaxing Assumptions : : : : : : : : : : : : : : : : : H2 and H1 Integral Control : : : : : : : : : : : : : : H1 Filtering : : : : : : : : : : : : : : : : : : : : : : Youla Parameterization Approach* : : : : : : : : : : Connections : : : : : : : : : : : : : : : : : : : : : : : State Feedback and Di erential Game : : : : : : : : Parameterization of State Feedback H1 Controllers : Notes and References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18 H1 Loop Shaping 18.1 18.2 18.3 18.4 Robust Stabilization of Coprime Factors : : : : : : : : Loop Shaping Using Normalized Coprime Stabilization Theoretical Justi cation for H1 Loop Shaping : : : : Notes and References : : : : : : : : : : : : : : : : : : : 19 Controller Order Reduction 19.1 19.2 19.3 19.4 19.5 Controller Reduction with Stability Criteria : : H1 Controller Reductions : : : : : : : : : : : : Frequency-Weighted L1 Norm Approximations An Example : : : : : : : : : : : : : : : : : : : : Notes and References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 405 405 406 412 416 423 423 425 426 431 436 438 439 441 442 445 449 450 454 456 460 462 465 468 469 469 478 481 487 489 490 497 504 509 513
- 9. CONTENTS xii 20 Structure Fixed Controllers 515 21 Discrete Time Control 527 20.1 20.2 20.3 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 Lagrange Multiplier Method : : : : : : : : : : : : : : : : : : : : : : 515 Fixed Order Controllers : : : : : : : : : : : : : : : : : : : : : : : : 520 Notes and References : : : : : : : : : : : : : : : : : : : : : : : : : : 525 Discrete Lyapunov Equations : : : : : : Discrete Riccati Equations : : : : : : : : Bounded Real Functions : : : : : : : : : Matrix Factorizations : : : : : : : : : : : Discrete Time H2 Control : : : : : : : : Discrete Balanced Model Reduction : : : Model Reduction Using Coprime Factors Notes and References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 527 528 537 542 548 553 558 561 Bibliography 563 Index 581
- 10. Preface This book is inspired by the recent development in the robust and H1 control theory, particularly the state-space H1 control theory developed in the paper by Doyle, Glover, Khargonekar, and Francis 1989] (known as DGKF paper). We give a fairly comprehensive and step-by-step treatment of the state-space H1 control theory in the style of DGKF. We also treat the robust control problems with unstructured and structured uncertainties. The linear fractional transformation (LFT) and the structured singular value (known as ) are introduced as the uni ed tools for robust stability and performance analysis and synthesis. Chapter 1 contains a more detailed chapter-by-chapter review of the topics and results presented in this book. We would like to thank Professor Bruce A. Francis at University of Toronto for his helpful comments and suggestions on early versions of the manuscript. As a matter of fact, this manuscript was inspired by his lectures given at Caltech in 1987 and his masterpiece { A Course in H1 Control Theory. We are grateful to Professor Andre Tits at University of Maryland who has made numerous helpful comments and suggestions that have greatly improved the quality of the manuscript. Professor Jakob Stoustrup, Professor Hans Henrik Niemann, and their students at The Technical University of Denmark have read various versions of this manuscript and have made many helpful comments and suggestions. We are grateful to their help. Special thanks go to Professor Andrew Packard at University of California-Berkeley for his help during the preparation of the early versions of this manuscript. We are also grateful to Professor Jie Chen at University of California-Riverside for providing material used in Chapter 6. We would also like to thank Professor Kang-Zhi Liu at Chiba University (Japan) and Professor Tongwen Chen at University of Calgary for their valuable comments and suggestions. In addition, we would like to thank G. Balas, C. Beck, D. S. Bernstein, G. Gu, W. Lu, J. Morris, M. Newlin, L. Qiu, H. P. Rotstein and many other people for their comments and suggestions. The rst author is especially grateful to Professor Pramod P. Khargonekar at The University of Michigan for introducing him to robust and H1 control and to Professor Tryphon Georgiou at University of Minnesota for encouraging him to complete this work. Kemin Zhou xiii
- 11. xiv PREFACE John C. Doyle Keith Glover
- 12. xv
- 13. NOTATION AND SYMBOLS xvi Notation and Symbols R C F R+ C ; and C ; C + and C + C 0 , jR D D @D eld of real numbers eld of complex numbers eld, either R or C nonnegative real numbers open and closed left-half plane open and closed right-half plane imaginary axis unit disk closed unit disk unit circle 2 belong to subset union intersection 2 3 ~ end of proof end of example end of remark := de ned as asymptotically greater than asymptotically less than much greater than much less than ' / j j Re ( ) (t) ij 1+(t) complex conjugate of 2 C absolute value of 2 C real part of 2 C unit impulse Kronecker delta function, ii = 1 and ij = 0 if i 6= j unit step function
- 14. NOTATION AND SYMBOLS In aij ] xvii n n identity matrix a matrix with aij as its i-th row and j -th column element diag(a1 : : : an ) an n n diagonal matrix with ai as its i-th diagonal element AT transpose A adjoint operator of A or complex conjugate transpose of A A;1 inverse of A A+ pseudo inverse of A A; shorthand for (A;1 ) det(A) determinant of A Trace(A) trace of A (A) eigenvalue of A (A) spectral radius of A (A) the set of spectrum of A (A) largest singular value of A (A) smallest singular value of A i-th singular value of A i (A) (A) condition number of A kAk spectral norm of A: kAk = (A) Im(A), R(A) image (or range) space of A Ker(A), N(A) kernel (or null) space of A X; (A) stable invariant subspace of A X+ (A) antistable invariant subspace of A Ric(H ) g f hi x?y D? S? L2 (;1 1) L2 0 1) L2 (;1 0] L2+ L2; l2+ l2; L2 (j R) L2 (@ D ) H2 (j R) the stabilizing solution of an ARE convolution of g and f Kronecker product direct sum or Kronecker sum angle inner product orthogonal, hx yi = 0 orthogonal complement of D, i.e., D D? D or D is unitary ? ? orthogonal complement of subspace S , e.g., H2 time domain Lebesgue space subspace of L2 (;1 1) subspace of L2 (;1 1) shorthand for L2 0 1) shorthand for L2 (;1 0] shorthand for l2 0 1) shorthand for l2 (;1 0) square integrable functions on C 0 including at 1 square integrable functions on @ D subspace of L2 (j R) with analytic extension to the rhp
- 15. NOTATION AND SYMBOLS xviii H2 (@ D ) ? H2 (j R) ? H2 (@ D ) L 1 ( j R) L1 (@ D ) H1 (j R) H1 (@ D ) ; H1 (j R) ; (@ D ) H1 subspace of L2 (@ D ) with analytic extension to the inside of @ D subspace of L2 (j R) with analytic extension to the lhp subspace of L2 (@ D ) with analytic extension to the outside of @ D functions bounded on Re(s) = 0 including at 1 functions bounded on @ D the set of L1 (j R) functions analytic in Re(s) > 0 the set of L1 (@ D ) functions analytic in jz j < 1 the set of L1 (j R) functions analytic in Re(s) < 0 the set of L1 (@ D ) functions analytic in jz j > 1 pre x B or B closed unit ball, e.g. BH1 and B pre x Bo open unit ball pre x R real rational, e.g., RH1 and RH2 , etc R s] Rp (s) polynomial ring rational proper transfer matrices G (s) G (z ) A B C D shorthand for GT (;s) (continuous time) shorthand for GT (z ;1 ) (discrete time) shorthand for state space realization C (sI ; A);1 B + D or C (zI ; A);1 B + D F` (M Q) F u ( M Q) S (M N ) lower LFT upper LFT star product
- 16. xix
- 17. LIST OF ACRONYMS xx List of Acronyms ARE BR CIF DF FC FDLTI FI HF i lcf LF LFT lhp or LHP LQG LQR LTI LTR MIMO nlcf NP nrcf NS OE OF OI rcf rhp or RHP RP RS SF SISO SSV SVD algebraic Riccati equation bounded real complementary inner factor disturbance feedforward full control nite dimensional linear time invariant full information high frequency if and only if left coprime factorization low frequency linear fractional transformation left-half plane Re(s) < 0 linear quadratic Gaussian linear quadratic regulator linear time invariant loop transfer recovery multi-input multi-output normalized left coprime factorization nominal performance normalized right coprime factorization nominal stability output estimation output feedback output injection right coprime factorization right-half plane Re(s) > 0 robust performance robust stability state feedback single-input single-output structured singular value ( ) singular value decomposition
- 18. 1 Introduction 1.1 Historical Perspective This book gives a comprehensive treatment of optimal H2 and H1 control theory and an introduction to the more general subject of robust control. Since the central subject of this book is state-space H1 optimal control, in contrast to the approach adopted in the famous book by Francis 1987]: A Course in H1 Control Theory, it may be helpful to provide some historical perspective of the state-space H1 control theory to be presented in this book. This section is not intended as a review of the literature in H1 theory or robust control, but rather only an attempt to outline some of the work that most closely touches on our approach to state-space H1 . Hopefully our lack of written historical material will be somewhat made up for by the pictorial history of control shown in Figure 1.1. Here we see how the practical but classical methods yielded to the more sophisticated modern theory. Robust control sought to blend the best of both worlds. The strange creature that resulted is the main topic of this book. The H1 optimal control theory was originally formulated by Zames 1981] in an input-output setting. Most solution techniques available at that time involved analytic functions (Nevanlinna-Pick interpolation) or operator-theoretic methods Sarason, 1967 Adamjan et al., 1978 Ball and Helton, 1983]. Indeed, H1 theory seemed to many to signal the beginning of the end for the state-space methods which had dominated control for the previous 20 years. Unfortunately, the standard frequency-domain approaches to H1 started running into signi cant obstacles in dealing with multi-input multi-output (MIMO) systems, both mathematically and computationally, much as the H2 (or LQG) theory of the 1950's had. 1
- 19. INTRODUCTION 2 Figure 1.1: A picture history of control Not surprisingly, the rst solution to a general rational MIMO H1 optimal control problem, presented in Doyle 1984], relied heavily on state-space methods, although more as a computational tool than in any essential way. The steps in this solution were as follows: parameterize all internally-stabilizing controllers via Youla et al., 1976] obtain realizations of the closed-loop transfer matrix convert the resulting model-matching problem into the equivalent 2 2-block general distance or best approximation problem involving mixed Hankel-Toeplitz operators reduce to the Nehari problem (Hankel only) solve the Nehari problem by the procedure of Glover 1984]. Both Francis, 1987] and Francis and Doyle, 1987] give expositions of this approach, which will be referred to as the 1984" approach. In a mathematical sense, the 1984 procedure solved" the general rational H1 optimal control problem and much of the subsequent work in H1 control theory focused on the 2 2-block problems, either in the model-matching or general distance forms. Unfortunately, the associated complexity of computation was substantial, involving several Riccati equations of increasing dimension, and formulae for the resulting controllers tended to be very complicated and have high state dimension. Encouragement came
- 20. 1.1. Historical Perspective 3 from Limebeer and Hung 1987] and Limebeer and Halikias 1988] who showed, for problems transformable to 2 1-block problems, that a subsequent minimal realization of the controller has state dimension no greater than that of the generalized plant G. This suggested the likely existence of similarly low dimension optimal controllers in the general 2 2 case. Additional progress on the 2 2-block problems came from Ball and Cohen 1987], who gave a state-space solution involving 3 Riccati equations. Jonckheere and Juang 1987] showed a connection between the 2 1-block problem and previous work by Jonckheere and Silverman 1978] on linear-quadratic control. Foias and Tannenbaum 1988] developed an interesting class of operators called skew Toeplitz to study the 2 2-block problem. Other approaches have been derived by Hung 1989] using an interpolation theory approach, Kwakernaak 1986] using a polynomial approach, and Kimura 1988] using a method based on conjugation. The simple state space H1 controller formulae to be presented in this book were rst derived in Glover and Doyle 1988] with the 1984 approach, but using a new 2 2-block solution, together with a cumbersome back substitution. The very simplicity of the new formulae and their similarity with the H2 ones suggested a more direct approach. Independent encouragement for a simpler approach to the H1 problem came from papers by Petersen 1987], Khargonekar, Petersen, and Zhou 1990], Zhou and Khargonekar 1988], and Khargonekar, Petersen, and Rotea 1988]. They showed that for the state-feedback H1 problem one can choose a constant gain as a (sub)optimal controller. In addition, a formula for the state-feedback gain matrix was given in terms of an algebraic Riccati equation. Also, these papers established connections between H1 -optimal control, quadratic stabilization, and linear-quadratic di erential games. The landmark breakthrough came in the DGKF paper (Doyle, Glover, Khargonekar, and Francis 1989]). In addition to providing controller formulae that are simple and expressed in terms of plant data as in Glover and Doyle 1988], the methods in that paper are a fundamental departure from the 1984 approach. In particular, the Youla parameterization and the resulting 2 2-block model-matching problem of the 1984 solution are avoided entirely replaced by a more purely state-space approach involving observer-based compensators, a pair of 2 1 block problems, and a separation argument. The operator theory still plays a central role (as does Redhe er's work Redhe er, 1960] on linear fractional transformations), but its use is more straightforward. The key to this was a return to simple and familiar state-space tools, in the style of Willems 1971], such as completing the square, and the connection between frequency domain inequalities (e.g kGk1 < 1), Riccati equations, and spectral factorizations. This book in some sense can be regarded as an expansion of the DGKF paper. The state-space theory of H1 can be carried much further, by generalizing timeinvariant to time-varying, in nite horizon to nite horizon, and nite dimensional to in nite dimensional. A ourish of activity has begun on these problems since the publication of the DGKF paper and numerous results have been published in the literature, not surprising, many results in DGKF paper generalize mutatis mutandis, to these cases, which are beyond the scope of this book.
- 21. 4 INTRODUCTION 1.2 How to Use This Book This book is intended to be used either as a graduate textbook or as a reference for control engineers. With the second objective in mind, we have tried to balance the broadness and the depth of the material covered in the book. In particular, some chapters have been written su ciently self-contained so that one may jump to those special topics without going through all the preceding chapters, for example, Chapter 13 on algebraic Riccati equations. Some other topics may only require some basic linear system theory, for instance, many readers may nd that it is not di cult to go directly to Chapters 9 11. In some cases, we have tried to collect some most frequently used formulas and results in one place for the convenience of reference although they may not have any direct connection with the main results presented in the book. For example, readers may nd that those matrix formulas collected in Chapter 2 on linear algebra convenient in their research. On the other hand, if the book is used as a textbook, it may be advisable to skip those topics like Chapter 2 on the regular lectures and leave them for students to read. It is obvious that only some selected topics in this book can be covered in an one or two semester course. The speci c choice of the topics depends on the time allotted for the course and the preference of the instructor. The diagram in Figure 1.2 shows roughly the relations among the chapters and should give the users some idea for the selection of the topics. For example, the diagram shows that the only prerequisite for Chapters 7 and 8 is Section 3.9 of Chapter 3 and, therefore, these two chapters alone may be used as a short course on model reductions. Similarly, one only needs the knowledge of Sections 13.2 and 13.6 of Chapter 13 to understand Chapter 14. Hence one may only cover those related sections of Chapter 13 if time is the factor. The book is separated roughly into the following subgroups: Basic Linear System Theory: Chapters 2 3. Stability and Performance: Chapters 4 6. Model Reduction: Chapters 7 8. Robustness: Chapters 9 11. H2 and H1 Control: Chapters 12 19. Lagrange Method: Chapter 20. Discrete Time Systems: Chapter 21. In view of the above classi cation, one possible choice for an one-semester course on robust control would cover Chapters 4 5 9 11 or 4 11 and an one-semester advanced course on H2 and H1 control would cover (parts of) Chapters 12 19. Another possible choice for an one-semester course on H1 control may include Chapter 4, parts of Chapter 5 (5:1 5:3 5:5 5:7), Chapter 10, Chapter 12 (except Section 12.6), parts of Chapter 13 (13:2 13:4 13:6), Chapter 15 and Chapter 16. Although Chapters 7 8 are very much independent of other topics and can, in principle, be studied at any
- 22. 1.2. How to Use This Book 5 2 3 4 ? 3:9 ? 5 - 6 - 7 A U A 11 - 8 ? 9 ? 10 ? 12 13 ? @ @ R @ ? I @ @ @ 15 16 ? 17 - ; ?; ; 19 14 ? 13:2 13:6 ? 13:4 18 20 21 Figure 1.2: Relations among the chapters
- 23. INTRODUCTION 6 stage with the background of Section 3.9, they may serve as an introduction to sources of model uncertainties and hence to robustness problems. Robust Control 4 5 6* 7* 8* 9 10 11 H1 Control Advanced Model & Controller Reductions 4 12 3.9 5.1 5.3,5.5,5.7 13.2,13.4,13.6 7 10 14 8 12 15 5.4,5.7 13.2,13.4,13.6 16 10.1 15 17* 16.1,16.2 16 18* 17.1 19* 19 H1 Control Table 1.1: Possible choices for an one-semester course (* chapters may be omitted) Table 1.1 lists several possible choices of topics for an one-semester course. A course on model and controller reductions may only include the concept of H1 control and the H1 controller formulas with the detailed proofs omitted as suggested in the above table. 1.3 Highlights of The Book The key results in each chapter are highlighted below. Note that some of the statements in this section are not precise, they are true under certain assumptions that are not explicitly stated. Readers should consult the corresponding chapters for the exact statements and conditions. Chapter 2 reviews some basic linear algebra facts and treats a special class of matrix dilation problems. In particular, we show min X X B C A = max C A B A and characterize all optimal (suboptimal) X . Chapter 3 reviews some system theoretical concepts: controllability, observability, stabilizability, detectability, pole placement, observer theory, system poles and zeros, and state space realizations. Particularly, the balanced state space realizations are studied in some detail. We show that for a given stable transfer function G(s) there A B is a state space realization G(s) = C D such that the controllability Gramian P and the observability Gramian Q de ned below are equal and diagonal: P = Q = = diag( 1 2 : : : n ) where AP + PA + BB = 0
- 24. 1.3. Highlights of The Book 7 A Q + QA + C C = 0: Chapter 4 de nes several norms for signals and introduces the H2 spaces and the H1 spaces. The input/output gains of a stable linear system under various input signals are characterized. We show that H2 and H1 norms come out naturally as measures of the worst possible performance for many classes of input signals. For example, let A G(s) = C B 2 RH1 0 kg uk g(t) = CeAt B Z 1 n X kg(t)k dt 2 i . Some state Then kGk1 = sup kuk 2 and 1 kGk1 0 2 i=1 space methods of computing real rational H2 and H1 transfer matrix norms are also presented: kGk2 = trace(B QB ) = trace(CPC ) 2 and kGk1 = maxf : H has an eigenvalue on the imaginary axisg where 2 A H = ;C C BBA= : ; Chapter 5 introduces the feedback structure and discusses its stability and performance properties. w1 + e e1 +6 - P ^ K + e2 ?+ w2 e We show that the above closed-loop system is internally stable if and only if ^ ^ ^ ^ ^ I ;K ;1 = I + K (I ; P K );1 P K (I ; P K );1 2 RH1 : ^ );1 P ^ );1 ;P I (I ; P K (I ; P K Alternative characterizations of internal stability using coprime factorizations are also presented. Chapter 6 introduces some multivariable versions of the Bode's sensitivity integral relations and Poisson integral formula. The sensitivity integral relations are used to study the design limitations imposed by bandwidth constraint and the open-loop unstable poles, while the Poisson integral formula is used to study the design constraints
- 25. INTRODUCTION 8 imposed by the non-minimum phase zeros. For example, let S (s) be a sensitivity function, and let pi be the right half plane poles of the open-loop system and i be the corresponding pole directions. Then we show that 1 Z 0 ln (S (j!))d! = max X i ! (Repi ) i i + H1 H1 0: This equality shows that the design limitations in multivariable systems are dependent on the directionality properties of the sensitivity function as well as those of the poles (and zeros), in addition to the dependence upon pole (and zero) locations which is known in single-input single-output systems. Chapter 7 considers the problem of reducing the order of a linear multivariable dynamical system using the balanced truncation method. Suppose 2 G(s) = 4 A11 A12 A21 A22 C1 C2 B1 B2 D 3 5 2 RH1 is a balanced realization with controllability and observability Gramians P = Q = = diag( 1 2 ) 1 = diag( 1 Is1 2 Is2 : : : r Isr ) 2 = diag( r+1 Isr+1 r+2 Isr+2 : : : N IsN ): Then the truncated system Gr (s) = A11 B1 is stable and satis es an additive C1 D error bound: N X kG(s) ; Gr (s)k1 2 i: i=r+1 On the other hand, if G;1 2 RH1 , and P and Q satisfy PA + AP + BB = 0 Q(A ; BD;1 C ) + (A ; BD;1 C ) Q + C (D;1 ) D;1 C = 0 such that P = Q = diag( 1 2 ) with G partitioned compatibly as before, then Gr (s) = A11 B1 C1 D is stable and minimum phase, and satis es respectively the following relative and multiplicative error bounds: G;1 (G ; Gr ) 1 G;1 (G ; Gr ) 1 r N Y i=r+1 N Y i=r+1 q 1 + 2 i ( 1 + i2 + i ) ; 1 q 1 + 2 i ( 1 + i2 + i ) ; 1:
- 26. 1.3. Highlights of The Book 9 Chapter 8 deals with the optimal Hankel norm approximation and its applications in L1 norm model reduction. We show that for a given G(s) of McMillan degree n ^ there is a G(s) of McMillan degree r < n such that ^ ^ G(s) ; G(s) H = inf G(s) ; G(s) H = r+1 : Moreover, there is a constant matrix D0 such that N X ^ G(s) ; G(s) ; D0 1 i: i=r+1 The well-known Nehari's theorem is also shown: inf kG ; Qk1 = kGkH = 1 : Q2RH; 1 Chapter 9 derives robust stability tests for systems under various modeling assumptions through the use of a small gain theorem. In particular, we show that an uncertain system described below with an unstructured uncertainty 2 RH1 with k k1 < 1 is robustly stable if and only if the transfer function from w to z has H1 norm no greater than 1. ∆ z w nominal system Chapter 10 introduces the linear fractional transformation (LFT). We show that many control problems can be formulated and treated in the LFT framework. In particular, we show that every analysis problem can be put in an LFT form with some structured (s) and some interconnection matrix M (s) and every synthesis problem can be put in an LFT form with a generalized plant G(s) and a controller K (s) to be designed. z z M w y G -K w u
- 27. INTRODUCTION 10 Chapter 11 considers robust stability and performance for systems with multiple sources of uncertainties. We show that an uncertain system is robustly stable for all i 2 RH1 with k i k1 < 1 if and only if the structured singular value ( ) of the corresponding interconnection model is no greater than 1. ∆1 ∆4 ∆2 nominal system ∆3 Chapter 12 characterizes all controllers that stabilize a given dynamical system G(s) using the state space approach. The construction of the controller parameterization is done via separation theory and a sequence of special problems, which are so-called full information (FI) problems, disturbance feedforward (DF) problems, full control (FC) problems and output estimation (OE). The relations among these special problems are established. FI - dual 6 equivalent FC 6 equivalent ? DF - dual ? OE For a given generalized plant 2 A B1 B2 G(s) = G11 (s) G12 (s) = 4 C1 D11 D12 G21 (s) G22 (s) C2 D21 D22 3 5 we show that all stabilizing controllers can be parameterized as the transfer matrix from y to u below where F and L are such that A + LC2 and A + B2 F are stable.
- 28. 1.3. Highlights of The Book 11 z y ?? c ; c u D22 C2 R -A - ;L u1 w G - c c 66 B2 c 6 -F y1 Q Chapter 13 studies the Algebraic Riccati Equation and the related problems: the properties of its solutions, the methods to obtain the solutions, and some applications. In particular, we study in detail the so-called stabilizing solution and its applications in matrix factorizations. A solution to the following ARE A X + XA + XRX + Q = 0 is said to be a stabilizing solution if A + RX is stable. Now let A H := ;Q ;R A and let X; (H ) be the stable H invariant subspace and X; (H ) = Im X1 X2 ; where X1 X2 2 C n n . If X1 is nonsingular, then X := X2 X1 1 is uniquely determined by H , denoted by X = Ric(H ). A key result of this chapter is the relationship between the spectral factorization of a transfer function and the solution of a corresponding ARE. Suppose (A B ) is stabilizable and suppose either A has no eigenvalues on j!-axis or P is sign de nite (i.e., P 0 or P 0) and (P A) has no unobservable modes on the j!-axis. De ne (s) = B (;sI ; A );1 I P S S R (sI ; A);1 B : I
- 29. INTRODUCTION 12 Then (j!) > 0 for all 0 ! 1 () 9 a stabilizing solution X to (A ; BR;1 S ) X + X (A ; BR;1 S ) ; XBR;1B X + P ; SR;1S = 0 () the Hamiltonian matrix ;1 S ;BR;1B H = ;A ; BR ;1S ) ;(A ; BR;1 S ) (P ; SR has no j!-axis eigenvalues. Similarly, () 9 a solution X to (j!) 0 for all 0 ! 1 (A ; BR;1 S ) X + X (A ; BR;1 S ) ; XBR;1B X + P ; SR;1S = 0 such that (A ; BR;1 S ; BR;1 B X ) C ; . Furthermore, there exists a M 2 Rp such that = M RM: with A M = ;F B I F = ;R;1(S + B X ): Chapter 14 treats the optimal control of linear time-invariant systems with quadratic performance criteria, i.e., LQR and H2 problems. We consider a dynamical system described by an LFT with 3 2 A B1 B2 G(s) = 4 C1 0 D12 5 : C2 D21 0 z y G - K w u
- 30. 1.3. Highlights of The Book De ne H2 := ;CA C ;0 A 1 1 13 ; ;CB2 12 1D D12 C1 B2 A C D21 B1 C2 J2 := ;B B ;0A ; ;B 2 1 1 1 D21 X2 := Ric(H2) 0 Y2 := Ric(J2 ) 0 F2 := ;(B2 X2 + D12 C1 ) L2 := ;(Y2 C2 + B1 D21 ): Then the H2 optimal controller, i.e. the controller that minimizes kTzw k2 , is given by L Kopt (s) := A + B2 F2 + L2C2 ;0 2 : F 2 Chapter 15 solves a max-min problem, i.e., a full information (or state feedback) H1 control problem, which is the key to the H1 theory considered in the next chapter. Consider a dynamical system x = Ax + B1 w + B2 u _ z = C1 x + D12 u D12 C1 D12 = 0 I : Then we show that sup umin kz k2 < if and only if H1 2 dom(Ric) and X1 = w2BL2+ 2L2+ Ric(H1 ) 0 where H1 := ;CA C 1 1 ;2B1 B1 ; B2 B2 ;A Furthermore, u = F1 x with F1 := ;B2 X1 is an optimal control. Chapter 16 considers a simpli ed H1 control problem with the generalized plant G(s) as given in Chapter 14. We show that there exists an admissible controller such that kTzw k1 < i the following three conditions hold: (i) H1 2 dom(Ric) and X1 := Ric(H1 ) 0 (ii) J1 2 dom(Ric) and Y1 := Ric(J1 ) 0 where A J1 := ;B B 1 1 (iii) (X1 Y1 ) < 2 . ;2 C1 C1 ; C2 C2 ;A :
- 31. INTRODUCTION 14 Moreover, the set of all admissible controllers such that kTzw k1 < equals the set of all transfer matrices from y to u in u M1 - y 2 ^ A1 ;Z1 L1 Z1 B2 4 F1 M1 (s) = 0 I I 0 ;C2 Q 3 5 where Q 2 RH1 , kQk1 < and ^ A1 := A + ;2B1 B1 X1 + B2 F1 + Z1 L1 C2 F1 := ;B2 X1 L1 := ;Y1 C2 Z1 := (I ; ;2Y1 X1 );1 : Chapter 17 considers again the standard H1 control problem but with some assumptions in the last chapter relaxed. We indicate how the assumptions can be relaxed to accommodate other more complicated problems such as singular control problems. We also consider the integral control in the H2 and H1 theory and show how the general H1 solution can be used to solve the H1 ltering problem. The conventional Youla parameterization approach to the H2 and H1 problems is also outlined. Finally, the general state feedback H1 control problem and its relations with full information control and di erential game problems are discussed. ~ ~ Chapter 18 rst solves a gap metric minimization problem. Let P = M ;1 N be a normalized left coprime factorization. Then we show that inf K stabilizing = K stabilizing inf K (I + PK );1 I P I K (I + PK );1M ;1 = ~ I 1 q 1; 1 ~ ~ N M 2 H ;1 : This implies that there is a robustly stabilizing controller for ~ ~ P = (M + ~ M );1 (N + ~ N ) with if and only if ~N ~M q 1< ~ ~ H 1; N M 2 : Using this stabilization result, a loop shaping design technique is proposed. The proposed technique uses only the basic concept of loop shaping methods and then a robust
- 32. 1.3. Highlights of The Book 15 stabilization controller for the normalized coprime factor perturbed system is used to construct the nal controller. Chapter 19 considers the design of reduced order controllers by means of controller reduction. Special attention is paid to the controller reduction methods that preserve the closed-loop stability and performance. In particular, two H1 performance preserving reduction methods are proposed: ^ a) Let K0 be a stabilizing controller such that kF` (G K0 )k1 < . Then K is also a ^ stabilizing controller such that F`(G K ) < if 1 ^ W ;1 (K ; K0)W ;1 2 1 1 <1 where W1 and W2 are some stable, minimum phase and invertible transfer matrices. b) Let K0 = 12 ;1 be a central H1 controller such that kF`(G K0 )k1 < and 22 ^ ^ let U V 2 RH1 be such that ;1 I 0 p ^ 12 ; U ;1 < 1= 2: ^ 0 I V 22 1 ^ ^^ ^ Then K = U V ;1 is also a stabilizing controller such that kF`(G K )k1 < . Thus the controller reduction problem is converted to weighted model reduction problems for which some numerical methods are suggested. Chapter 20 brie y introduces the Lagrange multiplier method for the design of xed order controllers. Chapter 21 discusses discrete time Riccati equations and some of their applications in discrete time control. Finally, the discrete time balanced model reduction is considered.
- 33. 16 INTRODUCTION
- 34. 2 Linear Algebra Some basic linear algebra facts will be reviewed in this chapter. The detailed treatment of this topic can be found in the references listed at the end of the chapter. Hence we shall omit most proofs and provide proofs only for those results that either cannot be easily found in the standard linear algebra textbooks or are insightful to the understanding of some related problems. We then treat a special class of matrix dilation problems which will be used in Chapters 8 and 17 however, most of the results presented in this book can be understood without the knowledge of the matrix dilation theory. 2.1 Linear Subspaces Let R denote the real scalar eld and C the complex scalar eld. For the interest of this chapter, let F be either R or C and let Fn be the vector space over F, i.e., Fn is either Rn or C n . Now let x1 x2 : : : xk 2 Fn . Then an element of the form 1 x1 + : : : + k xk with i 2 F is a linear combination over F of x1 : : : xk . The set of all linear combinations of x1 x2 : : : xk 2 Fn is a subspace called the span of x1 x2 : : : xk , denoted by spanfx1 x2 : : : xk g := fx = 1 x1 + : : : + k xk : i 2 Fg: A set of vectors x1 x2 : : : xk 2 Fn are said to be linearly dependent over F if there exists 1 : : : k 2 F not all zero such that 1 x2 + : : : + k xk = 0 otherwise they are said to be linearly independent. Let S be a subspace of Fn , then a set of vectors fx1 x2 : : : xk g 2 S is called a basis for S if x1 x2 : : : xk are linearly independent and S = spanfx1 x2 : : : xk g. However, 17
- 35. LINEAR ALGEBRA 18 such a basis for a subspace S is not unique but all bases for S have the same number of elements. This number is called the dimension of S , denoted by dim(S ). A set of vectors fx1 x2 : : : xk g in Fn are mutually orthogonal if xi xj = 0 for all i 6= j and orthonormal if xi xj = ij , where the superscript denotes complex conjugate transpose and ij is the Kronecker delta function with ij = 1 for i = j and ij = 0 for i 6= j . More generally, a collection of subspaces S1 S2 : : : Sk of Fn are mutually orthogonal if x y = 0 whenever x 2 Si and y 2 Sj for i 6= j . The orthogonal complement of a subspace S Fn is de ned by S ? := fy 2 Fn : y x = 0 for all x 2 S g: We call a set of vectors fu1 u2 : : : uk g an orthonormal basis for a subspace S 2 Fn if they form a basis of S and are orthonormal. It is always possible to extend such a basis to a full orthonormal basis fu1 u2 : : : un g for Fn . Note that in this case S ? = spanfuk+1 : : : un g and fuk+1 : : : un g is called an orthonormal completion of fu1 u2 : : : uk g. Let A 2 Fm n be a linear transformation from Fn to Fm , i.e., A : Fn 7;! Fm : (Note that a vector x 2 Fm can also be viewed as a linear transformation from F to Fm , hence anything said for the general matrix case is also true for the vector case.) Then the kernel or null space of the linear transformation A is de ned by KerA = N (A) := fx 2 Fn : Ax = 0g and the image or range of A is ImA = R(A) := fy 2 Fm : y = Ax x 2 Fn g: It is clear that KerA is a subspace of Fn and ImA is a subspace of Fm . Moreover, it can be easily seen that dim(KerA) + dim(ImA) = n and dim(ImA) = dim(KerA)? . Note that (KerA)? is a subspace of Fn . Let ai i = 1 2 : : : n denote the columns of a matrix A 2 Fm n , then ImA = spanfa1 a2 : : : an g: The rank of a matrix A is de ned by rank(A) = dim(ImA): It is a fact that rank(A) = rank(A ), and thus the rank of a matrix equals the maximal number of independent rows or columns. A matrix A 2 Fm n is said to have full row rank if m n and rank(A) = m. Dually, it is said to have full column rank if n m and rank(A) = n. A full rank square matrix is called a nonsingular matrix. It is easy
- 36. 2.1. Linear Subspaces 19 to see that rank(A) = rank(AT ) = rank(PA) if T and P are nonsingular matrices with appropriate dimensions. A square matrix U 2 F n n whose columns form an orthonormal basis for Fn is called an unitary matrix (or orthogonal matrix if F = R), and it satis es U U = I = UU . The following lemma is useful. Lemma 2.1 Let D = d1 : : : dk 2 Fn k (n > k) be such that D D = I , so di i = 1 2 : : : k are orthonormal. Then there exists a matrix D? 2 Fn (n;k) such that D D? is a unitary matrix. Furthermore, the columns of D?, di i = k + 1 : : : n, form an orthonormal completion of fd1 d2 : : : dk g. The following results are standard: Lemma 2.2 Consider the linear equation where A 2 Fn l and B AX = B 2 Fn m are given matrices. Then the following statements are equivalent: (i) there exists a solution X 2 Fl m . (ii) the columns of B 2 ImA. (iii) rank A B =rank(A). (iv) Ker(A ) Ker(B ). Furthermore, the solution, if it exists, is unique if and only if A has full column rank. The following lemma concerns the rank of the product of two matrices. Lemma 2.3 (Sylvester's inequality) Let A 2 Fm n and B 2 Fn k . Then rank (A) + rank(B ) ; n rank(AB ) minfrank (A) rank(B )g: For simplicity, a matrix M with mij as its i-th row and j -th column's element will sometimes be denoted as M = mij ] in this book. We will mostly use I as above to denote an identity matrix with compatible dimensions, but from time to time, we will use In to emphasis that it is an n n identity matrix. Now let A = aij ] 2 C n n , then the trace of A is de ned as Trace(A) := n X i=1 aii : Trace has the following properties: Trace( A) = Trace(A) 8 2 C A 2 C n n Trace(A + B ) = Trace(A) + Trace(B ) 8A B 2 C n n Trace(AB ) = Trace(BA) 8A 2 C n m B 2 C m n :
- 37. LINEAR ALGEBRA 20 2.2 Eigenvalues and Eigenvectors Let A 2 C n n then the eigenvalues of A are the n roots of its characteristic polynomial p( ) = det( I ; A). This set of roots is called the spectrum of A and is denoted by (A) (not to be confused with singular values de ned later). That is, (A) := f 1 2 : : : n g if i is a root of p( ). The maximal modulus of the eigenvalues is called the spectral radius, denoted by (A) := 1maxn j i j i where, as usual, j j denotes the magnitude. If 2 (A) then any nonzero vector x 2 C n that satis es Ax = x is referred to as a right eigenvector of A. Dually, a nonzero vector y is called a left eigenvector of A if y A= y : It is a well known (but nontrivial) fact in linear algebra that any complex matrix admits a Jordan Canonical representation: Theorem 2.4 For any square complex matrix A 2 C n n , there exists a nonsingular matrix T such that A = TJT ;1 where 2 Jij = with Pl i=1 Pmi 6 6 6 6 6 4 J = diagfJ1 J2 : : : Jl g Ji = diagfJi1 Ji2 : : : Jimi g i j =1 nij = n, and with 1 i 3 1 ... ... i 1 7 7 7 7 7 5 2 C nij i nij f i : i = 1 : : : lg as the distinct eigenvalues of A. The transformation T has the following form: T = T1 T2 Ti = Ti1 Ti2 Tij = tij1 tij2 where tij1 are the eigenvectors of A, Atij1 = : : : Tl : : : Timi : : : tijnij i tij 1
- 38. 2.2. Eigenvalues and Eigenvectors 21 and tijk 6= 0 de ned by the following linear equations for k 2 (A ; i I )tijk = tij(k;1) are called the generalized eigenvectors of A. For a given integer q nij , the generalized eigenvectors tijl 8l < q, are called the lower rank generalized eigenvectors of tijq . De nition 2.1 A square matrix A 2 Rn n is called cyclic if the Jordan canonical form of A has one and only one Jordan block associated with each distinct eigenvalue. More speci cally, a matrix A is cyclic if its Jordan form has mi = 1 i = 1 : : : l. Clearly, a square matrix A with all distinct eigenvalues is cyclic and can be diagonalized: 2 A x1 x2 xn = x1 x2 xn 6 6 6 4 1 3 2 ... 7 7 7 5 : n In this case, A has the following spectral decomposition: A= where yi 2 C n is given by 2 6 6 6 4 y1 y2 .. . yn n X i=1 i xi yi 3 7 7 7 5 = x1 x2 xn ;1 : In general, eigenvalues need not be real, and neither do their corresponding eigenvectors. However, if A is real and is a real eigenvalue of A, then there is a real eigenvector corresponding to . In the case that all eigenvalues of a matrix A are real1 , we will denote max (A) for the largest eigenvalue of A and min (A) for the smallest eigenvalue. In particular, if A is a Hermitian matrix, then there exist a unitary matrix U and a real diagonal matrix such that A = U U , where the diagonal elements of are the eigenvalues of A and the columns of U are the eigenvectors of A. The following theorem is useful in linear system theory. Theorem 2.5 (Cayley-Hamilton) Let A 2 C n n and denote det( I ; A) = n + a1 n;1 + + an : Then An + a1 An;1 + + an I = 0: 1 For example, this is the case if A is Hermitian, i.e., A = A .
- 39. LINEAR ALGEBRA 22 This is obvious if A has distinct eigenvalues. Since An + a1 An;1 + + an I = T ;1 diag f: : : n + a1 in;1 + i + an : : :g T = 0 and i is an eigenvalue of A. The proof for the general case follows from the following lemma. Lemma 2.6 Let A 2 C n n . Then 1 ( I ; A);1 = det( I ; A) (R1 n;1 + R2 n;2 + + Rn ) and det( I ; A) = n + a1 n;1 + + an where ai and Ri can be computed from the following recursive formulas: a1 a2 = = . . . = = ; Trace A 1 ; 2 Trace(R2 A) R1 = I R2 = R1 A + a1 I . . . 1 Trace(Rn;1 A) an;1 ; n;1 Rn = Rn;1 A + an;1 I 1 0 = Rn A + an I: an ; n Trace(Rn A) The proof is left to the reader as an exercise. Note that the Cayley-Hamilton Theorem follows from the fact that 0 = Rn A + an I = An + a1 An;1 + + an I: 2.3 Matrix Inversion Formulas Let A be a square matrix partitioned as follows A := A11 A12 A21 A22 where A11 and A22 are also square matrices. Now suppose A11 is nonsingular, then A has the following decomposition: I 0 A11 A12 = A21 A22 A21 A;1 I 11 A11 0 0 with := A22 ; A21 A;1 A12 , and A is nonsingular i 11 Dually, if A22 is nonsingular, then A11 A12 = I A12 A;1 22 A21 A22 0 I 0 0 A22 I A;1 A12 11 0 I is nonsingular. I 0 A;1 A21 I 22
- 40. 2.3. Matrix Inversion Formulas 23 with := A11 ; A12 A;1 A21 , and A is nonsingular i is nonsingular. The matrix 22 ( ) is called the Schur complement of A11 (A22 ) in A. Moreover, if A is nonsingular, then A11 A12 ;1 = A;1 + A;1 A12 ;1 A21 A;1 ;A;1 A12 ;1 11 11 11 11 ;1 A21 A22 ; ;1 A21 A;1 11 and # " ;1 ; ;1 A12 A;1 A11 A12 ;1 = 22 : A21 A22 ;A;1 A21 ;1 A;1 + A;1 A21 ;1 A12 A;1 22 22 22 22 The above matrix inversion formulas are particularly simple if A is block triangular: A11 0 ;1 = A;1 0 11 ;1 A21 A;1 A;1 A21 A22 ;A22 11 22 A11 A12 ;1 = A;1 ;A;1 A12 A;1 : 11 11 ;1 22 0 A22 0 A22 The following identity is also very useful. Suppose A11 and A22 are both nonsingular matrices, then (A11 ; A12 A;1 A21 );1 = A;1 + A;1 A12 (A22 ; A21 A;1 A12 );1 A21 A;1 : 22 11 11 11 11 As a consequence of the matrix decomposition formulas mentioned above, we can calculate the determinant of a matrix by using its sub-matrices. Suppose A11 is nonsingular, then det A = det A11 det(A22 ; A21 A;1 A12 ): 11 On the other hand, if A22 is nonsingular, then det A = det A22 det(A11 ; A12 A;1 A21 ): 22 In particular, for any B 2 C m n and C 2 C n m , we have I B det ;m I = det(In + CB ) = det(Im + BC ) C n and for x y 2 C n det(In + xy ) = 1 + y x:
- 41. LINEAR ALGEBRA 24 2.4 Matrix Calculus Let X = xij ] 2 C m n be a real or complex matrix and F (X ) 2 C be a scalar real or complex function of X then the derivative of F (X ) with respect to X is de ned as @ @ @X F (X ) := @xij F (X ) : Let A and B be constant complex matrices with compatible dimensions. Then the following is a list of formulas for the derivatives2: @ @X TracefAXB g @ T @X TracefAX B g @ @X TracefAXBX g @ T @X TracefAXBX g @ k @X TracefX g = AT B T @ k @X TracefAX g = @ ;1 @X TracefAX B g @ @X log det X @ T @X det X @ k @X detfX g = ;(X ;1 BAX ;1)T = BA = AT X T B T + B T X T AT = AT XB T + AXB = k(X k;1 )T Pk;1 i=0 X i AX k;i;1 T = (X T );1 @ = @X det X = (detX )(X T );1 = k(detX k )(X T );1 : And nally, the derivative of a matrix A( ) 2 C m n with respect to a scalar 2 C is de ned as dA := daij d d so that all the rules applicable to a scalar function also apply here. In particular, we have d(AB ) = dA B + A dB d d d dA;1 = ;A;1 dA A;1 : d d 2 Note that transpose rather than complex conjugate transpose should be used in the list even if the involved matrices are complex matrices.
- 42. 2.5. Kronecker Product and Kronecker Sum 25 2.5 Kronecker Product and Kronecker Sum Let A 2 C m n and B 2 C p q , then the Kronecker product of A and B is de ned as 2 6 A B := 6 6 4 a11 B a12 B a21 B a22 B a1n B a2n B am1 B am2 B amn B .. . .. . .. . 3 7 7 7 5 2 C mp nq : Furthermore, if the matrices A and B are square and A 2 C n n and B 2 C m m then the Kronecker sum of A and B is de ned as A B := (A Im ) + (In B ) 2 C nm nm : Let X 2 C m n and let vec(X ) denote the vector formed by stacking the columns of X into one long vector: 2 vec(X ) := 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x11 x21 .. . 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 xm1 x12 x22 : . .. x1n x2n .. . xmn Then for any matrices A 2 C k m , B 2 C n l , and X 2 C m n , we have vec(AXB ) = (B T A)vec(X ): Consequently, if k = m and l = n, then vec(AX + XB ) = (B T A)vec(X ): Let A 2 C n n and B 2 C m m , and let f i i = 1 : : : ng be the eigenvalues of A and f j j = 1 : : : mg be the eigenvalues of B . Then we have the following properties: The eigenvalues of A B are the mn numbers i j , i = 1 2 : : : n, j = 1 2 : : : m. The eigenvalues of A B = (A Im ) + (In B ) are the mn numbers i + j , i = 1 2 : : : n, j = 1 2 : : : m.
- 43. LINEAR ALGEBRA 26 Let fxi i = 1 : : : ng be the eigenvectors of A and let fyj j = 1 : : : mg be the eigenvectors of B . Then the eigenvectors of A B and A B correspond to the eigenvalues i j and i + j are xi yj . Using these properties, we can show the following Lemma. Lemma 2.7 Consider the Sylvester equation AX + XB = C (2:1) where A 2 Fn n , B 2 Fm m , and C 2 Fn m are given matrices. There exists a unique solution X 2 Fn m if and only if i (A)+ j (B ) 6= 0 8i = 1 2 : : : n and j = 1 2 : : : m. In particular, if B = A , (2.1) is called the Lyapunov Equation" and the necessary and su cient condition for the existence of a unique solution is that i (A) + j (A) 6= 0 8i j = 1 2 : : : n Proof. Equation (2.1) can be written as a linear matrix equation by using the Kronecker product: (B T A)vec(X ) = vec(C ): Now this equation has a unique solution i B T A is nonsingular. Since the eigenvalues of B T A have the form of i (A) + j (B T ) = i (A) + j (B ), the conclusion follows. 2 The properties of the Lyapunov equations will be studied in more detail in the next chapter. 2.6 Invariant Subspaces Let A : C n 7;! C n be a linear transformation, be an eigenvalue of A, and x be a corresponding eigenvector, respectively. Then Ax = x and A( x) = ( x) for any 2 C . Clearly, the eigenvector x de nes an one-dimensional subspace that is invariant with respect to pre-multiplication by A since Ak x = k x 8k. In general, a subspace S C n is called invariant for the transformation A, or A-invariant, if Ax 2 S for every x 2 S . In other words, that S is invariant for A means that the image of S under A is contained in S : AS S . For example, f0g, C n , KerA, and ImA are all A-invariant subspaces. As a generalization of the one dimensional invariant subspace induced by an eigenvector, let 1 : : : k be eigenvalues of A (not necessarily distinct), and let xi be the corresponding eigenvectors and the generalized eigenvectors. Then S = spanfx1 : : : xk g is an A-invariant subspace provided that all the lower rank generalized eigenvectors are included. More speci cally, let 1 = 2 = = l be eigenvalues of A, and
- 44. 2.6. Invariant Subspaces 27 let x1 x2 : : : xl be the corresponding eigenvector and the generalized eigenvectors obtained through the following equations: (A ; 1 I )x1 = 0 (A ; 1 I )x2 = x1 .. . (A ; 1 I )xl = xl;1 : Then a subspace S with xt 2 S for some t l is an A-invariant subspace only if all lower rank eigenvectors and generalized eigenvectors of xt are in S , i.e., xi 2 S 81 i t. This will be further illustrated in Example 2.1. On the other hand, if S is a nontrivial subspace3 and is A-invariant, then there is x 2 S and such that Ax = x. An A-invariant subspace S C n is called a stable invariant subspace if all the eigenvalues of A constrained to S have negative real parts. Stable invariant subspaces will play an important role in computing the stabilizing solutions to the algebraic Riccati equations in Chapter 13. Example 2.1 Suppose a matrix A has the following Jordan canonical form 2 A x1 x2 x3 x4 = x1 x2 x3 x4 with Re 1 < 0, 3 < 0, and 4 S1 = spanfx1 g S3 = spanfx3 g S4 = spanfx4 g 6 6 4 1 3 1 1 3 7 7 5 4 > 0. Then it is easy to verify that S12 = spanfx1 x2 g S13 = spanfx1 x3 g S14 = spanfx1 x4 g S123 = spanfx1 x2 x3 g S124 = spanfx1 x2 x4 g S34 = spanfx3 x4 g are all A-invariant subspaces. Moreover, S1 S3 S12 S13 , and S123 are stable A-invariant subspaces. However, the subspaces S2 = spanfx2 g, S23 = spanfx2 x3 g, S24 = spanfx2 x4 g, and S234 = spanfx2 x3 x4 g are not A-invariant subspaces since the lower rank generalized eigenvector x1 of x2 is not in these subspaces. To illustrate, consider the subspace S23 . Then by de nition, Ax2 2 S23 if it is an A-invariant subspace. Since Ax2 = x2 + x1 Ax2 2 S23 would require that x1 be a linear combination of x2 and x3 , but this is impossible since x1 is independent of x2 and x3 . 3 3 We will say subspace S is trivial if S = f0g.
- 45. LINEAR ALGEBRA 28 2.7 Vector Norms and Matrix Norms In this section, we will de ne vector and matrix norms. Let X be a vector space, a realvalued function k k de ned on X is said to be a norm on X if it satis es the following properties: (i) kxk 0 (positivity) (ii) kxk = 0 if and only if x = 0 (positive de niteness) (iii) k xk = j j kxk, for any scalar (homogeneity) (iv) kx + yk kxk + kyk (triangle inequality) for any x 2 X and y 2 X . A function is said to be a semi-norm if it satis es (i), (iii), and (iv) but not necessarily (ii). Let x 2 C n . Then we de ne the vector p-norm of x as kxkp := n X i=1 jxi jp !1=p for 1 p 1: In particular, when p = 1 2 1 we have kxk1 := kxk2 := n X i=1 jxi j v u n uX t i=1 jxi j2 kxk1 := 1maxn jxi j: i Clearly, norm is an abstraction and extension of our usual concept of length in 3dimensional Euclidean space. So a norm of a vector is a measure of the vector length", for example kxk2 is the Euclidean distance of the vector x from the origin. Similarly, we can introduce some kind of measure for a matrix. Let A = aij ] 2 C m n , then the matrix norm induced by a vector p-norm is de ned as kAxk kAkp := sup kxk p : x6=0 p In particular, for p = 1 2 1, the corresponding induced matrix norm can be computed as m X kAk1 = 1maxn jaij j (column sum) j i=1
- 46. 2.7. Vector Norms and Matrix Norms kAk2 = p kAk1 = 1max i m 29 max (A A) n X j =1 jaij j (row sum) : The matrix norms induced by vector p-norms are sometimes called induced p-norms. This is because kAkp is de ned by or induced from a vector p-norm. In fact, A can be viewed as a mapping from a vector space C n equipped with a vector norm k kp to another vector space C m equipped with a vector norm k kp . So from a system theoretical point of view, the induced norms have the interpretation of input/output ampli cation gains. We shall adopt the following convention throughout the book for the vector and matrix norms unless speci ed otherwise: let x 2 C n and A 2 C m n , then we shall denote the Euclidean 2-norm of x simply by kxk := kxk2 and the induced 2-norm of A by kAk := kAk2 : The Euclidean 2-norm has some very nice properties: Lemma 2.8 Let x 2 Fn and y 2 Fm . 1. Suppose n m. Then kxk = kyk i there is a matrix U 2 Fn and U U = I . 2. Suppose n = m. Then jx yj for some 2 F or y = 0. m such that x = Uy kxk kyk. Moreover, the equality holds i x = y 3. kxk kyk i there is a matrix 2 Fn m with k k Furthermore, kxk < kyk i k k < 1. 1 such that x = y. 4. kUxk = kxk for any appropriately dimensioned unitary matrices U . Another often used matrix norm is the so called Frobenius norm. It is de ned as p kAkF := Trace(A A) = v u m n uX X t i=1 j =1 jaij j2 : However, the Frobenius norm is not an induced norm. The following properties of matrix norms are easy to show: Lemma 2.9 Let A and B be any matrices with appropriate dimensions. Then
- 47. LINEAR ALGEBRA 30 1. (A) kAk (This is also true for F norm and any induced matrix norm). 2. kAB k kAk kB k. In particular, this gives A;1 kAk;1 if A is invertible. (This is also true for any induced matrix norm.) 3. kUAV k = kAk, and kUAV kF = kAkF , for any appropriately dimensioned unitary matrices U and V . 4. kAB kF kAk kB kF and kAB kF kB k kAkF . Note that although pre-multiplication or post-multiplication of a unitary matrix on a matrix does not change its induced 2-norm and F -norm, it does change its eigenvalues. For example, let A= 1 0 : 1 0 Then 1 (A) = 1 2 (A) = 0. Now let U= " 1 p2 1 ; p2 then U is a unitary matrix and # 1 p2 1 p2 p 2 0 0 0 UA = p with 1 (UA) = 2, 2 (UA) = 0. This property is useful in some matrix perturbation problems, particularly, in the computation of bounds for structured singular values which will be studied in Chapter 10. Lemma 2.10 Let A be a block partitioned matrix with 2A A 6 A11 A12 A=6 . 6 .21 ..22 4 . . A1q A2q Am1 Am2 . . . Amq 3 7 7 =: Aij ] 7 5 and let each Aij be an appropriately dimensioned matrix. Then for any induced matrix p-norm 2 3 kA11 k kAkp kA12 k 6 kA21 kp kA22 kp p 6 . p 6 . . . 4 . . kAm1 kp kAm2 kp kA1q kp kA2q kp 7 7 . . . kAmq kp 7 : 5 Further, the inequality becomes an equality if the F -norm is used. p (2:2)
- 48. 2.7. Vector Norms and Matrix Norms 31 Proof. It is obvious that if the F -norm is used, then the right hand side of inequality (2.2) equals the left hand side. Hence only the induced p-norm cases, 1 p 1, will be shown. Let a vector x be partitioned consistently with A as 2x 6 x1 x = 6 .. 6 2 4 . xq and note that kxkp = Then k Aij ]kp := = = 2 kx1k 6 kx2kp 6 .p 6 . 4 . kxq kp 3 7 7 : 7 5 p 2 Pq A1j xj 3 6 Pjq=1 A2j xj 7 7 sup k Aij ] xkp = sup 6 j=1.. 6 7 4P . 5 kxkp =1 kxkp=1 q A x mj j p 2 Pq 3 j=1 2 Pq kA1j k kxj k 6 Pjq=1 A1j xj p 7 6 6 j=1 A2j xj p 7 7 6 Pjq=1 kA2j kp kxj kp 7 sup 6 6 7 kxsup 6 j=1 .. p p 6 .. 7 kxkp =1 6 kp =1 4 P . 6 P . 7 q kA k kx k 4 q 5 j =1 mj p j p j =1 Amj xj p p 2 kA11 k kA12k kA1q kp 3 2 kx1 kp 3 p p 6 kA21 kp kA22kp kA2q kp 7 6 kx2 kp 7 76 7 sup 6 .. 6 .. .. 7 6 .. 7 4 . 54 . 5 kxkp =1 . . kxq kp p kAm1 kp kAm2 kp kA1q kp h i sup kxkp =1 = 3 7 7 7 5 h kAij kp kAij kp i p p 3 7 7 7 5 p kxkp : 2
- 49. LINEAR ALGEBRA 32 2.8 Singular Value Decomposition A very useful tool in matrix analysis is Singular Value Decomposition (SVD). It will be seen that singular values of a matrix are good measures of the size" of the matrix and that the corresponding singular vectors are good indications of strong/weak input or output directions. Theorem 2.11 Let A 2 Fm n . There exist unitary matrices U = u1 u2 : : : um] 2 Fm m V = v1 v2 : : : vn ] 2 Fn n such that A=U V = 01 0 0 2 6 6 1=6 4 where and 1 0 1 0 . . . 0 2 . . . 0 p 2 0 0 ... . . . p 3 7 7 7 5 0 p = minfm ng: Proof. Let = kAk and without loss of generality assume m n. Then from the de nition of kAk, there exists a z 2 Fn such that kAz k = kz k : ~ ~ ~ By Lemma 2.8, there is a matrix U 2 F m n such that U U = I and ~ Az = Uz: Now let We have Ax = y. Let and z x = kz k 2 Fn ~ y = Uz 2 Fm : ~ Uz V = x V1 2 Fn n U = y U1 2 Fm m be unitary.4 Consequently, U AV has the following structure: A1 := U AV = 0 w B 4 Recall that it is always possible to extend an orthonormal set of vectors to an orthonormal basis for the whole space.
- 50. 2.8. Singular Value Decomposition 33 where w 2 Fn;1 and B 2 F(m;1) (n;1) . Since 2132 607 A1 6 .. 7 = ( 2 + w w) 6.7 4 5 0 2 2 + w w. But since = kAk = kA1 k, we must have w = 0. it follows that kA1 k2 An obvious induction argument gives U AV = : 2 This completes the proof. The i is the i-th singular value of A, and the vectors ui and vj are, respectively, the i-th left singular vector and the j -th right singular vector. It is easy to verify that Avi = A ui = i ui i vi : The above equations can also be written as A Avi = AA ui = Hence i2 is an eigenvalue of AA or A A, ui eigenvector of A A. 2 i vi 2 i ui : is an eigenvector of AA , and vi is an The following notations for singular values are often adopted: (A) = max (A) = 1 = the largest singular value of A and (A) = min (A) = p = the smallest singular value of A : Geometrically, the singular values of a matrix A are precisely the lengths of the semiaxes of the hyperellipsoid E de ned by E = fy : y = Ax x 2 C n kxk = 1g: Thus v1 is the direction in which kyk is largest for all kxk = 1 while vn is the direction in which kyk is smallest for all kxk = 1. From the input/output point of view, v1 (vn ) is the highest (lowest) gain input direction, while u1 (um ) is the highest (lowest) gain observing direction. This can be illustrated by the following 2 2 matrix: cos 2 ; sin 2 : 1 A = cos 1 ; sin 1 sin 1 cos 1 sin 2 cos 2 2
- 51. LINEAR ALGEBRA 34 It is easy to see that A maps a unit disk to an ellipsoid with semi-axes of 1 and 2 . Hence it is often convenient to introduce the following alternative de nitions for the largest singular value : (A) := kmax kAxk xk=1 and for the smallest singular value of a tall matrix: (A) := kmin kAxk : xk=1 Lemma 2.12 Suppose A and are square matrices. Then (i) j (A + ) ; (A)j ( ) (ii) (A ) (A) ( ) (iii) (A;1 ) = 1 if A is invertible. (A) Proof. (i) By de nition (A + ) := kmin k(A + )xk xk=1 min fkAxk ; k xkg kxk=1 min kAxk ; max k xk = kxk=1 (A) ; ( ): kxk=1 Hence ; ( ) (A + ) ; (A). The other inequality (A + ) ; (A) follows by replacing A by A + and by ; in the above proof. (ii) This follows by noting that ( ) (A ) := kmin kA xk x rk=1 min x A A x = kxk=1 = (A) kmin k xk xk=1 (A) ( ): (iii) Let the singular value decomposition of A be A = U V , then A;1 = V ;1 U . Hence (A;1 ) = ( ;1 ) = 1= ( ) = 1= (A).
- 52. 2.8. Singular Value Decomposition 35 2 Some useful properties of SVD are collected in the following lemma. Lemma 2.13 Let A 2 Fm n and 1 2 r > r+1 = = 0 r minfm ng: Then 1. rank(A) = r 2. KerA = spanfvr+1 : : : vn g and (KerA)? = spanfv1 : : : vr g 3. ImA = spanfu1 : : : ur g and (ImA)? = spanfur+1 : : : um g 4. A 2 Fm n has a dyadic expansion: A= 5. 6. 7. 8. r X i=1 i ui vi = Ur r Vr where Ur = u1 : : : ur ], Vr = v1 : : : vr ], and r = diag ( 1 : : : r ) 2 2 2 kAk2 = 1 + 2 + + r F kAk = 1 i (U0 AV0 ) = i (A) i = 1 : : : p for any appropriately dimensioned unitary matrices U0 and V0 P Let k < r = rank(A) and Ak := k=1 i ui vi , then i min kA ; B k = kA ; Ak k = k+1 : rank(B) k Proof. We shall only give a proof for part 8. It is easy to see that rank(Ak ) k and kA ; Ak k = k+1 . Hence, we only need to show that min kA ; B k k+1 . Let rank(B) k B be any matrix such that rank(B ) k. Then kA ; B k = kU V ; B k = k ; U BV k +1 Ik+1 0 ( ; U BV ) Ik0 = ^ k+1 ; B +1 ^ ^ where B = Ik+1 0 U BV Ik0 2 F(k+1) (k+1) and rank(B ) k. Let x 2 Fk+1 ^ be such that Bx = 0 and kxk = 1. Then ^ ^ kA ; B k ( k+1 ; B )x = k k+1 xk k+1 : k+1 ; B Since B is arbitrary, the conclusion follows. 2
- 53. LINEAR ALGEBRA 36 2.9 Generalized Inverses Let A 2 C m n . A matrix X 2 C n m is said to be a right inverse of A if AX = I . Obviously, A has a right inverse i A has full row rank, and, in that case, one of the right inverses is given by X = A (AA );1 . Similarly, if Y A = I then Y is called a left inverse of A. By duality, A has a left inverse i A has full column rank, and, furthermore, one of the left inverses is Y = (A A);1 A . Note that right (or left) inverses are not necessarily unique. For example, any matrix in the form I is a right inverse of I 0 . ? More generally, if a matrix A has neither full row rank nor full column rank, then all the ordinary matrix inverses do not exist however, the so called pseudo-inverse, known also as the Moore-Penrose inverse, is useful. This pseudo-inverse is denoted by A+ , which satis es the following conditions: (i) AA+ A = A (ii) A+ AA+ = A+ (iii) (AA+ ) = AA+ (iv) (A+ A) = A+ A. It can be shown that pseudo-inverse is unique. One way of computing A+ is by writing A = BC so that B has full column rank and C has full row rank. Then A+ = C (CC );1 (B B );1 B : Another way to compute A+ is by using SVD. Suppose A has a singular value decomposition A=U V with 0 = 0r 0 r > 0: Then A+ = V +U with += ;1 0 r 0 0 : 2.10 Semide nite Matrices A square hermitian matrix A = A is said to be positive de nite (semi-de nite) , denoted by A > 0 ( 0), if x Ax > 0 ( 0) for all x 6= 0. Suppose A 2 Fn n and A = A 0, then there exists a B 2 Fn r with r rank(A) such that A = BB . Lemma 2.14 Let B 2 Fm n and C 2 Fk n . Suppose m k and B B = C C . Then there exists a matrix U 2 Fm k such that U U = I and B = UC .
- 54. 2.10. Semide nite Matrices 37 Proof. Let V1 and V2 be unitary matrices such that B1 = V1 B1 0 C1 = V2 C1 0 where B1 and C1 are full row rank. Then B1 and C1 have the same number of rows and V3 := B1 C1 (C1 C1 );1 satis es V3 V3 = I since B B = C C . Hence V3 is a unitary matrix and V3 B1 = C1 . Finally let 0 U = V1 V3 V V2 0 4 for any suitably dimensioned V4 such that V4 V4 = I . 2 by We can de ne square root for a positive semi-de nite matrix A, A1=2 = (A1=2 ) 0, A = A1=2 A1=2 : Clearly, A1=2 can be computed by using spectral decomposition or SVD: let A = U U , then A1=2 = U 1=2 U where p p = diagf 1 : : : n g 1=2 = diagf 1 : : : n g: Lemma 2.15 Suppose A = A > 0 and B = B 0. Then A > B i (BA;1 ) < 1. Proof. Since A > 0, we have A > B i 0 < I ; A;1=2 BA;1=2 = I ; A;1=2 (BA;1 )A1=2 : However, A;1=2 BA;1=2 and BA;1 are similar, and hence i (BA;1 ) = i (A;1=2 BA;1=2 ). Therefore, the conclusion follows by the fact that 0 < I ; A;1=2 BA;1=2 i (A;1=2 BA;1=2 ) < 1 i (BA;1 ) < 1. 2 Lemma 2.16 Let X = X Then KerX22 + X12 X22 solves and 0 be partitioned as X = X11 X12 : X12 X22 + KerX12 . Consequently, if X22 is the pseudo-inverse of X22 , then Y = + X11 X12 = I X12 X22 X12 X22 0 I Y X22 = X12 + X11 ; X12 X22 X12 0 0 X22 I 0 + X22 X12 I :
- 55. LINEAR ALGEBRA 38 Proof. Without loss of generality, assume X22 = U1 U2 with 1= 1 and U1 U2 1 0 0 0 > 0 and U = U1 U2 unitary. Then Ker X22 = spanfcolumns of U2 g + X22 = U1 U2 Moreover ;1 0 1 0 U1 : U2 0 I 0 X11 X12 I 0 0 0 U X12 X22 0 U gives X12 U2 = 0. Hence, Ker X22 Ker X12 and now + X12 X22 X22 = X12 U1 U1 = X12 U1 U1 + X12 U2 U2 = X12 : The factorization follows easily. 2 2.11 Matrix Dilation Problems* In this section, we consider the following induced 2-norm optimization problem: min X X B C A (2:3) where X , B , C , and A are constant matrices of compatible dimensions. The matrix X B is a dilation of its sub-matrices as indicated in the following C A diagram: d X B C A d 6 c c ? C A d c B A d - 6 A c ?
- 56. 2.11. Matrix Dilation Problems* 39 In this diagram, c" stands for the operation of compression and d" stands for dilation. Compression is always norm non-increasing and dilation is always norm non-decreasing. Sometimes dilation can be made to be norm preserving. Norm preserving dilations are the focus of this section. The simplest matrix dilation problem occurs when solving X A min X : (2:4) Although (2.4) is a much simpli ed version of (2.3), we will see that it contains all the essential features of the general problem. Letting 0 denote the minimum norm in (2.4), it is immediate that 0 = kAk : The following theorem characterizes all solutions to (2.4). Theorem 2.17 8 0, i there is a Y with kY k 1 such that X A X = Y ( 2 I ; A A)1=2 : Proof. X A i X X +A A 2I i X X ( 2 I ; A A): Now suppose X X ( 2 I ; A A) and let h Y := X ( 2 I ; A A)1=2 then X = Y ( 2 I ; A A)1=2 and Y Y Y Y I then X X ( 2 I ; A A): i+ I . Similarly if X = Y ( 2 I ; A A)1=2 and 2 This theorem implies that, in general, (2.4) has more than one solution, which is in contrast to the minimization in the Frobenius norm in which X = 0 is the unique solution. The solution X = 0 is the central solution but others are possible unless 2 A A = 0I.
- 57. LINEAR ALGEBRA 40 Remark 2.1 The theorem still holds if ( 2I ; A A)1=2 is replaced by any matrix R such that 2 I ; A A = R R. ~ A more restricted version of the above theorem is shown in the following corollary. Corollary 2.18 8 > 0 , i X A (< ) X ( 2 I ; A A);1=2 1(< 1): The corresponding dual results are Theorem 2.19 8 0 X A i there is a Y , kY k 1, such that X = ( 2 I ; AA )1=2 Y: Corollary 2.20 8 > i 0 X A (< ) ( 2 I ; AA );1=2 X 1 (< 1): Now, returning to the problem in (2.3), let 0 := min X X B C A : (2:5) The following so called Parrott's theorem will play an important role in many control related optimization problems. The proof is the straightforward application of Theorem 2.17 and its dual, Theorem 2.19. Theorem 2.21 (Parrott's Theorem) The minimum in (2.5) is given by 0 = max C A B A : (2:6)
- 58. 2.11. Matrix Dilation Problems* 41 Proof. Denote by ^ the right hand side of the equation (2.6). Clearly, 0 ^ since compressions are norm non-increasing, and that 0 ^ will be shown by using Theorem 2.17 and Theorem 2.19. Suppose A 2 C n m and n m (the case for m > n can be shown in the same fashion). Then A has the following singular value decomposition: A=U 0 m V n;m m Hence U 2 C n n V 2 C m m: ^2 I ; A A = V (^2 I ; 2 )V m and 2 2 ^2 I ; AA = U ^ I ; m ^2 I0 U: 0 n;m Now let (^2 I ; A A)1=2 := V (^2 I ; 2 )1=2 V m and 2 2 1=2 0 (^2 I ; AA )1=2 := U (^ I ;0 m ) ^In;m U : Then it is easy to verify that (^2 I ; A A)1=2 A = A (^2 I ; AA )1=2 : Using this equality, we can show that ;A (^2 I ; AA )1=2 (^2 I ; A A)1=2 A ;A (^2 I ; AA )1=2 (^2 I ; A A)1=2 A 2 = ^0I ^0I : 2 Now we are ready to show that 0 ^. From Theorem 2.17 we have that B = Y (^2 I ; A A)1=2 for some Y such that kY k 1. Similarly, Theorem 2.19 yields C = (^2 I ; AA )1=2 Z for some Z with ^ kZ k 1. Now let X = ;Y A Z . Then ^ ;Y A Z Y (^2 I ; A A)1=2 X B = 2 I ; AA )1=2 Z C A (^ A Y ;A (^2 I ; A A)1=2 Z = 2 I ; AA )1=2 I I (^ A 2 I ; A A)1=2 ;A (^ (^2 I ; AA )1=2 A = ^:
- 59. LINEAR ALGEBRA 42 Thus ^ 0, 2 so ^ = 0 . This theorem gives one solution to (2.3) and an expression for 0 . As in (2.4), there may be more than one solution to (2.3), although the proof of theorem 2.21 exhibits only one. Theorem 2.22 considers the problem of parameterizing all solutions. The ^ solution X = ;Y A Z is the central" solution analogous to X = 0 in (2.4). Theorem 2.22 Suppose 0 . The solutions X such that X B C A (2:7) are exactly those of the form X = ;Y A Z + (I ; Y Y )1=2 W (I ; Z Z )1=2 where W is an arbitrary contraction (kW k 1), and Y with kY k kZ k 1 solve the linear equations B = Y ( 2 I ; A A)1=2 C = ( 2 I ; AA )1=2 Z: Proof. Since that 0, (2:8) 1 and Z with again from Theorem 2.19 there exists a Z with kZ k (2.9) (2.10) 1 such C = ( 2 I ; AA )1=2 Z: Note that using the above expression for C we have 2I ; C A C A Z Z 1=2 0 (I ; Z Z )1=2 0 = (I ; A Z ) 2 I ; A A)1=2 ; ( ;A Z ( 2 I ; A A)1=2 : Now apply Theorem 2.17 (Remark 2.1) to inequality (2.7) with respect to the partitioned matrix X B to get C A ^ X B =W (I ; Z Z )1=2 ;A Z ( 0 2I ; A A)1=2 ^ ^ ^ ^ for some contraction W , W 1. Partition W as W = W1 Y to obtain the expression for X and B : X = ;Y A Z + W1 (I ; Z Z )1=2 B = Y ( 2 I ; A A)1=2 : Then kY k 1 and the theorem follows by noting that W1 Y 1 i there is a 1=2 W . W , kW k 1, such that W1 = (I ; Y Y ) 2 The following corollary gives an alternative version of Theorem 2.22 when > 0 .
- 60. 2.11. Matrix Dilation Problems* 43 Corollary 2.23 For > 0. X B C A i (< ) (I ; Y Y );1=2 (X + Y A Z )(I ; Z Z );1=2 (2:11) (< ) (2:12) where Y = B ( 2 I ; A A);1=2 (2.13) 2 I ; AA );1=2 C: Z = ( (2.14) Note that in the case of > 0 , I ; Y Y and I ; Z Z are invertible since Corollary 2.18 and 2.20 clearly show that kY k < 1 and kZ k < 1. There are many alternative characterizations of solutions to (2.11), although the formulas given above seem to be the simplest. As a straightforward application of the dilation results obtained above, consider the following matrix approximation problem: 0 = min kR + UQV k Q (2:15) where R, U , and V are constant matrices such that U U = I and V V = I . Corollary 2.24 The minimum achievable norm is given by 0 = max fkU? Rk kRV? kg and the parameterization for all optimal solutions Q can be obtained from Theorem 2.22 with X = Q + U RV , A = U? RV? , B = U RV? , and C = U?RV . V Proof. Let U? and V? be such that U U? and V? are unitary matrices. Then 0 = min kR + UQV k Q = min Q = min Q V U U? (R + UQV ) V ? U RV + Q U RV? : U?RV U? RV? The result follows after applying Theorem 2.21 and 2.22. A similar problem arises in H1 control theory. 2
- 61. 44 LINEAR ALGEBRA 2.12 Notes and References A very extensive treatment of most topics in this chapter can be found in Brogan 1991], Horn and Johnson 1990,1991] and Lancaster and Tismenetsky 1985]. Golub and Van Loan's book 1983] contains many numerical algorithms for solving most of the problems in this chapter. The matrix dilation theory can be found in Davis, Kahan, and Weinberger 1982].
- 62. 3 Linear Dynamical Systems This chapter reviews some basic system theoretical concepts. The notions of controllability, observability, stabilizability, and detectability are de ned and various algebraic and geometric characterizations of these notions are summarized. Kalman canonical decomposition, pole placement, and observer theory are then introduced. The solutions of Lyapunov equations and their connections with system stability, controllability, and so on, are discussed. System interconnections and realizations, in particular the balanced realization, are studied in some detail. Finally, the concepts of system poles and zeros are introduced. 3.1 Descriptions of Linear Dynamical Systems Let a nite dimensional linear time invariant (FDLTI) dynamical system be described by the following linear constant coe cient di erential equations: x = Ax + Bu x(t0 ) = x0 _ y = Cx + Du (3.1) (3.2) where x(t) 2 Rn is called the system state, x(t0 ) is called the initial condition of the system, u(t) 2 Rm is called the system input, and y(t) 2 Rp is the system output. The A B C and D are appropriately dimensioned real constant matrices. A dynamical system with single input (m = 1) and single output (p = 1) is called a SISO (single input and single output) system, otherwise it is called MIMO (multiple input and multiple 45
- 63. 46 LINEAR DYNAMICAL SYSTEMS output) system. The corresponding transfer matrix from u to y is de ned as Y (s) = G(s)U (s) where U (s) and Y (s) are the Laplace transform of u(t) and y(t) with zero initial condition (x(0) = 0). Hence, we have G(s) = C (sI ; A);1 B + D: Note that the system equations (3.1) and (3.2) can be written in a more compact matrix form: x = A B _ x : y C D u To expedite calculations involving transfer matrices, the notation A B ;1 C D := C (sI ; A) B + D will be used. Other reasons for using this notation will be discussed in Chapter 10. Note that A B C D is a real block matrix, not a transfer function. Now given the initial condition x(t0 ) and the input u(t), the dynamical system response x(t) and y(t) for t t0 can be determined from the following formulas: Zt x(t) = eA(t;t0 ) x(t0 ) + eA(t; )Bu( )d (3.3) t0 y(t) = Cx(t) + Du(t): (3.4) In the case of u(t) = 0 8t t0 , it is easy to see from the solution that for any t1 t0 and t t0 , we have x(t) = eA(t;t1 ) x(t1 ): Therefore, the matrix function (t t1 ) = eA(t;t1 ) acts as a transformation from one state to another, and thus (t t1 ) is usually called the state transition matrix. Since the state of a linear system at one time can be obtained from the state at another through the transition matrix, we can assume without loss of generality that t0 = 0. This will be assumed in the sequel. The impulse matrix of the dynamical system is de ned as g(t) = L;1 fG(s)g = CeAt B 1+ (t) + D (t) where (t) is the unit impulse and 1+ (t) is the unit step de ned as 1+ (t) := 1 t < 0: 0 t 0
- 64. 3.2. Controllability and Observability 47 The input/output relationship (i.e., with zero initial state: x0 = 0) can be described by the convolution equation y(t) = (g u)(t) := Z1 ;1 g(t ; )u( )d = Zt ;1 g(t ; )u( )d : 3.2 Controllability and Observability We now turn to some very important concepts in linear system theory. De nition 3.1 The dynamical system described by the equation (3.1) or the pair (A B ) is said to be controllable if, for any initial state x(0) = x0 , t1 > 0 and nal state x1 , there exists a (piecewise continuous) input u( ) such that the solution of (3.1) satis es x(t1 ) = x1 . Otherwise, the system or the pair (A B ) is said to be uncontrollable. The controllability (or observability introduced next) of a system can be veri ed through some algebraic or geometric criteria. Theorem 3.1 The following are equivalent: (i) (A B ) is controllable. (ii) The matrix Wc (t) := Zt 0 eA BB eA d is positive de nite for any t > 0. (iii) The controllability matrix C = B AB A2 B : : : An;1 B P has full row rank or, in other words, hA jImB i := n=1 Im(Ai;1 B ) = Rn . i (iv) The matrix A ; I B ] has full row rank for all in C . and x be any eigenvalue and any corresponding left eigenvector of A, i.e., x A = x , then x B 6= 0. (v) Let (vi) The eigenvalues of A + BF can be freely assigned (with the restriction that complex eigenvalues are in conjugate pairs) by a suitable choice of F .
- 65. LINEAR DYNAMICAL SYSTEMS 48 Proof. (i) , (ii): Suppose Wc (t1 ) > 0 for some t1 > 0, and let the input be de ned as u( ) = ;B eA (t1 ; )Wc (t1 );1 (eAt1 x0 ; x1 ): Then it is easy to verify using the formula in (3.3) that x(t1 ) = x1 . Since x1 is arbitrary, the pair (A B ) is controllable. To show that the controllability of (A B ) implies that Wc (t) > 0 for any t > 0, assume that (A B ) is controllable but Wc (t1 ) is singular for some t1 > 0. Since eAtBB eA t 0 for all t, there exists a real vector 0 6= v 2 Rn such that v eAtB = 0 t 2 0 t1 ]: Now let x(t1 ) = x1 = 0, and then from the solution (3.3), we have 0 = eAt1 x(0) + Zt 1 0 eA(t1 ; )Bu( )d : Pre-multiply the above equation by v to get 0 = v eAt1 x(0): If we chose the initial state x(0) = e;At1 v, then v = 0, and this is a contradiction. Hence, Wc (t) can not be singular for any t > 0. (ii) , (iii): Suppose Wc (t) > 0 for all t > 0 (in fact, it can be shown that Wc (t) > 0 for all t > 0 i , for some t1 , Wc (t1 ) > 0) but the controllability matrix C does not have full row rank. Then there exists a v 2 Rn such that v Ai B = 0 for all 0 i n ; 1. In fact, this equality holds for all i 0 by the CayleyHamilton Theorem. Hence, v eAt B = 0 for all t or, equivalently, v Wc (t) = 0 for all t this is a contradiction, and hence, the controllability matrix C must be full row rank. Conversely, suppose C has full row rank but Wc (t) is singular for some t1 . Then there exists a 0 6= v 2 Rn such that v eAt B = 0 for all t 2 0 t1 ]. Therefore, set t = 0, and we have v B = 0: Next, evaluate the i-th derivative of v eAt B = 0 at t = 0 to get v Ai B = 0 i > 0: Hence, we have v B AB A2 B : : : An;1 B = 0 or, in other words, the controllability matrix C does not have full row rank. This is again a contradiction.
- 66. 3.2. Controllability and Observability 49 (iii) ) (iv): Suppose, on the contrary, that the matrix A; I B does not have full row rank for some 2 C . Then there exists a vector x 2 C n such that x A; I B =0 i.e., x A = x and x B = 0. However, this will result in x B AB : : : An;1 B = x B x B ::: n;1 x B =0 i.e., the controllability matrix C does not have full row rank, and this is a contradiction. (iv) ) (v): This is obvious from the proof of (iii) ) (iv). (v) ) (iii): We will again prove this by contradiction. Assume that (v) holds but rank C = k < n. Then in section 3.3, we will show that there is a transformation T such that TAT ;1 = Ac A12 TB = Bc 0 A 0 with Ac 2 R(n;k) (n;k) . c Let 1 and xc be any eigenvalue and any corresponding left eigenvector of Ac , i.e., xc Ac = 1 xc . Then x (TB ) = 0 and 0 x= x c is an eigenvector of TAT ;1 corresponding to the eigenvalue 1 , which implies that (TAT ;1 TB ) is not controllable. This is a contradiction since similarity transformation does not change controllability. Hence, the proof is completed. (vi) ) (i): This follows the same arguments as in the proof of (v) ) (iii): assume that (vi) holds but (A B ) is uncontrollable. Then, there is a decomposition so that some subsystems are not a ected by the control, but this contradicts the condition (vi). (i) ) (vi): This will be clear in section 3.4. In that section, we will explicitly construct a matrix F so that the eigenvalues of A + BF are in the desired locations. 2 De nition 3.2 An unforced dynamical system x = Ax is said to be stable if all the _ eigenvalues of A are in the open left half plane, i.e., Re (A) < 0. A matrix A with such a property is said to be stable or Hurwitz.
- 67. LINEAR DYNAMICAL SYSTEMS 50 De nition 3.3 The dynamical system (3.1), or the pair (A B), is said to be stabilizable if there exists a state feedback u = Fx such that the system is stable, i.e., A + BF is stable. Therefore, it is more appropriate to call this stabilizability the state feedback stabilizability to di erentiate it from the output feedback stabilizability de ned later. The following theorem is a consequence of Theorem 3.1. Theorem 3.2 The following are equivalent: (i) (A B ) is stabilizable. (ii) The matrix A ; I B ] has full row rank for all Re (iii) For all and x such that x A = x 0. 0, x B 6= 0. and Re (iv) There exists a matrix F such that A + BF is Hurwitz. We now consider the dual notions of observability and detectability of the system described by equations (3.1) and (3.2). De nition 3.4 The dynamical system described by the equations (3.1) and (3.2) or by the pair (C A) is said to be observable if, for any t1 > 0, the initial state x(0) = x0 can be determined from the time history of the input u(t) and the output y(t) in the interval of 0 t1]. Otherwise, the system, or (C A), is said to be unobservable. Theorem 3.3 The following are equivalent: (i) (C A) is observable. (ii) The matrix Wo (t) := is positive de nite for any t > 0. (iii) The observability matrix Zt 0 eA C CeA d 2 C 3 6 CA 7 6 7 O = 6 CA2 7 6 . 7 6 .. 7 4 5 CAn;1 T has full column rank or n=1 Ker(CAi;1 ) = 0. i (iv) The matrix A; I C has full column rank for all in C .
- 68. 3.2. Controllability and Observability 51 (v) Let and y be any eigenvalue and any corresponding right eigenvector of A, i.e., Ay = y, then Cy 6= 0. (vi) The eigenvalues of A + LC can be freely assigned (with the restriction that complex eigenvalues are in conjugate pairs) by a suitable choice of L. (vii) (A C ) is controllable. Proof. First, we will show the equivalence between conditions (i) and (iii). Once this is done, the rest will follow by the duality or condition (vii). (i) ( (iii): Note that given the input u(t) and the initial condition x0 , the output in the time interval 0 t1] is given by y(t) = CeAt x(0) + Zt 0 CeA(t; ) Bu( )d + Du(t): Since y(t) and u(t) are known, there is no loss of generality in assuming u(t) = 0 8t. Hence, y(t) = CeAt x(0) t 2 0 t1 ]: From this equation, we have 3 2 C 3 CA 7 7 6 CA2 7 7=6 6 7 6 . 7 x(0) 7 5 6 .. 7 4 5 (n;1) (0) y 2 y(0) 6 y_ (0) 6 . 6 .. 4 CAn;1 where y(i) stands for the i-th derivative of y. Since the observability matrix O has full column rank, there is a unique solution x(0) in the above equation. This completes the proof. (i) ) (iii): This will be proven by contradiction. Assume that (C A) is observable but that the observability matrix does not have full column rank, i.e., there is a vector x0 such that Ox0 = 0 or equivalently CAi x0 = 0 8i 0 by the Cayley-Hamilton Theorem. Now suppose the initial state x(0) = x0 , then y(t) = CeAt x(0) = 0. This implies that the system is not observable since x(0) cannot be determined from y(t) 0. 2 De nition 3.5 The system, or the pair (C A), is detectable if A + LC is stable for some L.
- 69. LINEAR DYNAMICAL SYSTEMS 52 Theorem 3.4 The following are equivalent: (i) (C A) is detectable. A ; I has full column rank for all Re 0. C (iii) For all and x such that Ax = x and Re 0, Cx 6= 0. (iv) There exists a matrix L such that A + LC is Hurwitz. (v) (A C ) is stabilizable. (ii) The matrix The conditions (iv) and (v) of Theorem 3.1 and Theorem 3.3 and the conditions (ii) and (iii) of Theorem 3.2 and Theorem 3.4 are often called Popov-Belevitch-Hautus (PBH) tests. In particular, the following de nitions of modal controllability and observability are often useful. De nition 3.6 Let be an eigenvalue of A or, equivalently, a mode of the system. Then the mode is said to be controllable (observable) if x B 6= 0 (Cx 6= 0) for all left (right) eigenvectors of A associated with , i.e., x A = x (Ax = x) and 0 6= x 2 C n . Otherwise, the mode is said to be uncontrollable (unobservable). It follows that a system is controllable (observable) if and only if every mode is controllable (observable). Similarly, a system is stabilizable (detectable) if and only if every unstable mode is controllable (observable). For example, consider the following 4th order system: 2 6 A B =6 6 6 C D 4 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0 2 1 0 3 7 7 7 7 5 with 1 6= 2 . Then, the mode 1 is not controllable if = 0, and 2 is not observable if = 0. Note that if 1 = 2 , the system is uncontrollable and unobservable for any and since in that case, both 203 6 7 x1 = 6 0 7 415 0 and 203 6 7 x2 = 6 0 7 405 1 are the left eigenvectors of A corresponding to 1 . Hence any linear combination of x1 and x2 is still an eigenvector of A corresponding to 1 . In particular, let x = x1 ; x2 , then x B = 0, and as a result, the system is not controllable. Similar arguments can be applied to check observability. However, if the B matrix is changed into a 4 2 matrix
- 70. 3.3. Kalman Canonical Decomposition 53 with the last two rows independent of each other, then the system is controllable even if 1 = 2 . For example, the reader may easily verify that the system with 20 03 6 7 B=6 1 0 7 4 15 1 0 is controllable for any . In general, for a system given in the Jordan canonical form, the controllability and observability can be concluded by inspection. The interested reader may easily derive some explicit conditions for the controllability and observability by using Jordan canonical form and the tests (iv) and (v) of Theorem 3.1 and Theorem 3.3. 3.3 Kalman Canonical Decomposition There are usually many di erent coordinate systems to describe a dynamical system. For example, consider a simple pendulum, the motion of the pendulum can be uniquely determined either in terms of the angle of the string attached to the pendulum or in terms of the vertical displacement of the pendulum. However, in most cases, the angular displacement is a more natural description than the vertical displacement in spite of the fact that they both describe the same dynamical system. This is true for most physical dynamical systems. On the other hand, although some coordinates may not be natural descriptions of a physical dynamical system, they may make the system analysis and synthesis much easier. In general, let T 2 Rn n be a nonsingular matrix and de ne x = Tx: Then the original dynamical system equations (3.1) and (3.2) become _ x = TAT ;1x + TBu y = CT ;1 x + Du: These equations represent the same dynamical system for any nonsingular matrix T , and hence, we can regard these representations as equivalent. It is easy to see that the input/output transfer matrix is not changed under the coordinate transformation, i.e., G(s) = C (sI ; A);1 B + D = CT ;1 (sI ; TAT ;1);1 TB + D: In this section, we will consider the system structure decomposition using coordinate transformation if the system is not completely controllable and/or is not completely observable. To begin with, let us consider further the dynamical systems related by a similarity transformation: " # A B 7;! A B = TAT ;1 TB : C D CT ;1 D C D
- 71. LINEAR DYNAMICAL SYSTEMS 54 The controllability and observability matrices are related by C = T C O = O T ;1 : Moreover, the following theorem follows easily from the PBH tests or from the above relations. Theorem 3.5 The controllability (or stabilizability) and observability (or detectability) are invariant under similarity transformations. Using this fact, we can now show the following theorem. Theorem 3.6 If the controllability matrix C has rank k1 < n, then there exists a similarity transformation x = xc = Tx x c such that _ xc _ xc Ac A12 0 Ac = Cc Cc y = xc + Bc u xc 0 xc + Du xc with Ac 2 C k1 k1 and (Ac Bc) controllable. Moreover, G(s) = C (sI ; A);1 B + D = Cc (sI ; Ac );1 Bc + D: Proof. Since rank C = k1 < n, the pair (A B) is not controllable. Now let q1 q2 : : : qk1 be any linearly independent columns of C . Let qi i = k1 +1 : : : n be any n ; k1 linearly independent vectors such that the matrix Q := q1 is nonsingular. De ne qk1 qk1 +1 qn T := Q;1 : Then the transformation x = Tx will give the desired decomposition. To see that, note that for each i = 1 2 : : : k1 , Aqi can be written as a linear combination of qi i = 1 2 : : : k1 since Aqi is a linear combination of the columns of C by the Cayley-Hamilton Theorem. Therefore, we have AT ;1 = = Aq1 q1 = T ;1 Aqk1 Aqk1 +1 Aqn qk1 qk1 +1 qn Ac A12 0 Ac Ac A12 0 Ac
- 72. 3.3. Kalman Canonical Decomposition 55 for some k1 k1 matrix Ac . Similarly, each column of the matrix B is a linear combination of qi i = 1 2 : : : k1 , hence B = Q Bc = T ;1 Bc 0 0 for some Bc 2 C k1 m . To show that (Ac Bc) is controllable, note that rank C = k1 and C = T ;1 Bc Ac0Bc 0 Ak1 ;1 Bc c An;1 Bc : c 0 0 Since, for each j k1 , Aj is a linear combination of Aic i = 0 1 : : : (k1 ; 1) by Cayleyc Hamilton theorem, we have rank Bc Ac Bc Ak1 ;1 Bc = k1 c i.e., (Ac Bc ) is controllable. 2 A numerically reliable way to nd a such transformation T is to use QR factorization. For example, if the controllability matrix C has the QR factorization QR = C , then T = Q;1 . Corollary 3.7 If the system is stabilizable and the controllability matrix C has rank k1 < n, then there exists a similarity transformation T such that TAT ;1 CT ;1 3 2 Ac A12 Bc TB = 6 0 Ac 0 7 5 4 D Cc Cc D with Ac 2 C k1 k1 , (Ac Bc ) controllable and with Ac stable. Hence, the state space x is partitioned into two orthogonal subspaces xc 0 and 0 xc with the rst subspace controllable from the input and second completely uncontrollable from the input (i.e., the state xc are not a ected by the control u). Write these subspaces in terms of the original coordinates x, we have x = T ;1 x = q1 qk1 qk1 +1 qn xc : xc So the controllable subspace is the span of qi i = 1 : : : k1 or, equivalently, Im C . On the other hand, the uncontrollable subspace is given by the complement of the controllable subspace. By duality, we have the following decomposition if the system is not completely observable.
- 73. LINEAR DYNAMICAL SYSTEMS 56 Theorem 3.8 If the observability matrix O has rank k2 < n, then there exists a similarity transformation T such that TAT ;1 CT ;1 3 2 Ao 0 Bo TB = 6 A21 Ao Bo 7 5 4 D Co 0 D with Ao 2 C k2 k2 and (Co Ao ) observable. Corollary 3.9 If the system is detectable and the observability matrix C has rank k2 < n, then there exists a similarity transformation T such that TAT ;1 TB CT ;1 D 2 3 Ao 0 Bo = 6 A21 Ao Bo 7 4 5 Co 0 D with Ao 2 C k2 k2 , (Co Ao ) observable and with Ao stable. Similarly, we have G(s) = C (sI ; A);1 B + D = Co (sI ; Ao );1 Bo + D: Carefully combining the above two theorems, we get the following Kalman Canonical Decomposition. The proof is left to the reader as an exercise. Theorem 3.10 Let an LTI dynamical system be described by the equations (3.1) and (3.2). Then there exists a nonsingular coordinate transformation x = Tx such that 2x 3 2 A 0 A 0 32 x 3 2 B 3 _ co co 6 xco 7 = 6 A21 Aco A13 A24 7 6 xco 7 + 6 Bco 7 u _ 7 6x 5 6 0 0 A23 0 7 6 xco 7 6 0co 7 4 _ co 4 5 4 co 5 4 5 co _ co x 0 0 A43 Aco xco 0 2x 6 xco 6 xco 4 y = Cco 0 Cco 0 TAT ;1 CT ;1 2 Aco 0 6 A21 A TB = 6 0 0co 6 6 0 0 D 4 or equivalently Cco co xco 0 3 7 + Du 7 5 A13 0 Bco A23 A24 Bco Aco 0 0 A43 Aco 0 Cco 0 D 3 7 7 7 7 5
- 74. 3.3. Kalman Canonical Decomposition 57 where the vector xco is controllable and observable, xco is controllable but unobservable, xco is observable but uncontrollable, and xco is uncontrollable and unobservable. Moreover, the transfer matrix from u to y is given by G(s) = Cco (sI ; Aco );1 Bco + D: One important issue is that although the transfer matrix of a dynamical system A B C D is equal to its controllable and observable part " Aco Bco Cco D # their internal behaviors are very di erent. In other words, while their input/output behaviors are the same, their state space response with nonzero initial conditions are very di erent. This can be illustrated by the state space response for the simple system 2x 3 2 A 0 0 0 32 x 3 2 B 3 _ co 6 xco 7 = 6 0co Aco 0 0 7 6 xco 7 + 6 Bco 7 u _ 6x 7 6 0 0 A 0 7 6 xco 7 6 0co 7 4 _ co 5 4 5 4 co 5 4 5 co _ co x 0 0 0 Aco xco 0 y = Cco 0 Cco 0 2x 6 xco 6 xco 4 co xco 3 7 7 5 with xco controllable and observable, xco controllable but unobservable, xco observable but uncontrollable, and xco uncontrollable and unobservable. The solution to this system is given by 2 x (t) 3 6 xco(t) 7 6 xco(t) 7 4 5 co xco (t) = 3 2 Acot R e xco (0) + R0t eAco (t; )Bco u( )d 6 eAcotxco(0) + t eAco(t; )Bcou( )d 7 7 6 0 7 6 4 5 eAco t xco (0) eAco t xco (0) y(t) = Cco xco (t) + Cco xco note that xco (t) and xco (t) are not a ected by the input u, while xco (t) and xco (t) do not show up in the output y. Moreover, if the initial condition is zero, i.e., x(0) = 0, then the output Zt y(t) = Cco eAco(t; ) Bcou( )d : 0
- 75. LINEAR DYNAMICAL SYSTEMS 58 However, if the initial state is not zero, then the response xco (t) will show up in the output. In particular, if Aco is not stable, then the output y(t) will grow without bound. The problems with uncontrollable and/or unobservable unstable modes are even more profound than what we have mentioned. Since the states are the internal signals of the dynamical system, any unbounded internal signal will eventually destroy the system. On the other hand, since it is impossible to make the initial states exactly zero, any uncontrollable and/or unobservable unstable mode will result in unacceptable system behavior. This issue will be exploited further in section 3.7. In the next section, we will consider how to place the system poles to achieve desired closed-loop behavior if the system is controllable. 3.4 Pole Placement and Canonical Forms Consider a MIMO dynamical system described by x = Ax + Bu _ y = Cx + Du and let u be a state feedback control law given by u = Fx + v: This closed-loop system is as shown in Figure 3.1, and the closed-loop system equations are given by x = (A + BF )x + Bv _ y = (C + DF )x + Dv: D y ? e C R x - 6 B u e v 6 A - e F Figure 3.1: State Feedback Then we have the following lemma in which the proof is simple and is left as an exercise to the reader.
- 76. 3.4. Pole Placement and Canonical Forms 59 Lemma 3.11 Let F be a constant matrix with appropriate dimension then (A B) is controllable (stabilizable) if and only if (A + BF B ) is controllable (stabilizable). However, the observability of the system may change under state feedback. For example, the following system 2 3 A B =4 1 1 1 5 1 0 0 C D 1 0 0 is controllable and observable. But with the state feedback u = Fx = ;1 ;1 x 20 A + BF B 4 C + DF D = 1 1 the system becomes and is not completely observable. The dual operation of the dynamical system by 0 1 0 0 0 0 3 5 x = Ax + Bu 7;! x = Ax + Bu + Ly _ _ is called output injection which can be written as A + LC B + LD : A B C D 7;! C D By duality, the output injection does not change the system observability (detectability) but may change the system controllability (stabilizability). Remark 3.1 We would like to call attention to the diagram and the signals ow con- vention used in this book. It may not be conventional to let signals ow from the right to the left, however, the reader will nd that this representation is much more visually appealing than the traditional representation, and is consistent with the matrix manipulations. For example, the following diagram represents the multiplication of three matrices and will be very helpful in dealing with complicated systems: z = M1 M2 M3 w: z M1 M2 M3 w The conventional signal block diagrams, i.e., signals owing from left to right, will also be used in this book. ~
- 77. LINEAR DYNAMICAL SYSTEMS 60 We now consider some special state space representations of the dynamical system described by equations (3.1) and (3.2). First, we will consider the systems with single inputs. Assume that a single input and multiple output dynamical system is given by A b G(s) = C d b 2 Rn C 2 Rp n d 2 Rp and assume that (A b) is controllable. Let det( I ; A) = n + a1 n;1 + + an and de ne 2 ;a ;a 6 11 02 6 A1 := 6 0 6 . 1 6 .. ... 4 0 ;an;1 ;an 3 0 0 .. . 1 0 0 0 .. . 0 213 607 6 7 b1 := 6 0 7 6.7 6 .. 7 4 5 7 7 7 7 7 5 0 and C = C1 = b Ab b1 A1 b1 An;1 b An;1 b1 : 1 Then it is easy to verify that both C and C1 are nonsingular. Moreover, the transformation Tc = C1 C ;1 will give the equivalent system representation TcATc;1 Tcb = A1 b1 CTc;1 d CTc;1 d where CTc;1 = 1 2 n;1 n for some i 2 Rp . This state space representation is usually called controllable canonical form or controller canonical form. It is also easy to show that the transfer matrix is given by n;1 + + 2 sn;2 G(s) = C (sI ; A);1 b + d = 1 s n + + sn;1 + + + a n;1 s a n + d s a1 n;1 s + n which also shows that, given a column vector of transfer matrix G(s), a state space representation of the transfer matrix can be obtained as above. The quadruple (A b C d) is called a state space realization of the transfer matrix G(s).
- 78. 3.4. Pole Placement and Canonical Forms 61 Now consider a dynamical system equation given by x = A1 x + b1 u _ and a state feedback control law u = Fx = f1 f2 : : : fn;1 fn x: Then the closed-loop system is given by x = (A1 + b1 F )x _ and det( I ; (A1 + b1 F )) = n +(a1 ; f1) n;1 + +(an ; fn ). It is clear that the zeros of det( I ; (A1 + b1 F ) can be made arbitrary for an appropriate choice of F provided that the complex zeros appear in conjugate pairs, thus showing that the eigenvalues of A + bF can be freely assigned if (A b) is controllable. Dually, consider a multiple input and single output system G(s) = A B c d B 2 Rn m c 2 Rn d 2 Rm and assume that (c A) is observable then there is a transformation To such that ToATo;1 cTo;1 2 ;a1 6 ;a2 6 ToB = 6 ... 6 6 ;an;1 d 6 6 ;an 4 1 and 1 0 .. . 0 0 0 0 1 .. . 0 0 0 0 0 .. . 1 0 0 1 2 .. . n;1 n d 3 7 7 7 7 7 7 7 5 i 2 Rm sn;1 + 2 sn;2 + s G(s) = c(sI ; A);1 B + d = 1 sn + a sn;1 + + + a n;1+ + n + d: 1 n;1 s an This representation is called observable canonical form or observer canonical form. Similarly, we can show that the eigenvalues of A + Lc can be freely assigned if (c A) is observable. The pole placement problem for a multiple input system (A B ) can be converted into a simple single input pole placement problem. To describe the procedure, we need some preliminary results. Lemma 3.12 If an m input system pair (A B) is controllable and if A is cyclic, then for almost all v 2 Rm , the single input pair (A Bv) is controllable.
- 79. LINEAR DYNAMICAL SYSTEMS 62 Proof. Without loss of generality, assume that A is in the Jordan canonical form and that the matrix B is partitioned accordingly: 2J 6 1 J2 A=6 6 4 where Ji is in the form of 2 6 6 6 Ji = 6 6 6 4 ... i 1 i Jk 3 2B 7 6 B1 7 B = 6 .2 7 6 .. 5 4 Bk ... ... ... i 1 i 3 7 7 7 5 3 7 7 7 7 7 7 5 and i 6= j if i 6= j . By PBH test, the pair (A B ) is controllable if and only if, for each i = 1 : : : k, the last row of Bi is not zero. Let bi 2 Rm be the last row of Bi , and then we only need to show that, for almost all v 2 Rm , bi v 6= 0 for each i = 1 : : : k which is clear since for each i, the set v 2 Rm such that bi v = 0 has measure zero in Rm since bi 6= 0. 2 The cyclicity assumption in this theorem is essential. Without this assumption, the theorem does not hold. For example, the pair A= 1 0 B= 1 0 0 1 0 1 is controllable but there is no v 2 R2 such that (A Bv) is controllable since A is not cyclic. Since a matrix A with distinct eigenvalues is cyclic, by de nition we have the following lemma. Lemma 3.13 If (A B) is controllable, then for almost any K 2 Rm n , all the eigenvalues of A + BK are distinct and, consequently, A + BK is cyclic. A proof can be found in Brasch and Pearson 1970], Davison 1968], and Heymann 1968]. Now it is evident that given a multiple input controllable pair (A B ), there is a matrix K 2 Rm n and a vector v 2 Rm such that A + BK is cyclic and (A + BK Bv) is controllable. Moreover, from the pole placement results for the single input system, there is a matrix f 2 Rn so that the eigenvalues of (A + BK )+(Bv)f can be arbitrarily assigned. Hence, the eigenvalues of A + BF can be arbitrarily assigned by choosing a state feedback in the form of u = Fx = (K + vf )x:
- 80. 3.5. Observers and Observer-Based Controllers 63 A dual procedure can be applied for the output injection A + LC . The canonical form for single input or single output system can also be generalized to multiple input and multiple output systems at the expense of notation. The interested reader may consult Kailath 1980] or Chen 1984]. If a system is not completely controllable, then the Kalman controllable decomposition can be applied rst and the above procedure can be used to assign the system poles corresponding to the controllable subspace. 3.5 Observers and Observer-Based Controllers We have shown in the last section that if a system is controllable and, furthermore, if the system states are available for feedback, then the system closed loop poles can be assigned arbitrarily through a constant feedback. However, in most practical applications, the system states are not completely accessible and all the designer knows are the output y and input u. Hence, the estimation of the system states from the given output information y and input u is often necessary to realize some speci c design objectives. In this section, we consider such an estimation problem and the application of this state estimation in feedback control. Consider a plant modeled by equations (3.1) and (3.2). An observer is a dynamical system with input of (u y) and output of, say x, which asymptotically estimates the ^ state x. More precisely, a (linear) observer is a system such as q_ = Mq + Nu + Hy x = Qq + Ru + Sy ^ so that x(t) ; x(t) ! 0 as t ! 1 for all initial states x(0), q(0) and for every input u( ). ^ Theorem 3.14 An observer exists i (C A) is detectable. Further, if (C A) is detectable, then a full order Luenberger observer is given by q_ = Aq + Bu + L(Cq + Du ; y) x = q ^ where L is any matrix such that A + LC is stable. (3.5) (3.6) Proof. We rst show that the detectability of (C A) is su cient for the existence of an observer. To that end, we only need to show that the so-called Luenberger observer de ned in the theorem is indeed an observer. Note that equation (3.5) for q is a simulation of the equation for x, with an additional forcing term L(Cq + Du ; y), which is a gain times the output error. Equivalent equations are q_ = (A + LC )q + Bu + LDu ; Ly x = q: ^
- 81. 64 LINEAR DYNAMICAL SYSTEMS These equations have the form allowed in the de nition of an observer. De ne the error, e := x ; x, and then simple algebra gives ^ e = (A + LC )e _ therefore e(t) ! 0 as t ! 1 for every x(0), q(0), and u(:). To show the converse, assume that (C A) is not detectable. Take the initial state x(0) in the undetectable subspace and consider a candidate observer: q_ = Mq + Nu + Hy x = Qq + Ru + Sy: ^ Take q(0) = 0 and u(t) 0. Then the equations for x and the candidate observer are x = Ax _ q_ = Mq + HCx x = Qq + SCx: ^ Since an unobservable subspace is an A-invariant subspace containing x(0), it follows that x(t) is in the unobservable subspace for all t 0. Hence, Cx(t) = 0 for all t 0, and, consequently, q(t) 0 and x(t) 0. However, for some x(0) in the undetectable ^ subspace, x(t) does not converge to zero. Thus the candidate observer does not have the required property, and therefore, no observer exists. 2 The above Luenberger observer has dimension n, which is the dimension of the state x. It's possible to get an observer of lower dimension. The idea is this: since we can measure y ; Du = Cx, we already know x modulo Ker C , so we only need to generate the part of x in Ker C . If C has full row rank and p := dim y, then the dimension of Ker C equals n ; p, so we might suspect that we can get an observer of order n ; p. This is true. Such an observer is called a minimal order observer". We will not pursue this issue further here. The interested reader may consult Chen 1984]. Recall that, for a dynamical system described by the equations (3.1) and (3.2), if (A B ) is controllable and state x is available for feedback, then there is a state feedback u = Fx such that the closed-loop poles of the system can be arbitrarily assigned. Similarly, if (C A) is observable, then the system observer poles can be arbitrarily placed so that the state estimator x can be made to approach x arbitrarily fast. Now ^ let us consider what will happen if the system states are not available for feedback so that the estimated state has to be used. Hence, the controller has the following dynamics: _ x = (A + LC )^ + Bu + LDu ; Ly ^ x u = F x: ^
- 82. 3.6. Operations on Systems 65 Then the total system state equations are given by x = A _ BF _ ;LC A + BF + LC x ^ x : x ^ Let e := x ; x, then the system equation becomes ^ e _ A + LC 0 _ x = ;LC A + BF ^ e x ^ and the closed-loop poles consist of two parts: the poles resulting from state feedback (A + BF ) and the poles resulting from the state estimation (A + LC ). Now if (A B ) is controllable and (C A) is observable, then there exist F and L such that the eigenvalues of A + BF and A + LC can be arbitrarily assigned. In particular, they can be made to be stable. Note that a slightly weaker result can also result even if (A B ) and (C A) are only stabilizable and detectable. The controller given above is called an observer-based controller and is denoted as u = K (s)y and LC K (s) = A + BF +F + LDF ;L : 0 Now denote the open loop plant by A B G= C D then the closed-loop feedback system is as shown below: y G u K In general, if a system is stabilizable through feeding back the output y, then it is said to be output feedback stabilizable. It is clear from the above construction that a system is output feedback stabilizable if (A B ) is stabilizable and (C A) is detectable. The converse is also true and will be shown in Chapter 12. 3.6 Operations on Systems In this section, we present some facts about system interconnection. Since these proofs are straightforward, we will leave the details to the reader.
- 83. LINEAR DYNAMICAL SYSTEMS 66 Suppose that G1 and G2 are two subsystems with state space representations: B G1 = A1 D1 C1 1 B G2 = A2 D2 : C2 2 Then the series or cascade connection of these two subsystems is a system with the output of the second subsystem as the input of the rst subsystem as shown in the following diagram: G1 G2 This operation in terms of the transfer matrices of the two subsystems is essentially the product of two transfer matrices. Hence, a representation for the cascaded system can be obtained as A2 B2 A1 B1 C1 D1 2 2 A B C C2 BDD 1 1 2 1 2 B2 = 4 0 A2 C1 D1 C2 D1 D2 G1 G2 = 3 2 A 0 2 5 = 4 B1C2 A1 B2 B1 D2 D1 D2 3 5: D1 C2 C1 Similarly, the parallel connection or the addition of G1 and G2 can be obtained as 2 3 0 B1 A1 B1 + A2 B2 = 4 A1 A 0 B2 5 : G1 + G2 = C D 2 C2 D2 1 1 C1 C2 D1 + D2 Next we consider a feedback connection of G1 and G2 as shown below: y r f G1 6 ; - G2 Then the closed-loop transfer matrix from r to y is given by 2 A ; B D R;1C ; ; 3 ;B1 R211 C2 B1 R211 1 1 2 12 1 ; ; ; B2 R121 C1 A2 ; B2 D1 R211 C2 B2 D1 R211 5 T =4 ;1 ;1 ;1 R12 C1 ;R12 D1 C2 D1 R21 where R12 = I + D1 D2 and R21 = I + D2 D1 . Note that these state space representations may not be necessarily controllable and/or observable even if the original subsystems G1 and G2 are. For future reference, we shall also introduce the following de nitions.
- 84. 3.6. Operations on Systems 67 De nition 3.7 The transpose of a transfer matrix G(s) or the dual system is de ned as G 7;! GT (s) = B (sI ; A );1 C + D or equivalently A C A B C D 7;! B D : De nition 3.8 The conjugate system of G(s) is de ned as G 7;! G (s) := GT (;s) = B (;sI ; A );1 C + D or equivalently A B ;A ;C C D 7;! B D In particular, we have G (j!) := G(j!)] = G (j!). : ^ De nition 3.9 A real rational matrix G(s) is called a right (left) inverse of a transfer ^ ^ ^ matrix G(s) if G(s)G(s) = I ( G(s)G(s) = I ). Moreover, if G(s) is both a right inverse and a left inverse of G(s), then it is simply called the inverse of G(s). Lemma 3.15 Let Dy denote a right (left) inverse of D if D has full row (column) rank. Then y y Gy = A ; BD C ;BD yC y D D is a right (left) inverse of G. Proof. The right inverse case will be proven and the left inverse case follows by duality. Suppose DDy = I . Then GGy = = 2 A BDy C 4 0 A ; BDy C yC 2 C DDy C A BD 4 0 A ; BDy C C 3 BDy ;BDy 5 DDy 3 BDy ;BDy 5 : I C Performing similarity transformation T = I I on the above system yields 0 I 3 2A 0 0 GGy = 4 0 A ; BDy C ;BDy 5 C 0 I = I: 2
- 85. LINEAR DYNAMICAL SYSTEMS 68 3.7 State Space Realizations for Transfer Matrices In some cases, the natural or convenient description for a dynamical system is in terms of transfer matrices. This occurs, for example, in some highly complex systems for which the analytic di erential equations are too hard or too complex to write down. Hence, certain engineering approximation or identi cation has to be carried out for example, input and output frequency responses are obtained from experiments so that some transfer matrix approximating the system dynamics is available. Since the state space computation is most convenient to implement on the computer, some appropriate state space representation for the resulting transfer matrix is necessary. In general, assume that G(s) is a real-rational transfer matrix which is proper. Then we call a state-space model (A B C D) such that A B G(s) = C D a realization of G(s). We note that if the transfer matrix is either single input or single output, then the formulas in Section 3.4 can be used to obtain a controllable or observable realization. The realization for a general MIMO transfer matrix is more complicated and is the focus of this section. De nition 3.10 A state space realization (A B C D) of G(s) is said to be a minimal realization of G(s) if A has the smallest possible dimension. Theorem 3.16 A state space realization (A B C D) of G(s) is minimal if and only if (A B ) is controllable and (C A) is observable. Proof. We shall rst show that if (A B C D) is a minimal realization of G(s), then (A B ) must be controllable and (C A) must be observable. Suppose, on the contrary, that (A B ) is not controllable and/or (C A) is not observable. Then from Kalman decomposition, there is a smaller dimensioned controllable and observable state space realization that has the same transfer matrix this contradicts the minimality assumption. Hence (A B ) must be controllable and (C A) must be observable. Next we show that if an n-th order realization (A B C D) is controllable and observable, then it is minimal. But supposing it is not minimal, let (Am Bm Cm D) be a minimal realization of G(s) with order k < n. Since G(s) = C (sI ; A);1 B + D = Cm (sI ; Am );1 Bm + D we have This implies that CAi B = Cm Aim Bm 8i 0: OC = Om Cm (3:7)
- 86. 3.7. State Space Realizations for Transfer Matrices 69 where C and O are the controllability and observability matrices of (A B ) and (C A), respectively, and Cm := Om := An;1 Bm m m 2 BmC AmB3 6 Cmmm 7 6 .A 7 : 6 .. 7 4 5 Cm An;1 m By Sylvester's inequality, rank C + rank O ; n rank (OC ) minfrank C rank Og and, therefore, we have rank (OC ) = n since rank C = rank O = n by the controllability and observability assumptions. Similarly, since (Am Bm Cm D) is minimal, (Am Bm ) is controllable and (Cm Am ) is observable. Moreover, rank Om Cm = k < n which is a contradiction since rank OC = rank Om Cm by equality (3.7). 2 The following property of minimal realizations can also be veri ed, and this is left to the reader. Theorem 3.17 Let (A1 B1 C1 D) and (A2 B2 C2 D) be two minimal realizations of a real rational transfer matrix G(s), and let C1 , C2 , O1 , and O2 be the corresponding controllability and observability matrices, respectively. Then there exists a unique nonsingular T such that A2 = TA1T ;1 B2 = TB1 C2 = C1 T ;1: Furthermore, T can be speci ed as T = (O2 O2 );1 O2 O1 or T ;1 = C1 C2 (C2 C2 );1 . We now describe several ways to obtain a state space realization for a given multiple input and multiple output transfer matrix G(s). The simplest and most straightforward way to obtain a realization is by realizing each element of the matrix G(s) and then combining all these individual realizations to form a realization for G(s). To illustrate, let us consider a 2 2 (block) transfer matrix such as G(s) = G1 (s) G2 (s) G3 (s) G4 (s) and assume that Gi (s) has a state space realization of B Gi (s) = Ai Di Ci i i = 1 : : : 4:
- 87. LINEAR DYNAMICAL SYSTEMS 70 Note that Gi (s) may itself be a multiple input and multiple output transfer matrix. In particular, if Gi (s) is a column or row vector of transfer functions, then the formulas in Section 3.4 can be used to obtain a controllable or observable realization for Gi (s). Then a realization for G(s) can be given by 2A 0 0 0 B 0 3 6 01 A2 0 0 01 B2 7 6 0 0 A 0 B 0 7 6 7 G(s) = 6 0 0 03 A4 03 B4 7 : 6 6C C 0 0 D D 7 7 4 1 2 1 2 5 0 0 C3 C4 D3 D4 Alternatively, if the transfer matrix G(s) can be factored into the product and/or the sum of several simply realized transfer matrices, then a realization for G can be obtained by using the cascade or addition formulas in the last section. A problem inherited with these kinds of realization procedures is that a realization thus obtained will generally not be minimal. To obtain a minimal realization, a Kalman controllability and observability decomposition has to be performed to eliminate the uncontrollable and/or unobservable states. (An alternative numerically reliable method to eliminate uncontrollable and/or unobservable states is the balanced realization method which will be discussed later.) We will now describe one factorization procedure that does result in a minimal realization by using partial fractional expansion (The resulting realization is sometimes called Gilbert's realization due to Gilbert). Let G(s) be a p m transfer matrix and write it in the following form: G(s) = N((ss)) d with d(s) a scalar polynomial. For simplicity, we shall assume that d(s) has only real and distinct roots i 6= j if i 6= j and d(s) = (s ; 1 )(s ; 2 ) (s ; r ): Then G(s) has the following partial fractional expansion: G(s) = D + r X Wi s; i: i=1 Suppose rank Wi = ki and let Bi 2 Rki m and Ci 2 Rp ki be two constant matrices such that Wi = Ci Bi :
- 88. 3.8. Lyapunov Equations 71 Then a realization for G(s) is given by 2 I 6 1k G(s) = 6 6 4 1 C1 ... 3 .. 7 . 7: 7 Br 5 B1 r Ikr Cr D It follows immediately from PBH tests that this realization is controllable and observable. Hence, it is minimal. An immediate consequence of this minimal realization is that a transfer matrix with an r-th order polynomial denominator does not necessarily have an r-th order state space realization unless Wi for each i is a rank one matrix. This approach can, in fact, be generalized to more complicated cases where d(s) may have complex and/or repeated roots. Readers may convince themselves by trying some simple examples. 3.8 Lyapunov Equations Testing stability, controllability, and observability of a system is very important in linear system analysis and synthesis. However, these tests often have to be done indirectly. In that respect, the Lyapunov theory is sometimes useful. Consider the following Lyapunov equation A X + XA + Q = 0 (3:8) with given real matrices A and Q. It has been shown in Chapter 2 that this equation has a unique solution i i (A) + j (A) 6= 0 8i j . In this section, we will study the relationships between the stability of A and the solution of X . The following results are standard. Lemma 3.18 Assume that A is stable, then the following statements hold: R (i) X = 01 eA t QeAtdt. (ii) X > 0 if Q > 0 and X 0 if Q 0. (iii) if Q 0, then (Q A) is observable i X > 0. An immediate consequence of part (iii) is that, given a stable matrix A, a pair (C A) is observable if and only if the solution to the following Lyapunov equation is positive de nite: A Lo + Lo A + C C = 0: The solution Lo is called the observability Gramian. Similarly, a pair (A B ) is controllable if and only if the solution to ALc + LcA + BB = 0
- 89. LINEAR DYNAMICAL SYSTEMS 72 is positive de nite and Lc is called the controllability Gramian. In many applications, we are given the solution of the Lyapunov equation and need to conclude the stability of the matrix A. Lemma 3.19 Suppose X is the solution of the Lyapunov equation (3.8), then (i) Re i (A) 0 if X > 0 and Q 0. (ii) A is stable if X > 0 and Q > 0. (iii) A is stable if X 0, Q 0 and (Q A) is detectable. Proof. Let be an eigenvalue of A and v 6= 0 be a corresponding eigenvector, then Av = v. Pre-multiply equation (3.8) by v and postmultiply (3.8) by v to get 2Re (v Xv) + v Qv = 0: Now if X > 0 then v Xv > 0, and it is clear that Re 0 if Q 0 and Re < 0 if Q > 0. Hence (i) and (ii) hold. To see (iii), we assume Re 0. Then we must have v Qv = 0, i.e., Qv = 0. This implies that is an unstable and unobservable mode, which contradicts the assumption that (Q A) is detectable. 2 3.9 Balanced Realizations Although there are in nitely many di erent state space realizations for a given transfer matrix, some particular realizations have proven to be very useful in control engineering and signal processing. Here we will only introduce one class of realizations for stable transfer matrices that are most useful in control applications. To motivate the class of realizations, we rst consider some simple facts. A B Lemma 3.20 Let C D be a state space realization of a (not necessarily stable) transfer matrix G(s). Suppose that there exists a symmetric matrix P = P = P1 0 0 0 with P1 nonsingular such that AP + PA + BB = 0: Now partition the realization (A B C D) compatibly with P as 2A A B 3 12 4 A11 A22 B1 5 : 21 2 C1 C2 D
- 90. 3.9. Balanced Realizations 73 Then A11 B1 C1 D is also a realization of G. Moreover, (A11 B1 ) is controllable if A11 is stable. Proof. Use the partitioned P and (A B C ) to get A 0 = AP + PA + BB = A11 P1 +PP1+ 11 + B1 B1 P1 A21 + B1 B2 A21 1 B2 B1 B2 B2 which gives B2 = 0 and A21 = 0 since P1 is nonsingular. Hence, part of the realization is not controllable: 2A A B 3 2A A B 3 11 12 4 A21 A22 B1 5 = 4 011 A12 01 5 = A11 B1 : 2 22 C1 D C1 C2 D C1 C2 D Finally, it follows from Lemma 3.18 that (A11 B1 ) is controllable if A11 is stable. 2 We also have A B Lemma 3.21 Let C D be a state space realization of a (not necessarily stable) transfer matrix G(s). Suppose that there exists a symmetric matrix Q = Q = Q1 0 0 0 with Q1 nonsingular such that QA + A Q + C C = 0: Now partition the realization (A B C D) compatibly with Q as 2A A B 3 12 4 A11 A22 B1 5 : 21 2 C1 C2 D Then A11 B1 C1 D is also a realization of G. Moreover, (C1 A11 ) is observable if A11 is stable. The above two lemmas suggest that to obtain a minimal realization from a stable non-minimal realization, one only needs to eliminate all states corresponding to the zero block diagonal term of the controllability Gramian P and the observability Gramian Q. In the case where P is not block diagonal, the following procedure can be used to eliminate non-controllable subsystems:
- 91. LINEAR DYNAMICAL SYSTEMS 74 A B 1. Let G(s) = C D be a stable realization. 2. Compute the controllability Gramian P 0 from AP + PA + BB = 0: 3. Diagonalize P to get P = U1 U2 U1 U2 unitary. 1 1 AU 4. Then G(s) = UCU 1 UDB 1 1 0 0 0 U1 U2 with 1 > 0 and is a controllable realization. A dual procedure can also be applied to eliminate non-observable subsystems. Now assume that 1 > 0 is diagonal and 1 = diag( 11 12 ) such that max ( 12 ) min ( 11 ), then it is tempting to conclude that one can also discard those states corresponding to 12 without causing much error. However, this is not necessarily true as shown in the following example. Consider a stable transfer function + G(s) = s2 3s 3s 18 18 : + + Then G(s) has a state space realization given by 2 ;1 ;4= ;2 G(s) = 4 4 1 2 0 ;1 2= 3 5 where is any nonzero number. It is easy to check that the controllability Gramian of the realization is given by P = 0:5 2 : Since the last diagonal term of P can be made arbitrarily small by making small, the controllability of the corresponding state can be made arbitrarily weak. If the state corresponding to the last diagonal term of P is removed, we get a transfer function ;1 ^ G = ;1 1 = s;11 0 + which is not close to the original transfer function in any sense. The problem may be easily detected if one checks the observability Gramian Q, which is Q = 0:5 1= 2 :
- 92. 3.9. Balanced Realizations 75 Since 1= 2 is very large if is small, this shows that the state corresponding to the last diagonal term is strongly observable. This example shows that controllability (or observability) Gramian alone can not give an accurate indication of the dominance of the system states in the input/output behavior. This motivates the introduction of a balanced realization which gives balanced Gramians for controllability and observability. A B Suppose G = C D is stable, i.e., A is stable. Let P and Q denote the controllability Gramian and observability Gramian, respectively. Then by Lemma 3.18, P and Q satisfy the following Lyapunov equations AP + PA + BB = 0 (3:9) A Q + QA + C C = 0 (3:10) and P 0, Q 0. Furthermore, the pair (A B ) is controllable i P > 0, and (C A) is observable i Q > 0. Suppose the state is transformed by a nonsingular T to x = Tx to yield the realiza^ tion " ^ ^# ;1 A B G = ^ ^ = TAT;1 TB : CT D C D ^ ^ Then the Gramians are transformed to P = TPT and Q = (T ;1 ) QT ;1. Note that ^ Q = TPQT ;1, and therefore the eigenvalues of the product of the Gramians are ^ P invariant under state transformation. Consider the similarity transformation T which gives the eigenvector decomposition PQ = T ;1 T = diag( 1 : : : n ): Then the columns of T ;1 are eigenvectors of PQ corresponding to the eigenvalues f i g. Later, it will be shown that PQ has a real diagonal Jordan form and that 0, which are consequences of P 0 and Q 0. Although the eigenvectors are not unique, in the case of a minimal realization they can always be chosen such that ^ P = TPT = ^ Q = (T ;1) QT ;1 = where = diag( 1 2 : : : n ) and 2 = . This new realization with controllability ^ ^ and observability Gramians P = Q = will be referred to as a balanced realization (also called internally balanced realization). The decreasingly order numbers, 1 2 ::: n 0, are called the Hankel singular values of the system. More generally, if a realization of a stable system is not minimal, then there is a transformation such that the controllability and observability Gramians for the transformed realization are diagonal and the controllable and observable subsystem is balanced. This is a consequence of the following matrix fact.
- 93. LINEAR DYNAMICAL SYSTEMS 76 Theorem 3.22 Let P and Q be two positive semide nite matrices. Then there exists a nonsingular matrix T such that 2 6 TPT = 6 4 respectively, with 1 1 2 2 0 3 7 7 5 2 6 (T ;1) QT ;1 = 6 4 1 0 3 0 3 diagonal and positive de nite. 0 3 7 7 5 Proof. Since P is a positive semide nite matrix, there exists a transformation T1 such that Now let T1PT1 = I 0 : 0 0 (T1 );1 QT1;1 = Q11 Q12 Q Q 22 12 and there exists a unitary matrix U1 such that 2 U1 Q11 U1 = 01 0 0 Let 0 (T2 );1 = U1 I 0 and then 2 (T2 );1 (T1 );1 QT1;1(T2 );1 = 4 ^ But Q 0 implies Q122 = 0. So now let 2 (T3 );1 = 4 1 > 0: 2 1 0 0 0 ^ ^ Q121 Q122 I 0 ^ 121 ;2 ;Q 1 0 0 I 0 0 I 3 ^ Q121 ^ Q122 5 : Q22 3 5 giving 2 (T3 );1 (T2 );1 (T1 );1 QT1;1(T2 );1 (T3 );1 = 4 2 1 0 0 0 0 0 ^ 1 ^ 0 0 Q22 ; Q121 ;2 Q121 Next nd a unitary matrix U2 such that ^ 1 ^ U2 (Q22 ; Q121 ;2 Q121 )U2 = 3 0 0 0 3 > 0: 3 5:
- 94. 3.9. Balanced Realizations 77 2 (T4 );1 = 4 De ne and let Then with ;1=2 0 1 I 0 0 0 0 0 U2 3 5 T = T4 T3 T2 T1: 2 6 TPT = 6 4 1 2 0 0 3 7 7 5 2 6 (T );1 QT ;1 = 6 4 1 0 3 0 3 7 7 5 2 2 = I. Corollary 3.23 The product of two positive semi-de nite matrices is similar to a positive semi-de nite matrix. Proof. Let P and Q be any positive semi-de nite matrices. Then it is easy to see that with the transformation given above TPQT ;1 = 2 1 0 0 0 : 2 A B Corollary 3.24 For any stable system G = C D , there exists a nonsingular ;1 T such that G = TAT;1 TB CT D Gramian Q given by 2 6 P =6 4 respectively, with 1 1 2 2 3 0 has controllability Gramian P and observability 0 3 2 7 Q=6 7 6 5 4 1 0 3 0 3 7 7 5 diagonal and positive de nite. A B In the special case where C D is a minimal realization, a balanced realization can be obtained through the following simpli ed procedure:
- 95. LINEAR DYNAMICAL SYSTEMS 78 1. Compute the controllability and observability Gramians P > 0 Q > 0. 2. Find a matrix R such that P = R R. 3. Diagonalize RQR to get RQR = U 2 U . ;1 4. Let T ;1 = R U ;1=2 . Then TPT = (T );1 QT ;1 = and TAT;1 TB CT D is balanced. Assume that the Hankel singular values of the system is decreasingly ordered so that = diag( 1 2 : : : n ) and 1 2 : : : n and suppose r r+1 for some r then the balanced realization implies that those states corresponding to the singular values of r+1 : : : n are less controllable and observable than those states corresponding to 1 : : : r . Therefore, truncating those less controllable and observable states will not lose much information about the system. These statements will be made more concrete in Chapter 7 where error bounds will be derived for the truncation error. Two other closely related realizations are called input normal realization with P = I and Q = 2 , and output normal realization with P = 2 and Q = I . Both realizations can be obtained easily from the balanced realization by a suitable scaling on the states. 3.10 Hidden Modes and Pole-Zero Cancelation Another important issue associated with the realization theory is the problem of uncontrollable and/or unobservable unstable modes in the dynamical system. This problem is illustrated in the following example: Consider a series connection of two subsystems as shown in the following diagram y s;1 s+1 1 s;1 u The transfer function for this system, 1 1 g(s) = s ; 1 s ; 1 = s + 1 s+1 is stable and has a rst order minimal realization. On the other hand, let x1 = y x2 = u ; : Then a state space description for this dynamical system is given by x1 = 1 ;1 _ x1 + 1 u x2 _ 0 ;1 x2 2 y = 1 0 x1 x2
- 96. 3.10. Hidden Modes and Pole-Zero Cancelation 79 and is a second order system. Moreover, it is easy to show that the unstable mode 1 is uncontrollable but observable. Hence, the output can be unbounded if the initial state x1 (0) is not zero. We should also note that the above problem does not go away by changing the interconnection order: y s;1 s+1 1 s;1 u In the later case, the unstable mode 1 becomes controllable but unobservable. The unstable mode can still result in the internal signal unbounded if the initial state (0) is not zero. Of course, there are fundamental di erences between these two types of interconnections as far as control design is concerned. For instance, if the state is available for feedback control, then the latter interconnection can be stabilized while the former cannot be. This example shows that we must be very careful in canceling unstable modes in the procedure of forming a transfer function in control designs otherwise the results obtained may be misleading and those unstable modes become hidden modes waiting to blow. One observation from this example is that the problem is really caused by the unstable zero of the subsystem s;1 . Although the zeros of an SISO transfer function s+1 are easy to see, it is not quite so for an MIMO transfer matrix. In fact, the notion of system zero" cannot be generalized naturally from the scalar transfer function zeros. For example, consider the following transfer matrix 2 1 1 3 6 s+1 s+2 7 7 G(s) = 6 4 2 1 5 s+2 s+1 which is stable and each element of G(s) has no nite zeros. Let 2 s+2 1 3 p ; s +p 5 K =4 s; 2 s; 2 0 1 which is unstable. However, 3 2 p s+ 2 6 ; (s + 1)(s + 2) 0 7 7 KG = 6 7 6 4 2 1 5 s+2 s+1 p is stable. This implies that G(s) must have an unstable zero at 2 that cancels the unstable pole of K . This leads us to the next topic: multivariable system poles and zeros.
- 97. 80 LINEAR DYNAMICAL SYSTEMS 3.11 Multivariable System Poles and Zeros Let R s] denote the polynomial ring with real coe cients. A matrix is called a polynomial matrix if every element of the matrix is in R s]. A square polynomial matrix is called a unimodular matrix if its determinant is a nonzero constant (not a polynomial of s). It is a fact that a square polynomial matrix is unimodular if and only if it is invertible in R s], i.e., its inverse is also a polynomial matrix. De nition 3.11 Let Q(s) 2 R s] be a (p m) polynomial matrix. Then the normal rank of Q(s), denoted normalrank (Q(s)), is the rank in R s] or, equivalently, is the maximum dimension of a square submatrix of Q(s) with nonzero determinant in R s]. In short, sometimes we say that a polynomial matrix Q(s) has rank(Q(s)) in R s] when we refer to the normal rank of Q(s). To show the di erence between the normal rank of a polynomial matrix and the rank of the polynomial matrix evaluated at certain point, consider 1 Q(s) = ss2 1 : Then Q(s) has normal rank 2 since det Q(s) = s ; s2 6 0. However, Q(0) has rank 1. It is a fact in linear algebra that any polynomial matrix can be reduced to a socalled Smith form through some pre- and post- unimodular operations. cf. Kailath, 1984, pp.391]. Lemma 3.25 (Smith form) Let P (s) 2 R s] be any polynomial matrix, then there exist unimodular matrices U (s) V (s) 2 R s] such that 2 (s) 0 3 0 0 1 6 0 2 (s) 0 07 6 . 6 . . .. . . 7 . . 7 U (s)P (s)V (s) = S (s) := 6 . . . . . . 7 6 0 7 4 0 (s) 0 5 r 0 0 0 0 and i (s) divides i+1 (s). S (s) is called the Smith form of P (s). It is also clear that r is the normal rank of P (s). We shall illustrate the procedure of obtaining a Smith form by an example. Let 2 s + 1 (s + 1)(2s + 1) s(s + 1) 3 P (s) = 4 s + 2 (s + 2)(s2 + 5s + 3) s(s + 2) 5 : 1 2s + 1 s The polynomial matrix P (s) has normal rank 2 since + + det(P (s)) 0 det s + 1 (s (s 2)(1)(2s 5s 1) 3) = (s + 1)2 (s + 2)2 6 0: s + 2 + s2 + +
- 98. 3.11. Multivariable System Poles and Zeros 81 First interchange the rst row and the third row and use row elementary operation to zero out the s+1 and s+2 elements of P (s). This process can be done by pre-multiplying a unimodular matrix U to P (s): 20 0 1 3 U = 4 0 1 ;(s + 2) 5 : 1 0 ;(s + 1) Then 21 3 2s + 1 s P1 (s) := U (s)P (s) = 4 0 (s + 1)(s + 2)2 0 5 : 0 0 0 Next use column operation to zero out the 2s + 1 and s terms in P1 . This process can be done by post-multiplying a unimodular matrix V to P1 (s): 2 1 ;(2s + 1) ;s 3 1 0 5 V (s) = 4 0 0 0 1 and 21 3 0 0 P1 (s)V (s) = 4 0 (s + 1)(s + 2)2 0 5 : 0 0 0 Then we have 21 3 0 0 S (s) = U (s)P (s)V (s) = 4 0 (s + 1)(s + 2)2 0 5 : 0 0 0 Similarly, let Rp (s) denote the set of rational proper transfer matrices.1 Then any real rational transfer matrix can be reduced to a so-called McMillan form through some pre- and post- unimodular operations. Lemma 3.26 (McMillan form) Let G(s) 2 Rp(s) be any proper real rational transfer matrix, then there exist unimodular matrices U (s) V (s) 2 R s] such that 3 2 1 (s) 0 0 0 1 (s) 6 0 2((ss)) 0 07 7 6 7 6 . 2. . U (s)G(s)V (s) = M (s) := 6 . . .. . . 7 . . 7 6 . . . . 7 6 r (s) 5 4 0 0 r (s) 0 0 0 0 0 and i (s) divides i+1 (s) and i+1 (s) divides i (s). 1 It is obvious that a similar notion of normal rank can be extended to a matrix in R ( ). In particular, let ( ) 2 R ( ) be a rational matrix and write ( ) = ( ) ( ) such that ( ) is a p s G s p s G s N s =d s d s polynomial and ( ) is a polynomial matrix, then ( ) is said to have normal rank if ( ) has normal rank . N s r G s r N s
- 99. LINEAR DYNAMICAL SYSTEMS 82 Proof. If we write the transfer matrix G(s) as G(s) = N (s)=d(s) such that d(s) is a scalar polynomial and N (s) is a p m polynomial matrix and if let the Smith form of N (s) be S (s) = U (s)N (s)V (s), the conclusion follows by letting M (s) = S (s)=d(s). 2 De nition 3.12 The number Pi deg( i (s)) is called the McMillan degree of G(s) where deg( i (s)) denotes the degree of the polynomial i (s), i.e., the highest power of s in i (s). The McMillan degree of a transfer matrix is closely related to the dimension of a minimal realization of G(s). In fact, it can be shown that the dimension of a minimal realization of G(s) is exactly the McMillan degree of G(s). De nition 3.13 The roots of all the polynomials i(s) in the McMillan form for G(s) are called the poles of G. Let (A B C D) be a minimal realization of G(s). Then it is fairly easy to show that a complex number is a pole of G(s) if and only if it is an eigenvalue of A. De nition 3.14 The roots of all the polynomials i(s) in the McMillan form for G(s) are called the transmission zeros of G(s). A complex number z0 2 C is called a blocking zero of G(s) if G(z0 ) = 0. It is clear that a blocking zero is a transmission zero. Moreover, for a scalar transfer function, the blocking zeros and the transmission zeros are the same. We now illustrate the above concepts through an example. Consider a 3 3 transfer matrix: 3 2 2s + 1 s 1 6 (s + 1)(s + 2) (s + 1)(s + 2) (s + 1)(s + 2) 7 7 6 7 6 7: 6 1 s2 + 5s + 3 s 7 6 (s + 1)2 G(s) = 6 7 (s + 1)2 (s + 1)2 7 6 7 6 5 4 1 2s + 1 s (s + 1)2 (s + 2) (s + 1)2 (s + 2) (s + 1)2 (s + 2) Then G(s) can be written as 2 s + 1 (s + 1)(2s + 1) s(s + 1) G(s) = (s + 1)1(s + 2) 4 s + 2 (s + 2)(s2 + 5s + 3) s(s + 2) 2 1 2s + 1 s 3 5 := N (s) : d(s)
- 100. 3.11. Multivariable System Poles and Zeros 83 Since N (s) is exactly the same as the P (s) in the previous example, it is clear that the G(s) has the McMillan form 3 2 1 0 07 6 (s + 1)2(s + 2) 7 6 7 6 6 s+2 0 7 M (s) = U (s)G(s)V (s) = 6 7 0 6 s+1 7 7 6 4 0 0 0 5 and G(s) has McMillan degree of 4. The poles of the transfer matrix are f;1 ;1 ;1 ;2g and the transmission zero is f;2g. Note that the transfer matrix has pole and zero at the same location f;2g this is the unique feature of multivariable systems. To get a minimal state space realization for G(s), note that G(s) has the following partial fractional expansion: 2 3 2 3 20 0 03 1 4 1 5 1 ;1 ;1 + 1 4 0 5 0 3 1 G(s) = 4 0 1 0 5 + s + 1 0 s+1 1 0 0 0 0 0 2 3 2 3 1 405 5 ;1 3 2 + (s + 1)2 1 1 2 ;1 3 1 + s + 2 4 0 5 1 ;3 ;2 0 1 +s + 1 4 0 1 1 ;1 ;1 1 Since there are repeated poles at ;1, the Gilbert's realization procedure described in the last section cannot be used directly. Nevertheless, a careful inspection of the fractional expansion results in a 4-th order minimal state space realization: 2 ;1 0 1 0 0 3 1 3 6 0 ;1 1 0 ;1 3 2 7 6 0 0 ;1 0 1 ;1 ;1 7 6 7 6 0 0 0 ;2 1 ;3 ;2 7 : G(s) = 6 7 6 0 0 1 ;1 0 0 0 7 6 7 6 1 0 0 0 0 1 0 7 5 4 0 1 0 1 0 0 0 We remind readers that there are many di erent de nitions of system zeros. The de nitions introduced here are the most common ones and are useful in this book. Lemma 3.27 Let G(s) be a p m proper transfer matrix with full column normal rank. Then z0 2 C is a transmission zero of G(s) if and only if there exists a 0 6= u0 2 C m such that G(z0 )u0 = 0.
- 101. LINEAR DYNAMICAL SYSTEMS 84 Proof. We shall outline a proof of this lemma. We shall rst show that there is a vector u0 2 C m such that G(z0 )u0 = 0 if z0 2 C is a transmission zero. Without loss of generality, assume 2 6 6 6 G(s) = U1 (s) 6 6 6 4 1 1 (s) (s) 0 0 .. . 0 0 2 2 (s) (s) .. . 0 0 ... 0 0 .. . m (s) m (s) 0 3 7 7 7 V (s) 7 1 7 7 5 for some unimodular matrices U1 (s) and V1 (s) and suppose z0 is a zero of 1 (s), i.e., 1 (z0 ) = 0. Let u0 = V1;1 (z0 )e1 6= 0 where e1 = 1 0 0 : : :] 2 Rm . Then it is easy to verify that G(z0 )u0 = 0. On the other hand, suppose there is a u0 2 C m such that G(z0 )u0 = 0. Then 2 6 6 6 U1 (z0 ) 6 6 6 4 De ne Then 1 1 (z0 ) (z0 ) 0 0 .. . 0 0 2 2 (z0 ) (z0 ) .. . 0 0 ... 0 0 .. . m (z0 ) m (z0 ) 0 3 7 7 7 V (z )u = 0: 7 1 0 0 7 7 5 2u 3 6 u1 7 6 .2 7 = V1 (z0)u0 6= 0: 6 .. 7 4 5 um 2 6 6 6 4 1 (z0 )u1 2 (z0 )u2 .. . m (z0 )um 3 7 7 = 0: 7 5 This implies that z0 must be a root of one of polynomials i (s) i = 1 : : : m. 2 Note that the lemma may not be true if G(s) does not have full column normal rank. This can be seen from the following example. Consider 1 G(s) = s + 1 1 1 1 1 1 u0 = ;1 :
- 102. 3.11. Multivariable System Poles and Zeros 85 It is easy to see that G has no transmission zero but G(s)u0 = 0 for all s. It should also be noted that the above lemma applies even if z0 is a pole of G(s) although G(z0 ) is not de ned. The reason is that G(z0 )u0 may be well de ned. For example, s;1 0 G(s) = s+1 s+2 0 s;1 u0 = 1 : 0 Then G(1)u0 = 0. Therefore, 1 is a transmission zero. Similarly we have the following lemma: Lemma 3.28 Let G(s) be a p m proper transfer matrix with full row normal rank. Then z0 2 C is a transmission zero of G(s) if and only if there exists a 0 6= 0 2 C p such that 0 G(z0 ) = 0. In the case where the transmission zero is not a pole of G(s), we can give a useful alternative characterization of the transfer matrix transmission zeros. Furthermore, G(s) is not required to be full column (or row) rank in this case. The following lemma is easy to show from the de nition of zeros. Lemma 3.29 Suppose z0 2 C is not a pole of G(s). Then z0 is a transmission zero if and only if rank(G(z0 )) < normalrank(G(s)). Corollary 3.30 Let G(s) be a square m m proper transfer matrix and det G(s) 6 0. Suppose zo 2 C is not a pole of G(s). Then z0 2 C is a transmission zero of G(s) if and only if det G(z0 ) = 0. Using thep above corollary, we can con rm that the example in the last section does have a zero at 2 since 2 1 1 3 6 s + 1 s + 2 7 = 2 ; s2 : 7 det 6 4 2 1 5 (s + 1)2 (s + 2)2 s+2 s+1 Note that the above corollary may not be true if z0 is a pole of G. For example, s;1 0 G(s) = s+1 s+2 0 s;1 has a zero at 1 which is not a zero of det G(s). The poles and zeros of a transfer matrix can also be characterized in terms of its state space realizations. Let be a state space realization of G(s). A B C D
- 103. LINEAR DYNAMICAL SYSTEMS 86 De nition 3.15 The eigenvalues of A are called the poles of the realization of G(s). To de ne zeros, let us consider the following system matrix B Q(s) = A ; sI D : C De nition 3.16 A complex number z0 2 C is called an invariant zero of the system realization if it satis es B B rank A ; z0 I D < normalrank A ; sI D : C C The invariant zeros are not changed by constant state feedback since ; B B I 0 A ; z0 I B : rank A + BFDFz0 I D = rank A ; z0 I D C+ C F I = rank C D It is also clear that invariant zeros are not changed under similarity transformation. The following lemma is obvious. B Lemma 3.31 Suppose A ; sI D has full column normal rank. Then z0 2 C is C an invariant zero of a realization (A B C D) if and only if there exist 0 6= x 2 C n and u 2 C m such that A ; z0 I B x = 0: C D u Moreover, if u = 0, then z0 is also a non-observable mode. Proof. By de nition, z0 is an invariant zero if there is a vector x 6= 0 such that u A ; z0 I B C D x =0 u B since A ; sI D has full column normal rank. C On the other hand, suppose z0 is an invariant zero, then there is a vector x 6= 0 u such that A ; z0 I B x = 0: C D u We claim that x 6= 0. Otherwise, B A ; sI B has full D u = 0 or u = 0 since C D column normal rank, i.e., x = 0 which is a contradiction. u
- 104. 3.11. Multivariable System Poles and Zeros 87 Finally, note that if u = 0, then A ; z0 I x = 0 C 2 and z0 is a non-observable mode by PBH test. B Lemma 3.32 Suppose A ; sI D has full row normal rank. Then z0 2 C is an C invariant zero of a realization (A B C D) if and only if there exist 0 6= y 2 C n and v 2 C p such that A ; z0 I B = 0: y v C D Moreover, if v = 0, then z0 is also a non-controllable mode. B Lemma 3.33 G(s) has full column (row) normal rank if and only if A ; sI D C has full column (row) normal rank. Proof. This follows by noting that A ; sI B = C and D I 0 C (A ; sI );1 I A ; sI 0 B G(s) B normalrank A ; sI D = n + normalrank(G(s)): C 2 Theorem 3.34 Let G(s) be a real rational proper transfer matrix and let (A B C D) be a corresponding minimal realization. Then a complex number z0 is a transmission zero of G(s) if and only if it is an invariant zero of the minimal realization. Proof. We will give a proof only for the case that the transmission zero is not a pole of G(s). Then, of course, z0 is not an eigenvalue of A since the realization is minimal. Note that A ; sI B = I 0 A ; sI B : C D C (A ; sI );1 I 0 G(s) Since we have assumed that z0 is not a pole of G(s), we have B rank A ; z0 I D = n + rank G(z0 ): C
- 105. LINEAR DYNAMICAL SYSTEMS 88 Hence B B rank A ; z0 I D < normalrank A ; sI D C C if and only if rank G(z0 ) < normalrank G(s). Then the conclusion follows from Lemma 3.29. 2 Note that the minimality assumption is essential for the converse statement. For example, consider a transfer matrix G(s) = D (constant) and a realization of G(s) = A 0 where A is any square matrix with any dimension and C is any matrix with C D compatible dimension. Then G(s) has no poles or zeros but every eigenvalue of A is an A 0 invariant zero of the realization C D . Nevertheless, we have the following corollary if a realization of the transfer matrix is not minimal. Corollary 3.35 Every transmission zero of a transfer matrix G(s) is an invariant zero of all its realizations, and every pole of a transfer matrix G(s) is a pole of all its realizations. Lemma 3.36 Let G(s) 2 Rp(s) be a p m transfer matrix and let (A B C D) be a minimal realization. If the system input is of the form u(t) = u0 e t , where 2 C is not a pole of G(s) and u0 2 C m is an arbitrary constant vector, then the output due to the input u(t) and the initial state x(0) = ( I ; A);1 Bu0 is y(t) = G( )u0 e t 8t 0. Proof. The system response with respect to the input u(t) = u0e t and the initial condition x(0) = ( I ; A);1 Bu0 is (in terms of Laplace transform): Y (s) = C (sI ; A);1 x(0) + C (sI ; A);1 BU (s) + DU (s) = C (sI ; A);1 x(0) + C (sI ; A);1 Bu0 (s ; );1 + Du0 (s ; );1 = C (sI ; A);1 (x(0) ; ( I ; A);1 Bu0 ) + G( )u0 (s ; );1 = G( )u0 (s ; );1 : Hence y(t) = G( )u0 e t . 2 Combining the above two lemmas, we have the following results that give a dynamical interpretation of a system's transmission zero. Corollary 3.37 Let G(s) 2 Rp(s) be a p m transfer matrix and let (A B C D) be a minimal realization. Suppose that z0 2 C is a transmission zero of G(s) and is not a pole of G(s). Then for any nonzero vector u0 2 C m the output of the system due to the initial state x(0) = (z0 I ; A);1 Bu0 and the input u = u0 ez0 t is identically zero: y(t) = G(z0 )u0 ez0 t = 0.
- 106. 3.12. Notes and References 89 The following lemma characterizes the relationship between zeros of a transfer function and poles of its inverse. A B Lemma 3.38 Suppose that G = C D is a square transfer matrix with D nonsingular, and suppose z0 is not an eigenvalue of A (note that the realization is not necessarily minimal). Then there exists x0 such that (A ; BD;1 C )x0 = z0 x0 Cx0 6= 0 i there exists u0 6= 0 such that G(z0 )u0 = 0: Proof. (() G(z0)u0 = 0 implies that ;1 ;1 G;1 (s) = A ; BD C ;BD D;1 C D;1 has a pole at z0 which is observable. Then, by de nition, there exists x0 such that (A ; BD;1 C )x0 = z0 x0 and Cx0 6= 0: ()) Set u0 = ;D;1 Cx0 6= 0. Then (z0 I ; A)x0 = ;BD;1Cx0 = Bu0 : Using this equality, one gets G(z0 )u0 = C (z0 I ; A);1 Bu0 + Du0 = Cx0 ; Cx0 = 0: 2 The above lemma implies that z0 is a zero of an invertible G(s) if and only if it is a pole of G;1 (s). 3.12 Notes and References Readers are referred to Brogan 1991], Chen 1984], Kailath 1980], and Wonham 1985] for the extensive treatment of the standard linear system theory. The balanced realization was rst introduced by Mullis and Roberts 1976] to study the roundo noise in digital lters. Moore 1981] proposed the balanced truncation method for model reduction which will be considered in Chapter 7.
- 108. 4 Performance Speci cations The most important objective of a control system is to achieve certain performance speci cations in addition to providing the internal stability. One way to describe the performance speci cations of a control system is in terms of the size of certain signals of interest. For example, the performance of a tracking system could be measured by the size of the tracking error signal. In this chapter, we look at several ways of de ning a signal's size, i.e., at several norms for signals. Of course, which norm is most appropriate depends on the situation at hand. For that purpose, we shall rst introduce some normed spaces and some basic notions of linear operator theory, in particular, the Hardy spaces H2 and H1 are introduced. We then consider the performance of a system under various input signals and derive the worst possible outputs with the class of input signals under consideration. We show that H2 and H1 norms come out naturally as measures of the worst possible performance for many classes of input signals. Some state space methods of computing real rational H2 and H1 transfer matrix norms are also presented. 4.1 Normed Spaces Let V be a vector space over C (or R) and let k k be a norm de ned on V . Then V is a normed space. For example, the vector space C n with any vector p-norm, k kp , for 1 p 1, is a normed space. As another example, consider the linear vector space C a b] of all bounded continuous functions on the real interval a b]. Then C a b] 91
- 109. PERFORMANCE SPECIFICATIONS 92 becomes a normed space if a supremum norm is de ned on the space: kf k1 := sup jf (t)j: t2 a b] A sequence fxn g in a normed space V is called a Cauchy sequence if kxn ; xm k ! 0 as n m ! 1. A sequence fxn g is said to converge to x 2 V , written xn ! x, if kxn ; xk ! 0. A normed space V is said to be complete if every Cauchy sequence in V converges in V . A complete normed space is called a Banach space. For example, Rn and C n with the usual spatial p-norm, k kp for 1 p 1, are Banach spaces. (One should not be confused with the notation k kp used here and the same notation used below for function spaces because usually the context will make the meaning clear.) The following are some more examples of Banach spaces: lp 0 1) spaces for 1 p < 1: For each 1 p < 1, lp 0 1) consists of all sequences x = (x0 x1 : : :) such that 1 X p jxi j < 1. The associated norm is de ned as i=0 kxkp := 1 X i=0 jxi jp !1=p : l1 0 1) space: l1 0 1) consists of all bounded sequences x = (x0 x1 : : :), and the l1 norm is de ned as kxk1 := sup jxi j: i Lp (I ) spaces for 1 p 1: For each 1 p < 1, Lp (I ) consists of all Lebesgue measurable functions x(t) de ned on an interval I kxkp := and R Z I such that jx(t)jp dt 1=p < 1 for 1 p < 1 kx(t)k1 := ess sup jx(t)j: t2I Some of these spaces, for example, L2 (;1 0], L2 0 1) and L2 (;1 1), will be discussed in more detail later on. C a b] space: C a b] consists of all continuous functions on the real interval a b] with the norm de ned as kxk1 := sup jx(t)j: t2 a b]
- 110. 4.2. Hilbert Spaces 93 Note that if each component or function is itself a vector or matrix, then the corresponding Banach space can also be formed by replacing the absolute value j j of each component or function with its spatially normed component or function. For example, consider a vector space with all sequences in the form of x = (x0 x1 : : :) where each component xi is a k m matrix and each element of xi is bounded. Then xi is bounded in any matrix norm, and the vector space becomes a Banach space if the following norm is de ned kxk1 := sup i (xi ) i where i (xi ) := kxi k is any matrix norm. This space will also be denoted by l1 . Let V1 and V2 be two vector spaces and let T be an operator from S V1 into V2 . An operator T is said to be linear if for any x1 x2 2 S and scalars 2 C , the following holds: T ( x1 + x2 ) = (Tx1 ) + (Tx2 ): Moreover, let V0 be a linear subspace in V1 . Then the operator T0 : V0 7;! V2 de ned by T0 x = Tx for every x 2 V0 is called the restriction of T to V0 and is denoted as T jV0 = T0 . On the other hand, a linear operator T : V1 7;! V2 coinciding with T0 on V0 V1 is x1 : x 2 C n called an extension of T0. For example, let V0 := V1 = C n+m , 1 0 and let T = A1 A2 2 C (n+m) (n+m) T0 = A1 0 2 C (n+m) (n+m) : 0 A3 0 0 Then T jV0 = T0 . De nition 4.1 Two normed spaces V1 and V2 are said to be linearly isometric, denoted by V1 = V2 , if there exists a linear operator T of V1 onto V2 such that kTxk = kxk for all x in V1 . In this case, the mapping T is said to be an isometric isomorphism. 4.2 Hilbert Spaces Recall the inner product of vectors de ned on a Euclidean space C n : hx yi := x y = n X i=1 2x 3 2y 3 1 1 xi yi 8x = 6 ... 7 y = 6 ... 7 2 C n : 4 5 4 5 xn yn
- 111. PERFORMANCE SPECIFICATIONS 94 Note that many important metric notions and geometrical properties such as length, distance, angle, and the energy of physical systems can be deduced from this inner product. For instance, the length of a vector x 2 C n is de ned as p kxk := hx xi and the angle between two vectors x y 2 C n can be computed from yi cos (x y) = khx kyk (x y) 2 0 ]: xk The two vectors are said to be orthogonal if (x y) = 2 . We now consider a natural generalization of the inner product on C n to more general (possibly in nite dimensional) vector spaces. De nition 4.2 Let V be a vector space over C . An inner product on V is a complex valued function, h i : V V 7;! C such that for any x y z 2 V and 2C (i) 1 hx y + z i = hx yi + hx z i (ii) hx yi = hy xi (iii) hx xi > 0 if x 6= 0. A vector space V with an inner product is called an inner product space. p It is clear that the inner product de ned above induces a norm kxk := hx xi, so that the norm conditions in Chapter 2 are satis ed. In particular, the distance between vectors x and y is d(x y) = kx ; yk. Two vectors x and y in an inner product space V are said to be orthogonal if hx yi = 0, denoted x ? y. More generally, a vector x is said to be orthogonal to a set S V , denoted by x ? S , if x ? y for all y 2 S . The inner product and the inner-product induced norm have the following familiar properties. Theorem 4.1 Let V be an inner product space and let x y 2 V . Then (i) jhx yij kxk kyk (Cauchy-Schwarz inequality). Moreover, the equality holds if and only if x = y for some constant or y = 0. (ii) kx + yk2 + kx ; yk2 = 2 kxk2 + 2 kyk2 (Parallelogram law) . (iii) kx + yk2 = kxk2 + kyk2 if x ? y. 1 This is the other way round to the usual mathematical convention since we want to have h rather than for 2 C n. y x x y x y i= x y
- 112. 4.2. Hilbert Spaces 95 A Hilbert space is a complete inner product space with the norm induced by its inner product. Obviously, a Hilbert space is also a Banach space. For example, C n with the usual inner product is a ( nite dimensional) Hilbert space. More generally, it is straightforward to verify that C n m with the inner product de ned as hA B i := Trace A B = n m XX i=1 j =1 aij bij 8A B 2 C n m is also a ( nite dimensional) Hilbert space. Here are some examples of in nite dimensional Hilbert spaces: l2 (;1 1): l2 (;1 1) consists of the set of all real or complex square summable sequences x = (: : : x;2 x;1 x0 x1 x2 : : :) i.e, 1 X i=;1 jxi j2 < 1 with the inner product de ned as hx yi := 1 X i=;1 xi yi for x y 2 l2 (;1 1). The subspaces l2 (;1 0) and l2 0 1) of l2 (;1 1) are de ned similarly and consist of sequences of the form x = (: : : x;2 x;1 ) and x = (x0 x1 x2 : : :), respectively. Note that we can also de ne a corresponding Hilbert space even if each component xi is a vector or a matrix in fact, the following inner product will su ce: hx yi := 1 X i=;1 Trace(xi yi ): L2 (I ) for I R: L2 (I ) consists of all square integrable and Lebesgue measurable functions de ned on an interval I R with the inner product de ned as Z hf gi := f (t) g(t)dt I for f g 2 L2 (I ). Similarly, if the function is vector or matrix valued, the inner product is de ned correspondingly as Z hf gi := Trace f (t) g(t)] dt: I
- 113. PERFORMANCE SPECIFICATIONS 96 Some very often used spaces in this book are L2 0 1) L2 (;1 0] L2 (;1 1). More precisely, they are de ned as L2 = L2 (;1 1): Hilbert space of matrix-valued functions on R, with inner product hf gi := Z1 ;1 Trace f (t) g(t)] dt: L2+ = L2 0 1): subspace of L2 (;1 1) with functions zero for t < 0. L2; = L2 (;1 0]: subspace of L2 (;1 1) with functions zero for t > 0. Let H be a Hilbert space and M H a subset. Then the orthogonal complement of M , denoted by H M or M ?, is de ned as M ? = fx : hx yi = 0 8y 2 M x 2 Hg : It can be shown that M ? is closed (hence M ? is also a Hilbert space). For example, let M = L2+ L2 , then M ? = L2; is a Hilbert space. Let M and N be subspaces of a vector space V . V is said to be the direct sum of M and N , written V = M N , if M N = f0g, and every element v 2 V can be expressed as v = x + y with x 2 M and y 2 N . If V is an inner product space and M and N are orthogonal, then V is said to be the orthogonal direct sum of M and N . As an example, it is easy to see that L2 is the orthogonal direct sum of L2; and L2+ . Similarly, l2 (;1 1) is the orthogonal direct sum of l2 (;1 0) and l2 0 1). The following is a version of the so-called orthogonal projection theorem: Theorem 4.2 Let H be a Hilbert space, and let M be a closed subspace of H. Then for each vector v 2 H, there exist unique vectors x 2 M and y 2 M ? such that v = x + y, i.e., H = M M ?. Moreover, x 2 M is the unique vector such that d(v M ) = kv ; xk. Let H1 and H2 be two Hilbert spaces, and let A be a bounded linear operator from H1 into H2 . Then there exists a unique linear operator A : H2 7;! H1 such that for all x 2 H1 and y 2 H2 hAx yi = hx A yi: A is called the adjoint of A. Furthermore, A is called self-adjoint if A = A . Let H be a Hilbert space and M H be a closed subspace. A bounded operator P mapping from H into itself is called the orthogonal projection onto M if P (x + y) = x 8 x 2 M and y 2 M ? :
- 114. 4.3. Hardy Spaces H2 and H1 97 4.3 Hardy Spaces H2 and H1 Let S C be an open set, and let f (s) be a complex valued function de ned on S : f (s) : S 7;! C : Then f (s) is said to be analytic at a point z0 in S if it is di erentiable at z0 and also at each point in some neighborhood of z0 . It is a fact that if f (s) is analytic at z0 then f has continuous derivatives of all orders at z0 . Hence, a function analytic at z0 has a power series representation at z0 . The converse is also true, i.e., if a function has a power series at z0 , then it is analytic at z0 . A function f (s) is said to be analytic in S if it has a derivative or is analytic at each point of S . A matrix valued function is analytic in S if every element of the matrix is analytic in S . For example, all real rational stable transfer matrices are analytic in the right-half plane and e;s is analytic everywhere. A well known property of the analytic functions is the so-called Maximum Modulus Theorem. Theorem 4.3 If f (s) is de ned and continuous on a closed-bounded set S and analytic on the interior of S , then the maximum of jf (s)j on S is attained on the boundary of S , i.e., max jf (s)j = s2@S jf (s)j max s2S where @S denotes the boundary of S . Next we consider some frequently used complex (matrix) function spaces. L2 (j R) Space L2 (j R) or simply L2 is a Hilbert space of matrix-valued (or scalar-valued) func- tions on j R and consists of all complex matrix functions F such that the integral below is bounded, i.e., Z1 ;1 Trace F (j!)F (j!)] d! < 1: The inner product for this Hilbert space is de ned as 1 Z 1 Trace F (j!)G(j!)] d! hF Gi := 2 ;1 for F G 2 L2 , and the inner product induced norm is given by p kF k2 := hF F i: For example, all real rational strictly proper transfer matrices with no poles on the imaginary axis form a subspace (not closed) of L2 (j R) which is denoted by RL2 (j R) or simply RL2 .
- 115. PERFORMANCE SPECIFICATIONS 98 H2 Space2 H2 is a (closed) subspace of L2 (j R) with matrix functions F (s) analytic in Re(s) > 0 (open right-half plane). The corresponding norm is de ned as Z1 kF k2 := sup 21 Trace F ( + j!)F ( + j!)] d! : 2 >0 It can be shown3 that kF k2 = 21 2 ;1 Z1 ;1 Trace F (j!)F (j!)] d!: Hence, we can compute the norm for H2 just as we do for L2 . The real rational subspace of H2 , which consists of all strictly proper and real rational stable transfer matrices, is denoted by RH2 . ? H2 Space ? H2 is the orthogonal complement of H2 in L2 , i.e., the (closed) subspace of functions in L2 that are analytic in the open left-half plane. The real rational subspace ? of H2 , which consists of all strictly proper rational transfer matrices with all poles in the open right half plane, will be denoted by RH? . It is easy to see that if G 2 is a strictly proper, stable, and real rational transfer matrix, then G 2 H2 and ? G 2 H2 . Most of our study in this book will be focused on the real rational case. The L2 spaces de ned above in the frequency domain can be related to the L2 spaces de ned in the time domain. Recall the fact that a function in L2 space in the time domain admits a bilateral Laplace (or Fourier) transform. In fact, it can be shown that this bilateral Laplace (or Fourier) transform yields an isometric isomorphism between the L2 spaces in the time domain and the L2 spaces in the frequency domain (this is what is called Parseval's relations or Plancherel Theorem in complex analysis): L2 (;1 1) = L2 (j R) L2 0 1) = H2 ? L2 (;1 0] = H2 : As a result, if g(t) 2 L2 (;1 1) and if its Fourier (or bilateral Laplace) transform is G(j!) 2 L2 (j R), then kGk2 = kgk2 : 2 The H space and H space de ned below together with the H spaces, 1, which will not be 1 p 2 introduced in this book, are usually called Hardy spaces named after the mathematician G. H. Hardy (hence the notation of H). 3 See Francis 1987]. p
- 116. 4.3. Hardy Spaces H2 and H1 99 Hence, whenever there is no confusion, the notation of functions in the time domain and in the frequency domain will be used interchangeably. De ne an orthogonal projection P+ : L2 (;1 1) 7;! L2 0 1) such that, for any function f (t) 2 L2 (;1 1), we have g(t) = P+ f (t) with g(t) := f (t) for t < 0: 0 for t 0 In this book, P+ will also be used to denote the projection from L2 (j R) onto H2 . Similarly, de ne P; as another orthogonal projection from L2 (;1 1) onto L2 (;1 0] ? (or L2 (j R) onto H2 ). Then the relationships between L2 spaces and H2 spaces can be shown as in Figure 4.1. Laplace Transform L2 0 1) H2 Inverse Transform P+ 6 L2 (;1 1) P; ? L2 (;1 0] 6 P+ Laplace Transform Inverse Transform - L2 (j R) P; Laplace Transform Inverse Transform - ? ? H2 Figure 4.1: Relationships among function spaces Other classes of important complex matrix functions used in this book are those bounded on the imaginary axis. L1 (j R) Space L1 (j R) or simply L1 is a Banach space of matrix-valued (or scalar-valued) functions that are (essentially) bounded on j R, with norm kF k1 := ess sup F (j!)] : ! 2R
- 117. PERFORMANCE SPECIFICATIONS 100 The rational subspace of L1 , denoted by RL1 (j R) or simply RL1 , consists of all rational proper transfer matrices with no poles on the imaginary axis. H1 Space H1 is a (closed) subspace in L1 with functions that are analytic in the open right-half plane and bounded on the imaginary axis. The H1 norm is de ned as kF k1 := sup F (s)] = ess sup F (j!)] : Re(s)>0 ! 2R The second equality can be regarded as a generalization of the maximum modulus theorem for matrix functions. The real rational subspace of H1 is denoted by RH1 which consists of all proper and real rational stable transfer matrices. ; H1 Space ; H1 is a (closed) subspace in L1 with functions that are analytic in the open ; left-half plane and bounded on the imaginary axis. The H1 norm is de ned as kF k1 := sup F (s)] = ess sup F (j!)] : Re(s)<0 ! 2R ; The real rational subspace of H1 is denoted by RH; which consists of all proper 1 rational transfer matrices with all poles in the open right half plane. ; De nition 4.3 A transfer matrix G(s) 2 H1 is usually said to be antistable or anti- causal. Some facts about L1 and H1 functions are worth mentioning: (i) if G(s) 2 L1 , then G(s)L2 := fG(s)f (s) : f (s) 2 L2 g L2 . (ii) if G(s) 2 H1 , then G(s)H2 := fG(s)f (s) : f (s) 2 H2 g H2 . ; ? ? ? (ii) if G(s) 2 H1 , then G(s)H2 := fG(s)f (s) : f (s) 2 H2 g H2 . Remark 4.1 The notation for L1 is somewhat unfortunate it should be clear to the reader that the L1 space in the time domain and in the frequency domain denote completely di erent spaces. The L1 space in the time domain is usually used to denote signals, while the L1 space in the frequency domain is usually used to denote transfer functions and operators. ~ Let G(s) 2 L1 be a p q transfer matrix. Then a multiplication operator is de ned as MG : L2 7;! L2 MG f := Gf:
- 118. 4.3. Hardy Spaces H2 and H1 101 In writing the above mapping, we have assumed that f has a compatible dimension. A more accurate description of the above operator should be MG : Lq 7;! Lp 2 2 i.e., f is a q-dimensional vector function with each component in L2 . However, we shall suppress all dimensions in this book and assume all objects have compatible dimensions. It is easy to verify that the adjoint operator MG = MG . A useful fact about the multiplication operator is that the norm of a matrix G in L1 equals the norm of the corresponding multiplication operator. Theorem 4.4 Let G 2 L1 be a p q transfer matrix. Then kMGk = kGk1 . Remark 4.2 It is also true that this operator norm equals the norm of the operator ? restricted to H2 (or H2 ), i.e., kMGk = kMGjH2 k := sup fkGf k2 : f 2 H2 kf k2 1g : This will be clear in the proof where an f 2 H2 is constructed. ~ Proof. By de nition, we have kMGk = sup fkGf k2 : f 2 L2 kf k2 1g : First we see that kGk1 is an upper bound for the operator norm: Z1 f (j!)G (j!)G(j!)f (j!) d! kGf k2 = 21 2 ;1 Z1 kGk2 21 kf (j!)k2 d! 1 ;1 = kGk2 kf k2: 1 2 To show that kGk1 is the least upper bound, rst choose a frequency !o where G(j!)] is maximum, i.e., G(j!0 )] = kGk1 and denote the singular value decomposition of G(j!0 ) by G(j!0 ) = u1 (j!0 )v1 (j!0 ) + r X i=2 i ui (j!0 )vi (j!0 ) where r is the rank of G(j!0 ) and ui vi have unit length. If !0 < 1, write v1 (j!0 ) as 2 ej 3 6 1 ej 7 v1 (j!0 ) = 6 .. 7 6 2. 7 4 5 1 2 q ej q
- 119. PERFORMANCE SPECIFICATIONS 102 where i 2 R is such that i 2 ; 0) and q is the column dimension of G. Now let i 0 be such that i ; j!0 i= + j! i and let f be given by 2 6 6 f (s) = 6 6 4 0 ;s 1 1 +s 1 ;s 2 2 +s 2 .. . ;s q q +s q 3 7 7 f^(s) 7 7 5 where a scalar function f^ is chosen so that jf^(j!)j = c if j! ; !o j < or j! + !oj < 0 otherwise where is a small positive number and c is chosen so that f has unit 2-norm, i.e., p =2 . This in turn implies that f has unit 2-norm. Then^ c= kGf k2 2 = 1 2 h G(;j!o )]2 + G(j!o )]2 G(j!o )]2 = kGk2 : 1 i If !0 = 1, then G(j!0 ) is real and v1 is real. In this case, the conclusion follows by letting f (s) = v1 f^(s). 2 4.4 Power and Spectral Signals In this section, we introduce two additional classes of signals that have been widely used in engineering. These classes of signals have some nice statistical and frequency domain representations. Let u(t) be a function of time. De ne its autocorrelation matrix as Ruu ( ) := Tlim 21 !1 T ZT ;T u(t + )u (t)dt if the limit exists and is nite for all . It is easy to see from the de nition that Ruu ( ) = Ruu (; ). Assume further that the Fourier transform of the signal's autocorrelation matrix function exists (and may contain impulses). This Fourier transform is called the spectral density of u, denoted Suu (j!): Suu (j!) := Z1 ;1 Ruu ( )e;j! d :
- 120. 4.4. Power and Spectral Signals 103 Thus Ruu ( ) can be obtained from Suu (j!) by performing an inverse Fourier transform: Z1 Ruu ( ) := 21 Suu (j!)ej! d!: ;1 We will call signal u(t) a power signal if the autocorrelation matrix Ruu ( ) exists and is nite for all , and moreover, if the power spectral density function Suu (j!) exists (note that Suu (j!) need not be bounded and may include impulses). The power of the signal is de ned as s 1 Z T ku(t)k2 dt = pTrace R (0)] kukP = Tlim 2T uu !1 ;T where k k is the usual Euclidean norm and the capital script P is used to di erentiate this power semi-norm from the usual Lebesgue Lp norm. The set of all signals having nite power will be denoted by P . The power semi-norm of a signal can also be computed from its spectral density function Z1 2 = 1 Trace Suu (j!)]d!: kukP 2 ;1 This expression implies that, if u 2 P , Suu is strictly proper in the sense that Suu (1) = 0. We note that if u 2 P and ku(t)k1 := supt ku(t)k < 1, then kukP kuk1. However, not every signal having nite 1-norm is a power signal since the limit in the de nition may not exist. For example, let 2k 2k+1 for k = 0 1 2 : : : u(t) = 1 2 < t <:2 0 otherwise R T Then limT !1 21 ;T u2 dt does not exist. Note also that power signals are persistent T signals in time such as sines or cosines clearly, a time-limited signal has zero power, as does an L2 signal. Thus k kP is only a semi-norm, not a norm. Now let G be a linear system transfer matrix with convolution kernel g(t), input u(t), and output z (t). Then Rzz ( ) = g( ) Ruu ( ) g (; ) and Szz (j!) = G(j!)Suu (j!)G (j!). These properties are useful in establishing some input and output relationships in the next section. A signal u(t) is said to have bounded power spectral density if kSuu (j!)k1 < 1. The set of signals having bounded spectral density is denoted as S := fu(t) 2 Rm : kSuu (j!)k1 < 1g: p The quantity kuks := kSuu (j!)k is called the spectral density norm of u(t). The set S can be used to model signals with xed spectral characteristics by passing white noise signals through a weighting lter. Similarly, P could be used to model signals whose spectrum is not known but which are nite in power.
- 121. PERFORMANCE SPECIFICATIONS 104 4.5 Induced System Gains Many control engineering problems involve keeping some system signals small" under various conditions, for example, under a set of possible disturbances and system parameter variations. In this section we are interested in answering the following question: if we know how big" the input (disturbance) is, how big" is the output going to be for a given stable dynamical system? Consider a q-input and p-output linear nite dimensional system as shown in the following diagram with input u, output z , and transfer matrix G 2 RH1 : z u G We will further assume that G(s) is strictly proper, i.e, G(1) = 0 although most of the results derived here hold for the non-strictly proper case. In the time-domain an input-output model for such a system has the form of a convolution equation, z=g u i.e., Zt z (t) = g(t ; )u( ) d 0 where the p q real matrix g(t) is the convolution kernel. Let the convolution kernel and the corresponding transfer matrix be partitioned as 2 g (t ) 3 2 g (t) 3 g1q (t) 11 1 6 ... .. 7 = 6 .. 7 g(t) = 4 . 5 4 . 5 gp1 (t) gpq (t) gp (t) 2 G (s) 6 11... G(s) = 4 Gp1 (s) G1q (s) .. . Gpq (s) 3 2 G (s) 3 7 = 6 1... 7 5 4 5 Gp (s) where gi (t) is a q-dimensional row vector of the convolution kernel and Gi (s) is a row vector of the transfer matrix. Now if G is considered as an operator from the input space to the output space, then a norm is induced on G, which, loosely speaking, measures the size of the output for a given input u. These norms can determine the achievable performance of the system for di erent classes of input signals The various input/output relationships, given di erent classes of input signals, are summarized in two tables below. Table 4.1 summarizes the results for xed input signals. Note that the (t) in this table denotes the unit impulse and u0 2 Rq is a constant vector indicating the direction of the input signal. We now prove Table 4.1.
- 122. 4.5. Induced System Gains 105 Input u(t) Output z (t) kz k2 = kGu0 k2 = kgu0k2 u(t) = u0 (t) u0 2 Rq kz k1 = kgu0k1 kz kP = 0 kz k2 = 1 u(t) = u0 sin !0 t u0 2 Rq lim sup max jzi (t)j = kG(j!0 )u0 k1 = max jGi (j!0 )u0 j i i t!1 lim sup kz (t)k = kG(j!0 )u0 k t!1 1 kz kP = p kG(j!0 )u0 k 2 Table 4.1: Output norms with xed inputs Proof. u = u0 (t): If u(t) = u0 (t), then z (t) = g(t)u0, so kz k2 = kgu0k2 and kz k1 = kgu0k1 . But by Parseval's relation, kgu0k2 = kGu0k2 . On the other hand, 1 Z T u g (t)g(t)u dt 0 P = Tlim 2T ;T 0 !1 Z1 Tracefg (t)g(t)g dt ku0 k2 lim 1 T !1 2T ;1 = Tlim 21 kGk2 ku0k2 = 0: 2 !1 T kz k2 u = u0 sin !0 t: With the input u(t) = u0 sin(!0 t), the i-th output as t ! 1 is zi (t) = jGi (j!0 )u0 j sin !0 t + argfGi (j!0 )u0 g]: (4:1)
- 123. PERFORMANCE SPECIFICATIONS 106 The 2-norm of this signal is in nite as long as Gi (j!0 )u0 6= 0, i.e., the system's transfer function does not have a zero in every channel in the input direction at the frequency of excitation. The amplitude of the sinusoid (4.1) equals jGi (j!0 )u0 j. Hence lim sup max jzi (t)j = max jGi (j!0 )u0 j = kG(j!0 )u0 k1 i i and t!1 vp uX u lim sup kz (t)k = t jGi (j!0 )u0 j2 = kG(j!0 )u0 k : t!1 i=1 Finally, let i := arg Gi (j!)u0 . Then ZTX p kz k2 = Tlim 21 jG (j! )u j2 sin2 (!0 t + i ) dt P !1 T ;T i=1 i 0 0 ZT p X 2 lim 1 = jGi (j!0 )u0 j T !1 2T sin2 (!0 t + i ) dt ;T i=1 Z T 1 ; cos 2(!0t + i ) p X dt = jGi (j!0 )u0 j2 Tlim 21 !1 T ;T 2 i=1 p 1X = 2 jGi (j!0 )u0 j2 = 1 kG(j!0 )u0 k2 : 2 i=1 2 It is interesting to see what this table signi es from the control point of view. To focus our discussion, let us assume that u(t) = u0 sin !0 t is a disturbance or a command signal on a feedback system, and z (t) is the tracking error. Then we say that the system has good tracking behavior if z (t) is small in some sense, for instance, lim supt!1 kz (t)k is small. Note that lim sup kz (t)k = kG(j!0 )u0 k t!1 for any given !0 and u0 2 Rq . Now if we want to track signals from various channels, that is if u0 can be chosen to be any direction, then we would require that (G(j!0 )) be small. Furthermore, if, in addition, we want to track signals of many di erent frequencies, we then would require that (G(j!0 )) be small at all those frequencies. This interpretation enables us to consider the control system in the frequency domain even though the speci cations are given in the time domain. Table 4.2 lists the maximum possible system gains when the input signal u is not required to be a xed signal instead it can be any signal in a unit ball in some function space. Now we give a proof for Table 4.2. Note that the rst row (L2 7;! L2 ) has been shown in Section 4.3.
- 124. 4.5. Induced System Gains 107 Input u(t) Output z (t) Signal Norms kuk2 = 2 L2 L2 S S S P kuk2 = 21 P P P kuk2 = 21 P L1 Z1 0 Induced Norms kuk2 dt kGk1 kuk2 = kSuu k1 S Z1 ;1 Z1 ;1 kGk1 TracefSuu (j!)gd! kGk2 TracefSuu (j!)gd! kGk1 kuk1 = sup max jui (t)j i max kgi k1 i t L1 kuk1 = sup ku(t)k Z1 t Table 4.2: Induced System Gains Proof. S 7;! S : If u 2 S , then Szz (j!) = G(j!)Suu (j!)G (j!): Now suppose G(j!0 )] = kGk1 and take a signal u such that Suu (j!0 ) = I . Then kSzz (j!)k1 = kGk2 : 1 S 7;! P : By de nition, we have 1 Z 1 TracefG(j!)S (j!)G (j!)g d!: uu P = 2 ;1 kz k2 0 kg(t)k dt
- 125. PERFORMANCE SPECIFICATIONS 108 Now let u be white with unit spectral density, i.e., Suu = I . Then Z1 2 = 1 kz kP 2 TracefG(j!)G (j!)g d! = kGk2 : 2 ;1 P 7;! P : If u is a power signal, then Z1 2 = 1 kz kP 2 TracefG(j!)Suu (j!)G (j!)g d! ;1 and immediately, we get that kz kP kGk1 kukP : To achieve the equality, assume that !0 is such that G(j!0 )] = kGk1 and denote the singular value decomposition of G(j!0 ) by G(j!0 ) = u1 (j!0 )v1 (j!0 ) + r X i=2 i ui (j!0 )vi (j!0 ) where r is the rank of G(j!0 ) and ui vi have unit length. If !0 < 1, write v1 (j!0 ) as 2 ej 3 6 1ej 7 v1 (j!) = 6 .. 7 6 2. 7 4 5 1 2 q ej q where i 2 R is such that i 2 ; 0) and q is the column dimension of G. Now let i 0 be such that i ; j!0 i= + j! i 0 and let the input u be generated from passing u through a lter: ^ 2 6 6 u(t) = 6 6 4 ;s 1 1 1 +s 2 ;s 2 2 +s .. . ;s q q +s q 3 7 7 u(t) 7^ 7 5
- 126. 4.5. Induced System Gains 109 where p u(t) = 2 sin(!o t): ^ Then Ruu ( ) = cos(!o ), so ^^ kukP = Ruu (0) = 1: ^ ^^ Also, Then (! ; !o ) + (! + !o)] : Suu (j!) = ^^ 2 6 6 Suu (j!) = 6 6 4 and it is easy to show ;j! 1 1 1 +j! ;j! 2 2 +j! 2 .. . q ;j! q q +j! 3 2 7 7 S (j!) 6 7 uu 6 7 ^^ 6 6 5 4 ;j! 1 1 +j! 1 ;j! 2 2 +j! 2 .. . ;j! q q +j! q 3 7 7 7 7 5 kukP = 1: Finally, 1 kz k2 = 2 G(j!o )]2 + 1 G(;j!o )]2 P 2 2 = G(;j!o)] = kGk2 : 1 Similarly, if !0 = 1, then G(j!) is real. The conclusion follows by letting u = v1 u(t) and !o ! 1. ^ L1 7;! L1 : 1. First of all, maxi kgik1 is an upper bound on the 1-norm/1-norm system gain: jzi (t)j = = Zt 0 gi ( )u(t ; ) d Z tX q jgij ( )j d kuk1 0 j =1 kgi k1 kuk1 Zt 0 jgi ( )u(t ; )j d Z 1X q 0 j =1 and kz k1 := max sup jzi (t)j i t max kgi k1 kuk1 : i jgij ( )j d kuk1
- 127. PERFORMANCE SPECIFICATIONS 110 That maxi kgi k1 is the least upper bound can be seen as follows: Assume that the maximum maxi kgi k1 is achieved for i = 1. Let t0 be given and set u(t0 ; ) := sign(g1 ( )) 8 : Note that since g1( ) is a vector function, the sign function sign(g1 ( )) is a component-wise operation. Then kuk1 = 1 and Zt 0 z1 (t0 ) = 0 g1 ( )u(t0 ; ) d Zt X q 0 = 0 j =1 = kg1 k1 ; jg1j ( )j d Z 1X q t0 j =1 jg1j ( )j d : Hence, let t0 ! 1, and we have kz k1 = kg1 k1 . 2. If ku(t)k1 := supt ku(t)k, then kz (t)k = Zt Z t0 Zt 0 And, therefore, kz k1 0 g( )u(t ; )d kg( )k ku(t ; )k d kg( )k d kuk1 : R 1 kg( )k d kuk . 1 0 2 Next we shall derive some simple and useful bounds for the H1 norm and the L1 norm of a stable system. Suppose A G(s) = C B 2 RH1 0 is a balanced realization, i.e., there exists = diag( with 1 2 ::: n 1 2 : : : n) 0 0, such that A + A + BB = 0 Then we have the following theorem. A + A + C C = 0:
- 128. 4.5. Induced System Gains 111 Theorem 4.5 Z1 kGk1 1 0 kg(t)k dt 2 where g(t) = CeAt B . n X i=1 i Remark 4.3 It should be clear that the inequalities stated in the theorem do not depend on a particular state space realization of G(s). However, use of the balanced realization does make the proof simple. ~ Proof. The inequality kGk1 follows from the Nehari Theorem of Chapter 8. 1 We will now show the other inequalities. Since G(s) := Z1 0 g(t)e;st dt Re(s) > 0 by the de nition of H1 norm, we have kGk1 = sup Re(s)>0 sup Z1 0 Z1 Re(s)>0 0 Z1 0 g(t)e;st dt g(t)e;st dt kg(t)k dt: To prove the last inequality, let ui be the ith unit vector. Then n 1 if i = j and X u u = I: i i 0 if i 6= j i=1 De ne i (t) = ui eAt=2 B and i (t) = CeAt=2 ui . It is easy to verify that ui uj = ij = k 2 i k2 = k 2 i k2 = Z1 0 Z1 0 ui eAt=2 BB eA t=2 ui dt = 2ui ui = 2 i ui eA t=2 C CeAt=2 ui dt = 2ui ui = 2 i : Using these two equalities, we have Z1 0 kg(t)k dt = Z1 X n 0 n X i=1 i=1 i i k i k2 k i k2 dt n XZ 1 i=1 0 n X 2 i: i=1 k i i k dt
- 129. PERFORMANCE SPECIFICATIONS 112 2 It should be clear from the above two tables that many system performance criteria can be stipulated as requiring a certain closed loop transfer matrix have small H2 norm or H1 norm or L1 norm. Moreover, if L1 performance is satis ed, then the H1 norm performance is also satis ed. We will be most interested in H2 and H1 performance in this book. 4.6 Computing L2 and H2 Norms Let G(s) 2 L2 and recall that the L2 norm of G is de ned as kGk2 := s Z1 1 2 = kgk2 = sZ 1 ;1 ;1 TracefG (j!)G(j!)g d! Tracefg (t)g(t)g dt where g(t) denotes the convolution kernel of G. It is easy to see that the L2 norm de ned above is nite i the transfer matrix G is strictly proper, i.e., G(1) = 0. Hence, we will generally assume that the transfer matrix is strictly proper whenever we refer to the L2 norm of G (of course, this also applies to H2 functions). One straightforward way of computing the L2 norm is to use contour integral. Suppose G is strictly proper, and then we have Z1 TracefG (j!)G(j!)g d! kGk2 = 21 2 ;1 I = 21 j TracefG (s)G(s)g ds: The last integral is a contour integral along the imaginary axis, and around an in nite semi-circle in the left half-plane the contribution to the integral from this semi-circle equals zero because G is strictly proper. By the residue theorem, kGk2 equals the sum 2 of the residues of TracefG (s)G(s)g at its poles in the left half-plane. Although kGk2 can, in principle, be computed from its de nition or from the method suggested above, it is useful in many applications to have alternative characterizations and to take advantage of the state space representations of G. The computation of a RH2 transfer matrix norm is particularly simple. Lemma 4.6 Consider a transfer matrix A G(s) = C B 0
- 130. 4.6. Computing L2 and H2 Norms 113 with A stable. Then we have kGk2 = trace(B Lo B ) = trace(CLc C ) 2 (4:2) where Lo and Lc are observability and controllability Gramians which can be obtained from the following Lyapunov equations ALc + LcA + BB = 0 A Lo + LoA + C C = 0: Proof. Since G is stable, we have At g(t) = L;1 (G) = Ce B t < 0 0 t 0 and kGk2 2 = = Z1 Z01 0 Tracefg (t)g(t)g dt = TracefB eA t C Z1 0 Tracefg(t)g(t) g dt CeAt B g dt = Z1 0 TracefCeAt BB eA t C g dt: The lemma follows from the fact that the controllability Gramian of (A B ) and the observability Gramian of (C A) can be represented as Lo = Z1 0 eA t C CeAt dt Lc = which can also be obtained from ALc + LcA + BB = 0 Z1 0 eAt BB eA t dt A Lo + LoA + C C = 0: 2 To compute the L2 norm of a rational transfer function, G(s) 2 L2 , using state space approach. Let G(s) = G(s)]+ + G(s)]; with G+ 2 RH2 and G; 2 RH? . Then 2 2 = k G(s)] k2 + k G(s)] k2 kGk2 + 2 ; 2 where k G(s)]+ k2 and k G(s)]; k2 = k G(;s)]+ k2 can be computed using the above lemma. Still another useful characterization of the H2 norm of G is in terms of hypothetical input-output experiments. Let ei denote the ith standard basis vector of Rm where m is the input dimension of the system. Apply the impulsive input (t)ei ( (t) is the unit impulse) and denote the output by zi (t)(= g(t)ei ). Assume D = 0, and then zi 2 L2+ and m X kGk2 = kzi k2 : 2 2 i=1 Note that this characterization of the H2 norm can be appropriately generalized for nonlinear time varying systems, see Chen and Francis 1992] for an application of this norm in sampled-data control.
- 131. PERFORMANCE SPECIFICATIONS 114 4.7 Computing L1 and H1 Norms We shall rst consider, as in the L2 case, how to compute the 1 norm of an L1 transfer matrix. Let G(s) 2 L1 and recall that the L1 norm of a transfer function G is de ned as kGk1 := ess sup fG(j!)g: ! The computation of the L1 norm of G is complicated and requires a search. A control engineering interpretation of the in nity norm of a scalar transfer function G is the distance in the complex plane from the origin to the farthest point on the Nyquist plot of G, and it also appears as the peak value on the Bode magnitude plot of jG(j!)j. Hence the 1 of a transfer function can in principle be obtained graphically. To get an estimate, set up a ne grid of frequency points, f!1 !N g: Then an estimate for kGk1 is max fG(j!k )g: 1 k N This value is usually read directly from a Bode singular value plot. The L1 norm can also be computed in state space if G is rational. Lemma 4.7 Let > 0 and A B G(s) = C D 2 RL1 : (4:3) Then kGk1 < if and only if (D) < and H has no eigenvalues on the imaginary axis where BR;1 D C BR;1 H := ;CA(+ + DR;1D )C ;(A + BR;BD C ) (4.4) 1 I and R = 2I ; D D. Proof. Let (s) = 2I ; G (s)G(s). Then it is clear that kGk1 < if and only if (j!) > 0 for all ! 2 R. Since (1) = R > 0 and since (j!) is a continuous function of !, (j!) > 0 for all ! 2 R if and only if (j!) is nonsingular for all ! 2 R f1g, i.e., (s) has no imaginary axis zero. Equivalently, ;1 (s) has no imaginary axis pole. It is easy to compute by some simple algebra that 2 ;1 (s) = 4 H R;1 D C R;1B BR;1 ;C DR;1 R ;1 3 5:
- 132. 4.7. Computing L1 and H1 Norms 115 Thus the conclusion follows if the above realization has neither uncontrollable modes nor unobservable modes on the imaginary axis. Assume that j!0 is an eigenvalue of H but not a pole of ;1 (s). Then j!0 must be either an unobservable mode of BR;1 ( R;1 D C R;1 B H ) or an uncontrollable mode of (H ;C DR;1 ). Now suppose j!0 is an unobservable mode of ( R;1 D C R;1 B H ). Then there exists an x0 = x1 6= 0 such that x 2 Hx0 = j!0 x0 R;1 D C R;1 B x0 = 0: These equations can be simpli ed to (j!0 I ; A)x1 = 0 (j!0 I + A )x2 = ;C Cx1 D Cx1 + B x2 = 0: Since A has no imaginary axis eigenvalues, we have x1 = 0 and x2 = 0. This contradicts our assumption, and hence the realization has no unobservable modes on the imaginary axis. Similarly, a contradiction will also be arrived if j!0 is assumed to be an uncontrolBR;1 lable mode of (H ;C DR;1 ). 2 Bisection Algorithm Lemma 4.7 suggests the following bisection algorithm to compute RL1 norm: (a) select an upper bound u and a lower bound l such that l kGk1 u (b) if ( u ; l )= l speci ed level, stop kGk ( u + l )=2. Otherwise go to next step (c) set = ( l + u )=2 (d) test if kGk1 < by calculating the eigenvalues of H for the given (e) if H has an eigenvalue on j R set l = otherwise set u = go back to step (b). Of course, the above algorithm applies to H1 norm computation as well. Thus L1 norm computation requires a search, over either or !, in contrast to L2 (H2 ) norm computation, which does not. A somewhat analogous situation occurs for constant matrices with the norms kM k2 = trace(M M ) and kM k1 = M ]. In principle, kM k2 2 2 can be computed exactly with a nite number of operations, as can the test for whether (M ) < (e.g. 2 I ; M M > 0), but the value of (M ) cannot. To compute (M ), we must use some type of iterative algorithm.
- 133. 116 PERFORMANCE SPECIFICATIONS Remark 4.4 It is clear that kGk1 < i ;1 G 1 < 1. Hence, there is no loss of generality to assume = 1. This assumption will often be made in the remainder of this book. It is also noted that there are other fast algorithms to carry out the above norm computation nevertheless, this bisection algorithm is the simplest. ~ The H1 norm of a stable transfer function can also be estimated experimentally using the fact that the H1 norm of a stable transfer function is the maximum magnitude of the steady-state response to all possible unit amplitude sinusoidal input signals. 4.8 Notes and References The basic concept of function spaces presented in this chapter can be found in any standard functional analysis textbook, for instance, Naylor and Sell 1982] and Gohberg and Goldberg 1981]. The system theoretical interpretations of the norms and function spaces can be found in Desoer and Vidyasagar 1975]. The bisection H1 norm computational algorithm is rst developed in Boyd, Balakrishnan, and Kabamba 1989]. A more e cient L1 norm computational algorithm is presented in Bruinsma and Steinbuch 1990].
- 134. 5 Stability and Performance of Feedback Systems This chapter introduces the feedback structure and discusses its stability and performance properties. The arrangement of this chapter is as follows: Section 5.1 discusses the necessity for introducing feedback structure and describes the general feedback conguration. In section 5.2, the well-posedness of the feedback loop is de ned. Next, the notion of internal stability is introduced and the relationship is established between the state space characterization of internal stability and the transfer matrix characterization of internal stability in section 5.3. The stable coprime factorizations of rational matrices are also introduced in section 5.4. Section 5.5 considers feedback properties and discusses how to achieve desired performance using feedback control. These discussions lead to a loop shaping control design technique which is introduced in section 5.6. Finally, we consider the mathematical formulations of optimal H2 and H1 control problems in section 5.7. 5.1 Feedback Structure In designing control systems, there are several fundamental issues that transcend the boundaries of speci c applications. Although they may di er for each application and may have di erent levels of importance, these issues are generic in their relationship to control design objectives and procedures. Central to these issues is the requirement to provide satisfactory performance in the face of modeling errors, system variations, and 117
- 135. 118 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS uncertainty. Indeed, this requirement was the original motivation for the development of feedback systems. Feedback is only required when system performance cannot be achieved because of uncertainty in system characteristics. The more detailed treatment of model uncertainties and their representations will be discussed in Chapter 9. For the moment, assuming we are given a model including a representation of uncertainty which we believe adequately captures the essential features of the plant, the next step in the controller design process is to determine what structure is necessary to achieve the desired performance. Pre ltering input signals (or open loop control) can change the dynamic response of the model set but cannot reduce the e ect of uncertainty. If the uncertainty is too great to achieve the desired accuracy of response, then a feedback structure is required. The mere assumption of a feedback structure, however, does not guarantee a reduction of uncertainty, and there are many obstacles to achieving the uncertainty-reducing bene ts of feedback. In particular, since for any reasonable model set representing a physical system uncertainty becomes large and the phase is completely unknown at su ciently high frequencies, the loop gain must be small at those frequencies to avoid destabilizing the high frequency system dynamics. Even worse is that the feedback system actually increases uncertainty and sensitivity in the frequency ranges where uncertainty is signi cantly large. In other words, because of the type of sets required to reasonably model physical systems and because of the restriction that our controllers be causal, we cannot use feedback (or any other control structure) to cause our closed-loop model set to be a proper subset of the open-loop model set. Often, what can be achieved with intelligent use of feedback is a signi cant reduction of uncertainty for certain signals of importance with a small increase spread over other signals. Thus, the feedback design problem centers around the tradeo involved in reducing the overall impact of uncertainty. This tradeo also occurs, for example, when using feedback to reduce command/disturbance error while minimizing response degradation due to measurement noise. To be of practical value, a design technique must provide means for performing these tradeo s. We will discuss these tradeo s in more detail later in section 5.5 and in Chapter 6. To focus our discussion, we will consider the standard feedback con guration shown in Figure 5.1. It consists of the interconnected plant P and controller K forced by r-e ;6 - K u di - ? upe d P - ? ye ? en Figure 5.1: Standard Feedback Con guration
- 136. 5.2. Well-Posedness of Feedback Loop 119 command r, sensor noise n, plant input disturbance di , and plant output disturbance d. In general, all signals are assumed to be multivariable, and all transfer matrices are assumed to have appropriate dimensions. 5.2 Well-Posedness of Feedback Loop Assume that the plant P and the controller K in Figure 5.1 are xed real rational proper transfer matrices. Then the rst question one would ask is whether the feedback interconnection makes sense or is physically realizable. To be more speci c, consider a simple example where ;1 P = ;s + 2 K = 1 s are both proper transfer functions. However, u = (s + 2) (r ; n ; d) ; s ; 1 di 3 3 i.e., the transfer functions from the external signals r ; n ; d and di to u are not proper. Hence, the feedback system is not physically realizable! De nition 5.1 A feedback system is said to be well-posed if all closed-loop transfer matrices are well-de ned and proper. Now suppose that all the external signals r n d, and di are speci ed and that the closed-loop transfer matrices from them to u are respectively well-de ned and proper. Then, y and all other signals are also well-de ned and the related transfer matrices are proper. Furthermore, since the transfer matrices from d and n to u are the same and di er from the transfer matrix from r to u by only a sign, the system is well-posed if and only if the transfer matrix from di to u exists and is proper. d In order to be consistent with the notation used in the rest of the book, we shall denote ^ K := ;K and regroup the external input signals into the feedback loop as w1 and w2 and regroup the input signals of the plant and the controller as e1 and e2 . Then the feedback loop with the plant and the controller can be simply represented as in Figure 5.2 and the system is well-posed if and only if the transfer matrix from w1 to e1 exists and is w2 proper. Lemma 5.1 The feedback system in Figure 5.2 is well-posed if and only if ^ I ; K (1)P (1) (5:1) is invertible.
- 137. 120 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS w1 - e e1 +6 + - P + e2 ?+ w2 e ^ K Figure 5.2: Internal Stability Analysis Diagram Proof. The system in the above diagram can be represented in equation form as ^ e1 = w1 + Ke2 e2 = w2 + Pe1 : Then an expression for e1 can be obtained as ^ ^ (I ; KP )e1 = w1 + Kw2 : ^ Thus well-posedness is equivalent to the condition that (I ; KP );1 exists and is proper. But this is equivalent to the condition that the constant term of the transfer function ^ I ; KP is invertible. 2 It is straightforward to show that (5.1) is equivalent to either one of the following two conditions: ^ I ;K (1) is invertible (5:2) ;P (1) I ^ I ; P (1)K (1) is invertible: The well-posedness condition is simple to state in terms of state-space realizations. ^ Introduce realizations of P and K : A B P= C D " ^ A ^ K= ^ C (5:3) # ^ B ^ : D (5:4) ^ ^ Then P (1) = D and K (1) = D. For example, well-posedness in (5.2) is equivalent to the condition that ^ I ;D is invertible: (5:5) ;D I Fortunately, in most practical cases we will have D = 0, and hence well-posedness for most practical control systems is guaranteed.
- 138. 5.3. Internal Stability 121 5.3 Internal Stability Consider a system described by the standard block diagram in Figure 5.2 and assume ^ the system is well-posed. Furthermore, assume that the realizations for P (s) and K (s) given in equations (5.3) and (5.4) are stabilizable and detectable. ^ Let x and x denote the state vectors for P and K , respectively, and write the state ^ equations in Figure 5.2 with w1 and w2 set to zero: x _ e2 _ x ^ e1 = = = = Ax + Be1 Cx + De1 ^^ ^ Ax + Be2 ^^ ^ C x + De2 : (5.6) (5.7) (5.8) (5.9) De nition 5.2 The system of Figure 5.2 is said to be internally stable if the origin (x x) = (0 0) is asymptotically stable, i.e., the states (x x) go to zero from all initial ^ ^ states when w1 = 0 and w2 = 0. Note that internal stability is a state space notion. To get a concrete characterization of internal stability, solve equations (5.7) and (5.9) for e1 and e2 : ^ ^ e1 = I ;D ;1 0 C e2 ;D I C 0 x : x ^ Note that the existence of the inverse is guaranteed by the well-posedness condition. Now substitute this into (5.6) and (5.8) to get x =A x _ ~ _ x ^ x ^ where ^ ^ I ;D ;1 0 C : ;D I C 0 ~ Thus internal stability is equivalent to the condition that A has all its eigenvalues in 0 0 ~ A= A A + B B ^ 0 0 ^ the open left-half plane. In fact, this can be taken as a de nition of internal stability. Lemma 5.2 The system of Figure 5.2 with given stabilizable and detectable realizations ^ ~ for P and K is internally stable if and only if A is a Hurwitz matrix. It is routine to verify that the above de nition of internal stability depends only on ^ ^ P and K , not on speci c realizations of them as long as the realizations of P and K are both stabilizable and detectable, i.e., no extra unstable modes are introduced by the realizations. The above notion of internal stability is de ned in terms of state-space realizations ^ of P and K . It is also important and useful to characterize internal stability from the
- 139. 122 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS transfer matrix point of view. Note that the feedback system in Figure 5.2 is described, in term of transfer matrices, by ^ e1 = w1 : I ;K (5:10) e w ;P I 2 2 Now it is intuitively clear that if the system in Figure 5.2 is internally stable, then for all bounded inputs (w1 w2 ), the outputs (e1 e2) are also bounded. The following lemma shows that this idea leads to a transfer matrix characterization of internal stability. Lemma 5.3 The system in Figure 5.2 is internally stable if and only if the transfer matrix ^ ^ ^ ^ ^ I ;K ;1 = I + K (I ; P K );1 P K (I ; P K );1 ^ );1 P ^ );1 ;P I (I ; P K (I ; P K from (w1 w2 ) to (e1 e2 ) belongs to RH1 . " (5:11) # ^ ^ A B Proof. As above let C D and A B be stabilizable and detectable realiza^ ^ C D ^ ^ tions of P and K , respectively. Let y1 denote the output of P and y2 the output of K . Then the state-space equations for the system in Figure 5.2 are x _ _ x ^ y1 y2 e1 e2 = = = A 0 x + ^ x ^ 0 A C 0 x + ^ x ^ 0 C w1 + 0 I w2 I 0 The last two equations can be rewritten as ^ ^ e1 = 0 C I ;D e2 ;D I C 0 B 0 ^ 0 B D 0 ^ 0 D y1 : y2 e1 e2 e1 e2 x + w1 : x ^ w2 Now suppose that this system is internally stable. Then the well-posedness condition ^ ^ ^ implies that (I ; DD) = (I ; P K )(1) is invertible. Hence, (I ; P K ) is invertible. Furthermore, since the eigenvalues of 0 0 ~ A= A A + B B 0 ^ 0 ^ ^ ^ I ;D ;1 0 C ;D I C 0 are in the open left-half plane, it follows that the transfer matrix from (w1 w2 ) to (e1 e2 ) given in (5.11) is in RH1 .
- 140. 5.3. Internal Stability 123 ^ Conversely, suppose that (I ; P K ) is invertible and the transfer matrix in (5.11) is ^ ^ in RH1 . Then, in particular, (I ; P K );1 is proper which implies that (I ; P K )(1) = ^ ) is invertible. Therefore, (I ; DD ^ I ;D ;D I is nonsingular. Now routine calculations give the transfer matrix from w1 to e1 w2 e2 in terms of the state space realizations: ^ I ;D ;1 ;D I ^ ^ I ;D + 0 C (sI ; A);1 B 0 ~ ^ ;D I C 0 0 B ^ I ;D ;1 : ;D I Since the above transfer matrix belongs to RH1 , it follows that ^ 0 C (sI ; A);1 B 0 ~ ^ C 0 0 B ^ ^ ^ as a transfer matrix belongs to RH1 . Finally, since (A B C ) and (A B C ) are stabilizable and detectable, ^ 0 0 C ~ A B B ^ 0 C 0 ~ is stabilizable and detectable. It then follows that the eigenvalues of A are in the open left-half plane. 2 Note that to check internal stability, it is necessary (and su cient) to test whether each of the four transfer matrices in (5.11) is in RH1 . Stability cannot be concluded even if three of the four transfer matrices in (5.11) are in RH1 . For example, let an interconnected system transfer function be given by 1 ^ K = ;s ; 1: P = s;1 s+1 Then it is easy to compute 3 2 s+1 + ; (s ;s1)(s1+ 2) 7 e1 = 6 s + 2 7 w1 6 5 w2 4 s;1 e2 s+1 s+2 s+2 which shows that the system is not internally stable although three of the four transfer functions are stable. This can also be seen by calculating the closed-loop A-matrix with ^ any stabilizable and detectable realizations of P and K .
- 141. 124 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS Remark 5.1 It should be noted that internal stability is a basic requirement for a practical feedback system. This is because all interconnected systems may be unavoidably subject to some nonzero initial conditions and some (possibly small) errors, and it cannot be tolerated in practice that such errors at some locations will lead to unbounded signals at some other locations in the closed-loop system. Internal stability guarantees that all signals in a system are bounded provided that the injected signals (at any locations) are bounded. ~ However, there are some special cases under which determining system stability is simple. ^ Corollary 5.4 Suppose K 2 RH1. Then the system in Figure 5.2 is internally stable ^ );1 P 2 RH1 . i (I ; P K Proof. The necessity is obvious. To prove the su ciency, it is su cient to show that ^ (I ; P K );1 2 RH1 . But this follows from ^ ^ ^ (I ; P K );1 = I + (I ; P K );1 P K ^ ^ and (I ; P K );1 P K 2 RH1 . 2 This corollary is in fact the basis for the classical control theory where the stability is checked only for one closed-loop transfer function with the implicit assumption that the controller itself is stable. Also, we have Corollary 5.5 Suppose P 2 RH1. Then the system in Figure 5.2 is internally stable ^ ^ i K (I ; P K );1 2 RH1 . ^ Corollary 5.6 Suppose P 2 RH1 and K 2 RH1 . Then the system in Figure 5.2 is ^ internally stable i (I ; P K );1 2 RH1 . To study the more general case, de ne ^ nc := number of open rhp poles of K (s) np := number of open rhp poles of P (s): Theorem 5.7 The system is internally stable if and only if ^ (i) the number of open rhp poles of P (s)K (s) = nc + np ^ (ii) (s) := det(I ; P (s)K (s)) has all its zeros in the open left-half plane (i.e., (I ; ^ P (s)K (s));1 is stable).
- 142. 5.3. Internal Stability 125 ^ ^ Proof. It is easy to show that P K and (I ; P K );1 have the following realizations: 2 ^ 3 ^ A BC BD ^ ^ 5 ^ 4 PK = 6 0 A B 7 ^ ^ C DC DD " A B ^ (I ; P K );1 = C D where # ^ ^ A B C + B D (I ; DD);1 C DC ^ ^ ^ ^ 0 A B ^ D ^ B = B^ (I ; DD);1 B ^ ^ C = (I ; DD);1 C DC ^ D = (I ; DD);1 : ~ It is also easy to see that A = A. Hence, the system is internally stable i A is stable. ^ Now suppose that the system is internally stable, then (I ; P K );1 2 RH1 . This ^ (s)) must be in the left-half plane. So we only implies that all zeros of det(I ; P (s)K A = need to show that given condition (ii), condition (i) is necessary and su cient for the internal stability. This follows by noting that (A B ) is stabilizable i ^ ^ A BC BD (5:12) ^ ^ 0 A B is stabilizable and (C A) is detectable i ^ A BC ^ (5:13) C DC ^ 0 A ^ is detectable. But conditions (5.12) and (5.13) are equivalent to condition (i), i.e., P K has no unstable pole/zero cancelations. 2 With this observation, the MIMO version of the Nyquist stability theorem is obvious. Theorem 5.8 (Nyquist Stability Theorem) The system is internally stable if and only if condition (i) in Theorem 5.7 is satis ed and the Nyquist plot of (j!) for ;1 ! 1 encircles the origin, (0 0), nk + np times in the counter-clockwise direction. Proof. Note that by SISO Nyquist stability theorem, (s) has all zeros in the open left-half plane if and only if the Nyquist plot of (j!) for ;1 ! 1 encircles the origin, (0 0), nk + np times in the counter-clockwise direction. 2
- 143. 126 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS 5.4 Coprime Factorization over RH1 Recall that two polynomials m(s) and n(s), with, for example, real coe cients, are said to be coprime if their greatest common divisor is 1 (equivalent, they have no common zeros). It follows from Euclid's algorithm1 that two polynomials m and n are coprime i there exist polynomials x(s) and y(s) such that xm + yn = 1 such an equation is called a Bezout identity. Similarly, two transfer functions m(s) and n(s) in RH1 are said to be coprime over RH1 if there exists x y 2 RH1 such that xm + yn = 1: The more primitive, but equivalent, de nition is that m and n are coprime if every common divisor of m and n is invertible in RH1 , i.e., h mh;1 nh;1 2 RH1 =) h;1 2 RH1 : More generally, we have De nition 5.3 Two matrices M and N in RH1 are right coprime over RH1 if they have the same number of columns and if there exist matrices Xr and Yr in RH1 such that Xr Yr M = Xr M + Yr N = I: N ~ ~ Similarly, two matrices M and N in RH1 are left coprime over RH1 if they have the same number of rows and if there exist matrices Xl and Yl in RH1 such that ~ ~ M N Xl = MX + NY = I: ~ l ~ l Yl Note that these de nitions are equivalent to saying that the matrix M is leftN ~ ~ invertible in RH1 and the matrix M N is right-invertible in RH1 . These two equations are often called Bezout identities. Now let P be a proper real-rational matrix. A right-coprime factorization (rcf) of P is a factorization P = NM ;1 where N and M are right-coprime over RH1 . ~ ~ ~ ~ Similarly, a left-coprime factorization (lcf) has the form P = M ;1N where N and M are left-coprime over RH1 . A matrix P (s) 2 Rp (s) is said to have double coprime factorization if there exist a right coprime factorization P = NM ;1 , a left coprime ~ ~ factorization P = M ;1N , and Xr Yr Xl Yl 2 RH1 such that Xr Yr ~ ~ ;N M M ;Yl N Xl = I: (5:14) ~ Of course implicit in these de nitions is the requirement that both M and M be square and nonsingular. 1 See, e.g., Kailath, 1980, pp. 140-141].
- 144. 5.4. Coprime Factorization over RH1 127 Theorem 5.9 Suppose P (s) is a proper real-rational matrix and A B P= C D is a stabilizable and detectable realization. Let F and L be such that A + BF and A + LC are both stable, and de ne 2 A + BF B ;L 3 M ;Yl = 4 F I 0 5 N Xl C + DF D I 2 3 Xr Yr = 4 A + LC ;(B + LD) L 5 : F I 0 ~ ~ ;N M ;D C I (5:15) (5:16) ~ ~ Then P = NM ;1 = M ;1N are rcf and lcf, respectively, and, furthermore, (5.14) is satis ed. Proof. The theorem follows by verifying the equation (5.14). 2 Remark 5.2 Note that if P is stable, then we can take Xr = Xl = I , Yr = Yl = 0, ~ ~ N = N = P , M = M = I. ~ Remark 5.3 The coprime factorization of a transfer matrix can be given a feedback control interpretation. For example, right coprime factorization comes out naturally from changing the control variable by a state feedback. Consider the state space equations for a plant P : x = Ax + Bu _ y = Cx + Du: Next, introduce a state feedback and change the variable v := u ; Fx where F is such that A + BF is stable. Then we get x = (A + BF )x + Bv _ u = Fx + v y = (C + DF )x + Dv: Evidently from these equations, the transfer matrix from v to u is M (s) = A + BF B F I
- 145. 128 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS and that from v to y is Therefore B A N (s) = C + BF D : + DF u = Mv y = Nv so that y = NM ;1 u, i.e., P = NM ;1 . ~ We shall now see how coprime factorizations can be used to obtain alternative characterizations of internal stability conditions. Consider again the standard stability analysis ^ diagram in Figure 5.2. We begin with any rcf's and lcf's of P and K : ~ ~ P = NM ;1 = M ;1 N (5:17) ^ ~ ~ K = UV ;1 = V ;1 U: (5:18) Lemma 5.10 Consider the system in Figure 5.2. The following conditions are equivalent: 1. The feedback system is internally stable. M U is invertible in RH . 1 N V ~ ~ V U 3. ;N ;~ is invertible in RH1 . ~ M ~ ~ 4. MV ; NU is invertible in RH1 . ~ ~ 5. V M ; UN is invertible in RH1 . 2. Proof. As we saw in Lemma 5.3, internal stability is equivalent to ^ I ;K ;1 2 RH1 ;P I or, equivalently, Now so that ^ I K ;1 2 RH1 : P I ^ I UV ;1 = M U I K = ;1 NM I N V P I ^ I K ;1 = M 0 0 V P I (5:19) M ;1 0 M U ;1 : N V 0 V ;1
- 146. 5.4. Coprime Factorization over RH1 Since the matrices M 0 0 V 129 M U N V are right-coprime (this fact is left as an exercise for the reader), (5.19) holds i M U ;1 2 RH : 1 N V This proves the equivalence of conditions 1 and 2. The equivalence of 1 and 3 is proved similarly. The conditions 4 and 5 are implied by 2 and 3 from the following equation: ~ ~ ~ ~ V ;U M U = V M ; UN 0 ~ ~ ~ ~ N V ;N M 0 MV ; NU : Since the left hand side of the above equation is invertible in RH1 , so is the right hand side. Hence, conditions 4 and 5 are satis ed. We only need to show that either condition 4 or condition 5 implies condition 1. Let us show condition 5 ! 1 this is obvious since ~ ~ ^ I V ;1 U ;1 I K ;1 = ;1 NM I P I = M 0 0 I ~ ~ ~ V M U ;1 V 0 2 RH1 N I 0 I ~ M ~ ;1 if VN U 2 RH1 or if condition 5 is satis ed. I 2 Combining Lemma 5.10 and Theorem 5.9, we have the following corollary. ~ ~ Corollary 5.11 Let P be a proper real-rational matrix and P = NM ;1 = M ;1N be corresponding rcf and lcf over RH1 . Then there exists a controller ^ ~ ~ K0 = U0 V0;1 = V0;1 U0 ~ ~ with U0 V0 U0 , and V0 in RH1 such that ~ ~ V0 ;U0 M U0 = I 0 : (5:20) ~ M ~ N V0 0 I ;N Furthermore, let F and L be such that A + BF and A + LC are stable. Then a particular set of state space realizations for these matrices can be given by 2 A + BF B ;L 3 M U0 = 4 F (5:21) I 0 5 N V0 C + DF D I 2 A + LC ;(B + LD) L 3 ~0 ;U0 ~ V 4 F (5:22) I 0 5: ~ ~ ;N M = C ;D I
- 147. 130 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS Proof. The idea behind the choice of these matrices is as follows. Using the observer ^ theory, nd a controller K0 achieving internal stability for example LC ^ K0 := A + BF +F + LDF ;L : (5:23) 0 Perform factorizations ^ ~ ~ K0 = U0 V0;1 = V0;1 U0 which are analogous to the ones performed on P . Then Lemma 5.10 implies that each of the two left-hand side block matrices of (5.20) must be invertible in RH1 . In fact, (5.20) is satis ed by comparing it with the equation (5.14). 2 5.5 Feedback Properties In this section, we discuss the properties of a feedback system. In particular, we consider the bene t of the feedback structure and the concept of design tradeo s for con icting objectives { namely, how to achieve the bene ts of feedback in the face of uncertainties. r-e ;6 - K u di - ? upe d P - ? ye ? en Figure 5.3: Standard Feedback Con guration Consider again the feedback system shown in Figure 5.1. For convenience, the system diagram is shown again in Figure 5.3. For further discussion, it is convenient to de ne the input loop transfer matrix, Li , and output loop transfer matrix, Lo , as Li = KP Lo = PK respectively, where Li is obtained from breaking the loop at the input (u) of the plant while Lo is obtained from breaking the loop at the output (y) of the plant. The input sensitivity matrix is de ned as the transfer matrix from di to up : Si = (I + Li );1 up = Si di : And the output sensitivity matrix is de ned as the transfer matrix from d to y: So = (I + Lo );1 y = So d:
- 148. 5.5. Feedback Properties 131 The input and output complementary sensitivity matrices are de ned as Ti = I ; Si = Li (I + Li );1 To = I ; So = Lo(I + Lo);1 respectively. (The word complementary is used to signify the fact that T is the complement of S , T = I ; S .) The matrix I + Li is called input return di erence matrix and I + Lo is called output return di erence matrix. It is easy to see that the closed-loop system, if it is internally stable, satis es the following equations: y r;y u up = = = = To (r ; n) + So Pdi + So d So (r ; d) + Ton ; So Pdi KSo (r ; n) ; KSod ; Ti di KSo (r ; n) ; KSod + Si di : (5.24) (5.25) (5.26) (5.27) These four equations show the fundamental bene ts and design objectives inherent in feedback loops. For example, equation (5.24) shows that the e ects of disturbance d on the plant output can be made small" by making the output sensitivity function So small. Similarly, equation (5.27) shows that the e ects of disturbance di on the plant input can be made small by making the input sensitivity function Si small. The notion of smallness for a transfer matrix in a certain range of frequencies can be made explicit using frequency dependent singular values, for example, (So ) < 1 over a frequency range would mean that the e ects of disturbance d at the plant output are e ectively desensitized over that frequency range. Hence, good disturbance rejection at the plant output (y) would require that ;(I + PK );1 = 1 (So ) = (I + PK ) (for disturbance at plant output, d) ;(I + PK );1P = (PS ) (for disturbance at plant input, d ) (So P ) = i i be made small and good disturbance rejection at the plant input (up ) would require that ;(I + KP );1 = 1 (Si ) = (I + KP ) (for disturbance at plant input, di ) ;K (I + PK );1 = (KS ) (for disturbance at plant output, d) (Si K ) = o be made small, particularly in the low frequency range where d and di are usually signi cant. Note that (PK ) ; 1 (KP ) ; 1 (I + PK ) (I + KP ) (PK ) + 1 (KP ) + 1
- 149. 132 then STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS 1 (PK ) + 1 1 (KP ) + 1 These equations imply that (So ) (Si ) (So ) (Si ) 1 (PK ) ; 1 if (PK ) > 1 1 (KP ) ; 1 if (KP ) > 1: 1 () 1 () Now suppose P and K are invertible, then (PK ) 1 or (KP ) 1 () (So P ) = (PK ) 1 or (KP ) 1 () (KSo ) = (PK ) (KP ) 1 1: ;(I + PK );1P ;K (I + PK );1 (K ;1 ) = (1 ) K 1 : (P ;1 ) = (P ) Hence good performance at plant output (y) requires in general large output loop gain (Lo ) = (PK ) 1 in the frequency range where d is signi cant for desensitizing d and large enough controller gain (K ) 1 in the frequency range where di is signi cant for desensitizing di . Similarly, good performance at plant input (up ) requires in general large input loop gain (Li ) = (KP ) 1 in the frequency range where di is signi cant for desensitizing di and large enough plant gain (P ) 1 in the frequency range where d is signi cant, which can not changed by controller design, for desensitizing d. (It should be noted that in general So 6= Si unless K and P are square and diagonal which is true if P is a scalar system. Hence, small (So ) does not necessarily imply small (Si ) in other words, good disturbance rejection at the output does not necessarily mean good disturbance rejection at the plant input.) Hence, good multivariable feedback loop design boils down to achieving high loop (and possibly controller) gains in the necessary frequency range. Despite the simplicity of this statement, feedback design is by no means trivial. This is true because loop gains cannot be made arbitrarily high over arbitrarily large frequency ranges. Rather, they must satisfy certain performance tradeo and design limitations. A major performance tradeo , for example, concerns commands and disturbance error reduction versus stability under the model uncertainty. Assume that the plant model is perturbed to (I + )P with stable, and assume that the system is nominally stable, i.e., the closed-loop system with = 0 is stable. Now the perturbed closed-loop system is stable if det (I + (I + )PK ) = det(I + PK ) det(I + To) has no right-half plane zero. This would in general amount to requiring that k Tok be small or that (To ) be small at those frequencies where is signi cant, typically at
- 150. 5.5. Feedback Properties 133 high frequency range, which in turn implies that the loop gain, (Lo ), should be small at those frequencies. Still another tradeo is with the sensor noise error reduction. The con ict between the disturbance rejection and the sensor noise reduction is evident in equation (5.24). Large (Lo(j!)) values over a large frequency range make errors due to d small. However, they also make errors due to n large because this noise is passed through" over the same frequency range, i.e., y = To (r ; n) + So Pdi + So d (r ; n): Note that n is typically signi cant in the high frequency range. Worst still, large loop gains outside of the bandwidth of P , i.e., (Lo (j!)) 1 or (Li (j!)) 1 while (P (j!)) 1, can make the control activity (u) quite unacceptable, which may cause the saturation of actuators. This follows from u = KSo(r ; n ; d) ; Ti di = Si K (r ; n ; d) ; Ti di P ;1 (r ; n ; d) ; di : Here, we have assumed P to be square and invertible for convenience. The resulting equation shows that disturbances and sensor noise are actually ampli ed at u whenever the frequency range signi cantly exceeds the bandwidth of P , since for ! such that (P (j!)) 1, we have 1 P ;1 (j!)] = P (j!)] 1: Similarly, the controller gain, (K ), should also be kept not too large in the frequency range where the loop gain is small in order to not saturate the actuators. This is because for small loop gain (Lo (j!)) 1 or (Li (j!)) 1 u = KSo (r ; n ; d) ; Ti di K (r ; n ; d): Therefore, it is desirable to keep (K ) not too large when the loop gain is small. To summarize the above discussion, we note that good performance requires in some frequency range, typically some low frequency range (0 !l ): (PK ) 1 (KP ) 1 (K ) 1 and good robustness and good sensor noise rejection require in some frequency range, typically some high frequency range (!h 1) (PK ) 1 (KP ) 1 (K ) M where M is not too large. These design requirements are shown graphically in Figure 5.4. The speci c frequencies !l and !h depend on the speci c applications and the knowledge one has on the disturbance characteristics, the modeling uncertainties, and the sensor noise levels.
- 151. 134 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS HH H 6 @ HH @ H @ XXXX @@ @ @ (L) ;; Z ;;;;; Z @@ @ ;;;; ;;;ZZ @ @ ;; ;;;; ;Z S @ ; ; ;;;;; ; ZS @Z ;;;;; ;;;; S ;;; ;;;;; ! HHH ZZZ !h ;;; ; ;;;;;; l HH Z ;;;; log ! ; ;;;; ; ;; ;; H;; ; ;H ;; (L)SS @@HH;; ;; ;H SS @@ H;; ; HH S @@ S Figure 5.4: Desired Loop Gain 5.6 The Concept of Loop Shaping The analysis in the last section motivates a conceptually simple controller design technique: loop shaping. Loop shaping controller design involves essentially nding a controller K that shapes the loop transfer function L so that the loop gains, (L) and (L), clear the boundaries speci ed by the performance requirements at low frequencies and by the robustness requirements at high frequencies as shown in Figure 5.4. In the SISO case, the loop shaping design technique is particularly e ective and simple since (L) = (L) = jLj. The design procedure can be completed in two steps: SISO Loop Shaping (1) Find a rational strictly proper transfer function L which contains all the right half plane poles and zeros of P such that jLj clears the boundaries speci ed by the performance requirements at low frequencies and by the robustness requirements at high frequencies as shown in Figure 5.4. L must also be chosen so that 1+ L has all zeros in the open left half plane, which can usually be guaranteed by making L well-behaved in the crossover region, i.e., L should not be decreasing too fast in the frequency range of jL(j!)j 1. (2) The controller is given by K = L=P . The loop shaping for MIMO system can be done similarly if the singular values of the loop transfer functions are used for the loop gains.
- 152. 5.6. The Concept of Loop Shaping 135 MIMO Loop Shaping (1) Find a rational strictly proper transfer function L which contains all the right half plane poles and zeros of P so that the product of P and P ;1 L (or LP ;1 ) has no unstable poles and/or zeros cancelations, and (L) clears the boundary speci ed by the performance requirements at low frequencies and (L) clears the boundary speci ed by the robustness requirements at high frequencies as shown in Figure 5.4. L must also be chosen so that det(I + L) has all zeros in the open left half plane. (This is not easy for MIMO systems.) (2) The controller is given by K = P ;1 L if L is the output loop transfer function (or K = LP ;1 if L is the input loop transfer function). The loop shaping design technique can be quite useful especially for SISO control system design. However, there are severe limitations when it is used for MIMO system design. Limitations of MIMO Loop Shaping Although the above loop shaping design can be e ective in some of applications, there are severe intrinsic limitations. Some of these limitations are listed below: The loop shaping technique described above can only e ectively deal with problems with uniformly speci ed performance and robustness speci cations. More speci cally, the method can not e ectively deal with problems with di erent speci cations in di erent channels and/or problems with di erent uncertainty characteristics in di erent channels without introducing signi cant conservatism. To illustrate this di culty, consider an uncertain dynamical system P = (I + )P where P is the nominal plant and is the multiplicative modeling error. Assume that can be written in the following form = ~ Wt ( ~ ) < 1: Then, for robust stability, we would require ( To ) = ( ~ Wt To) < 1 or (Wt To) 1. If a uniform bound is required on the loop gain to apply the loop shaping technique, we would need to overbound (Wt To ): ) (Wt To ) (Wt ) (To ) (Wt ) 1 ;(L(oL ) if (Lo ) < 1 o and the robust stability requirement is implied by 1 (Lo ) (W1) + 1 (Wt ) if (Lo ) < 1: t
- 153. 136 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS Similarly, if the performance requirements, say output disturbance rejection, are not uniformly speci ed in all channels but by a weighting matrix Ws such that (Ws So ) 1, then it is also necessary to overbound (Ws So ) in order to apply the loop shaping techniques: ( (Ws So ) (Ws ) (So ) (L Ws ) 1 if (Lo) > 1 o) ; and the performance requirement is implied by (Lo ) (Ws ) + 1 (Ws ) if (Lo) > 1: It is possible that the bounds for the loop shape may contradict each other at some frequency range, as shown in the gure 5.5. However, this does not imply that there is no controller that will satisfy both nominal performance and robust stability except for SISO systems. This contradiction happens because the bounds 6 (Ws ) ; ; ; ;; Z ;; ; ; ;Z Z ; ; ; ; ;;; ;ZZ ; ; ;;; ; ; ;; ;; ZZ ;;; ; ;; ; ; ; ;Z ;; ; ; ; ; ; ; ; ;; ;; ;@;@;@;@ @ @ @ @ @ ; ;; ; ; P@;@ ;; @ @ @ @@ log ! ;P;;@ @ ;;; ; ; ; ; P @ P@ P ;; ; ; ; ; ; ; P@ P @ @ @ @ @ @@ ; 7 @ PP @ @ @ ;;;;;;; 1 (Wt ) Figure 5.5: Con ict Requirements do not utilize the structure of weights, Ws and Wt and the bounds are only su cient conditions for robust stability and nominal performance. This possibility can be further illustrated by the following example: Assume that a two-input and two-output system transfer matrix is given by 1 P (s) = s + 1 1 1 0 and suppose the weighting matrices are given by 2 Ws = 4 1 s+1 (s+1)(s+2) 0 s+1 1 3 2 5 Wt = 4 s+2 s+10 0 (s+1) s+10 s+2 s+10 3 5:
- 154. 5.7. Weighted H2 and H1 Performance 137 It is easy to show that for large , the weighting functions are as shown in Figure 5.5, and thus the above loop shaping technique cannot be applied. However, it is also easy to show that the system with controller K = I2 gives 1 1 Ws S = s+2 0 1 0 s+2 Wt T = s+10 0 1 0 s+10 and, therefore, the robust performance criterion is satis ed. Even if all of the above problems can be avoided, it may still be di cult to nd a matrix function Lo so that K = P ;1 Lo is stabilizing. This becomes much harder if P is non-minimum phase and/or unstable. Hence some new methodologies have to be introduced to solve complicated problems. The so-called LQG/LTR (Linear Quadratic Gaussian/Loop Transfer Recovery) procedure developed rst by Doyle and Stein 1981] and extended later by various authors can solve some of the problems, but it is essentially limited to nominally minimum phase and output multiplicative uncertain systems. For these reasons, it will not be introduced here. This motivates us to consider the closed-loop performance directly in terms of the closed-loop transfer functions instead of open loop transfer functions. The following section considers some simple closed-loop performance problem formulations. 5.7 Weighted H2 and H1 Performance In this section, we consider how to formulate some performance objectives into mathematically tractable problems. As shown in section 5.5, the performance objectives of a feedback system can usually be speci ed in terms of requirements on the sensitivity functions and/or complementary sensitivity functions or in terms of some other closedloop transfer functions. For instance, the performance criteria for a scalar system may be speci ed as requiring js(j!)j < 1 8! !0 js(j!)j > 1 8! > !0 1 where s(j!) = 1+p(j!)k(j!) . However, it is much more convenient to re ect the system performance objectives by choosing appropriate weighting functions. For example, the above performance objective can be written as jws (j!)s(j!)j 1 8! with jws (j!)j = ;1 ;1 8! !0 8! > !0 :
- 155. 138 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS u ~ 6 ~ di ? Wu - Wr r-e ;6 - 6 K ~ d u ? Wi di e -? - Wd d P e - ? y- We e- ? en Wn n ~ Figure 5.6: Standard Feedback Con guration with Weights In order to use ws in control design, a rational transfer function ws is usually used to approximate the above frequency response. The advantage of using weighted performance speci cations is obvious in multivariable system design. First of all, some components of a vector signal are usually more important than others. Secondly, each component of the signal may not be measured in the same metric for example, some components of the output error signal may be measured in terms of length, and the others may be measured in terms of voltage. Therefore, weighting functions are essential to make these components comparable. Also, we might be primarily interested in rejecting errors in a certain frequency range (for example low frequencies), hence some frequency dependent weights must be chosen. In general, we shall modify the standard feedback diagram in Figure 5.3 into Figure 5.6. The weighting functions in Figure 5.6 are chosen to re ect the design objectives and knowledge on the disturbances and sensor noise. For example, Wd and Wi may be chosen to re ect the frequency contents of the disturbances d and di or they may be used to model the disturbance power spectrum depending on the nature of signals involved in the practical systems. The weighting matrix Wn is used to model the frequency contents of the sensor noise while We may be used to re ect the requirements on the shape of certain closed-loop transfer functions, for example, the shape of the output sensitivity function. Similarly, Wu may be used to re ect some restrictions on the control or actuator signals, and the dashed precompensator Wr is an optional element used to achieve deliberate command shaping or to represent a non-unity feedback system in equivalent unity feedback form. It is, in fact, essential that some appropriate weighting matrices be used in order to utilize the optimal control theory discussed in this book, i.e., H2 and H1 theory. So a very important step in controller design process is to choose the appropriate weights, We Wd Wu , and possibly Wn Wi , and Wr . The appropriate choice of weights for a particular practical problem is not trivial. In many occasions, as in the scalar case, the weights are chosen purely as a design parameter without any physical bases, so
- 156. 5.7. Weighted H2 and H1 Performance 139 these weights may be treated as tuning parameters which are chosen by the designer to achieve the best compromise between the con icting objectives. The selection of the weighting matrices should be guided by the expected system inputs and the relative importance of the outputs. Hence, control design may be regarded as a process of choosing a controller K such that certain weighted signals are made small in some sense. There are many di erent ways to de ne the smallness of a signal or transfer matrix, as we have discussed in the last chapter. Di erent de nitions lead to di erent control synthesis methods, and some are much harder than others. A control engineer should make a judgment of the mathematical complexity versus engineering requirements. Below, we introduce two classes of performance formulations: H2 and H1 criteria. For the simplicity of presentation, we shall assume di = 0 and n = 0. H2 Performance ~ Assume, for example, that the disturbance d can be approximately modeled as an impulse with random input direction, i.e., ~ d(t) = (t) and E( ) = I where E denotes the expectation. We may choose to minimize the expected energy of ~ the error e due to the disturbance d: n o E kek2 = E 2 Z1 0 kek2 dt = kWe So Wd k2 : 2 ~ Alternatively, if we suppose that the disturbance d can be approximately modeled as white noise with Sdd = I , then ~~ See = (We So Wd )Sd~d~(We So Wd ) and we may chose to minimize the power of e: Z1 kek2 = 21 Trace See (j!) d! = kWe So Wd k2 : P 2 ;1 In general, a controller minimizing only the above criterion can lead to a very large control signal u that could cause saturation of the actuators as well as many other undesirable problems. Hence, for a realistic controller design, it is necessary to include the control signal u in the penalty function. Thus, our design criterion would usually be something like this n o E kek2 + 2 kuk2 = ~2 2 We So Wd Wu KSoWd 2 2
- 157. 140 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS with some appropriate choice of weighting matrix Wu and scalar . The parameter clearly de nes the tradeo we discussed earlier between good disturbance rejection at the output and control e ort (or disturbance and sensor noise rejection at the actuators). Note that can be set to = 1 by an appropriate choice of Wu . This problem can be viewed as minimizing the energy consumed by the system in order to reject the disturbance d. This type of problem was the dominant paradigm in the 1960's and 1970's and is usually referred to as Linear Quadratic Gaussian Control or simply as LQG. (They will also be referred to as H2 mixed sensitivity problems for the consistency with the H1 problems discussed next.) The development of this paradigm stimulated extensive research e orts and is responsible for important technological innovation, particularly in the area of estimation. The theoretical contributions include a deeper understanding of linear systems and improved computational methods for complex systems through state-space techniques. The major limitation of this theory is the lack of formal treatment of uncertainty in the plant itself. By allowing only additive noise for uncertainty, the stochastic theory ignored this important practical issue. Plant uncertainty is particularly critical in feedback systems. H1 Performance Although the H2 norm (or L2 norm) may be a meaningful performance measure and although LQG theory can give e cient design compromises under certain disturbance and plant assumptions, the H2 norm su ers a major de ciency. This de ciency is due to the fact that the tradeo between disturbance error reduction and sensor noise error reduction is not the only constraint on feedback design. The problem is that these performance tradeo s are often overshadowed by a second limitation on high loop gains { namely, the requirement for tolerance to uncertainties. Though a controller may be designed using FDLTI models, the design must be implemented and operated with a real physical plant. The properties of physical systems, in particular the ways in which they deviate from nite-dimensional linear models, put strict limitations on the frequency range over which the loop gains may be large. A solution to this problem would be to put explicit constraints on the loop gain in the penalty function. For instance, one may chose to minimize sup kek2 = kWe So Wd k1 kd~k2 1 subject to some restrictions on the control energy or control bandwidth: sup kuk2 = kWu KSo Wd k1 : ~ kd~k2 1 Or more frequently, one may introduce a parameter and a mixed criterion sup kd~k2 n 1 o kek2 + 2 kuk2 = ~2 2 We So Wd Wu KSo Wd 2 1 :
- 158. 5.8. Notes and References 141 This problem can also be regarded as minimizing the maximum power of the error subject to all bounded power disturbances: let e e := u ^ ~ then W W See = W e So Wd Sd~d~ W e So Wd ^^ KSoWd u u KSo Wd and sup kek2 = sup ^P ~ ~ kdkP 1 kdkP 1 1 Z 1 Trace S (j!) d! = ee ~~ 2 ;1 We SoWd Wu KSo Wd 2 1 : Alternatively, if the system robust stability margin is the major concern, the weighted complementary sensitivity has to be limited. Thus the whole cost function may be We So Wd W1 ToW2 1 where W1 and W2 are the frequency dependent uncertainty scaling matrices. These design problems are usually called H1 mixed sensitivity problems. For a scalar system, an H1 norm minimization problem can also be viewed as minimizing the maximum magnitude of the system's steady-state response with respect to the worst case sinusoidal inputs. 5.8 Notes and References The presentation of this chapter is based primarily on Doyle 1984]. The discussion of internal stability and coprime factorization can also be found in Francis 1987] and Vidyasagar 1985]. The loop shaping design is well known for SISO systems in the classical control theory. The idea was extended to MIMO systems by Doyle and Stein 1981] using LQG design technique. The limitations of the loop shaping design are discussed in detail in Stein and Doyle 1991]. Chapter 18 presents another loop shaping method using H1 control theory which has the potential to overcome the limitations of the LQG/LTR method.
- 159. 142 STABILITY AND PERFORMANCE OF FEEDBACK SYSTEMS
- 160. 6 Performance Limitations This chapter introduces some multivariable versions of the Bode's sensitivity integral relations and Poisson integral formula. The sensitivity integral relations are used to study the design limitations imposed by bandwidth constraints and the open-loop unstable poles, while the Poisson integral formula is used to study the design constraints imposed by the non-minimum phase zeros. These results display that the design limitations in multivariable systems are dependent on the directionality properties of the sensitivity function as well as those of the poles and zeros, in addition to the dependence upon pole and zero locations which is known in single-input single-output systems. These integral relations are also used to derive lower bounds on the singular values of the sensitivity function which display the design tradeo s. 6.1 Introduction One important problem that arises frequently is concerned with the level of performance that can be achieved in feedback design. It has been shown in the previous chapters that the feedback design goals are inherently con icting, and a tradeo must be performed among di erent design objectives. It is also known that the fundamental requirements such as stability and robustness impose inherent limitations upon the feedback properties irrespective of design methods, and the design limitations become more severe in the presence of right-half plane zeros and poles in the open-loop transfer function. An important tool that can be used to quantify feedback design constraints is furnished by the Bode's sensitivity integral relation and the Poisson integral formula. These integral formulae express design constraints directly in terms of the system's sensitivity 143
- 161. PERFORMANCE LIMITATIONS 144 and complementary sensitivity functions. A well-known theorem due to Bode states that for single-input single-output open-loop stable systems with more than one polezero excess, the integral of the logarithmic magnitude of the sensitivity function over all frequencies must equal to zero. This integral relation therefore suggests that in the presence of bandwidth constraints, the desirable property of sensitivity reduction in one frequency range must be traded o against the undesirable property of sensitivity increase at other frequencies. A result by Freudenberg and Looze 1985] further extends Bode's theorem to open-loop unstable systems, which shows that the presence of open-loop unstable poles makes the sensitivity tradeo a more di cult task. In the same reference, the limitations imposed by the open loop non-minimum phase zeros upon the feedback properties were also quanti ed using the Poisson integral. The results presented here are some multivariable extensions of the above mentioned integral relations. We shall now consider a linear time-invariant feedback system with an n n loop transfer matrix L. Let S (s) and T (s) be the sensitivity function and the complementary sensitivity function, respectively S (s) = (I + L(s));1 T (s) = L(s)(I + L(s));1 : (6:1) Before presenting the multivariable integral relations, recall that a point z 2 C is a transmission zero of L(s), which has full normal rank and a minimal state space realization (A B C D), if there exist vectors and such that the relation A ; zI B C D =0 holds, where = 1, and is called the input zero direction associated with z . Analogously, a transmission zero z of L(s) satis es the relation x w A ; zI B = 0 C D where x and w are some vectors with w satisfying the condition w w = 1. The vector w is called the output zero direction associated with z . Note also that p 2 C is a pole of L(s) if and only if it is a zero of L;1(s). By a slight abuse of terminology, we shall call the input and output zero directions of L;1 (s) the input and output pole directions of L(s), respectively. In the sequel we shall preclude the possibility that z is both a zero and pole of L(s). Then, by Lemma 3.27 and 3.28, z is a zero of L(s) if and only if L(z ) = 0 for some vector , = 1, or w L(z ) = 0 for some vector w, w w = 1. Similarly, p is a pole of L(s) if and only if L;1(p) = 0 for some vector , = 1, or w L;1(p) = 0 for some vector w, w w = 1. It is well-known that a non-minimum phase transfer function admits a factorization that consists of a minimum phase part and an all-pass factor. Let zi 2 C + , i = 1 k, be the non-minimum phase zeros of L(s) and let i i i = 1 be the input directions generated from the following iterative procedure
- 162. 6.2. Integral Relations 145 Let (A B C D) be a minimal realization of L(s) and B (0) := B Repeat for i = 1 to k A ; zi I B (i;1) C D i i =0 B (i) := B (i;1) ; 2(Rezi ) i i : Then, the input factorization of L(s) is given by L(s) = Lm (s)Bk (s) B1 (s) where Lm (s) denotes the minimum phase factor of L(s), and Bi (s) corresponds to the all-pass factor associated with zi : 2Rez (6:2) Bi (s) = I ; s + zi i i : i In fact, Lm(s) can be written as A (k) Lm(s) := C BD : For example, suppose z 2 C is a zero of L(s). Then it is easy to verify using state space calculation that L(s) can be factorized as 2Re A L(s) = C B ; 2(Rez ) I ; s+z : D z Note that a non-minimum phase transfer function admits an output factorization analogous to the input factorization, and the subsequent results can be applied to both types of factorizations. 6.2 Integral Relations In this section we provide extensions of the Bode's sensitivity integral relations and Poisson integral relations to multivariable systems. Consider a unit feedback system with a loop transfer function L. The following assumptions are needed: Assumption 6.1 The closed-loop system is stable. Assumption 6.2 lim R!1 sup s2C+ jsj R R (L(s)) = 0
- 163. PERFORMANCE LIMITATIONS 146 Each of these assumptions has important implications. Assumption 6.1 implies that the sensitivity function S (s) is analytic in C + . Assumption 6.2 states that the open-loop transfer function has a rollo rate of more than one pole-zero excess. Note that most of practical systems require a rollo rate of more than one pole-zero excess in order to maintain a su cient stability margin. One instance for this assumption to hold is that each element of L has a rollo rate of more than one pole-zero excess. Suppose that the open-loop transfer function L(s) has poles pi in the open right-half plane with input pole directions i , i = 1 k, which are obtained through a similar iterative procedure as in the last section. Lemma 6.1 Let L(s) and S (s) be de ned by (6.1). Then p 2 C is a zero of S (s) with zero direction if and only if it is a pole of L(s) with pole direction . Proof. Let p be a pole of L(s). Then there exists ;a vector such that ;1 and L;1(p) = 0. However, S (s) = (I + L(s));1 = I + L;1 (s) =1 L;1 (s). Hence, S (p) = 0. This establishes the su ciency part. The proof for necessity follows by reversing the above procedure. 2 Then the sensitivity function S (s) can be factorized as S (s) = Sm (s)B1 (s)B2 (s) Bk (s) (6:3) where Sm (s) has no zeros in C + , and Bi (s) is given by Bi (s) = I ; 2Repi i i : s + pi Theorem 6.2 Let S (s) be factorized in (6.3) and suppose that Assumptions 6.1-6.2 hold. Then Z1 0 ln (S (j!))d! 0k 1 X max @ (Repj ) j j A : j =1 (6:4) It is also instructive to examine the following two extreme cases. (i) If i j = 0 for all i j = 1 k, i 6= j , then Z1 0 (ii) If i = for all i = 1 ln (S (j!))d! max Repi : (6:5) k, then Z1 0 ln (S (j!))d! k X i=1 Repi : (6:6)
- 164. 6.2. Integral Relations 147 Note also that the following equality holds Z1 0 ln jS (j!)jd! = k X i=1 Repi if S (s) is a scalar function. Theorem 6.2 has an important implication toward the limitations imposed by the open-loop unstable poles on sensitivity properties. It shows that there will exist a frequency range over which the largest singular value of the sensitivity function exceeds one if it is to be kept below one at other frequencies. In the presence of bandwidth constraint, this imposes a sensitivity tradeo in di erent frequency ranges. Furthermore, this result suggests that unlike in single-input single-output systems, the limitations imposed by open-loop unstable poles in multivariable systems are related not only to the locations, but also to the directions of poles and their relative interaction. The case (i) of this result corresponds to the situation where the pole directions are mutually orthogonal, for which the integral corresponding to each singular value is related solely to one unstable pole with a corresponding distance to the imaginary axis, as if each channel of the system is decoupled from the others in e ects of sensitivity properties. The case (ii) corresponds to the situation where all the pole directions are parallel, for which the unstable poles a ect only the integral corresponding to the largest singular value, as if the channel corresponding to the largest singular value contains all unstable poles. Clearly, these phenomena are unique to multivariable systems. The following result further strengthens these observations and shows that the interaction between open-loop unstable poles plays an important role toward sensitivity properties. Corollary 6.3 Let k = 2 and let the Assumptions 6.1-6.2 hold. Then Z1 0 ln (S (j!))d! p 2 2 2 Re(p1 + p2 ) + (Re(p1 ; p2 )) + 4Rep1 Rep2 cos ( Proof. Note that max ((Rep1 ) 1 1 + (Rep2 ) 2 2 ) = max = max = amx = max 1 2 Rep1 0 0 Rep2 Rep1 0 1 0 Rep2 2 Rep1 0 1 0 Rep2 2 1 Rep1 (Rep1 ) 1 (Rep2 ) 2 1 Rep2 A straightforward calculation then gives 1 max ((Rep1 ) 1 1 + (Rep2 ) 2 2 ) = 2 Re(p1 + p2 ) 1 2) (6:7) 1 2 1 2 1 2 1 2 : :
- 165. PERFORMANCE LIMITATIONS 148 p + 1 (Re(p1 ; p2 ))2 + 4Rep1 Rep2 cos2 ( 2 1 2 ): 2 The proof is now completed. The utility of this corollary is clear. This result fully characterizes the limitation imposed by a pair of open-loop unstable poles on the sensitivity reduction properties. This limitation depends not only on the relative distances of the poles to the imaginary axis, but also on the principal angle between the two pole directions. Next, we investigate the design constraints imposed by open-loop non-minimum phase zeros upon sensitivity properties. The results below may be considered to be a matrix extension of the Poisson integral relation. Theorem 6.4 Let S (s) 2 H1 be factorized as in (6.3) and assume that lim max R!1 2 ; =2 =2] ln (S (Rej )) R = 0: (6:8) Then, for any non-minimum phase zero z = x0 + jy0 2 C + of L(s) with output direction w, w w = 1, Z1 Z1 x ln (S (j!)) x2 + (!0; y )2 d! 0 ;1 0 x ln (S (j!)) x2 + (!0; y )2 d! 0 ;1 0 ln (Sm (z )) ; ln w Bk 1 (z ) ln (S (z )): ; B1 1 ( z ) : (6:9) (6:10) Note that the condition (6.8) is satis ed if L(s) is a proper rational transfer matrix. Furthermore, for single-input single-output systems (n = 1), w and Z1 k Y i=1 Bi;1 (z ) = k Y z + pi i=1 z ; pi k Y z + pi x ln jS (j!)j x2 + (!0; y )2 d! = ln : 0 ;1 0 i=1 z ; pi The more interesting result, however, is the inequality (6.10). This result again suggests that the multivariable sensitivity properties are closely related to the pole and zero directions. This result implies that the sensitivity reduction ability of the system may be severely limited by the open-loop unstable poles and non-minimum phase zeros, especially when these poles and zeros are close to each other and the angles between their directions are small.
- 166. 6.3. Design Limitations and Sensitivity Bounds 149 6.3 Design Limitations and Sensitivity Bounds The integral relations derived in the preceding section are now used to analyze the design tradeo s and the limitations imposed by the bandwidth constraint and right-half plane poles and zeros upon sensitivity reduction properties. Similar to their counterparts for single-input single-output systems, these integral relations show that there will necessarily exist frequencies at which the sensitivity function exceeds one if it is to be kept below one over other frequencies, hence exhibiting a tradeo between the reduction of the sensitivity over one frequency range against its increase over another frequency range. Suppose that the feedback system is designed such that the level of sensitivity reduction is given by (S (j!)) ML < 1 8! 2 ;!L !L ] (6:11) where ML > 0 is a given constant. Let z = x0 + jy0 2 C + be an open right-half plane zero of L(s) with output direction w. De ne also (z ) := Z !L x0 2 + (! ; y0 )2 d!: ;!L x0 The following lower bound on the maximum sensitivity displays a limitation due to the open right-half plane zeros upon the sensitivity reduction properties. Corollary 6.5 Let the assumption in Theorem 6.4 holds. In addition, suppose that the condition (6.11) is satis ed. Then, for each open right-half plane zero z 2 C + of L(s) with output direction w, kS (s)k1 1 ML and kS (s)k1 (z ) ; (z) 1 ML ; w Bk 1 (z ) (z ) ; (z ) ; B1 1 (z ) ( (S (z ))) ; (z) ; (z) : (6:12) (6:13) Proof. Note that Z1 x ln (S (j!)) x2 + (!0; y )2 d! ( ; (z )) ln kS (j!)k1 + (z ) ln(ML): 0 ;1 0 Then the inequality (6.12) follows by applying inequality (6.10) and inequality (6.13) follows by applying inequality (6.9) 2 The interpretation of Corollary 6.5 is similar to that in single-input single-output systems. Roughly stated, this result shows that for a non-minimum phase system, its
- 167. PERFORMANCE LIMITATIONS 150 sensitivity must increase beyond one at certain frequencies if the sensitivity reduction is to be achieved at other frequencies. Of particular importance here is that the sensitivity function will in general exhibit a larger peak in multivariable systems than in singleinput single-output systems, due to the fact that (S (z )) 1. The design tradeo s and limitations on the sensitivity reduction which arise from bandwidth constraints as well as open-loop unstable poles can be studied using the extended Bode integral relations. However, these integral relations by themselves do not mandate a meaningful tradeo between the sensitivity reduction and the sensitivity increase, since the sensitivity function can be allowed to exceed one by an arbitrarily small amount over an arbitrarily large frequency range so as not to violate the Bode integral relations. However, bandwidth constraints in feedback design typically require that the open-loop transfer function be small above a speci ed frequency, and that it roll o at a rate of more than one pole-zero excess above that frequency. These constraints are commonly needed to ensure stability robustness despite the presence of modeling uncertainty in the plant model, particularly at high frequencies. One way of quantifying such bandwidth constraints is by requiring the open-loop transfer function to satisfy MH < 1 8! 2 !H 1) (6:14) !1+k where !H > !L , and MH > 0, k > 0 are some given constants. With the bandwidth (L(j!)) constraint given as such, the following result again shows that the sensitivity reduction speci ed by (6.11) can be achieved only at the expense of increasing the sensitivity at certain frequencies. Corollary 6.6 Suppose the Assumptions 6.1-6.2 hold. In addition, suppose that the conditions (6.11) and (6.14) are satis ed for some !H and !L such that !H > !L . Then !L ! 1 !H ;!L (1 ; ) k(!HH !L) ; (6:15) max (S (j!)) e ML !2 !L !H ] where = max Pk i=1 (Repi ) i i !H ; !L : Proof. Note rst that for ! !H , (S (j!)) = (I + 1 (j!)) L and Z1 1 1 ; (L(j!)) 1 MH d! = X Z 1 1 MH i d! ; ln 1 ; !1+k 1+k !H i=1 !H i ! 1 MH 1 ; !1+k
- 168. 6.4. Bode's Gain and Phase Relation 151 !i 1 X1 !H MH = 1+ i i(1 + k) ; 1 !H k i=1 !i ! 1 !H X 1 MH = ; !H ln 1 ; MH 1+ 1+ k i=1 i !H k k !H k ; !kH ln(1 ; ): Then Z1 0 ln (S (j!))d! = Z !L 0 ln (S (j!))d! + Z !H !L ln (S (j!))d! + Z1 Z1 !H ln (S (j!))d! MH ln 1 ; !1+k d! L H !H !H ln(1 ; ) !L ln ML + (!H ; !L ) !2max ] ln (S (j!)) ; k !L !H and the result follows from applying (6.4). !L ln ML + (!H ; !L ) !2max ] ln (S (j!)) ; ! ! 2 The above lower bound shows that the sensitivity can be very signi cant in the transition band. 6.4 Bode's Gain and Phase Relation In the classical feedback theory, the Bode's gain-phase integral relation (see Bode 1945]) has been used as an important tool to express design constraints in scalar systems. The following is an extended version of the Bode's gain and phase relationship for an openloop stable scalar system with possible right-half plane zeros, see Freudenberg and Looze 1988] and Doyle, Francis, and Tannenbaum 1992]: k 1 Z 1 d ln jLj ln coth j j d + Y j!0 + zi L(j!0 ) = 2 ;1 d i=1 j!0 ; zi where zi 's are assumed to be the right-half plane zeros of L(s) and := ln(!=!0 ). Note L that d ln jd (j!)j is the slope of the Bode plot which is almost always negative. It follows that L(j!0 ) will be large if the gain L attenuates slowly and small if it attenuates rapidly. The behavior of L(j!) is particularly important near the crossover frequency !c where jL(j!c )j = 1 since + L(j!c) is the phase margin of the feedback system, and further the return di erence is given by L j1 + L(j!c )j = j1 + L;1(j!c )j = 2 sin + 2 (j!c)
- 169. PERFORMANCE LIMITATIONS 152 which must not be too small for good stability robustness. If + L(j!c ) is forced to be very small by rapid gain attenuation, the feedback system will amplify disturbances and exhibit little uncertainty tolerance at and near !c . Since j!0 + z i 0 for each i, j!0 ; zi a non-minimum phase zero contributes an additional phase lag and imposes limitations upon the rollo rate of the open-loop gain. The con ict between attenuation rate and loop quality near crossover is thus clearly evident. A thorough discussion of the limitations these relations impose upon feedback control design is given by Bode 1945], Horowitz 1963], and Freudenberg and Looze 1988]. See also Freudenberg and Looze 1988] for some multivariable generalizations. In the classical feedback theory, it has been common to express design goals in terms of the shape" of the open-loop transfer function. A typical design requires that the open-loop transfer function have a high gain at low frequencies and a low gain at high frequencies while the transition should be well-behaviored. The same conclusion applies to multivariable system where the singular value plots should be well-behaviored between the transition band. 6.5 Notes and References The results presented in this chapter are based on Chen 1992a, 1992b, 1995]. Some related results can be found in Boyd and Desoer 1985] and Freudenberg and Looze 1988]. The related results for scalar systems can be found in Bode 1945], Horowitz 1963], Doyle, Francis, and Tannenbaum 1992], and Freudenberg and Looze 1988]. The study of analytic functions, harmonic functions1 , and various integral relations in the scalar case can be found in Garnett 1981] and Ho man 1962]. 1 A function : C 7;! R is said to be a harmonic function (subharmonic function) if 52 ( ) = 0 (52 ( ) 0) where the symbol 52 ( ) with = + denotes the Laplacian of ( ) and is de ned by 2 2 52 ( ) := (2 ) + (2 ) f f s f s f s s f s x jy f s @ f s @ f s @x @y :
- 170. 7 Model Reduction by Balanced Truncation In this chapter we consider the problem of reducing the order of a linear multivariable dynamical system. There are many ways to reduce the order of a dynamical system. However, we shall study only two of them: balanced truncation method and Hankel norm approximation method. This chapter focuses on the balanced truncation method while the next chapter studies the Hankel norm approximation method. A model order reduction problem can in general be stated as follows: Given a full order model G(s), nd a lower order model, say, an r-th order model Gr , such that G and Gr are close in some sense. Of course, there are many ways to de ne the closeness of an approximation. For example, one may desire that the reduced model be such that G = Gr + a and a is small in some norm. This model reduction is usually called an additive model reduction problem. On the other hand, one may also desire that the approximation be in relative form Gr = G(I + rel ) so that rel is small in some norm. This is called a relative model reduction problem. We shall be only interested in L1 norm approximation in this book. Once the norm is chosen, the additive model reduction problem can be formulated as inf deg(Gr ) r kG ; Gr k1 153
- 171. 154 MODEL REDUCTION BY BALANCED TRUNCATION and the relative model reduction problem can be formulated as inf G;1 (G ; Gr ) 1 deg(Gr ) r if G is invertible. In general, a practical model reduction problem is inherently frequency weighted, i.e., the requirement on the approximation accuracy at one frequency range can be drastically di erent from the requirement at another frequency range. These problems can in general be formulated as frequency weighted model reduction problems inf kW (G ; Gr )Wi k1 deg(G ) r o r with appropriate choice of Wi and Wo . We shall see in this chapter how the balanced realization can give an e ective approach to the above model reduction problems. 7.1 Model Reduction by Balanced Truncation A B Consider a stable system G 2 RH1 and suppose G = C D is a balanced realization, i.e., its controllability and observability Gramians are equal and diagonal. Denote the balanced Gramians by , then A + A + BB = 0 (7:1) A + A + C C = 0: (7:2) Now partition the balanced Gramian as = 01 0 and partition the system 2 accordingly as 2 3 A11 A12 G = 4 A21 A22 C1 C2 B1 B2 D 5: Then (7.1) and (7.2) can be written in terms of their partitioned matrices as A11 1 + 1 A11 + B1 B1 = 0 (7.3) A11 + A11 1 + C1 C1 = 0 (7.4) 1 A21 1 + 2 A12 + B2 B1 = 0 (7.5) (7.6) 2 A21 + A12 1 + C2 C1 = 0 A22 2 + 2 A22 + B2 B2 = 0 (7.7) A22 + A22 2 + C2 C2 = 0: (7.8) 2 The following theorem characterizes the properties of these subsystems. Theorem 7.1 Assume that 1 and 2 have no diagonal entries in common. Then both subsystems (Aii Bi Ci ) i = 1 2 are asymptotically stable.
- 172. 7.1. Model Reduction by Balanced Truncation 155 Proof. It is clearly su cient to show that A11 is asymptotically stable. The proof for the stability of A22 is similar. By Lemma 3.20 or Lemma 3.21, 1 can be assumed to be positive de nite without loss of generality. Then it is obvious that i (A11 ) 0 by Lemma 3.19. Assume that A11 is not asymptotically, then there exists an eigenvalue at j! for some !. Let V be a basis matrix for Ker(A11 ; j!I ). Then we have (A11 ; j!I )V = 0 which gives (7:9) V (A11 + j!I ) = 0: Equations (7.3) and (7.4) can be rewritten as (A11 ; j!I ) 1 + 1 (A11 + j!I ) + B1 B1 = 0 1 (A11 ; j!I ) + (A11 + j!I ) 1 + C1 C1 = 0: (7.10) (7.11) Multiplication of (7.11) from the right by V and from the left by V gives V C1 C1 V = 0, which is equivalent to C1 V = 0: Multiplication of (7.11) from the right by V now gives (A11 + j!I ) 1 V = 0: Analogously, rst multiply (7.10) from the right by obtain B1 1 V = 0: Then multiply (7.10) from the right by 1 V to get 1V and from the left by V 1 to (A11 ; j!I ) 2 V = 0: 1 It follows that the columns of matrix 1 such that 2V 1 are in Ker(A11 ; j!I ). Therefore, there exists a 2V 1 = V 2: 1 2 is the restriction of 2 to the space spanned by V , it follows that it is possible Since 1 1 to choose V such that 2 is diagonal. It is then also possible to choose 1 diagonal and 1 such that the diagonal entries of 1 are a subset of the diagonal entries of 1 . Multiply (7.5) from the right by 1 V and (7.6) by V to get A21 2 V + 2 A12 1 V = 0 1 2 A21 V + A12 1 V = 0 which gives (A21 V ) 2 = 2 (A V ) : 1 2 21
- 173. 156 MODEL REDUCTION BY BALANCED TRUNCATION 2 1 This is a Sylvester equation in (A21 V ). Because common it follows from Lemma 2.7 that and 2 2 have no diagonal entries in A21 V = 0 (7:12) is the unique solution. Now (7.12) and (7.9) implies that A11 A12 A21 A22 V = j! V 0 0 which means that the A-matrix of the original system has an eigenvalue at j!. This contradicts the fact that the original system is asymptotically stable. Therefore A11 must be asymptotically stable. 2 Corollary 7.2 If has distinct singular values, then every subsystem is asymptotically stable. The stability condition in Theorem 7.1 is only su cient. For example, 2 ;2 ;2:8284 ;2 3 (s ; 1)(s ; 2) = 4 0 ;1 ;1:4142 5 (s + 1)(s + 2) 2 is a balanced realization with On the other hand, 1:4142 1 = I and every subsystem of the realization is stable. 3 2 s2 ; s + 2 = 4 ;1;1 1:4142 1:4142 5 :4142 0 0 s2 + s + 2 ;1:4142 0 1 is also a balanced realization with = I but one of the subsystems is not stable. Theorem 7.3 Suppose G(s) 2 RH1 and 2 A A 11 12 G(s) = 4 A21 A22 B1 B2 D C1 C2 is a balanced realization with Gramian 1 2 and 1 = diag( 1 3 5 2) = diag( 1 Is1 2 Is2 : : : r Isr ) = diag( r+1 Isr+1 r+2 Isr+2 : : : N IsN ) > 2 > > r > r+1 > r+2 > > N
- 174. 7.1. Model Reduction by Balanced Truncation 157 where i has multiplicity si i = 1 2 : : : N and s1 + s2 + + sN = n. Then the truncated system Gr (s) = A11 B1 C1 D is balanced and asymptotically stable. Furthermore kG(s) ; Gr (s)k1 2( r+1 + r+2 + + N ) and the bound is achieved if r = N ; 1, i.e., kG(s) ; GN ;1 (s)k1 = 2 N : Proof. The stability of Gr follows from Theorem 7.1. We shall now prove the error bound for the case si = 1 for all i. The case where the multiplicity of i is not equal to one is more complicated and an alternative proof is given in the next chapter. Hence, we assume si = 1 and N = n. Let (s) := (sI ; A11 );1 (s) := sI ; A22 ; A21 (s)A12 ~ B (s) := A21 (s)B1 + B2 ~ C (s) := C1 (s)A12 + C2 then using the partitioned matrix results of section 2.3, G(s) ; Gr (s) = C (sI ; A);1 B ; C1 (s)B1 sI ; A11 ;A12 ;1 B1 ; C (s)B 1 1 ;A21 sI ; A22 B2 ~ ~ = C (s) ;1 (s)B (s) = C1 C2 computing this quantity on the imaginary axis to get h i ~ ~ ~ ~ G(j!) ; Gr (j!)] = 1=2 ;1 (j!)B (j!)B (j!) ; (j!)C (j!)C (j!) : (7:13) max ~ ~ ~ ~ Expressions for B (j!)B (j!) and C (j!)C (j!) are obtained by using the partitioned form of the internally balanced Gramian equations (7.3){(7.8). ~ ~ An expression for B (j!)B (j!) is obtained by using the de nition of B (s), substituting for B1 B1 , B1 B2 and B2 B2 from the partitioned form of the Gramian equations (7.3){(7.5), we get ~ ~ B (j!)B (j!) = (j!) 2 + 2 (j!): ~ ~ The expression for C (j!)C (j!) is obtained analogously and is given by ~ ~ C (j!)C (j!) = 2 (j!) + (j!) 2 :
- 175. 158 MODEL REDUCTION BY BALANCED TRUNCATION ~ ~ ~ ~ These expressions for B (j!)B (j!) and C (j!)C (j!) are then substituted into (7.13) to obtain G(j!) ; Gr (j!)] = 1=2 max 2+ ;1 (j!) 2 (j!) 2+ Now consider one-step order reduction, i.e., r = n ; 1, then ; (j!) 2 (j!) : 2= n and G(j!) ; Gr (j!)] = n 1=2 1 + ;1(j!) 1 + (j!)] (7:14) max where := ; (j!) (j!) = ; is an all pass" scalar function. (This is the only place we need the assumption of si = 1) Hence j (j!)j = 1. Using triangle inequality we get G(j!) ; Gr (j!)] n 1 + j (j! )j] = 2 n : (7:15) This completes the bound for r = n ; 1. The remainder of the proof is achieved by using the order reduction by one step results and by noting that Gk (s) = A11 B1 obtained by the k-th" order partiC1 D tioning is internally balanced with balanced Gramian given by 1 = diag( 1 Is1 2 Is2 : : : k Isk ): Let Ek (s) = Gk+1 (s) ; Gk (s) for k = 1 2 : : : N ; 1 and let GN (s) = G(s). Then Ek (j!)] 2 k+1 since Gk (s) is a reduced order model obtained from the internally balanced realization of Gk+1 (s) and the bound for one-step order reduction, (7.15) holds. Noting that G(s) ; Gr (s) = by the de nition of Ek (s), we have G(j!) ; Gr (j!)] N ;1 X k =r N ;1 X k=r Ek (s) Ek (j!)] 2 N ;1 X k=r k+1 : This is the desired upper bound. To see that the bound is actually achieved when r = N ; 1, we note that (0) = I . Then the right hand side of (7.14) is 2 N at ! = 0. 2 The singular values i are called the Hankel singular values. A useful consequence of the above theorem is the following corollary.
- 176. 7.1. Model Reduction by Balanced Truncation Corollary 7.4 Let i Then 159 i = 1 : : : N be the Hankel singular values of G(s) 2 RH1 . kG(s) ; G(1)k1 2( 1 + : : : + N ): The above bound can be tight for some systems. For example, consider an n-th order transfer function n X 2j G(s) = s + 2j j =1 with > 0. Then kG(s)k1 = G(0) = n and G(s) has the following state space realization 2 2 3 6 6 G=6 6 6 4 ; ; 4 2 ... 2 ; 2n n .. . n 7 7 7 7 7 5 0 and the controllability and observability Gramians of the realization are given by P =Q= i+j 2i + 2j 1 and P = Q ! 2 In as ! 1. So the Hankel singular values j ! 1 and 2( 1 + 2 + 2 + n ) ! n = kG(s)k1 as ! 1. The model reduction bound can also be loose for systems with Hankel singular values close to each other. For example, consider the balanced realization of a fourth order system 2 ;19:9579 ;5:4682 9:6954 0:9160 ;6:3180 3 6 5:4682 0 0 0:2378 0:0020 7 6 ;9:6954 7 6 0 0 ;4:0051 ;0:0067 7 G(s) = 6 4 0:9160 ;0:2378 4:0051 ;0:0420 0:2893 7 5 ;6:3180 ;0:0020 0:0067 0:2893 0 with Hankel singular values given by 1 =1 2 = 0:9977 3 = 0:9957 4 = 0:9952: The approximation errors and the estimated bounds are listed in the following table. The table shows that the actual error for an r-th order approximation is almost the same as 2 r+1 which would be the estimated bound if we regard r+1 = r+2 = = 4 . In general, it is not hard to construct an n-th order system so that the r-th order balanced model reduction error is approximately 2 r+1 but the error bound is arbitrarily close to 2(n ; r) r+1 . One method to construct such system is as follows: Let G(s) be a stable all-pass function, i.e., G(s) G(s) = I , then there is a balanced realization for G so that the controllability and observability Gramians are P = Q = I . Next make
- 177. 160 MODEL REDUCTION BY BALANCED TRUNCATION a very small perturbation to the balanced realization then the perturbed system has a balanced realization with distinct singular values and P = Q I . This perturbed system will have the desired properties and this is exactly how the above example is constructed. r kG ; Gr k1 P Bounds: 2 4=r+1 i 2 r+1 0 1 2 3 2 1.996 1.991 1.9904 i 7.9772 5.9772 3.9818 1.9904 2 1.9954 1.9914 1.9904 7.2 Frequency-Weighted Balanced Model Reduction This section considers the extension of the balanced truncation method to frequency weighted case. Given the original full order model G 2 RH1 , the input weighting matrix Wi 2 RH1 and the output weighting matrix Wo 2 RH1 , our objective is to nd a lower order model Gr such that kWo (G ; Gr )Wi k1 is made as small as possible. Assume that G Wi , and Wo have the following state space realizations A G= C B 0 B Wi = Ai Di Ci i B Wo = Ao Do Co o with A 2 Rn n . Note that there is no loss of generality in assuming D = G(1) = 0 since otherwise it can be eliminated by replacing Gr with D + Gr . Now the state space realization for the weighted transfer matrix is given by 2 A 0 BC BD 3 6 BoC Ao 0 i 0 i 7 =: " A B # : Wo GWi = 6 0 0 Ai Bi 7 4 5 C 0 Do C Co 0 0 Let P and Q be the solutions to the following Lyapunov equations AP + P A + B B = 0 QA + A Q + C C = 0: (7.16) (7.17) Then the input weighted Gramian P and the output weighted Gramian Q are de ned by n n P := In 0 P I0 Q := In 0 Q I0 :
- 178. 7.3. Relative and Multiplicative Model Reductions 161 It can be shown easily that P and Q satisfy the following lower order equations A BC 0 A i i P P12 P12 P22 + P P12 P12 P22 A BC 0 A i + i BD B i i BD B i =0 i : (7 18) Q Q12 Q12 Q22 A 0 BC A o o + A 0 BC A o o Q Q12 Q12 Q22 + CD C o o CD C o o : =0 : (7 19) The computation can be further reduced if Wi = I or Wo = I . In the case of Wi = I , P can be obtained from PA + AP + BB = 0 (7:20) while in the case of Wo = I , Q can be obtained from QA + A Q + C C = 0: (7:21) Now let T be a nonsingular matrix such that TPT = (T ;1 ) QT ;1 = 1 (i.e., balanced) with 1 = diag( 1 Is1 : : : r Isr ) and and partition the system accordingly as TAT ;1 TB CT ;1 0 2 2 = diag( r+1 Isr+1 : : : n Isn ) 2A A B 3 1 11 12 = 4 A21 A22 B2 5 : C1 C2 0 Then a reduced order model Gr is obtained as Gr = A11 B1 : C1 0 Unfortunately, there is generally no known a priori error bound for the approximation error and the reduced order model Gr is not guaranteed to be stable either. 7.3 Relative and Multiplicative Model Reductions A very special frequency weighted model reduction problem is the relative error model reduction problem where the objective is to nd a reduced order model Gr so that Gr = G(I + rel )
- 179. MODEL REDUCTION BY BALANCED TRUNCATION 162 and k rel k1 is made as small as possible. rel is usually called the relative error. In the case where G is square and invertible, this problem can be simply formulated as min G;1 (G ; Gr ) 1 : degG r r Of course the dual approximation problem Gr = (I + rel )G can be obtained by taking the transpose of G. We will show below that, as a bonus, the approximation Gr obtained below also serves as a multiplicative approximation: G = Gr (I + mul ) where mul is usually called the multiplicative error. Theorem 7.5 Let G G;1 2 RH1 be an n-th order square transfer matrix with a state space realization A B G(s) = C D : ;1 ;1 Let Wi = I and Wo = G;1 (s) = A ; BD C ;BD1 . D;1 C D; (a) Then the weighted Gramians P and Q for the frequency weighted balanced realization of G can be obtained as PA + AP + BB = 0 (7:22) Q(A ; BD;1 C ) + (A ; BD;1 C ) Q + C (D;1 ) D;1 C = 0: (7:23) (b) Suppose the realization for G is weighted balanced, i.e., P = Q = diag( 1 Is1 : : : r Isr r+1 Isr+1 : : : N IsN ) = diag( 1 2 ) with 1 > 2 > : : : > N 0 and let the realization of G be partitioned compatibly with 1 and 2 as 2 3 A11 A12 B1 G(s) = 4 A21 A22 B2 5 : C1 C2 D Then Gr (s) = A11 B1 C1 D is stable and minimum phase. Furthermore k k rel k1 mul k1 N Y i=r+1 N Y i=r+1 q 1 + 2 i ( 1 + i2 + i ) ; 1 q 1 + 2 i ( 1 + i2 + i ) ; 1:
- 180. 7.3. Relative and Multiplicative Model Reductions 163 Proof. Since the input weighting matrix Wi = I , it is obvious that the input weighted Gramian is given by P . Now the output weighted transfer matrix is given by G;1 (G ; D) = 2 A 3 " # 0 B ;1 C A ; BD;1 C 0 5 =: A B : 4 ;BD C 0 ;1 ;1 D C It is easy to verify that D C 0 Q := Q Q Q Q satis es the following Lyapunov equation QA + A Q + C C = 0: Hence Q is the output weighted Gramian. The proof for part (b) is more involved and needs much more work. We refer readers to the references at the end of the chapter for details. 2 In the above theorem, we have assumed that the system is square, we shall now extend the results to include non-square case. Let G(s) be a minimum phase transfer matrix with a minimal realization A B G(s) = C D and assume that D has full row rank. Without loss of generality, we shall also assume that D is normalized such that DD = I . Let D? be a matrix with full row rank such D that D is square and unitary. ? Lemma 7.6 A complex number z 2 C is a zero of G(s) if and only if z is an uncontrollable mode of (A ; BD C BD? ). A B Proof. Since D has full row rank and G(s) = C D is a minimal realization, z is a transmission zero of G(s) if and only if A ; zI B C D does not have full row rank. Now note that A ; zI B C D I 0 0 D D? ] 2 I 0 4 ;C I 0 0 0 0 I 3 5
- 181. 164 MODEL REDUCTION BY BALANCED TRUNCATION Then it is clear that = A ; BD C ; zI BD 0 I BD? : 0 A ; zI B C D does not have full row rank if and only if A ; BD C ; zI BD? does not have full row rank. By PBH test, this implies that z is a zero of G(s) if and only if it is an uncontrollable mode of (A ; BD C BD? ). 2 ~ Corollary 7.7 There exists a matrix C such that the augmented system 2 A B A B Ga := C D = 4 C D a a ~ C D? 3 5 = G(s) ~ G(s) is minimum phase. Proof. Note that the zeros of Ga are given by the eigenvalues of D ;1 C = A ; BD C ; BD C: A;B D ?~ ~ C ? ~ ~ Hence C can always be chosen so that A ; BD C ; BD? C is stable. 2 If the previous model reduction algorithms are applied to the augmented system Ga , the corresponding P and Q equations are given by PA + AP + BB = 0 ; ; ; ; Q(A ; BDa 1 Ca ) + (A ; BDa 1 Ca ) Q + Ca (Da 1 ) Da 1 Ca = 0: Moreover, we have " ^ # " ^ # G(s) = G(s) (I + ) G(s) = G(s) (I + rel mul ) ^ ^ ~ ~ ~ ~ G(s) G(s) G(s) G(s) and ^ ^ G(s) = G(s)(I + rel ) G(s) = G(s)(I + mul ): ~ However, there are in general in nitely many choices of C and the model reduction ~ results will in general depend on the speci c choice. Hence an appropriate choice of C is important. To motivate our choice, note that the equation for Q can be rewritten as ~ ~ Q(A;BD C )+(A;BD C ) Q;QBD?D? B Q+C C +(C ;D? B Q) (C ;D?B Q) = 0
- 182. 7.3. Relative and Multiplicative Model Reductions A natural choice might be 165 ~ C = D?B Q: The existence of a solution Q to the following so-called algebraic Riccati equation Q(A ; BD C ) + (A ; BD C ) Q ; QBD? D? B Q + C C = 0 such that ; ~ A ; BDa 1 Ca = A ; BD C ; BD? C = A ; BD C ; BD?D? B Q is stable will be shown in Chapter 13. In the case where the model is not stable and /or is not minimum phase, the following procedure can be used: Let G(s) be factorized as G(s) = Gap (s)Gmp (s) such that Gap ^ is an all-pass, i.e., Gap Gap = I , and Gmp is stable and minimum phase. Let Gmp be a relative/multiplicative reduced model of Gmp such that ^ Gmp = Gmp (I + and ^ Gmp = Gmp (I + rel ) mul ): ^ ^ Then G := Gap Gmp has exactly the same right half plane poles and zeros as that of G and ^ G = G(I + rel ) ^ G = G(I + mul): Unfortunately, this approach may be conservative if the transfer matrix has many nonminimum phase zeros or unstable poles. An alternative relative/multiplicative model reduction approach, which does not require that the transfer matrix be minimum phase but does require solving an algebraic Riccati equation, is the so-called Balanced Stochastic Truncation (BST) method. Let G(s) 2 RH1 be a square transfer matrix with a state space realization A B G(s) = C D and det(D) 6= 0. Let W (s) 2 RH1 be a minimum phase left spectral factor of G(s)G (s), i.e, W (s)W (s) = G(s)G (s): Then W (s) can be obtained as W (s) = CA BW W D
- 183. MODEL REDUCTION BY BALANCED TRUNCATION 166 with BW = PC + BD CW = D;1 (C ; BW X ) where P is the controllability Gramian given by AP + PA + BB = 0 (7.24) and X is the solution of a Riccati equation XAW + AW X + XBW (DD );1 BW X + C (DD );1 C = 0 (7:25) with AW := A ; BW (DD );1 C such that AW + BW (DD );1 BW X is stable. The realization G is said to be a balanced stochastic realization if 2 I 6 1s P =X =6 6 4 1 2 Is2 ... n Isn 3 7 7 7 5 with 1 > 2 > : : : > n 0. i is in fact the i-th Hankel singular value of the so-called phase matrix" (W (s));1 G(s). Theorem 7.8 Let G(s) 2 RH1 have the following balanced stochastic realization 2 A11 A12 B1 A B G(s) = C D = 4 A21 A22 B2 C1 C2 D 3 5 with det(D) 6= 0 and P = X = diag(M1 M2 ) where M1 = diag( 1 Is1 : : : r Isr ) M2 = diag( r+1 Isr+1 : : : n Isn ): Then is stable and ^ G = A11 B1 C1 D n Y 1+ i ^ G;1 (G ; G) 1 ;1 i=r+1 1 ; i n Y 1+ i ^ ^ G;1 (G ; G) 1 ; 1: i=r+1 1 ; i
- 184. 7.4. Notes and References 167 It can be shown that the balanced stochastic realization and the self-weighted balanced p realization in Theorem 7.5 are the same and i = i = 1 + i2 if G(s) is minimum phase. To illustrate, consider a third order stable and minimum phase transfer function s3 + 2s2 + 3s + 4 G(s) = s3 + 3s2 + 4s + 4 : It is easy to show that the Hankel singular values of the phase function is given by 1 = 0:55705372196966 2 = 0:53088390857035 3 = 0:03715882438832 and a rst order BST approximation is given by ^ s + 0 00375717470515 G = s + 0::00106883052786: The relative approximation error is 2:51522024045904 and the error bound is 3 Y 1+ i ; 1 = 2:51522024046226: i=2 1 ; i 7.4 Notes and References The balanced model reduction method was rst introduced by Moore 1981]. The stability properties of the reduced order model were shown by Pernebo and Silverman 1982]. The error bound for the balanced model reduction was shown by Enns 1984] and Glover 1984] subsequently gave an independent proof. The frequency weighted balanced model reduction method was also introduced by Enns 1984] from a somewhat di erent perspective. The error bounds for the relative and multiplicative approximations using the self-weighted balanced realization were shown by Zhou 1993]. The Balanced Stochastic Truncation (BST) method was proposed by Desai and Pal 1984] and generalized by Green 1988a,1988b] and many other people. The relative error bound for the Balanced Stochastic Truncation method was obtained by Green 1988a] and the multiplicative error bound for the BST was obtained by Wang and Safonov 1992]. It was also shown in Zhou 1993] that the frequency self-weighted method and the BST method are the same if the transfer matrix is minimum phase. Improved error bounds for the BST reduction were reported in Wang and Safonov 1990,1992] where it was claimed that the following error bounds hold n X i ^ G;1 (G ; G) 1 2 1; i i=r+1 ^ ^ G;1 (G ; G) 1 2 n X i 1; i: i=r+1
- 185. 168 MODEL REDUCTION BY BALANCED TRUNCATION However, the example in the last section gives 2 3 X i 1 ; i = 2:34052280324021 i=2 which is smaller than the actual error. Other weighted model reduction methods can be found in Al-Saggaf and Franklin 1988], Glover 1986,1989], Glover, Limebeer and Hung 1992], Hung and Glover 1986] and references therein. Discrete time balance model reduction results can be found in Al-Saggaf and Franklin 1987], Hinrichsen and Pritchard 1990], and references therein.
- 186. 8 Hankel Norm Approximation This chapter is devoted to the study of optimal Hankel norm approximation and its applications in L1 norm model reduction. The Hankel operator is introduced rst together with some time domain and frequency domain characterizations. The optimal Hankel norm approximation problem can be stated as follows: Given G(s) of McMillan ^ ^ degree n, nd G(s) of McMillan degree k < n such that G(s) ; G(s) is minimized. H The solution to this approximation problem relies on the all-pass dilation result of a square transfer function which will be given for a general class of transfer functions. The all-pass dilation results are then specialized to obtain the optimal Hankel norm approximations, which gives ^ inf G(s) ; G(s) = k+1 H where 1 > 2 : : : > k+1 : : : > n are the Hankel singular values of G(s). Moreover, we show that a square stable transfer function G(s) can be represented as G(s) = D0 + 1 E1 (s) + 2 E2 (s) + : : : + n En (s) where Ek (s) are all-pass functions and the partial sum D0 + 1 E1 (s) + 2 E2 (s) + : : : + k Ek (s) have McMillan degrees k . This representation is obtained by reducing the order one dimension at a time via optimal Hankel norm approximations. This representation also gives that kGk1 2( 1 + : : : + n ) and further that there exists a constant D0 such that kG(s) ; D0 k1 ( 1 + : : : + n ): 169
- 187. HANKEL NORM APPROXIMATION 170 The above bounds are then used to show that the k-th order optimal Hankel norm ^ approximation, G(s), together with some constant matrix D0 satis es ^ G(s) ; G(s) ; D0 1 ( k+1 + : : : + n ): We shall also provide an alternative proof for the error bounds derived in the last chapter for the truncated balanced realizations using the results obtained in this chapter. Finally we consider the Hankel operator in discrete time and o er some alternative proof of the well-known Nehari's theorem. 8.1 Hankel Operator Let G(s) 2 L1 be a matrix function. The Hankel operator associated with G will be denoted by ;G and is de ned as ? ;G : H2 7;! H2 ? ;G f := (P+ MG)f = P+ (Gf ) for f 2 H2 i.e., ;G = P+ MGjH? . This is shown in the following diagram: 2 MG ? H2 ;G > L2 P+ - ? H2 There is a corresponding Hankel operator in the time domain. Let g(t) denote the inverse (bilateral) Laplace transform of G(s). Then the time domain Hankel operator is ;g : L2 (;1 0] 7;! L2 0 1) ;g f := P+ (g f ) for f 2 L2 (;1 0]: Thus R 0 g(t ; )f ( )d t 0 (;g f )(t) = ;1 0 t < 0: Because of the isometric isomorphism property between the L2 spaces in the time domain and in the frequency domain, we have k;g k = k;G k :
- 188. 8.1. Hankel Operator 171 Hence, in this book we will use the time domain and the frequency domain descriptions for Hankel operators interchangeably. For the interest of this book, we will now further restrict G to be rational, i.e., G(s) 2 RL1 . Then G can be decomposed into strictly causal part and anticausal part, i.e., there are Gs (s) 2 RH2 and Gu (s) 2 RH? such that 2 G(s) = Gs (s) + G(1) + Gu (s): ? Now for any f 2 H2 , it is easy to see that ;Gf = P+ (Gf ) = P+ (Gs f ): Hence, the Hankel operator associated with G 2 RL1 depends only on the strictly causal part of G. In particular, if G is antistable, i.e., G (s) 2 RH1 , then ;G = 0. Therefore, there is no loss of generality in assuming G 2 RH1 and strictly proper. The adjoint operator of ;G can be computed easily from the de nition as below: let ? f 2 H2 g 2 H2 , then h;G f gi := hP+ Gf gi = hP+ Gf gi + hP; Gf gi (since P; Gf and g are orthogonal ) = hGf gi = hf G gi = hf P+ G gi + hf P; G gi = hf P; G gi (since f and P+ G g are orthogonal ): ? Hence ;Gg = P; (G g) : H2 7;! H2 or ;G = P; MG jH2 . Now suppose G 2 RH1 has a state space realization as given below: x = Ax + Bu _ y = Cx with A stable and x(;1) = 0. Then the Hankel operator ;g can be written as ;g u(t) = Z0 ;1 CeA(t; )Bu( )d for t 0 and has the interpretation of the system future output y(t) = ;g u(t) t 0 based on the past input u(t) t 0. In the state space representation, the Hankel operator can be more speci cally decomposed as the composition of maps from the past input to the initial state and then
- 189. HANKEL NORM APPROXIMATION 172 Γ g u(t) 0 y(t) t 0 t Figure 8.1: System theoretical interpretation of Hankel operators from the initial state to the future output. These two operators will be called the controllability operator, c , and the observability operator, o, respectively, and are de ned as c : L2 (;1 0] 7;! C n c u := and Z0 ;1 e;A Bu( )d o : C n 7;! L2 0 1) o x0 := CeAt x0 t 0: (If all the data are real, then the two operators become c : L2 (;1 0] 7;! Rn and o : Rn 7;! L2 0 1).) Clearly, x0 = c u(t) for u(t) 2 L2 (;1 0] is the system state at t = 0 due to the past input and y(t) = o x0 t 0, is the future output due to the initial state x0 with the input set to zero. It is easy to verify that ;g = o c : c L2 (;1 0] > Cn ;g ZZ Z o ZZ ~ Z - L2 0 1 ) The adjoint operators of c and o can also be obtained easily from their de nitions as follows: let u(t) 2 L2 (;1 0], x0 2 C n , and y(t) 2 L2 0 1), then h cu x0 iC n = Z0 u ( )B e;A x0 d = hu B e;A x0 iL2 (;1 0] = hu c x0 iL2 (;1 0] ;1
- 190. 8.1. Hankel Operator and h o x0 yiL2 0 1) = Z1 0 x 173 (t)eA t C y(t)dt = hx0 Z1 0 eA t C y(t)dtiC n = hx0 o yiC n where h iX denotes the inner product in the Hilbert space X . Therefore, we have n c : C 7;! L2 (;1 0] ;A x0 0 c x0 = B e and n o : L2 0 1) 7;! C o y (t) = Z1 0 eA t C y(t)dt: This also gives the adjoint of ;g : ;g = ( o c) = c o : L2 0 1) 7;! L2 (;1 0] ;g y = Z1 0 B eA (t; )C y(t)dt 0: Let Lc and Lo be the controllability and observability Gramians of the system, i.e., Lc = Lo = Then we have Z1 Z0 0 1 eAt BB eA t dt eA t C CeAt dt: c c x0 = o o x0 = Lcx0 Lox0 for every x0 2 C n . Thus Lc and Lo are the matrix representations of the operators c c and o o . Theorem 8.1 The operator ;g ;g (or ;G;G) and the matrix LcLo have the same p nonzero eigenvalues. In particular k;g k = (Lc Lo). Proof. Let 6= 0 be an eigenvalue of ;g ;g , and let 0 6= u 2 L2 (;1 0] be a corresponding eigenvector. Then by de nition ;g ;g u = c o o cu = 2 u: (8:1) Pre-multiply (8.1) by c and de ne x = cu 2 C n to get LcLo x = 2 x: (8:2) 2
- 191. HANKEL NORM APPROXIMATION 174 Note that x = cu 6= 0 since otherwise 2 u = 0 from (8.1) which is impossible. So 2 is an eigenvalue of LcLo . On the other hand, suppose 2 6= 0 and x 6= 0 are an eigenvalue and a corresponding eigenvector of LcLo . Pre-multiply (8.2) by c Lo and de ne u = c Lox to get (8.1). It is easy to see that u 6= 0 since c u = c c Lox = LcLo x = 2 x 6= 0. Therefore 2 is an eigenvalue of ;g ;g . Finally, since G(s) is rational, ;g ;g is compact and self-adjoint and has only discrete spectrum. Hence k;g k2 = ;g ;g = (Lc Lo). 2 Remark 8.1 Let 6= 0 be an eigenvalue of ;g ;g and 0 6= u 2 L2 (;1 0] be a corresponding eigenvector. De ne v := 1 ; u 2 L 0 1): 2 g 2 Then (u v) satisfy ;g u = ;g v = v u: This pair of vectors (u v) are called a Schmidt pair of ;g . The proof given above suggests a way to construct this pair: nd the eigenvalues and eigenvectors of LcLo , i.e., i2 and xi such that Lc Loxi = i2 xi : Then the pairs (ui vi ) given below are the corresponding Schmidt pairs: u = ( 1 L x ) 2 L (;1 0] v = x 2 L 0 1): i c i o i i 2 o i 2 ~ Remark 8.2 As seen in various literature, there are some alternative ways to write a Hankel operator. For comparison, let us examine some of the alternatives below: (i) Let v(t) = u(;t) for u(t) 2 L2 (;1 0], and then v(t) 2 L2 0 1). Hence, the Hankel operator can be written as ;g : L2 0 1) 7;! L2 0 1) or ;G : H2 7;! H2 (;g v)(t) = = R 1 g(t + )v( )d 0 t 0 t<0 1 CeA(t+ )Bv( )d for t 0: 0 Z 0
- 192. 8.2. All-pass Dilations 175 (ii) In some applications, it is more convenient to work with an anticausal operator G and view the Hankel operator associated with G as the mapping from the future input to the past output. It will be clear in later chapters that this operator is closely related to the problem of approximating an anticausal function by a causal function, which is the problem at the heart of the H1 Control Theory. A Let G(s) = C B be an antistable transfer matrix, i.e., all the eigenvalues of 0 A have positive real parts. Then the Hankel operator associated with G(s) can be written as ^ ;g : L2 0 1) 7;! L2 (;1 0] ^ (;g v)(t) = = R 1 g(t ; )v( )d 0 t 0 t>0 1 CeA(t; )Bv( )d for t 0 0 Z 0 or in the frequency domain ? ^ ;G = P; MGjH2 : H2 7;! H2 ^ ;Gv = P; (Gv) for v 2 H2 : ? Now for any v 2 H2 and u 2 H2 , we have hP; (Gv) ui = hGv ui = hv G ui = hv P+ (G u)i: ^ Hence, ;G = ;G . ~ 8.2 All-pass Dilations This section considers the dilation of a given transfer function to an all-pass transfer function. This transfer function dilation is the key to the optimal Hankel norm approximation in the next section. But rst we need some preliminary results and some state space characterizations of all-pass functions. De nition 8.1 The inertia of a general complex, square matrix A denoted In(A) is the triple ( (A) (A) (A)) where (A) = number of eigenvalues of A in the open right half-plane. (A) = number of eigenvalues of A in the open left half-plane. (A) = number of eigenvalues of A on the imaginary axis.
- 193. HANKEL NORM APPROXIMATION 176 Theorem 8.2 Given complex n n and n m matrices A and B, and hermitian matrix P = P satisfying then (1) If (P ) = 0 then (A) (2) If (A) = 0 then (P ) AP + PA + BB = 0 (P ) (A) (A) (P ) (8:3) (P ). (A). Proof. (1) If (P ) = 0 then observe that (8.3) implies AP + PA 0: Now suppose (A) > (P ) then there is an eigenvalue of A, 2 C with Re( ) > 0, and a corresponding eigenvector, x 2 C n , such that x A = x and x Px > 0, which implies x (AP + PA )x = ( + )x Px 0 i.e., Re( ) 0, a contradiction. Hence (A) (P ). Similarly, we can show that (A) (P ). (2) Assume (A) = 0 and that P = U P1 0 U with (P1 ) = 0 U U = I , and 0 0 de ne ~ ~ A = U AU = A11 A12 B = U B = B1 : A A B Then U (8:3)U gives 21 22 ~ ~ ~~ A P1 0 + P1 0 A + B B = 0 0 0 0 (8:4) ) B2 B2 = 0 ) B2 = (8:4) (8:5) ) A21 P1 = 0 ) A21 = (8:4) ) A11 P1 + P1 A11 + B1 B1 = ) (by part (1)) (A11 ) (A11 ) but since (A11 ) = (P1 ) = 0 (P1 ) = (A11 ) (A) (P1 ) = (A11 ) (A): 2 0 (8.4) 0 0 0 (8.5) (8.6) (8.7) (P1 ) (P1 ) 2 Theorem 8.3 Given a realization (A B C ) (not necessarily stable) with A 2 C n n , B 2 C n m , C 2 C m n , then
- 194. 8.2. All-pass Dilations 177 (1) If (A B C ) is completely controllable and completely observable the following two statements are equivalent: (a) there exists a D such that GG = 2 I where G(s) = D + C (sI ; A);1 B . (b) there exist P Q 2 C n n such that (i) P = P , Q = Q (ii) AP + PA + BB = 0 (iii) A Q + QA + C C = 0 (iv) PQ = 2 I (2) Given that part (1b) is satis ed then there exists a D satisfying D D = 2I D C +B Q = 0 DB + CP = 0 and any such D will satisfy part (1a) (note, observability and controllability are not assumed). Proof. Any systems satisfying part (1a) or (1b) can be transformed to the case = 1 p ^ p ^ ^ ^ ^ by B = B= , C = C= , D = D= , P = P= , Q = Q= . Hence, without loss of generality the proof will be given for the case = 1 only. (1a) ) (1b) This is proved by constructing P and Q to satisfy (1b) as follows. Given (1a), G(1) = D ) DD = I . Also GG = I ) G = G;1 , i.e., A ; BD;1 C ;BD;1 = A ; BD C ;BD D;1 C D;1 DC D = G = ;A ;C : B D These two transfer functions are identical and both minimal (since (A B C ) is assumed to be minimal), and hence there exists a similarity transformation T relating the stateG;1 (s) = space descriptions, i.e., Further ;A = T (A ; BD C )T ;1 C B (8:9) ) B (8:10) ) C (8:8) ) ;A = TBD = D CT ;1 : = = = = (8:9) and (8:10) ) = D C (T );1 T BD ;C DB + (T ;1A T ) T (A ; (T );1 C DB T )(T );1 T (A ; BD C )(T );1 : (8.8) (8.9) (8.10) (8.11) (8.12) (8.13)
- 195. HANKEL NORM APPROXIMATION 178 Hence, T and T satisfy identical equations, (8.8) to (8.10) and (8.11) to (8.13), and minimality implies these have a unique solution and hence T = T . Now setting Q = ;T P = ;T ;1 (8.14) (8.15) clearly satis es part (1b), equations (i) and (iv). Further, (8.8) and (8.9) imply TA + A T ; C C = 0 (8:16) which veri es (1b), equation (iii). Also (8.16) implies AT ;1 + T ;1A ; T ;1 C CT ;1 = 0 (8:17) which together with (8.10) implies part (1b), equation (ii). (1b) ) (1a) This is proved by rst constructing D according to part (2) and then verifying part (1a) by calculation. Firstly note that since Q = P ;1 , Q ((1b) equation (ii)) Q gives QA + A Q + QBB Q = 0 (8:18) which together with part (1b), equation (iii) implies that QBB Q = C C (8:19) and hence by Lemma 2.14 there exists a D such that D D = I and DB Q = ;C DB = ;CQ;1 = ;CP: (8.20) (8.21) Equations (8.20) and (8.21) imply that the conditions of part (2) are satis ed. Now note that = (sI ; A)P + P (;sI ; A ) = CP (;sI ; A );1 C + C (sI ; A);1 PC (8:21) ) = ;DB (;sI ; A );1 C ; C (sI ; A);1 BD : BB ) C (sI ; A);1 BB (;sI ; A );1 C Hence, on expanding G(s)G (s) we get G(s)G = I: Part (2) follows immediately from the proof of (1b) ) (1a) above. The following theorem dilates a given transfer function to an all-pass function. 2
- 196. 8.2. All-pass Dilations 179 A B Theorem 8.4 Let G(s) = C D with A 2 C n B 2 Cn m C 2 Cm n D 2 AP + PA + BB = 0 A Q + QA + C C = 0 (8.22) (8.23) P = P = diag( Q = Q = diag( Cm n (8.24) (8.25) m satisfy for 1 2 Ir ) Ir ) with 1 and 2 diagonal, 6= 0 and ( 1 2 ; 2 I ) = 0. Partition (A B C ) conformably with P , as A = A11 A12 A21 A22 ^ ^ and de ne W (s) := A B ^ D C ^ ^ A ^ B ^ C ^ D = = = = B = B1 B2 C = C1 C2 (8:26) with ;;1 ( 2 A11 + 2 A11 ;;1 ( 2 B1 + C1 U ) C1 1 + UB1 1; C1 UB1 ) D; U (8.27) (8.28) (8.29) (8.30) where U is a unitary matrix satisfying B2 = ;C2 U (8.31) and ; = 1 2; 2 I: (8.32) Also de ne the error system B E (s) = G(s) ; W (s) = Ae De Ce e with 0 Ae = A A 0 ^ Then Be = B ^ B ^ Ce = C ;C ^ De = D ; D: (8:33)
- 197. HANKEL NORM APPROXIMATION 180 (1) (Ae Be Ce ) satisfy Ae Pe + Pe Ae + Be Be = 0 Ae Qe + Qe Ae + Ce Ce = 0 with 2 Pe = 4 2 Qe = Pe Qe = 4 1 0 0 Ir I (8.34) (8.35) 3 0 5 I 0 2 ;;1 3 0 ;; 2 0 Ir 0 5 ;; 0 1 ; 2I (8.36) (8.37) (8.38) (2) E (s)E (s) = 2 I . (3) If (A) = 0 then ^ (a) (A) = 0 (b) If ( 1 2 ) = 0 then ^ In(A) = In(; 1 ;) = In(; 2 ;) ^ ^ ^ (c) If P > 0 Q > 0 then the McMillan degree of the stable part of (A B C ) equals ( 1 ;) = ( 2 ;). (d) If either (i) 1 ; > 0 and 2 ; > 0 or (ii) 1 ; < 0 and 2 ; < 0 then ^ ^ ^ (A B C ) is a minimal realization. Proof. For notational convenience it will be assumed that = 1 and this can be done without loss of generality since B C and can be simply rescaled to give = 1. It is rst necessary to verify that there exists a unitary matrix U satisfying (8.31). The (2,2) blocks of (8.22) and (8.23) give A22 + A22 + B2 B2 = 0 A22 + A22 + C2 C2 = 0 (8.39) (8.40) and hence B2 B2 = C2 C2 and by Lemma 2.14 there exists a unitary U satisfying (8.31). (1) The proof of equations (8.34) to (8.38) is by a straightforward calculation, as follows. To verify (8.34) and (8.36) we need (8.22) which is assumed, together with ^ ^ A11 + A + B1 B = 0 (8.41) ^ A21 + B2 B = 0 (8.42) ^ 2 ;;1 + 2 ;;1 A + B B = 0 ^ ^^ A (8.43)
- 198. 8.2. All-pass Dilations which will now be veri ed. ^ B2 B = (8.31) ) = = = 181 B2 (B1 2 + U C1 );;1 (B2 B1 2 ; C2 C1 );;1 ((;A21 1 ; A12 ) 2 + A12 2 + A21 );;1 ;A21 ) (8:42) (8.44) (8.45) (8.46) where B2 B1 and C2 C1 were substituted in (8.45) using the (2,1) blocks of (8.22) and (8.23), respectively. To verify (8.41) ^ B1 B = (B1 B1 2 + B1 U C1 );;1 = (;A11 1 2 ; 1 A11 2 + B1 U C1 );;1 ^ = ;A11 ; A ) (8:41) (8.47) (8.48) where B1 B1 was substituted using the (1 1) block of (8.22) and (8.27) substituting in (8.48). Finally to verify (8.43) consider ^ ^ ;A 2 + 2 A ; = (A11 + 2 A11 1 ; C1 UB1 ) 2 + 2 (A11 + 1 A11 2 ; B1 U C1 ) (8.49) = ;( 2 B1 + C1 U )(B1 2 + U C1 ) + (A11 2 + 2 A11 + C1 C1 ) + 2 (A11 1 + 1 A11 + B1 B1 ) 2 (8.50) ^^ = ;;BB ; ) (8:43) where (8.27)!(8.49) and (8.50) is a rearrangement of (8.49) and nally, the (1 1) blocks of (8.22) and (8.23) are used. Equations (8.34) and (8.36) are hence veri ed. Similarly in order to verify (8.35) and (8.37) we need (8.23) which is assumed together with ^ ^ A11 (;;) + (;;)A ; C1 C = 0 (8.51) ^ = 0 A12 (;;) ; C2 C (8.52) ^ ^ ^ ^ A 1 ; + 1 ;A + C C = 0: (8.53) Equations (8.51) to (8.53) are now veri ed in an analogous manner to equations (8.41) to (8.43) ^ C2 C = C2 (C1 1 + UB1 ) = (;A12 2 ; A21 ) 1 ; B2 B1 = ;A12 ; ) (8:52) ^ C1 C = C1 C1 1 + C1 UB1 = ;A11 2 1 ; 2 A11 1 + C1 UB1
- 199. 182 HANKEL NORM APPROXIMATION ^ = ;A11 ; ; ;A ) (8:51) ^ ^ A 1 ; + 1 ;A = (A11 + 1 A11 2 ; B1 U C1 ) 1 + 1 (A11 + 2 A11 1 ; C1 UB1 ) = ;( 1 C1 + B1 U )(C1 1 + UB1 ) + 1 (A11 2 + 2 A11 + C1 C1 ) 1 + (A11 1 + 1 A11 + B1 B1 ) ^ C ) (8:53) = ;C ^ Therefore, (8.35) and (8.37) have been veri ed, (8.38) is immediate, and the proof of part (1) is complete. (2) Equations (8.34), (8.35) and (8.38) ensure the conditions of Theorem 8.3, part (1b) are satis ed and Theorem 8.3, part (2) can be used to show that the De given in (8.33) makes E (s) all-pass. (Note it is still assumed that = 1.) We hence need to verify that De De = I (8.54) De Ce + Be Qe = 0 (8.55) De Be + Ce Pe = 0: (8.56) ^ ^ Equation (8.54) is immediate, (8.55) follows by substituting the de nitions of B C De and Q, and (8.56) follows from De (8:55) Pe . ^ (3) (a) To show that (A) = 0 if (A) = 0 we will assume that there exists x 2 C n;r ^ = x and + = 0, and show that this implies x = 0. From and 2 C such that Ax x (8:53)x, ^ ^ ( + )x 1 ;x + x C Cx = 0 (8.57) ^ ) Cx = 0: (8.58) Now (8:51)x gives ^ ;A11 ;x ; ; x + C1 Cx = 0 ) x ;A11 = ; x ;: (8.59) Also (8:52)x and (8.58) give A12 ;x = 0: (8:60) Equations (8.59) and (8.60) imply that (x ; 0)A = ; (x ; 0) but since it is assumed ^ that (A) = 0, + = 0 and ;;1 exists this implies that x = 0 and (A) = 0 is proven. ^ (b) Since (A) = 0 has been proved and ( 1 2 ) = 0 is assumed () ( 2 ;;1 ) = ( 1 ;) = 0) Theorem 8.2 can be applied since equations (8.43) and (8.53) have been veri ed. Hence ^ In(A) = In(; 1 ;;1 ) = In(; 1 ;) = In(; 2 ;) ^ (c) Assume that there exists x 6= 0 2 C n;r and 2 C such that Ax = x and ^ ^ ^ Cx = 0 (i.e., (C A) is not completely observable). Then (8:51)x and (8:52)x give ;A11 ;x ; ; x = 0 ;A12 ;x = 0
- 200. 8.2. All-pass Dilations 183 hence (; ) is an eigenvalue of A since ;x 6= 0. However, since P > 0 and (A) = 0 are assumed then In(A) = (0 n 0) and all the unobservable modes must be in the open ^ ^ right half plane. Similarly, if it is assumed that (A B ) is not completely controllable then (8.41) and (8.42) will give the analogous conclusion and therefore all the modes in the left half-plane are controllable and observable, and the condition in (3b) gives their number. ^ (d) (i) If 1 ; > 0 or 2 ; > 0 then by (3b) In(A) = (0 n ; r 0) and by (3c) the ^ B C ) is n ; r and the result is proven. ^ ^ McMillan degree of (A ^ ^ (ii) Assume there exists x such that Ax = x and Cx = 0. Then x (8:53)x gives ( + )x 1 ;x = 0 ^ ^ but ( + ) 6= 0 by (3a) and 1 ; < 0 is assumed so that x = 0. Hence (C A) is ^ B ) completely controllable. ^ completely observable. Similarly (8.43) gives (A 2 Example 8.1 Take 2 + 105 G(s) = 39(s + 2)(s s + 250 : s + 5)2 This has a balanced realization given by 2 3 2 3 ;2 4 ;4 2 A = 4 ;4 ;1 ;4 5 B = 4 1 5 C = 2 ;1 6 ;4 4 ;9 6 Now using the above construction with 1 2 =4 1 0 0 0 1 0 2 0 0 2 3 5 = 1 0 , = 2 gives 1 0 2 2 ^ ^ ^ ^ 1 2 B = 1 ;2 C = ;2 ;5=2 D = 2 A = 3 ;8 10 5 3 2 + 90 + 2)( + 2 (3 2 + + 30) W (s) = 63ss2; 13ss+ 30 G(s) ; W (s) = 2(;ss+ 2)(s;s 5)25) s2 s 7s 7s 30) ;7 ( + (3 ; + W (s) is an optimal anticausal approximation to G(s) with L1 error of 2. 3 Example 8.2 Let us also illustrate Theorem 8.4 when ( 2 ; 2 I ) is inde nite. Take 1 G(s) as in the above example and permute the rst and third states of the balanced 1 realization so that = diag(2 2 1), 1 = diag(2 1 ), = 1. The construction of 2 Theorem 8.4 now gives 2 ^ ^ ^ ^ A = ;3 2 B = ;2 C = 6 ;3=2 D = 1: 8 3 Theorem 8.4, part (2b) implies that ^ In(A) = In(; 1 ( 2 ; 2 I )) = In ;6 0 = (1 1 0) 3 1 0 8
- 201. HANKEL NORM APPROXIMATION 184 ^ which is veri ed by noting that A has eigenvalues of 5 and ;5. " # ^ ^ A B W (s) = ^ ^ = ss+ 20 +5 C D and we note that the stable part of W (s) has McMillan degree 1 as predicted by Theo^ ^ ^ rem 8.4, part (3c). However, this example has been constructed to show that (A B C ) itself may not be minimal when the conditions of part (3d) are not satis ed, and in this case the unstable pole at +5 is both uncontrollable and unobservable. ss+20 is in fact +5 an optimal Hankel norm approximation to G(s) of degree 1 and s + 2)(;s E (s) = (;s + 2)(s ++ 5) : ( 5)2 2 In general the error E (j!) will have modulus equal to but E (s) will contain unstable poles. 3 Example 8.3 Let us nally complete the analysis of this G(s) by permuting the second and third states in the balanced realization of the last example to obtain 1 = 2 . We will nd ^ A = ;15 ;4 ;20 ;6 ^ B= 4 4 " ^ C = 15 3 1 = diag(2 1), ^ D = ;1 2 # ^ ^ A B 1 2 + 110) W (s) = ^ ^ = 2 (;(s 2 + 123s+ 10) s + 21s C D s s 2 2 s E (s) = G(s) ; W (s) = ; 1 (;s + 2)(;++ 5) s(s+; 21++ 10) : 2 ( + 2)(s 5)2 ( 2 21s 10) ^ Note that ; 1 ; = diag(;15=2 ;3=4) so that A is stable by Theorem 8.4, part (3b). ^ ^ ^ jE (j!)j = 1 by Theorem 8.4, part (2), (A B C ) is minimal by Theorem 8.4, part (3d). 2 W (s) is in fact an optimal second-order Hankel norm approximation to G(s). 3 8.3 Optimal Hankel Norm Approximation We are now ready to give a solution to the optimal Hankel norm approximation problem based on Theorem 8.4. The following Lemma gives a lower bound on the achievable Hankel norm of the error ^ inf G(s) ; G(s) k+1 (G) ^ G H and then Theorem 8.6 shows that the construction of Theorem 8.4 can be used to achieve this lower bound.
- 202. 8.3. Optimal Hankel Norm Approximation 185 Lemma 8.5 Given a stable, rational, p m, transfer function matrix G(s) with Hankel singular values 1 2::: and of McMillan degree k k ^ i (G(s) ; G(s)) ^ i+k (G(s) ; G(s)) In particular, k+2 : : : k+1 n ^ > 0, then for all G(s) stable i+k (G(s)) i = 1 : : : n ; k i (G(s)) i = 1 : : : n: ^ G(s) ; G(s) H k+1 (G(s)): (8.61) (8.62) (8:63) Proof. We shall prove (8.61) only and the inequality (8.62) follows from (8.61) by setting ^ ^ G(s) = (G(s) ; G(s)) ; (;G(s)): ^ ^ ^ ^ Let (A B C ) be a minimal state space realization of G(s), then (Ae Be Ce ) given by ^ (s). Now let P = P and Q = Q (8.33) will be a state space realization of G(s) ; G satisfy (8.34) and (8.35) respectively (but not necessary (8.36) and (8.37) and write P = P11 P12 P12 P22 Since P Q = Q11 Q12 Q12 Q22 P11 Q11 2 Rn n : 0 it can be factorized as P = RR where 11 R = R0 R12 R22 with 1 ; R22 = P22=2 R12 = P12 P221=2 R11 R11 = P11 ; R12 R12 ^ ^ ^ (P22 > 0 since (A B C ) is a minimal realization.) ^ i (G(s) ; G(s)) = i (PQ) = i (RR Q) = i (R QR) n In 0 R QR I0 i R 0 Q R11 = i = = = = 11 0 i (R11 Q11 R11 ) = i (Q11 R11 R11 ) i (Q11 (P11 ; R12 R12 )) 1=2 1=2 1=2 i (Q11 P11 Q11 ; XX ) where X = Q11 R12 1=2 1=2 i+k (Q11 P11 Q11 ) 2 i+k (P11 Q11 ) = i+k (G) (8.64)
- 203. HANKEL NORM APPROXIMATION 186 where (8.64) follows from the fact that X is an n k matrix () rank(XX ) k). 2 We can now give a solution to the optimal Hankel norm approximation problem for square transfer functions. Theorem 8.6 Given a stable, rational, m m, transfer function G(s) then ^ ^ (1) k+1 (G(s)) = inf G(s) ; G(s) ; F (s) , McMillan degree (G) k. ^ G2H1 F 2H; 1 1 (2) If G(s) has Hankel singular values 1 2 : : : k > k+1 = k+2 : : : = k+r > ^ k+r+1 : : : n > 0 then G(s) of McMillan degree k is an optimal Hankel norm ; approximation to G(s) if and only if there exists F (s) 2 H1 (whose McMillan ^ degree can be chosen n + k ; 1) such that E (s) := G(s) ; G(s) ; F (s) satis es 2 E (s)E (s) = k+1 I: (8:65) In which case ^ G(s) ; G(s) = k+1 : (8:66) H (3) Let G(s) be as in (2) above, then an optimal Hankel norm approximation of McMil^ lan degree k, G(s), can be constructed as follows. Let (A B C ) be a balanced realization of G(s) with corresponding = diag( 1 2 : : : k k+r+1 : : : n k+1 : : : k+r ) ^ ^ ^ ^ and de ne (A B C D) from equations (8.26) to (8.31). Then # " ^ ^ ^ (s) + F (s) = A B (8:67) G ^ ^ C D ; ^ ^ where G(s) 2 H1 and F (s) 2 H1 with the McMillan degree of G(s) = k and the McMillan degree of F (s) = n ; k ; r. ; ^ Proof. By the de nition of L1 norm, for all F (s) 2 H1, and G(s) of McMillan degree k ^ G(s) ; G(s) ; F (s) 1 sup ^ (G(s) ; G(s) ; F (s))f sup ^ P+ (G ; G ; F )f sup ^ P+ (G ; G)f f 2H? kf k2 1 2 f 2H? kf k2 1 2 = = f 2H? kf k2 1 2 ^ G;G H k+1 (G(s)) 2 2 2 (8.68)
- 204. 8.3. Optimal Hankel Norm Approximation 187 where (8.68) follows from Lemma 8.5. ^ Now de ne G(s) and F (s) via equation (8.26), then Theorem 8.4, part (2) implies that (8.65) holds and hence kE (s)k1 = k+1 : (8:69) Also from Theorem 8.4, part (3b) 2 ^ In(A) = In(; 1 ( 2 ; k+1 I )) = (n ; k ; r k 0): (8:70) 1 ^ Hence, G has McMillan degree k and it in the correct class, and therefore (8.69) implies that the inequalities in (8.68) becomes equalities, and part (1) is proven, as in part (3). Clearly the su ciency of part (2) can be similarly veri ed by noting that (8.65) implies that (8.68) is satis ed with equality. ^ To show the necessity of part (2) suppose that G(s) is an optimal Hankel norm approximation to G(s) of McMillan degree k, i.e, equation (8.66) holds. Now Theorem 8.4 ^ can be applied to G(s) ; G(s) to produce an optimal anticausal approximation F (s), ^ (s) ; F (s))= k+1 (G) is all-pass since k+1 (G) = 1 (G ; G). Further, ^ such that (G(s) ; G ^ the McMillan degree of this F (s) will be, the McMillan degree of (G(s) ; G(s)) minus ^ ), n + k ; 1. the multiplicity of 1 (G ; G 2 The following corollary gives the solution to the well-known Nehari's problem. Corollary 8.7 Let G(s) be a stable, rational, m m, transfer function of McMillan degree n such that 1 (G) has multiplicity r1 . Also let F (s) be an optimal anticausal approximation of degree n ; r1 given by the construction of Theorem 8.4. Then (1) (G(s) ; F (s))= 1 is all-pass. (2) i;r1 (F (;s)) = i (G(s)) i = r1 + 1 : : : n. Proof. (1) is proved in Theorem 8.4, part (2). (2) is obtained from the forms of Pe and Qe in Theorem 8.4, part (1). F (;s) is used since it will be stable and have well-de ned Hankel singular values. 2 The optimal Hankel norm approximation for non-square case can be obtained by rst augmenting the function to form a square function. For example, consider a stable, rational, p m (p < m), transfer function G(s). Let Ga = G be an augmented square 0 ^ G be the optimal Hankel norm approximation of ^ transfer function and let Ga = ^ G2 Ga such that ^ Ga ; Ga H = k+1 (Ga ): Then ^ ^ G ; G H Ga ; Ga H = k+1 (Ga ) = k+1 (G) k+1 (G) ^ i.e., G is an optimal Hankel norm approximation of G(s).
- 205. HANKEL NORM APPROXIMATION 188 8.4 L1 Bounds for Hankel Norm Approximation The natural question that arise now is, does the Hankel norm being small imply that any other more familiar norms are also small? We shall have a de nite answer in this section. A B Lemma 8.8 Let an m m transfer matrix E = C D satisfy E (s)E (s) = 2 I and all equations of Theorem 8.3 and let A have dimension n1 + n2 with n1 eigenvalues strictly in the left half plane and n2 < n1 eigenvalues strictly in the right half plane. If E = Gs + F with Gs 2 RH1 and F 2 RH; then, 1 i = 1 2 : : : n1 ; n2 i (Gs ) = i = n1 ; n2 + 1 : : : n1 i;n1 +n2 (F (;s)) Proof. Firstly let the realization be transformed to, 2 3 A1 0 B1 A B E = 4 0 A2 B2 5 = C D Re i (A1 ) < 0 Re i (A2 ) > 0 C1 C2 D in which case G = A1 B1 , F = A2 B2 . The equations of Theorem 8.3 C1 D C2 0 (i)-(iv) are then satis ed by a transformed P and Q, partitioned as, P = P11 P12 Q = Q11 Q21 P12 P22 Q21 Q22 PQ = 2 I implies that, det( I ; P11 Q11 ) = det( I ; ( 2 I ; P12 Q21 )) = det(( ; 2 )I + P12 Q21 ) = ( ; 2 )n1 ;n2 det(( ; 2 )I + Q21 P12 ) = ( ; 2 )n1 ;n2 det( I ; Q22 P22 ): The result now follows on observing that i (G(s)) = i (P11 Q11 ) and i2 (F (;s)) = 2 i (Q22 P22 ): ^ Corollary 8.9 Let E (s) = G(s) ; G(s) ; F (s) be as de ned in part (3) of Theorem 8.6 ^ with G(s) G(s) 2 RH1 and F (s) 2 RH; . Then for i = 1 2 ::: 2k + r, 1 ^ i (G ; G) = k+1 (G) and for i = 1 2 ::: n ; k ; r, ^ i+3k+r (G) i (F (;s)) = i+2k+r (G ; G) i+k+r (G)
- 206. 8.4. L1 Bounds for Hankel Norm Approximation 189 Proof. The construction of E (s) ensures that the all-pass equations are satis ed and an inertia argument easily establishes that the A-matrix has precisely n1 = n + k eigenvalues in the open left half plane and n2 = n ; k ; r in the open right half plane. Hence Lemma 8.8 can be applied to give the equalities. The inequalities follow from Lemma 8.5. 2 The following lemma gives the properties of certain optimal Hankel norm approximations when the degree is reduced by the multiplicity of n . In this case some precise statements on the error and the approximating system can be made. Lemma 8.10 Let G(s) be a stable, rational m m, transfer function of McMillan ^ degree n and such that n (G) has multiplicity r. Also let G(s) be an optimal Hankel norm approximation of degree n ; r given by Theorem 8.6, part (3) (with F (s) 0) then ^ (1) (G(s) ; G(s))= n (G(s)) is all-pass. ^ (2) i (G(s)) = i (G(s)) i = 1 : : : n ; r. ^ Proof. Theorem 8.6 gives that A 2 R(n;r) (n;r) is stable and hence F (s) can be ^ ^ chosen to be zero and therefore (G(s) ; G(s))= n (G) is all-pass. The i (G(s)) are 2 obtained from Lemma 8.8. ^ ^ Applying the above reduction procedure again on G(s) and repeating until G(s) has zero McMillan degree gives the following new representation of stable systems. Theorem 8.11 Let G(s) be a stable, rational m m, transfer function with Hankel singular values 1 > 2 : : : > N where i has multiplicity ri and r1 + r2 + : : : + rN = n. Then there exists a representation of G(s) as G(s) = D0 + 1 E1 (s) + 2 E2 (s) + : : : + N EN (s) where (1) Ek (s) are all-pass and stable for all k. (2) For k = 1 2 : : : N ^ Gk (s) := D0 + has McMillan degree r1 + r2 + : : : + rk . k X i=1 i Ei (s) (8:71)
- 207. HANKEL NORM APPROXIMATION 190 ^ ^ Proof. Let Gk (s) be the optimal Hankel norm approximation to Gk+1 (s) (given by ^ Lemma 8.10) of degree r1 + r2 + : : : + rk , with GN (s) := G(s). Lemma 8.10 (2) applied ^ at each step then gives that the Hankel singular values of Gk (s) will be 1 2 : : : k ^ with multiplicities r1 r2 : : : rk , respectively. Hence Lemma 8.10 (1) gives that Gk (s) ; ^ k;1 (s) = k Ek (s) for some stable, all-pass Ek (s). Note also that Theorem 8.4, part G ^ (3d), relation (i) also ensures that each Gk (s) will have McMillan degree r1 + r2 + : : : + rk . ^ 0 (s) which will be a constant and combining the steps gives the Finally taking D0 = G result. 2 Note that the construction of Theorem 8.11 immediately gives an approximation ^ algorithm that will satisfy G(s) ; G(s) k+1 + k+2 + : : : + N . This will not be 1 an optimal Hankel norm approximation in general, but would involve less computation ^ since the decomposition into G(s) = F (s) need not be done, and at each step a balanced ^ ^ ^ ^ realization of Gk (s) is given by (Ak Bk Ck ) with a diagonal scaling. An upper-bound on the L1 norm of G(s) is now obtained as an immediate consequence of Theorem 8.11. Corollary 8.12 Let G(s) be a stable, rational p m, transfer function with Hankel singular values 1 > 2 > : : : > N , where each i has multiplicity ri , and such that G(1) = 0. Then (1) kG(s)k1 2( 1 + 2 + : : : + N ) (2) there exists a constant D0 such that kG(s) ; D0 k1 1 + 2 + : : : + N: Proof. For p = m consider the representation of G(s) given by Theorem 8.11 then kG(s) ; D0 k1 = k 1 E1 (s) + 2 E2 (s) + : : : + N EN (s)k1 1 + 2 +::: + N since Ek (s) are all-pass. Further setting s = 1, since G(1) = 0, gives kD0 k ) kG(s)k1 1 + 2 +:::+ N 2( 1 + 2 + : : : + N ): For the case p 6= m just augment G(s) by zero rows or columns to make it square, but will have the same L1 norm, then the above argument gives upper bounds on the L1 norm of this augmented system. 2 Theorem 8.13 Given a stable, rational, m m, transfer function G(s) with Hankel singular values 1 2 ::: k > k+1 = k+2 : : : = k+r > k+r+1 ::: n >0
- 208. 8.4. L1 Bounds for Hankel Norm Approximation 191 ^ and let G(s) 2 RH1 of McMillan degree k be an optimal Hankel norm approximation to G(s) obtained in Theorem 8.6, then there exists a D0 such that ^ k+1 (G) kG ; G ; D0 k1 k+1 (G) + n;k;r X i=1 i+k+r (G): Proof. The theorem follows from Corollary 8.9 and Corollary 8.12. 2 ^ It should be noted that if G is an optimal Hankel norm approximation of G then ^ + D for any constant matrix D is also an optimal Hankel norm approximation. Hence G ^ the constant term of G can not be determined from Hankel norm. An appropriate constant term D0 in Theorem 8.13 can be obtained in many ways. We shall mention three of them: ^ Apply Corollary 8.12 to G(s) ; G(s). This is usually complicated since (G(s) ; ^ G(s)) in general has McMillan degree of n + k. An alternative is to use the unstable part of the optimal Hankel norm approxi^ mation in Theorem 8.6. Let G + F be obtained from Theorem 8.6, part (3) such that F (s) 2 RH; has McMillan degree n ; k ; r then 1 ^ ^ kG ; G ; D0 k1 G ; G ; F + kF ; D0 k1 = k+1 (G) + kF ; D0 k1 1 Now Corollary 8.12 can be applied to F (;s) to obtain a D0 such that kF ; D0 k1 since by Corollary 8.9, n;k;r X i=1 i (F (;s)) i (F (;s)) n;k;r X i=1 i+k+r (G): i+k+r (G) for i = 1 2 ::: n ; k ; r. D0 can of course be obtained using any standard convex optimization algorithm: ^ D0 = arg min G ; G ; D0 : 1 D0 Note that Theorem 8.13 can also be applied to non-square systems. For example, consider a stable, rational, p m (p < m), transfer function G(s). Let Ga = G 0 ^ ^ be an augmented square transfer function and let Ga = G be the optimal Hankel ^ G2 norm approximation of Ga such that k+1 (Ga ) ^ Ga ; Ga ; Da 1 k+1 (Ga ) + n;k;r X i=1 i+k+r (Ga )
- 209. HANKEL NORM APPROXIMATION 192 with Da = D0 . Then D 2 k+1 (G) ^ G ; G ; D0 1 k+1 (G) + n;k;r X i=1 i+k+r (G) ^ ^ since i (G) = i (Ga ) and G ; G ; D0 Ga ; Ga ; Da 1 . 1 A less tighter error bound can be obtained without computing the appropriate D0 . Corollary 8.14 Given a stable, rational, m m, strictly proper transfer function G(s), with Hankel singular values 1 2 ::: k > k+1 = k+2 : : : = k+r > k+r+1 ^ : : : n > 0 and let G(s) 2 RH1 of McMillan degree k be a strictly proper optimal Hankel norm approximation to G(s) obtained in Theorem 8.6, then k+1 (G) ^ kG ; Gk1 2( k+1 (G) + n;k;r X i=1 i+k+r (G)) Proof. The result follows from Theorem 8.13. 2 8.5 Bounds for Balanced Truncation Very similar techniques to those of Theorem 8.11 can be used to bound the error obtained by truncating a balanced realization. We will rst need a lemma that gives a perhaps surprising relationship between a truncated balanced realization of degree (n ; rN ) and an optimal Hankel norm approximation of the same degree. Lemma 8.15 Let (A B C ) be a balanced realization of the stable, rational, m m transfer function G(s), and let A = A11 A12 A21 A22 B = B1 B2 C = C1 C2 1 0 0 I ^ ^ ^ ^ Let (A B C D) be de ned by equations (8.27) to (8.32) (where = Gb := Gh := then A11 B1 C1 0 ^ ^ A B ^ ^ C D 2 = 1) and de ne
- 210. 8.5. Bounds for Balanced Truncation (1) (Gb (s) ; Gh (s))= is all-pass. (2) kG(s) ; Gb (s)k1 2 . (3) If 1 > I then kG(s) ; Gb (s)kH 193 2 . Proof. (1) In order to prove that (Gb (s) ; Gh(s))= is all-pass we note that " ~ A Gb (s) ; Gh (s) = ~ C where 11 0 ~ A = A0 A ^ ~ B = B1 ^ B ~ B ~ D # ~ ^ C = C1 ;C ~ ^ D = ;D: Now Theorem 8.4, part (1) gives that the solutions to Lyapunov equations ~~ ~ ~ ~ ~ AP + P A + B B = 0 ~ ~ ~~ ~ ~ A Q + QA + C C = 0 are I 1 ~ ~ P = I1 Q = ;; ;; ;;1 ; 1 1 (8.72) (8.73) (8:74) (This is veri ed by noting that the blocks of equations (8.72) and (8.73) are also blocks ~~ of equations (8.34) and (8.35) for Pe and Qe .) Hence P Q = 2 I and by Theorem 8.3 ~ such that (Gb (s) ; Gh (s))= is all-pass. That D = ;D is an appropriate ~ ^ there exists D choice is veri ed from equations (8.54) to (8.56) and Theorem 8.3, part (2). (2) (G(s) ; Gb (s))= = (G(s) ; Gh (s))= + (Gb (s) ; Gh (s))= but the rst term on the right hand side is all-pass by Theorem 8.4, part (2) and the second term is all-pass by part (1) above. Hence kG(s) ; Gb (s)k1 2 . (3) Similarly using the fact that all-pass functions have unity Hankel norms gives that kG(s) ; Gb (s)kH kG(s) ; Gh (s)kH + kGb (s) ; Gh (s)kH = 2 (Note that Gh (s) is stable if 1 > I .) 2 Given the results of Lemma 8.15 bounds on the error in a truncated balanced realization are easily proved as follows. Theorem 8.16 Let G(s) be a stable, rational, p m, transfer function with Hankel ~ singular values 1 > 2 : : : > N , where each i has multiplicity ri and let Gk (s) be obtained by truncating the balanced realization of G(s) to the rst (r1 + r2 + : : : + rk ) states. Then ~ (1) G(s) ; Gk (s) 2( k+1 + k+2 + : : : + N ). ~ (2) G(s) ; Gk (s) 1 H 2( k+1 + k+2 + : : : + N ).
- 211. HANKEL NORM APPROXIMATION 194 Proof. If p 6= m then augmenting B or C by zero columns or rows, respectively, will still give a balanced realization and the same argument is valid. Hence assume p = m. Notice that since truncation of balanced realization are also balanced, satisfying the ~ truncated Lyapunov equations, the Hankel singular values of Gi (s) will be 1 2 : : : i ~ i (s) can be obtained by truncating with multiplicities r1 r2 : : : ri , respectively. Also G ~ ~ ~ the balanced realization of Gi+1 (s) and hence Gi+1 (s) ; Gi (s) 2 i+1 for both L1 and Hankel norms. Hence (GN (s) := G(s)) ~ G(s) ; Gk (s) = N ;1 X i=k ~ ~ (Gi+1 (s) ; Gi (s)) 2( k+1 + k+2 + : : : + N ) 2 for both norms, and the proof is complete. 8.6 Toeplitz Operators In this section, we consider another operator. Again let G(s) 2 L1 . Then a Toeplitz operator associated with G is denoted by TG and is de ned as TG : H2 7;! H2 TGf := (P+ MG)f = P+ (Gf ) for f 2 H2 i.e., TG = P+ MG jH2 . In particular if G(s) 2 H1 then TG = MGjH2 . > MG H2 L2 P+ TG ? - H2 Analogous to the Hankel operator, there is also a corresponding time domain description for the Toeplitz operator: Tg : L2 0 1) 7;! L2 0 1) Tg f := P+ (g f ) = for f (t) 2 L2 0 1). 1 Z 0 g(t ; )f ( )d t 0
- 212. 8.7. Hankel and Toeplitz Operators on the Disk* 195 It is seen that the multiplication operator (in frequency domain) or the convolution operator (in time domain) plays an important role in the development of the Hankel and Toeplitz operators. In fact, a multiplication operator can be decomposed as several ? Hankel and Toeplitz operators: let L2 = H2 H2 and G 2 L1 . Then the multiplication operator associated with G can be written as ? ? MG : H2 H2 7;! H2 H2 P M j ? P M j 2 2 MG = P; MGjH? P; MGjH2 = P; MGjH? ;G : ;G TG + G H2 + G H2 Note that P; MG jH? is actually a Toeplitz operator however, we will not discuss it 2 further here. A fact worth mentioning is that if G is causal, i.e., G 2 H1 , then ;G = 0 and MG is a lower block triangular operator. In fact, it can be shown that G is causal if and only if ;G = 0, i.e., the future input does not a ect the past output. This is yet another characterization of causality. 8.7 Hankel and Toeplitz Operators on the Disk* It is sometimes more convenient and insightful to study operators on the disk. From the control system point of view, some operators are much easier to interpret and compute in discrete time. The objective of this section is to examine the Hankel and Toeplitz operators in discrete time (i.e., on the disk) and, hopefully, to give the readers some intuitive ideas about these operators since they are very important in the H1 control theory. To start with, it is necessary to introduce some function spaces in respect to a unit disk. Let D denote the unit disk : D := f 2 C : j j < 1g and let D and @ D denote its closure and boundary, respectively: D := f 2 C : j j 1g @ D := f 2 C : j j = 1g : Let L2 (@ D ) denote the Hilbert space of matrix valued functions de ned on the unit circle @ D as Z 2 L2 (@ D ) = F ( ) : 21 Trace F (ej )F (ej ) d < 1 0 with inner product hF Gi := 21 Z 2 0 Trace F (ej )G(ej ) d :
- 213. HANKEL NORM APPROXIMATION 196 Furthermore, let H2 (@ D ) be the (closed) subspace of L2 (@ D ) with matrix functions F ( ) analytic in D , i.e., H2 (@ D ) = F ( ) 2 L2 (@ D ) : 21 Z 2 0 F (ej )ejn d = 0 for all n > 0 ? and let H2 (@ D ) be the (closed) subspace of L2 (@ D ) with matrix functions F ( ) analytic in C nD . It can be shown that the Fourier transform (or bilateral Z transform) gives the following isometric isomorphism: L2 (@ D ) = l2 (;1 1) H2 (@ D ) = l2 0 1) ? H2 (@ D ) = l2 (;1 0): Remark 8.3 It should be kept in mind that, in contrast to the variable z in the standard Z -transform, here denotes = z ;1 . ~ Analogous to the space in the half plane, L1 (@ D ) is used to denote the Banach space of essentially bounded matrix functions with norm kF k1 = ess sup F (ej ) : 202 ] The Hardy space H1 (@ D ) is the closed subspace of L1 (@ D ) with functions analytic in D and is de ned as H1 (@ D ) = F ( ) 2 L1 (@ D ) : The H1 norm is de ned as Z 2 0 F (ej )ejn d = 0 for all n > 0 : kF k1 := sup F ( )] = ess sup F (ej ) : 2D 202 ] It is easy to see that L1 (@ D ) L2 (@ D ) and H1 (@ D ) H2 (@ D ). (However, it should be pointed out that these inclusions are not true for functions in the half planes or for continuous time functions.) Example 8.4 Let F ( ) 2 L2 (@ D ), and let F ( ) have a power series representation as follows: F( ) = 1 X i=;1 Fi i : ? Then F ( ) 2 H2 (@ D ) if and only if Fi = 0 for all i < 0 and F ( ) 2 H2 (@ D ) if and only if Fi = 0 for all i 0. 3
- 214. 8.7. Hankel and Toeplitz Operators on the Disk* 197 Now let P; and P+ denote the orthogonal projections: P+ : l2 (;1 1) 7;! l2 0 1) or L2 (@ D ) 7;! H2 (@ D ) ? P; : l2 (;1 1) 7;! l2 (;1 0) or L2 (@ D ) 7;! H2 (@ D ): Suppose Gd ( ) 2 L1 (@ D ). Then the Hankel operator associated with Gd ( ) is de ned as ? ;Gd : l2 (;1 0) 7;! l2 0 1) or H2 (@ D ) 7;! H2 (@ D ) ;Gd = P+ MGd jH? (@ D) : 2 Similarly, a Toeplitz operator associated with Gd( ) is de ned as TGd : l2 0 1) 7;! l2 0 1) or H2 (@ D ) 7;! H2 (@ D ) TGd = P+ MGd jH2 (@ D) : P1 Now let Gd ( ) = i=;1 Gi i 2 L1 (@ D ) be a linear system transfer matrix, u( ) = P1 i i=;1 ui 2 L2 (@ D ) be the system input in the frequency domain, and ui be the input at time t = i. Then the system output in the frequency domain is given by y( ) = Gd ( )u( ) = 1 X 1 X i=;1 j =;1 Gi uj i+j =: 1 X i=;1 yi i where yi is the system time response at time t = i, and consequently, we have 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 y2 y1 y0 y;1 y;2 y;3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 32 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 = G0 G;1 G;2 G;3 G;4 G;5 G1 G0 G;1 G;2 G;3 G;4 G2 G1 G0 G;1 G;2 G;3 G4 G3 G2 G1 G0 G;1 G5 G4 G3 G2 G1 G0 G0 G;1 G;2 G1 G0 G;1 G2 G1 G0 G1 G2 G3 G2 G3 G3 G4 G4 G5 G;1 G;2 G;3 G;2 G;3 G;4 G;3 G;4 G;5 G0 G1 G2 G;1 G0 G1 G;2 G;1 G0 Equation (8.75) can be rewritten as 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 y0 y1 y2 y;1 y;2 y;3 3 2 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 = G3 G2 G1 G0 G;1 G;2 76 76 76 76 76 76 76 76 76 76 76 76 76 76 54 3 7 7 u2 7 7 u1 7 7 u0 7 : (8:75) 7 u;1 7 7 u;2 7 7 u;3 7 7 5 32 76 76 76 76 76 76 76 76 76 76 76 76 76 76 54 u0 u1 u2 u;1 u;2 u;3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
- 215. HANKEL NORM APPROXIMATION 198 2 =: u0 u1 u2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 T1 H1 H2 T2 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 u;1 : u;2 u;3 Matrices like T1 and T2 are called (block) Toeplitz matrices and matrices like H1 and H2 are called (block) Hankel matrices. In fact, H1 is the matrix representation of the Hankel operator ;Gd and T1 is the matrix representation of the Toeplitz operator TGd , and so on. Thus these operator norms can be computed from the matrix norms of their corresponding matrix representations. Lemma 8.17 kGd( )k1 = T1 H1 H2 T2 , k;Gd k = kH1 k, and kTGd k = kT1 k. ;1 I ; A);1 B + D be a state Example 8.5 Let Gd( ) 2 RH1(@ D ) and Gd( ) = C ( space realization of Gd ( ) and let A have all the eigenvalues in D . Then Gd ( ) = D + 1 X 1 i+1 CAi B = X Gi i i=0 i=0 with G0 = D and Gi = CAi;1 B 8i 1. The state space equations are given by xk+1 = Axk + Buk x;1 = 0 yk = Cxk + Duk : P Then for u 2 l2 (;1 0), we have x0 = 1 Ai;1 Bu;i , which de nes the controllability i=1 operator 2 3 6 6 6 4 x0 = cu = B AB A2 B u;1 u;2 u;3 7 7 7 5 2 Rn : .. . n , the output can be computed as On the other hand, given x0 and uk = 0 i 0, x 2 R 2 y0 6 y1 6 o x0 = 6 y2 4 .. . 3 2 3 C 7 6 CA 7 7 6 7 7 = 6 CA2 7 x0 2 l2 0 1) 5 4 5 .. .
- 216. 8.7. Hankel and Toeplitz Operators on the Disk* 199 which de nes the observability operator. Of course, the adjoint operators of c and o can be computed easily as 2 3 B 6 B A 7 x0 = 6 B (A )2 7 x0 2 l2 (;1 0) 6 7 c 4 5 .. . and 2 oy = C AC 6 6 6 4 (A )2 C y0 y1 y2 3 7 7 7 5 2 Rn : .. . Hence, the Hankel operator has the following matrix representation: 2 C 6 CA H = 6 CA2 6 4 .. . 3 7 7 7 5 2 G1 G2 G3 6 G2 G3 G4 = 6 G3 G4 G5 6 4 B AB A2 B .. . .. . .. . 3 .. . 7 7 7 5 : 3 It is interesting to establish some connections between a Hankel operator de ned on the unit disk and the one de ned on the half plane. First de ne the map as 1 (8:76) = s;1 s= 1+ s+ 1; which maps the closed right-half plane Re(s) 0 onto the closed unit disk, D . Suppose G(s) 2 H1 (j R) and de ne (8:77) Gd ( ) := G(s)js= 1; : 1+ Since G(s) is analytic in Re(s) > 0, including the point at 1, Gd ( ) is analytic in D , i.e., Gd ( ) 2 H1 (@ D ). Lemma 8.18 ;G(s) = ;Gd( ) . Proof. De ne the function p (s) = s +21 : The relation between a point j! on the imaginary axis and the corresponding point ej on the unit circle is, from (8.76), ej = j! ; 1 : j! + 1
- 217. HANKEL NORM APPROXIMATION 200 This yields d = ; !2 2 1 d! = ;j (j!)j2 d! + which implies that the mapping fd( ) 7;! (s)f (s) : H2 (@ D ) 7;! H2 (j R) where f (s) = fd( )j = s;1 s+1 is an isomorphism. Similarly, ? ? fd ( ) 7;! (s)f (s) : H2 (@ D ) 7;! H2 (j R) ? is an isomorphism note that if fd( ) 2 H2 (@ D ), then fd (1) = 0, so that f (;1) = 0, and hence (s)f (s) is analytic in Re(s) < 0. The lemma now follows from the commutative diagram ;Gd ( ) ? - H2 (@ D ) H2 (@ D ) ? ? H2 (j R) ;G(s) ? - H2 (j R ) 2 The above isomorphism between H2 (@ D ) and H2 (j R) has also the implication that every discrete time H2 control problem can be converted to an equivalent continuous time H2 control problem. It should be emphasized that the bilinear transformation is not an isomorphism between H2 (@ D ) and H2 (j R). 8.8 Nehari's Theorem* As we have mentioned at the beginning of this chapter, a Hankel operator is closely related to an analytic approximation problem. In this section, we shall be concerned with approximating G by an anticausal transfer matrix, i.e., one analytic in Re(s) < 0 (or j j > 1), where the approximation is with respect to the L1 norm. For convenience, ; let H1 (@ D ) denote the subspace of L1 (@ D ) with functions analytic in j j > 1. So ; G( ) 2 H1 (@ D ) if and only if G( ;1 ) 2 H1 (@ D ). Minimum Distance Problem In the following, we shall establish that the distance in L1 from G to the nearest ; matrix in H1 equals k;Gk. This is the classical Nehari Theorem. Some applications
- 218. 8.8. Nehari's Theorem* 201 and explicit construction of the approximation for the matrix case will be considered in the later chapters. Theorem 8.19 Suppose G 2 L1, then inf kG ; Qk1 = k;Gk Q2H; 1 and the in mum is achieved. Remark 8.4 Note that from the mapping (8.76) and Lemma 8.18, the above theorem is the same whether the problem is on the unit disk or on the half plane. Hence, we will only give the proof on the unit disk. ~ Proof. We shall give the proof on the unit disk. First, it is easy to establish that the ; Hankel norm is the lower bound: for xed Q 2 H1 (@ D ), we have ; kG ; Qk1 = sup k(G kf Q)f k2 k2 ? (@ D ) f 2H2 kP+ (G ; Q)f k2 sup kf k2 ? (@ D ) f 2H2 kP+ Gf k2 = sup ? (@ D ) kf k2 f 2H2 = k;G k : ; Next we shall construct a Q( ) 2 H1 (@ D ) such that kG ; Qk1 = k;G k : Let the power series expansion of G( ) and Q( ) be G( ) = Q( ) = 1 X i=;1 0 X i=;1 i Gi i Qi : The left-hand side of (8.78) equals the norm of the operator ? f 7;! (G( ) ; Q( ))f ( ) : H2 (@ D ) 7;! L2 (@ D ): (8:78)
- 219. HANKEL NORM APPROXIMATION 202 From the discussion of the last section, there is a matrix representation of this operator: 2 3 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 G3 G2 G1 G4 G3 G2 G1 G5 G4 G3 G2 G1 G0 ; Q0 G;1 ; Q;1 G0 ; Q0 G;2 ; Q;2 G;1 ; Q;1 G0 ; Q0 : (8:79) The idea in the construction of a Q to satisfy (8.78) is to select Q0 , Q;1 , : : : , in turn to minimize the matrix norm of (8.79). First choose Q0 to minimize 3 2 6 6 6 6 6 6 4 G3 G2 G1 G0 ; Q0 G4 G3 G2 G1 7 7 7 7 7 7 5 G5 G4 G3 G2 : By Parrott's Theorem (i.e., matrix dilation theory presented in Chapter 2), the minimum equals the norm of the following matrix 2 3 6 6 6 6 4 7 7 7 7 5 G3 G4 G5 G2 G3 G4 G1 G2 G3 which is the rotated Hankel matrix H1 . Hence, the minimum equals the Hankel norm k;Gk. Next, choose Q;1 to minimize 2 3 6 6 6 6 6 6 6 6 4 7 7 7 7 7 7 7 7 5 G3 G2 G1 G0 ; Q0 G;1 ; Q;1 G4 G3 G2 G1 G5 G4 G3 G2 G0 ; Q0 G1 : Again, the minimum equals k;G k. Continuing in this way gives a suitable Q. 2 As we have mentioned earlier we might be interested in approximating an L1 function by an H1 function. Then we have the following corollary.
- 220. 8.8. Nehari's Theorem* 203 Corollary 8.20 Suppose G 2 L1, then ^ inf kG ; Qk1 = ;G Q2H1 and the in mum is achieved. Proof. This follows from the fact that ^ inf kG ; Qk1 = inf ; kG ; Q k1 = k;G k = ;G : Q2H1 Q 2H1 2 Let G 2 RH1 and g be the impulse response of G and let 1 be the largest Hankel singular value of G, i.e., k;G k = 1 . Suppose u 2 L2 (;1 0] (or l2 (;1 0)) and v 2 L2 0 1) (or l2 0 1)) are the corresponding Schmidt pairs: ;g u = 1 v ;g v = 1 u: Now denote the Laplace transform (or Z -transform) of u and v as U 2 RH? and 2 V 2 RH2 . Lemma 8.21 Let G 2 RH1 and 1 be the largest Hankel singular value of G. Then inf; kG ; Qk1 = 1 Q2H1 and (G ; Q)U = 1 V: Moreover, if G is a scalar function, then Q = G ; 1 V=U is the unique solution and G ; Q is all-pass. Proof. Let H := (G ; Q)U and note that ;GU 2 RH2 and P+H = P+(GU ) = ;GU . Then 0 = = = = = = kH ; ;GU k2 2 2 + k; U k2 ; hH ; U i ; h; U H i kH k2 G 2 G G 2 + k; U k2 ; hP H ; U i ; h; U P H i kH k2 G 2 + G G + 2 ; h; U ; U i kH k2 G G kH k2 ; hU ;G ;GU i 2 2 kH k2 ; 1 hU U i 2 2 kH k2 ; 1 kU k2 2 2 2 kG ; Qk2 kU k2 ; 1 kU k2 1 2 2 = 0:
- 221. HANKEL NORM APPROXIMATION 204 Hence, H = ;G U , i.e., (G ; Q)U = ;GU = 1 V . Now if G is a scalar function, then Q is uniquely determined and Q = G ; 1 V=U . We shall prove through explicit construction below that V=U is an all-pass. 2 Formulas for Continuous Time Let G(s) 2 RH1 and let A G(s) = C B : 0 Let Lc and Lo be the corresponding controllability and observability Gramians: And let 2 1, ALc + LcA + BB = 0 A Lo + Lo A + C C = 0: be the largest eigenvalue and the corresponding eigenvector of Lc Lo: Lc Lo = De ne 2 1 : := 1 Lo : 1 Then u = v = ;A 2 L2 (;1 0] c =B e At 2 L2 0 1) o = Ce are the Schmidt pair and U (s) = V (s) = ;A ? ;B 0 2 RH2 A C 0 2 RH2 : It is easy to show that if G is a scalar, then U V are scalar transfer functions and U U = V V . Hence, V=U is an all-pass. The details are left as an exercise for the reader. Formulas for Discrete Time Similarly, let G be a discrete time transfer matrix and let A G( ) = C ( ;1 I ; A);1 B = C B : 0
- 222. 8.8. Nehari's Theorem* 205 Let Lc and Lo be the corresponding controllability and observability Gramians: ALcA ; Lc + BB = 0 A Lo A ; Lo + C C = 0: 2 , be the largest eigenvalue and a corresponding eigenvector of Lc Lo : And let 1 2 Lc Lo = 1 : De ne := 1 Lo : 1 Then 2 B 6 B A 6 c = 6 B (A )2 4 u = .. . 2 C 6 CA 6 o = 6 CA2 4 v = are the Schmidt pair and .. . 3 7 7 7 5 2 l2 (;1 0) 3 7 7 7 5 2 l2 0 1) 1 X B (A )i;1 ;i = A B 2 RH? 2 0 i=1 1 X A V( ) = CAi i = CA C 2 RH2 i=0 U( ) = where W ( ) := W T ( ;1 ). Alternatively, since the Hankel operator has a matrix representation H , u and v can be obtained from a singular value decomposition" (possibly in nite size): let G( ) = 1 X i=1 Gi i Gi 2 Rp q (in fact, Gi = CAi;1 B if a state space realization of G is available) and let 2 G1 G2 G3 6 G2 G3 G4 H = 6 G3 G4 G5 6 4 .. . .. . .. . 3 .. . 7 7 7 5 :
- 223. HANKEL NORM APPROXIMATION 206 Suppose H has a singular value decomposition" H = U V = diagf 1 U = u1 u2 V = v1 v2 2 with U U = UU = I and V V = V V = I . Then 2 u11 6 u12 u = u1 = 4 . .. 3 7 5 g 2 v11 6 v12 v = v1 = 4 . .. 3 7 5 where u and v are partitioned such that u1i 2 Rp and v1i 2 Rq . Finally, U ( ) and V ( ) can be obtained as U( ) = V( ) = 1 X i=1 1 X i=1 u1i ;i 2 RH? 2 v1i i;1 2 RH2 : In particular, if G( ) is an n-th order matrix polynomial, then matrix H has only a nite number of nonzero elements and H = Hn 0 0 0 with 2 6 6 =6 6 6 4 G1 G2 G2 G3 G3 G4 Gn;1 Gn Gn 0 3 7 7 7 7 7 5 0 0 : .. .. .. . . . Gn 0 0 0 Hence, u and v can be obtained from a standard SVD of Hn . Hn .. . 8.9 Notes and References The study of the Hankel operators and of the optimal Hankel norm approximation theory can be found in Adamjan, Arov, and Krein 1978], Bettayeb, Silverman, and Safonov 1980], Kung and Lin 1981], Power 1982], Francis 1987], Kavranoglu and Bettayeb 1994] and references therein. The presentation of this chapter is based on Francis 1987] and Glover 1984,1989].
- 224. 9 Model Uncertainty and Robustness In this chapter we brie y describe various types of uncertainties which can arise in physical systems, and we single out unstructured uncertainties" as generic errors which are associated with all design models. We obtain robust stability tests for systems under various model uncertainty assumptions through the use of the small gain theorem. And we also obtain some su cient conditions for robust performance under unstructured uncertainties. The di culty associated with MIMO robust performance design and the role of plant condition numbers for systems with skewed performance and uncertainty speci cations are revealed. We show by examples that the classical gain margin and phase margin are insu cient indicators for the system robustness. A simple example is also used to indicate the fundamental di erence between the robustness of an SISO system and that of an MIMO system. In particular, we show that applying the SISO analysis/design method to an MIMO system may lead to erroneous results. 9.1 Model Uncertainty Most control designs are based on the use of a design model. The relationship between models and the reality they represent is subtle and complex. A mathematical model provides a map from inputs to responses. The quality of a model depends on how closely its responses match those of the true plant. Since no single xed model can respond exactly like the true plant, we need, at the very least, a set of maps. However, the 207
- 225. 208 MODEL UNCERTAINTY AND ROBUSTNESS modeling problem is much deeper { the universe of mathematical models from which a model set is chosen is distinct from the universe of physical systems. Therefore, a model set which includes the true physical plant can never be constructed. It is necessary for the engineer to make a leap of faith regarding the applicability of a particular design based on a mathematical model. To be practical, a design technique must help make this leap small by accounting for the inevitable inadequacy of models. A good model should be simple enough to facilitate design, yet complex enough to give the engineer con dence that designs based on the model will work on the true plant. The term uncertainty refers to the di erences or errors between models and reality, and whatever mechanism is used to express these errors will be called a representation of uncertainty. Representations of uncertainty vary primarily in terms of the amount of structure they contain. This re ects both our knowledge of the physical mechanisms which cause di erences between the model and the plant and our ability to represent these mechanisms in a way that facilitates convenient manipulation. For example, consider the problem of bounding the magnitude of the e ect of some uncertainty on the output of a nominally xed linear system. A useful measure of uncertainty in this context is to provide a bound on the spectrum of the output's deviation from its nominal response. In the simplest case, this spectrum is assumed to be independent of the input. This is equivalent to assuming that the uncertainty is generated by an additive noise signal with a bounded spectrum the uncertainty is represented as additive noise. Of course, no physical system is linear with additive noise, but some aspects of physical behavior are approximated quite well using this model. This type of uncertainty received a great deal of attention in the literature during the 1960's and 1970's, and the attention is probably due more to the elegant theoretical solutions that are yielded (e.g., white noise propagation in linear systems, Wiener and Kalman ltering, LQG) than to the great practical signi cance o ered. Generally, the deviation's spectrum of the true output from the nominal will depend signi cantly on the input. For example, an additive noise model is entirely inappropriate for capturing uncertainty arising from variations in the material properties of physical plants. The actual construction of model sets for more general uncertainty can be quite di cult. For example, a set membership statement for the parameters of an otherwise known FDLTI model is a highly-structured representation of uncertainty. It typically arises from the use of linear incremental models at various operating points, e.g., aerodynamic coe cients in ight control vary with ight environment and aircraft con gurations, and equation coe cients in power plant control vary with aging, slag buildup, coal composition, etc. In each case, the amounts of variation and any known relationships between parameters can be expressed by con ning the parameters to appropriately de ned subsets of parameter space. However, for certain classes of signals (e.g., high frequency), the parameterized FDLTI model fails to describe the plant because the plant will always have dynamics which are not represented in the xed order model. In general, we are forced to use not just a single parameterized model but model sets that allow for plant dynamics which are not explicitly represented in the model structure.
- 226. 9.1. Model Uncertainty 209 A simple example of this involves using frequency-domain bounds on transfer functions to describe a model set. To use such sets to describe physical systems, the bounds must roughly grow with frequency. In particular, at su ciently high frequencies, phase is completely unknown, i.e., 180o uncertainties. This is a consequence of dynamic properties which inevitably occur in physical systems. This gives a less structured representation of uncertainty. Examples of less-structured representations of uncertainty are direct set membership statements for the transfer function matrix of the model. For instance, the statement (j!)] < 1 8! 0 (9:1) P (s) = P (s) + W1 (s) (s)W2 (s) where W1 and W2 are stable transfer matrices that characterize the spatial and frequency structure of the uncertainty, con nes the matrix P to a neighborhood of the nominal model P . In particular, if W1 = I and W2 = w(s)I where w(s) is a scalar function, then P describes a disk centered at P with radius w(j!) at each frequency as shown in Figure 9.1. The statement does not imply a mechanism or structure which gives rise to . The uncertainty may be caused by parameter changes, as mentioned above or by neglected dynamics or by a host of other unspeci ed e ects. An alternative statement to (9.1) is the so-called multiplicative form: P (s) = (I + W1 (s) (s)W2 (s))P (s): (9:2) This statement con nes P to a normalized neighborhood of the nominal model P . An advantage of (9.2) over (9.1) is that in (9.2) compensated transfer functions have the same uncertainty representation as the raw model (i.e., the weighting functions apply to PK as well as P ). Some other alternative set membership statements will be discussed later. P(j ω ) w(jω ) Figure 9.1: Nyquist Diagram of an Uncertain Model The best choice of uncertainty representation for a speci c FDLTI model depends, of course, on the errors the model makes. In practice, it is generally possible to represent
- 227. 210 MODEL UNCERTAINTY AND ROBUSTNESS some of these errors in a highly-structured parameterized form. These are usually the low frequency error components. There are always remaining higher frequency errors, however, which cannot be covered this way. These are caused by such e ects as in nite-dimensional electro-mechanical resonance, time delays, di usion processes, etc. Fortunately, the less-structured representations, e.g., (9.1) or (9.2), are well suited to represent this latter class of errors. Consequently, (9.1) and (9.2) have become widely used generic" uncertainty representations for FDLTI models. An important point is that the construction of the weighting matrices W1 and W2 for multivariable systems is not trivial. Motivated from these observations, we will focus for the moment on the multiplicative description of uncertainty. We will assume that P in (9.2) remains a strictly proper FDLTI system for all . More general perturbations (e.g., time varying, in nite dimensional, nonlinear) can also be covered by this set provided they are given appropriate conic sector" interpretations via Parseval's theorem. This connection is developed in Safonov, 1980] and Zames, 1966] and will not be pursued here. When used to represent the various high frequency mechanisms mentioned above, the weighting functions in (9.2) commonly have the properties illustrated in Figure 9.2. They are small ( 1) at low frequencies and increase to unity and above at higher frequencies. The growth with frequency inevitably occurs because phase uncertainties eventually exceed 180 degrees and magnitude deviations eventually exceed the nominal transfer function magnitudes. Readers who are skeptical about this reality are encouraged to try a few experiments with physical devices. nominal model log ω actual model Figure 9.2: Typical Behavior of Multiplicative Uncertainty: p (s) = 1 + w(s) (s)]p(s) Also note that the representation of uncertainty in (9.2) can be used to include perturbation e ects that are in fact certain. A nonlinear element, may be quite accurately
- 228. 9.2. Small Gain Theorem 211 modeled, but because our design techniques cannot e ectively deal with the nonlinearity, it is treated as a conic linearity1 . As another example, we may deliberately choose to ignore various known dynamic characteristics in order to achieve a simple nominal design model. This is the model reduction process discussed in the previous chapters. The following terminologies are used in this book. De nition 9.1 Given the description of an uncertainty model set and a set of performance objectives, suppose P 2 is the nominal design model and K is the resulting controller. Then the closed-loop feedback system is said to have Nominal Stability (NS): if K internally stabilizes the nominal model P . Robust Stability (RS): if K internally stabilizes every plant belong to . Nominal Performance (NP): if the performance objectives are satis ed for the nominal plant P . Robust Performance (RP): if the performance objectives are satis ed for every plant belong to . The nominal stability and performance can be easily checked using various standard techniques. The conditions for which the robust stability and robust performance are satis ed under various assumptions on the uncertainty set will be considered in the following sections. 9.2 Small Gain Theorem This section and the next section consider the stability test of a nominally stable system under unstructured perturbations. The basis for the robust stability criteria derived in the sequel is the so-called small gain theorem. Consider the interconnected system shown in Figure 9.3 with M (s) a stable p q transfer matrix. w1 - e e1 + 6 + M + e2 e + w2 ? Figure 9.3: Small Gain Theorem 1 See, for example, Safonov 1980] and Zames 1966].
- 229. 212 MODEL UNCERTAINTY AND ROBUSTNESS Theorem 9.1 (Small Gain Theorem) Suppose M 2 RH1 . Then the interconnected system shown in Figure 9.3 is well-posed and internally stable for all (s) 2 RH1 with (a) k k1 1= if and only if kM (s)k1 < (b) k k1 < 1= if and only if kM (s)k1 . Proof. We shall only prove part (a). The proof for part (b) is similar. Without loss of generality, assume = 1. (Su ciency) It is clear that M (s) (s) is stable since both M (s) and (s) are stable. Thus by Theorem 5.7 (or Corollary 5.6) the closed-loop system is stable if det(I ; M ) has no zero in the closed right-half plane for all 2 RH1 and k k1 1. Equivalently, the closed-loop system is stable if inf (I ; M (s) (s)) 6= 0 s2C + for all 2 RH1 and k k1 1. But this follows from inf (I ; M (s) (s)) 1; sup (M (s) (s)) = 1;kM (s) (s)k1 s2C + s2C + 1;kM (s)k1 > 0: (Necessity) This will be shown by contradiction. Suppose kM k1 1. We will show that there exists a 2 RH1 with k k1 1 such that det(I ; M (s) (s)) has a zero on the imaginary axis, so the system is unstable. Suppose !0 2 R+ f1g is such that (M (j!0 )) 1. Let M (j!0 ) = U (j!) (j!0 )V (j!0 ) be a singular value decomposition with up U (j!0 ) = u1 u2 vq V (j!0 ) = v1 v2 2 3 1 (j!0 ) = 6 4 2 ... 7 5 and kM k1 = (M (j!0)) = 1 . To obtain a contradiction, it now su ces to construct a 2 RH1 such that (j!0 ) = 11 v1 u1 and k k1 1. Indeed, for such (s), det(I ; M (j!0 ) (j!0 )) = det(I ; U V v1 u1 = 1 ) = 1 ; u1 U V v1 = 1 = 0 and thus the closed-loop system is either not well-posed (if !0 = 1) or unstable (if ! 2 R). There are two di erent cases: (1) !0 = 0 or 1: then U and V are real matrices. In this case, (s) can be chosen as = 1 v u 2 Rq p : 1 1 1
- 230. 9.2. Small Gain Theorem 213 (2) 0 < !0 < 1: write u1 and v1 in the following form: 2 6 u1 = 6 6 4 u11 ej 1 u12 ej 2 .. . u1p ej p 3 2 7 7 7 5 v1 = 6 6 4 6 v11 ej v12 ej .. . v1q ej 3 1 2 7 7 7 5 q where u1i 2 R and v1j 2 R are chosen so that i j 2 ; 0) for all i j . Choose i 0 and j 0 so that i ; j!0 = i i + j!0 for i = 1 2 : : : p and j = 1 2 : : : q. Let (s) = 1 1 Then k k1 = 1= 1 2 6 4 v11 1 ;s 1 +s .. . v1q q ;s q +s j ; j!0 = j j + j!0 3 7 5 h u11 1 ;s 1 +s 1 and (j!0 ) = 11 v1 u1 . s i u1p p ;s 2 RH1 : p+ 2 The theorem still holds even if and M are in nite dimensional. This is summarized as the following corollary. Corollary 9.2 The following statements are equivalent: (i) The system is well-posed and internally stable for all 2 H1 with k k1 < 1= (ii) The system is well-posed and internally stable for all 2 RH1 with k k1 < 1= (iii) The system is well-posed and internally stable for all 2 C q p with k k < 1= (iv) kM k1 . Remark 9.1 It can be shown that the small gain condition is su cient to guarantee internal stability even if is a nonlinear and time varying stable" operator with an appropriately de ned stability notion, see Desoer and Vidyasagar 1975]. ~ The following lemma shows that if kM k1 > , there exists a destabilizing with k k1 < 1= such that the closed-loop system has poles in the open right half plane. (This is stronger than what is given in the proof of Theorem 9.1.) Lemma 9.3 Suppose M 2 RH1 and kM k1 > . Then there exists a 0 > 0 such that for any given 2 0 0 ] there exists a 2 RH1 with k k1 < 1= such that det(I ; M (s) (s)) has a zero on the axis Re(s) = .
- 231. MODEL UNCERTAINTY AND ROBUSTNESS 214 Proof. Without loss of generality, assume = 1. Since M 2 RH1 and kM k1 > 1, there is a su ciently small 0 > 0 such that kM ( 0 + s)k1 := sup kM ( 0 + s)k > 1. Re(s)>0 Now let 2 0 0 ]. Then there exists a 0 < !0 < 1 such that kM ( + j!0)k > 1. Let M ( + j!0 ) = U V be a singular value decomposition with up U = u1 u2 V = v1 v2 2 =6 4 Write u1 and v1 in the following form: 2 6 u1 = 6 6 4 u11 ej 1 u12 ej 2 .. . u1p ej p 1 vq 3 2 ... 7 5 3 2 7 7 7 5 v1 = 6 6 4 6 : v11 ej v12 ej .. . v1q ej 3 1 2 7 7 7 5 q where u1i 2 R and v1j 2 R are chosen so that i j 2 ; 0) for all i j . Choose i 0 and j 0 so that i ; ; j!0 = i i + + j!0 for i = 1 2 : : : p and j = 1 2 : : : q. Let 2 (s) = 1 1 6 4 1 ;s v11 1 +s .. . s v1q q ;s q+ j ; ; j!0 = j j + + j!0 3 7 5 h ;s u11 1 +s 1 i u1p p;s 2 RH1 : p +s Then k k1 = 1= 1 < 1 and det (I ; M ( + j!0 ) ( + j!0 )) = 0. Hence s = + j!0 is a zero for the transfer function det (I ; M (s) (s)). 2 The above lemma plays a key role in the necessity proofs of many robust stability tests in the sequel. 9.3 Stability under Stable Unstructured Uncertainties The small gain theorem in the last section will be used here to derive robust stability tests under various assumptions of model uncertainties. The modeling error will again
- 232. 9.3. Stability under Stable Unstructured Uncertainties 215 be assumed to be stable. (Most of the robust stability tests discussed in the sequel can be easily generalized to unstable case with some mild assumptions on the number of unstable poles of the uncertain model, we encourage readers to ll in the details.) In addition, we assume that the modeling error is suitably scaled with weighting functions W1 and W2 , i.e., the uncertainty can be represented as W1 W2 . We shall consider the standard setup shown in Figure 9.4, where is the set of uncertain plants with P 2 as the nominal plant and with K as the internally stabilizing controller for P . The sensitivity and complementary sensitivity matrix functions are de ned as usual as So = (I + PK );1 To = I ; So and Si = (I + KP );1 Ti = I ; Si : Recall that the closed-loop system is well-posed and internally stable if and only if I K ;1 = (I + K );1 ;K (I + K );1 2 RH : 1 ; I (I + K );1 (I + K );1 -f 6 ; - K - - Figure 9.4: Unstructured Robust Stability Analysis 9.3.1 Additive Uncertainty We assume that the model uncertainty can be represented by an additive perturbation: = P + W1 W2 : Theorem 9.4 Let = fP + W1 W2 : 2 RH1g and let K be a stabilizing controller for the nominal plant P . Then the closed-loop system is well-posed and internally stable for all k k1 < 1 if and only if kW2 KSoW1 k1 1. Proof. Let = P + W1 W2. Then K ;1 = (I + KSo W1 W2 );1 Si ;KSo(I + W1 W2 KSo );1 ;1 So (P + W1 W2 ) So (I + W1 W2 KSo);1 ; I (I + So W1 W2 K ) is well-posed and internally stable if (I + W2 KSo W1 );1 2 RH1 since det(I + KSo W1 W2 ) = det(I + W1 W2 KSo) = (I + SoW1 W2 K ) = det(I + W2 KSoW1 ): I
- 233. MODEL UNCERTAINTY AND ROBUSTNESS 216 But (I + W2 KSo W1 );1 2 RH1 is guaranteed if k W2 KSoW1 k1 < 1. Hence kW2 KSoW1 k1 1 is su cient for robust stability. To show the necessity, note that robust stability implies that K (I + K );1 = KSo(I + W1 W2 KSo );1 2 RH1 for all admissible . This in turn implies that W2 K (I + K );1 W1 = I ; (I + W2 KSoW1 );1 2 RH1 2 RH1 with 2 Similarly, it is easy to show that the closed-loop system is stable for all 2 RH1 with k k1 1 if and only if kW2 KSoW1 k1 < 1. for all admissible . By small gain theorem, this is true for all k k1 < 1 only if kW2 KSoW1 k1 1. 9.3.2 Multiplicative Uncertainty In this section, we assume that the system model is described by the following set of multiplicative perturbation P = (I + W1 W2 )P with W1 W2 2 RH1 . Consider the feedback system shown in the Figure 9.5. z - W2 r - e- K ;6 - P dm we - ? W1 - ~ d ? Wd d ?- e y We ? -e e - Figure 9.5: Output Multiplicative Perturbed Systems Theorem 9.5 Let = f(I + W1 W2 )P : 2 RH1g and let K be a stabilizing controller for the nominal plant P . Then (i) the closed-loop system is well-posed and internally stable for all 2 RH1 with k k1 < 1 if and only if kW2 ToW1 k1 1. (ii) the closed-loop system is well-posed and internally stable for all 2 RH1 with k k1 1 if kW2 ToW1 k1 < 1.
- 234. 9.3. Stability under Stable Unstructured Uncertainties (iii) the robust stability of the closed-loop system for all does not necessarily imply kW2 To W1 k1 < 1. 217 2 RH1 with k k1 (iv) the closed-loop system is well-posed and internally stable for all k k1 1 only if kW2 To W1 k1 1. 1 2 RH1 with (v) In addition, assume that neither P nor K has poles on the imaginary axis. Then the closed-loop system is well-posed and internally stable for all 2 RH1 with k k1 1 if and only if kW2 ToW1 k1 < 1. Proof. We shall rst prove that the condition in (i) is necessary for robust stability. Suppose kW2 ToW1 k1 > 1. Then by Lemma 9.3, for any given su ciently small > 0, there is a 2 RH1 with k k1 < 1 such that (I + W2 ToW1 );1 has poles on the axis Re(s) = . This implies (I + K );1 = So (I + W1 W2 To);1 has poles on the axis Re(s) = since can always be chosen so that the unstable poles are not cancelled by the zeros of So . Hence kW2 To W1 k1 1 is necessary for robust stability. In fact, we have also proven part (iv). The su ciency parts of (i), (ii), and (v) follow from the small gain theorem. To show the necessity part of (v), suppose kW2 ToW1 k1 = 1. From the proof of the small gain theorem, there is a 2 RH1 with k k1 1 such that (I + W2 To W1 );1 has poles on the imaginary axis. This implies (I + K );1 = So (I + W1 W2 To);1 has poles on the imaginary axis since the imaginary axis poles of (I + W1 W2 To);1 are not cancelled by the zeros of So , which are the poles of P and K . Hence kW2 ToW1 k1 < 1 is necessary for robust stability. The proof of part (iii) is given below by exhibiting an example with kW2 ToW1 k1 = 1 but there is no destabilizing 2 RH1 with k k1 1. 2 Example 9.1 Let P (s) = 1 , K (s) = 1, and W1 = W2 = 1. It is easy to check that K s stabilizes P . We have and 1 To = s + 1 kTo k1 = 1 s (I + K );1 = (I + K );1 = K (I + K );1 = s + 1 1 (I + K );1 = s + 1 1 +1 : 1 + s+1 1 1 1 + s+1
- 235. MODEL UNCERTAINTY AND ROBUSTNESS 218 1 1 Since jTo (s)j = s+1 < 1 for all 0 6= s 2 C and Re(s) 0, 1 + s+1 6= 0 for all 1 0 6= s 2 C , Re(s) 0, 2 RH1 and k k 1. The only point where 1 + s+1 = 0 in the closed right half plane is s = 0. Then (0) = ;1. By assumption, is analytic in a neighborhood of the origin since 2 RH1 . Hence, we can write (s) = ;1 + 1 X i=1 an sn ai 2 R: ( = a1 < 0, along with (0) = ;1, We now claim that a1 0. Otherwise, d dss) s=0 implies k k1 > 1. Hence (s) = ;1 + a1 s + s2 g(s) for some g(s) and a1 0 and s (I + K );1 = (I + K );1 = K (I + K );1 = s + 1 1+ 1 1 s+1 = 1 + a1 + sg 1 a sg 1 (I + K );1 = s + 1 1 +1 = 1 +1a+ + sg 1 + s+1 1 both of which are bounded in the neighborhood of the origin. Hence there is no destabilizing 2 RH1 with k k1 1. 3 The gap between the necessity and the su ciency of the above robust stability conditions for the closed ball of uncertainty is only technique and will not be considered in the sequel. The reason for the existence of a such gap may be attributed to the fact that in the multiplicative perturbed case, the signals z , w, and dm in Figure 9.5 are arti cial and they are not physical signals. Indeed, = (I + W1 W2 )P is a single system, the internal stability of the closed-loop system does not necessarily imply the boundedness of the arti cial signals z or w with respect to the arti cial disturbance dm . This is the case for the above example where = (1 + )P = (a1 s + s2 g(s))=s = a1 + sg(s) and the pole s = 0 is cancelled. This cancelation is arti cial and is caused by the particular model representation (i.e., there is really no cancelation in the physical system.) Thus the closed-loop system is robustly stable although the transfer function from dm to z is unstable. 9.3.3 Coprime Factor Uncertainty As another example, consider a left coprime factor perturbed plant described in Figure 9.6. Theorem 9.6 Let ~ ~ = (M + ~ M );1 (N + ~ N ) ~ ~ ~ ~ with M N ~ M , ~ N 2 RH1 . The transfer matrices (M N ) are assumed to be a ~ ;1 N ), and K internally stabilizes ~ stable left coprime factorization of P (i.e., P = M
- 236. 9.3. Stability under Stable Unstructured Uncertainties z1 ~N r-e K ;6 - N ~ - e; ~M w - ? -M ;1 e ~ 219 z2 y - Figure 9.6: Left Coprime Factor Perturbed Systems the nominal system P . De ne := well-posed and internally stable for all k ~ N ~ M . Then the closed-loop system is k1 < 1 if and only if K (I + PK );1 M ;1 ~ 1: I 1 Proof. Let K = UV ;1 be a right coprime factorization over RH1. By Lemma 5.10, the closed-loop system is internally stable if and only if ~ ~ (N + ~ N )U + (M + ~ M )V ;1 2 RH1 : (9:3) ~ ~ Since K stabilizes P , (NU + MV );1 2 RH1 . Hence (9.3) holds if and only if ~ ~ I + ( ~ N U + ~ M V )(NU + MV );1 ;1 2 RH1 : By the small gain theorem, the above is true for all k k1 < 1 if and only if U ~ ~ ;1 V (NU + MV ) 1 = K (I + PK );1 M ;1 ~ 1: I 1 2 Similarly, one can show that the closed-loop system is well-posed and internally stable for all k k1 1 if and only if K (I + PK );1 M ;1 < 1: ~ I 1 9.3.4 Unstructured Robust Stability Tests Table 9.1 summaries robust stability tests on the plant uncertainties under various assumptions. All of the tests pertain to the standard setup shown in Figure 9.4, where is the set of uncertain plants with P 2 as the nominal plant and with K as the internally stabilizing controller of P .
- 237. MODEL UNCERTAINTY AND ROBUSTNESS 220 Table 9.1 should be interpreted as UNSTRUCTURED ANALYSIS THEOREM Given NS & Perturbed Model Sets Then Closed-Loop Robust Stability if and only if Robust Stability Tests The table also indicates representative types of physical uncertainties which can be usefully represented by cone bounded perturbations inserted at appropriate locations. For example, the representation P = (I + W1 W2 )P in the rst row is useful for output errors at high frequencies (HF), covering such things as unmodeled high frequency dynamics of sensors or plant, including di usion processes, transport lags, electro-mechanical resonances, etc. The representation P = P (I + W1 W2 ) in the second row covers similar types of errors at the inputs. Both cases should be contrasted with the third and the fourth rows which treat P (I + W1 W2 );1 and (I + W1 W2 );1 P . These representations are more useful for variations in modeled dynamics, such as low frequency (LF) errors produced by parameter variations with operating conditions, with aging, or across production copies of the same plant. Discussion of still other cases is left to the table. Note from the table that the stability requirements on do not limit our ability to represent variations in either the number or locations of rhp singularities as can be seen from some simple examples. Example 9.2 Suppose an uncertain system with changing numbers of right-half plane poles is described by 1 P = s; : 2 R j j 1 : 1 1 Then P1 = s;1 2 P has one right-half plane pole and P2 = s+1 2 P has no right-half plane pole. Nevertheless, the set of P can be covered by a set of feedback uncertain plants: with P = 1 . s P := P (1 + P );1 : 2 RH1 k k1 1 Example 9.3 As another example, consider the following set of plants: P = (s s + 1 ++ 2) j j 2: + 1)(s 3
- 238. 9.3. Stability under Stable Unstructured Uncertainties W1 2 RH1 W2 2 RH1 Perturbed Model Sets 2 RH1 k k1 < 1 (I + W1 W2 );1 P Representative Types of Uncertainty Characterized output (sensor) errors neglected HF dynamics changing # of rhp zeros input (actuators) errors neglected HF dynamics changing # of rhp zeros LF parameter errors changing # of rhp poles P (I + W1 W2 );1 LF parameter errors changing # of rhp poles (I + W1 W2 )P P (I + W1 W2 ) P (I + W1 W2 P );1 additive plant errors neglected HF dynamics uncertain rhp zeros LF parameter errors uncertain rhp poles ~ ~ (M + ~ M );1 (N + ~ N ) ~ ~ P = M ;1 N = ~N ~M (N + N )(M + M );1 P = NM ;1 N = LF parameter errors neglected HF dynamics uncertain rhp poles & zeros LF parameter errors neglected HF dynamics uncertain rhp poles & zeros P + W1 W2 M 221 Robust Stability Tests kW2 ToW1 k1 1 kW2 Ti W1 k1 1 kW2 So W1 k1 1 kW2 Si W1 k1 1 kW2 KSoW1 k1 1 kW2 So PW1 k1 1 K I ~ So M ;1 M ;1 Si K I ] 1 1 1 1 Table 9.1: Unstructured Robust Stability Tests (# stands for `the number'.)
- 239. MODEL UNCERTAINTY AND ROBUSTNESS 222 This set of plants have changing numbers of right-half plane zeros since the plant has no right-half plane zero when = 0 and has one right-half plane zero when = ;2. The uncertain plant can be covered by a set of multiplicative perturbed plants: P 1 := s + 2 (1 + s 2 1 ) + 2 RH1 k k1 1 : 3 It should be noted that this covering can be quite conservative. 9.3.5 Equivalence: Robust Stability vs. Nominal Performance A robust stability problem can be viewed as another nominal performance problem. For example, the output multiplicative perturbed robust stability problem can be treated as a sensor noise rejection problem shown in Figure 9.7 and vise versa. It is clear that the system with output multiplicative uncertainty as shown in Figure 9.5 is robustly stable for k k1 < 1 i the H1 norm of the transfer function from w to z , Tzw , is no greater than 1. Since Tzw = Ten , hence kTzw k1 1 i supknk2 1 kek2 = kW2 ToW1 k1 1. ~ ~ e ;6 - K - P - W2 ? e W1 e n ~ Figure 9.7: Equivalence Between Robust Stability With Output Multiplicative Uncertainty and Nominal Performance With Sensor Noise Rejection There is, in fact, a much more general result along this line of equivalence: any robust stability problem (with open ball of uncertainty) can be regarded as an equivalent performance problem. This will be considered in Chapter 11. 9.4 Unstructured Robust Performance Consider the perturbed system shown in Figure 9.8 with the set of perturbed models described by a set . Suppose the weighting matrices Wd We 2 RH1 and the performance criterion is to keep the error e as small as possible in some sense for all possible models belong to the set . In general, the set can be either parameterized set or unstructured set such as those described in Table 9.1. The performance speci cations are usually speci ed in terms of the magnitude of each component e in time domain, i.e., L1 norm, with respect to bounded disturbances, or alternatively and more conveniently some requirements on the closed-loop frequency response of the transfer matrix between ~ d and e, say, integral of square error or the magnitude of the steady state error with
- 240. 9.4. Unstructured Robust Performance r-e - K ;6 -P 2 223 ~ d ? Wd d ? y We -e - e - Figure 9.8: Diagram for Robust Performance Analysis respect to sinusoidal disturbances. The former design criterion leads to the so-called L1 -optimal control framework and the latter leads to H2 and H1 design frameworks, respectively. In this section, we will focus primarily on the H2 and H1 performance objectives with unstructured model uncertainty descriptions. The performance under structured uncertainty will be considered in Chapter 11. 9.4.1 Robust H2 Performance Although nominal H2 performance analysis is straightforward and involves only the computation of H2 norms, the H2 performance analysis with H1 norm bounded model uncertainty is much harder and little studied. Nevertheless, this problem is sensible and important for the reason that performance is sometimes more appropriately speci ed in terms of the H2 norm than the H1 norm. On the other hand, model uncertainties are more conveniently described using the H1 norm bounded sets and arise naturally in identi cation processes. ~ Let Ted~ denote the transfer matrix between d and e, then Ted~ = We (I + P K );1 Wd P 2 : (9:4) And the robust H2 performance analysis problem can be stated as nding sup Ted~ 2 : P 2 To simplify our analysis, consider a scalar system with Wd = 1, We = ws , P = p, and assume that the model is given by a multiplicative uncertainty set = f(1 + wt )p : 2 RH1 k k1 < 1g : Assume further that the system is robustly stabilized by a controller k. Then w sup Ted 2 = sup 1 + s " ~ wt k k1 <1 P 2 w = 1 ; js "w j t 2 2
- 241. 224 MODEL UNCERTAINTY AND ROBUSTNESS where " = (1 + pk);1 and = 1 ; ". The exact analysis for the matrix case is harder to determine although some upper bounds can be derived as we shall do for the H1 case below. However, the upper bounds do not seem to be insightful in the H2 setting, and, therefore, are omitted. It should be pointed out that synthesis for robust H2 performance is much harder even in the scalar case although synthesis for nominal performance is relatively easy and will be considered in Chapter 14. 9.4.2 Robust H1 Performance with Output Multiplicative Uncertainty Suppose the performance criterion is to keep the energy of error e as small as possible, i.e., sup kek2 kd~k2 1 for some small . By scaling the error e (i.e., by properly selecting We ) we can assume without loss of generality that = 1. Then the robust performance criterion in this case can be described as requiring that the closed-loop system be robustly stable and that Ted~ 1 1 8P 2 : (9:5) More speci cally, an output multiplicatively perturbed system will be analyzed rst. The analysis for other classes of models can be done analogously. The perturbed model can be described as := f(I + W1 W2 )P : 2 RH1 k k1 < 1g (9:6) with W1 W2 2 RH1 . The explicit system diagram is as shown in Figure 9.5. For this class of models, we have Ted~ = We So (I + W1 W2 To);1 Wd and the robust performance is satis ed i kW2 ToW1 k1 1 and Ted~ 1 1 8 2 RH1 k k1 < 1: The exact analysis for this robust performance problem is not trivial and will be given in Chapter 11. However, some su cient conditions are relatively easy to obtain by bounding these two inequalities, and they may shed some light on the nature of these problems. It will be assumed throughout that the controller K internally stabilizes the nominal plant P .
- 242. 9.4. Unstructured Robust Performance 225 Theorem 9.7 Suppose P 2 f(I + W1 W2)P : 2 RH1 k k1 < 1g and K internally stabilizes P . Then the system robust performance is guaranteed if either one of the following conditions is satis ed (i) for each frequency ! (Wd ) (We So) + (W1 ) (W2 To ) 1 (9:7) (W1;1 Wd ) (We So Wd ) + (W2 To W1 ) 1 (9:8) (ii) for each frequency ! where W1 and Wd are assumed to be invertible and (W1;1 Wd ) is the condition number. Proof. It is obvious that both condition (9.7) and condition (9.8) guarantee that kW2 To W1 k1 1. So it is su cient to show that Ted~ 1 1 8 2 RH1 k k1 < 1. Now for any frequency !, it is easy to see that (Ted~) (We So ) (I + W1 W2 To);1 ] (Wd ) S (W = (I(WeWo ) W d ) ) + 1 2 To (We So) (Wd ) 1 ; (W1 W2 To ) (We So) (Wd ) 1 ; (W1 ) (W2 To ) ( ) : Hence condition (9.7) guarantees (Ted ) 1 for all 2 RH1 with k k1 < 1 at all ~ frequencies. Similarly, suppose W1 and Wd are invertible write Ted~ = We So Wd (W1;1 Wd );1 (I + W2 ToW1 );1 (W1;1 Wd ) and then (Ted~) Hence by condition (9.8), (Ted ) ~ at all frequencies. (We So Wd ) (W1;1 Wd ) : 1 ; (W2 ToW1 ) ( ) 1 is guaranteed for all 2 RH1 with k k1 < 1 2 Remark 9.2 It is not hard to show that either one of the conditions in the theorem is also necessary for scalar valued systems.
- 243. MODEL UNCERTAINTY AND ROBUSTNESS 226 Remark 9.3 Suppose (W1;1Wd) 1 (weighting matrices satisfying this condition are usually called round weights). This is particularly the case if W1 = w1 (s)I and Wd = wd (s)I . Recall that (We So Wd ) 1 is the necessary and su cient condition for nominal performance and that (W2 To W1 ) 1 is the necessary and su cient condition for robust stability. Hence it is clear that in this case nominal performance plus robust stability almost guarantees robust performance, i.e., NP + RS RP. However, this is not necessarily true for other types of uncertainties as will be shown later. ~ Remark 9.4 Note that in light of the equivalence relation between the robust stability and nominal performance, it is reasonable to conjecture that the above robust performance problem is equivalent to the robust stability problem in Figure 9.4 with the uncertainty model set given by := (I + Wd e We );1 (I + W1 W2 )P and k e k1 < 1, k k1 < 1, as shown in Figure 9.9. This conjecture is indeed true however, the equivalent model uncertainty is structured, and the exact stability analysis for such systems is not trivial and will be studied in Chapter 11. ~ - W2 - K 6 ; e - P - W1 ? Wd e ? ? - e - e - We Figure 9.9: Robust Performance with Unstructured Uncertainty vs. Robust Stability with Structured Uncertainty Remark 9.5 Note that if W1 and Wd are invertible, then Ted~ can also be written as Ted~ = We So Wd I + (W1;1 Wd );1 W2 ToW1 (W1;1 Wd ) ;1 : So another alternative su cient condition for robust performance can be obtained as (We So Wd ) + (W1;1 Wd ) (W2 ToW1 ) 1: A similar situation also occurs in the skewed case below. We will not repeat all these variations. ~
- 244. 9.4. Unstructured Robust Performance 227 9.4.3 Skewed Speci cations and Plant Condition Number We now consider the system with skewed speci cations, i.e., the uncertainty and performance are not measured at the same location. For instance, the system performance is still measured in terms of output sensitivity, but the uncertainty model is in input multiplicative form: := fP (I + W1 W2 ) : 2 RH1 k k1 < 1g : For systems described by this class of models, see Figure 9.8, the robust stability condition becomes kW2 Ti W1 k1 1 and the nominal performance condition becomes kWe So Wd k1 1: ~ ~ To consider the robust performance, let Ted~ denote the transfer matrix from d to e. Then ~ Ted~ = We So (I + PW1 W2 KSo);1 Wd = We So Wd I + (Wd;1 PW1 ) (W2 Ti W1 )(Wd;1 PW1 );1 ;1 : The last equality follows if W1 , Wd , and P are invertible and, if W2 is invertible, can also be written as ~ Ted~ = We So Wd (W1;1 Wd );1 I + (W1;1 PW1 ) (W2 P ;1 W2;1 )(W2 ToW1 ) ;1 (W1;1 Wd ): Then the following results follow easily. Theorem 9.8 Suppose P 2 = fP (I + W1 W2 ) : 2 RH1 k k1 < 1g and K internally stabilizes P . Assume that P W1 W2 , and Wd are square and invertible. Then the system robust performance is guaranteed if either one of the following conditions is satis ed (i) for each frequency ! (We So Wd ) + (Wd;1 PW1 ) (W2 Ti W1 ) 1 (9:9) (ii) for each frequency ! (W1;1 Wd ) (We So Wd ) + (W1;1 PW1 ) (W2 P ;1 W2;1 ) (W2 ToW1 ) 1: (9:10) Remark 9.6 If the appropriate invertibility conditions are not satis ed, then an alter- native su cient condition for robust performance can be given by (Wd ) (We So ) + (PW1 ) (W2 KSo ) 1: Similar to the previous case, there are many di erent variations of su cient conditions although (9.10) may be the most useful one. ~
- 245. 228 MODEL UNCERTAINTY AND ROBUSTNESS Remark 9.7 It is important to note that in this case, the robust stability condition is given in terms of Li = KP while the nominal performance condition is given in terms of Lo = PK . These classes of problems are called skewed problems or problems with skewed speci cations.2 Since, in general, PK 6= KP , the robust stability margin or tolerances for uncertainties at the plant input and output are generally not the same. ~ Remark 9.8 It is also noted that the robust performance condition is related to the condition number of the weighted nominal model. So in general if the weighted nominal model is ill-conditioned at the range of critical frequencies, then the robust performance condition may be far more restrictive than the robust stability condition and the nominal performance condition together. For simplicity, assume W1 = I , Wd = I and W2 = wt I where wt 2 RH1 is a scalar function. Further, P is assumed to be invertible. Then robust performance condition (9.10) can be written as (We So ) + (P ) (wt To ) 1 8!: Comparing these conditions with those obtained for non-skewed problems shows that the condition related to robust stability is scaled by the condition number of the plant3 . Since (P ) 1, it is clear that the skewed speci cations are much harder to satisfy if the plant is not well conditioned. This problem will be discussed in more detail in section 11.3.3 of Chapter 11. ~ ~ Remark 9.9 Suppose K is invertible, then T ~ can be written as ed ~ Ted~ = We K ;1 (I + Ti W1 W2 );1 Si KWd: Assume further that We = I Wd = ws I W2 = I where ws 2 RH1 is a scalar function. Then a su cient condition for robust performance is given by (K ) (Si ws ) + (Ti W1 ) 1 8! with (K ) := (K ) (K ;1 ). This is equivalent to treating the input multiplicative plant uncertainty as the output multiplicative controller uncertainty. ~ These skewed speci cations also create problems for MIMO loop shaping design which has been discussed brie y in Chapter 5. The idea of loop shaping design is based on the fact that robust performance is guaranteed by designing a controller with a su cient nominal performance margin and a su cient robust stability margin. For example, if (W1;1 Wd ) 1, the output multiplicative perturbed robust performance is guaranteed by designing a controller with twice the required nominal performance and robust stability margins. See Stein and Doyle 1991]. Alternative condition can be derived so that the condition related to nominal performance is scaled by the condition number. 2 3
- 246. 9.5. Gain Margin and Phase Margin 229 The fact that the condition number appeared in the robust performance test for skewed problems can be given another interpretation by considering two sets of plants 1 and 2 as shown below. 1 2 - wt - := fP (I + wt ) : := f(I + wt )P : ~ 2 RH1 k k1 < 1g 2 RH1 k k1 < 1g : - wt ~ ? -e - P - - P ? -e - Figure 9.10: Converting Input Uncertainty to Output Uncertainty Assume that P is invertible, then 2 1 if jwt j jwt j (P ) 8! ~ since P (I + wt ) = (I + wt P P ;1 )P . The condition number of a transfer matrix can be very high at high frequency which may signi cantly limit the achievable performance. The example below, taken from the textbook Franklin, Powell, and Workman, 1990], shows that the condition number shown in Figure 9.11 may increase with the frequency: 2 ;0:2 0:1 1 0 1 3 6 ;0:05 0 0 0 0:7 7 6 7 s + 1)(s + 0 07) 6 0 ;1 1 0 7 = 1 ;0s:05 0:(7(s + 1)(s + :0:13) P (s) = 6 0 7 a(s) 4 1 0 0 0 0 5 0 1 0 0 0 where a(s) = (s + 1)(s + 0:1707)(s + 0:02929). 9.5 Gain Margin and Phase Margin In this section, we show that the gain margin and phase margin de ned in classical control theory may not be su cient indicators of a system's robustness. Let L(s) be a scalar transfer function and consider a unit feedback system such as the one shown in the following diagram:
- 247. MODEL UNCERTAINTY AND ROBUSTNESS 230 10 2 condition number 10 3 10 1 0 10 -3 10 -2 10 10 -1 0 10 1 10 10 2 frequency Figure 9.11: Condition Number (!) = (P (j!))= (P (j!)) -e ;6 - L(s) - Suppose that the above closed-loop feedback system with L(s) = L0 (s) is stable. Then the system is said to have Gain Margin kmin and kmax if the closed-loop system is stable for all L(s) = kL0(s) with kmin < k < kmax but unstable for L(s) = kmax L0 (s) and for L(s) = kmin L0 (s) where 0 kmin 1 and kmax 1. Phase Margin min and max if the closed-loop system is stable for all L(s) = e;j L0 (s) with min < < max but unstable for L(s) = e;j max L0 (s) and for L(s) = e;j min L0(s) where ; . min 0 and 0 max These margins can be easily read from the open-loop system Nyquist diagram as shown in Figure 9.12 where kmax and kmin represent how much the loop gain can be increased
- 248. 9.5. Gain Margin and Phase Margin 231 and decreased, respectively, without causing instability. Similarly max and min represent how much loop phase lag and lead can be tolerated without causing instability. 1/K min 1/K max φ min -1 -1 φ max L(jω) L(jω) Figure 9.12: Gain Margin and Phase Margin of A Scalar System However, gain margin or phase margin alone may not be a su cient indicator of a system's robustness. To be more speci c, consider a simple dynamical system a P = as; s1 a > 1 ; with a stabilizing controller K . Now let L = PK and consider a controller b K = bs+ s1 b > 0: + It is easy to show that the closed-loop system is stable for any 1 < b < a: a To compute the stability margins, consider three cases: (i) b = 1: in this case, K = 1 and the stability margins can be easily shown to be 1 ;1 a2 ; 1 =: : kmin = a kmax = a min = ; max = sin a2 + 1 It is easy to see that both gain margin and phase margin are very large for large a. 1 (ii) a < b < a and b ! a: in this case =; !0 k = 1 ! 1 k = ab ! a2 min ab a2 max min max i.e., very large gain margin but arbitrarily small phase margin.
- 249. MODEL UNCERTAINTY AND ROBUSTNESS 232 1 1 (iii) a < b < a and b ! a : in this case k = 1 ! 1 k = ab ! 1 min max ab min = ; max !2 i.e., very large phase margin but arbitrarily small gain margin. The open-loop frequency response of the system is shown in Figure 9.13 for a = 2 and b = 1 b = 1:9 and b = 0:55, respectively. Sometimes, gain margin and phase margin together may still not be enough to indicate the true robustness of a system. For example, it is possible to construct a (complicated) controller such that =; > k < 1 k >a min a max min max but the Nyquist plot is arbitrarily close to (;1 0). Such a controller is complicated to construct however, the following controller should give the reader a good idea of its construction: + + 0 55 7 2 + + Kbad = 3s:3s 3:31 0s:55s :+ 1 1s:2 s 1:51s:5s 1:71 : + + + The open-loop frequency response of the system with this controller is also shown in Figure 9.13 by the dotted line. It is easy to see that the system has at least the same gain margin and phase margin as the system with controller K = 1, but the Nyquist plot is closer to (;1 0). Therefore this system is less robust with respect to the simultaneous gain and phase perturbations. The problem is that the gain margin and phase margin do not give the correct indication of the system's robustness when the gain and phase are varying simultaneously. 9.6 De ciency of Classical Control for MIMO Systems In this section, we show through an example that the classical control theory may not be reliable when it is applied to MIMO system design. Consider a symmetric spinning body with torque inputs, T1 and T2 , along two orthogonal transverse axes, x and y, as shown in Figure 9.14. Assume that the angular velocity of the spinning body with respect to the z axis is constant, . Assume further that the inertia of the spinning body with respect to the x y, and z axes are I1 , I2 = I1 , and I3 , respectively. Denote by !1 and !2 the angular velocities of the body with respect to the x and y axes, respectively. Then the Euler's equation of the spinning body is given by I1 !1 ; !2 (I1 ; I3 ) = T1 _ I1 !2 ; !1 (I3 ; I1 ) = T2 : _
- 250. 9.6. De ciency of Classical Control for MIMO Systems 233 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -2 -1.5 -1 -0.5 0 0.5 Figure 9.13: Nyquist Plots of L with a = 2 and b = 1(solid), b = 1:9(dashed), b = 0:55(dashdot) and with Kbad (dotted) De ne u1 := T1 =I1 u2 T2 =I1 a := (1 ; I3 =I1 ) : Then the system dynamical equations can be written as !1 = 0 a _ !1 u1 !2 _ ;a 0 !2 + u2 : Now suppose that the angular rates !1 and !2 are measured in scaled and rotated coordinates: cos sin !1 = 1 a !1 y1 = 1 !2 ;a 1 !2 y2 cos ; sin cos where tan := a. (There is no speci c physical meaning for the measurements of y1 and y1 but they are assumed here only for the convenience of discussion.) Then the transfer matrix for the spinning body can be computed as Y (s) = P (s)U (s) with 1 s a2 ( P (s) = s2 + a2 ;a(; + 1) ass;+a1) : 2 s
- 251. MODEL UNCERTAINTY AND ROBUSTNESS 234 z x y Figure 9.14: Spinning Body Suppose the control law is chosen as u = K1 r ; y where 1 1 K1 = 1 + a2 a ;a : 1 y P u f 6 ; r ~ K1 r Figure 9.15: Closed-loop with a Bizarre" Controller Then the closed-loop transfer function is given by 1 Y (s) = s + 1 1 0 R(s) 0 1
- 252. 9.6. De ciency of Classical Control for MIMO Systems 235 and the sensitivity function and the complementary sensitivity function are given by 1 1 s 1 S = (I + P );1 = s + 1 a ;a T = P (I + P );1 = s + 1 ;a a : s 1 It is noted that this controller design has the property of decoupling the loops. Furthermore, each single loop has the open-loop transfer function as 1 s so each loop has phase margin max = ; min = 90o and gain margin kmin = 0 kmax = 1. Suppose one loop transfer function is perturbed, as shown in Figure 9.16. 6 z w y1 u1 ? e P u2 e 6 ; y2 e 6 ; Figure 9.16: One-Loop-At-A-Time Analysis Denote 1 z (s) w(s) = ;T11 = ; s + 1 : Then the maximum allowable perturbation is given by k k < 1 =1 1 kT11 k1 which is independent of a. Similarly the maximum allowable perturbation on the other loop is also 1 by symmetry. However, if both loops are perturbed at the same time, then the maximum allowable perturbation is much smaller, as shown below. Consider a multivariable perturbation, as shown in Figure 9.17, i.e., P = (I + )P , with = 11 12 2 RH1 21 22
- 253. MODEL UNCERTAINTY AND ROBUSTNESS 236 y1 f 6 o S S f 11 6 g12 12 21 y2 ? / f 22 g11 ; ~ ? r1 f g21 ? f g22 f r2 ~ 6 ; Figure 9.17: Simultaneous Perturbations a 2 2 transfer matrix such that k k1 < . Then by the small gain theorem, the system is robustly stable for every such i 1 1 kT k1 = p1 + a2 ( 1 if a 1): In particular, consider = d = 11 2 R2 2 : 22 Then the closed-loop system is stable for every such i 1 ; det(I + T d) = (s + 1)2 s2 + (2 + 11 + 22 )s + 1 + 11 + 22 + (1 + a2 ) 11 22 has no zero in the closed right-half plane. Hence the stability region is given by 2 + 11 + 22 > 0 1 + 11 + 22 + (1 + a2 ) 11 22 > 0: It is easy to see that the system is unstable with 1 : 11 = ; 22 = p 1 + a2
- 254. 9.7. Notes and References 237 δ 22 0.707 0.707 δ 11 Figure 9.18: Stability Region for a = 1 The stability region is drawn in the Figure 9.18. This clearly shows that the analysis of an MIMO system using SISO methods can be misleading and can even give erroneous results. Hence an MIMO method has to be used. 9.7 Notes and References The small gain theorem was rst presented by Zames 1966]. The book by Desoer and Vidyasagar 1975] contains a quite extensive treatment and applications of this theorem in various forms. Robust stability conditions under various uncertainty assumptions are discussed in Doyle, Wall, and Stein 1982]. It was rst shown in Kishore and Pearson 1992] that the small gain condition may not be necessary for robust stability with closed-ball perturbed uncertainties. In the same reference, some extensions of stability and performance criteria under various structured/unstructured uncertainties are given. Some further extensions are also presented in Tits and Fan 1994].
- 255. 238 MODEL UNCERTAINTY AND ROBUSTNESS
- 256. 10 Linear Fractional Transformation This chapter introduces a new matrix function: linear fractional transformation (LFT). We show that many interesting control problems can be formulated in an LFT framework and thus can be treated using the same technique. 10.1 Linear Fractional Transformations This section introduces the matrix linear fractional transformations. It is well known from the one complex variable function theory that a mapping F : C 7! C of the form a + bs F (s) = c + ds with a b c and d 2 C is called a linear fractional transformation. In particular, if c 6= 0 then F (s) can also be written as F (s) = + s(1 ; s);1 for some and 2 C . The linear fractional transformation described above for scalars can be generalized to the matrix case. De nition 10.1 Let M be a complex matrix partitioned as M = M11 M12 2 C (p1 +p2 ) (q1 +q2 ) M21 M22 239
- 257. LINEAR FRACTIONAL TRANSFORMATION 240 and let ` 2 C q2 p2 and u 2 C q1 p1 be two other complex matrices. Then we can formally de ne a lower LFT with respect to ` as the map F`(M ) : with C q2 p2 7! C p1 q1 ` ) = M11 + M12 ` (I ; M22 ` );1 M21 provided that the inverse (I ; M22 ` );1 exists. We can also de ne an upper LFT with respect to u as Fu (M ) : C q1 p1 7! C p2 q2 F` (M with u ) = M22 + M21 u (I ; M11 u );1 M12 provided that the inverse (I ; M11 u );1 exists. Fu (M The matrix M in the above LFTs is called the coe cient matrix. The motivation for the terminologies of lower and upper LFTs should be clear from the following diagram representations of F` (M ` ) and Fu (M u ): z1 y1 M - ` w1 u1 z2 - y2 u M u2 w2 The diagram on the left represents the following set of equations: z1 y1 u1 = M w1 = M11 M12 u1 M21 M22 = ` y1 w1 u1 while the diagram on the right represents y2 z2 u = M w2 = M11 M12 M21 M22 2 u = u y2 : u2 w2 It is easy to verify that the mapping de ned on the left diagram is equal to F`(M ` ) and the mapping de ned on the right diagram is equal to Fu (M u ). So from the above diagrams, F`(M ` ) is a transformation obtained from closing the lower loop on the left diagram similarly Fu (M u ) is a transformation obtained from closing the upper loop on the right diagram. In most cases, we will use the general term LFT in referring to both upper and lower LFTs and assume that the contents will distinguish the situations. The reason for this is that one can use either of these notations to express
- 258. 10.1. Linear Fractional Transformations 241 a given object. Indeed, it is clear that Fu (N ) = F` (M ) with N = M22 M21 . M12 M11 It is usually not crucial which expression is used however, it is often the case that one expression is more convenient than the other for a given problem. It should also be clear to the reader that in writing F` (M ) (or Fu (M )) it is implied that has compatible dimensions. A useful interpretation of an LFT, e.g., F` (M ), is that F` (M ) has a nominal mapping, M11 , and is perturbed by , while M12 M21 , and M22 re ect a prior knowledge as to how the perturbation a ects the nominal map, M11 . A similar interpretation can be applied to Fu (M ). This is why LFT is particularly useful in the study of perturbations, which is the focus of the next chapter. The physical meaning of an LFT in control science is obvious if we take M as a proper transfer matrix. In that case, the LFTs de ned above are simply the closed-loop transfer matrices from w1 7! z1 and w2 7! z2 , respectively, i.e., Tzw1 = F` (M ` ) Tzw2 = Fu (M u ) where M may be the controlled plant and tainties or the controllers. may be either the system model uncer- De nition 10.2 An LFT, F`(M ), is said to be well de ned (or well-posed) if (I ; M22 ) is invertible. Note that this de nition is consistent with the well-posedness de nition of the feedback system, which requires the corresponding transfer matrix be invertible in Rp (s). It is clear that the study of an LFT that is not well-de ned is meaningless, hence throughout the book, whenever an LFT is invoked, it will be assumed implicitly to be well de ned. It is also clear from the de nition that, for any M , F` (M 0) is well de ned hence any function that is not well de ned at the origin cannot be expressed as an LFT in terms of its variables. For example, f ( ) = 1= is not an LFT of . In some literature, LFT is used to refer to the following matrix functions: (A + BQ)(C + DQ);1 or (C + QD);1 (A + QB ) where C is usually assumed to be invertible due to practical consideration. The following is true. Lemma 10.1 Suppose C is invertible. Then (A + BQ)(C + DQ);1 = F`(M Q) (C + QD);1 (A + QB ) = F`(N Q) with ;1 ; AC ;1 M = AC;1 B ;C ;1 D D C C ;1 C ;1 N = B ; DCA 1 A ;DC ;1 : ;
- 259. LINEAR FRACTIONAL TRANSFORMATION 242 The converse also holds if M satis es certain conditions. Lemma 10.2 Let F`(M Q) be a given LFT with M = M11 M12 , then M21 M22 (a) if M21 is invertible, F` (M Q) = (A + BQ)(C + DQ);1 with ; ; ; ; A = M11 M211 B = M12 ; M11 M211 M22 C = M211 D = ;M211 M22 : (b) if M12 is invertible, F` (M Q) = (C + QD);1 (A + QB ) with ; ; ; ; A = M121 M11 B = M21 ; M22 M121 M11 C = M121 D = ;M22 M121 : However, for an arbitrary LFT F`(M Q), neither M21 nor M12 is necessarily square and invertible therefore, the alternative fractional formula is more restrictive. It should be pointed out that some seemingly similar functions do not have simple LFT representations. For example, (A + QB )(I + QD);1 cannot always be written in the form of F` (M Q) for some M however, it can be written as (A + QB )(I + QD);1 = F`(N ) with 3 2 A I A N = 4 ;B 0 ;B D 0 D 5 = Q Q : Note that the dimension of is twice as many as Q. The following lemma summarizes some of the algebraic properties of LFT s. Lemma 10.3 Let M Q, and G be suitably partitioned matrices: M = M11 M12 M21 M22 Q = Q11 Q12 Q21 Q22 2 3 A B1 B2 G = 4 C1 D11 D12 5 : C2 D21 D22
- 260. 10.1. Linear Fractional Transformations 243 (i) Fu (M ) = Fl (N ) with 0 0 N = I I M I I = M22 M21 0 0 M12 M11 where the dimensions of identity matrices are compatible with the partitions of M and N . (ii) Suppose Fu (M ) is square and well-de ned and M22 is nonsingular. Then the inverse of Fu (M ) exists and is also an LFT with respect to : (Fu (M ));1 = Fu (N ) with N given by M ;1 M ;1 N = M11 ; M12M 22 M21 ;M12;1 22 : ;1 21 M22 M22 (iii) Fu (M 1 ) + Fu (Q 2 ) = Fu (N ) with 2 M12 M11 0 Q12 N = 4 0 Q11 M21 Q21 M22 + Q22 (iv) Fu (M 1 )Fu (Q 2 ) = Fu (N ) with 2 M11 M12Q21 M12 Q22 M11 Q12 N =4 0 M21 M22Q21 M22 Q22 3 1 0 1 0 = 5 0 = 0 2 3 5 2 : : (v) Consider the following interconnection structure where the dimensions of 1 are compatible with A: z w Fu (G 1) - Fu (Q 2) Then the mapping from w to z is given by Fl (Fu (G 1) Fu (Q 2 )) = Fu (Fl (G Fu (Q 2 )) 1 ) = Fu (N )
- 261. LINEAR FRACTIONAL TRANSFORMATION 244 2 A + B2 Q22 L1 C2 B2 L2 Q21 B1 + B2 Q22 L1 D21 Q12 L1 C2 Q11 + Q12 L1 D22 Q21 Q12 L1 D21 N =4 C1 + D12 L2 Q22 C2 D12 L2Q21 D11 + D12 Q22 L1 D21 where L1 := (I ; D22 Q22 );1 , L2 := (I ; Q22 D22 );1 , and = 1 0 0 2 3 5 . Proof. These properties can be straightforwardly veri ed by the de nition of LFT , so the proofs are omitted. 2 Property (v) shows that if the open-loop system parameters are LFTs of some variables, then the closed-loop system parameters are also LFTs of the same variables. This property is particularly useful in perturbation analysis and in building the interconnection structure. Similar results can be stated for lower LFT. It is usually convenient to interpret an LFT Fu (M ) as a state space realization of a generalized system with frequency structure . In fact, all the above properties can be reduced to the standard transfer matrix operations if = 1 I . s The following proposition is concerned with the algebraic properties of LFT s in the general control setup. Lemma 10.4 Let P = P11 P12 and let K be rational transfer function matrices P21 P22 and let G = F` (P K ). Then (a) G is proper if P and K are proper with det(I ; P22 K )(1) = 0. 6 (b) F`(P K1 ) = F`(P K2 ) implies that K1 = K2 if P12 and P21 have normal full column and row rank in Rp (s), respectively. (c) If P and G are proper, det P (1) 6= 0, det P ; G 0 (1) 6= 0 and P12 and 0 0 P21 are square and invertible for almost all s, then K is proper and K = Fu (P ;1 G): Proof. (a) is immediate from the de nition of F`(P K ) (or well-posedness condition). (b) follows from the identity F`(P K1 ) ; F`(P K2 ) = P12 (I ; K2 P22 );1 (K1 ; K2)(I ; P22 K1 );1 P21 : (c) it is su cient to show that the formula for K is well-posed and K thus obtained is proper. Let Q = P ;1 , which will be proper since det P (1) 6= 0, and de ne K = Fu (Q G) = Q22 + Q21 G(I ; Q11 G);1 Q12 :
- 262. 10.1. Linear Fractional Transformations 245 This expression is well-posed and proper since at s = 1 det(I ; Q11 G) = det I ; I 0 P ;1 I G 0 = det P ;1 P ; G 0 6= 0: 0 0 We also need to ensure that F` (P K ) is well-posed: I ; P22 K = (I ; P22 Q22 ) ; P22 Q21 G(I ; Q11 G);1 Q12 = P21 Q12 + P21 Q11 G(I ; Q11 G);1 Q12 = P21 (I ; Q11 G);1 Q12 : The above form is obtained by using the fact that PQ = I . Then det(I ;P22 K ) 6= 0 ; ; since P211 exists and Q;1 = P21 ;P22 P121P11 . Hence the LFTs are both well-posed 12 and we immediately obtain that F`(P K ) = G as required upon substituting for K and (I ; P22 K ), as shown above. 2 Remark 10.1 This lemma shows that under certain conditions, an LFT of transfer matrices is a bijective map between two sets of proper and real rational matrices. When given proper transfer matrices P and G with compatible dimensions which satisfy conditions in (c), there exists a unique proper K such that G = Fl (P K ). On the other hand, the conditions of part (c) show that the feedback systems are well-posed. ~ Remark 10.2 A simple interpretation of the result (c) is given by considering the signals in the feedback systems, z y P - K w u assuming this structure is well-posed. And we have z y hence = P w u u = Ky ) z = F` (P K )w = Gw w u z = P ;1 y z = Gw ) u = Fu (P ;1 G)y or K = Fu (P ;1 G): ~
- 263. LINEAR FRACTIONAL TRANSFORMATION 246 10.2 Examples of LFTs LFT is a very convenient tool to formulate many mathematical objects. In this section and the sections to follow, some commonly encountered control or mathematical objects are given new perspectives, i.e., they will be examined from the LFT point of view. Polynomials A very commonly encountered object in control and mathematics is a polynomial function. For example, p( ) = a0 + a1 + + an n with indeterminate . It is easy to verify that p( ) can be written in the following LFT form: p( ) = F`(M In ) with 3 2 a0 a1 an;1 an 1 0 0 0 7 7 0 0 7: M= 0 1 7 . . . .. 7 0 .. . . .. . 5 1 0 0 0 Hence every polynomial is a linear fraction of their indeterminates. More generally, any multivariate (matrix) polynomials are also LFTs in their indeterminates for example, 6 6 6 6 6 4 p( Then with 2 2 1 2 ) = a1 1 + a2 2 + a3 1 2 + a4 1 + a5 2 + a6 : p( 2 6 6 6 6 4 1 2 ) = F` (N a6 1 0 1 0 a4 0 a1 0 a3 ) 3 7 7 N= 1 0 0 0 0 7 = 1 I2 I : 7 2 2 a5 0 0 0 a2 5 1 0 0 0 0 It should be noted that these representations or realizations of polynomials are neither unique nor necessarily minimal. Here a minimal realization refers to a realization with the smallest possible dimension of . As commonly known, in multidimensional systems and lter theory, it is usually very hard, if not impossible, to nd a minimal realization for even a two variable polynomial. In fact, the minimal dimension of depends also on the eld (real, complex, etc.) of the realization. More detailed discussion of this issue is beyond the scope of this book, the interested readers should consult the references in 2-d or n-d systems or lter theory.
- 264. 10.2. Examples of LFTs 247 Rational Functions As another example of LFT representation, we consider a rational matrix function (not necessarily proper), F ( 1 2 0) is m ), with a nite value at the origin: F (0 0 nite. Then F ( 1 2 m ) can be written as an LFT in ( 1 2 m ) (some i may be repeated). To see that, write N( 1 2 m) = N ( F( 1 2 1 2 m ) (d( 1 2 m )I );1 m ) = d( 1 2 m) where N ( 1 2 m ) is a multivariate matrix polynomial and d( 1 2 m ) is a scalar multivariate polynomial with d(0 0 0) 6= 0. Both N and dI can be represented as LFTs, and, furthermore, since d(0 0 0) 6= 0, the inverse of dI is also an LFT as shown in Lemma 10.3. Now the conclusion follows by the fact that the product of LFTs is also an LFT. (Of course, the above LFT representation problem is exactly the problem of state space realization for a multidimensional discrete transfer matrix.) However, this is usually not a nice way to get an LFT representation for a rational matrix since this approach usually results in a much higher dimensioned than required. For example, + f ( ) = 1 + = F`(M ) with M= 1 ; ; : By using the above approach, we would end up with f ( ) = F` (N I2 ) and 3 2 ; N =4 1 0 ; 5 : 1 0 ; Although the latter can be reduced to the former, it is not easy to see how to carry out such reduction for a complicated problem, even if it is possible. State Space Realizations We can use the LFT formulae to establish the relationship between transfer matrices and their state space realizations. A system with a state space realization as x = Ax + Bu _ y = Cx + Du has a transfer matrix of G(s) = D + C (sI ; A);1 B = F ( A B 1 I ): u C D s
- 265. LINEAR FRACTIONAL TRANSFORMATION 248 Now take = 1 I , the transfer matrix can be written as s A B G(s) = Fu ( C D ): More generally, consider a discrete time 2-D (or MD) system realized by the rst-order state space equation x1 (k1 + 1 k2 ) = A11 x1 (k1 k2 ) + A12 x2 (k1 k2 ) + B1 u(k1 k2 ) x2 (k1 k2 + 1) = A21 x1 (k1 k2 ) + A22 x2 (k1 k2 ) + B2 u(k1 k2 ) y(k1 k2 ) = C1 x1 (k1 k2 ) + C2 x2 (k1 k2 ) + Du(k1 k2 ): In the same way, take ;1 0 = z10 I z ;1I 2 =: 1I 0 0 2I where zi denotes the forward shift operator, and let A = A11 A12 A21 A22 B = B1 B2 C = C1 C2 then its transfer matrix is G(z1 z2 ) = D + C ( A =: Fu ( C z1 I 0 B D 0 ;1 ;1 z2 I ; A) B = D + C (I ; A) B ): Both formulations can correspond to the following diagram: - A B C D The following notation for a transfer matrix has already been adopted in the previous chapters: A B A B ): C D := Fu ( C D
- 266. 10.2. Examples of LFTs 249 It is easy to see that this notation can be adopted for general dynamical systems, e.g., multidimensional systems, as far as the structure is speci ed. This notation means that the transfer matrix can be expressed as an LFT of with the coe cient A B matrix C D . In this special case, we say the parameter matrix is the frequency structure of the system state space realization. This notation is deliberately somewhat ambiguous and can be viewed as both a transfer matrix and one of its realizations. The ambiguity is benign and convenient and can always be resolved from the context. Frequency Transformation The bilinear transformation between the z -domain and s-domain z+1 s= z;1 transforms the unit disk to the left-half plane and is the simplest example of an LFT. We may rewrite it in our standard form as 1 I = I ; p2I z ;1I (I + z ;1I );1 p2I = F (N z ;1I ) u s where p I p N = ; 2I ;2II : Now consider a continuous system A B G(s) = C D = Fu (M 1 I ) s where A B M= C D then the corresponding discrete time system realization is given by z;1 ~ ~ G(z ) = Fu (M z + 1 I ) = Fu (M Fu (N z ;1 I )) = Fu (M z ;1 I ) with p I ; ;1 I ; 2( );1 ~ M = ;(p2CAI) ; (A)+1A) C (I ; I ; AB +B : ; ;1 ( A) D The transformation from the z -domain to the s-domain can be obtained similarly.
- 267. LINEAR FRACTIONAL TRANSFORMATION 250 W1 uf i ;6 y K u d ? -i - P - W2 v- ? n F i Simple Block Diagrams A feedback system with the following block diagram can be rearranged as an LFT: z y with w= and w G u - K d n v uf z= 2 W2 P W1 G=4 0 ;FP ;F ;FP W2 P 0 0 3 5 : Constrained Structure Synthesis Using the properties of LFTs, we can show that constrained structure control synthesis problems can be converted to constrained structure constant output feedback problems. Consider the synthesis structure in the last example and assume 2 A B1 B2 G = 4 C1 D11 D12 C2 D21 D22 3 5 B K = AK DK : CK K Then it is easy to show that F`(G K ) = F` (M (s) F )
- 268. 10.2. Examples of LFTs where and 251 2 A 0 B1 0 B2 6 0 0 0 I 0 6 M (s) = 6 C1 0 D11 0 D12 6 4 0 I 0 0 0 C2 0 D21 0 D22 3 7 7 7 7 5 B F = AK DK : CK K Note that F is a constant matrix, not a system matrix. Hence if the controller structure is xed (or constrained), then the corresponding control problem becomes a constant (constrained) output feedback problem. Parametric Uncertainty: A Mass/Spring/Damper System One natural type of uncertainty is unknown coe cients in a state space model. To motivate this type of uncertainty description, we will begin with a familiar mass/spring/damper system, shown below in Figure 10.1. 6 F m X X k XX X X X c Figure 10.1: Mass/Spring/Damper System The dynamical equation of the system motion can be described by F c_ k x + mx + mx = m: Suppose that the 3 physical parameters m c, and k are not known exactly, but are believed to lie in known intervals. In particular, the actual mass m is within 10% of a nominal mass, m, the actual damping value c is within 20% of a nominal value of c, and the spring sti ness is within 30% of its nominal value of k. Now introducing perturbations m , c, and k , which are assumed to be unknown but lie in the interval ;1 1], the block diagram for the dynamical system can be shown in Figure 10.2.
- 269. LINEAR FRACTIONAL TRANSFORMATION 252 x x _ 1 s 1 s - x 1 m(1+0:1 m ) F d 6 ; -d 6 + c(1 + 0:2 c ) k(1 + 0:3 k ) Figure 10.2: Block Diagram of Mass/Spring/Damper Equation 1 It is easy to check that m can be represented as an LFT in m : 1 1 1 0:1 ;1 m = m(1 + 0:1 m) = m ; m m (1 + 0:1 m) = F` (M1 m ) : " # : ; 0m1 . Suppose that the input signals of the dynamical system are 1 ;0:1 1 m with M1 = selected as x1 = x x2 = x F , and the output signals are selected as x1 and x2 . To _ _ _ represent the system model as an LFT of the natural uncertainty parameters m c and k , we shall rst isolate the uncertainty parameters and denote the inputs and outputs of k c and m as yk yc ym and uk uc um, respectively, as shown in Figure 10.3. Then 2 6 6 6 6 6 6 4 i.e., x1 _ x2 _ yk yc ym 3 2 7 7 7 7 7 7 5 6 6 6 6 6 6 4 = 0 k ;m 1 c ;m 0 1 m 0:3k 0 0 0 0:2c 0 ;k ;c 1 " 0 ;1 m 0 0 ;1 x1 _ x2 _ # 0 ;1 m 0 3 2 6 6 ; 0:1 7 6 7 m 0 0 0 0 ;1 ;0:1 76 76 76 76 56 6 4 2 x1 6 = F` (M ) 4 x2 F x1 x2 F uk uc um 3 7 5 3 7 7 7 7 7 7 7 7 7 5 2 6 4 uk uc um 3 7 5 2 = 6 4 yk yc ym 3 7 5
- 270. 10.2. Examples of LFTs where 2 6 6 6 6 6 6 4 0 1 k c ;m ;m M = 0:3k 253 0 0 0 0 ;1 x2 1 s 3 0 1 1 : ; m ; m ; 0m1 1 m 0 0 0 0:2c 0 ;k ;c 1 x1 0 0 0 0 0 ;1 ;0:1 1 2 7 7 7 7 7 7 5 =6 0 4 0 ym - - c - k - 0:3 0 6 ; um m - 0:2 yc c e M1 s 0 k c yk k 0 0 3 7 5 : m F -e -e 6 6 uc -e 6 uk Figure 10.3: Mass/Spring/Damper System General A ne State-Space Uncertainty We will consider a special class of state space models with unknown coe cients and show how this type of uncertainty can be represented via the LFT formulae with respect to an uncertain parameter matrix so that the perturbations enter the system in a feedback form. This type of modeling will form the basic building blocks for components with parametric uncertainty. Consider a linear system G (s) that is parameterized by k uncertain parameters, 1 : : : k , and has the realization 2 G (s) 6 6 =6 6 4 A+ C+ k X i=1 k X i=1 ^ i Ai B + ^ i Ci D + k X i=1 k X i=1 ^ i Bi ^ i Di 3 7 7 7 7 5 :
- 271. LINEAR FRACTIONAL TRANSFORMATION 254 ^ ^ ^ ^ Here A Ai 2 Rn n B Bi 2 Rn nu C Ci 2 Rny n , and D Di 2 Rny nu . The various terms in these state equations are interpreted as follows: the nominal system description G(s), given by known matrices A B C and D, is (A B C D) and the parametric uncertainty in the nominal system is re ected by the k scalar uncertain parameters 1 : : : k , and we can specify them, say by i 2 ;1 1]. The structural ^ ^ ^ ^ knowledge about the uncertainty is contained in the matrices Ai Bi Ci , and Di . They re ect how the i'th uncertainty, i , a ects the state space model. Now, we consider the problem of describing the perturbed system via the LFT formulae so that all the uncertainty can be represented as a nominal system with the unknown parameters entering it as the feedback gains. This is shown in Figure 10.4. Since G (s) = Fu (M 1 I ) where s 2 M 6 6 =6 6 4 A+ C+ k X i=1 k X i=1 ^ i Ai B + ^ i Ci D+ k X i=1 k X i=1 3 ^ i Bi 7 7 7 7 5 ^ i Di we need to nd an LFT representation for the matrix M with respect to p = diag f 1 I 2 I : : : kIg : To achieved this with the smallest possibly size of repeated blocks, let qi denote the rank of the matrix " # ^ ^ Ai Bi 2 R( n+ny ) (n+nu ) Pi = ^ ^ Ci Di for each i. Then Pi can be written as " Pi = Li Wi #" Ri Zi # where Li 2 Rn qi , Wi 2 Rny qi , Ri 2 Rn qi and Zi 2 Rnu qi . Hence, we have " Li i Pi = Wi # " Ri i Iqi ] Zi # and M can be written as z" M = M11 }| A B C D #{ z" + M12 L1 W1 }| Lk Wk z #{ 2 6 6 4 p 1 Iq1 }| {2 3z ... 76 76 54 k Iqk M21 }| R1 Z1 .. . .. . Rk Zk { 3 7 7 5
- 272. 10.2. Examples of LFTs i.e. 255 " M = F`( M11 M12 M21 0 - # p ): 1I s A B B2 C D D12 C2 D21 D22 x _ y y2 1I - ... x u u2 kI Figure 10.4: LFT Representation of State Space Uncertainty Therefore, the matrices B2 C2 D12 D21 , and D22 in the diagram are B2 = D12 = h L1 L2 Lk i W1 W2 Wk R1 R2 Rk D21 = Z1 Z2 D22 = 0 Zk C2 = and h h h " G ( ) = Fu (F`( M11 M12 M21 0 i i i # p) 1 I ): s
- 273. 256 LINEAR FRACTIONAL TRANSFORMATION 10.3 Basic Principle We have studied several simple examples of the use of LFTs and, in particular, their role in modeling uncertainty. The basic principle at work here in writing a matrix LFT is often referred to as pulling out the s". We will try to illustrate this with another picture (inspired by Boyd and Barratt 1991]). Consider a structure with four substructures interconnected in some known way as shown in Figure 10.5. . . . . . . . . . . . . . . . . . . . . .......... . . . . .................................................................... .... . ............................ .. .. .. .. .... . .. 1 ..... . . . . .. . . . . . . . . . .................................... 2 .... . . . . . .... ...... ............ ..... ... ....... . . .... ...... ............ ..... ... ... ....... . . . . ......... ..... . . ................................................................. .... . . .. . . . . .. . . ... ... ..... 3 ... ............. ........ ... .. K .......... . ... . . .... .. .. ... .. .................................... . . . ........................................ . . . . . ........... . . . . . . . . . . . . . . . . . . . . Figure 10.5: Multiple Source of Uncertain Structure This diagram can be redrawn as a standard one via pulling out the s" in Figure 10.6. Now the matrix M " of the LFT can be obtained by computing the corresponding transfer matrix in the shadowed box. We shall illustrate the above principle with an example. Consider an input/output relation a+b c 2 z = 1 + d 2 + + 1 2 w =: Gw e2 1 2 1 where a b c d and e are given constants or transfer functions. We would like to write G as an LFT in terms of 1 and 2 . We shall do this in three steps: 1. Draw a block diagram for the input/output relation with each separated as shown in Figure 10.7. 2. Mark the inputs and outputs of the 's as y's and u's, respectively. (This is essentially pulling out the s). 3. Write z and y's in terms of w and u's with all 's taken out. (This step is equivalent to computing the transformation in the shadowed box in Figure 10.6.) 2 6 6 6 6 6 6 4 y1 y2 y3 y4 z 3 2 u1 7 6 7 6 u2 7 6 7 = M 6 u3 7 6 7 6 5 4 u4 w 3 7 7 7 7 7 7 5
- 274. 10.4. Redhe er Star-Products - 257 1 - 2 - 3 . . . . . . . . . . . . . . . . . . . . .......... . . . . .................................................................... .... . ............................ .. .. .. .. .... . .. ..... . . . . . . . . . . . . .................................... ? ......... . . . .... ...... ............ ..... ... ....... . . .... ...... ............ ..... ... ... ....... . . . . ......... ..... . . ................................................................. .... . . .. . . . . .. .. . . ........ ..... ................................ . .. . . . . .. .... .. .. ... .. ................................ ........... ........................................ . . . . . ....... . . . . . . . . . . . . . . . . . . . . - K Figure 10.6: Pulling out the s where 2 M= 6 6 6 6 6 6 4 0 1 1 0 0 e 0 0 be ae d 0 0 0 0 0 bd + c 0 ad 1 Then z = Fu (M )w " = 1 I2 0 3 1 07 7 7 0 7: 7 b a 7 5 0 2 I2 # : All LFT examples in the last section can be obtained following the above steps. 10.4 Redhe er Star-Products The most important property of LFTs is that any interconnection of LFTs is again an LFT. This property is by far the most often used and is the heart of LFT machinery. Indeed, it is not hard to see that most of the interconnection structures discussed early, e.g., feedback and cascade, can be viewed as special cases of the so-called star product.
- 275. LINEAR FRACTIONAL TRANSFORMATION 258 a z ?u4 e 2 y4 b e 6 c 6 u3 y3 u1 2 ? ;d y2 y1 1 1 e u2 ;e 6 w -e 6 Figure 10.7: Block diagram for G Suppose that P and K are compatibly partitioned matrices " # " # P = P11 P12 K = K11 K12 P21 P22 K21 K22 such that the matrix product P22 K11 is well de ned and square, and assume further that I ; P22 K11 is invertible. Then the star product of P and K with respect to this partition is de ned as " # Fl (P K11 ) P12 (I ; K11 P22 );1 K12 : S (P K ) := K21 (I ; P22 K11 );1 P21 Fu (K P22 ) Note that this de nition is dependent on the partitioning of the matrices P and K above. In fact, this star product may be well de ned for one partition and not well de ned for another however, we will not explicitly show this dependence because it is always clear from the context. In a block diagram, this dependence appears, as shown in Figure 10.8. Now suppose that P and K are transfer matrices with state space representations: 2 A B1 B2 P = 6 C1 D11 D12 4 C2 D21 D22 Then the transfer matrix 3 2 3 AK BK 1 BK 2 K = 6 CK 1 DK 11 DK 12 7 : 4 5 CK 2 DK 21 DK 22 7 5 " # " S (P K ) : w 7! z w ^ z ^ #
- 276. 10.4. Redhe er Star-Products z w P aa !!! a ! !! aaa K z ^ 259 z z ^ S (P K ) w w ^ w ^ Figure 10.8: Interconnection of LFTs has a representation 2 A B1 B2 6 6 C S (P K ) = 4 1 D11 D12 C2 D21 D22 where A = B = C = D = " " " " 3 7 7 5 # " A B = C D # ~ ~ A + B2 R;1 DK 11 C2 B2 R;1 CK 1 ;1 C2 BK 1 R AK + BK 1 R;1 D22 CK 1 # ~ ~ B1 + B2 R;1 DK 11 D21 B2 R;1 DK 12 BK 1 R;1 D21 BK 2 + BK 1 R;1 D22 DK 12 # ~ C1 + D12 DK 11 R;1 C2 D12 R;1 CK 1 DK 21 R;1C2 CK 2 + DK 21 R;1 D22 CK 1 ~ D11 + D12 DK 11 R;1D21 D12 R;1 DK 12 DK 21 R;1 D21 DK 22 + DK 21 R;1 D22 DK 12 R = I ; D22 DK 11 ~ R = I ; DK 11 D22 : In fact, it is easy to show that A = S B = S " " A C2 B1 D21 B2 D22 B2 D22 # " # DK 11 BK 1 " DK 11 BK 1 #! CK 1 AK DK 12 BK 2 #! #
- 277. LINEAR FRACTIONAL TRANSFORMATION 260 C = S D = S " " C1 C2 D11 D21 D12 D22 D12 D22 # " # DK 11 DK 21 " DK 11 DK 21 #! CK 1 CK 2 DK 12 DK 22 #! : 10.5 Notes and References This chapter is based on the lecture notes by Packard 1991] and the paper by Doyle, Packard, and Zhou 1991].
- 278. 11 Structured Singular Value It is noted that the robust stability and robust performance criteria derived in Chapter 9 vary with the assumptions on the uncertainty descriptions and performance requirements. We will show in this chapter that they can all be treated in a uni ed framework using the LFT machinery introduced in the last chapter and the structured singular value to be introduced in this chapter. This, of course, does not mean that those special problems and their corresponding results are not important on the contrary, they are sometimes very enlightening to our understanding of complex problems such as those in which complex problems are formed from simple problems. On the other hand, a uni ed approach may relieve the mathematical burden of dealing with speci c problems repeatedly. Furthermore, the uni ed framework introduced here will enable us to treat exactly the robust stability and robust performance problems for systems with multiple sources of uncertainties, which is a formidable problem in the standing point of Chapter 9, in the same fashion as single unstructured uncertainty. Indeed, if a system is subject to multiple sources of uncertainties, in order to use the results in Chapter 9 for unstructured cases, it is necessary to re ect all sources of uncertainties from their known point of occurrence to a single reference location in the loop. Such re ected uncertainties invariably have a great deal of structure which must then be covered up" with a large, arbitrarily more conservative perturbation in order to maintain a simple cone bounded representation at the reference location. Readers might have already had some idea about the conservativeness in such re ection from the skewed speci cation problem, where an input multiplicative uncertainty of the plant is re ected at the output and the size of the re ected uncertainty is proportional to the condition number of the plant. In general, the re ected uncertainty may be proportional to the condition 261
- 279. 262 STRUCTURED SINGULAR VALUE number of the transfer matrix between its original location and the re ected location. Thus it is highly desirable to treat the uncertainties as they are and where they are. The structured singular value is de ned exactly for that purpose. 11.1 General Framework for System Robustness As we have illustrated in the last chapter, any interconnected system may be rearranged to t the general framework in Figure 11.1. Although the interconnection structure can become quite complicated for complex systems, many software packages, such as SIMULINK1 and -TOOLS2, are available which could be used to generate the interconnection structure from system components. Various modeling assumptions will be considered and the impact of these assumptions on analysis and synthesis methods will be explored in this general framework. Note that uncertainty may be modeled in two ways, either as external inputs or as perturbations to the nominal model. The performance of a system is measured in terms of the behavior of the outputs or errors. The assumptions which characterize the uncertainty, performance, and nominal models determine the analysis techniques which must be used. The models are assumed to be FDLTI systems. The uncertain inputs are assumed to be either ltered white noise or weighted power or weighted Lp signals. Performance is measured as weighted output variances, or as power, or as weighted output Lp norms. The perturbations are assumed to be themselves FDLTI systems which are norm-bounded as input-output operators. Various combinations of these assumptions form the basis for all the standard linear system analysis tools. Given that the nominal model is an FDLTI system, the interconnection system has the form 2 3 P11 (s) P12 (s) P13 (s) P (s) = 6 P21 (s) P22 (s) P23 (s) 7 4 5 P31 (s) P32 (s) P33 (s) and the closed-loop system is an LFT on the perturbation and the controller given by z = Fu (F` (P K ) ) w = F` (Fu (P ) K ) w: We will focus our discussion in this section on analysis methods therefore, the controller may be viewed as just another system component and absorbed into the interconnection structure. Denote " M (s) = F` (P (s) K (s)) = M11(s) M12 (s) M21(s) M22 (s) 1 2 SIMULINK is a trademark of The MathWorks, Inc. -TOOLS is a trademark of MUSYN Inc. #
- 280. 11.1. General Framework for System Robustness 263 z w P - K Figure 11.1: General Framework and then the general framework reduces to Figure 11.2, where z = Fu (M )w = M22 + M21 (I ; M11 );1 M12 w: z w M Figure 11.2: Analysis Framework Suppose K (s) is a stabilizing controller for the nominal plant P . Then M (s) 2 RH1 . In general, the stability of Fu (M ) does not necessarily imply the internal stability of the closed-loop feedback system. However, they can be made equivalent with suitably chosen w and z . For example, consider again the multiplicatively perturbed system shown in Figure 11.3. Now let " w := d1 d2 # " z := e1 e2 # then the system is robustly stable for all (s) 2 RH1 with k k1 < 1 if and only if Fu (M ) 2 RH1 for all admissible , which is guaranteed by kM11 k1 1. (Note that this is not necessarily equivalent to (I ; M11 );1 2 RH1 if belongs to a closed ball as shown in Theorem 9.5.) The analysis results presented in the previous chapters together with the associated synthesis tools are summarized in Table 11.1 with various uncertainty modeling assumptions. However, the analysis is not so simple for systems with multiple sources of model uncertainties, including the robust performance problem for systems with unstructured
- 281. STRUCTURED SINGULAR VALUE 264 d2 d1 e1 ?e-e - K -e 2 P ;6 e - W2 3 - d3 - W1 ? -e Figure 11.3: Multiplicatively Perturbed Systems uncertainty. As we have shown in the last chapter, if a system is built from components which are themselves uncertain, then, in general, the uncertainty in the system level is structured involving typically a large number of real parameters. The stability analysis involving real parameters is much more involved and is beyond the scope of this book and that, instead, we shall simply cover the real parametric uncertainty with norm bounded dynamical uncertainty. Moreover, the interconnection model M can always be chosen so that (s) is block diagonal, and, by absorbing any weights, k k1 < 1. Thus we shall assume that (s) takes the form of (s) = fdiag : : : s IrS 1 : : : F ] : i (s) 2 RH1 j 2 RH1 g with k i k1 < 1 and k j k1 < 1. Then the system is robustly stable i the intercon1 Ir1 nected system in Figure 11.4 is stable. - F (s) .. . - 1 (s)I .. . M11 (s) .. . Figure 11.4: Robust Stability Analysis Framework The results of Table 11.1 can be applied to the analyses of the system's robust stability in two ways:
- 282. 11.1. General Framework for System Robustness Input Assumptions Performance Speci cations Perturbation Assumptions 265 Analysis Tests Synthesis Methods E (w(t)w(t) ) = I E (z (t) z (t)) 1 w = U0 (t) E (U0 U0 ) = I E (kz k2 ) 1 2 kwk2 1 kz k2 1 kwk2 1 Internal Stability LQG =0 kM22 k2 1 Wiener-Hopf H2 =0 k k1 < 1 kM22 k1 1 Singular Value Loop Shaping kM11 k1 1 H1 Table 11.1: General Analysis for Single Source of Uncertainty (1) kM11 k1 1 implies stability, but not conversely, because this test ignores the known block diagonal structure of the uncertainties and is equivalent to regarding as unstructured. This can be arbitrarily conservative3 in that stable systems can have arbitrarily large kM11k1 . (2) Test for each i ( j ) individually (assuming no uncertainty in other channels). This test can be arbitrarily optimistic because it ignores interaction between the i ( j ). This optimism is also clearly shown in the spinning body example. The di erence between the stability margins(or bounds on ) obtained in (1) and (2) can be arbitrarily far apart. Only when the margins are close can conclusions be made about the general case with structured uncertainty. The exact stability and performance analysis for systems with structured uncertainty requires a new matrix function called the structured singular value (SSV) which is denoted by . 3 By arbitrarily conservative," we mean that examples can be constructed where the degree of conservatism is arbitrarily large. Of course, other examples exist where it is quite reasonable, see for example the spinning body example.
- 283. STRUCTURED SINGULAR VALUE 266 11.2 Structured Singular Value 11.2.1 Basic Concept Conceptually, the structured singular value is nothing but a straightforward generalization of the singular values for constant matrices. To be more speci c, it is instructive at this point to consider again the robust stability problem of the following standard feedback interconnection with stable M (s) and (s). w1 - e e1 + 6 + + e2 e + w2 ? M One important question one might ask is how large (in the sense of k k1 ) can be without destabilizing the feedback system. Since the closed-loop poles are given by det(I ; M ) = 0, the feedback system becomes unstable if det(I ; M (s) (s)) = 0 for some s 2 C + . Now let > 0 be a su ciently small number such that the closed-loop system is stable for all stable k k1 < . Next increase until max so that the closedloop system becomes unstable. So max is the robust stability margin. By small gain theorem, 1 = kM k := sup (M (s)) = sup (M (j!)) 1 max ! s2C + is unstructured. Note that for any xed s 2 C + , (M (s)) can be written as (M (s)) = min f ( ) : det (I ; M (s) 1 ) = 0 is unstructuredg : (11:1) In other words, the reciprocal of the largest singular value of M is a measure of the smallest unstructured that causes instability of the feedback system. To quantify the smallest destabilizing structured , the concept of singular values needs to be generalized. In view of the characterization of the largest singular value of a matrix M (s) given by (11.1), we shall de ne (11:2) (M (s)) = min f ( ) : det (I ; M (s1 ) = 0 is structuredg ) as the largest structured singular value of M (s) with respect to the structured . Then it is obvious that the robust stability margin of the feedback system with structured uncertainty is 1 = sup (M (s)) = sup (M (j!)): if max s2C + ! The last equality follows from the following lemma.
- 284. 11.2. Structured Singular Value 267 Lemma 11.1 Let be a structured set and M (s) 2 RH1. Then sup s2C + (M (s)) = sup (M (s)) = sup (M (j!)): (M (s)) = sup (M (s)) sup (M (j!)): s2C + ! Proof. It is clear that sup s2C + s2C + ! Now suppose sups2C + (M (s)) > 1= , then by the de nition of , there is an so 2 f1g and a complex structured such that ( ) < and det(I ; M (so ) ) = 0. This implies that there is a 0 ! 1 and 0 < ^ 1 such that det(I ; M (j !) ) = ^ 0. This in turn implies that (M (j !)) > 1= since ( ) < . In other words, ^ sups2C + (M (s)) sup! (M (j!)). The proof is complete. 2 C+ The formal de nition and characterization of the structured singular value of a constant matrix will be given below. 11.2.2 De nitions of This section is devoted to de ning the structured singular value, a matrix function denoted by ( ). We consider matrices M 2 C n n . In the de nition of (M ), there is an underlying structure (a prescribed set of block diagonal matrices) on which everything in the sequel depends. For each problem, this structure is, in general, di erent it depends on the uncertainty and performance objectives of the problem. De ning the structure involves specifying three things: the type of each block, the total number of blocks, and their dimensions. There are two types of blocks: repeated scalar and full blocks. Two nonnegative integers, S and F , represent the number of repeated scalar blocks and the number of full blocks, respectively. To bookkeep their dimensions, we introduce positive integers r1 : : : rS m1 : : : mF . The i'th repeated scalar block is ri ri , while the j 'th full block is mj mj . With those integers given, we de ne C n n as = diag 1 Ir1 : : : s IrS 1 : : : F ] : i 2 C j 2 C mj mj : (11:3) For consistency among all the dimensions, we must have S X i=1 ri + F X j =1 mj = n: Often, we will need norm bounded subsets of , and we introduce the following notation: B = f 2 : ( ) 1g (11:4) Bo = f 2 : ( ) < 1g (11:5)
- 285. STRUCTURED SINGULAR VALUE 268 where the superscript o" symbolizes the open ball. To keep the notation as simple as possible in (11.3), we place all of the repeated scalar blocks rst in actuality, they can come in any order. Also, the full blocks do not have to be square, but restricting them as such saves a great deal in terms of notation. De nition 11.1 For M 2 C n n (M ) is de ned as (M ) := min f ( ) : 2 1 det (I ; M ) = 0g (11:6) 2 makes I ; M singular, in which case (M ) := 0. Remark 11.1 Without a loss in generality, the full blocks in the minimal norm can unless no each be chosen to be dyads (rank = 1). To see this, assume S = 0, i.e., all blocks are full blocks. Suppose that I ; M is singular for some 2 . Then there is an x 2 C n such that M x = x. Now partition x conformably with : 2 6 6 4 3 x1 x2 . 7 7 7 7 5 x = 6 .. 6 xF xi 2 C mi i = 1 : : : F and de ne D = diag(d1 Im1 : : : dF ImF ) where di = 1 if xi = 0 and di = 1= kxi k if xi 6= 0. Next de ne 2 6 6 x1 ~ x2 ~ x = 6 .. ~ 6 4 and . xF ~ 2 6 6 y1 y2 4 . y = 6 .. 6 3 2 7 7 7 7 5 := Dx = 6 6 3 2 7 7 7 7 5 := x = 6 ~ 6 6 6 4 6 6 d1 x1 d2 x2 .. . dm xF 3 7 7 7 7 5 3 ~ 1 x1 ~ 7 2 x2 7 7 .. 7 : . 5 yF xF ~ It follows that kxi k = 1 if xi 6= 0, kxi k = 0 if xi = 0 , and y 6= 0. Hence, de ne a new ~ ~ perturbation ~ 2 C n n as ~ := diag y1 x1 : : : yF xF ] : ~ ~ 4
- 286. 11.2. Structured Singular Value 269 Obviously, ( ~ ) ( ) and y = ~ x. Note that D = D and D ~ = ~ D, we have ~ M x = x =) MD;1 x = x =) MD;1 y = x ~ ;1 ~ x = x =) M ~ x = x =) MD ~ ~ is also singular. Hence we have replaced a general perturbation which i.e., I ; M satis es the singularity condition with a perturbation ~ that is no larger (in the ( ) sense) and has rank 1 for each blocks but still satis es the singularity condition. ~ An alternative expression for (M ) follows from the de nition. Lemma 11.2 (M ) = max (M ) 2B In view of this lemma, continuity of the function : C n n ! R is apparent. In general, though, the function : C n n ! R is not a norm, since it doesn't satisfy the triangle inequality however, for any 2 C , ( M ) = j j (M ), so in some sense, it is related to how big" the matrix is. We can relate (M ) to familiar linear algebra quantities when is one of two extreme sets. If = f I : 2 C g (S = 1 F = 0 r1 = n), then (M ) = (M ), the spectral radius of M . Proof. The only 's in which satisfy the det (I ; M ) = 0 constraint are reciprocals of nonzero eigenvalues of M . The smallest one of these is associated with the largest (magnitude) eigenvalue, so, (M ) = (M ). 2 If = C n n (S =0 F =1 m1 = n), then Proof. If ( ) < (M ) = (M ). (M ) < 1, so I ; M is nonsingular. Applying (M ). On the other hand, let u and v be unit equation (11.6) implies (M ) vectors satisfying Mv = (M ) u, and de ne := (1 ) vu . Then ( ) = (1 ) M M 2 and I ; M is obviously singular. Hence, (M ) (M ). 1 (M ) , then Obviously, for a general as in (11.3) we must have f In : 2 C g C n n: (11:7) Hence directly from the de nition of and from the two special cases above, we conclude that (M ) (M ) (M ) : (11:8) These bounds alone are not su cient for our purposes"because the gap between and # 1 0 can be arbitrarily large. For example, suppose = and consider 0 2
- 287. STRUCTURED SINGULAR VALUE 270 " # 0 (1) M = for any > 0. Then (M ) = 0 and (M ) = . But det(I ;M ) = 0 0 1 so (M ) = 0. " # ;1=2 1=2 . Then (M ) = 0 and (M ) = 1. Since det(I ; M ) = (2) M = ;1=2 1=2 1 ; 2 , it is easy to see that min maxi j i j 1 + 1 ; 2 = 0 = 1, so (M ) = 1. 1+ 2 2 Thus neither nor provide useful bounds even in simple cases. The only time they do provide reliable bounds is when . However, the bounds can be re ned by considering transformations on M that do not a ect (M ), but do a ect and . To do this, de ne the following two subsets of C n n : U = fU 2 : UU = In g ( ) diag D1 : : : DS d1 Im1 : : : dF ;1 ImF ;1 ImF : : Di 2 C ri ri Di = Di > 0 dj 2 R dj > 0 Note that for any 2 U 2 U , and D 2 D, (U ) = ( U ) = ( ) U 2U U 2 U2 D = D: Consequently, Theorem 11.3 For all U 2 U and D 2 D ; (MU ) = (UM ) = (M ) = DMD;1 : D = (11:9) (11:10) (11:11) (11:12) (11:13) Proof. For all D 2 D and 2 , ; ; det (I ; M ) = det I ; MD;1 D = det I ; DMD;1 ; since D commutes with . Therefore (M ) = DMD;1 . Also, for each U 2 U , det (I ; M ) = 0 if and only if det (I ; MUU ) = 0. Since U 2 and (U ) = ( ), we get (MU ) = (M ) as desired. The argument for UM is the same. 2 Therefore, the bounds in (11.8) can be tightened to max (UM ) max ( M ) = (M ) D2D inf U 2U 2B ; DMD;1 (11:14) where the equality comes from Lemma 11.2. Note that the last element in the D matrix is normalized to 1 since for any nonzero scalar , DMD;1 = ( D) M ( D);1 .
- 288. 11.2. Structured Singular Value 271 Remark 11.2 Note that the scaling set D in Theorem 11.3 and in the inequality (11.14) does not necessarily be restricted to Hermitian. In fact, they can be replaced by any set of nonsingular matrices that satisfy (11.12). However, enlarging the set of scaling matrices does not improve the upper bound in inequality (11.14). This can be shown as follows: Let D be any nonsingular matrix such that D = D. Then there exist a Hermitian matrix 0 < R = R 2 D and a unitary matrix U such that D = UR and inf D ; DMD;1 = inf D ; ; URMR;1U = R2D RMR;1 : inf Therefore, there is no loss of generality in assuming D to be Hermitian. ~ 11.2.3 Bounds In this section we will concentrate on the bounds max (UM ) U 2U (M ) inf D2D ; DMD;1 : The lower bound is always an equality Doyle, 1982]. Theorem 11.4 max (MU ) = (M ). U 2U Unfortunately, the quantity (UM ) can have multiple local maxima which are not global. Thus local search cannot be guaranteed to obtain , but can only yield a lower bound. For computation purposes one can derive a slightly di erent formulation of the lower bound as a power algorithm which is reminiscent of power algorithms for eigenvalues and singular values Packard, Fan, and Doyle, 1988]). While there are open questions about convergence, the algorithm usually works quite well and has proven to be an e ective method to compute . The upper bound can be reformulated as a convex optimization problem, so the global minimum can, in principle, be found. Unfortunately, the upper bound is not always equal to . For block structures satisfying 2S + F 3, the upper bound is always equal to (M ), and for block structures with 2S + F > 3, there exist matrices for which is less than the in mum. This can be summarized in the following diagram, which shows for which cases the upper bound is guaranteed to be equal to . Theorem 11.5 (M ) = D2D (DMD;1) if 2S + F 3 inf S= 0 1 2 F= 0 yes no 1 2 3 4 yes yes yes no yes no no no no no no no
- 289. STRUCTURED SINGULAR VALUE 272 Several of the boxes have connections with standard results. S = 0, F = 1 : (M ) = (M ). ; (M ) = (M ) = D2D DMD;1 . This is a standard result in inf linear algebra. In fact, without a loss in generality, the matrix M can be assumed in Jordan Canonical form. Now let S = 1, F = 0 : 2 J1 = 6 6 6 6 6 6 6 4 1 3 1 ... ... 1 2 7 7 7 7 7 7 7 5 6 6 6 =6 6 6 6 4 D1 1 3 k ... kn1 ;2 kn1 ;1 7 7 7 7 7 7 7 5 2 C n1 n1 : ; ; Then inf1 n1 (D1 J1 D1 1 ) = klim (D1 J1 D1 1 ) = j j. (Note that by Re!1 D1 2C n mark 11.2, the scaling matrix does not need to be Hermitian.) The conclusion follows by applying this result to each Jordan block. It is also equivalent to the fact that Lyapunov asymptotic stability and exponential stability are equivalent for discrete time systems. This is because (M ) < 1 (exponential stability of a discrete time system matrix M ) implies for some nonsingular D 2 C n n (DMD;1 ) < 1 or (D;1 ) M D DMD;1 ; I < 0 which in turn is equivalent to the existence of a P = D D > 0 such that M PM ; P < 0 (Lyapunov asymptotic stability). S = 0, F = 2 : This case was studied by Redhe er 1959]. S = 1, F = 1 : This is equivalent to a state space characterization of the H1 norm of a discrete time transfer function, see Chapter 21. S = 2, F = 0 : This is equivalent to the fact that for multidimensional systems (2-d, in fact), exponential stability is not equivalent to Lyapunov stability. S = 0, F 4 : For this case, the upper bound is not always equal to . This is important, as these are the cases that arise most frequently in applications. Fortunately, the bound seems to be close to . The worst known example has a ratio of over the bound of about :85, and most systems are close to 1.
- 290. 11.2. Structured Singular Value 273 The above bounds are much more than just computational schemes. They are also theoretically rich and can unify a number of apparently quite di erent results in linear systems theory. There are several connections with Lyapunov asymptotic stability, two of which were hinted at above, but there are further connections between the upper bound scalings and solutions to Lyapunov and Riccati equations. Indeed, many major theorems in linear systems theory follow from the upper bounds and from some results of Linear Fractional Transformations. The lower bound can be viewed as a natural generalization of the maximum modulus theorem. Of course one of the most important uses of the upper bound is as a computational scheme when combined with the lower bound. For reliable use of the theory, it is essential to have upper and lower bounds. Another important feature of the upper bound is that it can be combined with H1 controller synthesis methods to yield an ad-hoc -synthesis method. Note that the upper bound when applied to transfer functions is simply a scaled H1 norm. This is exploited in the D ; K iteration procedure to perform approximate -synthesis (Doyle 1982]), which will be brie y introduced in section 11.4. 11.2.4 Well Posedness and Performance for Constant LFTs Let M be a complex matrix partitioned as " M = M11 M12 M21 M22 # (11:15) and suppose there are two de ned block structures, 1 and 2 , which are compatible in size with M11 and M22 , respectively. De ne a third structure as (" = 1 0 0 2 # : 1 2 1 22 ) 2 : (11:16) Now, we may compute with respect to three structures. The notations we use to keep track of these computations are as follows: 1 ( ) is with respect to 1 , 2 ( ) is with respect to 2 , and ( ) is with respect to . In view of these notations, 1 (M11 ), (M ) all make sense, though, for instance, 1 (M ) does not. 2 (M22 ) and This section is interested in following constant matrix problems: determine whether the LFT Fl (M 2 ) is well posed for all 2 2 2 with ( 2 ) (< ), and, if so, then determine how large" Fl (M 2 ) can get for this norm-bounded set of perturbations. Let 2 2 2 . Recall that Fl (M 2 ) is well posed if I ; M22 2 is invertible. The rst theorem is nothing more than a restatement of the de nition of . Theorem 11.6 The linear fractional transformation Fl (M 2) is well posed
- 291. STRUCTURED SINGULAR VALUE 274 (a) for all 2 2 B 2 if and only if 2 (M22) < 1. (b) for all 2 2 Bo 2 if and only if 2 (M22 ) 1. As the perturbation" 2 deviates from zero, the matrix Fl (M 2 ) deviates from M11 . The range of values that 1 (Fl (M 2 )) takes on is intimately related to (M ), as shown in the following theorem: Theorem 11.7 (MAIN LOOP THEOREM) The following are equivalent: () (M ) < 1 () (M ) 1 8 > > < > > : 8 > > < > > : 2 (M22 ) < 1 max 22 B 1 (Fl (M 2 2 (M22 ) max o 22 B and 2 )) < 1: 1 and 2 1 ( Fl ( M 1: 2 )) Proof. We shall only prove the rst part of the equivalence. The proof for the second part is similar. ( Let i 2 i be given, with ( i ) 1, and de ne = diag 1 2 ]. Obviously 2 . Now " # I ; M11 1 ;M12 2 : det (I ; M ) = det (11:17) ;M I ;M By hypothesis I ; M22 21 1 2 is invertible, and hence, det (I ; M ) becomes det (I ; M22 2 ) det I ; M11 Collecting the 22 2 1 ; M12 2 (I ; M22 2 ) ;1 M 21 1 : terms leaves det (I ; M ) = det (I ; M22 2 ) det (I ; Fl (M 2 ) 1 ) : (11:18) But, 1 (Fl (M 2 )) < 1 and 1 2 B 1 , so I ; Fl (M 2 ) 1 must be nonsingular. Therefore, I ; M is nonsingular and, by de nition, (M ) < 1. ) Basically, the argument above is reversed. Again let 1 2 B 1 and 2 2 B 2 be given, and de ne = diag 1 2]. Then 2 B and, by hypothesis, det (I ; M ) 6= 0. It is easy to verify from the de nition of that (always) (M ) max f 1 (M11 ) 2 (M22 )g : We can see that 2 (M22 ) < 1, which gives that I ;M22 2 is also nonsingular. Therefore, the expression in (11.18) is valid, giving det (I ; M22 2 ) det (I ; Fl (M 2 ) 1 ) = det (I ; M ) 6= 0: 1
- 292. 11.2. Structured Singular Value Obviously, I ; Fl (M the claim is true. 275 is nonsingular for all i 2 B i , which indicates that 2) 1 2 Remark 11.3 This theorem forms the basis for all uses of in linear system robustness analysis, whether from a state-space, frequency domain, or Lyapunov approach. ~ The role of the block structure 2 in the MAIN LOOP theorem is clear - it is the structure that the perturbations come from however, the role of the perturbation structure 1 is often misunderstood. Note that 1 ( ) appears on the right hand side of the theorem, so that the set 1 de nes what particular property of Fl (M 2 ) is considered. As an example, consider the theorem applied with the two simple block structures considered right after Lemma 11.2. De ne 1 := f 1 In : 1 2 C g . Hence, for A 2 C n n 1 (A) = (A). Likewise, de ne 2 = C m m then for D 2 C m m 2 (D) = (D). Now, let be the diagonal augmentation of these two sets, namely (" := 1 In 0m n 0n m 2 # : 1 2C m m 2 2C ) C (n+m) (n+m) : Let A 2 C n n B 2 C n m C 2 C m n , and D 2 C m m be given, and interpret them as the state space model of a discrete time system xk+1 = Axk + Buk yk = Cxk + Duk : And let M 2 C (n+m) (n+m) be the block state space matrix of the system " # M= A B : C D Applying the theorem with this data gives that the following are equivalent: The spectral radius of A satis es (A) < 1, and max j 1 2C 1j 1 D + C 1 (I ; A 1 );1 B < 1: (11:19) The maximum singular value of D satis es (D) < 1, and max m m 2 2C ( 2) 1 A+B 2 (I ; D 2 ) The structured singular value of M satis es (M ) < 1: ;1 C < 1: (11:20) (11:21)
- 293. STRUCTURED SINGULAR VALUE 276 The rst condition is recognized by two things: the system is stable, and the jj jj1 norm on the transfer function from u to y is less than 1 (by replacing 1 with 1 ) z kGk1 := max D + C (zI ; A);1 B = max 2C z2C jzj 1 j 1 1j 1 D + C 1 (I ; A 1 );1 B : The second condition implies that (I ; D 2 );1 is well de ned for all ( 2 ) that a robust stability result holds for the uncertain di erence equation xk+1 = A + B 2 (I ; D 2 ) 1 and ;1 C x k where 2 is any element in C m m with ( 2 ) 1, but otherwise unknown. This equivalence between the small gain condition, kGk1 < 1, and the stability robustness of the uncertain di erence equation is well known. This is the small gain theorem, in its necessary and su cient form for linear, time invariant systems with one of the components norm-bounded, but otherwise unknown. What is important to note is that both of these conditions are equivalent to a condition involving the structured singular value of the state space matrix. Already we have seen that special cases of are the spectral radius and the maximum singular value. Here we see that other important linear system properties, namely robust stability and input-output gain, are also related to a particular case of the structured singular value. 11.3 Structured Robust Stability and Performance 11.3.1 Robust Stability The most well-known use of as a robustness analysis tool is in the frequency domain. Suppose G(s) is a stable, multi-input, multi-output transfer function of a linear system. For clarity, assume G has q1 inputs and p1 outputs. Let be a block structure, as in equation (11.3), and assume that the dimensions are such that C q1 p1 . We want to consider feedback perturbations to G which are themselves dynamical systems with the block-diagonal structure of the set . Let M ( ) denote the set of all block diagonal and stable rational transfer functions that have block structures such as . M ( ) := ( ) 2 RH1 : (so ) 2 for all so 2 C + Theorem 11.8 Let > 0. The loop shown below is well-posed and internally stable for all ( ) 2 M ( ) with k k1 < 1 if and only if sup !2R (G(j!))
- 294. 11.3. Structured Robust Stability and Performance w1 - e e1 + 6 + 277 G(s) + e2 ?+ w2 e Proof. ((=) By Lemma 11.1, sups2C + (G(s)) = sup!2R (G(j!)) . Hence det(I ; G(s) (s)) 6= 0 for all s 2 C + f1g whenever k k1 < 1= , i.e., the system is robustly stable. (=)) Suppose sup!2R (G(j!)) > . Then there is a 0 < !o < 1 such that (G(j!o )) > . By Remark 11.1, there is a complex c 2 that each full block has rank 1 and ( c ) < 1= such that I ; G(j!o ) c is singular. Next, using the same construction used in the proof of the small gain theorem (Theorem 9.1), one can nd a rational (s) such that k (s)k1 = ( c ) < 1= , (j!o ) = c , and (s) destabilizes the system. 2 Hence, the peak value on the plot of the frequency response determines the size of perturbations that the loop is robustly stable against. Remark 11.4 The internal stability with closed ball of uncertainties is more complicated. The following example is shown in Tits and Fan 1994]. Consider " # 0 ;1 1 G(s) = s + 1 1 0 and = (s)I2 . Then sup (G(j!)) = sup jj! 1 1j = (G(j 0)) = 1: + ! 2R ! 2R On the other hand, (G(s)) < 1 for all s 6= 0 s 2 C + , and the only matrices in the form of ; = I2 with j j 1 for which det(I ; G(0);) = 0 are the complex matrices jI2 . Thus, clearly, (I ; G(s) (s));1 2 RH1 for all real rational (s) = (s)I2 with k k1 1 since (0) must be real. This shows that sup!2R (G(j!)) < 1 is not necessary for (I ; G(s) (s));1 2 RH1 with the closed ball of structured uncertainty k k1 1. Similar examples with no repeated blocks are 1 generated by setting G(s) = s+1 M where M is any real matrix with (M ) = 1 for which there is no real 2 with ( ) = 1 such that det(I ; M ) = 0. For example, let 82 9 2 3 3 " # 0 > > 1 < = 6 7 ; 6 7 M =4 = >4 5 5 i 2 C> 2 0 ; : ; 3
- 295. STRUCTURED SINGULAR VALUE 278 with 2 = 1 and 2 + 2 2 = 1. Then it is shown in Packard and Doyle 1993] that 2 (M ) = 1 and all 2 with ( ) = 1 that satisfy det(I ; M ) = 0 must be complex. ~ Remark 11.5 Let 2 RH1 be a structured uncertainty and " # G(s) = G11 (s) G12 (s) 2 RH1 G21 (s) G22 (s) then Fu (G ) 2 RH1 does not necessarily imply (I ; G11 );1 2 RH1 whether in an open ball or is in a closed ball. For example, consider 2 1 s+1 G(s) = 6 0 4 1 " and = admissible 0 10 s+1 0 1 0 0 is 3 7 5 # 1 with k k1 < 1. Then Fu (G ) = 1 2 RH1 for all 1 ; 1 s+1 2 (k k1 < 1) but (I ; G11 );1 2 RH1 is true only for k k1 < 0:1. ~ 1 11.3.2 Robust Performance Often, stability is not the only property of a closed-loop system that must be robust to perturbations. Typically, there are exogenous disturbances acting on the system (wind gusts, sensor noise) which result in tracking and regulation errors. Under perturbation, the e ect that these disturbances have on error signals can greatly increase. In most cases, long before the onset of instability, the closed-loop performance will degrade to the point of unacceptability, hence the need for a robust performance" test. Such a test will indicate the worst-case level of performance degradation associated with a given level of perturbations. Assume Gp is a stable, real-rational, proper transfer function with q1 + q2 inputs and p1 + p2 outputs. Partition Gp in the obvious manner " # Gp (s) = G11 G12 G21 G22 so that G11 has q1 inputs and p1 outputs, and so on. Let C q1 structure, as in equation (11.3). De ne an augmented block structure (" 0 # f 2 C q2 p2 ) p1 be a block : 2 : 0 f The setup is to theoretically address the robust performance questions about the loop shown below P :=
- 296. 11.3. Structured Robust Stability and Performance 279 - (s) z w Gp (s) The transfer function from w to z is denoted by Fu (Gp ). Theorem 11.9 Let > 0. For all (s) 2 M ( ) with k k1 < 1 , the loop shown above is well-posed, internally stable, and kFu (Gp )k1 if and only if sup !2R P (Gp (j!)) : Note that by internal stability, sup!2R (G11 (j!)) , then the proof of this theorem is exactly along the lines of the earlier proof for Theorem 11.8, but also appeals to Theorem 11.7. This is a remarkably useful theorem. It says that a robust performance problem is equivalent to a robust stability problem with augmented uncertainty as shown in Figure 11.5. - f - Gp (s) Figure 11.5: Robust Performance vs Robust Stability 11.3.3 Two Block : Robust Performance Revisited Suppose that the uncertainty block is given by " = 1 2 # 2 RH1
- 297. STRUCTURED SINGULAR VALUE 280 with k k1 < 1 and that the interconnection model G is given by " # G(s) = G11 (s) G12 (s) 2 RH1 : G21 (s) G22 (s) (G(j!)) 1. Then the closed-loop system is well-posed and internally stable i sup! Let " # d! I D! = d! 2 R+ I then # " ; D! G(j!)D! 1 = 1G11 (j!) d! G12 (j!) : d! G21 (j! ) G22 (j! ) Hence by Theorem 11.5, at each frequency ! " (G(j!)) = d inf 2R ! G11 (j!) d! G12 (j!) 1 G21 (j! ) G22 (j! ) d! + #! : (11:22) Since the minimization is convex in log d! see, Doyle, 1982], the optimal d! can be found by a search however, two approximations to d! can be obtained easily by approximating the right hand side of (11.22): (1) From Lemma 2.10, we have (G(j!)) d inf ! 2R+ s inf d! 2R+ = q " kG11 (j!)k d! kG12 (j!)k 1 kG21 (j! )k kG22 (j! )k d! #! kG11 (j!)k2 + d2 kG12 (j!)k2 + d12 kG21 (j!)k2 + kG22 (j!)k2 ! ! kG11 (j!)k2 + kG22 (j!)k2 + 2 kG12 (j!)k kG21 (j!)k with the minimizing d! given by ^ d! = 8 q kG21 (j!)k > > < kG12 (j!)k > > : 0 1 if G12 6= 0 & G21 6= 0 if G21 = 0 if G12 = 0: (11:23) (2) Alternative approximation can be obtained by using the Frobenius norm (G(j!)) d inf ! 2R+ " G11 (j!) d! G12 (j!) 1 G21 (j! ) G22 (j! ) d! # F
- 298. 11.3. Structured Robust Stability and Performance s = = inf d! 2R+ q kG11 281 kG11 (j!)k2 + d2 kG12 (j!)k2 + d12 kG21 (j!)k2 + kG22 (j!)k2 ! F F F F ! (j!)k2 F + kG22 (j!)k2 F + 2 kG12 (j! )kF kG21 (j! )kF with the minimizing d! given by 8 q kG21 (j!)kF > > < kG12 (j!)k F ~ d! = > 0 > : 1 if G12 6= 0 & G21 6= 0 if G21 = 0 if G12 = 0: (11:24) It can be shown that the approximations for the scalar d! obtained above are exact for a 2 2 matrix G. For higher dimensional G, the approximations for d! are still reasonably good. Hence an approximation of can be obtained as (G(j!)) " ^ G11 (j!) d! G12 (j!) 1 G21 (j! ) G22 (j! ) ^ d! #! " ~ G11 (j!) d! G12 (j!) 1 G21 (j! ) G22 (j! ) ~ d! #! (11:25) or, alternatively, as (G(j!)) We can now see how these approximated conditions obtained in Chapter 9. : (11:26) tests are compared with the su cient Example 11.1 Consider again the robust performance problem of a system with output multiplicative uncertainty in Chapter 9 (see Figure 9.5): P = (I + W1 W2 )P k k1 < 1: Then it is easy to show that the problem can be put in the general framework by selecting " # G(s) = ;W2 To W1 ;W2 ToWd We So W1 We So Wd and that the robust performance condition is satis ed if and only if kW2 ToW1 k1 1 and (11:27) kFu (G )k1 1 (11:28)
- 299. STRUCTURED SINGULAR VALUE 282 for all 2 RH1 with k k1 < 1. But (11.27) and (11.28) are satis ed i for each frequency ! " (G(j!)) = d inf 2R ! ;W2 ToW1 ;d! W2 To Wd 1 d! We So W1 + #! We So Wd 1: Note that, in contrast to the su cient condition obtained in Chapter 9, this condition is an exact test for robust performance. To compare the test with the criteria obtained in Chapter 9, some upper bounds for can be derived. Let s W d! = kWe SoW1 k : kW2 To d k Then, using the rst approximation for , we get (G(j!)) q kW2 To W1 k2 + kWe So Wd k2 + 2 kW2 ToWd k kWe So W1 k kW2 To W1 k2 + kWe So Wd k2 + 2 (W1;1 Wd ) kW2 ToW1 k kWe SoWd k kW2 To W1 k + (W1;1 Wd ) kWe So Wd k q where W1 is assumed to be invertible in the last two inequalities. The last term is exactly the su cient robust performance criteria obtained in Chapter 9. It is clear that any term preceding the last forms a tighter test since (W1;1 Wd ) 1. Yet another alternative su cient test can be obtained from the above sequence of inequalities: (G(j!)) q (W1;1 Wd )(kW2 ToW1 k + kWe SoWd k): Note that this su cient condition is not easy to get from the approach taken in Chapter 9 and is potentially less conservative than the bounds derived there. 3 Next we consider the skewed speci cation problem, but rst the following lemma is needed in the sequel. Lemma 11.10 Suppose = 1 2 ::: m= inf max (d i )2 + (d 1 )2 = d2R+ i i > 0, then + : Proof. Consider a function y = x + 1=x, then y is a convex function and the maximization over a closed interval is achieved at the boundary of the interval. Hence for any xed d max (d i )2 + (d 1 )2 = max (d )2 + (d1 )2 (d )2 + (d1 )2 : i i
- 300. 11.3. Structured Robust Stability and Performance 283 Then the minimization over d is obtained i (d )2 + (d1 )2 = (d )2 + (d1 )2 which gives d2 = 1 2 . The result then follows from substituting d. Example 11.2 As another example, consider again the skewed speci cation problem from Chapter 9. Then the corresponding G matrix is given by " # G = ;W2 Ti W1 ;W2 KSoWd : We So PW1 We So Wd So the robust performance speci cation is satis ed i " (G(j!)) = d inf 2R ! + ;W2 Ti W1 1 d! We So PW1 ;d! W2 KSoWd We So Wd #! 1 for all ! 0. As in the last example, an upper bound can be obtained by taking s kW So PW d! = kW eKS W1 k : 2 o dk Then q (Wd;1 PW1 )(kW2 Ti W1 k + kWe So Wd k): (G(j!)) In particular, this suggests that the robust performance margin is inversely proportional to the square root of the plant condition number if Wd = I and W1 = I . This can be further illustrated by considering a plant-inverting control system. To simplify the exposition, we shall make the following assumptions: We = ws I Wd = I W1 = I W2 = wt I and P is stable and has a stable inverse (i.e., minimum phase) (P can be strictly proper). Furthermore, we shall assume that the controller has the form K (s) = P ;1 (s)l(s) where l(s) is a scalar loop transfer function which makes K (s) proper and stabilizes the closed-loop. This compensator produces diagonal sensitivity and complementary sensitivity functions with identical diagonal elements, namely l( So = Si = 1 +1l(s) I To = Ti = 1 + s)s) I: l(
- 301. STRUCTURED SINGULAR VALUE 284 Denote "(s) = 1 +1l(s) l( (s) = 1 + s)s) l( and substitute these expressions into G we get " # ;1 G = ;wt I ;wt P : ws "P ws "I The structured singular value for G at frequency ! can be computed by " (G(j!)) = dinf 2R ;wt I ;wt (dP );1 ws "dP + #! ws "I : Let the singular value decomposition of P (j!) at frequency ! be P (j!) = U V = diag( 1 2 : : : m ) with 1 = and m = where m is the dimension of P . Then " (G(j!)) = dinf 2R ;wt I ;wt (d );1 ws "d + #! ws "I since unitary operations do not change the singular values of a matrix. Note that " # ;wt I ;wt (d );1 = P diag(M M : : : M )P 1 1 2 m 2 ws "d ws "I where P1 and P2 are permutation matrices and where Mi = " ;wt ws "d i Hence " (G(j!)) = dinf max 2R+ i # ;wt (d i );1 : ws " ;wt ;wt (d i );1 #! ws "d i ws " # ! ;wt h 1 (d );1 i = dinf max i 2R+ i ws "d i p = dinf max (1 + jd i j;2 )(jws "d i j2 + jwt j2 ) 2R i + " s 2 = dinf max jws "j2 + jwt j2 + jws "d i j2 + wt : 2R+ i di
- 302. 11.3. Structured Robust Stability and Performance 285 Using Lemma 11.10, it is easy to show that the maximum is achieved at either or and that optimal d is given by j t d2 = jww"j j s so the structured singular value is s (G(j!)) = jws "j2 + jwt j2 + jws "jjwt j (P ) + (1 ) ]: P (11:29) Note that if jws "j and jwt j are not too large, which are guaranteed if the nominal performance and robust stability conditions are satis ed, then the structured singular value is proportional to the square root of the plant condition number: (G(j!)) p jws "jjwt j (P ) : (11:30) 3 This con rms our intuition that an ill-conditioned plant with skewed speci cations is hard to control. 11.3.4 Approximation of Multiple Full Block The approximations given in the last subsection can be generalized to the multiple block problem by assuming that M is partitioned consistently with the structure of = diag( so that 2 M= and 6 6 6 6 4 1 2 ::: F) M11 M12 M21 M22 M1F M2F MF 1 MF 2 MFF .. . .. . .. . 3 7 7 7 7 5 D = diag(d1 I : : : dF ;1 I I ): Now d DMD;1 = Mij di j And hence (M ) dF := 1: inf (DMD;1 ) = D2D inf D2D d Mij d i j
- 303. STRUCTURED SINGULAR VALUE 286 d kMij k di j inf D2D inf D2D v u F F uX X t i=1 j =1 inf D2D v u F F uX X t i=1 j =1 d kMij k2 di2 2 j 2 kMij k2 di2 : Fd j An approximate D can be found by solving the following minimization problem: inf D2D F F XX 2 di kMij k2 d2 j i=1 j =1 or, more conveniently, by minimizing inf D2D F F XX i=1 j =1 2 kMij k2 di2 Fd j with dF = 1. The optimal di minimizing of the above two problems satisfy, respectively, d4 k= and P 2 2 i6=k kMik k di 2 2 j 6=k kMkj k =dj k = 1 2 ::: F ;1 (11:31) kMik k2 d2 F i k = 1 2 : : : F ; 1: kMkj k2 =d2 j 6=k F j (11:32) P P i6=k k=P d4 Using these relations, dk can be obtained by iterations. Example 11.3 Consider a 3 3 complex matrix 2 1 + j 10 ; 2j ;20j 6 M = 4 5j 3 + j ;1 + 3j ;2 j 4;j 3 7 5 with structured = diag( 1 2 3 ). The largest singular value of M is (M ) = 22:9094 and the structured singular value of M computed using the -TOOLS is equal to its upper bound: (M ) = D2D (DMD;1 ) = 11:9636 inf with the optimal scaling Dopt = diag(0:3955 0:6847 1). The optimal D minimizing inf D2D F F XX i=1 j =1 2 di kMij k2 d2 j
- 304. 11.4. Overview on Synthesis 287 is Dsubopt = diag(0:3212 0:4643 1) which is solved from equation (11.31). Using this Dsubopt , we obtain another upper bound for the structured singular value: ;1 (Dsubopt MDsubopt ) = 12:2538: (M ) One may also use this Dsubopt as an initial guess for the exact optimization. 3 11.4 Overview on Synthesis This section brie y outlines various synthesis methods. The details are somewhat complicated and are treated in the other parts of the book. At this point, we simply want to point out how the analysis theory discussed in the previous sections leads naturally to synthesis questions. From the analysis results, we see that each case eventually leads to the evaluation of kM k = 2 1 or (11:33) for some transfer matrix M . Thus when the controller is put back into the problem, it involves only a simple linear fractional transformation as shown in Figure 11.6 with " M = F` (G K ) = G11 + G12 K (I ; G22 K );1 G21 : G11 G12 where G = G21 G22 # is chosen, respectively, as " P P nominal performance only ( = 0): G = 22 23 P32 P33 " P P robust stability only: G = 11 13 P31 P33 # # 3 2 P11 P12 P13 robust performance: G = P = 6 P21 P22 P23 7. 5 4 P31 P32 P33 Each case then leads to the synthesis problem min kF` (G K )k K for = 2 1 or (11:34) which is subject to the internal stability of the nominal. The solutions of these problems for = 2 and 1 are the focus of the rest of this book. The solutions presented in this book unify the two approaches in a common synthesis framework. The = 2 case was already known in the 1960's, and the results
- 305. STRUCTURED SINGULAR VALUE 288 z G w -K Figure 11.6: Synthesis Framework are simply a new interpretation. The two Riccati solutions for the = 1 case were new products of the late 1980's. The synthesis for the = case is not yet fully solved. Recall that may be obtained by scaling and applying k k1 (for F 3 and S = 0), a reasonable approach is to solve" min ;1 inf DF`(G K )D;1 1 (11:35) K D D 2H1 by iteratively solving for K and D. This is the so-called D-K Iteration. The stable and minimum phase scaling matrix D(s) is chosen such that D(s) (s) = (s)D(s) (Note that D(s) is not necessarily belong to D since D(s) is not necessarily Hermitian, see Remark 11.2). For a xed scaling transfer matrix D, minK DF` (G K )D;1 1 is a standard H1 optimization problem which will be solved in the later part of the book. For a given stabilizing controller K , inf D D;1 2H1 DF`(G K )D;1 1 is a standard convex optimization problem and it can be solved pointwise in the frequency domain: ; sup Dinf D! F` (G K )(j!)D! 1 : 2D ! ! Indeed, inf D D;1 2H1 ; DF` (G K )D;1 1 = sup Dinf D! F`(G K )(j!)D! 1 : ! ! 2D This follows intuitively from the following arguments: the left hand side is always no smaller than the right hand side, and, on the other hand, given the minimizing D! from the right hand side across the frequency, there is always a rational function D(s) uniformly approximating the magnitude frequency response D! . Note that when S = 0, (no scalar blocks) D! = diag(d! I : : : d! ;1 I I ) 2 D 1 F which is a block-diagonal scaling matrix applied pointwise across frequency to the frequency response F` (G K )(j!). D-K Iterations proceed by performing this two-parameter minimization in sequential fashion: rst minimizing over K with D! xed, then minimizing pointwise over D! with K xed, then again over K , and again over D! , etc. Details of this process are summarized in the following steps:
- 306. 11.4. Overview on Synthesis 289 D D;1 G - K Figure 11.7: -Synthesis via Scaling (i) Fix an initial estimate of the scaling matrix D! 2 D pointwise across frequency. (ii) Find scalar transfer functions di (s) d;1 (s) 2 RH1 for i = 1 2 : : : (F ; 1) such i that jdi (j!)j d! . This step can be done using the interpolation theory Youla i and Saito, 1967] however, this will usually result in very high order transfer functions, which explains why this process is currently done mostly by graphical matching using lower order transfer functions. (iii) Let D(s) = diag(d1 (s)I : : : dF ;1 (s)I I ): Construct a state space model for system " ^ G(s) = D(s) # I " ;1 G(s) D (s) # I : (iv) Solve an H1 -optimization problem to minimize ^ F` (G K ) 1 over all stabilizing K 's. Note that this optimization problem uses the scaled ^ version of G. Let its minimizing controller be denoted by K . ^ ; (v) Minimize D! F` (G K )D! 1 ] over D! , pointwise across frequency.4 Note that ^ this evaluation uses the minimizing K from the last step, but that G is unscaled. The minimization itself produces a new scaling function. Let this new function ^ be denoted by D! . ^ (vi) Compare D! with the previous estimate D! . Stop if they are close, but, otherwise, ^ replace D! with D! and return to step (ii). With either K or D xed, the global optimum in the other variable may be found using the and H1 solutions. Although the joint optimization of D and K is not convex and 4 The approximate solutions given in the last section may be used.
- 307. 290 STRUCTURED SINGULAR VALUE the global convergence is not guaranteed, many designs have shown that this approach works very well see e.g. Balas, 1990]. In fact, this is probably the most e ective design methodology available today for dealing with such complicated problems. The detailed treatment of analysis is given in Packard and Doyle 1991]. The rest of this book will focus on the H1 optimization which is a fundamental tool for synthesis. 11.5 Notes and References This chapter is partially based on the lecture notes given by Doyle 1984] in Honeywell and partially based on the lecture notes by Packard 1991] and the paper by Doyle, Packard, and Zhou 1991]. Parts of section 11.3.3 come from the paper by Stein and Doyle 1991]. The small theorem for systems with non-rational plants and uncertainties is proven in Tits 1994]. Other results on can be found in Fan and Tits 1986], Fan, Tits, and Doyle 1991], Packard and Doyle 1993], Packard and Pandey 1993], Young 1993], and references therein.
- 308. 12 Parameterization of Stabilizing Controllers The basic con guration of the feedback systems considered in this chapter is an LFT as shown in Figure 12.1, where G is the generalized plant with two sets of inputs: the exogenous inputs w, which include disturbances and commands, and control inputs u. The plant G also has two sets of outputs: the measured (or sensor) outputs y and the regulated outputs z . K is the controller to be designed. A control problem in this setup is either to analyze some speci c properties, e.g., stability or performance, of the closedloop or to design the feedback control K such that the closed-loop system is stable in some appropriate sense and the error signal z is speci ed, i.e., some performance condition is satis ed. In this chapter we are only concerned with the basic internal stabilization problems. We will see again that this setup is very convenient for other general control synthesis problems in the coming chapters. Suppose that a given feedback system is feedback stabilizable. In this chapter, the problem we are mostly interested in is parameterizing all controllers that stabilize the system. The parameterization of all internally stabilizing controllers was rst introduced by Youla et al 1976] in their parameterization, the coprime factorization technique is used. All of the existing results are mainly in the frequency domain although they can also be transformed to state-space descriptions. In this chapter, we consider this issue in the general setting and directly in state space without adopting coprime factorization technique. The construction of the controller parameterization is done via considering a sequence of special problems, which are so-called full information (FI) problems, disturbance feedforward (DF) problems, full control (FC) problems and output estimation 291
- 309. 292 PARAMETERIZATION OF STABILIZING CONTROLLERS z y G w u - K Figure 12.1: General System Interconnection (OE) problems. On the other hand, these special problems are also of importance in their own right. In addition to presenting the controller parameterization, this chapter also aims at introducing the synthesis machinery, which is essential in some control syntheses (the H2 and H1 control in Chapter 14 and Chapter 16), and at seeing how it works in the controller parameterization problem. The structure of this chapter is as follows: in section 12.1, the conditions for the existence of a stabilizing controller are examined. In section 12.2, we shall examine the stabilization of di erent special problems and establish the relations among them. In section 12.3, the construction of controller parameterization for the general output feedback problem will be considered via the special problems FI , DF , FC and OE . In section 12.4, the structure of the controller parameterization is displayed. Section 12.5 shows the closed-loop transfer matrix in terms of the parameterization. Section 12.6 considers an alternative approach to the controller parameterization using coprime factorizations and establishes the connections with the state space approach. This section can be either studied independently of all the preceding sections or skipped over without loss of continuity. 12.1 Existence of Stabilizing Controllers Consider a system described by the standard block diagram in Figure 12.1. Assume that G(s) has a stabilizable and detectable realization of the form 2 3 A B1 B2 G11 (s) G12 (s) = 6 7 G(s) = (12:1) 4 C1 D11 D12 5 : G21 (s) G22 (s) C2 D21 D22 The stabilization problem is to nd feedback mapping K such that the closed-loop " # system is internally stable the well-posedness is required for this interconnection. This general synthesis question will be called the output feedback (OF) problem. De nition 12.1 A proper system G is said to be stabilizable through output feedback if there exists a proper controller K internally stabilizing G in Figure 12.1. Moreover, a proper controller K (s) is said to be admissible if it internally stabilizes G.
- 310. 12.1. Existence of Stabilizing Controllers 293 The following result is standard and follows from Chapter 3. Lemma 12.1 There exists a proper K achieving internal stability i (A B2) is stabilizable and (C2 A) is detectable. Further, let F and L be such that A + B2 F and A + LC2 are stable, then an observer-based stabilizing controller is given by # " K (s) = A + B2 F + LC2 + LD22 F ;L : F 0 Proof. (() By the stabilizability and detectability assumptions, there exist F and L such that A + B2 F and A + LC2 are stable. Now let K (s) be the observer-based controller given in the lemma, then the closed-loop A-matrix is given by ~ A= " # A B2 F : ;LC2 A + B2 F + LC2 It is easy to check that this matrix is similar to the matrix " # A + LC2 0 : ;LC2 A + B2 F ~ Thus the spectrum of A equals the union of the spectra of A + LC2 and A + B2 F . In ~ is stable. particular, A ()) If (A B2 ) is not stabilizable or if (C2 A) is not detectable, then there are some ~ eigenvalues of A which are xed in the right half-plane, no matter what the compensator is. The details are left as an exercise. 2 The stabilizability and detectability conditions of (A B2 C2 ) are assumed throughout the remainder of this chapter1. It follows that the realization for G22 is stabilizable and detectable, and these assumptions are enough to yield the following result: y G22 u - K Figure 12.2: Equivalent Stabilization Diagram 1 It should be clear that the stabilizability and detectability of a realization for G do not guarantee the stabilizability and/or detectability of the corresponding realization for G22 .
- 311. 294 PARAMETERIZATION OF STABILIZING CONTROLLERS " # Lemma 12.2 Suppose (A B2 C2 ) is stabilizable and detectable and G22 = A B2 . C2 D22 Then the system in Figure 12.1 is internally stable i the one in Figure 12.2 is internally stable. In other words, K (s) internally stabilizes G(s) if and only if it internally stabilizes G22 . Proof. The necessity follows from the de nition. To show the su ciency, it is su cient to show that the system in Figure 12.1 and that in Figure 12.2 share the same A-matrix, which is obvious. 2 From Lemma 12.2, we see that the stabilizing controller for G depends only on G22 . Hence all stabilizing controllers for G can be obtained by using only G22 , which is how it is usually done in the conventional Youla parameterization. However, it will be shown that the general setup is very convenient and much more useful since any closed-loop system information can also be considered in the same framework. Remark 12.1 There should be no confusion between a given realization for a transfer matrix G22 and the inherited realization from G where G22 is a submatrix. A given realization for G22 may be stabilizable and detectable while the inherited realization may be not. For instance, " 1 G22 = s + 1 = ;1 1 1 0 # is a minimal realization but the inherited realization of G22 from 2 " # ;1 0 0 1 6 G11 G12 = 6 0 1 1 0 6 6 G21 G22 1 0 0 4 0 1 0 0 0 is 2 ;1 0 1 G22 = 6 0 1 0 4 1 0 0 which is neither stabilizable nor detectable. 3 7 7 7 7 5 3 7 5 1 = s+1 ~
- 312. 12.2. Duality and Special Problems 295 12.2 Duality and Special Problems In this section, we will discuss four problems from which the output feedback solutions are constructed via a separation argument. These special problems are fundamental to the approach taken for synthesis in this book, and, as we shall see, they are also of importance in their own right. 12.2.1 Algebraic Duality and Special Problems Before we get into the study of the algebraic structure of control systems, we now introduce the concept of algebraic duality which will play an important role. It is well known that the concepts of controllability (stabilizability) and observability (detectability) of a system (C A B ) are dual because of the duality between (C A B ) and (B T AT C T ). So, to deal with the issues related to a system's controllability and/or observability, we only need to examine the issues related to the observability and/or controllability of its dual system, respectively. The notion of duality can be generalized to a general setting. Consider a standard system block diagram z y G - K w u where the plant G and controller K are assumed to be linear time invariant. Now consider another system shown below z ~ w ~ GT y ~ - KT u ~ whose plant and controller are obtained by transposing G and K . We can check easily T that Tzw = F`(G K )]T = F`(GT K T ) = Tzw . It is not di cult to see that K internally ~~ stabilizes G i K T internally stabilizes GT . And we say that these two control structures are algebraically dual, especially, GT and K T which are dual objects of G and K , respectively. So as far as stabilization or other synthesis problems are concerned, we can obtain the results for GT from the results for its dual object G if they are available. Now, we consider some special problems which are related to the general OF problems stated in the last section and which are important in constructing the results for OF problems. The special problems considered here all pertain to the standard block diagram, but to di erent structures than G. The problems are labeled as
- 313. 296 PARAMETERIZATION OF STABILIZING CONTROLLERS FI. Full information, with the corresponding plant 2 A 6 6 C1 GFI = 6 " # 6 I 4 3 B1 D " 11# 0 B2 7 D 7 " 12# 7 : 7 0 0 0 I 5 FC. Full control, with the corresponding plant 2 A B1 6 GFC = 6 C1 D11 6 4 C2 D21 h h h i 3 I 0 0 I i 7 7 7 i 5 0 0 : DF. Disturbance feedforward, with the corresponding plant 3 2 A B1 B2 GDF = 6 C1 D11 D12 7 : 5 4 C2 I 0 OE. Output estimation, with the corresponding plant 2 3 A B1 B2 6 GOE = 4 C1 D11 I 7 : 5 C2 D21 0 The motivations for these special problems will be given later when they are considered. There are also two additional structures which are standard and which will not be considered in this chapter they are SF. State feedback OI. Output injection 2 3 A B1 B2 6 GSF = 4 C1 D11 D12 7 : 5 I 0 0 2 3 A B1 I 6 GOI = 4 C1 D11 0 7 : 5 C2 D21 0
- 314. 12.2. Duality and Special Problems 297 Here we assume that all physical variables have compatible dimensions. We say that these special problems are special cases of OF problems in the sense that their structures are speci ed in comparison to OF problems. The structure of transfer matrices shows clearly that FC, OE (and OI) are duals of FI, DF (and SF), respectively. These relationships are shown in the following diagram: dual FI 6 - FC 6 equivalent equivalent ? ? dual - OE DF The precise meaning of equivalent" in this diagram will be explained below. 12.2.2 Full Information and Disturbance Feedforward " # x . In the FI problem, the controller is provided with Full Information since y = w For the FI problem, we only need to assume that (A B2 ) is stabilizable to guarantee the solvability. It is clear that if any output feedback control problem is to be solvable then the corresponding FI problem has to be solvable, provided FI is of the same structure with OF except for the speci ed parts. To motivate the name Disturbance Feedforward, consider the special case with C2 = 0. Then there is no feedback and the measurement is exactly w, where w is generally regarded as disturbance to the system. Only the disturbance, w, is fed through directly to the output. As we shall see, the feedback caused by C2 6= 0 does not a ect the transfer function from w to the output z , but it does a ect internal stability. In fact, the conditions for the solvability of the DF problem are that (A B2 ) is stabilizable and (C2 A) is detectable. Now we examine the connection between the DF problem and the FI problem and show the meaning of their equivalence. Suppose that we have controllers KFI and KDF and let TFI and TDF denote the closed-loop Tzw s in z yFI GFI - KFI w z u yDF GDF - KDF w u
- 315. PARAMETERIZATION OF STABILIZING CONTROLLERS 298 The question of interest is as follows: given either the KFI or the KDF controller, can we construct the other in such a way that TFI = TDF ? The answer is positive. Actually, we have the following: Lemma 12.3 Let GFI and GDF be given as above. Then " # I 0 0 G (s). (i) GDF (s) = FI 0 C2 I (ii) GFI = S (GDF PDF ) (where S ( ) denotes the star-product) GDF aa !! ! a !!aa ! a PDF 2 6 6 PDF (s) = 6 6 4 A ; B1 C2 " 0 I ; C2 3 B1 # 0 0 " # B2 7 I # 7: 7 " 7 0 0 I 5 Proof. Part (i) is obvious. Part (ii) follows from applying the star product formula. Nevertheless, we will give a direct proof to show the system structure. Let x and x ^ denote the state of GDF and PDF , respectively. Take e := x ; x and x as the states of ^ ^ the resulting interconnected system then its realization is 2 6 6 6 6 6 6 4 A ; B1 C2 B1 C2 C " 1 # 0 0 A C " 1# I 0 B1 D " 11# 0 I 0 which is exactly GFI , as claimed. C2 0 3 7 2 7 7 12# 7 7 " 0 7 5 B D 2 A 6 6 C1 =6 " # 6 I 4 B1 D " 11# 0 0 B2 D " 12# 0 0 0 I 3 7 7 7 7 5 2 Remark 12.2 There is an alternative way to see part (ii). The fact is that in the DF problems, the disturbance w and the system states x can be solved in terms of y and u: 2 A ; B1 C2 B1 B2 x =6 I 0 0 4 w ;C2 I 0 Now connect V up with GDF as shown below " # 3 7 5 " # y =: V u " # y : u
- 316. 12.2. Duality and Special Problems z 299 w u GDF y x w V Then it is easy to show that the transfer function from (w u) to (z x w) is GFI , and, furthermore, that the internal stability is not changed if A ; B1 C2 is stable. ~ The following theorem follows immediately: Theorem 12.4 Let GFI , GDF , and PDF be given as above. h (i) KFI := KDF C2 I GDF . Furthermore, i internally stabilizes GFI if KDF internally stabilizes h i F`(GFI KDF C2 I ) = F` (GDF KDF ): (ii) Suppose A ; B1 C2 is stable. Then KDF := F` (PDF KFI ) as shown below u y ^ PDF - KFI yDF u internally stabilizes GDF if KFI internally stabilizes GFI . Furthermore, F`(GDF F` (PDF KFI )) = F` (GFI KFI ): Remark 12.3 This theorem shows that if A ; B1C2 is stable, then problems FI and DF are equivalent in the above sense. Note that the transfer function from w to yDF is " # G21 (s) = A B1 : C2 I Hence this stability condition implies that G21 (s) has neither right half plane invariant zeros nor hidden unstable modes. ~
- 317. 300 PARAMETERIZATION OF STABILIZING CONTROLLERS 12.2.3 Full Control and Output Estimation For the FC problem, the term Full Control is used because the controller has full access to both the state through output injection and the output z . The only restriction on the controller is that it must work with the measurement y. This problem is dual to the FI case and has the dual solvability condition to the FI problem, which is also guaranteed by the assumptions on OF problems. The solutions to this kind of control problem can be obtained by rst transposing GFC , and solving the corresponding FI problem, and then transposing back. On the other hand, problem OE is dual to DF . Thus the discussion of the DF problem is relevant here, when appropriately dualized. And the solvability conditions for the OE problem are that (A B2 ) is stabilizable and (C2 A) is detectable. To examine the physical meaning of output estimation, rst note that z = C1 x + D11 w + u where z is to be controlled by an appropriately designed control u. In general, our control objective will be to specify z in some well-de ned mathematical sense. To put it in other words, it is desired to nd a u that will estimate C1 x + D11 w in such de ned mathematical sense. So this kind of control problem can be regarded as an estimation problem. We are focusing on this particular estimation problem because it is the one that arises in solving the output feedback problem. A more conventional estimation problem would be the special case where no internal stability condition is imposed and B2 = 0. Then the problem would be that of estimating the output z given the measurement y. This special case motivates the term output estimation and can be obtained immediately from the results obtained for the general case. The following discussion will explain the meaning of equivalence between FC and OE problems. Consider the following FC and OE diagrams: z yFC GFC - KFC w z u yOE GOE - KOE We have similar results to the ones in the last subsection: Lemma 12.5 Let GFC and GOE be given as above. Then 2 I 0 6 (i) GOE (s) = GFC (s) 4 0 B2 0 I 3 7 5 w u
- 318. 12.3. Parameterization of All Stabilizing Controllers (ii) GFC = S (GOE POE ), where POE is given by 2 A ; B2 C1 0 6 6 POE (s) = 6 C1 0 4 C2 I h I ;B2i h 0 I i h 0 0 301 i 3 7 7 7 5 : Theorem 12.6 Let GFC GOE , and POE be given as above. " # B2 K internally stabilizes G if K internally stabilizes G . (i) KFC := OE FC OE OE I Furthermore, " # F` (GFC B2 KOE ) = F`(GOE KOE ): I (ii) Suppose A ; B2 C1 is stable. Then KOE := F` (POE KFC ), as shown below y u POE y ^ - KFC u ^ internally stabilizes GOE if KFC internally stabilizes GFC . Furthermore, F` (GOE F`(POE KFC )) = F`(GFC KFC ): Remark 12.4 It is seen that if A ; B2C1 is stable, then FC and OE problems are equivalent in the above sense. This condition implies that the transfer matrix G12 (s) from u to z has neither right half-plane invariant zeros nor hidden unstable modes, which indicates that it has a stable inverse. ~ 12.3 Parameterization of All Stabilizing Controllers 12.3.1 Problem Statement and Solution Consider again the standard system block diagram in Figure 12.1 with 2 3 " # A B1 B2 G11 (s) G12 (s) : 7 6 G(s) = 4 C1 D11 D12 5 = G21 (s) G22 (s) C2 D21 D22 Suppose (A B2 ) is stabilizable and (C2 A) is detectable. In this section we discuss the following problem:
- 319. PARAMETERIZATION OF STABILIZING CONTROLLERS 302 Given a plant G, parameterize all controllers K that internally stabilize G. This parameterization for all stabilizing controllers is usually called Youla parameterization. As we have mentioned early, the stabilizing controllers for G will depend only on G22 . However, it is more convenient to consider the problem in the general framework as will be shown. The parameterization of all stabilizing controllers is easy when the plant itself is stable. Theorem 12.7 Suppose G 2 RH1 then the set of all stabilizing controllers can be described as K = Q(I + G22 Q);1 for any Q 2 RH1 and I + D22 Q(1) nonsingular. (12:2) Remark 12.5 This result is very natural considering Corollary 5.5, which says that a controller K stabilizes a stable plant G22 i K (I ; G22 K );1 is stable. Now suppose Q = K (I ; G22 K );1 is a stable transfer matrix, then K can be solved from this equation which gives exactly the controller parameterization in the above theorem. ~ Proof. Note that G22 (s) is stable by the assumptions on G. Now use straightforward algebra to verify that the controllers given above stabilize G22 . On the other hand, suppose K0 is a stabilizing controller then Q0 := K0(I ; G22 K0 );1 2 RH1 , so K0 can be expressed as K0 = Q0 (I + G22 Q0 );1 . Note that the invertibility in the last equation is guaranteed by the well posedness of the interconnected system with controller K0 since I + D22 Q0 (1) = (I ; D22 K0 (1));1 . 2 However, if G is not stable, the parameterization is much more complicated. The results can be more conveniently stated using state space representations. Theorem 12.8 Let F and L be such that A + LC2 and A + B2F are stable, and then all controllers that internally stabilize G can be parameterized as the transfer matrix from y to u below u J - Q y 2 A + B2 F + LC2 + LD22F ;L B2 + LD22 6 J =4 F 0 I ;(C2 + D22 F ) I ;D22 3 7 5 with any Q 2 RH1 and I + D22 Q(1) nonsingular. A non-constructive proof of the theorem can be given by using the same argument as in the proof of Theorem 12.7, i.e., rst verify that any controller given by the formula indeed stabilizes the system G, and then show that we can construct a stable Q for any
- 320. 12.3. Parameterization of All Stabilizing Controllers 303 given stabilizing controller K . This approach, however, does not give much insight into the controller construction and thus can not be generalized to other synthesis problems. The conventional Youla approach to this problem is via coprime factorization Youla et al, 1976, Vidyasagar, 1985, Desoer et al, 1982], which will be adopted in the later part of this chapter as an alternative approach. In the following sections, we will present a novel approach to this problem without adopting coprime factorizations. The idea of this approach is to reduce the output feedback problem into some simpler problems, such as FI and OE or FC and DF which admit simple solutions, and then to solve the output feedback problem by the separation argument. The advantages of this approach are that it is simple and that many other synthesis problems, such as H2 and H1 optimal control problems in Chapters 14 and 16, can be solved by using the same machinery. Readers should bear in mind that our objective here is to nd all admissible controllers for the OF problem. So at rst, we will try to build up enough tools for this objective by considering the special problems. We will see that it is not necessary to parameterize all stabilizing controllers for these special problems to get the required tools. Instead, we only parameterize some equivalent classes of controllers which generate the same control action. ^ De nition 12.2 Two controllers K and K are of equivalent control actions if their ^ corresponding closed loop transfer matrices are identical, i.e. Fl (G K ) = Fl (G K ), ^. written as K = K Algebraically, the controller equivalence is an equivalence relation. We will see that for di erent special problems we have di erent re ned versions of this relation. We will also see that the characterizations of equivalent classes of stabilizing controllers for special problems are good enough to construct the parameterization of all stabilizing controllers for the OF problem. In the next two subsections, we mainly consider the stabilizing controller characterizations for special problems. Also, we use the solutions to these special problems and the approach provided in the last section to characterize all stabilizing controllers of OF problems. 12.3.2 Stabilizing Controllers for FI and FC Problems In this subsection, we rst examine the FI structure z yFI GFI - KFI w u where the transfer matrix GFI is given in section 12.2. The purpose of this subsection is to characterize the equivalent classes of stabilizing controllers KFI that stabilize
- 321. 304 PARAMETERIZATION OF STABILIZING CONTROLLERS internally GFI and to build up enough tools to be used later. For this problem, we ^ say two controllers KFI and KFI are equivalent if they produce the same closed-loop transfer function from w to u. Obviously, this also guarantees that Fl (GFI KFI ) = ^ Fl (GFI KFI ). The characterization of equivalent classes of controllers can be suitably called the control parameterization in contrast with controller parameterization. Note that the same situation will occur in FC problems by duality. Since we have full information for feedback, our controller will have the following general form: h i KFI = K1 (s) K2(s) with K1 (s) stabilizing internally (sI ; A);1 B2 and arbitrary K2 (s) 2 RH1 . Note that the stability of K2 is required to guarantee the internal stability of the whole system since w is fed directly through K2. Lemma 12.9 Let F be a constant matrix such that A + B2F is stable. Then all stabilizing controllers, in the sense of generating all admissible closed-loop transfer functions, for FI can be parameterized as h i KFI = F Q with any Q 2 RH1 . Note that for the parameter matrix Q 2 RH1 , it is reasonable to assume that the realization of Q(s) is stabilizable and detectable. Proof. It is easy to see that the controller given in the above formula stabilizes the system GFI . Hence we only need to show that the given set of controllers parameterizes all stabilizing control action, u, i.e., there is a choice of Q 2 RH1 such that the transfer h i functions from w to u for any stabilizing controller KFI = K1 (s) K2 (s) and for h i 0 KFI = F Q are the same. To show this, make a change of control variable as v = u ; Fx, where x denotes the state of the system GFI then the system with the controller KFI will be as shown in the following diagram: z yFI with 2 A + B2 F 6 6 C1 + D12 F ~ GFI = 6 " # 6 I 4 0 w ~ GFI v - KFI ~ B1 D11# " 0 I B2 D12# " 0 0 3 7 7 7 7 5 ~ KFI = KFI ; F 0]:
- 322. 12.3. Parameterization of All Stabilizing Controllers 305 Let Q be the transfer matrix from w to hv Q belongs to RH1 by internal stability. i Then u = Fx + v = Fx + Qw, so KFI = F Q . 2 0 Remark 12.6 The equivalence of KFI = KFI in the above equation can actually be shown by directly computing Q from equating the transfer matrices from w to u for the 0 0 cases of KFI and KFI . In fact, the transfer matrices from w to u with KFI and KFI are given by I ; K1(sI ; A);1 B2 ;1 K1(sI ; A);1 B1 + I ; K1 (sI ; A);1 B2 ;1 K2 (12:3) I ; F (sI ; A);1 B2 ;1 F (sI ; A);1 B1 + I ; F (sI ; A);1 B2 ;1 Q (12:4) and respectively. We can verify that Q = K2 + (K1 ; F )(sI ; A ; B2 K1 );1 (B2 K2 + B1 ) is stable and makes the formulas in (12.3) and (12.4) equal. ~ Now we consider the dual FC problem the system diagram pertinent to this case is z yFC GFC - KFC w u ^ Dually, we say controllers KFC and KFC are equivalent in the sense that the same injection inputs yFC 's produce the same outputs z 's. This also guarantees the identity of their resulting closed-loop transfer matrices from w to z . And we also have Lemma 12.10 Let L be a constant matrix such that A + LC2 is stable. Then the set of equivalent classes of all stabilizing controllers for FC in the above sense can be parameterized as " # KFC = L Q with any Q 2 RH1 .
- 323. 306 PARAMETERIZATION OF STABILIZING CONTROLLERS 12.3.3 Stabilizing Controllers for DF and OE Problems In the DF case we have the following system diagram: z yDF GDF - KDF w u The transfer matrix is given as in section 12.2.1. We will further assume that A ; B1 C2 is stable in this subsection. It should be pointed out that the existence of a stabilizing controller for this system is guaranteed by the stabilizability of (A B2 ) and detectability of (C2 A). Hence this assumption is not necessary for our problem to be solvable however, it does simplify the solution. We will now parameterize stabilizing controllers for GDF by invoking the relationship between the FI problem and DF problem established in section 12.2. We say that the ^ controllers KDF and KDF are equivalent for the DF problem if the two transfer matrices from w to u in the above diagram are the same. Of course, the resulting two closed-loop transfer matrices from w to z are identical. Remark 12.7 By the equivalence between FI and DF problems, it is h to show easy h i i ^ DF in the DF structure, then KDF C2 I = KDF C2 I in ^ that if KDF = K ^ the corresponding FI structure. We also have that if KFI = KFI , then Fl (PDF KFI ) = ^ FI ). Fl (PDF K ~ Now we construct the parameterization of equivalent classes of stabilizing controllers in DF structure via the tool we have developed in section 12.2. h i Let KDF (s) be a stabilizing control for DF then KFI (s) = KDF (s) C2 I h i ^ stabilizes the corresponding GFI . Assume KFI = KFI = F Q for some Q 2 ^ ^ RH1 then KFI stabilizes GFI and F` (JDF Q) = F` (PDF KFI ) where 2 A + B2 F ; B1 C2 B1 B2 JDF = 6 F 0 I 4 I 0 ;C2 3 7 5 ^ with F such that A + B2 F is stable. Hence by Theorem 12.4, KDF := F` (JDF Q) ^ FI , by Remarks 12.7 we have stabilizes GDF for any Q 2 RH1 . Since KFI = K ^ KDF = KDF = F`(JDF Q). This equation characterizes an equivalent class of all controllers for the DF problem by the equivalence of FI and DF . In fact, we have the following lemma which shows that the above construction of parameterization characterizes all stabilizing controllers for the DF problem.
- 324. 12.3. Parameterization of All Stabilizing Controllers 307 Lemma 12.11 All stabilizing controllers for the DF problem can be characterized by KDF = F` (JDF Q) with Q 2 RH1 , where JDF is given as above. Proof. We have already shown that the controller KDF = F`(JDF Q) for any given Q 2 RH1 does internally stabilize GDF . Now let KDF be any stabilizing controller for ^ GDF then F` (JDF KDF ) 2 RH1 where 2 3 A B1 B2 ^DF = 6 ;F 0 I 7 : J 4 5 C2 I 0 ^ (JDF is stabilized by KDF since it has the same `G22 ' matrix as GDF .) ^ ^ Let Q0 := F` (JDF KDF ) 2 RH1 then F`(JDF Q0 ) = F`(JDF F`(JDF KDF )) =: F` (Jtmp KDF ), where Jtmp can be obtained by using the state space star product formula given in Chapter 10: 3 2 A ; B1 C2 + B2 F ;B2F B1 B2 7 6 6 ;B1 C2 A B1 B2 7 7 Jtmp = 6 6 F ;F 0 I 7 5 4 I 0 ;C2 C2 3 2 A ; B1 C2 ;B2F B1 B2 7 6 6 0 A + B2 F 0 0 7 7 = 6 6 0 ;F 0 I 7 5 4 I 0 0 C2 " # 0 I = : I 0 Hence F` (JDF Q0 ) = F`(Jtmp KDF ) = KDF . This shows that any stabilizing controller can be expressed in the form of F` (JDF Q0 ) for some Q0 2 RH1 . 2 Now we turn to the dual OE case. The corresponding system diagram is shown as below: z yOE GOE - KOE w u We will assume that A ; B2 C1 is stable. Again this assumption is made only for the simplicity of the solution, it is not necessary for the stabilization problem to be
- 325. PARAMETERIZATION OF STABILIZING CONTROLLERS 308 solvable. The parameterization of an equivalent class of stabilizing controllers for OE can be obtained by invoking Theorem 12.6 and the equivalent classes of controllers for ^ FC . Here we say controllers KOE and KOE are equivalent if the transfer matrices from yOE to z are the same. This also guarantees that the resulting closed-loop transfer matrices are identical. Now we construct the parameterization for the OE structure as a dual case to DF . " # B2 K Assume that KOE is an admissible controller for OE then KFC = OE = " I # L =: K for some Q 2 RH , and K stabilizes G and F (J Q) = ^ FC ^ FC 1 FC ` OE Q ^ F` (POE KFC ), where 2 A ; B2 C1 + LC2 L ;B2 6 JOE = 4 C1 0 I C2 I 0 3 7 5 ^ with L such that A + LC2 is stable. Hence by Theorem 12.6, KOE = F`(JOE Q) ^ stabilizes GOE for any Q 2 RH1 . Since KOE = KOE , KOE = F` (JOE Q). In fact, we have the following lemma. Lemma 12.12 All admissible controllers for the OE problem can be characterized as F` (JOE Q0 ) with any Q0 2 RH1 , where JOE is de ned as above. Proof. The controllers in the form as stated in the theorem are admissible since the corresponding FC controllers internally stabilize resulting GFC . ^ Now assume KOE is any stabilizing controller for GOE then F`(JOE KOE ) 2 RH1 where 2 3 A ;L B2 ^ JOE = 6 C1 4 C2 0 I I 0 7 5 : ^ ^ Let Q0 := F` (JOE KOE ) 2 RH1 . Then F`(JOE Q0 ) = F` (JOE F` (JOE KOE )) = KOE , by using again the state space star product formula given in Chapter 10. This shows that any stabilizing controller can be expressed in the form of F` (JOE Q0 ) for some Q0 2 RH1 . 2 12.3.4 Output Feedback and Separation We are now ready to give a complete proof for Theorem 12.8. We will assume the results of the special problems and show how to construct all admissible controllers for the OF problem from them. And we can also observe the separation argument as the
- 326. 12.3. Parameterization of All Stabilizing Controllers 309 byproduct this essentially involves reducing the OF problem to the combination of the simpler FI and FC problems. Moreover, we can see from the construction why the stability conditions of A ; B1 C2 and A ; B2 C1 in DF and OE problems were reasonably assumed and are automatically guaranteed in this case. Again we assume that the system has the following standard system block diagram: z y with G - K w u 3 2 A B1 B2 G(s) = 6 C1 D11 D12 7 5 4 C2 D21 D22 and that (A B2 ) is stabilizable and (C2 A) is detectable. Proof of Theorem 12.8. Without loss of generality, we shall assume D22 = 0. For more general cases, i.e. D22 6= 0, the mapping ^ K (s) = K (s)(I ; D22 K (s));1 is well de ned if the closed-loop system is assumed to be well posed. Then the system ^ in terms of K has the structure z y where ^ G - K ^ w u 2 3 A B1 B2 ^ (s) = 6 C1 D11 D12 7 : G 4 5 C2 D21 0 Now we construct the controllers for the OF problem with D22 = 0. Denote x the state of system G then the open-loop system can be written as 2 6 4 3 2 x _ A B1 B2 7 6 z 5 = 4 C1 D11 D12 y C2 D21 0 32 76 54 3 x w 7: 5 u
- 327. 310 PARAMETERIZATION OF STABILIZING CONTROLLERS Since (A B2 ) is stabilizable, there is a constant matrix F such that A + B2 F is stable. h i Note that F 0 is actually a special FI stabilizing controller. Now let v = u ; Fx: Then the system can be broken into two subsystems: " # " x = A + B2 F B1 B2 _ z C1 + D12 F D11 D12 and 2 6 4 3 2 A B1 B2 x _ 7 6 v 5 = 4 ;F 0 I C2 D21 0 y 32 76 54 # 2 6 4 x w v 3 7 5 3 x w 7: 5 u This can be shown pictorially below: z v G1 y w u Gtmp - K with and " # G1 = A + B2 F B1 B2 2 RH1 C1 + D12 F D11 D12 2 3 A B1 B2 6 Gtmp = 4 ;F 0 I 7 : 5 C2 D21 0 Obviously, K stabilizes G if and only if K stabilizes Gtmp however, Gtmp is of OE structure. Now let L be such that A + LC2 is stable. Then by Lemma 12.12 all controllers stabilizing Gtmp are given by K = F`(J Q)
- 328. 12.3. Parameterization of All Stabilizing Controllers 311 where 2 A + B2 F + LC2 L ;B2 J =6 ;F 0 I 4 C2 I 0 3 2 3 A + B2 F + LC2 ;L B2 7 6 =4 F 0 I 7: 5 5 ;C2 I 0 2 This concludes the proof. Remark 12.8 We can also get the same result by applying the dual procedure to the above construction, i.e., rst use an output injection to reduce the OF problem to a DF problem. The separation argument is obvious since the synthesis of the OF problem can be reduced to FI and FC problems, i.e. the latter two problems can be designed independently. ~ Remark 12.9 Theorem 12.8 shows that any stabilizing controller K (s) can be characterized as an LFT of a parameter matrix Q 2 RH1 , i.e., K (s) = F` (J Q). Moreover, using the same argument as in the proof of Lemma 12.11, a realization of Q(s) in terms of K can be obtained as ^ Q := F` (J K ) where 3 2 A ;L B2 0 I C2 I D22 ^ 4 J = 6 ;F 7 5 and where K (s) has the stabilizable and detectable realization. ~ Now we can reconsider the characterizations of all stabilizing controllers for the special problems with some reasonable assumptions, i.e. the stability conditions of A ; B1 C2 and A ; B2 C1 for DF and OE problems which were assumed in the last section can be dropped. If we specify " # " # " # I D = 0 D = 0 C2 = 21 22 0 I 0 the OF problem, in its general setting, becomes the FI problem. We know that the solvability conditions for the FI problem are reduced, because of its special structure, to (A B2 ) as stabilizable. By assuming this, we can get the following result from the OF problem. Corollary 12.13 Let L1 and F be such that A + L1 and A + B2 F are stable then all controllers that stabilize GFI can be characterized as F`(JFI Q) with any Q 2 RH1 ,
- 329. PARAMETERIZATION OF STABILIZING CONTROLLERS 312 where 2 A + B2 F + L1 L ;B2 6 6 ;F # 0 I " JFI = 6 6 I 4 I 0 3 7 7 7 7 5 0 and where L = (L1 L2) is the injection matrix for any L2 with compatible dimensions. In the same way, we can consider the FC problem as the special OF problem by specifying h i h i h i B2 = I 0 D12 = 0 I D22 = 0 0 : The DF (OE ) problem can also be considered as the special case of OF by simply setting D21 = I (D12 = I ) and D22 = 0. Corollary 12.14 Consider the DF problem, and assume that (C2 A B2 ) is stabilizable and detectable. Let F and L be such that A + LC2 and A + B2 F are stable, and then all controllers that internally stabilize G can be parameterized as Fl (JDF Q) for some Q 2 RH1 , i.e. the transfer function from y to u is shown as below u JDF - Q y 2 3 A + B2 F + LC2 ;L B2 6 JDF = 4 F 0 I 7: 5 ;C2 I 0 Remark 12.10 It would be interesting to compare this result with Lemma 12.11. It can be seen that Lemma 12.11 is a special case of this corollary. The condition that A ; B1 C2 is stable, which is required in Lemma 12.11, provides the natural injection matrix L = ;B1 which satis es a partial condition in this corollary. ~ 12.4 Structure of Controller Parameterization Let us recap what we have done. We begin with a stabilizable and detectable realization of G22 # " A B2 : G22 = C2 D22 We choose F and L so that A + B2 F and A + LC2 are stable. De ne J by the formula in Theorem 12.8. Then the proper K 's achieving internal stability are precisely those representable in Figure 12.3 and K = F` (J Q) where Q 2 RH1 and I + D22 Q(1) is invertible.
- 330. 12.4. Structure of Controller Parameterization z 313 w G y u D22 ? c ? ; c R C2 c c 66 -A c 6 -F - ;L u1 B2 y1 - Q Figure 12.3: Structure of Stabilizing Controllers It is interesting to note that the system in the dashed box is an observer-based stabilizing controller for G (or G22 ). Furthermore, it is easy to show that the transfer function between (y y1 ) and (u u1) is J , i.e., " # " # u =J y : u1 y1 It is also easy to show that the transfer matrix from y1 to u1 is zero. This diagram of all stabilizing controller parameterization also suggests an interesting interpretation: every internal stabilization amounts to adding stable dynamics to the plant and then stabilizing the extended plant by means of an observer. The precise statement is as follows: for simplicity of the formulas, only the cases of strictly proper G22 and K are treated. Theorem 12.15 Assume that G22 and K are strictly proper and the system is Figure 12.1 is internally stable. Then G22 can be embedded in a system " where " Ae = A 0 0 Aa # Ae Be Ce 0 " Be = B2 0 # # h Ce = C2 0 i (12:5)
- 331. 314 PARAMETERIZATION OF STABILIZING CONTROLLERS and where Aa is stable, such that K has the form " K = Ae + Be Fe + Le Ce ;Le Fe 0 where Ae + Be Fe and Ae + Le Ce are stable. # (12:6) Proof. K is representable as in Figure 12.3 for some Q in RH1 . For K to be strictly proper, Q must be strictly proper. Take a minimal realization of Q: " # Q = Aa Ba : Ca 0 Since Q 2 RH1 , Aa is stable. Let x and xa denote state vectors for J and Q, respectively, and write the equations for the system in Figure 12.3: x _ u u1 xa _ y1 (A + B2 F + LC2 )x ; Ly + B2 y1 Fx + y1 ;C2 x + y Aa xa + Ba u1 = = = = = Ca xa These equations yield xe = (Ae + Be Fe + Le Ce )xe ; Le y _ u = Fe xe where " # h xe := x Fe := F Ca xa and where Ae Be Ce are as in (12.5). i " Le := L ; Ba # 2 12.5 Closed-Loop Transfer Matrix Recall that the closed-loop transfer matrix from w to z is a linear fractional transformation F`(G K ) and that K stabilizes G if and only if K stabilizes G22 . Elimination of the signals u and y in Figure 12.3 leads to Figure 12.4 for a suitable transfer matrix T . Thus all closed-loop transfer matrices are representable as in Figure 12.4. z = F` (G K )w = F`(G F` (J Q))w = F`(T Q)w: It remains to give a realization of T . (12:7)
- 332. 12.6. Youla Parameterization via Coprime Factorization* z T 315 w - Q Figure 12.4: Closed loop system Theorem 12.16 Let F and L be such that A + BF and A + LC are stable. Then the set of all closed-loop transfer matrices from w to z achievable by an internally stabilizing proper controller is equal to F` (T Q) = fT11 + T12 QT21 : Q 2 RH1 I + D22 Q(1) invertibleg where T is given by 2 " T = T11 T21 ;B2 F 3 B1 B2 # 6 7 6 0 A + LC2 B1 + LD21 0 7 : T12 = 6 7 6 T22 ;D12F D11 D12 7 4 C1 + D12 F 5 0 C2 D21 0 A + B2 F Proof. This is straightforward by using the state space star product formula and follows from some tedious algebra. Hence it is left for the readers to verify. 2 An important point to note is that the closed-loop transfer matrix is simply an a ne function of the controller parameter matrix Q since T22 = 0. 12.6 Youla Parameterization via Coprime Factorization* In this section, all stabilizing controller parameterization will be derived using the conventional coprime factorization approach. Readers should be familiar with the results presented in Section 5.4 of Chapter 5 before proceeding further. ~ ~ Theorem 12.17 Let G22 = NM ;1 = M ;1N be the rcf and lcf of G22 over RH1 , respectively. Then the set of all proper controllers achieving internal stability is parameterized either by K = (U0 + MQr )(V0 + NQr );1 det(I + V0;1 NQr )(1) 6= 0 (12:8) for Qr 2 RH1 or by ~ ~ ~ ~ ~~ K = (V0 + Ql N );1 (U0 + Ql M ) det(I + Ql N V0;1 )(1) 6= 0 (12:9)
- 333. 316 PARAMETERIZATION OF STABILIZING CONTROLLERS ~ ~ for Ql 2 RH1 where U0 V0 U0 V0 2 RH1 satisfy the Bezout identities: ~ ~ ~ ~ V0 M ; U0 N = I MV0 ; NU0 = I: ~ ~ ~ ~ Moreover, if U0 V0 U0 , and V0 are chosen such that U0V0;1 = V0;1 U0 , i.e., " #" # " # ~ ~ V0 ;U0 M U0 = I 0 : ~ ~ ;N M N V0 0 I Then K = (U0 + MQy )(V0 + NQy );1 ~ ~ ~ ~ = (V0 + Qy N );1 (U0 + Qy M ) = F` (Jy Qy ) where " # ~ U0 V0;1 V0;1 Jy := ;1 ;1 V0 ;V0 N (12.10) (12:11) and where Qy ranges over RH1 such that (I + V0;1 NQy )(1) is invertible Proof. We shall prove the parameterization given in (12.8) rst. Assume that K has the form indicated, and de ne U := U0 + MQr V := V0 + NQr : Then ~ ~ ~ ~ ~ ~ ~ ~ MV ; NU = M (V0 + NQr ) ; N (U0 + MQr ) = MV0 ; NU0 + (MN ; NM )Qr = I: Thus K achieves internal stability by Lemma 5.10. Conversely, suppose K is proper and achieves internal stability. Introduce an rcf of ~ ~ K over RH1 as K = UV ;1 . Then by Lemma 5.10, Z := MV ; NU is invertible in RH1 . De ne Qr by the equation U0 + MQr = UZ ;1 (12:12) so Qr = M ;1 (UZ ;1 ; U0 ): Then using the Bezout identity, we have V0 + NQr = V0 + NM ;1(UZ ;1 ; U0 ) ~ ~ = V0 + M ;1 N (UZ ;1 ; U0 ) ~ ;1(MV0 ; NU0 + NUZ ;1 ) ~ ~ = M ~ ~ ~ = M ;1(I + NUZ ;1) ;1(Z + NU )Z ;1 ~ ~ = M ;1MV Z ;1 ~ = M ~ = V Z ;1: (12.13)
- 334. 12.6. Youla Parameterization via Coprime Factorization* 317 Thus, K = UV ;1 = (U0 + MQr )(V0 + NQr );1 : To see that Qr belongs to RH1 , observe rst from (12.12) and then from (12.13) that both MQr and NQr belong to RH1 . Then ~ ~ ~ ~ Qr = (V0 M ; U0 N )Qr = V0 (MQr ) ; U0 (NQr ) 2 RH1 : Finally, since V and Z evaluated at s = 1 are both invertible, so is V0 + NQr from (12.13), hence so is I + V0;1 NQr . Similarly, the parameterization given in (12.9) can be obtained. To show that the controller can be written in the form of equation (12.10), note that (U0 + MQy )(V0 + NQy );1 = U0 V0;1 + (M ; U0 V0;1 N )Qy (I + V0;1 NQy );1 V0;1 ~ ~ and that U0 V0;1 = V0;1 U0 . We have ~ ~ ~ ~ ~ ~ (M ; U0 V0;1 N ) = (M ; V0;1 U0N ) = V0;1 (V0 M ; U0 N ) = V0;1 and ~ K = U0 V0;1 + V0;1 Qy (I + V0;1 NQy );1 V0;1 : (12:14) 2 Corollary 12.18 Given an admissible controller K with coprime factorizations K = ~ ~ UV ;1 = V ;1 U , the free parameter Qy 2 RH1 in Youla parameterization is given by Qy = M ;1 (UZ ;1 ; U0 ) where ~ ~ Z := MV ; NU: Next, we shall establish the precise relationship between the above all stabilizing controller parameterization and the parameterization obtained in the previous sections via LFT framework. Theorem 12.19 Let the doubly coprime factorizations of G22 be chosen as 3 2 " # A + B2 F B2 ;L M U 0 N V0 =6 4 F I C2 + D22 F D22 0 I 7 5
- 335. 318 PARAMETERIZATION OF STABILIZING CONTROLLERS 2 3 A + LC2 ;(B2 + LD22) L 6 =4 F I 0 7 5 C2 ;D22 I where F and L are chosen such that A + B2 F and A + LC2 are both stable. Then Jy can be computed as " ~ ~ V0 ;U0 ~ ~ ;N M # 2 3 A + B2 F + LC2 + LD22F ;L B2 + LD22 6 7 Jy = 4 F 0 I 5: ;(C2 + D22 F ) I ;D22 Proof. This follows from some tedious algebra. 2 Remark 12.11 Note that Jy is exactly the same as the J in Theorem 12.8 and that ;1 K0 := U0V0 is an observer-based stabilizing controller with " # K0 := A + B2 F + LC2 + LD22 F ;L : F 0 ~ 12.7 Notes and References The special problems FI, DF, FC, and OE were rst introduced in Doyle, Glover, Khargonekar, and Francis 1989] for solving the H1 problem, and they have been since used in many other papers for di erent problems. The new derivation of all stabilizing controllers was reported in Lu, Zhou, and Doyle 1991]. The paper by Moore et al 1990] contains some other related interesting results. The conventional Youla parameterization can be found in Youla et al 1976], Desoer et al 1980], Doyle 1984], Vidyasagar 1985], and Francis 1987]. The parameterization of all two-degree-of-freedom stabilizing controllers is given in Youla and Bongiorno 1985] and Vidyasagar 1985].
- 336. 13 Algebraic Riccati Equations We have studied the Lyapunov equation in Chapter 3 and have seen the roles it played in some applications. A more general equation than the Lyapunov equation in control theory is the so-called Algebraic Riccati Equation or ARE for short. Roughly speaking, Lyapunov equations are most useful in system analysis while AREs are most useful in control system synthesis particularly, they play the central roles in H2 and H1 optimal control. Let A, Q, and R be real n n matrices with Q and R symmetric. Then an algebraic Riccati equation is the following matrix equation: A X + XA + XRX + Q = 0: (13:1) Associated with this Riccati equation is a 2n 2n matrix: " H := A R ;Q ;A # : (13:2) A matrix of this form is called a Hamiltonian matrix. The matrix H in (13.2) will be used to obtain the solutions to the equation (13.1). It is useful to note that (H ) (the spectrum of H) is symmetric about the imaginary axis. To see that, introduce the 2n 2n matrix: " # 0 ;I J := I 0 having the property J 2 = ;I . Then J ;1 HJ = ;JHJ = ;H 319
- 337. ALGEBRAIC RICCATI EQUATIONS 320 so H and ;H are similar. Thus is an eigenvalue i ; is. This chapter is devoted to the study of this algebraic Riccati equation and related problems: the properties of its solutions, the methods to obtain the solutions, and some applications. In Section 13.1, we will study all solutions to (13.1). The word all" means that any X , which is not necessarily real, not necessarily hermitian, not necessarily nonnegative, and not necessarily stabilizing, satis es equation (13.1). The conditions for a solution to be hermitian, real, and so on, are also given in this section. The most important part of this chapter is Section 13.2 which focuses on the stabilizing solutions. This section is designed to be essentially self-contained so that readers who are only interested in the stabilizing solution may go to Section 13.2 directly without any di culty. Section 13.3 presents the extreme (i.e., maximal or minimal) solutions of a Riccati equation and their properties. The relationship between the stabilizing solution of a Riccati equation and the spectral factorization of some frequency domain function is established in Section 13.4. Positive real functions and inner functions are introduced in Section 13.5 and 13.6. Some other special rational matrix factorizations, e.g., inner-outer factorizations and normalized coprime factorization, are given in Sections 13.7-13.8. 13.1 All Solutions of A Riccati Equation The following theorem gives a way of constructing solutions to (13.1) in terms of invariant subspaces of H . Theorem 13.1 Let V C 2n be an n-dimensional invariant subspace of H , and let X1 X2 2 C n n be two complex matrices such that " # V = Im X1 : X2 ; If X1 is invertible, then X := X2 X1 1 is a solution to the Riccati equation (13.1) and (A + RX ) = (H jV ). Furthermore, the solution X is independent of a speci c choice of bases of V . Proof. Since V is an H invariant subspace, there is a matrix 2 C n n such that " A R ;Q ;A #" # ; Postmultiply the above equation by X1 1 to get " A R ;Q ;A #" " X1 = X1 X2 X2 # " # # : I = I X X ;1: X X 1 1 (13:3)
- 338. 13.1. All Solutions of A Riccati Equation h Now pre-multiply (13.3) by ;X I i 321 to get " #" A R 0 = ;X I ;Q ;A = ;XA ; A X ; XRX ; Q h i I X # which shows that X is indeed a solution of (13.1). Equation (13.3) also gives ; A + RX = X1 X1 1 therefore, (A + RX ) = ( ). But, by de nition, is a matrix representation of the map H jV , so (A + RX ) = (H jV ). Finally note that any other basis spanning V can be represented as " # " # X1 P = X1 P X2 X2 P for some nonsingular matrix P . The conclusion follows from the fact (X2 P )(X1 P );1 = ; X2 X1 1. 2 As we would expect, the converse of the theorem also holds. Theorem 13.2 If X 2 C n n is a solution to the Riccati equation (13.1),;then there exist matrices X1#X2 2 C n n , with X1 invertible, such that X = X2 X1 1 and the " columns of X1 X2 form a basis of an n-dimensional invariant subspace of H . Proof. De ne := A + RX . Multiplying this by X gives X = XA + XRX = ;Q ; A X with the second equality coming from the fact that X is a solution to (13.1). Write these two relations as " " A R ;Q ;A I X # #" # " I = I X X # : Hence, the columns of span an n-dimensional invariant subspace of H , and de ning X1 := I , and X2 := X completes the proof. 2
- 339. ALGEBRAIC RICCATI EQUATIONS 322 Remark 13.1 It is now clear that to obtain solutions to the Riccati equation, it is necessary to be able to construct bases for those invariant subspaces of H . One way of constructing those invariant subspaces is to use eigenvectors and generalized eigenvectors of H . Suppose i is an eigenvalue of H with multiplicity k (then i+j = i for all j = 1 : : : k ; 1), and let vi be a corresponding eigenvector and vi+1 : : : vi+k;1 be the corresponding generalized eigenvectors associated with vi and i . Then vj are related by ( H ; i I )v i = 0 (H ; i I )vi+1 = vi .. . (H ; i I )vi+k;1 = vi+k;2 and the spanfvj j = i : : : i + k ; 1g is an invariant subspace of H . ~ Example 13.1 Let " # " # " # A = ;3 2 R = 0 0 Q= 0 0 : ;2 1 0 ;1 0 0 The eigenvalues of H are 1 1 ;1 ;1, and the corresponding eigenvector and generalized eigenvector of 1 are 2 3 2 3 2 7 7 7 7 5 6 6 =6 6 4 3 1 ;1 7 6 7 6 6 2 7 6 ;3=2 7 7 6 7: v1 = 6 6 2 7 v2 = 6 17 4 5 4 5 ;2 0 The corresponding eigenvector and generalized eigenvector of ;1 are 2 v3 = 6 6 6 6 4 1 1 0 0 v4 3 1 7 3=2 7 7: 0 7 5 0 All solutions of the Riccati equation under various combinations are given below: " # " X1 = v v ], then X = X X ;1 = spanfv1 v2 g is H -invariant: let 1 2 2 1 X2 is a solution and (A + RX ) = f1 1g " # " X1 = v v ], then X = X X ;1 = spanfv1 v3 g is H -invariant: let 1 3 2 1 X2 is also a solution and (A + RX ) = f1 ;1g ;10 6 6 ;4 ;2 2 2 ;2 # #
- 340. 13.1. All Solutions of A Riccati Equation " 323 # X1 = v v ], then X = 0 is a solution and spanfv3 v4 g is H -invariant: let 3 4 X2 (A + RX ) = f;1 ;1g spanfv1 v4 g spanfv2 v3 g, and spanfv2 v4 g are not H -invariant subspaces. Read- ers can verify that the matrices constructed from those vectors are not solutions to the Riccati equation (13.1). 3 Up to this point, we have said nothing about the structure of the solutions given by Theorem 13.1 and 13.2. The following theorem gives a su cient condition for a Riccati solution to be hermitian (not necessarily real symmetric). Theorem 13.3 Let V be an n-dimensional H -invariant subspace and let X1 X2 2 Cn n be such that " # V = Im X1 : X2 Then i + j 6= 0 for all i j = 1 : : : n, i j 2 (H jV ) implies that X1 X2 is hermi; tian, i.e., X1 X2 = (X1 X2 ) . Furthermore, if X1 is nonsingular, then X = X2 X1 1 is hermitian. Proof. Since V is an invariant subspace of H , there is a matrix representation for H jV such that ( ) = (H jV ) and " # " H X1 = X1 X2 X2 " X1 Pre-multiply this equation by X2 " X1 X2 # # " # : J to get # " JH X1 = X1 X2 X2 # " J X1 X2 # : (13:4) Note that JH is hermitian (actually symmetric since H is real) therefore, the left-hand side of (13.4) is hermitian as well as the right-hand side: " i.e., X1 X2 # " J X1 X2 # " = X1 X2 # J " # X1 = ; X2 " X1 X2 (;X1 X2 + X2 X1 ) = ; (;X1 X2 + X2 X1 ): # " J X1 X2 #
- 341. ALGEBRAIC RICCATI EQUATIONS 324 This is a Lyapunov equation. Since i + j 6= 0, the equation has a unique solution: ;X1 X2 + X2 X1 = 0: This implies that X1 X2 is hermitian. ; ; That X is hermitian is easy to see by noting that X = (X1 1 ) (X1 X2 )X1 1 . Remark 13.2 It is clear from Example 13.1 that the condition necessary for the existence of a hermitian solution. 2 i + j 6= 0 is not ~ The following theorem gives necessary and su cient conditions for a solution to be real. Theorem 13.4 Let V be an n-dimensional H -invariant subspace, and let X1 X2 2 " # n n be such that X1 is nonsingular and the columns of X1 form a basis of V . C X2 ;1 is real if and only if V is conjugate symmetric, i.e., v 2 V implies Then X := X2 X1 that v 2 V . Proof. (() Since V is conjugate symmetric, there is a nonsingular matrix P 2 C n such that " # " n # X1 = X1 P X2 X2 ; where the over bar X denotes the complex conjugate. Therefore, X = X2 X1 1 = ;1 = X2 X ;1 = X is real as desired. X2 P (X1 P ) 1 ; ()) De ne X := X2 X1 1. By assumption, X 2 Rn n and " # I =V Im X therefore, V is conjugate symmetric. 2 Example 13.2 This example is intended to show that there are non-real, non-hermitian solutions to equation (13.1). It is also designed to show that the condition i + j 6= 0 8 i j , which excludes the possibility of having imaginary axis eigenvalues since if l = j! then l + l = 0, is not necessary for the existence of a hermitian solution. Let 2 A=6 4 1 0 0 0 0 ;1 0 1 0 3 7 5 2 ;1 0 ;2 R=6 0 0 0 4 ;2 0 ;4 3 2 7 5 Q=6 4 3 0 0 0 0 0 0 7: 5 0 0 0
- 342. 13.2. Stabilizing Solution and Riccati Operator Then H has eigenvalues that 1 =1=; 2 3 = 5 325 = j = ; 4 = ; 6 . It is easy to show 2 3 0 0:5 + 0:5j 0:5 ; 0:5j 6 7 X=4 0 0 0 5 0 0 0 satis es equation (13.1) and that X is neither real nor hermitian. On the other hand, 2 X=6 4 2 0 0 0 0 0 0 0 0 3 7 5 is a real symmetric nonnegative de nite solution corresponding to the eigenvalues ;1, j , ;j . 3 13.2 Stabilizing Solution and Riccati Operator In this section, we discuss when a solution is stabilizing, i.e., (A + RX ) C ; and the properties of such solutions. This section is the central part of this chapter and is designed to be self-contained. Hence some of the material appearing in this section may be similar to that seen in the previous section. Assume H has no eigenvalues on the imaginary axis. Then it must have n eigenvalues in Re s < 0 and n in Re s > 0. Consider the two n-dimensional spectral subspaces, X; (H ) and X+ (H ): the former is the invariant subspace corresponding to eigenvalues in Re s < 0 and the latter corresponds to eigenvalues in Re s > 0. By nding a basis for X; (H ), stacking the basis vectors up to form a matrix, and partitioning the matrix, we get " # X; (H ) = Im X1 X2 where X1 X2 2 C n n . (X1 and X2 can be chosen to be real matrices.) If X1 is nonsingular or, equivalently, if the two subspaces X; (H ) Im " 0 I # (13:5) ; are complementary, we can set X := X2 X1 1. Then X is uniquely determined by H , i.e., H 7;! X is a function, which will be denoted Ric. We will take the domain of Ric, denoted dom(Ric), to consist of Hamiltonian matrices H with two properties: H has no eigenvalues on the imaginary axis and the two subspaces in (13.5) are complementary. For ease of reference, these will be called the stability property and the complementarity
- 343. ALGEBRAIC RICCATI EQUATIONS 326 property, respectively. This solution will be called the stabilizing solution. Thus, X = Ric(H ) and Ric : dom(Ric) R2n 2n 7;! Rn n : The following well-known results give some properties of X as well as veri able conditions under which H belongs to dom(Ric). Theorem 13.5 Suppose H 2 dom(Ric) and X = Ric(H ). Then (i) X is real symmetric (ii) X satis es the algebraic Riccati equation A X + XA + XRX + Q = 0 (iii) A + RX is stable . Proof. (i) Let X1 X2 be as above. It is claimed that X1 X2 is symmetric: To prove this, note that there exists a stable matrix H; in Rn n such that " # " (13:6) # H X1 = X1 H; : X2 X2 (H; is a matrix representation of H jX; (H ) .) Pre-multiply this equation by " to get " X1 X2 # " # X1 X2 " # J JH X1 = X1 X2 X2 # " # J X1 H; : X2 (13:7) Since JH is symmetric, so is the left-hand side of (13.7) and so is the right-hand side: (;X1 X2 + X2 X1 )H; = H; (;X1 X2 + X2 X1 ) = ;H;(;X1 X2 + X2 X1 ): This is a Lyapunov equation. Since H; is stable, the unique solution is ;X1 X2 + X2 X1 = 0: This proves (13.6). ; ; Since X1 is nonsingular and X = (X1 1 ) (X1 X2 )X1 1 , X is symmetric.
- 344. 13.2. Stabilizing Solution and Riccati Operator 327 (ii) Start with the equation " # " # H X1 = X1 H; X2 X2 ; and post-multiply by X1 1 to get " # " # ; H I = I X1 H; X1 1: X X Now pre-multiply by X ; I ]: " (13:8) # X ; I ]H I = 0: X This is precisely the Riccati equation. (iii) Pre-multiply (13.8) by I 0] to get ; A + RX = X1 H; X1 1: 2 Thus A + RX is stable because H; is. Now, we are going to state one of the main theorems of this section which gives the necessary and su cient conditions for the existence of a unique stabilizing solution of (13.1) under certain restrictions on the matrix R. Theorem 13.6 Suppose H has no imaginary eigenvalues and R is either positive semide nite or negative semi-de nite. Then H 2 dom(Ric) if and only if (A R) is stabilizable. Proof. (() To prove that H 2 dom(Ric), we must show that X; (H ) Im " 0 # I are complementary. This requires a preliminary step. As in the proof of Theorem 13.5 de ne X1 X2 H; so that " # X; (H ) = Im X1 " # " X2 # H X1 = X1 H; : X2 X2 (13:9)
- 345. ALGEBRAIC RICCATI EQUATIONS 328 We want to show that X1 is nonsingular, i.e., Ker X1 = 0. First, it is claimed that Ker X1 is H; -invariant. To prove this, let x 2 Ker X1 . Pre-multiply (13.9) by I 0] to get AX1 + RX2 = X1 H; : (13:10) Pre-multiply by x X2 , post-multiply by x, and use the fact that X2 X1 is symmetric (see (13.6)) to get x X2 RX2 x = 0: Since R is semide nite, this implies that RX2 x = 0. Now post-multiply (13.10) by x to get X1 H; x = 0, i.e. H; x 2 Ker X1 . This proves the claim. Now to prove that X1 is nonsingular, suppose, on the contrary, that Ker X1 6= 0. Then H; jKer X1 has an eigenvalue, , and a corresponding eigenvector, x: H; x = x (13:11) Re < 0 0 6= x 2 Ker X1 : Pre-multiply (13.9) by 0 I ]: ; QX1 ; A X2 = X2 H; : (13:12) Post-multiply the above equation by x and use (13.11): (A + I )X2 x = 0: Recall that RX2 x = 0, we have x X2 A + I R] = 0: Then stabilizability of (A R) implies X2 x = 0. But if both X1 x = 0 and X2 x = 0, then " # x = 0 since X1 has full column rank, which is a contradiction. X2 ()) This is obvious since H 2 dom(Ric) implies that X is a stabilizing solution and that A + RX is asymptotically stable. It also implies that (A R) must be stabilizable. 2 Theorem 13.7 Suppose H has the form " # A ;BB : H= ;C C ;A Then H 2 dom(Ric) i (A B ) is stabilizable and (C A) has no unobservable modes on the imaginary axis. Furthermore, X = Ric(H ) 0 if H 2 dom(Ric), and Ker(X ) = 0 if and only if (C A) has no stable unobservable modes. Note that Ker(X ) Ker(C ), so that the equation XM = C always has a solution for M , and a minimum F -norm solution is given by X yC .
- 346. 13.2. Stabilizing Solution and Riccati Operator 329 Proof. It is clear from Theorem 13.6 that the stabilizability of (A B) is necessary, and it is also su cient if H has no eigenvalues on the imaginary axis. So we only need to show that, assuming (A B ) is stabilizable, H has no imaginary eigenvalues i (C A) has no unobservable modes on the imaginary axis. Suppose that j! is an eigenvalue " # x is a corresponding eigenvector. Then and 0 6= z Ax ; BB z = j!x ;C Cx ; A z = j!z: Re-arrange: (A ; j!I )x = BB z ; (A ; j!I ) z = C Cx: Thus (13:13) (13:14) hz (A ; j!I )xi = hz BB z i = kB z k2 ;hx (A ; j!I ) z i = hx C Cxi = kCxk2 so hx (A ; j!I ) z i is real and ;kCxk2 = h(A ; j!I )x z i = hz (A ; j!I )xi = kB z k2: Therefore B z = 0 and Cx = 0. So from (13.13) and (13.14) (A ; j!I )x = 0 (A ; j!I ) z = 0: Combine the last four equations to get z A ; j!I B ] = 0 " # A ; j!I x = 0: C The stabilizability of (A B ) gives z = 0. Now it is clear that j! is an eigenvalue of H i j! is an unobservable mode of (C A). Next, set X := Ric(H ). We'll show that X 0. The Riccati equation is A X + XA ; XBB X + C C = 0 or equivalently (A ; BB X ) X + X (A ; BB X ) + XBB X + C C = 0: (13:15)
- 347. ALGEBRAIC RICCATI EQUATIONS 330 Noting that A ; BB X is stable (Theorem 13.5), we have X= 1 Z 0 e(A;BB X ) t (XBB X + C C )e(A;BB X )t dt: (13:16) Since XBB X + C C is positive semi-de nite, so is X . Finally, we'll show that KerX is non-trivial if and only if (C A) has stable unobservable modes. Let x 2 KerX , then Xx = 0. Pre-multiply (13.15) by x and post-multiply by x to get Cx = 0: Now post-multiply (13.15) again by x to get XAx = 0: We conclude that Ker(X ) is an A-invariant subspace. Now if Ker(X ) 6= 0, then there is a 0 6= x 2 Ker(X ) and a such that x = Ax = (A ; BB X )x and Cx = 0. Since (A ; BB X ) is stable, Re < 0 thus is a stable unobservable mode. Conversely, suppose (C A) has an unobservable stable mode , i.e., there is an x such that Ax = x Cx = 0. By pre-multiplying the Riccati equation by x and post-multiplying by x, we get 2Re x Xx ; x XBB Xx = 0: Hence x Xx = 0, i.e., X is singular. 2 Example 13.3 This example shows that the observability of (C A) is not necessary for the existence of a positive de nite stabilizing solution. Let " A= 1 0 0 2 # " B= 1 1 # h i C= 0 0 : Then (A B ) is stabilizable, but (C A) is not detectable. However, " # X = 18 ;24 > 0 ;24 36 3 is the stabilizing solution. Corollary 13.8 Suppose that (A B) is stabilizable and (C A) is detectable. Then the Riccati equation A X + XA ; XBB X + C C = 0 has a unique positive semide nite solution. Moreover, the solution is stabilizing.
- 348. 13.2. Stabilizing Solution and Riccati Operator 331 Proof. It is obvious from the above theorem that the Riccati equation has a unique stabilizing solution and that the solution is positive semide nite. Hence we only need to show that any positive semide nite solution X 0 must also be stabilizing. Then by the uniqueness of the stabilizing solution, we can conclude that there is only one positive semide nite solution. To achieve that goal, let us assume that X 0 satis es the Riccati equation but that it is not stabilizing. First rewrite the Riccati equation as (A ; BB X ) X + X (A ; BB X ) + XBB X + C C = 0 (13:17) and let and x be an unstable eigenvalue and the corresponding eigenvector of A ; BB X , respectively, i.e., (A ; BB X )x = x: Now pre-multiply and postmultiply equation (13.17) by x and x, respectively, and we have ( + )x Xx + x (XBB X + CC )x = 0: This implies B Xx = 0 Cx = 0 since Re( ) 0 and X 0. Finally, we arrive at Ax = x Cx = 0 i.e., (C A) is not detectable, which is a contradiction. Hence Re( ) < 0, i.e., X the stabilizing solution. 0 is 2 Lemma 13.9 Suppose D has full column rank and let R = D D > 0 then the following statements are equivalent: " (i) A ; j!I B C D # has full column rank for all !. ; (ii) (I ; DR;1 D )C A ; BR;1 D C has no unobservable modes on j!-axis. ; Proof. Suppose j! is an unobservable mode of (I ; DR;1D )C A ; BR;1D C then there is an x = 0 such that 6 (A ; BR;1 D C )x = j!x (I ; DR;1 D )Cx = 0 i.e., " A ; j!I B C D #" I 0 ;R;1D C I #" # x = 0: 0
- 349. ALGEBRAIC RICCATI EQUATIONS 332 But this implies that " A ; j!I B C D # (13:18) does not have full column rank. Conversely, suppose (13.18) does not have full column " # u 6= 0 such that v rank for some ! then there exists " Now let " Then " and A ; j!I B C D # #" " # u = 0: v u = I 0 v ;R;1D C I # " x = I 0 y R;1D C I #" #" # x : y # u 6= 0 v (A ; BR;1 D C ; j!I )x + By = 0 (I ; DR;1D )Cx + Dy = 0: Pre-multiply (13.20) by D to get y = 0. Then we have (A ; BR;1 D C )x = j!x (I ; DR;1 D )Cx = 0 ; i.e., j! is an unobservable mode of (I ; DR;1 D )C A ; BR;1 D C . (13:19) (13:20) 2 h i Remark 13.3 If D is not square, then there is a D? such that D? DR;1=2 is unitary and that D? D? = I ;DR;1 D . Hence, in some cases we will write the condition (ii) in the above lemma as (D? C A ; BR;1 D C ) having no imaginary unobservable modes. Of course, if D is square, the condition is simpli ed to A ; BR;1 D C with no imaginary eigenvalues. Note also that if D C = 0, condition (ii) becomes (C A) with no imaginary unobservable modes. ~ Corollary 13.10 Suppose D has full column rank and denote R = D D > 0. Let H have the form H = " " = # " # h A 0 ; B R;1 D C B ;C C ;A ;C D # A ; BR;1 D C ;BR;1 B : ;C (I ; DR;1 D )C ;(A ; BR;1 D C ) i
- 350. 13.3. Extreme Solutions and Matrix Inequalities " 333 # A ; j!I B has full column rank Then H 2 dom(Ric) i (A B ) is stabilizable and C D for all !. Furthermore, X = Ric(H ) 0 if H 2 dom(Ric), and Ker(X ) = 0 if and only if (D? C A ; BR;1 D C ) has no stable unobservable modes. Proof. This is the consequence of the Lemma 13.9 and Theorem 13.7. 2 Remark 13.4 It is easy to see that the detectability (observability) of (D?C A ; BR;1 D C ) implies the detectability (observability) of (C A) however, the converse is in general not true. Hence the existence of a stabilizing solution to the Riccati equation in the above corollary is not guaranteed by the stabilizability of (A B ) and detectability of (C A). Furthermore, even if a stabilizing solution exists, the positive de niteness of the solution is not guaranteed by the observability of (C A) unless D C = 0. As an example, consider " # " # " # " # 0 1 0 1 0 1 A= B= C= D= : 0 0 ;1 0 0 0 Then (C A) is observable, (A B ) is controllable, and " # h i 0 1 A ; BD C = D? C = 0 0 : 1 0 A Riccati equation with the above data has a nonnegative de nite stabilizing solution since (D? C A ; BR;1 D C ) has no unobservable modes on the imaginary axis. However, the solution is not positive de nite since (D? C A ; BR;1 D C ) has a stable unobservable mode. On the other hand, if the B matrix is changed to " # 0 B= 1 then the corresponding Riccati equation has no stabilizing solution since, in this case, (A ; BD C ) has eigenvalues on the imaginary axis although (A B ) is controllable and (C A) is observable. ~ 13.3 Extreme Solutions and Matrix Inequalities We have shown in the previous sections that given a Riccati equation, there are generally many possible solutions. Among all solutions, we are most interested in those which are real, symmetric, and, in particular, the stabilizing solutions. There is another class of solutions which are interesting they are called extreme (maximal or minimal) solutions. Some properties of the extreme solutions will be studied in this section. The connections between the extreme solutions and the stabilizing solutions will also be established in this section. To illustrate the idea, let us look at an example.
- 351. ALGEBRAIC RICCATI EQUATIONS 334 Example 13.4 Let 2 A=6 4 1 0 0 0 2 0 0 0 ;3 3 2 7 5 B=6 4 " 1 1 1 A ;BB The eigenvalues of matrix H = ;C C ;A 3 h 7 5 i C= 0 0 0 : # are = 1 2 = ;1 3 = 2 4 = ;2 5 = ;3 6 = 3 and their corresponding eigenvectors are 3 2 3 2 3 2 3 2 3 2 3 2 ;3 7 0 4 0 3 1 6 7 6 7 6 7 6 7 6 7 6 6 ;6 7 6 0 7 6 3 7 6 1 7 6 2 7 6 0 7 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 6 0 7 6 ;3 7 6 0 7 6 ;1 7 6 1 7 6 ;12 7 7 6 7 6 7 6 v1 = 6 7 v2 = 6 7: 7 v6 = 6 7 v5 = 6 7 v4 = 6 7 v3 = 6 6 0 7 6 0 7 6 0 7 6 0 7 6 6 7 6 0 7 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 6 0 7 6 0 7 4 0 5 4 0 5 4 12 5 4 0 5 5 4 5 4 6 0 0 0 0 0 There are four distinct nonnegative de nite symmetric solutions depending on the chosen invariant subspaces: 2 3 " # 0 0 0 X1 = v v v ] Y = X X ;1 = 6 7 4 0 0 0 5 1 3 5 1 2 1 X2 0 0 0 1 " " # 2 X1 = v v v ] Y = X X ;1 = 6 4 2 3 5 2 2 1 X2 # 2 X1 = v v v ] Y = X X ;1 = 6 4 1 4 5 3 2 1 X2 2 2 0 0 0 0 0 0 0 0 3 0 0 0 0 4 0 0 0 0 3 7 5 7 5 3 18 ;24 0 X1 = v v v ] Y = X X ;1 = 6 36 0 7. 4 ;24 5 2 4 5 4 2 1 X2 0 0 0 These solutions can be ordered as Y4 Yi Y1 i = 2 3. Of course, this is only a partial ordering since Y2 and Y3 are not comparable. Note also that only Y4 is a stabilizing solution, i.e., A ; BB Y4 is stable. Furthermore, Y4 and Y1 are the maximal" and minimal" solutions, respectively. 3 " #
- 352. 13.3. Extreme Solutions and Matrix Inequalities 335 The partial ordering concept shown in Example 13.4 can be stated in a much more general setting. To do that, again consider the Riccati equation (13.1). We shall call a hermitian solution X+ of (13.1) a maximal solution if X+ X for all hermitian solutions X of (13.1). Similarly, we shall call a hermitian solution X; of (13.1) a minimal solution if X; X for all hermitian solutions X of (13.1). Clearly, maximal and minimal solutions are unique if they exist. To study the properties of the maximal and minimal solutions, we shall introduce the following quadratic matrix: Q(X ) := A X + XA + XRX + Q: (13:21) Theorem 13.11 Assume R 0 and assume there is a hermitian matrix X = X such that Q(X ) 0. (i) If (A R) is stabilizable, then there exists a unique maximal solution X+ to the Riccati equation (13.1). Furthermore, X+ X 8 X such that Q(X ) 0 and (A + RX+ ) C ; . (ii) If (;A R) is stabilizable, then there exists a unique minimal solution X; to the Riccati equation (13.1). Furthermore, X; X 8 X such that Q(X ) 0 and (A + RX; ) C + . (iii) If (A R) is controllable, then both X+ and X; exist. Furthermore, X+ > X; i (A + RX+ ) C ; i (A + RX;) C + . In this case, X+ ; X; = = 1 Z Z 0 0 ;1 e(A+RX+ )t Re(A+RX+ ) t dt e(A+RX; )t Re(A+RX; ) t dt ;1 ;1 : (iv) If Q(X ) > 0, the results in (i) and (ii) can be respectively strengthened to X+ > X , (A + RX+ ) C ; , and X; < X , (A + RX; ) C + . Proof. Let R = ;BB for some B. Note the fact that (A R) is stabilizable (controllable) i (A B ) is. (i): Let X be such that Q(X ) 0. Since (A B ) is stabilizable, there is an F0 such that A0 := A + BF0
- 353. ALGEBRAIC RICCATI EQUATIONS 336 is stable. Now let X0 be the unique solution to the Lyapunov equation X0 A0 + A0 X0 + F0 F0 + Q = 0: Then X0 is hermitian. De ne ^ F0 := F0 + B X and we have the following equation: ^ ^ (X0 ; X )A0 + A0 (X0 ; X ) = ;F0 F0 ; Q(X ) 0: The stability of A0 implies that X0 X: Starting with X0 , we shall de ne a non-increasing sequence of hermitian matrices fXi g. Associated with fXi g, we shall also de ne a sequence of stable matrices fAi g and a sequence of matrices fFi g. Assume inductively that we have already de ned matrices fXi g, fAi g, and fFi g for i up to n ; 1 such that Xi is hermitian and X0 X1 Next, introduce Xn;1 X Ai = A + BFi is stable i = 0 : : : n ; 1 Fi = ;B Xi;1 i = 1 : : : n ; 1 Xi Ai + Ai Xi = ;Fi Fi ; Q i = 0 1 : : : n ; 1: (13:22) Fn = ;B Xn;1 An = A + BFn : First we show that An is stable. Then, using (13.22), with i = n, we de ne a hermitian matrix Xn with Xn;1 Xn X . Now using (13.22), with i = n ; 1, we get Xn;1 An + An Xn;1 + Q + Fn Fn + (Fn ; Fn;1 ) (Fn ; Fn;1 ) = 0: Let (13:23) ^ Fn := Fn + B X then ^ ^ (Xn;1 ; X )An + An (Xn;1 ; X ) = ;Q(X ) ; Fn Fn ; (Fn ; Fn;1 ) (Fn ; Fn;1 ): (13:24) Now assume that An is not stable, i.e., there exists an with Re 0 and x 6= 0 such that An x = x. Then pre-multiply (13.24) by x and postmultiply by x, and we have ^ ^ 2Re x (Xn;1 ; X )x = ;x fQ(X ) + Fn Fn + (Fn ; Fn;1 ) (Fn ; Fn;1 )gx:
- 354. 13.3. Extreme Solutions and Matrix Inequalities 337 Since it is assumed Xn;1 X , each term on the right-hand side of the above equation has to be zero. So we have x (Fn ; Fn;1 ) (Fn ; Fn;1 )x = 0: This implies (Fn ; Fn;1 )x = 0: But now An;1 x = (A + BFn;1 )x = (A + BFn )x = An x = x which is a contradiction with the stability of An;1 . Hence An is stable as well. Now we introduce Xn as the unique solution of the Lyapunov equation Xn An + A Xn = ;Fn Fn ; Q: (13:25) Then Xn is hermitian. Next, we have ^ ^ (Xn ; X )An + An (Xn ; X ) = ;Q(X ) ; Fn Fn 0 and, by using (13.23), (Xn;1 ; Xn )An + An (Xn;1 ; Xn ) = ;(Fn ; Fn;1 ) (Fn ; Fn;1 ) 0: Since An is stable, we have Xn;1 Xn X: We have a non-increasing sequence fXi g, and the sequence is bounded below by Xi X . Hence the limit Xf := nlim Xn !1 exists and is hermitian, and we have Xf X . Passing the limit n ! 1 in (13.25), we get Q(Xf ) = 0. So Xf is a solution of (13.1). Since X is an arbitrary element satisfying Q(X ) 0 and Xf is independent of the choice of X , we have Xf X 8 X such that Q(X ) 0: In particular, Xf is the maximal solution of the Riccati equation (13.1), i.e., Xf = X+ . To establish the stability property of the maximal solution, note that An is stable for any n. Hence, in the limit, the eigenvalues of A ; BB Xf will have non-positive real parts. The uniqueness follows from the fact that the maximal solution is unique. (ii): The results follow by the following substitutions in the proof of part (i): A ;A X ;X X+ ;X; :
- 355. ALGEBRAIC RICCATI EQUATIONS 338 (iii): The existence of X+ and X; follows from (i) and (ii). Let A+ := A + RX+ we now show that X+ ; X; > 0 i (A+ ) C ; . It is easy to verify that A+ (X+ ; X; ) + (X+ ; X; )A+ ; (X+ ; X; )R(X+ ; X; ) = 0: ()): Suppose X+ ; X; > 0 then (A+ R(X+ ; X; )) is controllable. Therefore, from Lyapunov theorem, we conclude that A+ is stable. ((): Assuming now that A+ is stable and that X is any other solution to the Riccati equation (13.1), A+ (X+ ; X ) + (X+ ; X )A+ ; (X+ ; X )R(X+ ; X ) = 0: This equation has an invertible solution X+ ; X = ; Z 0 1 e(A+RX+ )t Re(A+RX+ ) t dt ;1 > 0: A simple rearrangement of terms gives (A + RX ) (X+ ; X ) + (X+ ; X )(A + RX ) + (X+ ; X )R(X+ ; X ) = 0: Since X+ ; X > 0, we conclude (A + RX ) C + . This in turn implies X = X; . Therefore, X+ > X; . That X+ ; X; > 0 i (A + RX; ) C + follows by analogy. (iv): We shall only show the case for X+ the case for X; follows by analogy. Note that from (i) we have (X+ ; X )A+ + A+ (X+ ; X ) = ;Q(X ) + (X+ ; X )R(X+ ; X ) < 0 (13:26) and X+ ; X 0. Now suppose X+ ; X is singular and there is an x 6= 0 such that (X+ ; X )x = 0. By pre-multiplying (13.26) by x and post-multiplying by x, we get x Q(X )x = 0, a contradiction. The stability of A + RX+ then follows from the Lyapunov theorem. 2 Remark 13.5 The proof given above also gives an iterative procedure to compute the maximal and minimal solutions. For example, to nd the maximal solution, the following procedures can be used: (i) nd F0 such that A0 = A + BF0 is stable (ii) solve Xi : Xi Ai + Ai Xi + Fi Fi + Q = 0 (iii) if kXi ; Xi;1 k =speci ed accuracy, stop. Otherwise go to (iv) (iv) let Fi+1 = ;B Xi and Ai+1 = A + BFi+1 go to (ii). This procedure will converge to the stabilizing solution if the solution exists. ~
- 356. 13.3. Extreme Solutions and Matrix Inequalities 339 Corollary 13.12 Let R 0 and suppose (A R) is controllable and X1 , X2 are two solutions to the Riccati equation (13.1). Then X1 > X2 implies that X+ = X1 , X; = X2 , (A + RX1) C ; and that (A + RX2 ) C + . The following example illustrates that the stabilizability of (A R) is not su cient to guarantee the existence of a minimal hermitian solution of equation (13.1). Example 13.5 Let " A= 0 0 0 ;1 # " R = ;1 0 0 # " # Q= 1 0 : 0 0 0 Then it can be shown that all the hermitian solutions of (13.1) are given by " 1 0 0 0 # " The maximal solution is clearly ;1 " # ; 1 j j2 2 X+ = 1 0 0 0 2 C: # however, there is no minimal solution. 3 The Riccati equation appeared in H1 control, which will be considered in the later part of this book, often has R 0. However, these conditions are only a dual of the above theorem. Corollary 13.13 Assume that R 0 and that there is a hermitian matrix X = X such that Q(X ) 0. (i) If (A R) is stabilizable, then there exists a unique minimal solution X; to the Riccati equation (13.1). Furthermore, X; X 8 X such that Q(X ) 0 and (A + RX; ) C ; . (ii) If (;A R) is stabilizable, then there exists a unique maximal solution X+ to the Riccati equation (13.1). Furthermore, X+ X 8 X such that Q(X ) 0 and (A + RX+ ) C +.
- 357. ALGEBRAIC RICCATI EQUATIONS 340 (iii) If (A R) is controllable, then both X+ and X; exist. Furthermore, X+ > X; i (A + RX; ) C ; i (A + RX+) C + . In this case, X+ ; X; = = 1 Z Z 0 0 ;1 ;1 e(A+RX; )t Re(A+RX; ) t dt ;1 e(A+RX+ )t Re(A+RX+ ) t dt : (iv) If Q(X ) < 0, the results in (i) and (ii) can be respectively strengthened to X+ > X , (A + RX+ ) C + , and X; < X , (A + RX; ) C ; . Proof. The proof is similar to the proof for Theorem 13.11. 2 Theorem 13.11 can be used to derive some comparative results for some Riccati equations. More speci cally, let " and ;1 Hs := A ; BRs;1S ;P + SRs S ; ;BRs 1 B ; ;(A ; BRs 1 S ) " # # ~ ~ Rs 1 ~ ~; ~ ~; ~ ;B Rs 1 B ~ Hs := A~; B~ ~ ;1S~ ~ ~ ~; ~ ;P + S Rs S ;(A ; B Rs 1S ) ~ ~ ~ where P P Rs , and Rs are real symmetric and Rs > 0, Rs > 0. We shall also make use of the following matrices: " # " # ~ ~ P S ~ T := P S T := ~ ~ : S Rs S Rs ~ We denote by X+ and X+ the maximal solution to the Riccati equation associated with ~ s , respectively: Hs and H ; ; ; ; (A ; BRs 1 S ) X + X (A ; BRs 1 S ) ; XBRs 1B X + (P ; SRs 1S ) = 0 (13:27) ~ ~ ~; ~ ~ ~ ~ ~ ~; ~ ~ ~ ~; ~ ~ ~ ~ ~; ~ (A ; B Rs 1 S ) X + X (A ; B Rs 1 S ) ; X B Rs 1B X + (P ; S Rs 1 S ) = 0: (13:28) ~ Recall that JHs and J Hs are hermitian where J is de ned as " # 0 ;I J= : I 0 Let " ;1 ;1 K := JHs = P ; SRs 1S (A ; BRs1 S ) ;S ;B A ; BRs ;BRs " ~ ~ ~ ;1 ~ ~ B ~ ;1S ~ ~ ~ ~ K := J Hs = P ; S Rs 1S (A ;~ ~Rs1 ~ ) ; ~ ; B Rs S ~ ~; ~ A ;BRs B # # :
- 358. 13.3. Extreme Solutions and Matrix Inequalities 341 ~ ~ Theorem 13.14 Suppose (A B) and (A B) are stabilizable. ~ ~ (i) Assume that (13.28) has a hermitian solution and that K K (K > K ) then ~ ~ ~ X+ and X+ exist, and X+ X+ (X+ > X+ ). ~ ~ ~ ~ (ii) Let A = A, B = B , and T T (T > T ). Assume that (13.28) has a hermitian ~+ exist, and X+ X+ (X+ > X+ ). ~ ~ solution. Then X+ and X (iii) If T 0 (T > 0), then X+ exists, and X+ 0 (X+ > 0). Proof. (i): Let X be a hermitian solution of (13:28) then we have " I X # " # ~ K I = 0: X ~ K , we have " # " # " I K I = I Qs (X ) := X X X Then, since K # " # ~ I (K ; K ) 0: X Now we use Theorem 13.11 to obtain the existence of X+ , and X+ X . Since X is ~ arbitrary, we get X+ X for all hermitian solutions X of (13.28). The existence of X+ follows immediately by applying Theorem 13.11. Moreover, ~ X+ X+ : ~ ~ (ii): Let A = A, B = B and let X be a hermitian solution of (13.28). Denote ;1 (S + B X ) and L = ;Rs ;1 (S + B X ). Then ~ ~ ~ L = ;Rs " Qs (X ) = I X while (13.28) becomes It is easy to show that # " # K I = A X + XA ; L Rs L + P X ~ ~ ~ ~ A X + XA ; L Rs L + P = 0: " I X # " # ~ I Qs (X ) = (K ; K ) X ~ ~ = P ; P ; L Rs L " L # s L +~ R ~ " # I (T ; T ) I + (L ; L) R (L ; L) 0: ~ ~ ~s ~ = L L ~ ~ Now as in part (i), there exist X+ and X+ , and X+ X+ . ;1 S (iii): The condition T 0 implies P ; SRs 0, so we have Qs (0) = P ; ; SRs 1 S 0. Apply Theorem 13.11 to get the existence of X+ , and X+ 0. 2
- 359. ALGEBRAIC RICCATI EQUATIONS 342 13.4 Spectral Factorizations Let A B P S R be real matrices of compatible dimensions such that P = P , R = R , and de ne a parahermitian rational matrix function (s) = R + S (sI ; A);1 B + B (;sI ; A );1 S + B (;sI ; A );1 P (sI ; A);1 B " #" # h i P S (sI ; A);1 B = B (;sI ; A );1 I : (13.29) S R I Lemma 13.15 Suppose R is nonsingular and either one of the following assumptions is satis ed: (A1) A has no eigenvalues on j!-axis (A2) P is sign de nite, i.e., P 0 or P 0, (A B ) has no uncontrollable modes on j!-axis, and (P A) has no unobservable modes on the j!-axis. Then the following statements are equivalent: (i) (j!0 ) is singular for some 0 !0 1. (ii) The Hamiltonian matrix " # ;1 ;BR;1B H = A ; BR ;1S ;(P ; SR S ) ;(A ; BR;1 S ) has an eigenvalue at j!0. Proof. (i) ) (ii): Let " ^ A (s) = ^ C 2 # A 0 ^ B := 6 ;P ;A 4 ^ D B ;S R " ;BR;1 S B ^ ^^ ^ Then H = A ; B D;1 C and 2 ;1 (s) = 6 6 4 h H R;1S R;1 B i SR;1 R;1 3 7 5 : # 3 7 7 5 : If (j!0 ) is singular, then j!0 is a zero of (s). Hence j!0 is a pole of ;1 (s), and j!0 is an eigenvalue of H .
- 360. 13.4. Spectral Factorizations 343 (ii) ) (i): Suppose j!0 is an eigenvalue ofhH but is not a pole of ;1 (s). Then i j!0 must be either an unobservable mode of ( R;1S R;1 B H ) or an uncon" # ;BR;1 ). Now suppose j! is an unobservable mode of trollable mode of (H 0 ;1 h ( R;1 S R;1 B i SR " # H ). Then there exists an x0 = x1 6= 0 such that x2 Hx0 = j!0 x0 h R ;1 S R;1 B i x0 = 0: These equations can be simpli ed to (j!0 I ; A)x1 = 0 (13:30) (j!0 I + A )x2 = ;Px1 (13:31) S x1 + B x2 = 0: (13:32) We now consider two cases under the di erent assumptions: (a) If assumption (A1) is true, then x1 = 0 from (13.30), and this, in turn, implies x2 = 0 from (13.31), which is a contradiction. (b) If assumption (A2) is true, since (13.30) implies x1 (j!0 I + A ) = 0, from (13.31) we have x1 Px1 = 0. This gives Px1 = 0 since P is sign de nite (P 0 or P 0). This implies, along with (13.30) that j!0 is an unobservable mode of (P A) if x1 6= 0. On the other hand, if x1 = 0, then (13.31) and (13.32) imply that (A B ) has an uncontrollable mode at j!0 , again a contradiction. Similarly, a contradiction will also be derived if j!0 is assumed to be an uncontrol" # ;BR;1 ). lable mode of (H 2 ;1 SR Corollary 13.16 Suppose R > 0 and either one of the assumptions (A1) or (A2) de ned in Lemma 13.15 is true. Then the following statements are equivalent: (i) (j!) > 0 for all 0 ! 1. (ii) The Hamiltonian matrix " ;1 ;BR;1B H = A ; BR ;1S ;(P ; SR S ) ;(A ; BR;1 S ) has no eigenvalue on j!-axis. #
- 361. ALGEBRAIC RICCATI EQUATIONS 344 Proof. (i) ) (ii) follows easily from Lemma 13.15. To prove (ii) ) (i), note that (j 1) = R > 0 and det (j!) 6= 0 for all ! from Lemma 13.15. Then the continuity of 2 (j!) gives (j!) > 0. Lemma 13.17 Suppose A is stable and P 0. Then (i) the matrix 0 (s) = h B (;sI ; A );1 I satis es i " P 0 0 I #" (sI ; A);1 B # I for all 0 ! 1 if and only if there exists a unique real X = X 0 such that A X + XA ; XBB X + P = 0 and (A ; BB X ) C ; . (ii) 0 (j!) 0 for all 0 ! 1 if and only if there exists a unique real X = X 0 such that A X + XA ; XBB X + P = 0 (13:33) and (A ; BB X ) C ; . 0 (j! ) > 0 Proof. (i): ()) From Corollary 13.16, 0(j!) > 0 implies that a Hamiltonian matrix " # A ;BB ;P ;A has no eigenvalues on the imaginary axis. This in turn implies from Theorem 13.6 that (13.33) has a stabilizing solution. Furthermore, since " we have A ;BB ;P ;A " # " ;I = Ric A ;BB ;P ;A # #" I A BB P ;A " #" A BB = ;Ric P ;A # ;I # I 0: (() The su ciency proof is omitted here and is a special case of (b) ) (a) of Theorem 13.19 below. (ii): ()) Let P = ;C C and G(s) := C (sI ; A);1 B . Then 0 (s) = I ; G (s)G(s). Let 0 < < 1 and de ne (s) := I ; 2 G (s)G(s):
- 362. 13.4. Spectral Factorizations 345 Then (j!) > 0 8!. Thus from part (i), there is an X = X 0 such that A ; BB X is stable and A X + X A ; X BB X ; 2 C C = 0: It is easy to see from Theorem 13.14 that X is monotone-decreasing with , i.e., X 1 X 2 if 1 2 . To show that lim !1 X exists, we need to show that X is bounded below for all 0 < < 1. In the following, it will be assumed that (A B ) is controllable. The controllability assumption will be removed later. Let Y be the destabilizing solution to the following Riccati equation: A Y + Y A ; Y BB Y = 0 with (A ; BB Y ) C + (note that the existence of such a solution is guaranteed by the controllability assumption). Then it is easy to verify that (A ; BB Y ) (X ; Y ) + (X ; Y )(A ; BB Y ) ; (X ; Y )BB (X ; Y ) ; 2C C = 0: This implies that X ;Y = 1 Z 0 e;(A;BB Y ) t (X ; Y )BB (X ; Y ) + 2 C C ]e;(A;BB Y )t dt 0: Thus X is bounded below with Y as the lower bound, and lim !1 X exists. Let X := lim !1 X then from continuity argument, X satis es the Riccati equation A X + XA ; XBB X + P = 0 and (A ; BB X ) C ; . Now suppose (A B ) is not controllable, and then assume without loss of generality that " # " # h i A11 A12 B1 A= B= C = C1 C2 0 A22 0 so that (A11 B1 ) is controllable and A11 and A22 are stable. Then the Riccati equation for # " X11 X12 X = (X12 ) X22 can be written as three equations A11 X11 + X11 A11 ; X11 B1 B1 X11 ; 2 C1 C1 = 0 (A11 ; B1 B1 X11 ) X12 + X12 A22 + X11 A12 ; 2 C1 C2 = 0 A22 X22 + X22 A22 + (X12 ) A12 + A12 X12 ; (X12 ) B1 B1 X12 ; 2 C2 C2 = 0
- 363. ALGEBRAIC RICCATI EQUATIONS 346 and " A ; BB X = A11 ; B1 B1 X11 A12 ; B1 B1 X12 0 A22 # is stable. Let Y11 be the anti-stabilizing solution to A11 Y11 + Y11 A11 ; Y11 B1 B1 Y11 = 0 with (A11 ; B1 B1 Y11 ) C + . Then it is clear that X11 ; Y11 lim !1 X11 exists and satis es the following Riccati equation: 0. Moreover, X11 = A11 X11 + X11 A11 ; X11 B1 B1 X11 ; C1 C1 = 0 with (A11 ; B1 B1 X11 ) C ;. Consequently, the following Sylvester equation (A11 ; B1 B1 X11 ) X12 + X12 A22 + X11 A12 ; C1 C2 = 0 has a unique solution X12 since i (A11 ; B1 B1 X11 ) + j (A22 ) 6= 0 8i j . Furthermore, the following Lyapunov equation has a unique nonnegative de nite solution X22 : A22 X22 + X22 A22 + X12 A12 + A12 X12 ; X12 B1 B1 X12 ; C2 C2 = 0: We have proven that there exists a unique X such that A X + XA ; XBB X ; C C = 0 and (A ; BB X ) C ; . (() same as in part (i). 2 Lemma 13.18 Let R > 0, h (s) = B (;sI ; A );1 I and where h ^ (s) = B (;sI ; A );1 I ^ ^ i " i " P S S R ^ P 0 0 I #" (sI ; A);1 B # I #" ^ ^ (sI ; A);1 B I ^ ^ ^ A := A ; BR;1 S B := BR;1=2 P := P ; SR;1 S : Then (j!) 0 i ^ (j!) 0. #
- 364. 13.4. Spectral Factorizations Proof. Note that " # " P S = I SR;1=2 S R 0 R1=2 347 #" ^ P 0 0 I #" I # 0 R;1=2 S R 1= 2 : Hence the function (s) can be written as h (s) = B (;sI ; A );1 ' (s) i " ^ P 0 0 I #" (sI ; A);1 B '(s) # with '(s) = R1=2 + R;1=2 S (sI ; A);1 B . It is easy to verify that ^ (s) = ';1 (s)] (s)';1 (s): 2 Hence the conclusion follows. Now we are ready to state and prove one of the main results of this section. The following theorem and corollary characterize the relations among spectral factorizations, Riccati equations, and decomposition of Hamiltonians. Theorem 13.19 Let A B P S R be matrices of compatible dimensions such that P = P , R = R > 0, with (A B ) stabilizable. Suppose either one of the following assump- tions is satis ed: (A1) A has no eigenvalues on j!-axis (A2) P is sign de nite, i.e., P 0 or P 0 and (P A) has no unobservable modes on the j!-axis. Then (I) The following statements are equivalent: (a) The parahermitian rational matrix h (s) = B (;sI ; A );1 I i " P S S R #" (sI ; A);1 B # I satis es (j!) > 0 for all 0 ! 1: (b) There exists a unique real X = X such that (A ; BR;1 S ) X + X (A ; BR;1 S ) ; XBR;1B X + P ; SR;1S = 0 and that A ; BR;1 S ; BR;1 B X is stable.
- 365. ALGEBRAIC RICCATI EQUATIONS 348 (c) The Hamiltonian matrix " ;1 S ;BR;1B H = A ; BR ;1 ;(P ; SR S ) ;(A ; BR;1 S ) # has no j!-axis eigenvalues. (II) The following statements are also equivalent: (d) (j!) 0 for all 0 ! 1. (e) There exists a unique real X = X such that (A ; BR;1 S ) X + X (A ; BR;1 S ) ; XBR;1B X + P ; SR;1S = 0 and that (A ; BR;1 S ; BR;1 B X ) C ;. The following corollary is usually referred to as the spectral factorization theory. Corollary 13.20 If either one of the conditions, (a){(e), in Theorem 13.19 is satis ed, then there exists an M 2 Rp such that = M RM: A particular realization of one such M is " M= A B ;F I # where F = ;R;1(S + B X ). Furthermore, if X is the stabilizing solution, then M ;1 2 RH1 . Remark 13.6 If the stabilizability of (A B) is changed into the stabilizability of (;A B), then the theorem still holds except that the solutions X in (b) and (e) are changed into the destabilizing solution ( (A ; BR;1 S ; BR;1 B X ) C + ) and the weakly destabilizing solution ( (A ; BR;1 S ; BR;1 B X ) C + ), respectively. ~ Proof. (a) ) (c) follows from Corollary 13.16. (c) ) (b) follows from Theorem 13.6 and Theorem 13.5. (b) ) (a) Suppose 9X = X such that A ; BR;1S ; BR;1 B X = A ; BR;1 (S + B X ) is stable. Let F = ;R;1(S + B X ) and " # M= A B : ;F I
- 366. 13.4. Spectral Factorizations 349 It is easily veri ed by use of the Riccati equation for X and by routine algebra that = M RM . Since " # A + BF B M ;1 = F I M ;1 2 RH1 . Thus M (s) has no zeros on the imaginary axis and (j!) > 0. (e) ) (d) follows the same procedure as the proof of (b) ) (a). (d) ) (e): Assume S = 0 and R = I otherwise use Lemma 13.18 rst to get a new function with such properties. Let P = C1 C1 ; C2 C2 with C1 and C2 square nonsingular. Note that this decomposition always exists since P = I ; ( I ; P ), with > 0 su ciently large, de nes one such possibility. Let X1 be the positive de nite solution to A X1 + X1 A ; X1 BB X1 + C1 C1 = 0: By Theorem 13.7, X1 indeed exists and is stabilizing, i.e., A1 := A ; BB X1 is stable. Let = X ; X1 . Then the equation in becomes A1 + A1 ; BB ; C2 C2 = 0: To show that this equation has a solution, recall Lemma 13.17 and note that A1 is stable then it is su cient to show that I ; B (;j!I ; A1 );1 C2 C2 (j!I ; A1 );1 B is positive semi-de nite. Notice rst that C2 (sI ; A + BB X1 );1 B = C2 (sI ; A);1 B I + B X1 (sI ; A);1 B ;1 : From the de nition of X1 and Corollary 13.20, we also have that I + B (;sI ; A );1 C1 C1 (sI ; A);1 B Now, by assumption = I + B X1 (sI ; A);1 B I + B X1 (sI ; A);1 B : (j!) = I + B (;j!I ; A );1 C1 C1 (j!I ; A);1 B ;B (;j!I ; A );1 C2 C2 (j!I ; A);1 B 0: It follows that n o I ; I + B (;j!I ; A );1 X1 B ;1 B (;j!I ; A );1 C2 n o C2 (j!I ; A);1 B I + B X1 (j!I ; A);1 B ;1 0:
- 367. ALGEBRAIC RICCATI EQUATIONS 350 Consequently, I ; B (;j!I ; A1 );1 C2 C2 (j!I ; A1 );1 B 0: We may now apply Lemma 13.17 to the equation. Consequently, there exists a unique solution such that (A1 ; BB ) = (A ; BB X ) C ; . This shows the existence and uniqueness of X . 2 We shall now illustrate the proceeding results through a simple example. Note in particular that the function can have poles on the imaginary axis. Example 13.6 Let A = 0 B = 1 R = 1 S = 0 and P = 1. Then (s) = 1 ; s12 and (j!) > 0. It is easy to check that the Hamiltonian matrix " H = 0 ;1 ;1 0 # does not have eigenvalues on the imaginary axis and X = 1 is the stabilizing solution to the corresponding Riccati equation and the spectral factor is given by # " M (s) = 0 1 = s + 1 : s 1 1 3 Some frequently used special spectral factorizations are now considered. " A B C D # Corollary 13.21 Assume that G(s) := 2 RL1 is a stabilizable and detectable realization and > kG(s)k1 . Then, there exists a transfer matrix M 2 RL1 such that M M = 2 I ; G G and M ;1 2 RH1 . A particular realization of M is " B A M (s) = 1=2 F R1=2 ;R # where R = 2I ; D D F = R;1"B X + D C ) ( A + BR;1 D C BR;1B X = Ric ;C (I + DR;1 D )C ;(A + BR;1 D C ) and X 0 if A is stable. #
- 368. 13.4. Spectral Factorizations 351 Proof. This is a special case of Theorem 13.19. In fact, the theorem follows by letting P = ;C C S = ;C D R = 2I ; D D in Theorem 13.19 and by using the fact that " A + BR;1 D C BR;1 B Ric ;1 D )C ;(A + BR;1 D C ) ;C (I + DR # = " ;1 ;BR;1 B ;Ric A + BR ;D C 1 D )C ;(A + BR;1 D C ) C (I + DR # : 2 The spectral factorization for the dual case is also often used and follows by taking the transpose of the corresponding transfer matrices. " # Corollary 13.22 Assume G(s) := A B 2 RL1 and > kG(s)k1. Then, there C D exists a transfer matrix M 2 RL1 such that MM = 2 I ; GG and M ;1 2 RH1 . A particular realization of M is " 1=2 M (s) = A ;LR 2 1= C R # where and Y R = 2 I ; DD L = (Y C" + BD )R;1 ;1 Y = Ric (A + BD R;1 C ) ;B (I + D R D)B 0 if A is stable. C R;1 C ;(A + BD R;1 C ) # For convenience, we also include the following spectral factorization results which are again special cases of Theorem 13.19. " # Corollary 13.23 Let G(s) = A B be a stabilizable and detectable realization. C D " # A ; j! B has full column rank for (a) Suppose G (j!)G(j!) > 0 for all ! or C D all !. Let " # A ; BR;1 D C ;BR;1 B X = Ric ;C (I ; DR;1D )C ;(A ; BR;1 D C )
- 369. ALGEBRAIC RICCATI EQUATIONS 352 with R := D D > 0. Then we have the following spectral factorization W W =G G where W ;1 2 RH1 and # " B : W = ;1=2 A R (D C + B X ) R1=2 " # A ; j! B has full row rank for all (b) Suppose G(j!)G (j!) > 0 for all ! or C D !. Let " # ~ ~ (A ; BD R;1 C ) ;C R;1 C Y = Ric ~ ~ ;B (I ; D R;1 D)B ;(A ; BD R;1 C ) ~ with R := DD > 0. Then we have the following spectral factorization ~ ~ W W = GG ~ where W ;1 2 RH1 and # " Y ~ ;1=2 : ~ W = A (BD + ~ 1C2 )R R= C Theorem 13.19 also gives some additional characterizations of a transfer matrix H1 norm. " # Corollary 13.24 Let > 0, G(s) = A B 2 RH1 and C D " A + BR;1 D C BR;1 B H := ;C (I + DR;1D )C ;(A + BR;1 D C ) # where R = 2 I ; D D. Then the following conditions are equivalent: (i) kGk1 < . (ii) (D) < and H has no eigenvalues on the imaginary axis. (iii) (D) < and H 2 dom(Ric) . (iv) (D) < and H 2 dom(Ric) and Ric(H ) able). 0 (Ric(H ) > 0 if (C A) is observ-
- 370. 13.5. Positive Real Functions 353 Proof. This follows from the fact that kGk1 < is equivalent to that the following function is positive de nite for all !: (j!) := 2 I ; GT (;j!)G(j!) h i = B (;j!I ; A );1 I " ;C C ;D C ;C D 2I ; D D #" (j!I ; A);1 B # I >0 and the fact that " # " # ;I 0 H ;I 0 = 0 I 0 I " A + BR;1D C ;BR;1 B C (I + DR;1 D )C ;(A + BR;1 D C ) # : 2 The equivalence between (i) and (iv) in the above corollary is usually referred as Bounded Real Lemma. 13.5 Positive Real Functions A square (m m) matrix function G(s) 2 RH1 is said to be positive real (PR) if G(j!) + G (j!) 0 for all nite !, i.e., ! 2 R, and G(s) is said to be strictly positive real (SPR) if G(j!) + G (j!) > 0 for all ! 2 R. " # Theorem 13.25 Let A B be a state space realization of G(s) with A stable (not C D necessarily a minimal realization). Suppose there exist an X 0, Q, and W such that XA + A X = ;Q Q B X +W Q = C D+D = W W (13.34) (13.35) (13.36) Then G(s) is positive real and G(s) + G (s) = M (s)M (s) " # A B . Furthermore, if M (j!) has full column rank for all ! 2 R, with M (s) = Q W then G(s) is strictly positive real.
- 371. ALGEBRAIC RICCATI EQUATIONS 354 Proof. 2 A B 6 G(s)+G (s) = 4 0 ;A ;C C B D+D 0 3 2 7 5 =6 4 " A 0 0 ;A B X +W Q B B 3 ;(XB + Q W ) 7 : 5 W W # I 0 to the last realization to get Apply a similarity transformation X I 2 B 6 G(s) + G (s) = 4 XA + A X ;A ;Q W W Q B W W #" # " A B : ;A ;Q = B W Q W A 0 3 2 B A 0 7 6 ;Q W 5 = 4 ;Q Q ;A W Q B W W 3 7 5 This implies that G(j!) + G (j!) = M (j!)M (j!) 0 i.e., G(s) is positive real. Finally note that if M (j!) has full column rank for all ! 2 R, then M (j!)x 6= 0 for all x 2 C m and ! 2 R. Thus G(s) is strictly positive real. 2 Theorem 13.26 Suppose (A B C D) is a minimal realization of G(s) with A stable and G(s) is positive real. Then there exist an X 0, Q, and W such that XA + A X = ;Q Q B X +W Q = C D + D = W W: and " # G(s) + G (s) = M (s)M (s) A B . Furthermore, if G(s) is strictly positive real, then M (j!) Q W given above has full column rank for all ! 2 R. with M (s) = Proof. Since G(s) is assumed to be positive real, there exists a transfer matrix M (s) = # " A1 B1 C1 D1 with A1 stable such that G(s) + G (s) = M (s)M (s)
- 372. 13.5. Positive Real Functions 355 where A and A1 have the same dimensions. Now let X1 following Lyapunov equation: X1 A1 + A1 X1 = ;C1 C1 : Then M (s)M (s) = " ;A1 ;C1 B1 2 D1 #" A1 B1 C1 D1 A1 # 0 be the solution of the 2 B1 A1 0 6 = 4 ;C1 C1 ;A1 ;C1 D1 D1 C1 B1 D1 D1 3 7 5 3 B1 = 6 X1 A1 + A1 X1 ;A1 ;C1 D1 7 5 4 D1 C1 B1 D1 D1 2 3 A1 0 B1 = 6 0 ;A1 ;(X1 B1 + C1 D1 ) 7 4 5 B1 X + D1 C1 B1 D1 D1 " # " # A1 B1 + ;A1 ;(B1 X + D1 C1 ) : = D1 D1 + B1 X + D1 C1 0 B1 0 But the realization for G(s) + G (s) is given by # " # " A B + ;A ;C : G(s) + G (s) = D + D + C 0 B 0 Since the realization for G(s) is minimal, there exists a nonsingular matrix T such that A = TA1T ;1 B = TB1 C = (B1 X + D1 C1 )T ;1 D + D = D1 D1 : 0 Now the conclusion follows by de ning X = (T ;1) X1 T ;1 " W = D1 Q = C1 T ;1: # 2 Corollary 13.27 Let A B be a state space realization of G(s) 2 RH1 with A C D stable and R := D + D > 0. Then G(s) is strictly positive real if and only if there exists a stabilizing solution to the following Riccati equation: X (A ; BR;1 C ) + (A ; BR;1 C ) X + XBR;1 B X + C R;1 C = 0: " A # B Moreover, M (s) = 1 ; 1 (C ; B X ) R 2 is minimal phase and 2 R G(s) + G (s) = M (s)M (s):
- 373. ALGEBRAIC RICCATI EQUATIONS 356 Proof. This follows from Theorem 13.19 and from the fact that G(j!) + G (j!) > 0 for all ! including 1. 2 The above corollary also leads to the following special spectral factorization. " # Corollary 13.28 Let G(s) = A B 2 RH1 with D full row rank and G(j!)G (j!) > C D 0 for all !. Let P be the controllability grammian of (A B ): PA + AP + BB = 0: De ne BW = PC + BD : Then there exists an X 0 such that XA + A X + (C ; BW X ) (DD );1 (C ; BW X ) = 0: Furthermore, there is an M (s) 2 RH1 such that M ;1 (s) 2 RH1 and G(s)G (s) = M (s)M (s) " A BW where M (s) = CW DW # with a square matrix DW such that DW DW = DD and CW = DW (DD );1 (C ; BW X ): Proof. This corollary follows from Corollary 13.27 and the fact that G(j!)G (j!) > 0 and " G(s)G (s) = A BW C 0 # " ;A ;C + BW 0 A dual spectral factorization can also be obtained easily. # + DD : 2
- 374. 13.6. Inner Functions 357 13.6 Inner Functions A transfer function N is called inner if N 2 RH1 and N N = I and co-inner if N 2 RH1 and NN = I . Note that N need not be square. Inner and co-inner are dual notions, i.e., N is an inner i N T is a co-inner. A matrix function N 2 RL1 is called all-pass if N is square and N N = I clearly a square inner function is all-pass. We will focus on the characterizations of inner functions here and the properties of co-inner functions follow by duality. Note that N inner implies that N has at least as many rows as columns. For N inner and any q 2 C m , v 2 L2 , kN (j!)qk = kqk, 8! and kNvk2 = kvk2 since N (j!) N (j!) = I for all !. Because of these norm preserving properties, inner matrices will play an important role in the control synthesis theory in this book. In this section, we present a state-space characterization of inner transfer functions. " # Lemma 13.29 Suppose N = A B 2 RH1 and X = X C D A X + XA + C C = 0: 0 satis es (13:37) Then (a) D C + B X = 0 implies N N = D D. (b) (A B ) controllable, and N N = D D implies D C + B X = 0. Proof. Conjugating the states of 2 B A 0 6 6 ;C C ;A ;C D N N =4 DC B DD " # " I 0 on the left and I 0 by ;X I ;X I 2 N N = 6 6 4 2 = 6 6 4 #;1 " 3 7 7 5 # I 0 on the right yields = X I A 0 B ;(A X + XA + C C ) ;A ;(XB + C D) B X +D C B DD 3 A 0 B 7 0 ;A ;(XB + C D) 7 : 5 B X +D C B DD Then (a) and (b) follow easily. 3 7 7 5 2
- 375. ALGEBRAIC RICCATI EQUATIONS 358 This lemma immediately leads to one characterization of inner matrices in terms of their state space representations. Simply add the condition that D D = I to Lemma 13.29 to get N N = I . Corollary 13.30 Suppose N = " A B C D # is stable and minimal, and X is the ob- servability grammian. Then N is inner if and only if (a) D C + B X = 0 (b) D D = I . A transfer matrix N? is called a complementary inner factor (CIF) of N if N N? ] is square and inner. The dual notion of the complementary co-inner factor is de ned in the obvious way. Given an inner N , the following lemma gives a construction of its CIF. The proof of this lemma follows from straightforward calculation and from the fact that CX y X = C since Im(I ; X + X ) Ker(X ) Ker(C ). " # Lemma 13.31 Let N = A B be an inner and X be the observability grammian. C D Then a CIF N? is given by " y N? = A ;X C D? C D? # where D? is an orthogonal complement of D such that D D? ] is square and orthogonal. 13.7 Inner-Outer Factorizations In this section, some special form of coprime factorizations will be developed. In particular, explicit realizations are given for coprime factorizations G = NM ;1 with inner numerator N and inner denominator M , respectively. The former factorization in the case of G 2 RH1 will give an inner-outer factorization1. The results will be proven for the right coprime factorizations, while the results for left coprime factorizations follow by duality. Let G 2 Rp be a p m transfer matrix and denote R1=2 R1=2 = R. For a given full column rank i matrix D, let D? denote for any orthogonal complement of D so that h ;1=2 D? (with R = D D > 0) is square and orthogonal. To obtain an rcf of DR G with N inner, we note that if NM ;1 is an rcf, then (NZr )(MZr );1 is also an rcf for any nonsingular real matrix Zr . We simply need to use the formulas in Theorem 5.9 to solve for F and Zr . 1 A p m (p m) transfer matrix Go 2 RH1 is called an outer if Go (s) has full row rank in the open right half plane, i.e., Go (s) has full row normal rank and has no open right half plane zeros.
- 376. 13.7. Inner-Outer Factorizations 359 Theorem 13.32 Assume p m. Then there exists an rcf G = NM ;1 such that N is inner if and only if G G > 0 on the j!-axis, including at 1. This factorization is unique up to a #constant unitary multiple. Furthermore, # assume that the realization of " " G = A B is stabilizable and that A ; j!I B has full column rank for all C D C D ! 2 R. Then a particular realization of the desired coprime factorization is " # 2 M := 6 4 N where and A + BF F C + DF BR;1=2 R;1=2 DR;1=2 3 7 5 2 RH1 R=D D>0 F = ;R;1(B X + D C ) " A ; BR;1 D C ;BR;1B X = Ric ;C (I ; DR;1 D )C ;(A ; BR;1 D C ) # 0: Moreover, a complementary inner factor can be obtained as " y N? = A + BF ;X C D? C + DF D? # if p > m. Proof. ()) Suppose G = NM ;1 is an rcf and N N = I . Then G G = (M ;1) M ;1 > 0 on the j!-axis since M 2 RH1 . (() This can be shown by using Corollary 13.20 rst to get a factorization G G = (M ;1 ) (M ;1 ) and then to compute N = GM . The following proof is more direct. It will be proven by showing that the de nition of the inner and coprime factorization formula given in Theorem 5.9 lead directly to the above realization of the rcf of G with an inner numerator. That G = NM ;1 is an rcf follows immediately once it is established that F is a stabilizing state feedback. Now suppose " N = A + BF BZr C + DF DZr # and let F and Zr be such that (DZr ) (DZr ) = I (13:38) (BZr ) X + (DZr ) (C + DF ) = 0 (13:39)
- 377. ALGEBRAIC RICCATI EQUATIONS 360 (A + BF ) X + X (A + BF ) + (C + DF ) (C + DF ) = 0: (13:40) ;1=2 U where R = D D > 0 and where U is any orthogonal Clearly, we have that Zr = R matrix. Take U = I and solve (13.39) for F to get F = ;R;1(B X + D C ): Then substitute F into (13.40) to get 0 = (A + BF ) X + X (A + BF ) + (C + DF ) (C + DF ) = (A ; BR;1 D C ) X + X (A ; BR;1 D C ) ; XBR;1B X + C D?D? C where D? D? = I ; DR;1 D . To show that such choices indeed make sense, we need to show that H 2 dom(Ric), where " # ;1 ;BR;1B H = A ; BR D C ;C D?D? C ;(A ; BR;1 D C ) so X " Ric(H ). However, by Theorem 13.19, H 2 dom(Ric) is guaranteed by the fact = # A ; j! B has full column rank (or G (j!)G(j!) > 0. that C D The uniqueness of the factorization follows from coprimeness and N inner. Suppose ; ; that G = N1M1 1 = N2 M2 1 are two right coprime factorizations and that both nu- merators are inner. By coprimeness, these two factorizations are unique up to a right multiple which is a unit2 in RH1 . That is, there exists a unit 2 RH1 such that " # " # M1 M2 . Clearly, is inner since = = N1 N1 = N2 N2 = I . N1 N2 The only inner units in RH1 are constant matrices, and thus the desired uniqueness property is established. Note that the non-uniqueness is contained entirely in the choice of a particular square root of R. Finally, the formula for N? follows from Lemma 13.31. 2 Note that the important inner-outer factorization formula can be obtained from this inner numerator coprime factorization if G 2 RH1 . Corollary 13.33 Suppose G 2 RH1 then the denominator matrix M in Theorem 13.32 is an outer. Hence, the factorization G = N (M ;1) given in Theorem 13.32 is an innerouter factorization. Remark 13.7 It is noted that the above inner-outer factorization procedure does not apply to the strictly proper transfer matrix even if the factorization exists. For example, s;1 1 G(s) = s+1 s+2 has inner-outer factorizations but the above procedure cannot be used. The inner-outer factorization for the general transfer matrices can be done using the method adopted in Section 6.1 of Chapter 6. ~ 2 A function ;1 2 RH1 . is called a unit in RH1 if
- 378. 13.7. Inner-Outer Factorizations 361 Suppose that the system G is not stable then a coprime factorization with an inner denominator can also be obtained by solving a special Riccati equation. The proof of this result is similar to the inner numerator case and is omitted. " # Theorem 13.34 Assume that G = A B 2 Rp and (A B) is stabilizable. Then C D there exists a right coprime factorization G = NM ;1 such that M 2 RH1 is inner if and only if G has no poles on j!-axis. A particular realization is 3 2 " # B A + BF M := 6 F I 7 2 RH1 5 4 N C + DF D where F = ;B X # " X = Ric A ;BB 0: 0 ;A Dual results can be obtained when p m by taking the transpose of the transfer function matrix. In these factorizations, output injection using the dual Riccati solution replaces state feedback to obtain the corresponding left factorizations. ~ ~ ~ Theorem 13.35 Assume p m. Then there exists an lcf G = M ;1N such that N is a co-inner if and only if GG > 0 on the j!-axis, including at 1. This factorization is unique up to a#constant unitary multiple. Furthermore, assume that the realization of " # " A B is detectable and that A ; j!I B has full row rank for all ! 2 R. G= C D C D Then a particular realization of the desired coprime factorization is " # h i A + LC L B + LD 2 RH ~ ~ M N := ~ ;1=2 ~ ;1=2 ~ ;1=2 1 R C R R D where and ~ R = DD > 0 ~ L = ;(BD + Y C )R;1 " # ~ ;1 ~ ~ ;C R;1 C Y = Ric (A ; BD R ;1C ) 0: ~ ~ ;B (I ; D R D)B ;(A ; BD R;1 C ) Moreover, a complementary co-inner factor can be obtained as # " ~? = A + LC B + LD N ~ ~ ;D? B Y y D? ~ ~ ~ ~ if p < m, where D? is a full row rank matrix such that D?D? = I ; D R;1D.
- 379. ALGEBRAIC RICCATI EQUATIONS 362 # " Theorem 13.36 Assume that G = A B 2 Rp and (C A) is detectable. Then C D ~ ~ ~ there exists a left coprime factorization G = M ;1N such that M 2 RH1 is inner if and only if G has no poles on j!-axis. A particular realization is " # h i A + LC L B + LD 2 RH ~ ~ M N := 1 C I D where L = ;Y C Y = Ric A ;C C 0 ;A " # 0: 13.8 Normalized Coprime Factorizations A right coprime factorization of G = NM ;1 with N M 2 RH1 is called a normalized right coprime factorization if M M +N N =I " i.e., if M N # ~ ~ is an inner. Similarly, an lcf G = M ;1N is called a normalized left h i ~ ~ coprime factorization if M N is a co-inner. The normalized coprime factorization is easy to obtain from the de nition. The following theorem can be proven using the same procedure as in the proof for the coprime factorization with inner numerator. In this case, the proof involves choosing F " # M is an inner. and Zr such that N Theorem 13.37 Let a realization of G be given by " G= A B C D and de ne # ~ R = I + D D > 0 R = I + DD > 0: (a) Suppose (A B ) is stabilizable and (C A) has no unobservable modes on the imaginary axis. Then there is a normalized right coprime factorization G = NM ;1 3 2 " # BR;1=2 A + BF M := 6 F R;1=2 7 2 RH1 5 4 N ;1=2 C + DF DR
- 380. 13.8. Normalized Coprime Factorizations where 363 F = ;R;1 (B X + D C ) and " ;1 ;BR;1B X = Ric A ; BR~ ;1D C ;C R C ;(A ; BR;1 D C ) # 0: (b) Suppose (C A) is detectable and (A B ) has no uncontrollable modes on the imag~ ~ inary axis. Then there is a normalized left coprime factorization G = M ;1 N h " A + LC L B + LD ~ ~ M N := ~ ;1=2 ~ ;1=2 ~ ;1=2 R C R R D i where # ~ L = ;(BD + Y C )R;1 and Y = Ric " ~ (A ; BD R;1C ) ;BR;1B ~ ;C R;1 C ~ ;(A ; BD R;1 C ) # 0: " # M are given (c) The controllability grammian P and observability grammian Q of N by P = (I + Y X );1Y Q = X h ~ ~ ~ ~ while the controllability grammian P and observability grammian Q of M N are given by ~ P =Y i ~ Q = (I + XY );1 X: Proof. We shall only prove the rst part of (c). It is obvious that Q = X since the Riccati equation for X can be written as " F X (A + BF ) + (A + BF ) X + C + DF # " F C + DF while the controllability grammian solves the Lyapunov equation (A + BF )P + P (A + BF ) + BR;1 B = 0 or equivalently P = Ric " (A + BF ) ;BR;1B # 0 : ;(A + BF ) # =0
- 381. ALGEBRAIC RICCATI EQUATIONS 364 " I X Now let T = 0 I " # then # (A + BF ) 0 =T ;(A + BF ) ;BR;1 B " # ~ ~ A ; BD R;1 C ;C R;1 C ;1 B ~ ;BR ;(A ; BD R;1 C ) T ;1: This shows that the stable invariant subspaces for these two Hamiltonian matrices are related by X; or " (A + BF ) ;BR;1B " # # " ~ ~ 0 A ; BD R;1 C ;C R;1 C = T X; ;1 B ~ ;(A + BF ) ;BR ;(A ; BD R;1 C ) " # " I = Im I + XY = T Im Y Y Hence we have P = Y (I + XY );1 . I Im P # " # # I : = Im Y (I + XY );1 2 13.9 Notes and References The general solutions of a Riccati equation are given by Martensson 1971]. The iterative procedure for solving ARE was rst introduced by Kleinman 1968] for a special case and was further developed by Wonham 1968]. It was used by Coppel 1974], Ran and Vreugdenhil 1988], and many others for the proof of the existence of maximal and minimal solutions. The comparative results were obtained in Ran and Vreugdenhil 1988]. The paper by Wimmer 1985] also contains comparative results for some special cases. The paper by Willems 1971] contains a comprehensive treatment of ARE and the related optimization problems. Some matrix factorization results are given in Doyle 1984]. Numerical methods for solving ARE can be found in Arnold and Laub 1984], Dooren 1981], and references therein. The state space spectral factorization for functions singular at 1 or on imaginary axis is considered in Clements and Glover 1989] and Clements 1993].
- 382. 14 H2 Optimal Control In this chapter we treat the optimal control of linear time-invariant systems with a quadratic performance criterion. The material in this chapter is standard, but the treatment is somewhat novel and lays the foundation for the subsequent chapters on H1 -optimal control. 14.1 Introduction to Regulator Problem Consider the following dynamical system: x = Ax + B2 u x(t0 ) = x0 _ (14:1) where x0 is given but arbitrary. Our objective is to nd a control function u(t) de ned on t0 T ] which can be a function of the state x(t) such that the state x(t) is driven to a (small) neighborhood of origin at time T . This is the so-called Regulator Problem. One might suggest that this regulator problem can be trivially solved for any T > t0 if the system is controllable. This is indeed the case if the controller can provide arbitrarily large amounts of energy since, by the de nition of controllability, one can immediately construct a control function that will drive the state to zero in an arbitrarily short time. However, this is not practical since any physical system has the energy limitation, i.e., the actuator will eventually saturate. Furthermore, large control action can easily drive the system out of the region where the given linear model is valid. Hence certain limitations have to be imposed on the control in practical engineering implementation. 365
- 383. H2 OPTIMAL CONTROL 366 The constraints on control u may be measured in many di erent ways for example, Z T t0 T Z kuk dt t0 kuk2 dt sup kuk t2 t0 T ] i.e., in terms of L1 -norm, L2 -norm, and L1 -norm, or more generally, weighted L1 -norm, L2 -norm, and L1 -norm T Z t0 kWu uk dt T Z t0 kWu uk2 dt sup kWu uk t2 t0 T ] for some constant weighting matrix Wu . Similarly, one might also want to impose some constraints on the transient response x(t) in a similar fashion Z T t0 kWx xk dt T Z t0 kWx xk2 dt sup kWx xk t2 t0 T ] for some weighting matrix Wx . Hence the regulator problem can be posed as an optimal control problem with certain combined performance index on u and x, as given above. In this chapter, we shall be concerned exclusively with the L2 performance problem or quadratic performance problem. Moreover, we will focus on the in nite time regulator problem, i.e., T ! 1, and, without loss of generality, we shall assume t0 = 0. In this case, our problem is as follows: nd a control u(t) de ned on 0 1) such that the state x(t) is driven to the origin at t ! 1 and the following performance index is minimized: min u 1 " x(t) Z 0 # " u(t) Q S S R #" # x(t) dt u(t) (14:2) for some Q = Q , S , and R = R > 0. This problem is traditionally called a Linear Quadratic Regulator problem or simply an LQR problem. Here we have assumed R > 0 to emphasis that the control energy has to be nite, i.e., u(t) 2 L2 0 1). So this is the space over which the integral is minimized. Moreover, it is also generally assumed that " Q S S R # 0: (14:3) Since R is positive de nite, it has a square root, R1=2 , which is also positive-de nite. By the substitution R1=2 u we may as well assume at the start that R = I . In fact, we can even assume S = 0 by using a pre-state feedback u = ;S x + v provided some care is exercised however, this u
- 384. 14.2. Standard LQR Problem 367 will not be assumed in the sequel. Since the matrix in (14.3) is positive semi-de nite with R = I , it can be factored as " # " Q S = C1 S I D12 # h i C1 D12 : And (14.2) can be rewritten as min kC1 x + D12 uk2 : 2 u2L2 0 1) In fact, the LQR problem is posed traditionally as the minimization problem min kC1 x + D12 uk2 2 u2L2 0 1) (14:4) x = Ax + Bu x(0) = x0 _ (14:5) without explicitly mentioning the condition that the control should drive the state to the origin. Instead some assumptions are imposed on Q S , and R (or equivalently C1 and D12 ) to ensure that the optimal control law u has this property. To see what assumption one needs to make in order to ensure that the minimization problem formulated in (14.4) and (14.5) has a sensible solution, let us consider a simple example with A = 1 B = 1 Q = 0 S = 0, and R = 1: min Z u2L2 0 1) 0 1 u2 dt x = x + u x(0) = x0 : _ It is clear that u = 0 is the optimal solution. However, the system with u = 0 is unstable and x(t) diverges exponentially to in nity, x(t) = et x0 . The problem with this example is that the performance index does not see" the unstable state x. This is true in general, and the proof of this fact is left as an exercise to the reader. Hence in order to ensure that the minimization problem in (14.4) and (14.5) is sensible, we must assume that all unstable states can be seen" from the performance index, i.e., (C1 A) must be detectable. This will be called a standard LQR problem. On the other hand, if the closed-loopRstability is imposed on the above minimization, then it can be shown that minu2L2 0 1) 01 u2 dt = 2x2 and u(t) = ;2x(t) is the optimal 0 control. This can also be generalized to a more general case where (C1 A) is not necessarily detectable. This problem will be referred to as an Extended LQR problem. 14.2 Standard LQR Problem In this section, we shall consider the LQR problem as traditionally formulated. Standard LQR Problem
- 385. H2 OPTIMAL CONTROL 368 Let a dynamical system be described by x = Ax + B2 u x(0) = x0 given but arbitrary _ (14.6) z = C1 x + D12 u (14.7) and suppose that the system parameter matrices satisfy the following assumptions: (A1) (A B2 ) is stabilizable h i (A2) D12 has full column rank with D12 D? unitary (A3) (C1 A) is detectable " # A ; j!I B2 has full column rank for all !. C1 D12 Find an optimal control law u 2 L2 0 1) such that the performance criterion kz k2 is minimized. 2 (A4) Remark 14.1 Assumption (A1) is clearly necessary for the existence of a stabilizing control function u. The assumption (A2) is made for simplicity of notation and is actually a restatement that R = D12 D12 = I . Note also that D? drops out when D12 is square. It is interesting to point out that (A3) is not needed in the Extended LQR problem. The assumption (A3) enforces that the unconditional optimization problem will result in a stabilizing control law. In fact, the assumption (A3) together with (A1) guarantees that the input/output stability implies the internal stability, i.e., u 2 L2 and z 2 L2 imply x 2 L2 , which will be shown in Lemma 14.1. Finally note that (A4) is equivalent to the condition that (D? C1 A ; B2 D12 C1 ) has no unobservable modes on the imaginary axis and is weaker than the popular assumption of detectability of (D? C1 A ; B2 D12 C1 ). (A4), together with the stabilizability of (A B2 ), guarantees by Corollary 13.10 that the following Hamiltonian matrix belongs to dom(Ric) and that X = Ric(H ) 0: H = " " = # " # h A 0 B2 ; ;C1 C1 ;A ;C1 D12 D12 C1 B2 # A ; B2 D12 C1 ;B2B2 : ;C1 D? D? C1 ;(A ; B2 D12 C1 ) i (14.8) Note also that if D12 C1 = 0, then (A4) is implied by the detectability of (C1 A), while the detectability of (C1 A) is implied by the detectability of (D? C1 A ; B2 D12 C1 ). The above implication is not true if D12 C1 6= 0, for example, " # " # h i 2 0 1 A= B2 = C1 = 1 0 D12 = 1: 0 ;2 0
- 386. 14.2. Standard LQR Problem 369 Then (C1 A) is detectable and A ; B2 D12 C1 = imaginary axis but is not stable. " 1 0 0 ;2 # has no eigenvalue on the ~ Note also that the Riccati equation corresponding to (14.8) is (A ; B2 D12 C1 ) X + X (A ; B2 D12 C1 ) ; XB2B2 X + C1 D? D?C1 = 0: (14:9) Now let X be the corresponding stabilizing solution and de ne F := ;(B2 X + D12 C1 ): (14:10) Then A + B2 F is stable. Denote AF := A + B2 F CF := C1 + D12 F and re-arrange equation (14.9) to get AF X + XAF + CF CF = 0: (14:11) Thus X is the observability Gramian of (CF AF ). Consider applying the control law u = Fx to the system (14.6) and (14.7). The controlled system is x = AF x x(0) = x0 _ z = CF x or equivalently x = AF x + x0 (t) x(0; ) = 0 _ z = CF x: The associated transfer matrix is " Gc (s) = AF I CF 0 and # kGc x0 k2 = x0 Xx0 : 2 The proof of the following theorem requires a preliminary result about internal stability given input-output stability. Lemma 14.1 If u z 2 Lp 0 1) for p 1 and (C1 A) is detectable in the system described by equations (14.6) and (14.7), then x 2 Lp 0 1). Furthermore, if p < 1, then x(t) ! 0 as t ! 1.
- 387. H2 OPTIMAL CONTROL 370 Proof. Since (C1 A) is detectable, there exists L such that A + LC1 is stable. Let x ^ be the state estimate of x given by _ x = (A + LC1 )^ + (LD12 + B2 )u ; Lz: ^ x Then x 2 Lp 0 1) since z and u are in Lp 0 1). Now let e = x ; x then ^ ^ e = (A + LC1 )e _ and e 2 Lp 0 1). Therefore, x = e + x 2 Lp 0 1). It is easy to see that e(t) ! 0 ^ as t ! 1 for any initial condition e(0). Finally, x(t) ! 0 follows from the fact that if p < 1 then x ! 0. ^ 2 Theorem 14.2 There exists a unique optimal control for the LQR problem, namely u = Fx. Moreover, min kz k2 = kGc x0 k2 : u2L2 0 1) Note that the optimal control strategy is constant gain state feedback, and this gain is independent of the initial condition x0 . Proof. With the change of variable v = u ; Fx, the system can be written as " # " x = AF B2 _ z CF D12 #" # x v x(0) = x0 : (14:12) Now if v 2 L2 0 1), then x z 2 L2 0 1) and x(1) = 0 since AF is stable. Hence u = Fx + v 2 L2 0 1). Conversely, if u z 2 L2 0 1), then from Lemma 14.1 x 2 L2 0 1). So v 2 L2 0 1). Thus the mapping v = u ; Fx between v 2 L2 0 1) and those u 2 L2 0 1) that make z 2 L2 0 1) is one-to-one and onto. Therefore, min kz k2 = v2L 0 1) kz k2 : min u2L2 0 1) 2 By di erentiating x(t) Xx(t) with respect to t along a solution of the di erential equation (14.12) and by using (14.9) and the fact that CF D12 = ;XB2, we see that d x Xx = x Xx + x X x _ _ dt = x (AF X + XAF )x + 2x XB2 v = ;x CF CF x + 2x XB2 v = ;(CF x + D12 v) (CF x + D12 v) + 2x CF D12 v + v v + 2x XB2 v = ; kz k2 + kvk2 : (14.13)
- 388. 14.2. Standard LQR Problem 371 Now integrate (14.13) from 0 to 1 to get kz k2 = x0 Xx0 + kvk2 : 2 2 2 Clearly, the unique optimal control is v = 0, i.e., u = Fx. This method of proof, involving change of variables and the completion of the square, is a standard technique and variants of it will be used throughout this book. An alternative proof can be given in frequency domain. To do that, let us rst note the following fact: Lemma 14.3 Let a transfer matrix be de ned as " # U := AF B2 2 RH1 : CF D12 Then U is inner and U Gc 2 RH? . 2 Proof. The proof uses standard manipulations of state space realizations. From U we get # " U (s) = ;AF ;CF : B2 Then it is easy to compute 2 ;AF ;CF CF ;CF D12 U U =6 0 4 B2 B2 I AF D12 CF D12 3 2 7 5 U Gc = 6 0 4 B2 Now do the similarity transformation " ;AF ;CF CF 0 I ;X 0 I I 7: AF 5 D12 CF 0 # on the states of the transfer matrices and use (14.11) to get 2 ;AF U U =6 0 4 B2 2 U Gc = 6 4 ;AF 0 B2 0 ;X 0 0 AF I 0 0 AF B2 0 I 3 7 5 3 7 5 =I " # ;AF ;X 2 RH? : = 2 B2 0 3
- 389. H2 OPTIMAL CONTROL 372 2 An alternative proof of Theorem 14.2 We have in the frequency domain z = Gc x0 + Uv: Let v 2 H2 . By Lemma 14.3, Gc x0 and Uv are orthogonal. Hence kz k2 = kGc x0 k2 + kUvk2 : 2 2 2 Since U is inner, we get kz k2 = kGc x0 k2 + kvk2 : 2 2 2 This equation immediately gives the desired conclusion. 2 Remark 14.2 It is clear that the LQR problem considered above is essentially equiva- lent to minimizing the 2-norm of z with the input w = x0 (t) in the following diagram: z A I B2 C1 0 D12 I 0 0 - w K But this problem is a special H2 norm minimization problem considered in a later section. ~ 14.3 Extended LQR Problem This section considers the extended LQR problem where no detectability assumption is made for (C1 A). Extended LQR Problem Let a dynamical system be given by x = Ax + B2 u x(0) = x0 given but arbitrary _ z = C1 x + D12 u with the following assumptions: (A1) (A B2 ) is stabilizable
- 390. 14.4. Guaranteed Stability Margins of LQR h 373 i (A2) D12 has full column rank with D12 D? unitary " # A ; j!I B2 has full column rank for all !. (A3) C1 D12 Find an optimal control law u 2 L2 0 1) such that the system is internally stable, i.e., x 2 L2 0 1) and the performance criterion kz k2 is minimized. 2 Assume the same notation as above, and we have Theorem 14.4 There exists a unique optimal control for the extended LQR problem, namely u = Fx. Moreover, min kz k = kGc x0 k2 : u2L 0 1) 2 2 Proof. The proof of this theorem is very similar to the proof of the standard LQR problem except that, in this case, the input/output stability may not necessarily imply the internal stability. Instead, the internal stability is guaranteed by the way of choosing control law. Suppose that u 2 L2 0 1) is such a control law that the system is stable, i.e., x 2 L2 0 1). Then v = u ; Fx 2 L2 0 1). On the other hand, let v 2 L2 0 1) and consider " # " #" # x = AF B2 _ x x(0) = x0 : z CF D12 v Then x z 2 L2 0 1) and x(1) = 0 since AF is stable. Hence u = Fx + v 2 L2 0 1). Again the mapping v = u ; Fx between v 2 L2 0 1) and those u 2 L2 0 1) that make z 2 L2 0 1) and x 2 L2 0 1) is one to one and onto. Therefore, min kz k2 = min kz k2 : u2L2 0 1) v2L2 0 1) Using the same technique as in the proof of the standard LQR problem, we have kz k2 = x0 Xx0 + kvk2 : 2 2 And the unique optimal control is v = 0, i.e., u = Fx. 2 14.4 Guaranteed Stability Margins of LQR Now we will consider the system described by equation (14.6) with the LQR control law u = Fx. The closed-loop block diagram is as shown in Figure 14.1. The following result is the key to stability margins of an LQR control law. Lemma 14.5 Let F = ;(B2 X + D12C1 ) and de ne G12 = D12 + C1(sI ; A);1B2 . Then ; ; I ; B2 (;sI ; A );1 F I ; F (sI ; A);1 B2 = G12 (s)G12 (s):
- 391. H2 OPTIMAL CONTROL 374 - F u - x_ = Ax + B2u x- Figure 14.1: LQR closed-loop system Proof. Note that the Riccati equation (14.9) can be written as XA + A X ; F F + C1 C1 = 0: Add and subtract sX to the equation to get ;X (sI ; A) ; (;sI ; A )X ; F F + C1 C1 = 0: Now multiply the above equation from the left by B2 (;sI ; A );1 and from the right by (sI ; A);1 B2 to get ;B2 (;sI ; A );1 XB2 ; B2 X (sI ; A);1 B2 ; B2 (;sI ; A );1 F F (sI ; A);1 B2 +B2 (;sI ; A );1 C1 C1 (sI ; A);1 B2 = 0: Using ;B2 X = F + D12 C1 in the above equation, we have B2 (;sI ; A );1 F + F (sI ; A);1 B2 ; B2 (;sI ; A );1 F F (sI ; A);1 B2 +B2 (;sI ; A );1 C1 D12 + D12 C1 (sI ; A);1 B2 +B2 (;sI ; A );1 C1 C1 (sI ; A);1 B2 = 0: Then the result follows from completing the square and from the fact that D12 D12 = I . 2 Corollary 14.6 Suppose D12C1 = 0. Then ; ; I ; B2 (;sI ; A );1 F I ; F (sI ; A);1 B2 = I +B2 (;sI ;A );1 C1 C1 (sI ;A);1 B2 : In particular, ; ; I ; B2 (;j!I ; A );1 F I ; F (j!I ; A);1 B2 I (14:14) and ; I + B2 (;j!I ; A ; F B2 );1 F ; I + F (j!I ; A ; B2 F );1 B2 I: (14:15)
- 392. 14.5. Standard H2 Problem 375 Note that the inequality (14.15) follows from taking the inverse of inequality (14.14). De ne G(s) = ;F (sI ; A);1 B2 and assume for the moment that the system is single input. Then the inequality (14.14) shows that the open-loop Nyquist diagram of the system G(s) in Figure 14.1 never enters the unit disk centered at (;1 0) of the complex plane. Hence the system has at least the following stability margins: kmin 1 kmax = 1 min ;60o max 60o 2 i.e., the system has at least a 6dB (= 20 log 2) gain margin and a 60o phase margin in both directions. A similar interpretation may be generalized to multiple input systems. Next, it is noted that the inequality (14.15) can also be given some robustness interpretation. In fact, it implies that the closed-loop system in Figure 14.1 is stable even if the open-loop system G(s) is perturbed additively by a 2 RH1 as long as k k1 < 1. This can be seen from the following block diagram and small gain theorem where the transfer matrix from w to z is exactly I + F (j!I ; A ; B2 F );1 B2 . z - w - - x_ = Ax + B2u - ? e F 14.5 Standard H2 Problem The system considered in this section is described by the following standard block diagram: z y G - K w u The realization of the transfer matrix G is taken to be of the form 2 3 A B1 B2 6 G(s) = 4 C1 0 D12 7 : 5 C2 D21 0
- 393. H2 OPTIMAL CONTROL 376 Notice the special o -diagonal structure of D: D22 is assumed to be zero so that G22 is strictly proper1 also, D11 is assumed to be zero in order to guarantee that the H2 problem properly posed.2 The case for D11 6= 0 will be discussed in Section 14.7. The following additional assumptions are made for the output feedback H2 problem in this chapter: (i) (A B2 ) is stabilizable and (C2 A) is detectable h i (ii) D12 has full column rank with D12 D? unitary, and D21 has full row rank " # D with ~21 unitary D? " A ; j!I B2 C1 D12 # " A ; j!I B1 C2 D21 # (iii) (iv) has full column rank for all ! has full row rank for all !. The rst assumption is for the stabilizability of G by output feedback, and the third and the fourth assumptions together with the rst guarantee that the two Hamiltonian matrices associated with the H2 problem below belong to dom(Ric). The rank assumptions (ii) are necessary to guarantee that the H2 optimal controller is a nite dimensional linear time invariant one, while the unitary assumptions are made for the simplicity of the nal solution they are not restrictions (see e.g., Chapter 17). H2 Problem The H2 control problem is to nd a proper, real-rational controller K which stabilizes G internally and minimizes the H2 -norm of the transfer matrix Tzw from w to z . In the following discussions we shall assume that we have state models of G and K . Recall that a controller is said to be admissible if it is internally stabilizing and proper. We now state the solution of the problem and then take up its derivation in the next several sections. By Corollary 13.10 the two Hamiltonian matrices " A 0 H2 := ;C1 C1 ;A " # # " B2 ; ;C1 D12 " A 0 C2 J2 := ; ;B1 B1 ;A ;B1D21 # # h h D12 C1 B2 D21 B1 C2 i i 1 As we have discussed in Section 12.3.4 of Chapter 12 there is no loss of generality in making this assumption since the controller for D22 nonzero case can be recovered from the zero case. 2 Recall that a rational proper stable transfer function is an RH function i it is strictly proper. 2
- 394. 14.5. Standard H2 Problem 377 belong to dom(Ric), and, moreover, X2 := Ric(H2 ) 0 and Y2 := Ric(J2 ) 0. De ne F2 := ;(B2 X2 + D12 C1 ) L2 := ;(Y2 C2 + B1 D21 ) and AF2 := A + B2 F2 C1F2 := C1 + D12 F2 AL2 := A + L2C2 B1L2 := B1 + L2 D21 ^ A2 := A + B2 F2 + L2C2 # " # " AL2 B1L2 : AF2 I Gf (s) := Gc (s) := C1F2 0 I 0 Theorem 14.7 There exists a unique optimal controller # " ^ A2 ;L2 : K (s) := opt F2 0 Moreover, min kTzw k2 = kGc B1 k2 + kF2 Gf k2 = kGc L2k2 + kC1 Gf k2 . 2 2 2 2 2 The controller Kopt has the well-known separation structure, which will be discussed in more detail in Section 14.9. For comparison with the H1 results, it is useful to describe all suboptimal controllers. Theorem 14.8 The family of all admissible controllers such that kTzw k2 < equals the set of all transfer matrices from y to u in u M2 - Q y 2 ^ A2 ;L2 B2 6 M2 (s) = 4 F2 0 I ;C2 I 0 3 7 5 where Q 2 RH2 , kQk2 < 2 ; (kGc B1 k2 + kF2 Gf k2 ). 2 2 2 Thus, the suboptimal controllers are parameterized by a xed (independent of ) linear-fractional transformation with a free parameter Q. With Q = 0, we recover Kopt . It is worth noting that the parameterization in Theorem 14.8 makes Tzw a ne in Q and yields the Youla parameterization of all stabilizing controllers when the conditions on Q are replaced by Q 2 RH1 .
- 395. H2 OPTIMAL CONTROL 378 14.6 Optimal Controlled System In this section, we look at the controller, K (s) = F`(M2 Q) Q 2 RH1 connected to G. (Keep in mind that all admissible controllers are parameterized by the above formula). We will give a brief analysis of the closed-loop system. It will be seen that a direct consequence from this analysis is the results of Theorem 14.7 and 14.8. The proof given here is not our emphasis. The reason is that this approach does not generalize nicely to other control problems and is often very involved. An alternative proof will be given in the later part of this chapter by using the FI and OE results discussed in section 14.8 and the separation argument. The idea of separation is the main theme for synthesis. We shall now analyze the system structure under the control of a such controller. In particular, we will compute explicitly kTzw k2 . 2 Consider the following system diagram with controller K (s) = F`(M2 Q): z y y1 G XXXX XXX M2 - u1 Q Then Tzw = F` (N Q) with 2 3 ;B2 F2 B1 B2 7 6 6 0 AL2 B1L2 0 7 : 7 N =6 6 0 D12 7 5 4 C1F2 ;D12 F2 D21 0 0 C2 De ne " We have AF2 w u U = AF2 B2 C1F2 D12 # " # V = AL2 B1L2 : C2 D21 Tzw = Gc B1 ; UF2 Gf + UQV: It follows from Lemma 14.3 that Gc B1 and U are orthogonal. Thus kTzw k2 = kGc B1 k2 + kUF2 Gf ; UQV k2 = kGc B1 k2 + kF2 Gf ; QV k2 : 2 2 2 2 2
- 396. 14.7. H2 Control with Direct Disturbance Feedforward* 379 It can also be shown easily by duality that Gf and V are orthogonal, i.e., Gf V 2 RH? , 2 and V is a co-inner, so we have kTzw k2 = kGc B1 k2 + kF2 Gf ; QV k2 = kGc B1 k2 + kF2 Gf k2 + kQk2 : 2 2 2 2 2 2 This shows clearly that Q = 0 gives the unique optimal control, so K = F`(M2 0) is the unique optimal controller. Note also that kTzw k2 is nite if and only if Q 2 RH2 . Hence Theorem 14.7 and 14.8 follow easily. It is interesting to examine the structure of Gc and Gf . First of all the transfer matrix Gc can be represented as a xed system with the feedback matrix F2 wrapped around it: A I B2 C1 0 D12 I 0 0 - F2 F2 is, in fact, an optimal LQR controller and minimizes the H2 norm of Gc . Similarly, Gf can be represented as A B1 I I 0 0 C2 D21 0 - L2 and L2 minimizes the H2 norm of Gf and solves a special ltering problem. 14.7 H2 Control with Direct Disturbance Feedforward* Let us consider the generalized system structure again with D11 not necessarily zero: 2 A B1 B2 6 G(s) = 4 C1 D11 D12 C2 D21 0 3 7 5
- 397. H2 OPTIMAL CONTROL 380 We shall consider the following question: what will happen and under what condition will the H2 optimal control problem make sense if D11 6= 0? Recall that F`(M2 Q) with Q 2 RH1 parameterizes all stabilizing controllers for G regardless of D11 = 0 or not. Now again consider the closed loop transfer matrix with the controller K = F`(M2 Q) then Tzw = Gc B1 ; UF2 Gf + UQV + D11 and Tzw (1) = D12 Q(1)D21 + D11 : Hence the H2 optimal control problem will make sense, i.e., having nite H2 norm, if and only if there is a constant Q(1) such that D12 Q(1)D21 + D11 = 0: This requires that and that Q(1) = ;D12 D11 D21 ; D12 D12 D11 D21 D21 + D11 = 0: (14:16) Note that the equation (14.16) is a very restrictive condition. For example, suppose D12 = " # 0 I h D21 = 0 I i and D11 is partitioned accordingly " # D11 = D1111 D1112 : D1121 D1122 Then equation (14.16) implies that " # " D1111 D1112 = 0 0 D1121 0 0 0 # and that Q(1) = ;D1122 . So only D1122 can be nonzero for a sensible H2 problem. Hence from now on in this section we shall assume that (14.16) holds and denotes DK := ;D12 D11 D21 . To nd the optimal control law for the system G with D11 6= 0, let us consider the following system con guration:
- 398. 14.8. Special Problems 381 z w G - DK -f 6 y - ;DK Then and ^ G u1 -f 6 ^ K K 2 A + B2 DK C2 B1 + B2 DK D21 B2 ^ = 6 C1 + D12 DK C2 G 4 0 D12 C2 D21 0 3 7 5 ^ K = DK + K: ^ It is easy to check that the system G satis es all assumptions in Section 14.5 hence the controller formula in Section 14.5 can be used. A little bit of algebra will show that # " ^ A2 ; B2 DK C2 ;(L2 ; B2 DK ) ^ K= F2 ; DK C2 0 ^ is the H2 optimal controller for G. Hence the controller K for the original system G will be given by " # ^ A2 ; B2 DK C2 ;(L2 ; B2 DK ) = F (M D ): K= ` 2 K F2 ; DK C2 14.8 Special Problems DK In this section we look at various H2 -optimization problems from which the output feedback solutions of the previous sections will be constructed via a separation argument. All the special problems in this section are to nd K stabilizing G and minimizing the H2 -norm from w to z in the standard setup, but with di erent structures for G. As
- 399. H2 OPTIMAL CONTROL 382 in Chapter 12, we shall call these special problems, respectively, state feedback (SF), output injection (OI), full information (FI), full control (FC), disturbance feedforward (DF), and output estimation (OE). OI, FC, and OE are natural duals of SF, FI, and DF, respectively. The output feedback solutions will be constructed out of the FI and OE results. The special problems SF, OI, FI, and FC are not, strictly speaking, special cases of the output feedback problem since they do not satisfy all of the assumptions for output feedback (while DF and OE do). Each special problem inherits some of the assumptions (i)-(iv) from the output feedback as appropriate. The assumptions will be discussed in the subsections for each problem. In each case, the results are summarized as a list of three items (in all cases, K must be admissible) 1. the minimum of kTzw k2 2. the unique controller minimizing kTzw k2 3. the family of all controllers such that kTzw k2 < , where is greater than the minimum norm. Warning: we will be more speci c below about what we mean about the uniqueness and all controllers in the second and third item. In particular, the controllers characterized here for SF, OI, FI and FC problems are neither unique nor all-inclusive. Once again we regard all controllers that give the same control signal u, i.e., having the same transfer function from w to u, as an equivalence class. In other words, if K1 and K2 generate the same control signal u, we will regard them as the same, denoted as K1 = K2. Hence the unique controller" here means one of the controllers from the unique equivalence class. The same comments apply to the all" situation. This will be much clearer in section 14.8.1 when we consider the state feedback problem. In that case we actually give a parameterization of all the elements in the equivalence class of the unique" optimal controllers. Thus the unique controller is really not unique. We chose not to give the parameterization of all the elements in an equivalence class in this book since it is very messy, as can be seen in section 14.8.1, and not very useful. However, it will be seen that this equivalence class problem will not occur in the general output feedback case including DF and OE problems. 14.8.1 State Feedback Consider an open-loop system transfer matrix 2 A B1 B2 6 GSF (s) = 4 C1 0 D12 I 0 0 with the following assumptions: 3 7 5
- 400. 14.8. Special Problems 383 (i) (A B2 ) is stabilizable h i (ii) D12 has full column rank with D12 D? unitary " (iii) # A ; j!I B2 C1 D12 has full column rank for all !. This is very much like the LQR problem except that we require from the start that u be generated by state feedback and that the detectability of (C1 A) is not imposed since the controllers are restricted to providing internal stability. The controller is allowed to be dynamic, but it turns out that dynamics are not necessary. State Feedback: 1. min kTzw k2 = kGc B1 k2 = (trace(B1 X2 B1 ))1=2 2. K (s) = F2 Remark 14.3 The class of all suboptimal controllers for state feedback are messy and are not very useful in this book, so they are omitted, as are the OI problems. ~ Proof. Let K be a stabilizing controller, u = K (s)x. Change control variables by de ning v := u ; F2 x and then write the system equations as 2 6 4 3 2 x _ AF2 B1 B2 7 6 z 5 = 4 C1F2 0 D12 v (K ; F2 ) 0 0 32 76 54 3 x w 7: 5 v The block diagram is z AF2 B1 B2 C1F2 0 D12 I 0 0 x w v - K ; F2 Let Tvw denote the transfer matrix from w to v. Notice that Tvw 2 RH2 because K stabilizes G. Then Tzw = Gc B1 + UTvw
- 401. H2 OPTIMAL CONTROL 384 " # AF2 B2 , and by Lemma 14.3 U is inner and U G is in RH? . We where U = c 2 C1F2 D12 get kTzw k2 = kGc B1 k2 + kTvw k2 : 2 2 2 Thus min kTzw k2 = kGc B1 k2 and the minimum is achieved i Tvw = 0. Furthermore, 2 2 K = F2 is a controller achieving this minimum, and any other controllers achieving minimum are in the equivalence class of F2 . 2 Note that the above proof actually yields a much stronger result than what is needed. The proof that the optimal Tvw is Tvw = 0 does not depend on the restriction that the controller measures just the state. We only require that the controller produce v as a causal stable function Tvw of w. This means that the optimal state feedback is also optimal for the full information problem as well. We now give some further explanation about the uniqueness of the optimal controller that we commented on before. The important observation for this issue is that the controllers making Tvw = 0 are not unique. The controller given above, F2 , is only one of them. We will now try to nd all of those controllers that stabilize the system and give Tvw = 0, i.e., all K (s) = F2 . Proposition 14.9 Let Vc be a matrix whose columns form a basis for KerB1 (Vc B1 = 0). Then all H2 optimal state feedback controllers can be parameterized as Kopt = F` (Msf ) with 2 RH2 and " # F2 I Msf = : Vc (sI ; AF2 ) ;Vc B2 Proof. Since " Tvw = I ; AF2 B2 K ; F2 0 #!;1 " # AF2 B1 = 0 K ; F2 0 we get (K ; F2 )(sI ; AF2 );1 B1 = 0 (14:17) which is achieved if K = F2 . Clearly, this is the only solution if B1 is square and nonsingular or if K is restricted to be constant and (AF2 B1 ) is controllable. To parameterize all elements in the equivalence class (Tvw = 0) to which K = F2 belongs, let " # Pc (s) := AF2 B2 : I 0
- 402. 14.8. Special Problems 385 Then all state feedback controllers stabilizing G can be parameterized as K (s) = F2 + (I + QPc );1 Q Q(s) 2 RH1 where Q is free. Substitute K in equation (14.17), and we get Q(s)(sI ; AF2 );1 B1 = 0: Hence we have (14:18) Q(s)(sI ; AF2 );1 = (s)Vc (s) 2 RH2 : Thus all Q(s) 2 RH1 satisfying (14.18) can be written as Q(s) = (s)Vc (sI ; AF2 ) (s) 2 RH2 : (14:19) Therefore, Kopt(s) = F2 + (I + (s)Vc (sI ; AF2 )Pc );1 (s)Vc (sI ; AF2 ) = F2 + (I + (s)Vc B2 );1 (s)Vc (sI ; AF2 ) (s) 2 RH2 2 parameterizes the equivalence class of K = F2 . 14.8.2 Full Information and Other Special Problems We shall consider the FI problem rst. 2 A 6 6 C1 GFI (s) = 6 " # 6 I 4 0 B1 " 0 0 # I B2 D12# " 0 0 3 7 7 7 7 5 The assumptions relevant to the FI problem are the same as the state feedback problem. This is similar to the state feedback problem except that the controller now has more information (w). However, as was pointed out in the discussion of the state feedback problem, this extra information is not used by the optimal controller. Full Information: 1. min kTzw k2 = kGc B1 k2 = (trace(B1 X2 B1 ))1=2 h 2. K (s) = F2 0 h i i 3. K (s) = F2 Q(s) , where Q 2 RH2 , kQk2 < 2 ; kGcB1 k2 2 2
- 403. H2 OPTIMAL CONTROL 386 Proof. Items 1 and 2 follow immediately from the proof of the state feedback results because the argument that Tvw = 0 did not depend on the restriction to state feedback only. Thus we only need to prove item 3. Let K be an admissible controller such that kTzw k2 < . As in the SF proof, de ne a new control variable v = u ; F2 x then the closed-loop system is as shown below z y with 2 AF2 6 6 C1F2 ~ GFI = 6 " # 6 I 4 ~ GFI " 0 0 v - ~ K # B2 D12# " B1 w 0 0 3 7 7 7 7 5 ~ K = K ; F2 0]: I 0 Denote by Q the transfer matrix from w to v it belongs to RH2 by internal stability and the fact that D12 has full column rank and Tzw with zi= C1F2 x + D12 Qw hasi nite h h H2 norm. Then u = F2 x + v = F2 x + Qw = F2 Q y so K = F2 Q , and kTzw k2 = kGc B1 + UQk2 = kGc B1 k2 + kQk2 hence, 2 2 2 2 kQk2 = kTzw k2 ; kGcB1 k2 < 2 ; kGc B1 k2 : 2 2 2 2 Likewise, one can show that every controller of the form given in item no.3 is admissible and suboptimal. 2 The results for DF, OI, FC, and OE follow from the parallel development of Chapter 12. Disturbance Feedforward: 2 A B1 B2 6 GDF (s) = 4 C1 0 D12 C2 I 0 3 7 5 This problem inherits the same assumptions (i)-(iii) as in the state feedback problem, in addition to the stability condition of A ; B1 C2 . 1. min kTzw k2 = kGc B1 k2 " A + B2 F2 ; B1 C2 B1 2. K (s) = F2 0 #
- 404. 14.8. Special Problems 387 3. the set of all transfer matrices from y to u in u y M2D - 2 A + B2 F2 ; B1 C2 B1 B2 6 M2D (s) = 4 F2 0 I ;C2 I 0 Q where Q 2 RH2 , kQk2 < 2 ; kGcB1 k2 2 2 Output Injection: 2 A B1 I 6 GOI (s) = 4 C1 0 0 C2 D21 0 with the following assumptions: (i) (C2 A) is detectable " D (ii) D21 has full row rank with ~21 D? " (iii) A ; j!I B1 C2 D21 # 3 7 5 # unitary has full row rank for all !. 1. min kTzw k2 = kC1 Gf k2 = (trace(C1 Y2 C1 ))1=2 2. K (s) = " L2 # 0 Full Control: 2 A B1 6 6 GFC (s) = 6 C1 0 4 C2 D21 h h h I 0 0 I i 3 i 7 7 7 i 5 0 0 with the same assumptions as an output injection problem. 1. min kTzw k2 = kC1 Gf k2 = (trace(C1 Y2 C1 ))1=2 2. K (s) = " L2 0 # 3 7 5
- 405. H2 OPTIMAL CONTROL 388 " # L2 , where Q 2 RH , kQk2 < 3. K (s) = 2 2 Q(s) Output Estimation: 2 2 ; kC1 Gf k2 2 A B1 B2 GOE (s) = 6 C1 0 I 4 C2 D21 0 3 7 5 The assumptions are taken to be those in the output injection problem plus an additional assumption that A ; B2 C1 is stable. 1. min kTzw k2 = kC1 Gf k2 " A + L2 C2 ; B2 C1 L2 2. K (s) = C1 0 # 3. the set of all transfer matrices from y to u in u M2O - Q y 2 A + L2 C2 ; B2 C1 L2 ;B2 M2O (s) = 6 C1 0 I 4 C2 I 0 3 7 5 where Q 2 RH2 , kQk2 < 2 ; kC1 Gf k2 2 2 14.9 Separation Theory Given the results for the special problems, we can now prove Theorem 14.7 using separation arguments. This essentially involves reducing the output feedback problem to a combination of the Full Information and the Output Estimation problems. 14.9.1 H2 Controller Structure Recall that the unique H2 optimal controller is " # " ^ A2 ;L2 = A + B2 F2 + L2C2 Y2 C2 K2 (s) := F2 0 ;B2 X2 0 2 = kGc B1 k2 + kF2 Gf k2 and min kTzw k2 2 2 # where X2 := Ric(H2) and Y2 := Ric(J2 ) and the min is over all stabilizing controllers. Note that F2 is the optimal state feedback in the Full Information problem and L2 is the
- 406. 14.9. Separation Theory 389 optimal output injection in the Full Control case. The well-known separation property of the H2 solution is re ected in the fact that K2 is exactly the optimal output estimate of F2 x and can be obtained by setting C1 = F2 in OE.2. Also, the minimum cost is the sum of the FI cost (FI.1) and the OE cost for estimating F2 x (OE.1). The controller equations can be written in standard observer form as _ x = Ax + B2 u + L2(C2 x ; y) ^ ^ ^ u = F2 x ^ where x is the optimal estimate of x. ^ 14.9.2 Proof of Theorem 14.7 As before we de ne a new control variable, v := u ; F2 x, and the transfer function to z becomes #" # " z= " AF2 B1 B2 C1F2 0 D12 # w = G B w + Uv c 1 v " (14:20) # AF2 I and U (s) := AF2 B2 . Furthermore, U is inner where Gc (s) := C1F2 0 C1F2 D12 (i.e., U U = I ) and U Gc belongs to RH? from Lemma 14.3. 2 Let K be any admissible controller and notice how v is generated: v w 2 3 A B1 B2 Gv Gv = 6 ;F2 0 I 7 4 5 y u C2 D21 0 -K Note that K stabilizes G i K stabilizes Gv (the two closed-loop systems have identical A-matrices) and that Gv has the form of the Output Estimation problem. From (14.20) and the properties of U we have that min kTzw k2 = kGcB1 k2 + min kTvw k2 : 2 2 2 But from item OE.2, kTvw k2 is minimized by the controller " A + B2 F2 + L2C2 ;L2 F2 0 and then from OE.1 min kTvw k2 = kF2 Gf k2 . # (14:21)
- 407. H2 OPTIMAL CONTROL 390 14.9.3 Proof of Theorem 14.8 Continuing with the development in the previous proof, we see that the set of all suboptimal controllers equals the set of all K 's such that kTvw k2 < 2 ; kGc B1 k2 . Apply 2 2 item OE.3 to get that such K 's are parameterized by u y M2 with Q 2 RH2 , kQk2 < 2 2 ^ A2 ;L2 B2 6 M2 (s) = 4 F2 0 I ;C2 I 0 Q 3 7 5 2 ; kGc B1 k2 ; kF2 Gf k2 . 2 2 14.10 Stability Margins of H2 Controllers We have shown that the system with LQR controller has at least 60o phase margin and 6dB gain margin. However, it is not clear whether these stability margins will be preserved if the states are not available and the output feedback H2 (or LQG) controller has to be used. The answer is provided here through a counterexample: there are no guaranteed stability margins for a H2 controller. Consider a single input and single output two state generalized dynamical system: 2 G(s) = " 1 1 0 1 6 6 6 6 " 6 6 6 6 0 6 4 h # pq pq 0 1 0 p p " # 0 0 # " " 0 i h 0 1 i 0 1 0 1 0 It can be shown analytically that X2 = " and F2 = ; where 2 h p # 1 1 =2+ 4+q Y2 = i " # 2 L2 = ; p " 1 1 # =2+ 4+ : # 3 7 7 7 # 7 7 7 7 7 7 5 :
- 408. 14.10. Stability Margins of H2 Controllers Then the optimal output H2 controller is given by 2 1; 1 Kopt = 6 ;( + ) 1 ; 4 ; ; 391 3 7 5 : 0 Suppose that the resulting closed-loop controller (or plant G22 ) has a scalar gain k with a nominal value k = 1. Then the controller implemented in the system is actually K = kKopt and the closed-loop system A-matrix becomes 2 3 1 1 0 0 6 7 ;k 7 : ~ 6 7 A = 6 0 1 ;k 6 0 1; 1 7 4 5 0 ; ; 1; It can be shown that the characteristic polynomial has the form ~ det(sI ; A) = a4 s4 + a3 s3 + a2 s2 + a1 s + a0 with a1 = + ; 4 + 2(k ; 1) a0 = 1 + (1 ; k) : Note that for closed-loop stability it is necessary to have a0 > 0 and a1 > 0. Note also that a0 (1 ; k) and a1 2(k ; 1) for su ciently large and if k 6= 1. It is easy to see that for su ciently large and (or q and ), the system is unstable for arbitrarily small perturbations in k in either direction. Thus, by choice of q and , the gain margins may be made arbitrarily small. It is interesting to note that the margins deteriorate as control weight (1=q) gets small (large q) and/or system deriving noise gets large (large ). In modern control folklore, these have often been considered ad hoc means of improving sensitivity. It is also important to recognize that vanishing margins are not only associated with open-loop unstable systems. It is easy to construct minimum phase, open-loop stable counterexamples for which the margins are arbitrarily small. The point of these examples is that H2 (LQG) solutions, unlike LQR solutions, provide no global system-independent guaranteed robustness properties. Like their more classical colleagues, modern LQG designers are obliged to test their margins for each speci c design. It may, however, be possible to improve the robustness of a given design by relaxing the optimality of the lter (or FC controller) with respect to error properties. A successful approach in this direction is the so called LQG loop transfer recovery (LQG/LTR) design technique. The idea is to design a ltering gain (or FC control law) in such way so that the LQG (or H2 ) control law will approximate the loop properties of the regular LQR control. This will not be explored further here interested reader may consult related references.
- 409. 392 14.11 Notes and References H2 OPTIMAL CONTROL The detailed treatment of H2 related theory, LQ optimal control, Kalman ltering, etc., can be found in Anderson and Moore 1990] or Kwakernaak and Sivan 1972].
- 410. 15 Linear Quadratic Optimization This chapter considers time domain characterizations of Hankel operators and Toeplitz operators by means of some related quadratic optimizations. These characterizations will be used to prove a max-min problem which is the key to the H1 theory considered in the next chapter. 15.1 Hankel Operators Let G(s) be a stable real rational transfer matrix with a state space realization x = Ax + Bw _ z = Cx + Dw: (15.1) Consider rst the problem of using an input w 2 L2; to maximize kP+ z k2. This is 2 exactly the standard problem of computing the Hankel norm of G, i.e., the induced norm of the Hankel operator ? P+ MG : H2 ! H2 and the norm can be expressed in terms of the controllability Gramian Lc and observability Gramian Lo: ALc + LcA + BB = 0 A Lo + LoA + C C = 0: 393
- 411. LINEAR QUADRATIC OPTIMIZATION 394 Although this result is well-known, we will include a time-domain proof similar in technique to the proofs of the optimal H2 and H1 control results. Lemma 15.1 w2L2; kwk2 x(0) = x0 = x0 y0 where y0 solves Lcy0 = x0. inf 2 Proof. Assume (A B) is controllable otherwise, factor out the uncontrollable subspace. Then Lc is invertible and y0 = L;1x0 . Moreover, w 2 L2; can be used to c produce any x(0) = x0 given x(;1) = 0. We need to show inf kwk2 x(0) = x0 = x0 L;1 x0 : (15:2) 2 c w2L2; To show this, we can di erentiate x(t) L;1 x(t) along the solution of (15.1) for any given c input w as follows: d (x L;1 x) = x L;1 x + x L;1 x = x (A L;1 + L;1 A)x + 2hw B L;1 xi: _ c c c _ c c c dt Using Lc equation to substitute for A L;1 + L;1 A and completion of the squares gives c c d ;1 2 ;1 2 dt (x Lc x) = kwk ; kw ; B Lc xk : Integration from t = ;1 to t = 0 with x(;1) = 0 and x(0) = x0 gives x0 L;1x0 = kwk2 ; kw ; B L;1 xk2 kwk2 : c 2 c 2 2 c If w(t) = B e;A t L;1x0 = B L;1e(A+BB L;1)t x0 on (;1 0], then w 2 L2; , w = c ;1 x and equality is achieved, cthus proving (15.2). B Lc 2 Lemma 15.2 supw2BL2; kP+z k2 = supw2BH? kP+ MG wk2 = (LoLc): 2 2 2 Proof. Given x(0) = x0 and w = 0, for t 0 the norm of z(t) = CeAtx0 can be found from kP+z k2 = 2 1 Z 0 x0 eA t C CeAt x0 dt = x0 Lo x0 : Combine this result with Lemma 15.1 to give + z k2 sup kP+ z k2 = sup kPwk2 2 = max x0 Lo1x0 = (LoLc ): 2 x0 6=0 x0 L; x0 w2BL2; 06=w2L2; k 2 c 2
- 412. 15.2. Toeplitz Operators 395 Remark 15.1 Another useful way to characterize the Hankel norm is to examine the following quadratic optimization with initial condition x(;1) = 0: sup kP+ z k2 ; 2 kwk2 : 2 2 w2L2; It is easy to see from the de nition of the Hankel norm that +z sup kPwkk2 k 2 06=w2L2; i sup 06=w2L2; kP+ z k2 ; 2 kwk2 2 2 So the Hankel norm is equal to the smallest Now sup kP+ z k2 ; 2 kwk2 2 2 w2L2; = such that the above inequality holds. sup (x0 Lo x0 ; 2 x0 L;1 x0 ) c x0 2Rn ( = 0: 0 +1 (Lo Lc) 2 (Lo Lc) > 2 : ~ Hence the Hankel norm is equal to the square root of (Lo Lc). 15.2 Toeplitz Operators If a transfer matrix G 2 RH1 and kGk1 < 1, then by Corollary 13.24, the Hamiltonian matrix " A + BR;1 D C BR;1 B H= ;1D )C ;(A + BR;1 D C ) ;C (I + DR # R = 2I ; D D with = 1 is in dom(Ric), X = Ric(H ) 0, A + BR;1 (B X + D C ) is stable and A X + XA + (XB + C D)R;1 (B X + D C ) + C C = 0: (15:3) The following lemma o ers yet another consequence of kGk1 < 1. (Recall that the H1 norm of a stable matrix is the Toeplitz operator norm.) Lemma 15.3 Suppose kGk1 < 1 and x(0) = x0 . Then sup (kz k2 ; kwk2 ) = x0 Xx0 2 2 w2L2+ and the sup is achieved.
- 413. LINEAR QUADRATIC OPTIMIZATION 396 Proof. We can di erentiate x(t) Xx(t) as in the last section, use the Riccati equation (15.3) to substitute for A X + XA, and complete the squares to get d (x Xx) = ;kz k2 + kwk2 ; kR;1=2 Rw ; (B X + D C )x]k2 : dt If w 2 L2+ , then x 2 L2+ , so integrating from t = 0 to t = 1 gives kz k2 ; kwk2 = x0 Xx0 ; kR;1=2 Rw ; (B X + D C )x]k2 x0 Xx0 : (15:4) 2 2 2 ;1 (B X +D C )]t If we let w = R;1 (B X + D C )x = R;1 (B X + D C )e A+BR x0 , then w 2 L2+ because A + BR;1 (B X + D C ) is stable. Thus the inequality in (15.4) can be made an equality and the proof is complete. Note that the sup is achieved for a w which is a linear function of the state. 2 As a direct consequence of this lemma, we have the following Corollary 15.4 Let x(0) = x0 and X0 = X0 > 0. (i) if kGk1 < and X = Ric(H ). Then sup kz k2 ; 2 (kwk2 + x0 X0 x0 ) 2 2 n 06=(x0 w)2R L2 0 1) = sup 8 > < > : 06=x0 2Rn <0 =0 = +1 x0 Xx0 ; 2 x0 X0 x0 max (X ; 2 X0 ) < 0 max (X ; 2 X0 ) = 0 max (X ; 2 X0 ) > 0: kP+z k2 < 2 if and only if (D) < , H 2 dom(Ric), sup 2 n L2 0 1) kw k2 + x0 X0 x0 06=(x0 w)2R and max (X ; 2 X0 ) < 0. Remark 15.2 The matrix X0 has the interpretation of the con dence on the initial condition x0 . So if (X0 ) is small, then the initial condition is probably not known very well. In that case min will be large where min denotes the smallest such that max (X ; 2 X0 ) 0. On the other hand, a large (X0 ) implies that the initial condition is known very well and that min is determined essentially by the condition H 2 dom(Ric). ~ A dual version of Lemma 15.3 can also be obtained and is useful in characterizing the so-called 2 2 block mixed Hankel-Toeplitz operator. To do that, we rst note that1 (ii) 2 GT (s) = " A B C D # 1 Note that since the system matrices are real, AT = A B T = B , etc. The conjugate transpose is used here for the transpose for the sake of consistency in notation.
- 414. 15.3. Mixed Hankel-Toeplitz Operators 397 and kGk1 < 1 i kGT k1 < 1. Let J denote the following Hamiltonian matrix " J= A ;BB # " C + ;A ;BD 0 # h ~ R;1 DB C i ~ ~ where R := I ; DD . Then J 2 dom(Ric), Y = Ric(J ) 0, A + (Y C + BD )R;1 C is stable and ~ AY + Y A + (Y C + BD )R;1 (CY + DB ) + BB = 0 (15:5) if kGk1 < 1. For simplicity, we shall assume that (A B ) is controllable hence, Y > 0. The case in which Y is singular can be obtained by restricting x0 2 Im(Y ) and replacing Y ;1 by Y + . Lemma 15.5 Suppose kGk1 < 1 and (A B) is controllable. Then sup kP;z k2 ; kwk2 x(0) = x0 = ;x0 Y ;1 x0 2 2 w2L2; and the sup is achieved. Proof. Analogous to the proof of Lemma 15.3, we can di erentiate x(t) Y ;1x(t), use the Riccati equation (15.5) to substitute for AY + Y A , and complete the squares to get d ;1 2 2 ;1=2 ;1 2 dt (x Y x) = ;kz k + kwk ; kR Rw ; (B Y + D C )x]k where R = I ; D D > 0. If w 2 L2; , then x 2 L2; and x(;1) = 0 so integrating from t = ;1 to t = 0 gives kz k2 ; kwk2 = ;x0 Y ;1 x0 ; kR;1=2 Rw ; (B Y ;1 + D C )x]k2 ;x0 Y ;1 x0 : (15:6) 2 2 2 If we let w = R;1 (B Y ;1 + D C )x = R;1 (B Y ;1 + D C )e A+BR;1 (B Y ;1 +D C )]tx0 , ~ then w 2 L2; because A + BR;1(B Y ;1 + D C ) = ;Y fA +(Y C + BD )R;1 C g Y ;1 ~ ;1 C is stable. Thus the inequality can be made an equality and A + (Y C + BD )R and the proof is complete. 2 15.3 Mixed Hankel-Toeplitz Operators h i h i Now suppose that the input is partitioned so that B = B1 B2 , D = D1 D2 , h i G(s) = G1 (s) G2 (s) 2 RH1
- 415. LINEAR QUADRATIC OPTIMIZATION 398 and w is partitioned conformably. Then kG2 k1 < 1 i HW := " A 0 ;C C ;A # " + # (D2 ) < 1, and h B2 ; R2 1 D2 C B2 ;C D2 i is in dom(Ric) where R2 := I ; D2 D2 > 0. For HW 2 dom(Ric), de ne W = Ric(HW ). Let (" # ) w1 w 2 H ? w 2 L : ? L2 = w 2 W := H2 (15:7) 1 2 2 2 w2 We are interested in a test for supw2BW kP+ z k2 < 1, or, equivalently, sup k;wk2 < 1 w2BW (15:8) where ; = P+ MG1 MG2 ] : W ! H2 is a mixed Hankel-Toeplitz operator: " w ; 1 w2 # = P+ = P+ h h " # w1 ? w1 2 H2 w2 2 L2 G1 G2 w2 i G1 G2 P; w + P+ G2 P+ w2 : i ? ? Thus ; is the sum of the Hankel operator P+ MG : H2 H2 ! H2 and the Toeplitz operator P+ MG2 : H2 ! H2 . The following lemma generalizes Corollary 13.24 (B1 = 0 D1 = 0) and Lemma 15.2 (B2 = 0 D2 = 0). Lemma 15.6 supw2BW k;wk2 < 1 i the following two conditions hold: (i) (D2 ) < 1 and HW 2 dom(Ric) (ii) (WLc) < 1. Proof. By Corollary 13.24, condition (i) is necessary for (15.8), so we will prove that given condition (i), (15.8) holds i condition (ii) holds. We will do this by showing, equivalently, that (WLc) 1 i supw2BW k;wk2 1. By de nition of W , if w 2 W then kP+ z k2 ; kwk2 = kP+ z k2 ; kP+ w2 k2 ; kP; wk2 : 2 2 2 2 2 2 through x(0). Thus if Lc is invertible, Note that the last term only contributes to kP+ z k2 then Lemma 15.1 and 15.3 yield sup kP+ z k2 ; kwk2 x(0) = x0 = x0 Wx0 ; x0 L;1 x0 2 2 c w2W (15:9) and the supremum is achieved for some w 2 W that can be constructed from the previous lemmas. Since (WLc ) 1 i 9 x0 6= 0 such that the right-hand side of
- 416. 15.4. Mixed Hankel-Toeplitz Operators: The General Case* (15.9) is 399 1 i 9 w 2 W , w 6= 0 such that 0, we have, by (15.9), that (WLc) kP+ z k2 kwk2 . But this is true i supw2BW k;wk2 1. 2 2 If Lc is not invertible, we need only restrict x0 in (15.9) to Im(Lc ), and then the above argument generalizes in a straightforward way. 2 In the di erential game problem considered later and in the H1 optimal control problem, we will make use of the adjoint ; : H2 ! W , which is given by " # " P (G z ) = P; G1 ; z= ; 1 G2 z G2 # z (15:10) where P; Gz := P; (Gz ) = (P; MG)z . That the expression in (15.10) is actually the adjoint of ; is easily veri ed from the de nition of the inner product on vector-valued L2 space. The adjoint of ; : W ! H2 is the operator ; : H2 ! W such that < z ;w >=< ; z w > for all w 2 W , z 2 H2 . By de nition, we have < z ;w > = < z P+ (G1 w1 + G2 w2 ) >=< z G1 w1 > + < z G2w2 > = < P; (G1 z ) w1 > + < G2 z w2 > = <; z w>: The mixed Hankel-Toeplitz operator just studied is the so-called 2 1-block mixed Hankel-Toeplitz operator. There is a 2 2-block generalization. 15.4 Mixed Hankel-Toeplitz Operators: The General Case* Historically, the mixed Hankel-Toeplitz operators have played important roles in H1 theory, so it is interesting to consider the 2 2-block generalization of Lemma 15.6. In fact, the whole H1 control theory can be developed using these tools. See Section 17.7 in Chapter 17. The proof of Lemma 15.7 below is completely straightforward and fairly short, given the other results in the previous sections. Suppose that 2 A B1 B2 G(s) = G11 (s) G12 (s) = 6 C1 D11 D12 4 G21 (s) G22 (s) C2 D21 D22 # " Denote " # 3 7 5 " # A B 2 RH : =: 1 C D h D 2 := D12 D2 := D21 D22 D22 Rx := I ; D 2 D 2 Ry := I ; D2 D2 i
- 417. LINEAR QUADRATIC OPTIMIZATION 400 " # " # h i A 0 B2 ; HX := + Rx 1 D 2 C B2 ;C C ;A ;C D 2 " # " # h i A 0 C2 ; HY := + Ry 1 D2 B C2 : ;BB ;A ;BD2 ? De ne W = H2 L2 , Z = H2 L2 , and ; : W ! Z as " # " #" w P 0 ; 1 = + w2 0 I G11 G12 G21 G22 #" # w1 : w2 Lemma 15.7 supw2BW k;wk2 < 1 holds i the following three conditions hold: (i) (D 2 ) < 1 and HX 2 dom(Ric) (ii) (D2 ) < 1 and HY 2 dom(Ric) (iii) (XY ) < 1 for X = Ric(HX ) and Y = Ric(HY ). Proof. The mixed Hankel-Toeplitz operator can be written as " ; # " # 2 0 w1 = P G w1 + 6 h 4 + P; G21 G22 w2 w2 Hence h i " 3 w1 w2 # 7 5 : i k;wk2 = kP+ Gwk2 + kP; G21 G22 wk2 : 2 2 2 So supw2BW k;wk < 1 implies sup kP+ Gwk < 1 w2BW and But and h G21 G22 h i sup kP; G21 G22 wk < 1: w2BW " i G12 G22 # 1 = sup P+ G w2 2H2 h " 0 # w2 i = sup P; G21 G22 w 1 ? w2H2 2 sup kP+ Gwk < 1 w2BW h 2 i sup kP; G21 G22 wk < 1: w2BW These two inequalities then imply that (i) and (ii) are necessary. Analogous to the proof of Lemma 15.6, we will show that given conditions (i) and (ii), supw2BW k;wk2 < 1
- 418. 15.5. Linear Quadratic Max-Min Problem 401 holds i condition (iii) holds. We will do this by showing, equivalently, that (XY ) 1 i supw2BW k;wk2 1. By de nition of W , if w 2 W then h ; i k;wk2 ; kwk2 = kP+ z k2 ; kP+ w2 k2 + kP; G21 G22 wk2 ; kP;wk2 : 2 2 2 2 2 2 Thus if Y is invertible, then Lemma 15.3 and 15.5 yield sup k;wk2 ; kwk2 x(0) = x0 = x0 Xx0 ; x0 Y ;1 x0 : 2 2 w2W Now the same arguments as in the proof of Lemma 15.6 give the desired conclusion. 2 15.5 Linear Quadratic Max-Min Problem Consider the dynamical system x = Ax + B1 w + B2 u _ z = C1 x + D12 u with the following assumptions: (i) (C1 A) is observable (ii) (A B2 ) is stabilizable h i h (15.11) (15.12) i (iii) D12 C1 D12 = 0 I . In this section, we are interested in answering the following question: when sup umin kz k2 < 1? 2L w2BL2+ 2+ Remark 15.3 This max-min problem is a game problem in the sense that u is chosen to minimize the quadratic norm of z and that w is chosen to maximize the norm. In other words, inputs u and w act as opposing players". This linear quadratic max-min problem can be reformulated in the traditional fashion as in the previous sections: sup umin 2L w2L2+ Z 2+ 0 1 kz k2 ; kwk2 dt = sup umin kz k2 ; kwk2 : 2 2 2L2+ w2L2+ A conventional game problem setup would be to consider the min-max problem, i.e., switching the order of sup and min. However, it will be seen that they are equivalent and that a saddle point exists. By saying that, we would like to warn readers that this may not be true in the general case where z = C1 x + D11 w + D12 u and D11 6= 0. In that case, it is possible that supw inf u < inf u supw . This will be elaborated in Chapter 17. It should also be pointed out that the results presented here still hold, subject to some minor modi cations, if the assumptions (i) and (iii) on the dynamical system are relaxed to:
- 419. LINEAR QUADRATIC OPTIMIZATION 402 " (i)' A ; j!I B2 C1 D12 # has full column rank for all ! 2 R, and ~ (iii)' D12 has full column rank. It is clear from the assumptions that H2 2 dom(Ric) and X2 = Ric(H2 ) > 0, where " # A ;B2 B2 : H2 = ;C1 C1 ;A Let F2 = ;B2 X2 and D? be such that D12 D? ] is an orthogonal matrix. De ne " Gc (s) := AF2 I C1F2 0 and " # " U (s) := AF2 B2 C1F2 D12 # # ;1 (15:13) U? = AF2 ;X2 C1 D? C1F2 D? where AF2 = A + B2 F2 and C1F2 = C1 + D12 F2 . The following is easily proven using Lemma 13.29 by obtaining a state-space realization and by eliminating uncontrollable states using a little algebra involving the Riccati equation for X2 . h i Lemma 15.8 U U?] is square and inner and a realization for Gc U U? is " # ;1 Gc U U? = AF2 ;B2 X2 C1 D? 2 RH2 : X2 0 0 h i (15:14) This implies that U and U? are each inner and that both U? Gc and U Gc are in RH? . To answer our earlier question, de ne a Hamiltonian matrix H1 and the 2 associated Riccati equation as " A B1 B1 ; B2 B2 H1 := ;C1 C1 ;A # A X1 + X1 A + X1 B1 B1 X1 ; X1 B2 B2 X1 + C1 C1 = 0: Theorem 15.9 sup umin kzk2 < 1 if and only if H1 2 dom(Ric) and X1 = 2L2+ w2BL2+ Ric(H1 ) > 0. Furthermore, if the condition is satis ed, then u = F1 x with F1 := ;B2 X1 is an optimal control.
- 420. 15.5. Linear Quadratic Max-Min Problem 403 Proof. ()) As in Chapter 14, de ne := u ; F2x to get z = Gc B1 w + U : Then the hypothesis implies that sup min kz k2 < 1: (15:15) w2BH2 2H2 Since by Lemma 15.8 U U? ] is square and inner, kz k2 = k U U?] z k2, and h U U? i " z = U G c B1 w + U? Gc B1 w Thus sup min kz k2 = sup min w2BH2 2H2 # " w2BH2 2H2 " # P (U Gc B1 w) + P+ (U Gc B1 w + ) : = ; U? Gc B1 w P; (U Gc B1 w) + P+ (U Gc B1 w + ) U? Gc B1 w # 2 and the right hand of the above equation is minimized by = ;P+ (U Gc B1 w) we have " P; (U Gc B1 w) sup min kz k2 = sup U? Gc B1 w w2BH2 2H2 w2BH2 =: sup k; wk2 < 1 # 2 w2BH2 where ; : L2+ ! W is de ned as " # " P (U Gc B1 w) = P; U ; w= ; U? Gc B1 w U? with W := (" q1 q2 # # Gc B1 w ) ? q1 2 H2 q2 2 L2 : Note that from equation (15.10) the adjoint operator ; : W ! H2 is given by " # h q ; 1 = P+ (B1 Gc (Uq1 + U? q2 )) = P+ B1 Gc U U? q2 where Gc U U? ] 2 RH2 has the realization in (15.14). So we have sup k;qk2 < 1: q2BW i " q1 q2 #
- 421. 404 LINEAR QUADRATIC OPTIMIZATION This is just the condition (15.8), so from Lemma 15.6 and equation (15.14) we have that " ; ; # AF2 X2 1C1 C1 X2 1 2 dom(Ric) HW := ;X2B1 B1 X2 ;AF2 and W = Ric(HW ) 0. Note that the observability of (C1 A) implies X2 > 0. ; Furthermore, the controllability Gramian for the system (15.14) is X2 1 since ; ; ; ; AF2 X2 1 + X2 1 AF2 + B2 B2 + X2 1 C1 C1 X2 1 = 0: ; Lemma 15.6 also implies (WX2 1) < 1 or, equivalently, X2 # W . Using the Ric> " ;I X2;1 provides a similarity ;X2 0 cati equation for X2 , one can verify that T := transformation between H1 and HW , i.e., H1 = THW T ;1. Then " # " # ;1 X; (H1 ) = T X; (HW ) = T Im I = Im I ; X2 W W X2 so H1 2 dom(Ric) and X1 = X2 (X2 ; W );1 X2 > 0. (() If H1 2 dom(Ric) and X1 = Ric(H1 ) > 0, A + B1 B1 X1 ; B2 B2 X1 is stable. De ne AF1 := A + B2 F1 C1F1 := C1 + D12 F1 : Then the Riccati equation can be written as AF1 X1 + X1 AF1 + C1F1 C1F1 + X1 B1 B1 X1 = 0: We conclude from Lyapunov theory that AF1 is stable since (AF1 B1 X1 ) is detectable and X1 > 0. Now with the given control law u = F1 x, the dynamical system becomes x = AF1 x + B1 w _ z = C1F1 x: So by Corollary 13.24, kTzw k1 < 1, i.e., supw2L2+ kz k2 < 1 for the given control law. 2 Theorem 15.9 will be used in the next chapter to solve the FI H1 control problem. 15.6 Notes and References Di erential game is a well-studied topic, see e.g., Bryson and Ho 1975]. The paper by Mageirou and Ho 1977] is one of the early papers that are relevant to the topics covered in this chapter. The current setup and proof are taken from Doyle, Glover, Khargonekar, and Francis 1989]. The application of game theoretic results to H1 problems can be found in Basar and Bernhard 1991], Khargonekar, Petersen, and Zhou 1990], Limebeer, Anderson, Khargonekar, and Green 1992], and references therein.
- 422. 16 H1 Control: Simple Case In this chapter we consider H1 control theory. Speci cally, we formulate the optimal and suboptimal H1 control problems in section 16.1. However, we will focus on the suboptimal case in this book and discuss why we do so. In section 16.2 all suboptimal controllers are characterized for a class of simpli ed problems while leaving the more general problems to the next chapter. Some preliminary analysis is given in section 16.3 for the output feedback results. The analysis suggested the need for solving the Full Information and Output Estimation problems, which are the topics of sections 16.4-16.7. Section 16.8 discusses the H1 separation theory and presents the proof of the output feedback results. The behavior of the H1 controller as a function of performance level is considered in section 16.9. The optimal controllers are also brie y considered in this section. Some other interpretations of the H1 controllers are given in section 16.10. Finally, section 16.11 presents the formulas for an optimal H1 controller. 16.1 Problem Formulation Consider the system described by the block diagram z y G - K 405 w u
- 423. H1 CONTROL: SIMPLE CASE 406 where the plant G and controller K are assumed to be real-rational and proper. It will be assumed that state space models of G and K are available and that their realizations are assumed to be stabilizable and detectable. Recall again that a controller is said to be admissible if it internally stabilizes the system. Clearly, stability is the most basic requirement for a practical system to work. Hence any sensible controller has to be admissible. Optimal H1 Control: nd all admissible controllers K (s) such that kTzw k1 is minimized. It should be noted that the optimal H1 controllers as de ned above are generally not unique for MIMO systems. Furthermore, nding an optimal H1 controller is often both numerically and theoretically complicated, as shown in Glover and Doyle 1989]. This is certainly in contrast with the standard H2 theory, in which the optimal controller is unique and can be obtained by solving two Riccati equations without iterations. Knowing the achievable optimal (minimum) H1 norm may be useful theoretically since it sets a limit on what we can achieve. However, in practice it is often not necessary and sometimes even undesirable to design an optimal controller, and it is usually much cheaper to obtain controllers that are very close in the norm sense to the optimal ones, which will be called suboptimal controllers. A suboptimal controller may also have other nice properties over optimal ones, e.g., lower bandwidth. Suboptimal H1 Control: Given > 0, nd all admissible controllers K (s) if there is any such that kTzw k1 < . For the reasons mentioned above, we focus our attention in this book on suboptimal control. When appropriate, we brie y discuss what will happen when approaches the optimal value. 16.2 Output Feedback H1 Control 16.2.1 Internal Stability and Input/output Stability Now suppose K is a stabilizing controller for the system G. Then the internal stability guarantees Tzw = F` (G K ) 2 RH1 , but the latter does not necessarily imply the internal stability. The following lemma provides the additional (mild) conditions to the equivalence of Tzw = F`(G K ) 2 RH1 and internal stability of the closed-loop system. To state the lemma, we shall assume that G and K have the following stabilizable and detectable realizations, respectively: 2 A B1 B2 6 G(s) = 4 C1 D11 D12 C2 D21 D22 3 7 5 " # ^ ^ K (s) = A B : ^ D C ^
- 424. 16.2. Output Feedback H1 Control 407 Lemma 16.1 Suppose that the realizations for G and K are both stabilizable and detectable. Then the feedback connection Tzw = F` (G K ) of the realizations for G and K is " # A ; I B2 (a) detectable if C1 D12 0 has full column rank for all Re " A ; I B1 (b) stabilizable if C2 D21 # has full row rank for all Re 0: Moreover, if (a) and (b) hold, then K is an internally stabilizing controller i Tzw 2 RH1 . Proof. The state-space equations for the closed-loop are: 3 2 ^ ^ ^ B1 + B2 DL1 D21 7 B2 L2 C A + B2 DL1 C2 6 ^ ^ ^ ^ ^ 7 BL1 C2 A + BL1 D22 C BL1 D21 F` ( G K ) = 6 5 4 ^ ^ ^ C1 + D12 L2 DC2 D12 L2 C D11 + D12 DL1 D21 " # Ac Bc =: Cc Dc ^ ^ where L1 := (I ; D22 D);1 , L2 := (I ; DD22 );1 . Suppose F`(G K ) has undetectable state (x0 y0)0 and mode Re 0 then the PBH test gives " This can be simpli ed as " and Now if Ac ; I Cc A ; I B2 C1 D12 #" #" # x = 0: y # x ^ 1 C2 x + L2 Cy = 0 ^ DL ^ ^ ^ BL1(C2 x + D22 Cy) + Ay ; y = 0: " # A ; I B2 C1 D12 ^ ^ ^ ^ has full column rank, then x = 0 and Cy = 0. This implies Ay = y. Since (C A) is detectable, we get y = 0, which is a contradiction. Hence part (a) is proven, and part (b) is a dual result. 2 These relations will be used extensively below to simplify our development and to enable us to focus on input/output stability only.
- 425. H1 CONTROL: SIMPLE CASE 408 16.2.2 Contraction and Stability One of the keys to the entire development of H1 theory is the fact that the contraction and internal stability is preserved under an inner linear fractional transformation. Theorem 16.2 Consider the following feedback system: z r P - Q w v " # P = P11 P12 2 RH1 P21 P22 ; Suppose that P P = I , P211 2 RH1 and that Q is a proper rational matrix. Then the following are equivalent: (a) The system is internally stable, well-posed, and kTzw k1 < 1. (b) Q 2 RH1 and kQk1 < 1. Proof. (b) ) (a). Note that since P Q 2 RH1, the system internal stability is guaranteed if (I ; P22 Q);1 2 RH1 . Therefore, internal stability and well-posedness follow easily from kP22 k1 1, kQk1 < 1, and a small gain argument. (Note that kP22 k1 1 follows from the fact that P22 is a compression of P .) To show that kTzw k1 < 1, consider the closed-loop system at any frequency s = j! with the signals xed as complex constant vectors. Let kQk1 =: < 1 and note that ; Twr = P211 (I ; P22 Q) 2 RH1 . Also let := kTwr k1 . Then kwk krk, and P inner implies that kz k2 + krk2 = kwk2 + kvk2. Therefore, kz k2 kwk2 + ( 2 ; 1)krk2 1 ; (1 ; 2 ) ;2 ]kwk2 which implies kTzw k1 < 1 . (a) ) (b). To show that kQk1 < 1, suppose there exist a (real or in nite) frequency ! and a constant nonzero vector r such that at s = j! , kQrk krk. Then setting ; w = P211 (I ; P22 Q)r, v = Qr gives v = Tvw w. But as above, P inner implies that 2 + krk2 = kwk2 + kv k2 and, hence, that kz k2 kwk2 , which is impossible since kz k kTzw k1 < 1. It follows that max (Q(j!)) < 1 for all !, i.e., kQk1 < 1 since Q is rational. To show Q 2 RH1 , let Q = NM ;1 with N M 2 RH1 be a right coprime factorization, i.e., there exist X Y 2 RH1 such that XN + Y M = I . We shall show that M ;1 2 RH1 . By internal stability we have Q(I ; P22 Q);1 = N (M ; P22 N );1 2 RH1 and (I ; P22 Q);1 = M (M ; P22 N );1 2 RH1 :
- 426. 16.2. Output Feedback H1 Control 409 Thus XQ(I ; P22 Q);1 + Y (I ; P22 Q);1 = (M ; P22 N );1 2 RH1 : This implies that the winding number of det(M ; P22 N ), as s traverses the Nyquist contour, equals zero. Now note the fact that, for all s = j!, det M ;1 6= 0, det(I ; P22 Q) 6= 0 for all in 0,1] (this uses the fact that kP22 k1 1 and kQk1 < 1). Also, det(I ; P22 Q) = det(M ; P22 N ) det M ;1, and we have det(M ; P22 N ) 6= 0 for all in 0,1] and all s = j!. We conclude that the winding number of det M also equals zero. Therefore, Q 2 RH1 , and the proof is complete. 2 16.2.3 Simplifying Assumptions In this chapter, we discuss a simpli ed version of H1 theory. The general case will be considered in the next chapter. The main reason for doing so is that the general case has its unique features but is much more involved algebraically. Involved algebra may distract attention from the essential ideas of the theory and therefore lose insight into the problem. Nevertheless, the problem considered below contains the essential features of the H1 theory. The realization of the transfer matrix G is taken to be of the form 2 3 A B1 B2 6 G(s) = 4 C1 0 D12 7 : 5 C2 D21 0 The following assumptions are made: (i) (A B1 ) is stabilizable and (C1 A) is detectable (ii) (A B2 ) is stabilizable and (C2 A) is detectable h i h " # (iii) D12 C1 D12 = 0 I " (iv) # i B1 D = 0 . D21 21 I Assumption (i) is made for a technical reason: together with (ii) it guarantees that the two Hamiltonian matrices (H2 and J2 in Chapter 14) in the H2 problem belong to dom(Ric). This assumption simpli es the theorem statements and proofs, but if it is relaxed, the theorems and proofs can be modi ed so that the given formulae are still correct, as will be seen in the next chapter. An important simpli cation that is a consequence of the assumption (i) is that internal stability is essentially equivalent to input-output stability (Tzw 2 RH1 ). This equivalence enables us to focus only on input/output stability and is captured in Corollary 16.3 below. Of course, assumption (ii) is necessary and su cient for G to be internally stabilizable, but is not needed
- 427. H1 CONTROL: SIMPLE CASE 410 to prove the equivalence of internal stability and Tzw 2 RH1 . (Readers should be clear that this does not mean that the realization for G need not be stabilizable and detectable. In point of fact, the internal stability and input-output stability can never be equivalent if either G or K has unstabilizable or undetectable modes.) Corollary 16.3 Suppose that assumptions (i), (iii), and (iv) hold. Then a controller K is admissible i Tzw 2 RH1 . Proof. The realization for plant G is stabilizable and detectable by assumption (i). We only need to verify that the rank conditions of the two matrices in Lemma 16.1 are satis ed. Now suppose assumptions (i) and (iii) are satis ed and let D? be such that i h D12 D? is a unitary matrix. Then " A ; I B2 rank C1 D12 2 # rank 6 4 = 2 = rank 6 4 So has full column rank for all Re " I 3 " # 7 5 D12 D? A; I # " 0 0 D? C1 A ; I B2 C1 D12 0i " 0 " A; I D? C1 A ; I B2 C1 D12 # 3 " B2 # I 7: 5 0 # # has full column rank. However, the last matrix has full rank for all Re 0 i (D? C1 A) is detectable. Since D? (D? C1 ) = (I ; D12 D12 )C1 = C1 , (D? C1 A) is detectable i (C1 A) is detectable. The rank condition for the other matrix follows by duality. 2 Two additional assumptions that are implicit in the assumed realization for G(s) are that D11 = 0 and D22 = 0. As we have mentioned many times, D22 6= 0 does not pose any problem since it is easy to form an equivalent problem with D22 = 0 by a linear fractional transformation on the controller K (s). However, relaxing the assumption D11 = 0 complicates the formulae substantially, as will be seen in the next chapter. 16.2.4 Suboptimal H1 Controllers In this subsection, we present the necessary and su cient conditions for the existence of an admissible controller K (s) such that kTzw k1 < for a given , and, furthermore,
- 428. 16.2. Output Feedback H1 Control 411 if the necessary and su cient conditions are satis ed, we characterize all admissible controllers that satisfy the norm condition. The proofs of these results will be given in the later sections. Let opt := min fkTzw k1 : K (s) admissibleg, i.e., the optimal level. Then, clearly, must be greater than opt for the existence of suboptimal H1 controllers. In Section 16.9 we will brie y discuss how to nd an admissible K to minimize kTzw k1 . Optimal H1 controllers are more di cult to characterize than suboptimal ones, and this is one major di erence between the H1 and H2 results. Recall that similar di erences arose in the norm computation problem as well. The H1 solution involves the following two Hamiltonian matrices: " A H1 := ;C1 C1 ;2 B1 B1 ; B2 B2 # ;A " A J1 := ;B1 B1 ;2 C1 C1 ; C2 C2 ;A # : The important di erence here from the H2 problem is that the (1,2)-blocks are not sign de nite, so we cannot use Theorem 13.7 in Chapter 13 to guarantee that H1 2 dom(Ric) or Ric(H1 ) 0. Indeed, these conditions are intimately related to the existence of H1 suboptimal controllers. Note that the (1,2)-blocks are a suggestive combination of expressions from the H1 norm characterization in Chapter 4 (or bounded real ARE in Chapter 13) and from the H2 synthesis of Chapter 14. It is also clear that if approaches in nity, then these two Hamiltonian matrices become the corresponding H2 control Hamiltonian matrices. The reasons for the form of these expressions should become clear through the discussions and proofs for the following theorem. Theorem 16.4 There exists an admissible controller such that kTzw k1 < i the following three conditions hold: (i) H1 2 dom(Ric) and X1 := Ric(H1) 0 (ii) J1 2 dom(Ric) and Y1 := Ric(J1 ) 0 (iii) (X1 Y1 ) < 2 . Moreover, when these conditions hold, one such controller is " ^ Ksub (s) := A1 ;Z1 L1 F1 0 where # ^ A1 := A + ;2B1 B1 X1 + B2 F1 + Z1 L1 C2 F1 := ;B2 X1 L1 := ;Y1 C2 Z1 := (I ; ;2 Y1 X1 );1 : The H1 controller displayed in Theorem 16.4, which is often called the central controller or minimum entropy controller, has certain obvious similarities to the H2 controller as well as some important di erences. Although not as apparent as in the H2 case, the H1 controller also has an interesting separation structure. Furthermore,
- 429. H1 CONTROL: SIMPLE CASE 412 each of the conditions in the theorem can be given a system-theoretic interpretation in terms of this separation. These interpretations, given in Section 16.8, require the ltering and full information (i.e., state feedback) results in sections 16.7 and 16.4. The proof of Theorem 16.4 is constructed out of these results as well. The term central controller will be obvious from the parameterization of all suboptimal controllers given below, while the meaning of minimum entropy will be discussed in Section 16.10.1. The following theorem parameterizes the controllers that achieve a suboptimal H1 norm less than . Theorem 16.5 If conditions (i) to (iii) in Theorem 16.4 are satis ed, the set of all admissible controllers such that kTzw k1 < equals the set of all transfer matrices from y to u in u M1 - y Q 2 ^ A1 ;Z1 L1 Z1 B2 6 M1 (s) = 4 F1 0 I I 0 ;C2 3 7 5 where Q 2 RH1 , kQk1 < . As in the H2 case, the suboptimal controllers are parameterized by a xed linearfractional transformation with a free parameter Q. With Q = 0 (at the center" of the set kQk1 < ), we recover the central controller Ksub (s). 16.3 Motivation for Special Problems Although the proof for output feedback results will be given later, we shall now try to give some ideas for approaching the problem. Speci cally, we try to motivate the study of the OE (and hence other special problems) and show how this problem arises naturally in proving the output feedback results. The key is to use the fact that contraction and internal stability are preserved under an inner linear fractional transformation, which is Theorem 16.2. Assuming that X1 exists, we will show that the general output feedback problem boils down to an output estimation problem which can be solved easily if a state feedback or full information control problem can be solved. Suppose X1 := Ric(H1 ) exists. Then X1 satis es the following Riccati equation: A X1 + X1 A + C1 C1 + ;2 X1 B1 B1 X1 ; X1 B2 B2 X1 = 0: (16:1) Let x denote the state of G with respect to a given input w, and then we can di erentiate x(t) X1 x(t): d _ _ dt (x X1 x) = x X1 x + x X1 x
- 430. 16.3. Motivation for Special Problems 413 = x (A X1 + X1 A)x + 2hw B1 X1 xi + 2hu B2 X1 xi: Using the Riccati equation for X1 to substitute in for A X1 + X1 A gives d (x X x) = ;kC xk2 ; ;2 kB X xk2 + kB X xk2 +2hw B X xi +2hu B X xi: 1 1 1 1 2 1 1 1 2 1 dt Finally, completion of the squares along with orthogonality of C1 x and D12 u gives the key equation d 2 2 2 2 ;2 2 2 dt (x X1 x) = ;kz k + kwk ; kw ; B1 X1 xk + ku + B2 X1 xk : (16:2) Assume x(0) = x(1) = 0, w 2 L2+ , and integrate (16.2) from t = 0 to t = 1: kz k2 ; 2 kwk2 = ku + B2 X1 xk2 ; 2 kw ; ;2B1 X1 xk2 = kvk2 ; 2 krk2 (16:3) 2 2 2 2 2 2 where r := w ; ;2 B1 X1 x: (16:4) With these new de ned variables, the closed-loop system can be expressed as two interconnected subsystems below: 2 6 4 and x _ z r v := u + B2 X1 x 3 2 A F1 7 6 C1F1 5=4 ; ;1B1 X1 2 6 4 3 B2 D12 0 I 32 32 ;1 B1 x r u 0 2 x _ Atmp B1 B2 7 6 = 4 ;F1 0 I v5 y C2 D21 0 76 54 76 54 x w v 3 AF1 := A + B2 F1 C1F1 := C1 + D12 F1 7 5 3 Atmp := A + ;2 B1 B1 X1 7 5 where F1 is de ned as in Section 16.2.4. This is shown in the following diagram: z r w P XXXXX XX Gtmp y - v r u K w 1
- 431. H1 CONTROL: SIMPLE CASE 414 where " P := P11 P12 P21 P22 2 and AF1 6 = 4 C1F1 ; ;1B1 X1 ;1 B1 2 # 3 3 0 I B2 D12 0 7 5 (16:5) Atmp B1 B2 6 Gtmp = 4 ;F1 0 I 7 : (16:6) 5 C2 D21 0 The equality (16.3) motivates the change of variables to r and v as in (16.4), and these variables provide the connection between Tzw and Tvr . Note that Tz( w) = ;1 Tzw and Tv( r) = ;1Tvr . It is immediate from equality (16.3) that kTzw k1 i kTvr k1 . While this is the basic idea behind the proof of Lemma 16.8 below, the details needed for strict inequality and internal stability require a bit more work. Note that wworst := ;2 B1 X1 x is the worst disturbance input in the sense that it maximizes the quantity kz k2 ; 2 kwk2 in (16.3) for the minimizing value of u = 2 2 ;B2 X1 x that is, the u making v = 0 and the w making r = 0 are values satisfying a saddle point condition. (See Section 17.8 for more interpretations.) It is also interesting to note that wworst is the optimal strategy for w in the corresponding LQ game problem (see the di erential game problem in the last chapter). Equation (16.3) also suggests that u = ;B2 X1 x is a suboptimal control for a full information (FI) problem if the state x is available. This will be shown later. In terms of the OE problem for Gtmp , the output being estimated is the optimal FI control input F1 x and the new disturbance r is o set by the worst case" FI disturbance input wworst . Notice the structure of Gtmp : it is an OE problem. We will show below that the output feedback can indeed be transformed into the OE problem. To show this we rst need to prove some preliminary facts. Lemma 16.6 Suppose H1 2 dom(Ric) and X1 = Ric(H1). Then AF1 = A + B2F1 is stable i X1 0. Proof. Re-arrange the Riccati equation for X1 and use the de nition of F1 and C1F1 to get " # " # C1F1 C1F1 AF1 X1 + X1 AF1 + = 0: (16:7) ; ;1 B1 X1 ; ;1B1 X1 Since H1 2 dom(Ric), (AF1 + ;2 B1 B1 X1 ) is stable and hence (B1 X1 AF1 ) is detectable. Then from standard results on Lyapunov equations (see Lemma 3.19), AF1 is stable i X1 0. 2 Equation (16.3) can be written as kz k2 + k rk2 = k wk2 + kvk2 : 2 2 2 2
- 432. 16.3. Motivation for Special Problems This suggests that P might be inner when X1 lemma. 415 0, which is veri ed by the following Lemma 16.7 If H1 2;dom(Ric) and X1 = Ric(H1) RH1 and inner, and P211 2 RH1 . 0, then P in (16.5) is in Proof. By Lemma 16.6, AF1 is stable. So P 2 RH1 . That P is inner (P P = I ) follows from Lemma 13.29 upon noting that the observability Gramian of P is X1 (see (16.7)) and " #" # " # ;1 B1 C1F1 0 I + X1 = 0: D12 0 ; ;1B1 X1 B2 ; Finally, the state matrix for P211 is (AF1 + ;2 B1 B1 X1 ), which is stable by de nition. ;1 2 RH1 . Thus, P21 2 The following lemma connects these two systems Tzw and Tvr , which is the central part of the separation argument in Section 16.8.2. Recall that internal and inputoutput stability are equivalent for admissibility of K in the output feedback problem by Corollary 16.3 . Lemma 16.8 Assume H1 2 dom(Ric) and X1 = Ric(H1) 0. Then K is admissible for G and kTzw k1 < i K is admissible for Gtmp and kTvr k1 < . Proof. We may assume without loss of generality that the realization of K is stabilizable and detectable. Recall from Corollary 16.3 that internal stability for Tzw is equivalent to Tzw 2 RH1 . Similarly since " # " #" # Atmp ; I B1 = A ; I B1 I 0 C2 D21 C2 D21 ;2 B1 X1 I has full row rank for all Re( ) 0 and since " # A ; I B2 = det(A + B F ; I ) 6= 0 det tmp tmp 2 1 ;F1 I for all Re( ) 0 by the stability of Atmp + B2 F1 , we have that " Atmp ; I B2 ;F1 I # has full column rank. Hence by Lemma 16.1 the internal stability of Tvr , i.e., the internal stability of the subsystem consisting of Gtmp and controller K , is also equivalent to Tvr 2 RH1 . Thus internal stability is equivalent to input-output stability for both
- 433. H1 CONTROL: SIMPLE CASE 416 G and Gtmp . This shows that K is an admissible controller for G if and only if it is admissible for Gtmp . Now it follows from Theorem 16.2 and Lemma 16.7 along with the above block diagram that Tz( w) 1 < 1 i Tv( r) 1 < 1 or, equivalently, kTzw k1 < i kTvr k1 < . 2 From the previous analysis, it is clear that to solve the output feedback problem we need to show (a) H1 2 dom(Ric) and X1 = Ric(H1 ) 0 (b) kTvr k1 < . To show (a), we need to solve a FI problem. The problem (b) is an OE problem which can be solved by using the relationship between FC and OE problems in Chapter 12, while the FC problem can be solved by using the FI solution through duality. So in the sections to follow, we will focus on these special problems. 16.4 Full Information Control Our system diagram in this section is standard as before z y with w G - u K 2 A 6 6 G(s) = 6 " C1 # 6 I 4 3 B1 " 0 0 # B2 7 D 7 " 12# 7 : 7 0 5 0 I 0 The H1 problem corresponding to this setup again is not, strictly speaking, a special case of the output feedback problem because it does not satisfy all of the assumptions. In particular, it should be noted that for the FI (and FC in the next section) problem, internal stability is not equivalent to Tzw 2 RH1 since 2 6 4 A; # I " I 0 B " 1# 0 I 3 7 5
- 434. 16.4. Full Information Control 417 can never have full row rank, although this presents no di culties in solving this problem. We simply must remember that in the FI case, K admissible means internally stabilizing, not just Tzw 2 RH1 . We have seen that in the H2 FI case, the optimal controller uses just the state x even though the controller is provided with full information. We will show below that, in the H1 case, a suboptimal controller exists which also uses just x. This case could have been restricted to state feedback, which is more traditional, but we believe that, once one gets outside the pure H2 setting, the full information problem is more fundamental and more natural than the state feedback problem. One setting in which the full information case is more natural occurs when the parameterization of all suboptimal controllers is considered. It is also appropriate when studying the general case when D11 6= 0 in the next chapter or when H1 optimal (not just suboptimal) controllers are desired. Even though the optimal problem is not studied in detail in this book, we want the methods to extend to the optimal case in a natural and straightforward way. The assumptions relevant to the FI problem which are inherited from the output feedback problem are (i) (C1 A) is detectable (ii) (A B2 ) is stabilizable h i h i (iii) D12 C1 D12 = 0 I . Assumptions (iv) and the second part of (ii) for the general output feedback case have been e ectively strengthened because of the assumed structure for C2 and D21 . Theorem 16.9 There exists an admissible controller K (s) for the FI problem such that kTzw k1 < if and only if H1 2 dom(Ric) and X1 = Ric(H1 ) 0. Furthermore, if these conditions are satis ed, then the equivalence class of all admissible controllers satisfying kTzw k1 < can be parameterized as h K (s) = F1 ; ;2 Q(s)B1 X1 Q(s) i (16:8) where Q 2 RH1 , kQk1 < . It is easy to see by comparing the H1 solution with the corresponding H2 solution that a fundamental di erence between H2 and H1 controllers is that the H1 controller depends on the disturbance through B1 whereas the H2 controller does not. This di erence is essentially captured by the necessary and su cient conditions for the existence of a controller given in Theorem 16.9. Note that these conditions are the same as condition (i) in Theorem 16.4. The two individual conditions in Theorem 16.9 may each be given their own interpretations. The condition that H1 2 dom(Ric) implies that X1 := Ric(H1 ) exists
- 435. H1 CONTROL: SIMPLE CASE 418 h i and K (s) = F1 0 gives Tzw as " Tzw = AF1 B1 C1F1 0 # AF1 = A + B2 F1 C1F1 = C1 + D12 F1 : (16:9) Furthermore, since Tzw = P11 and P11 P11 = I ; P21 P21 by P P = I , we have kP11 k1 < 1 and kTzw k1 = kP11 k1 < . The further condition that X1 0 is equivalent, by Lemma 16.6, to this K stabilizing Tzw . Remark 16.1 Given that H1 2 dom(Ric) and X1 = Ric(H1) 0, there is an intuitive way to see why all the equivalence classes of controllers can be parameterized in the form of (16.8). Recall the following equality from equation (16.3): kz k2 ; 2 kwk2 = ku + B2 X1 xk2 ; 2 kw ; 2 2 2 ;2 B X1 xk2 = kvk2 ; 2 krk2 : 1 2 2 2 kTzw k1 < means that kz k2 ; 2 kwk2 < 0 for w 6= 0. This implies that r 6= 0 and 2 2 kvk2 ; 2 krk2 < 0. Now all v satisfying this inequality can be written as v = Qr 2 2 for Q 2 RH1 and kQk1 < or as u + B2 X1 x = Q(w ; ;2 B1 X1 x). This gives equation (16.8). ~ We shall now prove the theorem. Proof. ()) For simplicity, in the proof to follow we assume that the system is normalized such that = 1. Further, we will show that we can, without loss of generality, strengthen the assumption on (C1 "A) from detectable to observable. Suppose there ex# ^ ^ ^ ^ = A B1 B2 such that kTzw k1 < 1. If (C1 A) is ists an admissible controller K ^ ^ ^ C D1 D2 " # x detectable but not observable, then change coordinates for the state of G to 1 with x2 x2 unobservable, (C11 A11 ) observable, and A22 stable, giving the following closed-loop state equations: 2 6 6 6 6 6 6 4 x1 _ x2 _ _ x ^ z u 3 2 A11 7 6 7 6 A21 7 6 7 = 6 B11 ^ 7 6 7 6 5 4 C11 ^ D11 0 A22 ^ B12 0 ^ 12 D 0 0 ^ A 0 ^ C B11 B21 B12 B22 ^ B2 0 0 D12 ^2 0 D 32 76 76 76 76 76 76 54 3 x1 x2 7 7 7 x 7: ^ 7 w 7 5 u
- 436. 16.4. Full Information Control 419 If we take a new plant Gobs with state x1 and output z and group the rest of the equations as a new controller Kobs with the state made up of x2 and x, then ^ 2 A11 6 6 Gobs (s) = 6 "C11# 6 I 4 B11 " 0 0 # B21 D " 12# 0 0 I 0 3 7 7 7 7 5 still satis es the assumptions of the FI problem and is stabilized by Kobs with the closed-loop H1 norm kTzw k1 < 1 where 2 3 ^ ^ ^ ^ A22 + B22 D12 B22 C A21 + B22 D11 B12 + B22 D2 6 7 ^ ^ ^ ^ Kobs = 4 B12 A B11 B2 5: ^ 12 ^ ^ 11 ^2 D C D D ^ If we now show that there exists X1 > 0 solving the H1 Riccati equation for Gobs , i.e., ^ X1 = Ric then " A11 B11 B11 ; B21 B21 ;C11 C11 ;A11 " ^ Ric(H1 ) = X1 = X1 0 0 0 # # 0 exists for G. We can therefore assume without loss of generality that (C1 A) is observable. We shall suppose that there exists an admissible controller such that kTzw k1 < 1. But note that the existence of an admissible controller such that kTzw k1 < 1 is equivalent to that the admissible controller makes sup kz k2 < 1 hence, it is necessary w2BL2+ that sup umin kz k2 < 1 2L w2BL2+ 2+ since the latter is always no greater than the former by the fact that the set of signals u generated from admissible controllers is a subset of L2+ . But from Theorem 15.9, the latter is true if and only if H1 2 dom(Ric) and X1 = Ric(H1 ) > 0. Hence the necessity is proven. (() Suppose H1 2 dom(Ric) and X1 = Ric(H1 ) 0 and suppose K (s) is an admissible controller such that kTzw k1 < 1. Again change variables to v := u ; F1 x
- 437. H1 CONTROL: SIMPLE CASE 420 and r := w ; B1 X1 x, so that the closed-loop system is as shown below: z r w P " P = P11 P12 P21 P22 v - Tvr # 2 AF1 B1 B2 = 6 C1F1 0 D12 4 ;B1 X1 I 0 3 7 5 ; By Lemma 16.7, P is inner and P211 2 RH1 . By Theorem 16.2 the system is internally stable and kTzw k1 < 1 i Tvr 2 RH1 and kTvr k1 < 1. Now denote Q := Tvr , then v = Qr and " # x v = u ; F1 x = K ; F1 0 w " # h i x r = w ; B1 X1 x = ;B1 X1 I : w h Hence we have h K ; F1 0 or h i " i # x = Qh ;B1 X1 I w i h i " x w # i K ; F1 0 = Q ;B1 X1 I : Thus h K (s) = F1 ; Q(s)B1 X1 Q(s) i Q 2 RH1 kQk1 < 1: Hence all suboptimal controllers are in the stated equivalence class. 2 Remark 16.2 It should be emphasized that the set of controllers given above does not parameterize all controllers although it is su cient for the purpose of deriving the output feedback results, and that is why equivalence class" is used. It is clear that h i there is a suboptimal controller K1 = F1 0 with F1 6= F1 however, there is no choice of Q such that K1 belongs to the set. Nevertheless, this problem will not occur in output feedback case. ~ The following theorem gives all full information controllers. Theorem 16.10 Suppose the condition in Theorem 16.9 is satis ed then all admissible controllers satisfying kTzw k1 < can be parameterized as K = F` (MFI Q):
- 438. 16.4. Full Information Control u MFI - Q h 421 y 2 MFI (s) = 6 6 6 6 6 4 h A + B2 F1 " 0 ;I # " 0 i h 0 B1 F1 0 i i I 0 ;2B1 X1 I ; # B2 I 0 3 7 7 7 7 7 5 where Q = Q1 Q2 2 RH1 and kQ2k1 < . Remark 16.3 It is simple to verify that for Q1 = 0, we have h i K = F1 ; ;2Q2 B1 X1 Q2 which is the parameterization given in Theorem 16.9. The parameterization of all suboptimal FC controllers follows by duality and, therefore, are omitted. ~ Proof. We only need to show that F`(MFI Q) with kQ2k1 < parameterizes all FI H1 suboptimal controllers. To show that, we shall make a change of variables as before: v = u + B2 X1 x r = w ; ;2 B1 X1 x: Then the system equations can be written as follows: 2 3 2 ;1 B1 B2 3 2 x 3 x _ AF1 6 7 6 C1F1 0 D12 7 6 w 7 4 z 5=4 54 5 ;1B1 X1 r ; I 0 v and 2 3 2 3 Atmp B1 B2 2 3 x _ 6 7 ;F1 # " 0 # I 7 6 x 7 6 7 6 " 6 74 r 5 4 v 5=6 7 I 0 4 0 5 u y ;2 where Atmp := A + ;2 B1 B1 X1 . B1 X1 I This is shown pictorially in the following diagram: z r y P XXXXX X ^ GFI - K w w v r u 1
- 439. H1 CONTROL: SIMPLE CASE 422 where P is as given in equation (16.5) and 2 Atmp ;F1 I ;2 B1 X1 6 6 ^ GFI = 6 6 " 4 3 B1 # " 0 0 B2 7 I 7: 7 7 # 5 0 I So from Theorem 16.2 and Lemma 16.8, we conclude that K is an admissible controller ^ for G and kTzw k1 < i K is an admissible" controller for GFI and kTvr k1 < . # h Now let L = B2 F1 ;B1 i I = A + B2 F1 is stable. ^ Also note that Atmp + B2 F1 is stable. Then all icontrollers that stabilize GFI can be h parameterized as K = F` (M ) = 1 2 2 RH1 where 2 M= 6 6 6 6 6 6 4 then Atmp + L Atmp + B2 F1 + L ; " " ;2B1 X1 I ;2 B1 X1 F1 I ;2B1 X1 # ;L B2 0 I I # 0 3 7 7 7 7 7 7 5 : With this parameterization of all controllers, the transfer matrix from r to v is Tvr = ^ F` (GFI F` (M )) =: F` (N ). It is easy to show that 2 N =6 4 " 0 0 # I 0 I 3 7 5 and Tvr = F` (N ) = 2 . Hence kTvr k1 < if and only if k 2 k1 < . This implies that all FI H1 controllers can be parameterized as K = F`(M ) with = h i 1 2 2 RH1 k 2 k1 < and h 2 A + 2B2 F1 6 6 M = 6 " F1 6 4 ; ;2 I B1 X1 Now let 1 = F1 ; ; B2 F1 ;B1 # ;2 Q2 B X1 + Q1 1 0 I Then it is easy to show that F` (M ) = F`(MFI Q). i 3 B2 7 I 7: 7 7 0 2 = Q2 : 5 2
- 440. 16.5. Full Control 423 16.5 Full Control 2 A B1 6 6 G(s) = 6 C1 0 4 C2 D21 h h h I 0 0 I 0 0 i 3 i 7 7 7 i 5 This problem is dual to the Full Information problem. The assumptions that the FC problem inherits from the output feedback problem are just the dual of those in the FI problem: (i) (A B1 ) is stabilizable (ii) (C2 A) is detectable " (iv) # " # B1 D = 0 . I D21 21 Theorem 16.11 There exists an admissible controller K (s) for the FC problem such that kTzw k1 < if and only if J1 2 dom(Ric) and Y1 = Ric(J1 ) 0. Moreover, if these conditions are satis ed then an equivalence class of all admissible controllers satisfying kTzw k1 < can be parameterized as " ;2 K (s) = L1 ; Y1 C1 Q(s) Q(s) # where Q 2 RH1 , kQk1 < . As expected, the condition in Theorem 16.11 is the same as that in (ii) of Theorem 16.4. 16.6 Disturbance Feedforward 2 A B1 B2 G(s) = 6 C1 0 D12 4 C2 I 0 3 7 5 This problem inherits the same assumptions (i)-(iii) as in the FI problem, but for internal stability we shall add that A ; B1 C2 is stable. With this assumption, it is easy to check that the condition in Lemma 16.1 is satis ed so that the internal stability is again equivalent to Tzw 2 RH1 , as in the output feedback case.
- 441. H1 CONTROL: SIMPLE CASE 424 Theorem 16.12 There exists an admissible controller for the DF problem such that kTzw k1 < if and only if H1 2 dom(Ric) and X1 = Ric(H1 ) 0. Moreover, if these conditions are satis ed then all admissible controllers satisfying kTzw k1 < can be parameterized as the set of all transfer matrices from y to u in u y M1 - 2 A + B2 F1 ; B1 C2 B1 B2 6 M1 (s) = 4 F1 0 I ;2 B1 X1 I 0 ;C2 ; Q 3 7 5 with Q 2 RH1 , kQk1 < Proof. Suppose there is a controller KDF solving the habove problem, i.e., with kTzw k1 < i . Then by Theorem 12.4, the controller KFI = KDF C2 I solves the corresponding H1 FI problem. Hence the conditions H1 2 dom(Ric) and X1 = Ric(H1) 0 are necessarily satis ed. On the other hand, if these conditions are satis ed then FI h is solvable. It is easy to verify that F` (M1 Q) = F` (PDF KFI ) with KFI = i F1 ; ;2 Q(s)B1 X1 Q(s) where PDF is as de ned in section 12.2 of Chapter 12. So again by Theorem 12.4, the controller F`(M1 Q) solves the DF problem. To show that F` (M1 Q) with kQk1 < parameterizes all DF H1 suboptimal controllers, we shall make a change of variables as in equation (16.4): v = u + B2 X1 x r = w ; ;2 B1 X1 x: Then the system equations can be written as follows: 2 6 4 and 2 6 4 x _ z r 3 2 AF1 7 6 C1F1 5=4 ;1B1 X1 ; 3 ;1 B1 0 I 2 B2 D12 0 x _ Atmp B1 B2 v 7=6 ;F1 0 I 5 4 ;2 B1 X1 I 0 y C2 + This is shown pictorially in the following diagram: 32 76 54 32 76 54 x w v 3 3 7 5 x r 7: 5 u
- 442. 16.7. Output Estimation 425 z r XXX ^ GDF - and P XXXX y where w w v r 1 u K 2 AF1 6 P = 4 C1F1 ; ;1B1 X1 ;1B1 0 I B2 D12 0 2 3 7 5 3 B1 B2 Atmp ^ DF = 6 G ;F1 0 I 7: 4 5 C2 + ;2 B1 X1 I 0 Since Atmp ; B1 (C2 + ;2 B1 X1 ) = A ; B1 C2 and Atmp + B2 F1 are stable, the rank ^ conditions of Lemma 16.1 for system GDF are satis ed. So from Theorem 16.2 and Lemma 16.7, we conclude that K is an admissible controller for G and kTzw k1 < i K ^ is an admissible controller for GDF and kTvr k1 < . Now it is easy to see by comparing this formula with the controller parameterization in Theorem 12.8 that F` (M1 Q) ^ with Q 2 RH1 (no norm constraint) parameterizes all stabilizing controllers for GDF ^ DF F`(M1 Q)) = Q. So kTvr k1 < however, simple algebra shows that Tvr = F` (G i kQk1 < , and F` (M1 Q) with Q 2 RH1 and kQk1 < parameterizes all suboptimal controllers for G. 2 16.7 Output Estimation 2 A B1 B2 6 G(s) = 4 C1 0 I C2 D21 0 3 7 5
- 443. H1 CONTROL: SIMPLE CASE 426 This problem is dual to DF, just as FC was to FI. Thus the discussion of the DF problem is relevant here, when appropriately dualized. The OE assumptions are (i) (A B1 ) is stabilizable and A ; B2 C1 is stable (ii) (C2 A) is detectable " (iv) # " # B1 D = 0 . D21 21 I Assumption (i), together with (iv), imply that internal stability is again equivalent to Tzw 2 RH1 , as in the output feedback case. Theorem 16.13 There exists an admissible controller for the OE problem such that kTzw k1 < if and only if J1 2 dom(Ric) and Ric(J1 ) 0. Moreover, if these conditions are satis ed then all admissible controllers satisfying kTzw k1 < can be parameterized as the set of all transfer matrices from y to u in u M1 - Q y 2 A + L1 C2 ; B2 C1 L1 ;B2 ; ;2Y1 C1 6 M1 (s) = 4 C1 0 I C2 I 0 3 7 5 with Q 2 RH1 , kQk1 < . It is interesting to compare H1 and H2 in the context of the OE problem, even though, by duality, the essence of these remarks was made before. Both optimal estimators are observers with the observer gain determined by Ric(J1 ) and Ric(J2 ). Optimal H2 output estimation consists of multiplying the optimal state estimate by the output map C1 . Thus optimal H2 estimation depends only trivially on the output z that is being estimated, and state estimation is the fundamental problem. In contrast, the H1 estimation problem depends very explicitly and importantly on the output being estimated. This will have implications for the separation properties of the H1 output feedback controller. 16.8 Separation Theory If we assume the results of the special problems, which are proven in the previous sections, we can now prove Theorems 16.4 and 16.5 using separation arguments. This essentially involves reducing the output feedback problem to a combination of the Full Information and the Output Estimation problems. The separation properties of the H1 controller are more complicated than the H2 controller, although they are no less interesting. The notation and assumptions for this section are as in Section 16.2.
- 444. 16.8. Separation Theory 427 16.8.1 H1 Controller Structure The H1 controller formulae from Theorem 16.4 are " # ^ A1 ;Z1 L1 Ksub (s) := F1 0 ^ A1 := A + ;2B1 B1 X1 + B2 F1 + Z1 L1 C2 F1 := ;B2 X1 L1 := ;Y1 C2 Z1 := (I ; ;2 Y1 X1 );1 where X1 := Ric(H1 ) and Y1 := Ric(J1 ). The necessary and su cient conditions for the existence of an admissible controller such that kTzw k1 < are (i) H1 2 dom(Ric) and X1 := Ric(H1 ) 0 (ii) J1 2 dom(Ric) and Y1 := Ric(J1 0 (iii) (X1 Y1 ) < 2 . We have seen that condition (i) corresponds to the Full Information condition and that (ii) corresponds to the Full Control condition. It is easily shown that, given the FI and FC results, these conditions are necessary for the output feedback case as well. Lemma 16.14 Suppose there exists an admissible controller making kTzw k1 < . Then conditions (i) and (ii) hold. Proof. Let K be an admissible controller for which kTzw k1 < . The controller K C2 D21 ] solves the FI problem hence from Theorem 16.9, H1 2 dom(Ric) and X1 := Ric(H1 ) 0. Condition (ii) follows by the dual argument. 2 We would also expect some condition beyond these two, and that is provided by (iii), which is an elegant combination of elements from FI and FC. Note that all the conditions of Theorem 16.4 are symmetric in H1 , J1 , X1 , and Y1 , but the formula for the controller is not. Needless to say, there is a dual form that can be obtained by inspection from the above formula. For a symmetric formula, the state equations above ; can be multiplied through by Z11 and put in descriptor form. A simple substitution from the Riccati equation for X1 will then yield a symmetric, though more complicated, formula: _ (I ; ;2 Y1 X1 )x = As x ; L1 y ^ ^ (16.10) u = F1 x ^ (16.11) where As := A + B2 F1 + L1 C2 + ;2Y1 A X1 + ;2 B1 B1 X1 + ;2Y1 C1 C1 . To emphasize its relationship to the H2 controller formulae, the H1 controller can be written as _ x = Ax + B1 wworst + B2 u + Z1 L1 (C2 x ; y) ^ ^ ^ ^
- 445. H1 CONTROL: SIMPLE CASE 428 u = F1 x ^ wworst = ;2 B1 X1 x: ^ ^ These equations have the structure of an observer-based compensator. The obvious questions that arise when these formulae are compared with the H2 formulae are 1) Where does the term B1 wworst come from? ^ 2) Why Z1 L1 instead of L1 ? 3) Is there a separation interpretation of these formulae analogous to that for H2 ? The proof of Theorem 16.4 reveals that there is a very well-de ned separation interpretation of these formulae and that wworst := ;2 B1 X1 x is, in some sense, a worst-case input for the Full Information problem. Furthermore, Z1 L1 is actually the optimal lter gain for estimating F1 x, which is the optimal Full Information control input, in the presence of this worst-case input. It is therefore not surprising that Z1 L1 should enter in the controller equations instead of L1 . The term wworst may be thought ^ of loosely as an estimate for wworst . 16.8.2 Proof of Theorem 16.4 It has been shown from Lemma 16.14 that conditions (i) and (ii) are necessary for kTzw k1 < . Hence we only need to show that if conditions (i) and (ii) are satis ed, condition (iii) is necessary and su cient for kTzw k1 < . As in section 16.3, we de ne new disturbance and control variables r := w ; ;2 B1 X1 x Then " 2 B B A v = 6 tmp 1 2 0 I 4 ;F1 y C2 D21 0 # z y G - K 3 7 5 " # v := u + B2 X1 x: " # r =G r tmp u u w v u y Atmp := A + ;2 B1 B1 X1 : Gtmp - K r u Recall from Lemma 16.8 that K is admissible for G and kTzw k1 < i K is admissible for Gtmp and kTvr k1 < . While Gtmp has the form required for the OE problem, to actually use the OE results, we will need to verify that Gtmp satis es the following assumptions for the OE problem:
- 446. 16.8. Separation Theory 429 (i) (Atmp B1 ) is stabilizable and Atmp + B2 F1 is stable (ii) (C2 Atmp ) is detectable " # " # B1 D = 0 . (iv) D21 21 I Assumption (iv) and that (Atmp B1 ) is stabilizable follow immediately from the corresponding assumptions for Theorem 16.4. The stability of Atmp + B2 F1 follows from the de nition of H1 2 dom(Ric). The following lemma gives conditions for assumption (ii) to hold. Of course, the existence of an admissible controller for Gtmp immediately implies that assumption (ii) holds. Note that the OE Hamiltonian matrix for Gtmp is " ;2 F1 F1 ; C2 C2 Jtmp := Atmp ;B1 B1 ;Atmp Lemma 16.15 If Jtmp 2 dom(Ric) and Ytmp := Ric(Jtmp) detectable. # : 0, then (C2 Atmp ) is Proof. The lemma follows from the dual to Lemma 16.6, which gives that (Atmp ; 2 Ytmp C2 C2 ) is stable. Proof of Theorem 16.4 (Su ciency) Assume the conditions (i) through (iii) in the theorem statement hold. Using the Riccati equation for X1 , one can easily verify that " # I ; ;2X1 provides a similarity transformation between J and J , i.e., T := tmp 1 0 I T ;1Jtmp T = J1 . So " # " I = Im I ; ;2 X1 Y1 X; (Jtmp ) = T X; (J1 ) = T Im Y1 Y1 # and (X1 Y1 ) < 2 implies that Jtmp 2 dom(Ric) and Ytmp := Ric(Jtmp ) = Y1 (I ; ;2 X1 Y1 );1 = Z1 Y1 0. Thus by Lemma 16.15 the OE assumptions hold for Gtmp , and by Theorem 16.13 the OE problem is solvable. From Theorem 16.13 with Q = 0, one solution is " A + ;2 B1 B1 X1 ; Ytmp C2 C2 + B2 F1 Ytmp C2 F1 0 # but this is precisely Ksub de ned in Theorem 16.4. We conclude that Ksub stabilizes Gtmp and that kTvr k1 < . Then by Lemma 16.8, Ksub stabilizes G and that kTzw k1 < .
- 447. H1 CONTROL: SIMPLE CASE 430 (Necessity) Let K be an admissible controller for which kTzw k1 < . By Lemma 16.14, H1 2 dom(Ric), X1 := Ric(H1 ) 0, J1 2 dom(Ric), and Y1 := Ric(J1) 0. From Lemma 16.8, K is admissible for Gtmp and kTvr k1 < . This implies that the OE assumptions hold for Gtmp and that the OE problem is solvable. Therefore, from Theorem 16.13 applied to Gtmp , we have that Jtmp 2 dom(Ric) and Ytmp = Ric(Jtmp) 0. Using the same similarity transformation formula as in the su ciency part, we get that Ytmp = (I ; ;2 Y1 X1 );1 Y1 0. We shall now show that Ytmp 0 implies that (X1 Y1 ) < 2 . We shall consider two cases: 1 1 Y1 is nonsingular: in this case Ytmp 0 implies that I ; ;2 Y1=2 X1 Y1=2 > 0. 1=2 X Y 1=2 ) < 2 or (X Y ) < 2 . So (Y1 1 1 1 1 Y1 is singular: there is a unitary matrix U such that Y1 = U " # Y11 0 U 0 0 with Y11 > 0. Let UX1 U be partitioned accordingly, " # UX1 U = X11 X12 : X21 X22 Then by the same argument as in the Y1 nonsingular case, " # (I ; ;2Y11 X11 );1 Y11 0 Ytmp = U U 0 0 I implies that 2 > (X11 Y11 ) (= (X1 Y1 )). 2 We now see exactly why the term involving wworst appears and why the observer" ^ gain is Z1 L1. Both terms are consequences of estimating the optimal Full Information (i.e., state feedback) control gain. While an analogous output estimation problem arises in the H2 output feedback problem, the resulting equations are much simpler. This is because there is no worst-case" disturbance for the H2 Full Information problem and because the problem of estimating any output, including the optimal state feedback, is equivalent to state estimation. We now present a separation interpretation for H1 suboptimal controllers. It will be stated in terms of the central controller, but similar interpretations could be made for the parameterization of all suboptimal controllers (see the proofs of Theorems 16.4 and 16.5). The H1 output feedback controller is the output estimator of the full information control law in the presence of the worst-case" disturbance wworst . Note that the same statement holds for the H2 optimal controller, except that wworst = 0.
- 448. 16.9. Optimality and Limiting Behavior 431 16.8.3 Proof of Theorem 16.5 From Lemma 16.8, the set of all admissible controllers for G such that kTzw k1 < equals the set of all admissible controllers for Gtmp such that kTvr k1 < . Apply Theorem 16.13. 2 16.9 Optimality and Limiting Behavior In this section, we will discuss, without proof, the behavior of the H1 suboptimal solution as varies, especially as approaches the in mal achievable norm, denoted by opt . Since Theorem 16.4 gives necessary and su cient conditions for the existence of an admissible controller such that kTzw k1 < , opt is the in mum over all such that conditions (i)-(iii) are satis ed. Theorem 16.4 does not give an explicit formula for opt , but, just as for the H1 norm calculation, it can be computed as closely as desired by a search technique. Although we have not focused on the problem of H1 optimal controllers, the assumptions in this book make them relatively easy to obtain in most cases. In addition to describing the qualitative behavior of suboptimal solutions as varies, we will indicate why the descriptor version of the controller formulae from Section 16.8.1 can usually provide formulae for the optimal controller when = opt . Most of these results can be obtained relatively easily using the machinery that is developed in the previous sections. The reader interested in lling in the details is encouraged to begin by strengthening ; assumption (i) to controllable and observable and considering the Hamiltonians for X11 ;1 . and Y1 As ! 1, H1 ! H2 , X1 ! X2 , etc., and Ksub ! K2 . This fact is the result of the particular choice for the central controller (Q = 0) that was made here. While it could be argued that Ksub is a natural choice, this connection with H2 actually hints at deeper interpretations. In fact, Ksub is the minimum entropy solution (see next section) as well as the minimax controller for kz k2 ; 2kwk2 . 2 2 If 2 1 > opt , then X1 ( 1 ) X1 ( 2 ) and Y1 ( 1 ) Y1 ( 2 ). Thus X1 and Y1 are decreasing functions of , as is (X1 Y1 ). At = opt , anyone of the three conditions in Theorem 16.4 can fail. If only condition (iii) fails, then it is relatively straightforward to show that the descriptor formulae for = opt are optimal, i.e., the optimal controller is given by ;2 _ (I ; opt Y1 X1 )x = As x ; L1 y ^ ^ (16.12) u = F1 x ^ (16.13) ;2 ;2 ;2 where As := A + B2 F1 + L1 C2 + opt Y1 A X1 + opt B1 B1 X1 + opt Y1 C1 C1 . See the example below. The formulae in Theorem 16.4 are not well-de ned in the optimal case because the ;2 term (I ; opt X1 Y1 ) is not invertible. It is possible but far less likely that conditions (i) or (ii) would fail before (iii). To see this, consider (i) and let 1 be the largest
- 449. H1 CONTROL: SIMPLE CASE 432 for which H1 fails to be in dom(Ric) because the H1 matrix fails to have either the stability property or the complementarity property. The same remarks will apply to (ii) by duality. If complementarity fails at = 1 , then (X1 ) ! 1 as ! 1 . For < 1 , H1 may again be in dom(Ric), but X1 will be inde nite. For such , the controller u = ;B2 X1 x would make kTzw k1 < but would not be stabilizing. See part 1) of the example below. If the stability property fails at = 1 , then H1 62 dom(Ric) but Ric can be extended to obtain X1 and the controller u = ;B2 X1 x is stabilizing and makes kTzw k1 = 1 . The stability property will also not hold for any 1 , and no controller whatsoever exists which makes kTzw k1 < 1 . In other words, if stability breaks down rst, then the in mum over stabilizing controllers equals the in mum over all controllers, stabilizing or otherwise. See part 2) of the example below. In view of this, we would typically expect that complementarity would fail rst. Complementarity failing at = 1 means (X1 ) ! 1 as ! 1 , so condition (iii) would fail at even larger values of , unless the eigenvectors associated with (X1 ) as ! 1 are in the null space of Y1 . Thus condition (iii) is the most likely of all to fail rst. If condition (i) or (ii) fails rst because the stability property fails, the formulae in Theorem 16.4 as well as their descriptor versions are optimal at = opt . This is illustrated in the example below for the output feedback. If the complementarity condition fails rst, (but (iii) does not fail), then obtaining formulae for the optimal controllers is a more subtle problem. Example 16.1 Let an interconnected dynamical system realization be given by 2 G(s) = 6 " 6 6 6 6 4 a 1 0 h # h c2 i 1 0 0 i 0 1 b2 # " 0 1 0 3 7 7 7 7 7 5 with jc2 j jb2 j > 0. Then all assumptions for the output feedback problem are satis ed and " " 2 2 # 2 2 # 2 2 a 1;b2 a 1;c2 H1 = J1 = : ;1 ;a ;1 ;a The eigenvalues of H1 and J1 are given, respectively, by ( p ) ( p ) (a2 + b2 ) 2 ; 1 (a2 + c2 ) 2 ; 1 : 2 2 (H ) = (J ) = 1 If 2> 1 a2 + b2 2 1 1 a2 + c2 , then X; (H1 ) and X; (J1 ) exist and 2 X; (H1 ) = Im " p(a2 +b2 ) 2 2 ;1;a 1 #
- 450. 16.9. Optimality and Limiting Behavior " p 2 2 2 (a +c2 ) ;1;a X; (J1 ) = Im 433 # 1 : We shall consider two cases: 1) a > 0: In this case, the complementary property of dom(Ric) will fail before the stability property fails since q (a2 + b2 ) 2 ; 1 ; a = 0 2 when 2= 1 1 > a2 + b2 . b2 2 2 Nevertheless, if 2> 1 a2 + b2 and 2 2 6= 1 b2 , then H1 2 dom(Ric) and 2 ( X1 = p (a2 + b2 ) 2 ; 1 ; a 2 = > 0 if 2 > b12 2 1 < 0 if a2 +b2 < 2 2 < 12 : b2 Let F1 = ;B2 X1 then p 2 2 + b2 2 A+B2 F1 = ; a + b2 b2(a2 ; 12 ) ; 1 = 2 ( < 0 (stable) if 2 > b12 2 1 > 0 (unstable) if a2 +b2 < 2 2< 1: 2 b2 { Suppose full information (or states) are available for feedback and let u = F1 x: Then the closed-loop transfer matrix is given by 2 Tzw = " # A + B2 F1 B1 = 6 6 6 4 C1 + D12 F1 0 2 ; a+b2 2 4 p(a2 +b2 ) 2 ;1 h 2 2 ;12 b2 3 1 ; p(a2 +b2b)2 2 ;1;a 1 0 5 2 0 i 3 7 7 7 5 1 1 1 2 b2 and is not stable for a2 + b2 < < b2 . Fur2 2 2 1 thermore, it can be shown that kTzw k < for all 2 > a2 + b2 and 2 6= b12 . 2 2 and Tzw is stable for all 2> It is clear that the optimal H1 norm is b12 but is not achievable. 2
- 451. H1 CONTROL: SIMPLE CASE 434 { Suppose the states are not available then output feedback must be considered. Note that if 2 > b12 , then H1 2 dom(Ric), J1 2 dom(Ric), and 2 X1 = p (a2 + b2 ) 2 ; 1 ; a 2 >0 > 0: (a2 + c2 ) 2 ; 1 ; a 2 Hence conditions (i) and (ii) in Theorem 16.4 are satis ed, and need to check condition (iii). Since Y1 = p (X1 Y1 ) = 2 p p ( (a2 + b2 ) 2 ; 1 ; a )( (a2 + c2 ) 2 ; 1 ; a ) 2 2 it is clear that (X1 Y1 ) ! 1 when 2 ! b12 . So condition (iii) will fail 2 before condition (i) or (ii) fails. 2) a < 0: In this case, complementary property is always satis ed, and, furthermore, H1 2 dom(Ric), J1 2 dom(Ric), and X1 = p >0 Y1 = p >0 (a2 + b2) 2 ; 1 ; a 2 for 2> (a2 + c2 ) 2 ; 1 ; a 2 1 a2 + b2 . 2 1 a2 + b2 , H1 62 dom(Ric) since stability property fails. Never2 1 2= theless, in this case, if 0 a2 + b2 , we can extend the dom(Ric) to include those 2 However, for 2 matrices H1 with imaginary axis eigenvalues as " X; (H1 ) = Im ;a # 1 1 such that X1 = ; a is a solution to the Riccati equation ; A X1 + X1 A + C1 C1 + 0 2X1 B1 B1 X1 ; X1 B2 B2 X1 = 0 ; and A + 0 2B1 B1 X1 ; B2 B2 X1 = 0.
- 452. 16.9. Optimality and Limiting Behavior 435 2 2 and F1 = ;B2 X1 , then A + B2 F1 = a + ba < 0. So if states are available for feedback and u = F1 x, we have { For = 0 2 6 " Tzw = 6 4 h 2 a + ba2 1 # 1 0 0 b2 a i 3 7 7 5 2 RH1 = . Hence the optimum is achieved. and kTzw k1 = p 21 a + b2 0 { If states are not available, the output feedback is considered, and jb2j = jc2j, then it can be shown that (X1 Y1 ) = p ( 2 (a2 + b2 ) 2 ; 1 ; a )2 2 < 2 is satis ed if and only if ! p 2 22 > a +b2 b2 + a > p 21 2 : a + b2 2 So condition (iii) of Theorem 16.4 will fail before either (i) or (ii) fails. In both a > 0 and a < 0 cases, the optimal for the output feedback is given by opt = p a2 + 2b2 + a 2 b2 2 if jb2 j = jc2 j and the optimal controller given by the descriptor formula in equations (16.12) and (16.13) is a constant. In fact, uopt = ; q opt 2 (a2 + b2 ) opt ; 1 ; a opt 2 y: p For instance, let a = ;1 and b2 = 1 = c2 . Then opt = 3 ; 1 = 0:7321 and uopt = ;0:7321 y. Further, 3 2 ;1:7321 1 ;0:7321 Tzw = 6 1 0 0 7: 5 4 ;0:7321 0 ;0:7321 It is easy to check that kTzw k1 = 0:7321. 3
- 453. H1 CONTROL: SIMPLE CASE 436 16.10 Controller Interpretations This section considers some additional connections with the minimum entropy solution and the work of Whittle and will be of interest primarily to readers already familiar with them. The connection with the Q-parameterization approach will be considered in the next chapter for the general case. Section 16.10.2 gives another separation interpretation of the central H1 controller of Theorem 16.4 in the spirit of Whittle (1981). It has been shown in Glover and Doyle 1988] that the central controller corresponds exactly to the steady state version of the optimal risk sensitive controller derived by Whittle, 1981], who also derives a separation result and a certainty equivalence principle (see also Whittle, 1986]). 16.10.1 Minimum Entropy Controller Let T be a transfer matrix with kT k1 < . Then the entropy of T (s) is de ned by I (T ) = ; 2 2Z 1 ;1 ; ln det I ; ;2T (j!)T (j!) d!: It is easy to see that I (T ) = ; 2 2Z 1X ;1 i ln 1 ; ;2 i2 (T (j!)) d! and I (T ) show that 0, where i (T (j!)) is the ith singular value of T (j!). It is also easy to Z 1 X 2 2 lim I (T ) = 21 i (T (j! )) d! = kT k2 : !1 ;1 i Thus the entropy I (T ) is in fact a performance index measuring the tradeo between the H1 optimality ( ! kT k1 ) and the H2 optimality ( ! 1). It has been shown in Glover and Mustafa 1989] that the central controller given in Theorem 16.4 is actually the controller that satis es the norm condition kTzw k1 < and minimizes the following entropy: ;2 2Z 1 ; ln det I ; ;2 Tzw (j!)Tzw (j!) d!: ;1 Therefore, the central controller is also called the minimum entropy controller (maxi~ mum entropy controller if the entropy is de ned as I (T ) = ;I (T )). 16.10.2 Relations with Separation in Risk Sensitive Control Although Whittle, 1981] treats a nite horizon, discrete time, stochastic control problem, his separation result has a clear interpretation for the present in nite horizon,
- 454. 16.10. Controller Interpretations 437 continuous time, deterministic control problem, as given below and it is an interesting exercise to compare the two separation statements. This discussion will be entirely in the time-domain. We will consider the system at time, t = 0, and evaluate the past stress, S; , and future stress, S+ , as functions of the current state, x. First de ne the future stress as S+ (x) := sup inf (kP+ z k2 ; 2kP+ wk2 ) 2 2 u w then by the completion of the squares and by the saddle point argument of Section 16.3, where u is not constrained to be a function of the measurements (FI case), we obtain S+ (x) = x X1 x: The past stress, S; (x), is a function of the past inputs and observations, u(t) y(t) for ;1 < t < 0, and of the present state, x, and is produced by the worst case disturbance, w, that is consistent with the given data: S; (x) := sup(kP; z k2 ; 2kP; wk2 ): 2 2 In order to evaluate S; we see that w can be divided into two components, D21 w ? ? and D21 w , with x dependent only on D21 w (since B1 D21 = 0) and D21 w = y ; C2 x. The past stress is then calculated by a completion of the square and in terms of a lter output. In particular, let x be given by the stable di erential equation _ x = Ax + B2 u + L1 (C2 x ; y) + Y1 C1 C1 x with x(;1) = 0: Then it can be shown that the worst case w is given by ? ? ; D21 w = D21 B1 Y1 1 (x(t) ; x(t)) for t < 0 and that this gives, with e := x ; x, ; S; (x) = ; 2e(0) Y1 1 e(0) ; 2 kP; (y ; C2 x)k2 + kP;(C1 x)k2 + kP; uk2: 2 2 2 The worst case disturbance will now reach the value of x to maximize the total stress, S; (x) + S+ (x) , and this is easily shown to be achieved at the current state of x = Z1 x(0): ^ The de nitions of X1 and Y1 can be used to show that the state equations for the central controller can be rewritten with the state x := Z1 x and with x as de ned ^ above. The control signal is then u = F1 x = F1 Z1 x: ^ The separation is between the evaluation of future stress, which is a control problem with an unconstrained input, and the past stress, which is a ltering problem with known control input. The central controller then combines these evaluations to give a worst case estimate, x, and the control law acts as if this were the perfectly observed ^ state.
- 455. H1 CONTROL: SIMPLE CASE 438 16.11 An Optimal Controller To o er a general idea about the appearance of an optimal controller, we shall give in the following without proof the conditions under which an optimal controller exists and an explicit formula for an optimal controller. Theorem 16.16 There exists an admissible controller such that kTzw k1 i the following three conditions hold: (i) there exists a full column rank matrix " such that and # " # X11 2 R2n n X12 # " H1 X11 = X11 TX Re i (TX ) 0 8i X12 X12 X11 X12 = X12 X11 (ii) there exists a full column rank matrix " such that and (iii) " # # Y11 2 R2n n Y12 " # J1 Y11 = Y11 TY Re i (TY ) 0 8i Y12 Y12 Y11 Y12 = Y12 Y11 " X12 X11 ;1 Y12 X12 ;1X12 Y12 Y12 Y11 # 0: Moreover, when these conditions hold, one such controller is Kopt (s) := CK (sEK ; AK )+ BK where EK := Y11 X11 ; ;2Y12 X12 BK := Y12 C2 CK := ;B2 X12 AK := EK TX ; BK C2 X11 = TY EK + Y11 B2 CK :
- 456. 16.12. Notes and References 439 Remark 16.4 It is simple;to show that if X11 and Y11 are nonsingular and if X1 = ;1 1 X12 X11 and Y1 = Y12 Y11 , then condition (iii) in the above theorem is equivalent to X1 0, Y1 0, and (Y1 X1 ) 2 . So in this case, the conditions for the existence of an optimal controller can be obtained from taking the limit" of the corresponding conditions in Theorem 16.4. Moreover, the controller given above is reduced to the descriptor form given in equations (16.12) and (16.13). ~ 16.12 Notes and References This chapter is based on Doyle, Glover, Khargonekar, Francis 1989], and Zhou 1992]. The minimum entropy controller is studied in detail in Glover and Mustafa 1989] and Mustafa and Glover 1990]. The risk sensitivity problem is treated in Whittle 1981, 1986]. The connections between the risk sensitivity controller and the central H1 controller are explored in Doyle and Glover 1988]. The complete characterization of optimal controllers for the general setup can be found in Glover, Limebeer, Doyle, Kasenally, and Safonov 1991].
- 457. 440 H1 CONTROL: SIMPLE CASE
- 458. 17 H1 Control: General Case In this chapter we will consider again the standard H1 control problem but with some assumptions in the last chapter relaxed. Since the proof techniques in the last chapter can be applied to this general case except with some more involved algebra, the detailed proof for the general case will not be given only the formulas are presented. However, some procedures to carry out the proof will be outlined together with some alternative approaches to solve the standard H1 problem and some interpretations of the solutions. We will also indicate how the assumptions in the general case can be relaxed further to accommodate other more complicated problems. More speci cally, Section 17.1 presents the solutions to the general H1 problem. Section 17.2 discusses the techniques to transform a general problem to a standard problem which satis es the assumptions in the last chapter. The problems associated with relaxing the assumptions for the general standard problems and techniques for dealing with them will be considered in Section 17.3. Section 17.4 considers the integral control in the H2 and H1 theory and Section 17.5 considers how the general H1 solution can be used to solve the H1 ltering problem. Section 17.6 considers an alternative approach to the standard H2 and H1 problems using Youla controller parameterizations, and Section 17.7 gives an 2 2 Hankel-Toeplitz operator interpretations of the H1 solutions presented here and in the last chapter. Finally, the general state feedback H1 control problem and its relations with full information control and di erential game problems are discussed in section 17.8 and 17.9. 441
- 459. H1 CONTROL: GENERAL CASE 442 17.1 General H1 Solutions Consider the system described by the block diagram z y G - K w u where, as usual, G and K are assumed to be real rational and proper with K constrained to provide internal stability. The controller is said to be admissible if it is real-rational, proper, and stabilizing. Although we are taking everything to be real, the results presented here are still true for the complex case with some obvious modi cations. We will again only be interested in characterizing all suboptimal H1 controllers. The realization of the transfer matrix G is taken to be of the form 2 A B1 B2 G(s) = 6 C1 D11 D12 4 C2 D21 0 3 7 5 " A B = C D # which is compatible with the dimensions z (t) 2 Rp1 , y(t) 2 Rp2 , w(t) 2 Rm1 , u(t) 2 Rm2 , and the state x(t) 2 Rn . The following assumptions are made: (A1) (A B2 ) is stabilizable and (C2 A) is detectable (A2) D12 = " 0 I # h and D21 = 0 I " A ; j!I B2 C1 D12 # " A ; j!I B1 C2 D21 # (A3) (A4) i has full column rank for all ! has full row rank for all !. Assumption (A1) is necessary for the existence of stabilizing controllers. The assumptions in (A2) mean that the penalty on z = C1 x + D12 u includes a nonsingular, normalized penalty on the control u, that the exogenous signal w includes both plant disturbance and sensor noise, and that the sensor noise weighting is normalized and nonsingular. Relaxation of (A2) leads to singular control problems see Stroorvogel 1990]. For those problems that have D12 full column rank and D21 full row rank but do not satisfy assumption (A2), a normalizing procedure is given in the next section so that an equivalent new system will satisfy this assumption.
- 460. 17.1. General H1 Solutions 443 Assumptions (A3) and (A4) are made for a technical reason: together with (A1) they guarantee that the two Hamiltonian matrices in the corresponding H2 problem belong to dom(Ric), as we have seen in Chapter 14. It is tempting to suggest that (A3) and (A4) can be dropped, but they are, in some sense, necessary for the methods presented in the last chapter to be applicable. A further discussion of the assumptions and their possible relaxation will be discussed in Section 17.3. The main result is now stated in terms of the solutions of the X1 and Y1 Riccati equations together with the state feedback" and output injection" matrices F and L. R := D1 D1 ; " " 2 Im 0 2 Ip 1 # 0 0 # 0 0 where D1 := D11 D12 ] # " D11 ~ where D 1 := R := D 1 D 1 ; D21 0 " # " # h i A 0 B H1 := ; R;1 D1 C1 B ;C1 C1 ;A ;C1 D1 " # " # h i A 0 C ~ J1 := ; R;1 D 1 B1 C ;B1 B1 ;A ;B1D 1 1 X1 := Ric(H1) F := L := " h # F11 := ;R;1 D C + B X ] 1 1 1 F21 i ~ L11 L21 := ; B1 D 1 + Y1 C ]R;1 Partition D, F11 , and L11 are as follows: 2 " Y1 := Ric(J1 ) 3 F111 F121 F21 0 # 6 L 7 F = 6 111 D1111 D1112 0 7 : 7 6 6 0 D L I 7 5 4 L121 D1121 D1122 0 I 0 L21 Remark 17.1 In the above matrix partitioning, some matrices may not exist depending on whether D12 or D21 is square. This issue will be discussed further later. For the time being, we shall assume all matrices in the partition exist. ~ Theorem 17.1 Suppose G satis es the assumptions (A1){(A4). (a) There exists an admissible controller K (s) such that jjF` (G K )jj1 < (i.e. kTzw k1 < ) if and only if
- 461. H1 CONTROL: GENERAL CASE 444 (i) > max( D1111 D1112 ] D1111 D1121 ]) (ii) H1 2 dom(Ric) with X1 = Ric(H1 ) 0 (iii) J1 2 dom(Ric) with Y1 = Ric(J1 ) 0 (iv) (X1 Y1 ) < 2 . (b) Given that the conditions of part (a) are satis ed, then all rational internally stabilizing controllers K (s) satisfying jjF` (G K )jj1 < are given by where K = F`(M1 Q) for arbitrary Q 2 RH1 2 such that kQk1 < 3 ^ ^ ^ A B1 B2 7 6 ^ ^ ^ 5 M1 = 6 C1 D11 D12 7 4 ^ ^ C2 D21 0 ^ D11 = ;D1121 D1111 ( 2 I ; D1111 D1111 );1 D1112 ; D1122 ^ 12 2 Rm2 m2 and D21 2 Rp2 p2 are any matrices (e.g. Cholesky factors) satisfying ^ D ^ ^ D12 D12 = I ; D1121 ( 2 I ; D1111 D1111 );1 D1121 ^ ^ D21 D21 = I ; D1112 ( 2 I ; D1111 D1111 );1 D1112 and where ^ ^ B2 = Z1 (B2 + L121 )D12 ^ ^ C2 = ;D21 (C2 + F121 ) ^ ^ ^; ^ B1 = ;Z1L21 + B2 D121 D11 ^ ^ ^; ^ C1 = F21 + D11 D211 C2 ^ ^ ^; ^ A = A + BF + B1 D211 C2 Z1 = (I ; ;2 Y1 X1 );1 : (Note that if D11 = 0 then the formulae are considerably simpli ed.) Some Special Cases: Case 1: D12 = I In this case 1. in part (a), (i) becomes > (D1121 ). 2. in part (b) ^ D11 = ;D1122 ^ ^ D12 D12 = I ; ;2 D1121 D1121 ^ ^ D21 D21 = I:
- 462. 17.2. Loop Shifting 445 Case 2: D21 = I In this case 1. in part (a), (i) becomes > (D1112 ). 2. in part (b) ^ D11 = ;D1122 ^ ^ D12 D12 = I ^ ^ D21 D21 = I ; ;2 D1112 D1112 : Case 3: D12 = I & D21 = I In this case 1. in part (a), (i) drops out. 2. in part (b) ^ D11 = ;D1122 ^ ^ D12 D12 = I ^ ^ D21 D21 = I: 17.2 Loop Shifting Let a given problem have the following diagram where zp (t) 2 Rp1 , yp (t) 2 Rp2 , wp (t) 2 Rm1 , and up (t) 2 Rm2 : zp yp P - Kp wp up The plant P has the following state space realization with Dp12 full column rank and Dp21 full row rank: 2 3 Ap Bp1 Bp2 P (s) = Cp1 Dp11 Dp12 Cp2 Dp21 Dp22 6 4 7 5 The objective is to nd all rational proper controllers Kp(s) which stabilize P and jjF` (P Kp )jj1 < . To solve this problem we rst transfer it to the standard one treated in the last section. Note that the following procedure can also be applied to the H2 problem (except the procedure for the case D11 6= 0).
- 463. H1 CONTROL: GENERAL CASE 446 Normalize D12 and D21 Perform singular value decompositions to obtain the matrix factorizations Dp12 = Up " 0 # I h i ~ ~ Dp21 = Rp 0 I Up Rp ~ such that Up and Up are square and unitary. Now let ~ ~ zp = Up z wp = Up w yp = Rp y up = Rp u and ~ K (s) = Rp Kp (s)Rp G(s) = " 2 = 6 6 4 2 =: Then 6 4 # " # ~ 0 P (s) Up 0 1 ; ~; 0 Rp 1 0 Rp ; ~ Bp1 Up Bp2 Rp 1 Ap ; ~ Up Cp1 Up Dp11 Up Up Dp12 Rp 1 ; ~; ~ ~; ~; Rp 1 Cp2 Rp 1Dp21 Up Rp 1Dp22 Rp 1 Up A B1 B2 C1 D11 D12 C2 D21 D22 D12 = " 0 # I and the new system is shown below: z y 7 5 A B : = C D h K 7 7 5 # " D21 = 0 I G - 3 3 i w u ~ Furthermore, jjF` (P Kp )jj = jjUp F` (G K )Up jj = kF`(G K )k for ~ since Up and Up are unitary. Remove the Assumption D22 = 0 = 2 or 1 Suppose K (s) is a controller for G with D22 set to zero. Then the controller for D22 6= 0 is K (I + D22 K );1 . Hence there is no loss of generality in assuming D22 = 0.
- 464. 17.2. Loop Shifting 447 Remove the Assumption D11 = 0 We can even assume that D11 = 0. In fact, Theorem 17.1 can be shown by rst transforming the general problem to the standard problem considered in the last chapter using Theorem 16.2. This transformation is called loop-shifting. Before we go into the detailed description of the transformation, let us rst consider a simple unitary matrix dilation problem. Lemma 17.2 Suppose D is a constant matrix such that kDk < 1 then N= " ;D (I ; DD )1=2 # (I ; D D)1=2 D is a unitary matrix, i.e., N N = I . This result can be easily veri ed, and the matrix N is called a unitary dilation of D (of course, there are many other ways to dilate a matrix to a unitary matrix). Consider again the feedback system shown at the beginning of this chapter without loss of generality, we shall assume that the system G has the following realization: 2 6 6 6 6 4 A G = C1 with " C2 D12 = " B1 D1111 D1112 D1121 D1122 h i 0 I 0 I # h # " B2 # 0 I 0 3 7 7 7 7 5 i D21 = 0 I : Suppose there is a stabilizing controller K such that kF` (G K )k1 < 1. (Suppose we have normalized to 1). In the following, we will show how to construct a new system ~ transfer matrix M (s) and a new controller K such that the D11 matrix for M (s) is zero, ~ and, furthermore, kF` (G K )k1 < 1 if and only if F`(M K ) < 1. To begin with, 1 note that kF` (G K )(1)k < 1 by the assumption kF`(G K )(1)k = D1111 D1121 Let ( D1112 : D1122 + K (1) ) D1 2 X : D1111 D1112 <1 : D1121 D1122 + X For example, let D1 = ;D1122 ; D1121 (I ; D1111 D1111 );1 D1111 D1112 and de ne " # D1112 ~ 11 := D1111 D D1121 D1122 + D1
- 465. H1 CONTROL: GENERAL CASE 448 ~ then D11 < 1. Let to get ~ ~ ~ F`(G K ) = F` (G K + D1 ) = F` (G K ) where 2 ~ G = 6 6 6 6 4 2 =: Let ~ K (s) = K (s) + D1 6 6 4 A + B2 D1 C2 " C1 + D12 D1 C2 ~ A ~ C1 ~ C2 C2 ~ B1 ~ D11 ~ D21 " 3 B1 + B2 D1 D21 D1111 D1112 D1121h D1122i+ D1 0 I # " B2 # 0 I 0 3 7 7 7 7 5 ~ B2 7 ~ 5 D12 7 : 0 # ~ ;D11 ~ ~ (I ; D11 D11 )1=2 N= ~ ~ ~ (I ; D11 D11 )1=2 D11 and then N N = I . Furthermore, by Theorem 16.2, we have that K stabilizes G and that kF` (G K )k1 < 1 if and only if F`(G K ) internally stabilizes N and kF`(N F` (G K ))k1 < 1: Note that ~ F`(N F` (G K )) = F`(M K ) with 3 2 ~ ; ~ ~ ; ~ ~ ~ ~ ; ~ ~ A + B1 R1 1D11 C1 B1 R1 1=2 B2 + B1 R1 1D11 D12 7 6 7 ~; ~ M (s) = 6 ~; ~ R1 1=2 C1 0 R1 1=2 D12 5 4 ~ ; ~ ~ ~ ~ ; ~ ~ ~ ; C2 + D21 R1 1 D11 C1 D21 R1 1=2 D21 R1 1 D11 D12 ~ ~ ~ ~ ~ where R1 = I ; D11 D11 and R1 = I ; D11 D11 . In summary, we have the following lemma. Lemma 17.3 There is a controller K that internally stabilizes G and kF`(G K )k1 < 1 ~ ~ if and only if there is a K that stabilizes M and F`(M K ) < 1. " # 1 Corollary 17.4 Let G(s) = A B 2 RH1 . Then kG(s)k1 < 1 if and only if C D kDk < 1, # " A + B (I ; D D);1 D C B (I ; D D);1=2 2 RH M (s) = 1 (I ; DD );1=2 C 0
- 466. 17.3. Relaxing Assumptions 449 and kM (s)k1 < 1. 17.3 Relaxing Assumptions In this section, we indicate how the results of section 17.1 can be extended to more general cases. Relaxing A3 and A4 Suppose that 2 G=6 4 0 0 1 0 0 1 1 1 0 3 7 5 which violates both A3 and A4 and corresponds to the robust stabilization of an integrator. If the controller u = ; x where > 0 is used, then Tzw = s; s with kTzw k1 = : + Hence the norm can be made arbitrarily small as ! 0, but = 0 is not admissible since it is not stabilizing. This may be thought of as a case where the H1 -optimum is not achieved on the set of admissible controllers. Of course, for this system, H1 optimal control is a silly problem, although the suboptimal case is not obviously so. If one simply drops the requirement that controllers be admissible and removes assumptions A3 and A4, then the formulae presented above will yield u = 0 for both the optimal controller and the suboptimal controller with = 0. This illustrates that assumptions A3 and A4 are necessary for the techniques used in the last chapter to be directly applicable. An alternative is to develop a theory which maintains the same notion of admissibility but relaxes A3 and A4. The easiest way to do this would be to pursue the suboptimal case introducing perturbations so that A3 and A4 are satis ed. Relaxing A1 If assumption A1 is violated, then it is obvious that no admissible controllers exist. Suppose A1 is relaxed to allow unstabilizable and/or undetectable modes on the j! axis and internal stability is also relaxed to also allow closed-loop j! axis poles, but A2-A4 is still satis ed. It can be easily shown that under these conditions the closedloop H1 norm cannot be made nite and, in particular, that the unstabilizable and/or undetectable modes on the j! axis must show up as poles in the closed-loop system, see Lemma 16.1.
- 467. 450 H1 CONTROL: GENERAL CASE Violating A1 and either or both of A3 and A4 Sensible control problems can be posed which violate A1 and either or both of A3 and A4. For example, cases when A has modes at s = 0 which are unstabilizable through B2 and/or undetectable through C2 arise when an integrator is included in a weight on a disturbance input or an error term. In these cases, either A3 or A4 are also violated, or the closed-loop H1 norm cannot be made nite. In many applications, such problems can be reformulated so that the integrator occurs inside the loop (essentially using the internal model principle) and is hence detectable and stabilizable. We will show this process in the next section. An alternative approach to such problems which could potentially avoid the problem reformulation would be to pursue the techniques in the last chapter but to relax internal stability to the requirement that all closed-loop modes be in the closed left half plane. Clearly, to have nite H1 norms, these closed-loop modes could not appear as poles in Tzw . The formulae given in this chapter will often yield controllers compatible with these assumptions. The user would then have to decide whether closed-loop poles on the imaginary axis were due to weights and hence acceptable or due to the problem being poorly posed, as in the above example. A third alternative is to again introduce perturbations so that A1, A3, and A4 are satis ed. Roughly speaking, this would produce sensible answers for sensible problems, but the behavior as ! 0 could be problematic. Relaxing A2 In the cases that either D12 is not full column rank or that D21 is not full row rank, improper controllers can give a bounded H1 -norm for Tzw , although the controllers will not be admissible by our de nition. Such singular ltering and control problems have been well-studied in H2 theory and many of the same techniques go over to the H1 -case (e.g. Willems 1981], Willems, Kitapci, and Silverman 1986], and Hautus and Silverman 1983]). In particular, the structure algorithm of Silverman 1969] could be used to make the terms D12 and D21 full rank by the introduction of suitable di erentiators in the controller. A complete solution to the singular problem can be found in Stroorvogel, 1990]. 17.4 H2 and H1 Integral Control It is interesting to note that the H2 and H1 design frameworks do not in general produce integral control. In this section we show how to introduce integral control into the H2 and H1 design framework through a simple disturbance rejection problem. We consider a feedback system shown in Figure 17.1. We shall assume that the frequency contents of the disturbance w are e ectively modeled by the weighting Wd 2 RH1 and the constraints on control signal are limited by an appropriate choice of Wu 2 RH1 . In order to let the output y track the reference signal r, we require K contain an integral,
- 468. 17.4. H2 and H1 Integral Control 451 i.e., K (s) has a pole at s = 0. (In general, K is required to have poles on the imaginary axis.) z 62 w ? Wu r -g ;6 K 6 u Wd - P ? - g y- We -z1 Figure 17.1: A Simple Disturbance Rejection Problem There are several ways to achieve the integral design. One approach is to introduce an integral in the performance weight We . Then the transfer function between w and z1 is given by z1 = We (I + PK );1 Wd w: Now if the resulting controller K stabilizes the plant P and makes the norm (2-norm or 1-norm) between w and z1 nite, then K must have a pole at s = 0 which is the zero of the sensitivity function (Assuming Wd has no zeros at s = 0). (This follows from the well-known internal model principle.) The problem with this approach is that the H2 (or H1 ) control theory presented in this chapter and in the previous chapters can not be applied to this problem formulation directly because the pole s = 0 of We becomes an uncontrollable pole of the feedback system and the very rst assumption for the H2 (or H1 ) theory is violated. However, the obstacles can be overcome by appropriately reformulating the problem. Suppose We can be factorized as follows ~ We = We (s)M (s) where M (s) is proper, containing all the imaginary axis poles of We , and M ;1 (s) 2 ~ RH1 , We (s) is stable and minimum phase. Now suppose there exists a controller K (s) which contains the same imaginary axis poles that achieves the performance. Then without loss of generality, K can be factorized as ^ K (s) = ;K (s)M (s) ^ where there is no unstable pole/zero cancelation in forming the product K (s)M (s). Now the problem can be reformulated as in Figure 17.2. Figure 17.2 can be put in the general LFT framework as in Figure 17.3. ~ Let We Wu Wd M , and P have the following stabilizable and detectable state space
- 469. H1 CONTROL: GENERAL CASE 452 z 62 w ? Wu - 6 ^ K u Wd - - ?g P y1- ~ We M z -1 Figure 17.2: Disturbance Rejection with Imaginary Axis Poles z1 ~ We z2 w Wu Wd ? g M y1 - P u ^ K Figure 17.3: LFT Framework for the Disturbance Rejection Problem realizations: " ~ We = Ae Be # Ce De " Wu = Au Bu Cu Du " M = Am Bm Cm Dm # # " Wd = Ad Bd Cd Dd " # P = Ap Bp : Cp Dp Then a realization for the generalized system is given by 2 " G(s) = 6 4 ~ We MWd 0 MWd # " ~ We MP Wu MP # 3 7 5 #
- 470. 17.4. H2 and H1 Integral Control 453 3 2 Ae 0 Be Cm Be DmCd Be Dm Cp Be Dm Dd Be Dm Dp 6 7 6 0 Au 0 Bu 7 0 0 0 7 6 6 0 0 Am Bm Cd Bm Cp Bm Dd Bm Dp 7 7 6 7 6 7 6 0 Bd 0 0 0 Ad 0 7: =6 6 0 0 0 0 Ap 0 Bp 7 7 6 7 6 6 Ce 0 De Cm De Dm Cd De DmCp De DmDd De Dm Dp 7 7 6 6 0 0 0 0 Du 7 5 4 0 Cu 0 0 Cm Dm Cd Dm Cp Dm Dd Dm Dp We shall illustrate the above design through a simple numerical example. Let 2 3 0 1 0 ; P = (s +s1)(s2; 3) = 6 3 2 1 7 Wd = 1 4 5 ;2 1 0 " + 10 Wu = ss+ 100 = ;100 ;90 1 1 # We = 1 : s Then we can choose without loss of generality that 1 ~ > 0: M = s + We = s + s This gives the following generalized system 2 3 ; 0 1 ;2 1 1 0 6 6 0 7 ;100 0 0 0 0 ;90 7 6 7 6 0 0 0 ;2 0 7 6 7 6 7 6 0 0 0 0 1 0 0 7 7 G(s) = 6 6 0 0 0 3 2 0 1 7 6 7 6 7 6 1 0 0 0 0 0 0 7 6 7 6 1 0 0 0 0 1 7 4 0 5 0 0 1 ;2 1 1 0 ^ and suboptimal H1 controller K1 for each can be computed easily as )(s + 100)( ; 0 ^ K1 = ;2060381:(4(s + 1)(s +s + 262343)(ss; 19::1557) 2 s + 32:17)( (s + ) 89) ^ which gives the closed-loop 1 norm 7:854. Hence the controller K1 = ;K1(s)M (s) is given by 1)(s ; 0:1557) + 100)(s ; 0: K1 (s) = 2060381:4(s +s1)(s262343)(s ; 191557) 7:85(ss(++ 32:+ 100)(s19:89) s(s + 32:17)( + :89) s 17)(s ;
- 471. H1 CONTROL: GENERAL CASE 454 which is independent of as expected. Similarly, we can solve an optimal H2 controller + 1)(s + + ; 0 069) ^ K2 = ;s43:487((s 2 + 30:94s )(s411100)(ss; 7::964) ( + )2 s + :81)( and + 100)( ^ K2 (s) = ;K2(s)M (s) = s43:2487(s :+ 1)(s 411:81)(ss; 0::069) : (s + 30 94s + ; 7 964) An approximate integral control can also be achieved without going through the above process by letting 1 ~ We = We = s + M (s) = 1 for a su ciently small > 0. For example, a controller for = 0:001 is given by s+ 7:85(s + 1)(s + 100)(s ; 0:1545) K1 = (s316880(s + 1)(32)(s100)(s ; 0:1545) + 0:001)(s + + 40370)(s ; 20) s(s + 32)(s ; 20) which gives the closed-loop H1 norm of 7:85. Similarly, an approximate H2 integral controller is obtained as + 1)(s + 100)(s 0679) K2 = (s + 043:47(s 2 + 30:93s + 411; 0:s ; 7:9718) : :001)(s :7)( 17.5 H1 Filtering In this section we show how the ltering problem can be solved using the H1 theory developed early. Suppose a dynamic system is described by the following equations x = Ax + B1 w(t) x(0) = 0 _ (17.1) y = C2 x + D21 w(t) (17.2) z = C1 x + D11 w(t) (17.3) The ltering problem is to nd an estimate z of z in some sense using the measurement ^ of y. The restriction on the ltering problem is that the lter has to be causal so that it can be realized, i.e., z has to be generated by a causal system acting on the ^ measurements. We will further restrict our lter to be unbiased, i.e., given T > 0 the estimate z(t) = 0 8t 2 0 T ] if y(t) = 0 8t 2 0 T ]. Now we can state our H1 ltering ^ problem. H1 Filtering: Given a > 0, nd a causal lter F (s) 2 RH1 if it exists such that ^2 J := sup kz ; z2k2 < 2 with z = F (s)y. ^ w2L2 0 1) kwk2
- 472. 17.5. H1 Filtering 455 z ? j z ; z ^ 2 6 4 y F (s) A B1 C1 D11 C2 D21 w 3 7 5 Figure 17.4: Filtering Problem Formulation A diagram for the ltering problem is shown in Figure 17.4. The above ltering problem can also be formulated in an LFT framework: given a system shown below z y w G(s) 2 A B1 0 G(s) = 6 C1 D11 ;I 4 C2 D21 0 z ^ - F (s) nd a lter F (s) 2 RH1 such that 3 7 5 2 sup kz k22 < 2 : w2L2 (17:4) kwk2 Hence the ltering problem can be regarded as a special H1 problem. However, comparing with control problems there is no internal stability requirement in the ltering problem. Hence the solution to the above ltering problem can be obtained from the H1 solution in the previous sections by setting B2 = 0 and dropping the internal stability requirement. Theorem 17.5 Suppose (C2 A) is detectable and " A ; j!I B1 C2 D21 # has full row rank for all !. Let D21 be normalized and D11 partitioned conformably as " # " # D11 = D111 D112 : D21 0 I Then there exists a causal lter F (s) 2 RH1 such that J < and J1 2 dom(Ric) with Y1 = Ric(J1) 0 where ~ R := " D11 D21 #" D11 D21 # ; " 2I 0 0 0 # 2 if and only if (D111 ) <
- 473. H1 CONTROL: GENERAL CASE 456 J1 := " # " # " # A 0 C1 C2 ~ ; R;1 D11 B1 C1 : ;B1 B1 ;A ;B1D11 ;B1D21 D21 B1 C2 Moreover, if the above conditions are satis ed, then a rational causal lter F (s) satisfying J < 2 is given by " # A + L21 C2 + L11 D112 C2 ;L21 ; L11 D112 z = F (s)y = ^ y C1 ; D112 C2 D112 where h i h i ~ L11 L21 := ; B1 D11 + Y1 C1 B1 D21 + Y1 C2 R;1 : In the case where D11 = 0 and B1 D21 = 0 the lter becomes much simpler: # " z = A ; Y1 C2 C2 Y1 C2 y ^ C1 0 where Y1 is the stabilizing solution to Y1 A + AY1 + Y1 ( ;2 C1 C1 ; C2 C2 )Y1 + B1 B1 = 0: 17.6 Youla Parameterization Approach* In this section, we shall brie y outline an alternative approach to solving the standard H2 and H1 control problems using the Q parameterization (Youla parameterization) approach. This approach was the main focus in the early 1980's and is still very useful in solving many interesting problems. We will see that this approach may suggest additional interpretations for the results presented in the last chapter and applies to both H2 and H1 problems. The H2 problem is very simple and involves a projection. While the H1 problem is much more di cult, it bears some similarities to the constant matrix dilation problem but with the restriction of internal stability. Nevertheless, we have built enough tools to give a fairly complete solution using this Q parameterization approach. Assume again that the G has the realization 2 A B1 B2 G = 6 C1 D11 D12 4 C2 D21 D22 3 7 5 with the same assumptions as before. But for convenience, we will allow the assumption (A2) to be relaxed to the following
- 474. 17.6. Youla Parameterization Approach* h 457 i (A20 ) D12 is full column rank with D12 D? unitary, and D21 is full row rank with " # D21 ~ D? unitary. Next, we will outline the steps required to solve the H2 and H1 control problems. Because of the similarity between the H2 and H1 problems, they are developed in parallel below. Parameterization: Recall that all controllers stabilizing the plant G can be expressed as K = F` (M2 Q) I + D22 Q(1) invertible with Q 2 RH1 and 2 A + B2 F2 + L2 C2 + L2 D22 F2 ;L2 B2 + L2 D22 M2 = 6 F2 0 I 4 ;(C2 + D22 F2 ) I ;D22 3 7 5 where F2 = ;(B2 X2 + D12 C1 ) L2 = ;(Y"C2 + B1 D21 ) 2 # A ; B2 D12 C1 ;B2 B2 X2 = Ric 0 ;C1 D? D?C1 ;(A ; B2 D12 C1 ) " # (A ; B1 D21 C2 ) ;C2 C2 Y2 = Ric 0: ~ ~ ;B1 D? D? B1 ;(A ; B1 D21 C2 ) Then the closed-loop transfer matrix from w to z can be written as F`(G K ) = To + UQV I + D22 Q(1) invertible where and 2 AF2 ;B2 F2 3 B1 To = 6 0 AL2 B1L2 7 2 RH1 4 5 C1F2 ;D12F2 D11 # " # " AL2 B1L2 AF2 B2 V= U= C1F2 D12 C2 D21 AF2 := A + B2 F2 AL2 := A + L2 C2 C1F2 = C1 + D12 F2 B1L2 = B1 + L2 D21 : It is easy to show that U is an inner, U U = I , and V is a co-inner, V V = I .
- 475. H1 CONTROL: GENERAL CASE 458 " V Unitary Invariant: There exist U? and V? such that U U? and V ? h square and inner: " + U? = AF2 ;X2 C1 D? C1F2 D? # V? = " i # are # AL2 B1L2 : ~ ~ ; D?B1 Y2+ D? Since the multiplication of square inner matrices do not change H2 and H1 norms, we have for = 2 or 1 kF` (P K )k = kTo + UQV k = = where " V U U? (To + UQV ) V? " # R11 + Q R12 R21 R22 h " i # h R = U To V U? V? # i with the obvious partitioning. It can be shown through some long and tedious algebra that R is antistable and has the following representation: 2 3 ~ ;AF2 EB1L2 ;ED21 ;E D? 6 ~ 7 6 C2 ;Y2+ B1 D? 7 0 ;AL2 6 7 R=6 ~ 5 B2 F2 Y2 ; D12 D11 B1L2 D12 D11 D21 D12 D11 D? 7 4 ~ ;D?C1 X2+ ;D?D11 B1L2 D? D11 D21 D? D11 D? where E := X2 B1 + C1F2 D11 . Projection/Dilation: At this point the = 2 and = 1 cases have to be treated separately. ~ H2 case: In this case, we will assume D?D11 = 0 and D11 D? = 0 otherwise, the 2-norm of the closed loop transfer matrix will be unbounded since " # ~ D12 D11 D21 D12 D11 D? : R(1) = ~ D? D11 D21 D? D11 D? Now from the de nition of 2-norm, the problem can be rewritten as kF`(G K )k2 2 " = R11 + Q R12 R21 R22 # 2 2
- 476. 17.6. Youla Parameterization Approach* = 459 kR11 + Qk2 + 2 " 0 R21 R12 R22 # 2 2 Furthermore, : kR11 + Qk2 = kR11 ; D12 D11 D21 k2 + kQ + D12 D11 D21 k2 2 2 2 ? since (R11 ; D12 D11 D21 ) 2 H2 and (Q + D12 D11 D21 ) 2 H1 . In fact, (Q + D12 D11 D21 ) has to be in H2 to guarantee that the 2-norm be nite. Hence the unique optimal solution is given by a projection to H1 : Qopt = ;R11 ]+ = ;D12 D11 D21 : The optimal controller is given by Kopt = F` (M2 Qopt): In particular, if D11 = 0 then Kopt = F` (M2 0). H1 case: Recall the de nition of the 1-norm: " " # #! R11 + Q R12 (j!): = sup R21 R22 ! 1 Consider the minimization problem with respect to Q frequency by frequency R11 + Q R12 R21 R22 it becomes a constant matrix dilation problem if no causality restrictions are imposed on Q. The H1 optimal control follows this idea. However, it should be noted that since Q is restricted to be an RH1 matrix, the term R11 + Q cannot be made into an arbitrary matrix. Therefore, the problem will be signi cantly di erent from the constant matrix case, and, generically, " min Q2H1 where ( 0 := max h R11 + Q R12 R21 R22 R21 R22 # 1 " i 1 > R12 R22 0 # ) 1 : It is convenient to use the characterization in Corollary 2.23. Recall that " i R11 + Q R12 R21 R22 # (< ) for > 0 (I ; WW );1=2 (Q + R11 + WR22 Z )(I ; Z Z );1=2 1 1 (< )
- 477. H1 CONTROL: GENERAL CASE 460 where W = R12 ( 2 I ; R22 R22 );1=2 Z = ( 2 I ; R22 R22 );1=2 R21 : The key here is to nd the spectral factors (I ; WW )1=2 and (I ; Z Z )1=2 with stable inverses such that if M = (I ; WW )1=2 and N = (I ; Z Z )1=2 , then M M ;1 N N ;1 2 H1 MM = I ; WW , and N N = (I ; Z Z ). Now let ^ Q := M ;1QN ;1 G := M ;1 (R11 + WR22 Z )N ;1 ^ then the problem is reduced to nding Q 2 H1 such that ^ G+Q 1 (< ): ^ ^ Note that Q = M QN 2 H1 i Q 2 H1 . (17:5) ^ The nal step in the H1 problem involves solving (17.5) for Q 2 H1 . This is a standard Nehari problem and is solved in Chapter 8. The optimal control law will be given by Kopt = F` (J Qopt ): 17.7 Connections This section considers the connections between the Youla parameterization approach and the current state space approach discussed in the last chapter. The key is Lemma 15.7 and its connection with the formulae given in the last section. To see how Lemma 15.7 might be used in the last section to prove Theorems 16.4 and 16.5 or Theorem 17.1, we begin with G having state dimension n. For simplicity, we assume further that D11 = 0. Then from the last section, we have kTzw k1 = " R+ Q 0 " = 0 0 1 # R11 + Q R12 R21 R22 " = R + Q where R has state dimension 2n. Now de ne " # 0 # 0 0 N11 N12 := R : N21 N22 # 1 1
- 478. 17.7. Connections Then 461 2 " N11 N12 N21 N22 # + AF2 0 B2 ;X2 C1 D? 6 6 ;B BX AL2 Y2 F2 0 = 6 1L2 1 2 6 ;C2 0 0 4 D21 B1 X2 ~ ? B1 X2 D? B1 Y2+ 0 ~ 0 D 3 7 7 7 7 5 2 RH1 (17:6) and the H1 problem becomes to nd an antistable Q such that " " w Let w = 1 w2 " N11 + Q N21 N11 + Q N21 N12 N22 # 1 # < : and note that N12 N22 # 2 1 " := sup w2BL2 = N11 + Q N21 sup fw2BL2 gfw1 2H? g 2 #" " # 2 N12 w1 N22 w2 2 #" N11 + Q N12 w1 N21 N22 w2 # 2 2 : The right hand side of the above equation can be written as sup 8 " < fw2BL2 gfw1 2H? g : 2 P+ 0 0 I #" N11 N12 N21 N22 #" w1 w2 # 2 2 o + kQ w1 + P; (N11 w1 + N12 w2 )k2 : 2 It is clear that the last term can be made zero for an appropriate choice of an antistable Q . Hence the H1 problem has a solution if and only if " sup fw2BL2 gfw1 2H? g 2 P+ 0 0 I #" N11 N12 N21 N22 #" w1 w2 # 2 < : But this is exactly the same operator as the one in Lemma 15.7. Lemma 15.7 may be applied to derive the solvability conditions and some additional arguments to construct a Q 2 RH1 from X and Y such that kTzw (Q)k1 < . In fact, it turns out that X in Lemma 15.7 for N is exactly W in the FI proof or in the di erential game problem. The nal step is to obtain the controller from M2 and Q. Since M2 has state dimension n and Q has 2n, the apparent state dimension of K is 3n, but some tedious state space manipulations produce cancelations resulting in the n dimensional controller
- 479. H1 CONTROL: GENERAL CASE 462 formulae in Theorems 16.4 and 16.5. This approach is exactly the procedure used in Doyle 1984] and Francis 1987] with Lemma 15.7 used to solve the general distance problem. Although this approach is conceptually straightforward and was, in fact, used to obtain the rst proof of the current state space results in this chapter, it seems unnecessarily cumbersome and indirect. 17.8 State Feedback and Di erential Game It has been shown in Chapters 15 and 16 that a (central) suboptimal full information H1 control law is actually a pure state feedback if D11 = 0. However, this is not true in general if D11 6= 0, as will be shown below. Nevertheless, the state feedback H1 control is itself a very interesting problem and deserves special treatment. This section and the section to follow are devoted to the study of this state feedback H1 control problem and its connections with full information control and di erential game. Consider a dynamical system x = Ax + B1 w + B2 u _ z = C1 x + D11 w + D12 u (17.7) (17.8) where z (t) 2 Rp1 , y(t) 2 Rp2 , w(t) 2 Rm1 , u(t) 2 Rm2 , and x(t) 2 Rn . The following assumptions are made: (AS1) (A B2 ) is stabilizable h i (AS2) There is a matrix D? such that D12 D? is unitary " (AS3) A ; j!I B2 C1 D12 # has full column rank for all !. We are interested in the following two related quadratic min-max problems: given > 0, check if sup min kz k2 < and w2B L2 0 1) u2L2 0 1) min sup u2L2 0 1) w2B L2 0 1) kz k2 < : The rst problem can be regarded as a full information control problem since the control signal u can be a function of the disturbance w and the system state x. On the other hand, the optimal control signal in the second problem cannot depend on the disturbance w (the worst disturbance w can be a function of u and x). In fact, it will be shown that the control signal in the latter problem depends only on the system state hence, it is equivalent to a state feedback control.
- 480. 17.8. State Feedback and Di erential Game 463 Theorem 17.6 Let > 0 be given and de ne R := D1 D1 ; " 2 Im 0 " A 0 H1 := ;C1 C1 ;A h 0 0 1 # i # where D1 := D11 D12 ] " B ; ;C1 D1 # h R;1 D1 C1 B i where B := B1 B2 . (a) sup min kz k2 < if and only if w2B L2 0 1) u2L2 0 1) (D? D11 ) < H1 2 dom(Ric) X1 = Ric(H1 ) 0: Moreover, an optimal u is given by h i u = ;D12D11 w + D12 D11 I Fx and a worst w worst is given by w where (b) min " worst = F11 x # F := F11 := ;R;1 D1 C1 + B X1 ] : F21 sup u2L2 0 1) w2B L2 0 1) kz k2 < if and only if (D11 ) < H1 2 dom(Ric) X1 = Ric(H1 0: Moreover, an optimal u is given by u = F21 x = ;D12D11 w worst + h i D12 D11 I Fx and the worst wsfworst is given by wsfworst = ( 2 I ; D11 D11 );1 f(D11 C1 + B1 X1 )x + D11 D12 ug : Proof. The condition for part (a) can be shown in the same way as in Chapter 15 and is, in fact, the solution to the FI problem for the general case. We now prove the condition for part (b). It is not hard to see that (D11 ) < is necessary since control u cannot feed back w directly, so the D11 term cannot be (partially) eliminated as in
- 481. H1 CONTROL: GENERAL CASE 464 the FI problem. The conditions H1 2 dom(Ric) and X1 = Ric(H1 0 can be easily seen as necessary since sup min kz k2 u2L 0 1) sup kz k2: min 2 w2B L2 0 1) w2B L2 0 1) u2L2 0 1) It is easy to verify directly that the optimal control and the worst disturbance can be chosen in the given form. 2 De ne R0 = I ; D11 D? D? D11 = 2 ~ R0 = I + D12 D11 ( 2 I ; D11 D11 );1 D11 D12 ^ R0 = I ; D11 D11 = 2: Then it is easy to show that d kz k2 ; 2 kwk2 + dt (x Xx) = u + D12 D11 w ; D12 D11 I Fx h ; i 1 R0=2 (w ; F11 x) 2 2 2 if conditions in (a) are satis ed. On the other hand, we have d ~1 kz k2 ; 2 kwk2 + dt (x Xx) = R0=2 (u ; F21 x) ; 2 2 ^1 R0=2 (w ; wsfworst ) 2 if conditions in (b) are satis ed. Integrating both equations from t = 0 to 1 with x(0) = x(1) = 0 gives h i 2 kz k2 ; 2 kwk2 = u + D12 D11 w ; D12 D11 I Fx 2 ; 2 2 2 1 R0=2 (w ; F11 x) 2 2 if conditions in (a) are satis ed, and 2 ~1 kz k2 ; 2 kwk2 = R0=2 (u ; F21 x) 2 ; 2 2 2 ^1 R0=2 (w ; wsfworst ) 2 2 if conditions in (b) are satis ed. These relations suggest that an optimal control law and a worst disturbance for problem (a) are h i u = ;D12 D11 w + D12 D11 I Fx w = F11 x and that an optimal control law and a worst disturbance for problem (b) are u = F21 x w = wsfworst :
- 482. 17.9. Parameterization of State Feedback H1 Controllers 465 Moreover, if problem (b) has a solution for a given , then the corresponding di erential game problem min sup kz k2 ; 2 kwk2 2 2 u2L 0 1) 2 w2L2 0 1) has a saddle point, i.e, sup min kz k2 ; 2 kwk2 = u2L 0 1) sup kz k2 ; 2 kwk2 min 2 2 2 2 2 w2L2 0 1) u2L2 0 1) w2L2 0 1) since the solvability of problem (b) implies the solvability of problem (a). However, the converse may not be true. In fact, it is easy to construct an example so that the problem (a) has a solution for a given and problem (b) does not. On the other hand, the problems (a) and (b) are equivalent if D11 = 0. 17.9 Parameterization of State Feedback H1 Controllers In this section, we shall consider the parameterization of all state feedback control laws. We shall rst assume for simplicity that D11 = 0 and show later how to reduce the general D11 6= 0 case to an equivalent problem with D11 = 0. We shall assume 2 3 A B1 B2 G = 6 C1 0 D12 7 : 4 5 I 0 0 Note that the state feedback H1 problem is not a special case of the output feedback problem since D21 = 0. Hence the parameterization cannot be obtained from the output feedback. Theorem 17.7 Suppose that the assumptions (AS 1) ; (AS 3) are satis ed and that B1 has the following SVD: " # 0 B1 = U V UU = In V V = Im1 0 < 2 Rr r : 0 0 There exists an admissible controller K (s) for the SF problem such that kTzw k1 < if and only if H1 2 dom(Ric) and X1 = Ric(H1) 0. Furthermore, if these conditions are satis ed, then all admissible controllers satisfying kTzw k1 < can be parameterized as ( " # );1 " # ;1 ;1 0 0 ;1 B2 ^ K = F1 + Im2 + Q U Q U ;1 (sI ; A) 0 Im1 ;r 0 Im1 ;r h i ^ where F1 = ;(D12 C1 +B2 X1 ), A = A+ ;2B1 B1 X1 +B2 F1 , and Q = Q1 Q2 2 RH2 with kQ1 k1 < . The dimensions of Q1 and Q2 are m2 r and m2 (m1 ; r), respectively.
- 483. H1 CONTROL: GENERAL CASE 466 Proof. The conditions for the existence of the state feedback control law have been shown in the last section and in Chapter 15. We only need to show that the above parameterization indeed satis es the H1 norm condition and gives all possible state feedback H1 control laws. As in the proof of the FI problem in Chapter 16, we make ^ ^ the same change of variables to get GSF instead of GFI : z r w P XXXXX X ^ GSF y - w v r 1 u K 2 3 Atmp B1 B2 ^ SF = 6 ;F1 0 I 7 : G 4 5 I 0 0 So again from Theorem 16.2 and Lemma 16.8, we conclude that K is an admissible ^ controller for G and kTzw k1 < i K is an admissible controller for GSF and kTvr k1 < . Now let L = B2 F1 , and then Atmp + L = Atmp + B2 F1 is stable. All controllers ^ that stabilize GSF can be parameterized as K = F`(Mtmp ) 2 RH1 where 2 3 Atmp + B2 F1 + L ;L B2 Mtmp = 6 F1 0 I 7: 4 5 ;I I 0 ^ Then Tvr = F` (GSF F` (Mtmp )) =: F` (Ntmp ). It is easy to show that 3 2 Atmp + B2 F1 B1 0 6 Ntmp = 4 ;F1 0 I 7: 5 0 0 I Now let = F1 + ^ , and we have " # F`(Ntmp ) = ^ Atmp + B2 F1 B1 : I 0
- 484. 17.9. Parameterization of State Feedback H1 Controllers Let h ^ = Q1 Q2 i h " ;1 0 i 0 I # 467 ^ U ;1 (sI ; A): Then the mapping from Q = Q1 Q2 2 RH2 to ^ 2 RH1 is one-to-one. Hence we have h i F`(Ntmp ) = Q1 0 V and kF`(Ntmp )k1 = kQ1 k1 , which in turn implies that kTvr k1 = kF`(Ntmp )k1 < if and only if kQ1 k1 < . Finally, substituting = F1 + ^ into K = F` (Mtmp ), we get the desired controller parameterization. 2 The controller parameterization for the general case can also be obtained: 2 3 A B1 B2 6 Gg (s) = 4 C1 D11 D12 7 : 5 I 0 0 Let N= " ;D11 (I ; D11 D11 )1=2 # (I ; D11 D11 )1=2 D11 and then N N = I . Furthermore, by Theorem 16.2, we have that K stabilizes Gg and kF`(Gg K )k1 < 1 if and only if F`(Gg K ) internally stabilizes N and kF` (N F` (Gg K ))k1 < 1: Note that with ~ F`(N F` (Gg K )) = F` (M K ) 2 ; ; A + B1 D11 R1 1 C1 B1 (I ; D11 D11 );1=2 B2 + B1 D11 R1 1D12 6 ; ; M (s) = 6 R1 1=2 C1 0 R1 1=2 D12 4 I 0 0 3 7 7 5 where R1 := I ; D11 D11 . In summary, we have the following lemma. Lemma 17.8 There is a K that internally stabilizes Gg and kF`(Gg K )k1 < 1 if and ~ ~ only if kD11 k < 1 and there is a K that stabilizes M and F` (M K ) < 1. 1 Now Theorem 17.7 can be applied to the new system M (s) to obtain the controller parameterization for the general problem with D11 6= 0.
- 485. 468 H1 CONTROL: GENERAL CASE 17.10 Notes and References The detailed derivation of the H1 solution for the general case is treated in Glover and Doyle 1988, 1989]. The loop-shifting procedures are given in Safonov, Limebeer, and Chiang 1989]. The idea is also used in Zhou and Khargonekar 1988] for state feedback problems. A fairly complete solution to the singular H1 problem is obtained in Stoorvogel 1992]. The H1 ltering and smoothing problems are considered in detail in Nagpal and Khargonekar 1991]. The Youla parameterization approach is treated very extensively in Doyle 1984] and Francis 1987] and in the references therein. The presentation of the state feedback H1 control in this chapter is based on Zhou 1992].
- 486. 18 H1 Loop Shaping This chapter introduces a design technique which incorporates loop shaping methods to obtain performance/robust stability tradeo s, and a particular H1 optimization problem to guarantee closed-loop stability and a level of robust stability at all frequencies. The proposed technique uses only the basic concept of loop shaping methods and then a robust stabilization controller for the normalized coprime factor perturbed system is used to construct the nal controller. This chapter is arranged as follows: The H1 theory is applied to solve the stabilization problem of a normalized coprime factor perturbed system in Section 18.1. The loop shaping design procedure is described in Section 18.2. The theoretical justi cation for the loop shaping design procedure is given in Section 18.3. 18.1 Robust Stabilization of Coprime Factors In this section, we use the H1 control theory developed in the previous chapters to solve the robust stabilization of a left coprime factor perturbed plant given by ~ ~ P = (M + ~ M );1 (N + ~ N ) h i ~ ~ ~ ~ with M N ~ M ~ N 2 RH1 and ~ N ~ M < . The transfer matrices (M N ) 1 ~ ~ are assumed to be a left coprime factorization of P (i.e., P = M ;1N ), and K internally stabilizes the nominal system. 469
- 487. H1 LOOP SHAPING 470 z1 r -f ;6 - K z2 ~N - f; ~ N - ? - M ;1 f ~ ~M w y - Figure 18.1: Left Coprime Factor Perturbed Systems It has been shown in Chapter 9 that the system is robustly stable i " # K (I + PK );1 M ;1 ~ 1= : I 1 Finding a controller such that the above norm condition holds is an H1 norm minimization problem which can be solved using H1 theory developed in the previous chapters. Suppose P has a stabilizable and detectable state space realization given by " # P= A B C D and let L be a matrix such that A + LC is stable then a left coprime factorization of ~ ~ P = M ;1N is given by # " h i A + LC B + LD L : ~ ~ N M = C D I Denote ^ K = ;K then the system diagram can be put in an LFT form as in Figure 18.2 with the generalized plant 2 " G(s) = 6 4 0 ~ ;1 M ~ M ;1 # " I P P 2 # 3 7 5 = 6 " 6 6 6 4 2 =: 6 4 A 0 C C # A B1 C1 D11 C2 D21 ;L # " 0 " I I 3 B2 D12 7 : 5 D22 B I D D 3 # 7 7 7 7 5
- 488. 18.1. Robust Stabilization of Coprime Factors 471 w z1 z2 ? i ~ M ;1 ~ N y u - ^ K Figure 18.2: LFT Diagram for Coprime Factor Stabilization To apply the H1 control formulae in Chapter 17, we need to normalize the D12" matrix rst. Note that " " # " # 1 # 2 I = U 0 (I + D D) 2 where U = D (I + DD ); 1 I + D D); 2 1 2 2 D I ;(I + DD ); 1 D(I + D D); 1 and U is a unitary matrix. Let " ^ K = (I " D D); 1 K + # 2~ # z1 = U z1 : ^ z2 z2 ^ Then kTzw k1 = kU Tzw k1 = kTzw k1 and the problem becomes of nding a controller ^ ~ K so that kTzw k1 < with the following generalized plant ^ " ^ G= U 0 # 6 " 6 6 6 4 A 2 ;(I + DD ); 1 C ;1 D C (I + D D) 2 C # G I # 0 1 0 (I + D D); 2 0 I 2 " " ;L 2 ;(I + DD ); 1 # B# " 0 3 7 7 7 7 5 : I 2 I D(I + D D); 1 ^ ~ Now the formulae in Chapter 17 can be applied to1 G to obtain a controller K and then ; 2 K . We shall leave the detail to the ~ the K can be obtained from K = ;(I + D D) = 2 (I + D D); 1 D
- 489. H1 LOOP SHAPING 472 reader. In the sequel, we shall consider the case D = 0. In this case, we have > 1 and 2 LC LC X1 (A ; 2 ; 1 ) + (A ; 2 ; 1 ) X1 ; X1 (BB ; LL 1 )X1 + 2C C = 0 2; ;1 Y1 (A + LC ) + (A + LC )Y1 ; Y1 C CY1 = 0: It is clear that Y1 = 0 is the stabilizing solution. Hence by the formulae in Chapter 17 we have h i h i L11 L21 = 0 L and ^ ^ ^ Z1 = I D11 = 0 D12 = I D21 = p 2;1 I: The results are summarized in the following theorem. Theorem 18.1 Let D = 0 and let L be such that A + LC is stable then there exists a controller K such that " # K (I + PK );1 M ;1 < ~ I 1 i > 1 and there exists a stabilizing solution X1 0 solving 2 LC LC X1 (A ; 2 ; 1 ) + (A ; 2 ; 1 ) X1 ; X1 (BB ; LL 1 )X1 + 2C C = 0: 2; ;1 Moreover, if the above conditions hold a central controller is given by " # A ; BB X1 + LC L K= : ; B X1 0 It is clear that the existence of a robust stabilizing controller depends upon the choice of the stabilizing matrix L, i.e., the choice of the coprime factorization. Now let Y 0 be the stabilizing solution to AY + Y A ; Y C CY + BB = 0 ~ ~ and let L = ;Y C . Then the left coprime factorization (M N ) given by h " A ; Y C C B ;Y C ~ ~ N M = C 0 I i is a normalized left coprime factorization (see Chapter 13). #
- 490. 18.1. Robust Stabilization of Coprime Factors 473 Corollary 18.2 Let D = 0 and L = ;Y C where Y 0 is the stabilizing solution to AY + Y A ; Y C CY + BB = 0: ~ ~ Then P = M ;1 N is a normalized left coprime factorization and " inf K stabilizing # K (I + PK );1 M ;1 ~ = I 1 p 1 ; max (Y Q) 1; = 1 h ~ ~ N M i 2 ;1=2 H =: min where Q is the solution to the following Lyapunov equation Q(A ; Y C C ) + (A ; Y C C ) Q + C C = 0: Moreover, if the above conditions hold then for any > min a controller achieving " is given by where # K (I + PK );1 M ;1 < ~ I 1 # " A ; BB X1 ; Y C C ;Y C K (s) = ; B X1 0 X1 = 2 2 ; 1Q I ; ;1 2 2 ; 1Y Q : Proof. Note that the Hamiltonian matrix associated with X1 is given by " # A + 21;1 Y C C ;BB + 21;1 Y C CY H1 = : 2 ;(A + 21;1 Y C C ) ; 2 ;1 C C Straightforward calculation shows that " where 2 I ; 2 ;1 Y H1 = 2 0 2 ;1 I " # " #;1 2 I ; 2;1 Y Hq 2 0 2 ;1 I # 0 Hq = A ; Y C C : ;C C ;(A ; Y C C ) It is clear that the stable invariant subspace of Hq is given by " # I X; (Hq ) = Im Q
- 491. H1 LOOP SHAPING 474 and the stable invariant subspace of H1 is given by " 2 X; (H1 ) = I ; 22;1 Y 0 2 ;1 I # " # 2 X; (Hq ) = Im I ; 22;1 Y Q : 2 ;1 Q Hence there is a nonnegative de nite stabilizing solution to the algebraic Riccati equation of X1 if and only if 2 I ; 2 ; 1Y Q > 0 or 1 1 ; max (Y Q) and the solution, if it exists, is given by >p X1 = 2 2 2 ; 1Q I ; 2 ; 1Y Q ;1 : Note that iY and Q are the controllability Gramian and the observability Gramian of h h i ~ ~ ~ ~ N M respectively. Therefore, we also have that the Hankel norm of N M p is max (Y Q). 2 ~ ~ Corollary 18.3 Let P = M ;1N be a normalized left coprime factorization and ~ ~ P = (M + ~ M );1 (N + ~ N ) with h i ~N ~M < : 1 Then there is a robustly stabilizing controller for P if and only if p r 1 ; max (Y Q) = 1 ; h ~ ~ N M i 2 H : The solutions to the normalized left coprime factorization stabilization problem are also solutions to a related H1 problem which is shown in the following lemma. ~ ~ Lemma 18.4 Let P = M ;1N be a normalized left coprime factorization. Then " # K (I + PK );1M ;1 = ~ I 1 " # K (I + PK );1 h I P I i 1 :
- 492. 18.1. Robust Stabilization of Coprime Factors 475 ~ ~ Proof. Since (M N ) is a normalized left coprime factorization of P , we have h and h ~ ~ M N Using these equations, we have " " = " " = " " = This implies " K I K I K I K I K I # # # # # ih ~ ~ M N i 1 = i ~ ~ M N h ~ ~ M N =I i 1 # K (I + PK );1M ;1 ~ I 1 h ih h i ~ ~ ~ (I + PK );1 M ;1 M N ~ ~ ~ (I + PK );1 M ;1 M N h h ~ (I + PK );1 M ;1 1 ~ (I + PK );1 M ;1 K (I + PK );1M ;1 = ~ I 1 1 " i ~ ~ M N h 1 i (I + PK );1 I P # = 1: 1 1 ~ ~ M N i 1 i ~ ~ M N 1 : # K (I + PK );1 h I P I i 1 : 2 Corollary 18.5 A controller solves the normalized left coprime factor robust stabilization problem if and only if it solves the following H1 control problem " and inf K stabilizing " # # I (I + PK );1 h I P K I (I + PK );1 h I P K i 1 = = i 1 p < 1 1 ; max (Y Q) 1; h ~ ~ N M i 2 ;1=2 H :
- 493. H1 LOOP SHAPING 476 The solution Q can also be obtained in other ways. Let X solution to XA + A X ; XBB X + C C = 0 then it is easy to verify that Q = (I + XY );1 X: Hence min = p 1 h = 1; ~ ~ N M i 2 ;1=2 0 be the stabilizing p = 1 + max (XY ): H 1 ; max (Y Q) Similar results can be obtained if one starts with a normalized right coprime factorization. In fact, a rather strong relation between the normalized left and right coprime factorization problems can be established using the following matrix fact. Lemma 18.6 Let M and N be any compatibly dimensioned complex matrices such that MM = M , NN = N , and M + N = I . Then i (M ) = i (N ) for all i such that 0 < i (M ) = 1. 6 Proof. We rst show that the eigenvalues of M and N are either 0 or 1 and M and N are diagonalizable. In fact, assume that is an eigenvalue of M and x is a corresponding eigenvector, then x = Mx = MMx = M (Mx) = Mx = 2 x, i.e., (1 ; )x = 0. This implies that either = 0 or = 1. To show that M is diagonalizable, assume M = TJT ;1 where J is a Jordan canonical form, it follows immediately that J must be diagonal by the condition M = MM . The proof for N is similar. Next, assume that M is diagonalized by a nonsingular matrix T such that " # M = T I 0 T ;1: 0 0 Then " # N = I ; M = T 0 0 T ;1: 0 I De ne " # A B := T T B D and assume 0 < 6= 1. Then A > 0 and , det(M M ;# I ) =" 0 " # I 0 T T I 0 ; T T) = 0 det( 0 0 0 0
- 494. 18.1. Robust Stabilization of Coprime Factors " (1 ; )A ; B 477 # , det , det(; D ; 1 ; B A;1 B ) = 0 det( 1 ; D + B A;1 B ) = 0 , , , ;B 2 ;D " =0 # ; A ; B =0 det ; B (1 ; )D det(N N ; I ) = 0: This implies that all nonzero eigenvalues of M M and N N that are not equal to 1 are equal, i.e., i (M ) = i (N ) for all i such that 0 < i (M ) 6= 1. 2 Using this matrix fact, we have the following corollary. Corollary 18.7 Let K and P be any compatibly dimensioned complex matrices. Then " # I (I + PK );1 h I P K Proof. De ne " # h M = I (I + PK );1 I P K i i " = # I (I + KP );1 h I K P " i : # h i N = ;P (I + KP );1 ;K I : I Then it is easy to verify that MM = M , NN = N and M + N = I . By Lemma 18.6, we have kM k = kN k. The corollary follows by noting that " # " # " # i I (I + KP );1 h 0 I N 0 ;I : I K = P ;I 0 I 0 2 ~ ~ Corollary 18.8 Let P = M ;1N = NM ;1 be respectively the normalized left and right coprime factorizations. Then " # K (I + PK );1M ;1 = M ;1 (I + KP );1 h ~ I K I 1 i 1 :
- 495. H1 LOOP SHAPING 478 Proof. This follows from Corollary 18.7 and the fact that h " i M ;1 (I + KP );1 I K 1 = # I (I + KP );1 h I K P i 1 : 2 This corollary says that any H1 controller for the normalized left coprime factorization is also an H1 controller for the normalized right coprime factorization. Hence one can work with either factorization. 18.2 Loop Shaping Using Normalized Coprime Stabilization This section considers the H1 loop shaping design. The objective of this approach is to incorporate the simple performance/robustness tradeo obtained in the loop shaping, with the guaranteed stability properties of H1 design methods. Recall from Section 5.5 of Chapter 5 that good performance controller design requires that ; (I + PK );1 ; (I + PK );1 P ; (I + KP );1 ; K (I + PK );1 (18:1) be made small, particularly in some low frequency range. And good robustness requires that ; ; PK (I + PK );1 KP (I + KP );1 (18:2) be made small, particularly in some high frequency range. These requirements in turn imply that good controller design boils down to achieving the desired loop (and controller) gains in the appropriate frequency range: (PK ) 1 (KP ) 1 (K ) 1 (KP ) 1 (K ) M 1 in some low frequency range and (PK ) in some high frequency range where M is not too large. The design procedure is stated below. Loop Shaping Design Procedure (1) Loop Shaping: Using a precompensator W1 and/or a postcompensator W2 , the singular values of the nominal plant are shaped to give a desired open-loop shape. The nominal plant G and the shaping functions W1 W2 are combined to form the shaped plant, Gs where Gs = W2 GW1 . We assume that W1 and W2 are such that Gs contains no hidden modes.
- 496. 18.2. Loop Shaping Using Normalized Coprime Stabilization r-e ;6 - di K -? e u - 479 d P - ? ye ? en Figure 18.3: Standard Feedback Con guration (2) Robust Stabilization: a) Calculate max , where " # I (I + G K );1 M ;1 ~s inf max = s K stabilizing K 1 r h i 2 ~ ~ = 1 ; Ns Ms <1 H !;1 ~ ~ ~ ~ and Ms Ns de ne the normalized coprime factors of Gs such that Gs = Ms;1 Ns and ~ ~ ~ ~ Ms Ms + Ns Ns = I: If max 1 return to (1) and adjust W1 and W2 . b) Select max , then synthesize a stabilizing controller K1 , which satis es " # I (I + G K );1 M ;1 ;1 : ~s s 1 K1 1 (3) The nal feedback controller K is then constructed by combining the H1 controller K1 with the shaping functions W1 and W2 such that K = W1 K1 W2 : A typical design works as follows: the designer inspects the open-loop singular values of the nominal plant, and shapes these by pre- and/or postcompensation until nominal performance (and possibly robust stability) speci cations are met. (Recall that the open-loop shape is related to closed-loop objectives.) A feedback controller K1 with associated stability margin (for the shaped plant) max , is then synthesized. If max is small, then the speci ed loop shape is incompatible with robust stability requirements, and should be adjusted accordingly, then K1 is reevaluated. In the above design procedure we have speci ed the desired loop shape by W2 GW1 . But, after Stage (2) of the design procedure, the actual loop shape achieved is in fact
- 497. H1 LOOP SHAPING 480 - W1 e ;6 - K1 G - W2 - - W1 - G - W2 Gs e ;6 - W2 - K1 - W1 - G K Figure 18.4: The Loop Shaping Design Procedure given by W1 K1W2 G at plant input and GW1 K1 W2 at plant output. It is therefore possible that the inclusion of K1 in the open-loop transfer function will cause deterioration in the open-loop shape speci ed by Gs . In the next section, we will show that the degradation in the loop shape caused by the H1 controller K1 is limited at frequencies where the speci ed loop shape is su ciently large or su ciently small. In particular, we show in the next section that can be interpreted as an indicator of the success of the loop shaping in addition to providing a robust stability guarantee for the closed-loop systems. A small value of max ( max 1) in Stage (2) always indicates incompatibility between the speci ed loop shape, the nominal plant phase, and robust closed-loop stability. Remark 18.1 Note that, in contrast to the classical loop shaping approach, the loop shaping here is done without explicit regard for the nominal plant phase information. That is, closed-loop stability requirements are disregarded at this stage. Also, in contrast with conventional H1 design, the robust stabilization is done without frequency weighting. The design procedure described here is both simple and systematic, and only assumes knowledge of elementary loop shaping principles on the part of the designer. ~ Remark 18.2 The above robust stabilization objective can also be interpreted as the more standard H1 problem formulation of minimizing the H1 norm of the frequency weighted gain from disturbances on the plant input and output to the controller input
- 498. 18.3. Theoretical Justi cation for H1 Loop Shaping and output as follows. " # I (I + G K );1 M ;1 ~s = s 1 K1 1 = = = " " " " 481 # i I (I + G K );1 h I Gs s 1 K1 1 # i W2 (I + GK );1 h ;1 W2 GW1 W1;1 K 1 # h i I (I + K G );1 I K1 1 s Gs 1 # i W1;1 (I + KG);1 h W1 GW2;1 W2 G 1 This shows how all the closed-loop objectives in (18.1) and (18.2) are incorporated. ~ 18.3 Theoretical Justi cation for H1 Loop Shaping The objective of this section is to provide justi cation for the use of parameter as a design indicator. We will show that is a measure of both closed-loop robust stability and the success of the design in meeting the loop shaping speci cations. We rst examine the possibility of loop shape deterioration at frequencies of high loop gain (typically low frequency). At low frequency (in particular, ! 2 (0 !l )), the deterioration in loop shape at plant output can be obtained by comparing (GW1 K1 W2 ) to (Gs ) = (W2 GW1 ). Note that (GK ) = (GW1 K1 W2 ) = (W2;1 W2 GW1 K1 W2 ) (W2 GW1 ) (K1 )= (W2 ) (18:3) where ( ) denotes condition number. Similarly, for loop shape deterioration at plant input, we have (KG) = (W1 K1W2 G) = (W1 K1 W2 GW1 W1;1 ) (W2 GW1 ) (K1 )= (W1 ): (18:4) In each case, (K1 ) is required to obtain a bound on the deterioration in the loop shape at low frequency. Note that the condition numbers (W1 ) and (W2 ) are selected by the designer. Next, recalling that Gs denotes the shaped plant, and that K1 robustly stabilizes the normalized coprime factorization of Gs with stability margin , then we have " # I (I + G K );1 M ;1 ~s s 1 K1 1 ;1 := (18:5) ~ ~ where (Ns Ms ) is a normalized left coprime factorization of Gs , and the parameter is de ned to simplify the notation to follow. The following result shows that (K1 )
- 499. H1 LOOP SHAPING 482 is explicitly bounded by functions of and (Gs ), the minimum singular value of the shaped plant, and hence by (18.3) and (18.4) K1 will only have a limited e ect on the speci ed loop shape at low frequency. Theorem 18.9 Any controller K1 satisfying (18.5), where Gs is assumed square, also satis es (K1 (j!)) for all ! such that p (Gs (j!)) ; 2 ; 1 p 2 ; 1 (G (j!)) + 1 s p (G (j!)) > 2 ; 1: p 2 ; 1, sthen (K (j!)) ' 1=p 2 ; 1, where ' denotes Furthermore, if (Gs ) 1 asymptotically greater than or equal to as (Gs ) ! 1. Proof. First note that (Gs ) > p 2 ; 1 implies that I + Gs Gs > 2 I: ~ ~ Further since (Ns Ms ) is a normalized left coprime factorization of Gs , we have ~ ~ ~ ~ ~ ~ Ms Ms = I ; Ns Ns = I ; Ms Gs Gs Ms : Then " Now can be rewritten as ~ ~ Ms Ms = (I + Gs Gs );1 < ;2 I: # I (I + G K );1 M ;1 ~s s 1 K1 1 (I + K1 K1 ) 2(I ; K G )(M Ms )(I ; Gs K1 ): 1 s ~s ~ (18:6) We will next show that K1 is invertible. Suppose that there exists an x such that K1 x = 0, then x (18:6) x gives ;2 x x x M Ms x ~s ~ ~ ~ which implies that x = 0 since Ms Ms < ;2I , and hence K1 is invertible. Equation (18.6) can now be written as ; ; ; ~ ~ ; (K1 K11 + I ) 2 (K1 ; Gs )Ms Ms (K11 ; Gs ): (18:7) De ne W such that ~ ~ (WW );1 = I ; 2 Ms Ms = I ; 2 (I + Gs Gs );1
- 500. 18.3. Theoretical Justi cation for H1 Loop Shaping 483 ; and completing the square in (18.7) with respect to K11 yields ; ; (K1 + N )(WW );1 (K11 + N ) ( 2 ; 1)R R where N = 2Gs ((1 ; 2 )I + Gs Gs );1 R R = (I + Gs Gs )((1 ; 2 )I + Gs Gs );1 : Hence we have ; ; R; (K1 + N )(WW );1 (K11 + N )R;1 ( 2 ; 1)I and Use ;W ;1(K ;1 + N )R;1 ;W ;1(K ;1 + N )R;1 1 ; and use (K11 + N ) p 2 ; 1: 1 ; (W ;1 ) (K11 + N ) (R;1 ) to get ; (K11 + N ) p 2 ; 1 (W ) (R) (K1 ) ; (N ) to get (K1 ) n o;1 ( 2 ; 1)1=2 (W ) (R) + (N ) : (18:8) Next, note that the eigenvalues of WW N N , and R R can be computed as follows 1 ( sG ) (WW ) = 1 ; + + G(G sG ) 2 s s 4 ( (N N ) = (1 ; 2 +Gs Gs ) ))2 (Gs Gs therefore (W ) = 1 ( sG ) (R R) = 1 ; + + G(G sG ) 2 s s p 1 min s G ) 1=2 = 1 + 2 (Gs ) 1=2 (WW ) = 1 ; + + (G(G sG ) max 2 min s s 1 ; 2 + 2 (Gs ) 2p min (Gs Gs ) 2 ( (N ) = max (N N ) = 1 ; 2 + = 1 ; 2 +Gs ) G ) 2( s min (Gs Gs ) p 1 + 2 (Gs ) 1=2 : 1 min s G ) 1=2 = (R) = max (R R) = 1 ; + + (G(G sG ) 2 min s s 1 ; 2 + 2 (Gs ) p
- 501. H1 LOOP SHAPING 484 Substituting these formulas into (18.8), we have p ( 2 ; 1)1=2 (1 + 2 (Gs )) + 2 (Gs ) ;1 = p Gs )) ; 2 ; 1 : ( (K1 ) 2 (Gs ) ; ( 2 ; 1) 2 ; 1 (Gs ) + 1 2 The main implication of Theorem 18.9 is that the bound on (K1 ) depends only on the selected loop shape, and the stability margin of the shaped plant. The value of (= ;1) directly determines the frequency range over which this result is valid{a small (large ) is desirable, as we would expect. Further, Gs has a su ciently large loop gain, then so also will Gs K1 provided (= ;1) is su ciently small. In an analogous manner, we now examine the possibility of deterioration in the loop shape at high frequency due to the inclusion of K1 . Note that at high frequency (in particular, ! 2 (!h 1)) the deterioration in plant output loop shape can be obtained by comparing (GW1 K1W2 ) to (Gs ) = (W2 GW1 ). Note that, analogously to (18.3) and (18.4) we have (GK ) = (GW1 K1W2 ) (W2 GW1 ) (K1 ) (W2 ): Similarly, the corresponding deterioration in plant input loop shape is obtained by comparing (W1 K1 W2 G) to (W2 GW1 ) where (KG) = (W1 K1 W2 G) (W2 GW1 ) (K1 ) (W1 ): Hence, in each case, (K1 ) is required to obtain a bound on the deterioration in the loop shape at high frequency. In an identical manner to Theorem 18.9, we now show that (K1 ) is explicitly bounded by functions of , and (Gs ), the maximum singular value of the shaped plant. Theorem 18.10 Any controller K1 satisfying (18.5) also satis es p 2 ; 1 + (G (j!)) s p (K1 (j!)) 1 ; 2 ; 1 (Gs (j!)) for all ! such that 1 (Gs (j!)) < p 2 : ;1 p 2 ; 1, then (K (j!)) / Furthermore, if (Gs ) 1= 1 asymptotically less than or equal to as (Gs ) ! 0. p 2 ; 1, where / denotes Proof. The proof of Theorem 18.10 is similar to that of Theorem 18.9, and is only ~ ~ sketched here: As in the proof of Theorem 18.9, we have Ms Ms = (I + Gs Gs );1 and ~ ~ (I + K1 K1 ) 2(I ; K1 Gs )(Ms Ms )(I ; Gs K1 ): (18:9)
- 502. 18.3. Theoretical Justi cation for H1 Loop Shaping Since (Gs ) < p 12 ;1 , 485 I ; 2 Gs (I + Gs Gs );1 Gs > 0 and there exists a spectral factorization V V = I ; 2 Gs (I + Gs Gs );1 Gs : Now completing the square in (18.9) with respect to K1 yields (K1 + M )V V (K1 + M ) ( 2 ; 1)Y Y where M = 2 Gs (I + (1 ; 2)Gs Gs );1 Y Y = ( 2 ; 1)(I + Gs Gs )(I + (1 ; 2 )Gs Gs );1 : Hence we have ;V (K which implies 1 + M )Y ;1 p p 2;1 2;1 (V ) (Y ;1 ) + (M ): As in the proof of Theorem 18.9, it is easy to show that (K1 ) (18:10) 1=2 2 2 ;1 ) = 1 ; ( ; 1) (Gs ) (V ) = (Y 1 + 2 (Gs ) 2 (Gs ) : 1 ; ( 2 ; 1) 2 (Gs ) Substituting these formulas into (18.10), we have p 2 ; 1 + (G ) ( 2 ; 1)1=2 (1 + 2 (Gs )) + 2 (Gs ) = s : p (K1 ) 1 ; ( 2 ; 1) 2 (Gs ) 1 ; 2 ; 1 (Gs ) (M ) = 2 The results in Theorem 18.9 and 18.10 con rm that (alternatively ) indicates the compatibility between the speci ed loop shape and closed-loop stability requirements. Theorem 18.11 Let G be the nominal plant and let K = W1K1W2 be the associated controller obtained from loop shaping design procedure in the last section. Then if " # I (I + G K );1 M ;1 ~s s 1 K1 1
- 503. H1 LOOP SHAPING 486 we have ;K (I + GK );1 ;(I + GK );1 ~ (Ms ) (W1 ) (W2 ) n ~ min (Ms ) (W2 ) 1 + (Ns ) n ~ min (Ns ) (W1 ) 1 + (Ms ) ~ (Ns ) (W1 ) (W2 ) n ~ min 1 + (Ns ) (W1 ) (Ms ) n ~ min 1 + (Ms ) (W2 ) (Ns ) ;K (I + GK );1G ;(I + GK );1G ;(I + KG);1 ;G(I + KG);1K where o (W2 ) o (W1 ) o o (18.11) (18.12) (18.13) (18.14) (W1 ) (18.15) (W2 ) (18.16) 2 (W2 GW1 ) 1=2 (18:17) 1 + 2 (W2 GW1 ) 1=2 1 ~ (18:18) (Ms ) = (Ms ) = 1 + 2 (W2 GW1 ) ~ ~ and (Ns Ms ), respectively, (Ns Ms ), is a normalized left coprime factorization, respectively, right coprime factorization, of Gs = W2 GW1 . ~ (Ns ) = (Ns ) = Proof. Note that and ~ ~ Ms Ms = (I + Gs Gs );1 ~ ~ ~ ~ Ms Ms = I ; Ns Ns : Then 1 2 (Ms ) = max (Ms Ms ) = ~ ~ ~ 1 + max (Gs Gs ) = 1 + 2 (Ns ) = 1 ; 2 (Ms ) = 2 (Gs ) : ~ 1 + 2 (Gs ) The proof for the normalized right coprime factorization is similar. follow from noting " # I (I + G K );1 M ;1 ~s s 1 and " K1 # I (I + G K );1 M ;1 ~s = s 1 K1 1 = " " # 1 2 (Gs ) All other inequalities 1 i W2 (I + GK );1 h ;1 W2 GW1 W1;1 K 1 ;1 # h i W1 (I + KG);1 W1 GW2;1 W2 G 1
- 504. 18.4. Notes and References 487 2 This theorem shows that all closed-loop objectives are guaranteed to have bounded magnitude and the bounds depend only on , W1 W2 , and G. 18.4 Notes and References The H1 loop shaping using normalized coprime factorization was developed by McFarlane and Glover 1990, 1992]. In the same references, some design examples were also shown. The method has been applied to the design of scheduled controllers for a VSTOL aircraft in Hyde and Glover 1993]. The robust stabilization of normalized coprime factors is closely related to the robustness in the gap metric and graph topology, see El-Sakkary 1985], Georgiou and Smith 1990], Glover and McFarlane 1989], McFarlane, Glover, and Vidyasagar 1990], Qiu and Davison 1992a, 1992b] Vinnicombe 1993], Vidyasagar 1984, 1985], Zhu 1989], and references therein.
- 505. 488 H1 LOOP SHAPING
- 506. 19 Controller Order Reduction We have shown in the previous chapters that the H1 control theory and synthesis can be used to design robust performance controllers for highly complex uncertain systems. However, since a great many physical plants are modeled as high order dynamical systems, the controllers designed with these methodologies typically have orders comparable to those of the plants. Simple linear controllers are normally preferred over complex linear controllers in control system designs for some obvious reasons: they are easier to understand and computationally less demanding they are also easier to implement and have higher reliability since there are fewer things to go wrong in the hardware or bugs to x in the software. Therefore, a lower order controller should be sought whenever the resulting performance degradation is kept within an acceptable magnitude. There are usually three ways in arriving at a lower order controller. A seemingly obvious approach is to design lower order controllers directly based on the high order models. However, this is still largely an open research problem. The Lagrange multiplier method developed in the next chapter is potentially useful for some problems. Another approach is to rst reduce the order of a high order plant, and then based on the reduced plant model a lower order controller may be designed accordingly. A potential problem associated with this approach is that such a lower order controller may not even stabilize the full order plant since the error information between the full order model and the reduced order model is not considered in the design of the controller. On the other hand, one may seek to design rst a high order, high performance controller and subsequently proceed with a reduction of the designed controller. This approach is usually referred to as controller reduction. A crucial consideration in controller order reduction is to take into account the closed-loop so that the closed-loop stability is guaranteed and the 489
- 507. CONTROLLER ORDER REDUCTION 490 performance degradation is minimized with the reduced order controllers. The purpose of this chapter is to introduce several controller reduction methods that can guarantee the closed-loop stability and possibly the closed-loop performance as well. 19.1 Controller Reduction with Stability Criteria We consider a closed-loop system shown in Figure 19.1 where the n-th order generalized plant G is given by # 2 A B1 B2 3 G12 = 6 4 C1 D11 D12 7 5 G22 " G = G11 G21 " A B2 C2 D22 C2 D21 D22 # is a p q transfer matrix. Suppose K is an m-th order and G22 = controller which stabilizes the closed-loop system. We are interested in investigating controller reduction methods that can preserve the closed-loop stability and minimize the performance degradation of the closed-loop systems with reduced order controllers. z y G(s) - K (s) w u Figure 19.1: Closed-loop System Diagram ^ ^ Let K be a reduced order controller and assume for the sake of argument that K has the same number of right half plane poles. Then it is obvious that the closed-loop stabil^ ity is guaranteed and the closed-loop performance degradation is limited if K ; K 1 is su ciently small. Hence a trivial controller reduction approach is to apply the model reduction procedure to the full order controller K . Unfortunately, this approach has only limited applications. One simple reason is that a reduced order controller that stabilizes the closed-loop system and gives satisfactory performance does not necessarily ^ make the error (K ;K )(j!) su ciently small uniformly over all frequencies. Therefore, the approximation error only has to be made small over those critical frequency ranges that a ect the closed-loop stability and performance. Since stability is the most basic requirement for a feedback system, we shall rst derive controller reduction methods that guarantee this property.
- 508. 19.1. Controller Reduction with Stability Criteria 491 19.1.1 Additive Reduction The following lemma follows from small gain theorem. ^ Lemma 19.1 Let K be a stabilizing controller and K be a reduced order controller. ^ and K have the same number of right half plane poles and de ne Suppose K ^ := K ; K Wa := (I ; G22 K );1 G22 : ^ Then the closed-loop system with K is stable if either kWa k1 < 1 k Wa k1 < 1: or (19:1) (19:2) Proof. Since ^ I ; G22 K = I ; G22 K ; G22 = (I ; G22 K )(I ; (I ; G22 K );1 G22 ) by small gain theorem, the system is stable if kWa k1 < 1. On the other hand, ^ I ; KG22 = I ; KG22 ; G22 = (I ; (I ; G22 K );1 G22 )(I ; KG22 ) so the system is stable if k Wa k1 < 1. 2 Now suppose K has the following state space realization # " K = Ak Bk : Ck Dk Then Wa = = 3" 7 A B2 S 0 Ip G22 5 C2 D22 Iq K Ck Iq Dk 2 3 ~ ~ A + B2 Dk R;1 C2 B2 R;1 Ck B2 R;1 7 6 B R;1C ~ 5 ~ 6 Ak + Bk D22 R;1 Ck Bk D22 R;1 7 k 2 4 " # ! 02 A k = S B6 0 @4 R;1 C2 0 Bk 0 Ip R;1 D22 Ck #1 C A ~ D22 R;1 ~ where R := I ; D22 Dk and R := I ; Dk D22 . Hence in general the order of Wa is equal to the sum of the orders of G and K . In view of the above lemma, the controller K should be reduced in such a way so ^ ^ ^ that the weighted error Wa (K ; K ) or (K ; K )Wa is small and K and K have 1 1
- 509. CONTROLLER ORDER REDUCTION 492 the same number of unstable poles. Suppose K is unstable, then in order to make sure ^ that K and K have the same number of right half plane poles, K is usually separated into stable and unstable parts as K = K+ + K; ^ where K+ is stable and a reduction is done on K+ to obtain a reduced K+ , the nal ^ = K+ + K; . ^ reduced order controller is given by K We shall illustrate the above procedure through a simple example. Example 19.1 Consider a system with 2 ;1 0 4 3 A=6 0 2 0 4 0 0 ;3 20 7 B1 = C1 = 6 1 5 4 0 0 0 0 " # D11 = 0 D12 = D21 = 0 A controller minimizing kTzw k2 is given by 1 3 213 7 B2 = C2 = 6 1 7 5 4 5 1 D22 = 0: 148:79( + 3) K = ; (s + 31:74)(ss+ 31)(s + ; 9:19) :85)(s with kTzw k2 = 55:09. Since K is unstable, we need to separate K into stable part and antistable part K = K+ + K; with 3 ;34:64 K+ = ; (s 114:15(s + +:61) + 31:74)(s 3:85) K; = s ; 9:19 : Next apply the frequency weighted balanced model reduction in Chapter 7 to ^ Wa (K+ ; K+ ) 1 we have and ^ K+ = ; s 117:085 + 34:526 :788) ^ ^ K := K+ + K; = ; (s 151:72(s +s0; 9:19) : + 34:526)( ^ The kTzw k2 with the reduced order controller K is 61:69. On the other hand, if K+ is reduced directly by balanced truncation without stability weighting Wa , then the reduced order controller does not stabilize the closed-loop system. The results are summarized in Table 19.1 where both weighted and unweighted errors are listed.
- 510. 19.1. Controller Reduction with Stability Criteria Methods BT WBT ^ K;K 1 0.1622 0.1461 ^ Wa (K ; K ) 1 2.5295 0.471 493 kTzw k2 unstable 61.69 Table 19.1: BT: Balance and Truncate, WBT: Weighted Balance and Truncate ^ It may seem somewhat strange that the unweighted error K ; K resulted from 1 weighted balanced reduction is actually smaller than that from unweighted balanced reduction. This happens because the balanced model reduction is not optimal in L1 ^ norm. We should also point out that the stability criterion Wa (K ; K ) < 1 (or 1 ^ ^ (K ; K )Wa < 1) is only su cient. Hence having Wa (K ; K ) 1 does not 1 1 ^ necessarily imply that K is not a stabilizing controller. 3 19.1.2 Coprime Factor Reduction It is clear that the additive controller reduction method in the last section is somewhat restrictive, in particular, this method can not be used to reduce the controller order if the controller is totally unstable, i.e., K has all poles in the right half plane. This motivates the following coprime factorization reduction approach. Let G22 and K have the following left and right coprime factorizations, respectively ~ ~ ~ ~ G22 = M ;1 N = NM ;1 K = V ;1 U = UV ;1 and de ne h and i h i h ~ ~ ~ ~ ~ ~ Nn Mn := (MV ; NU );1 N M = V ;1 (I ; G22 K );1 G22 I " # " # " i # Nn := N (V M ; UN );1 = G22 (I ; KG );1 V ;1 : ~ ~ 22 ~ Mn M I ~ ~ Note that Mn Nn Mn Nn do not depend upon the speci c coprime factorizations of G22 . ^ ^ Lemma 19.2 Let U V 2 RH1 be the reduced right coprime factors of U and V . Then ^ ^^ K := U V ;1 stabilizes the system if h ~ ~ ;Nn Mn i " U # " U #! ^ ; ^ < 1: V V 1 (19:3)
- 511. CONTROLLER ORDER REDUCTION 494 ^ ^ ~ ~ ~ ~ Similarly, let U V 2 RH1 be the reduced left coprime factors of U and V . Then ;1 ^ ^ U stabilizes the system if ^ ~ K := V ~ h i h ^ ^ ~ ~ ~ ~ U V ; U V i " ;Nn # < 1: M n 1 (19:4) Proof. We shall only show the results for the right coprime controller reduction. The ^ ^^ case for the left coprime factorization is analogous. It is well known that K := U V ;1 ~^ ~^ stabilizes the system if and only if (M V ; N U );1 2 RH1 . Since " h i " U ; U ## ^ ~ V ; N U = (MV ; NU ) I ; ;Nn Mn ^ ~^ ~ ~ ~ ~ M ^ V ;V the stability of the closed-loop system is guaranteed if h i" U ;U # ^ ~ ~ < 1: ;Nn Mn V ; V ^ 1 ^ ^ Now suppose U and V have the following state space realizations ^ ^ "^# 2 A B 3 U =6 ^ ^ 7 4 C1 D1 5 ^ V ^ ^ C2 D2 ^ and suppose D2 is nonsingular. Then the reduced order controller is given by " ^ ^ ^ ;1 ^ ^ ^ ;1 # C ^ = A ; B D2 ;1 2 B D2;1 : K ^1 ; D1 D2 C2 D1 D2 ^ ^ ^ ^ ^ C ^ ^ ~ ~ Similarly, suppose U and V have the following state space realizations h ^ ^ i " A B1 B2 # ^ ^ ^ ~ ~ U V = C D D ^ ^1 ^2 ^ and suppose D2 is nonsingular. Then the reduced order controller is given by " ^ ^ ^ ;1 ^ ^ ^ ^ ;1 ^ # B2 ^ K = A ; ^ ;1D^2 C B1 ;^B21D2 D1 : ; ^ D2 C D2 D1 2 (19:5) (19:6) (19:7) (19:8) It is clear from this lemma that the coprime factors of the controller should be reduced so that the weighted errors in (19.3) and (19.4) are small. Note that there is no restriction
- 512. 19.1. Controller Reduction with Stability Criteria 495 ^ on the right half plane poles of the controller. In particular, K and K may have di erent number of right half plane poles. It is also not hard to see that the additive reduction method in the last subsection may be regarded as a special case of the coprime factor ~ reduction if the controller K is stable by taking V = I or V = I . Let L F Lk and Fk be any matrices such that A + LC2 A + B2 F Ak + Lk Ck and Ak + Bk Fk are stable. De ne 3 2 i " A + LC2 B2 + LD22 L # " N # 6 A + B2 F B2 7 ~ ~ = 4 C2 + D22 F D22 5 N M = M C2 D22 I F I " # 2 Ak + Bk Fk Bk 3 h " # U =6 7 U V i = Ak + Lk Ck Bk + Lk Dk Lk : 4 Ck + Dk Fk Dk 5 ~ ~ V Ck Dk I h Fk I ~ ~ ~ ~ Then K = UV ;1 = V ;1 U and G22 = NM ;1 = M ;1 N are right and left coprime factorizations over RH1 , respectively. Moreover, 3 2 ~ ~ B2 R;1 ;B2 Dk R;1 ;B2 R;1 Ck h i 6 A + B2 Dk R;1C2 ~ ~ Ak + Bk R;1 D22 Ck ;Bk R;1 D22 Bk R;1 7 ;Nn Mn = 4 ;Bk R;1 C2 5 ;R;1 C2 R;1 D22 Ck ; Fk ;R;1 D22 R;1 2 ~ ~ 3 Ak + Bk D22 R;1 Ck ;Bk R;1 C2 ;Lk + Bk D22 R;1 " # 6 7 ~ ~ ~ 7 ;B2 R;1 A + B2 R;1 Dk C2 ;Nn = 6 ;B2R;1 Ck 6 7 6 ; D22R;1Ck 7 ~ ~ Mn R;1C2 ;D22 R;1 4 5 ~ ~ ~ R ;1 R;1 Ck ;R;1 Dk C2 ~ where R := I ; D22 Dk and R := I ; Dk D22 . Note that the orders of the weighting h i " ;Nn # ~ ~ matrices ;Nn Mn and are in general equal to the sum of the orders of Mn G and K . However, if K is an observer-based controller, i.e., " K = A + B2 F + LC2 + LD22 F ;L F 0 letting Fk = ;(C2 + D22 F ) and Lk = ;(B2 + LD22), we get " # " # # 2 A + B2 F ;L 3 h U =6 7 U V i = A + LC2 ;L ;(B2 + LD22) ~ ~ 0 5 4 F V F 0 I C2 + D22 F I ~ ~ ~ ~ MV ; NU = I V M ; UN = I:
- 513. CONTROLLER ORDER REDUCTION 496 " h # " i h i ;Nn = ;N ~ ~ ~ ~ Therefore, we can chose ;Nn Mn = ;N M and Mn M which have the same orders as that of the plant G. We shall also illustrate the above procedure through a simple example. Example 19.2 Consider a system with 2 ;1 0 4 3 20 03 213 A = 6 0 ;2 0 7 B1 = C1 = 6 1 0 7 B2 = C2 = 6 1 7 4 5 4 5 4 5 0 0 ;3 0 0 1 " # D11 = 0 D12 = D21 = 0 A controller minimizing kTzw k2 is given by 1 # D22 = 0: 3 2 ;1 ;8:198 4 0 6 ;8:198 ;18:396 ;8:198 8:198 7 7=; 6 67:2078(s + 1)(s + 3) 7 K =6 7 (s + 23:969)(s + 3:7685)(s ; 5:3414) 6 0 ;8:198 ;3 0 5 4 0 ;8:198 0 0 with kTzw k2 = 37:02. Since the controller is an observer-based controller, a natural coprime factorization of K is given by 2 ;1 ;8:198 4 0 3 " # 6 0 ;10:198 0 8:198 7 6 7 U =6 6 0 ;8:198 ;3 0 7 : 7 6 V 6 0 ;8:198 0 0 7 7 4 5 1 1 1 1 Furthermore, we have 3 2 ;1 0 4 ;1 0 7 h i 6 7 6 ~ M = 6 ;8:198 ;10:198 ;8:198 ;1 ;8:198 7 : ~ ;N 6 0 0 ;3 ;1 0 7 5 4 1 1 1 0 0 Applying the frequency weighted balanced model reduction in Chapter 7 to the weighted coprime factors, we obtain a rst order approximation " ^ # 2 ;0:1814 1:0202 3 U =6 4 1:2244 0 7 5 ^ V 1 6:504
- 514. 19.2. H1 Controller Reductions which gives h 497 i " U # " U #! ^ ~ ~ ; ^ ;Nn Mn V V 1 = 2:0928 > 1 1 ^ K = s +:2491 kTzw k2 = 52:29 6:8165 and a second order approximation 2 3 ;0:1814 14:5511 1:0202 7 "^# 6 U = 6 ;0:5288 ;4:1434 ;1:2642 7 6 6 1:2244 29:5182 0 7 7 ^ V 4 5 6:504 which gives h ;1:3084 i " U # " U #! ^ ~ ~ ; ^ ;Nn Mn V V 1 1 = 0:024 < 1 36:069(s + :1102) ^ K = ; (s + 17:3741)(s1; 4:7601) kTzw k2 = 39:14: Note that the rst order reduced controller does not have the same number of right half plane poles as that of the full order controller. Moreover, the su cient stability condition is not satis ed nevertheless the controller is a stabilizing controller. It is also interesting to note that the unstable pole of the second order reduced controller is not at the same location as that of the full order controller. 3 19.2 H1 Controller Reductions In this section, we consider H1 performance preserving controller order reduction problem. Again we consider the feedback system shown in Figure 19.1 with a generalized plant realization given by 2 A B B 3 1 2 G(s) = 6 C1 D11 D12 7 : 4 5 C2 D21 D22 The following assumptions are made: (A1) (A B2 ) is stabilizable and (C2 A) is detectable (A2) D12 has full column rank and D21 has full row rank (A3) " A ; j!I B2 C1 D12 # has full column rank for all !
- 515. CONTROLLER ORDER REDUCTION 498 (A4) " A ; j!I B1 C2 D21 # has full row rank for all !. It is shown in Chapter 17 that all stabilizing controllers satisfying kTzw k1 < can be parameterized as K = F`(M1 Q) Q 2 RH1 kQk1 < (19:9) where M1 is of the form M1 = " ^ ^ ^ # 2 A B1 B2 3 M11 (s) M12 (s) = 6 ^ ^ ^ 7 4 C1 D11 D12 5 M21 (s) M22 (s) ^ ^ ^ C D D 2 21 22 ^ ^ ^ ^ ^; ^ ^ ^ ^; ^ such that D12 and D21 are invertible and A ; B2 D121 C1 and A ; B1 D211 C2 are both ;1 and M ;1 are both stable. stable, i.e., M12 21 ^ The problem to be considered here is to nd a controller K with a minimal possible ^ order such that the H1 performance requirement F`(G K ) < is satis ed. This 1 is clearly equivalent to nding a Q so that it satis es the above constraint and the order ^ of K is minimized. Instead of choosing Q directly, we shall approach this problem from a di erent perspective. The following lemma is useful in the subsequent development and can be regarded as a special case of Theorem 11.7 (main loop theorem). Lemma 19.3 Consider a feedback system shown below z y N - Q where N is a suitably partitioned transfer matrix " w u # N (s) = N11 N12 : N21 N22 Then, the closed-loop transfer matrix from w to z is given by Tzw = F` (N Q) = N11 + N12 Q(I ; N22 Q);1 N21 : Assume that the feedback loop is well-posed, i.e., det(I ; N22 (1)Q(1)) 6= 0, and either N21 (j!) has full row rank for all ! 2 R 1 or N12 (j!) has full column rank for all ! 2 R 1 and kN k1 1 then kF` (N Q)k1 < 1 if kQk1 < 1.
- 516. 19.2. H1 Controller Reductions 499 Proof. We shall assume N21 has full row rank. The case when N12 has full column rank can be shown in the same way. To show that kTzw k1 < 1, consider the closed-loop system at any frequency s = j! with the signals xed as complex constant vectors. Let kQk1 =: < 1 and note that + + Twy = N21 (I ; N22 Q) where N21 is a right inverse of N21 . Also let := kTwy k1 . Then kwk2 kyk2, and kN k1 1 implies that kz k2 + kyk2 kwk2 + kuk2. Therefore, 2 2 2 2 2 kwk2 + ( 2 ; 1)kyk2 1 ; (1 ; 2 ) ;2 ]kwk2 kz k2 2 2 2 which implies kTzw k1 < 1 . 2 19.2.1 Additive Reduction Consider the class of (reduced order) controllers that can be represented in the form ^ K = K0 + W2 W1 where K0 may be interpreted as a nominal, higher order controller, is a stable perturbation, with stable, minimum phase, and invertible weighting functions W1 and W2 . Suppose that kF` (G K0 )k1 < . A natural question is whether it is possible to obtain ^ a reduced order controller K in this class such that the H1 performance bound remains ^ is in place of K0 . Note that this is somewhat a special case of the above valid when K ^ ^ general problem the speci c form of K restricts that K and K0 must possess the same right half plane poles, thus to a certain degree limiting the set of attainable reduced order controllers. ^ Suppose K is a suboptimal H1 controller, i.e., there is a Q 2 RH1 with kQk1 < ^ = F` (M1 Q). It follows from simple algebra that such that K ; ^ Q = F`(Ka 1 K ) where " # " # 0 I ; M11 0 I : a I 0 I 0 Furthermore, it follows from straightforward manipulations that ; ^ kQk1 < () F` (Ka 1 K ) 1 < ; () F` (Ka 1 K0 + W2 W1 ) 1 < ~ () F` (R ) < 1 where K ;1 := ~ R= " ;1=2 I 0 0 W1 #" 1 R11 R12 R21 R22 #" ;1=2 I 0 0 W2 #
- 517. CONTROLLER ORDER REDUCTION 500 and R is given by the star product " # R11 R12 = S (K ;1 a R21 R22 " # Ko I ): I 0 ~ ~ It is easy to see that R12 and R21 are both minimum phase and invertible, and hence have full column and full row rank, respectively for all ! 2 R 1. Consequently, ~ by invoking Lemma 19.3, we conclude that if R is a contraction and k k1 < 1 then ~ F` (R ) 1 < 1. This guarantees the existence of a Q such that kQk1 < , or ^ ^ equivalently, the existence of a K such that F` (G K ) < . This observation leads 1 to the following theorem. Theorem 19.4 Suppose W1 and W2 are stable, minimum phase and invertible transfer ~ matrices such that R is a contraction. Let K0 be a stabilizing controller such that ^ ^ kF`(G K0 )k1 < . Then K is also a stabilizing controller such that F` (G K ) 1 < if ^ k k1 = W2;1 (K ; K0 )W1;1 1 < 1: ~ Since R can always be made contractive for su ciently small W1 and W2 , there are in nite many W1 and W2 that satisfy the conditions in the theorem. It is obvious that ^ ^ to make W2;1 (K ; K0 )W1;1 < 1 for some K , one would like to select the largest" 1 W1 and W2 . Lemma 19.5 Assume kR22 k1 < and de ne 2 0 " # 6 ;R ;R11 R0 R012 6 L = L1 L2 = F`(6 11 0 6 0 R21 021 ;R22 L2 L3 4 R12 ;R22 0 ~ Then R is a contraction if W1 and W2 satisfy " (W1 W1 );1 0 0 (W2 W2 );1 # " Proof. See Goddard and Glover 1993]. 0 3 7 7 7 7 5 ;1 I ): # L1 L2 : L2 L 3 2 An algorithm that maximizes det(W1 W1 ) det(W2 W2 ) has been developed by Goddard and Glover 1993]. The procedure below, devised directly from the above theorem, can be used to generate a required reduced order controller which will preserve the ^ closed-loop H1 performance bound F`(G K ) < . 1
- 518. 19.2. H1 Controller Reductions 501 1. Let K0 be a full order controller such that kF`(G K0 )k1 < ~ 2. Compute W1 and W2 so that R is a contraction ^ ^ 3. Using model reduction method to nd a K so that W2;1 (K ; K0 )W1;1 1 < 1. 19.2.2 Coprime Factor Reduction The H1 controller reduction problem can also be considered in the coprime factor framework. For that purpose, we need the following alternative representation of all admissible H1 controllers. Lemma 19.6 The family of all admissible controllers such that kTzw k1 < can also be written as K (s) = F` (M1 Q) = ( 11 Q + 12 )( 21 Q + 22 );1 := UV ;1 ~ ~ = (Q ~ 12 + ~ 22 );1 (Q ~ 11 + ~ 21) := V ;1 U ~ ~ where Q 2 RH1 , kQk1 < , and UV ;1 and V ;1 U are respectively right and left coprime factorizations over RH1 , and ^ ^ ^; ^ ^ ^; ^ ^ ^; ^ " # 2 A ; B1D211C2 B2 ; B1D211D22 B1D211 3 11 12 = 6 C ; D D;1 C D ; D D;1 D D D;1 7 = 4 ^1 ^ 11 ^ 21 ^2 ^ 12 ^ 11 ^ 21 ^ 22 ^ 11 ^ 21 5 21 22 ^; ^ ^; ^ ^; ;D211C2 ;D211 D22 D211 ~= " ~ 11 ~ 12 ~ 21 ~ 22 ^ ^ ^; ^ ^ ^ ^; ^ ^ ^; # 2 A ; B2 D121C1 B1 ; B2 D121D11 ;B2D121 3 ^ ^; 5 = 6 C2 ; D22 D121 C1 D21 ; D22 D121 D11 ;D22D121 7 4 ^ ^ ^; ^ ^ ^ ^; ^ ^; ^ D121 C1 ^; ^ D121 D11 ^; D121 2 A ; B D;1C B D;1 B ; B D;1D 3 ^ ^2 ^ 12 ^1 ^2 ^ 12 ^1 ^2 ^ 12 ^ 11 7 ;1 = 6 ;1 C1 ;1 ^ ^ ^; ^ D12 ;D121 D11 4 ;D12 ^ 5 ^ ^ ^; ^ ^ ^; ^ ^ ^; ^ C2 ; D22 D121 C1 D22 D121 D21 ; D22 D121 D11 2 A ; B D;1C ;B D;1 B ; B D;1D 3 ^1 ^ 21 ^2 ^1 ^ 21 ^ 22 ^ ^1 ^ 21 ^2 7: ;1 C2 ;1 ~ ;1 = 6 ^ ^ ^; ^ D21 D211 D22 5 4 D21 ^ ^ ^ ^; ^ ^ ^; ^ ^ ^; ^ C1 ; D11 D211 C2 ;D11 D211 D12 ; D11 D211 D22 Proof. Note that all admissible H1 controllers are given by K (s) = F`(M1 Q)
- 519. CONTROLLER ORDER REDUCTION 502 " M11 M12 M21 M22 where M1 = Lemma 10.2, we have = " " 11 21 # 12 22 is given in the last subsection. Next note that from # " # # " # ; ; M12 ; M11 M211 M22 M11 M211 = ;1M22 ;1 ;M21 M21 ; ; M21 ; M22 M121M11 ;M22 M121 ~= = ; ; M121 M11 M121 " ;1 # ; M12 ;M121 M11 ;1 = ; ; M22 M121 M21 ; M22 M121 M11 " # ; ; M211 M211M22 ~ ;1 = : ; ; ;M11M211 M12 ; M11 M211 M22 ~ 11 ~ 12 ~ 21 ~ 22 2 Then the results follow from some algebra. Theorem 19.7 Let K0 = 12 ;1 be a central H1 controller such that kF`(G K0)k1 < 22 ^ ^ and let U V 2 RH1 be such that " ;1 # " # " ^ #! p I 0 ;1 12 ; U (19:10) < 1= 2: ^ 0 I V 22 1 ^ ^^ ^ Then K = U V ;1 is also a stabilizing controller such that kF`(G K )k1 < . Proof. Note that by Lemma 19.6, K is an admissible controller such that kTzw k1 < if and only if there exists a Q 2 RH1 with kQk1 < such that " U V # " := 11 Q + 12 21 Q + 22 # = " Q I # and (19:11) K = UV ;1 : ^ ^^ ^ ^ Hence, to show that K = U V ;1 with U and V satisfying equation (19.10) is also ^ a stabilizing controller such that kF`(G K )k1 < , we need to show that there is ^ another coprime factorization for K = UV ;1 and a Q 2 RH1 with kQk1 < such that equation (19.11) is satis ed. De ne " ;1 I 0 := 0 I # ;1 " 12 22 # " ^ ; U ^ V #!
- 520. 19.2. H1 Controller Reductions and partition 503 as := Then and " ^ U ^ V # " = " 12 22 # ; " # " U V I 0 0 I # : # = " ; I; " U V # # ^ U (I ; V );1 = ; U (I ; V );1 : ^ V (I ; V );1 I ^ ^ De ne U := U (I ; V );1 , V := V (I ; V );1 and Q := ; U (I ; V );1 . Then UV ;1 ^ ^ ^^ is another coprime factorization for K . To show that K = UV ;1 = U V ;1 is a stabilizing ^ )k1 < , we need to show that controller such that kF`(G K U (I ; V );1 1 < , or equivalently U (I ; V );1 1 < 1. Now U (I ; V );1 = = and by Lemma 19.3 h I 0 02 F` @4 i h i I; 0 I h i 0 I 0 p h p I= 2 0 I= 2 ;1 3 1 p A i5 2 U (I ; V );1 1 < 1 since p h i 3 2 0 h I 0 i5 4 p p I= 2 0 I= 2 2 1 < 1. 2 Similarly, we have the following theorem. Theorem 19.8 Let K0 = ~ ;1 ~ 21 be a central H1 controller such that kF`(G K0)k1 < 22 ^ V 2 RH1 be such that ~ ^ ~ and let U " ~ # " ^ #! " ;1 # ~ p I 0 21 ; U ~ ;1 < 1= 2: ^ ~ 22 ~ 0 I 1 V ^ ^ ;1 ^ ~ ~ ^ Then K = V U is also a stabilizing controller such that kF`(G K )k1 < . The above two theorems show that the H1 controller reduction problem is equivalent to a frequency weighted H1 model reduction problem. is a contraction and H1 Controller Reduction Procedures
- 521. CONTROLLER ORDER REDUCTION 504 (i) Let K0 = 12 ;1(= ~ ;1 ~ 21) be a suboptimal H1 central controller (Q = 0) such 22 22 that kTzw k1 < . ^ ;1 ^ ^ ^^ ~ ~ (ii) Find a reduced order controller K = U V ;1 (or V U ) such that the following frequency weighted H1 error " ;1 # " # " ^ #! p I 0 ;1 12 ; U < 1= 2 ^ 0 I V 22 or " # "^ ~ 21 ~ ; U ^ ~ 22 ~ V #! ~ ;1 " ;1 I 0 # I 0 1 p 1 < 1= 2: ^ (iii) The closed-loop system with the reduced order controller K is stable and the performance is maintained with the reduced order controller, i.e., ^ kTzw k1 = F`(G K ) 1 < : 19.3 Frequency-Weighted L1 Norm Approximations We have shown in the previous sections that controller reduction problems are equivalent to frequency weighted model reduction problems. To that end, the frequency weighted balanced model reduction approach in Chapter 7 can be applied. In this section, we propose another method based on the frequency weighted Hankel norm approximation method. Theorem 19.9 Let W1(s) 2 RH; and W2(s) 2 RH; with minimal state space real1 1 izations " W1 (s) = A1w B1w C1w D1w # " W2 (s) = A2w B2w C2w D2w " # # ^ ^ A B ^ and let G(s) 2 RH1 . Suppose that G1 (s) = ^1 ^ 1 2 RH1 is an r-th order C1 D1 optimal Hankel norm approximation of W1 GW2 ]+ , i.e., ^ G1 = arg degQ r W1 GW2 ]+ ; Q H inf and assume " A1w ; I B1w C1w D1w # " A2w ; I B2w C2w D2w #
- 522. 19.3. Frequency-Weighted L1 Norm Approximations 505 ^ have respectively full row rank and full column rank for all = i (A1 ) i = 1 : : : r. Then there exist matrices X Y Q, and Z such that ^ A1w X ; X A1 + B1w Y = 0 (19.12) ^1 C1w X + D1w Y = C (19.13) ^1 Q + ZC2w = 0 QA2w ; A (19.14) ^ QB2w + ZD2w = B1 : (19.15) # " ^ A1 Z is the frequency weighted optimal Hankel norm approxFurthermore, Gr := Y 0 imation, i.e., ^ inf W1 (G ; G)W2 = kW1 (G ; Gr )W2 kH = r+1 ( W1 GW2 ]+ ) : ^ H degG r Proof. We shall assume W2 = I for simplicity. The general case can be proven ^ similarly. Assume without loss of generality that A1 has a diagonal form ^ A1 = diag 1 2 : : : r ]: ^ (The proof can be easily modi ed if A1 has a general Jordan canonical form). Partition ^ X , Y , and C1 as ^ ^ ^ ^ X = X1 X2 : : : Xr ] Y = Y1 Y2 : : : Yr ] C1 = C11 C12 : : : C1r ]: Then the equations (19.12) and (19.13) can be rewritten as " A1w ; i I B1w C1w D1w By the assumption, the matrix " #" # " Xi = 0 ^ Yi C1i A1w ; i I B1w C1w D1w # i = 1 2 : : : r: # has full row rank for all i and thus the existence of X and Y is guaranteed. Let " # ^ A1w ;X B1 2 RH; : ^ W1 = 1 ^ C1w ;D1 Then using equations (19.12) and (19.13), we get 2 A B Y 0 3 2 A A X ; X A + B Y ;X B ^ ^1 1 1 1 6 0w Aw1 B1 7 = 6 0w 1w A1 1 1w ^ ^ 5 4 ^ ^1 W1 Gr = 4 B C1w D1w Y 0 C1w C1w X + D1w Y 0 3 7 5
- 523. CONTROLLER ORDER REDUCTION 506 = 2 A 0 ;X B 3 ^ 1 6 0w A1 B1 1 7 = W1 + G1: ^ ^ 5 ^ ^ 4 ^ C1w C1 0 Using this expression, we have kW1 (G ; Gr )kH = = ^ ^ ^ W1 G]+ + W1 G]; ; W1 ; G1 H = W1 G]+ ; G1 H ^ inf W1 (G ; G) kW1 (G ; Gr )kH : r+1 ( W1 G]+ ) ^ H degG r 2 ^ ^ Note that Y = C1 if W1 = I , Z = B1 if W2 = I , and the rank conditions in the ^ above theorem are actually equivalent to the statements that the poles of G1 are not zeros of W1 and W2 . These conditions will of course be satis ed automatically if W1 (s) and W2 (s) have all zeros in the right half plane. Numerical experience shows that if the weighted Hankel approximation is used to obtain a L1 norm approximation, then choosing W1 (s) and W2 (s) to have all poles and zeros in the right half plane may reduce the L1 norm approximation error signi cantly. Corollary 19.10 Let W1 (s) 2 RH; W2 (s) 2 RH; and G(s) 2 RH1. Then 1 1 ^ ^ inf W1 (G ; G)W2 inf W (G ; G)W2 = r+1 ( W1 GW2 ]+ ) : ^ ^ 1 degG r 1 H degG r The lower bound in the above corollary is not necessarily achievable. To make the 1norm approximation error as small as possible, a suitable constant matrix Dr should ^ be chosen so that W1 (G ; G ; Dr )W2 is made as small as possible. This Dr can 1 usually be obtained using any standard convex optimization algorithm. To further reduce the approximation error, the following optimization is suggested. Weighted L1 Model Reduction Procedures Let W1 and W2 be any antistable transfer matrices with all zeros in the right half plane. (i) Let # " ^ A1 Z ^ G1 = Y 0 be a weighted optimal Hankel norm approximation of G. ^ (ii) Let the reduced order model G be parameterized as " ^ # " ^ # ^ ( ) = A1 B ^ ( ) = A1 Z : G or G Y D C D
- 524. 19.3. Frequency-Weighted L1 Norm Approximations 507 (iii) Find C (or B ) and D from the following convex optimization: ^ min W1 (G ; G( ))W2 : 2Rm 1 It is noted that the weighted Hankel singular values can be used to predict the approximation error and hence to determine the order of the reduced model as in the unweighted Hankel approximation problem although we do not have an explicit L1 norm error bound in the weighted case. If the given W1 and W2 do not have all poles and zeros in the right half plane, factorizations must be performed rst to obtain the equivalent W1 (s) and W2 (s) so that W1 (s) and W2 (s) have all poles and zeros in the right half plane and W1 (s)W1 (s) = W1 (s)W1 (s) W2 (s)W2 (s) = W2 (s)W2 (s) Then we have ^ ^ W1 (G ; G)W2 1 = W1 (G ; G)W2 1 : These factorizations can be easily done using Corollary 13.28 if W1 and W2 are stable and W1 (1) and W"(1) have respectively full column rank and full row rank. For example, 2 # A B 2 RH with D full row rank and W (j!)W (j!) > 0 for all assume W2 = 1 2 2 C D " # BW 2 RH1 such that M ;1 (s) 2 RH1 !. Then there is a M (s) = A CW (DD )1=2 and where and W2 (s)W2 (s) = M (s)M (s) BW = PC + BD CW = (DD );1=2 (C ; BW X ) PA + AP + BB = 0 XA + A X + (C ; BW X ) (DD );1 (C ; BW X ) = 0: Finally, take W2 (s) = M (s). Then W2 (s) has all the poles and zeros in the right half plane and ^ ^ W1 (G ; G)W2 1 = W1 (G ; G)W2 1 : In the case where W1 and W2 are not necessarily stable, the following procedures can be applied to accomplish this task. Spectral Factorization Procedures Let W1 2 L1 and W2 2 L1 . (i) Let W1n := W1 (;s) and W2n := W2n(;s).
- 525. CONTROLLER ORDER REDUCTION 508 (ii) Let W1n = M1;1N1 and W2n = N2M2;1 be respectively the left and right coprime factorizations such that M1 and M2 are inners. (This step can be done using Theorem 13.34.) (iii) Perform the following spectral factorizations N1 N1 = V1 V1 N2 N2 = V2 V2 so that V1 and V2 have all zeros in the left half plane. (Corollary 13.23 may be used here if N1 (1) has full column rank and N2 (1) has full row rank. Otherwise, the factorization in Section 6.1 may be used to factor out the undesirable poles and zeros in the weights.) (iv) Let W1 (s) = V1 (;s) and W2 (s) = V2 (;s). We shall summarize the state space formulas for the above factorizations as a lemma. " # Lemma 19.11 Let W (s) = A B 2 L1 be a controllable and observable realizaC D tion. " # A ; j! B has full column rank for (a) Suppose W (j!)W (j!) > 0 for all ! or C D all !. Let " # Y = Ric ;A ;C C 0 0 A " ;1 ;BR;1B X = Ric ;(A ; BR ;D C ) ;C (I ; DR 1 D )C (A ; BR;1 D C ) # 0 with R := D D > 0. Then we have the following spectral factorization W W =W W where W W ;1 2 RH; and 1 " # B+YC D : W = ;1=2 A + Y C C ;1 R (D C ; B X )(I + Y X ) R1=2 " # A ; j! B has full row rank for all (b) Suppose W (j!)W (j!) > 0 for all ! or C D !. Let " # ;A ;BB X = Ric 0 0 A
- 526. 19.4. An Example 509 " ~ ;1 ~ ;C R;1 C Y = Ric ;(A ; BD ~R 1 C ) ~ ;B (I ; D R; D)B (A ; BD R;1 C ) # 0 ~ with R := DD > 0. Then we have the following spectral factorization WW = W W where W W ;1 2 RH; and 1 # " ;1 ~;1=2 : W = A + BB X (I + Y X ) (BD=2; Y C )R ~ R1 C + DB X 19.4 An Example We consider a four-disk control system studied by Enns 1984]. We shall set up the dynamical system in the standard linear fractional transformation form x = Ax + B1 w#+ B2" # _ u "p q1 H x + 0 u z = 0 I h i y = C2 x + 0 I w where q1 = 1 10;6 q2 = 1 and 2 ;0:161 ;6:004 ;0:58215 ;9:9835 ;0:40727 ;3:982 6 1 0 0 0 0 0 6 6 0 6 1 0 0 0 0 6 6 0 1 0 0 0 A=6 0 6 0 0 0 1 0 0 6 6 0 0 0 0 1 0 6 6 0 4 0 0 0 0 1 0 h 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 0 0 0 0 0 h i h B1 = pq2 B2 0 H = 0 0 0 0 0:55 11 1:32 18 213 607 6 7 607 6 7 6 7 6 7 B2 = 6 0 7 607 6 7 607 6 7 607 4 5 i 0 i C2 = 0 0 6:4432 10;3 2:3196 10;3 7:1252 10;2 1:0002 0:10455 0:99551 : The optimal H1 norm for Tzw is opt = 1:1272. We choose = 1:2 to compute an 8th order suboptimal controller Ko . The coprime factorizations of Ko are obtained using Lemma 19.6 as Ko = 12 ;1 = ~ ;1 ~ 21 . The controller is reduced using several 22 22 methods and the results are listed in Table 19.2 where the following abbreviations are made:
- 527. CONTROLLER ORDER REDUCTION 510 UWA Unweighted additive reduction: ^ Ko ; K 1 UWRCF Unweighted right coprime factor reduction: " 12 22 # " ^ ; U ^ V # 1 UWLCF Unweighted left coprime factor reduction: h SWA i h ^ ^ ~ 21 ~ 22 ; U V ~ ~ i 1 Stability weighted additive reduction: ^ Wa (Ko ; K ) 1
- 528. 19.4. An Example 511 SWRCF Stability weighted right coprime factor reduction: h SWLCF ~ ~ ;Nn Mn 12 22 # " ^ ; U ^ V #! 1 Stability weighted left coprime factor reduction: h PWA i " i h ^ ^ ~ 12 ~ 22 ; U V ~ ~ i " ;Nn # Mn 1 Performance weighted additive reduction: ^ W2;1 (Ko ; K )W1;1 1 PWRCF Performance weighted right coprime factor reduction: " ;1 I 0 0 I # ;1 " 12 22 # " ^ ; U ^ V #! 1 PWLCF Performance weighted left coprime factor reduction: h B H/O i h ^ ^ ~ 12 ~ 22 ; U V ~ ~ i ~ ;1 " ;1I 0 I 0 # 1 Balance reduction with or without weighting Hankel/convex optimization reduction with or without weighting Table 19.3 lists the performance weighted right coprime factor reduction errors and their lower bounds obtained using Corollary 19.10. The in Table 19.3 is de ned as " ;1 # " # " ^ #! I 0 ;1 12 ; U : := ^ 0 I V 22 1 " ^ U r+1 if the McMillan degree of ^ V # By Corollary 19.10, is no greater than r. Similarly, Table 19.4 lists the stability weighted right coprime factor reduction errors
- 529. CONTROLLER ORDER REDUCTION 512 ^ Order of K PWA PWRCF PWLCF UWA UWRCF UWLCF SWA SWRCF SWLCF B H/O B H/O B H/O B H/O B H/O B H/O B H/O B H/O B H/O 7 1.196 1.196 1.2 1.196 1.197 1.197 U 23.15 1.198 1.197 1.985 5.273 1.327 1.375 1.236 2.401 1.417 1.267 6 1.196 1.197 1.196 1.198 1.196 1.197 1.321 U 1.196 1.197 1.258 U 1.199 2.503 1.197 1.893 1.217 1.485 5 1.199 1.195 1.207 1.196 U 1.198 U U 1.199 1.282 27.04 U 2.27 2.802 1.251 1.612 48.04 2.259 4 1.197 1.199 1.195 1.199 1.197 1.198 U U 1.196 1.218 5.059 U 1.47 4.341 1.201 1.388 3.031 1.849 3 2 1 0 U 4.99 U U 2.73 1.94 U U 2.98 1.674 U U 2.036 1.981 U U U U U U 1.586 2.975 3.501 U U U U U U U U U U U U U U U U U U U U U U U U U 23.5 U U U 1.488 15.12 2.467 U 13.91 1.415 U U 2.622 3.527 U U U U U U 4.184 27.965 3.251 U ^ Table 19.2: F`(G K ) with reduced order controller: U{closed-loop system is unstable and their lower bounds obtained using Corollary 19.10. The s and u in Table 19.3 are de ned as h i " 12 # " U #! ^ ~ ~ ; ^ ;Nn Mn s := V 22 1 " # "^# 12 ; U u := ^ V 1 22 ^ ^ where u is obtained by taking the same U and V as in s , not from the unweighted model reduction. Table 19.2 shows that performance weighted controller reduction methods work very well. In particular, the PWRCF and PWLCF are easy to use and e ective and there is in general no preference of using either the right coprime method or left coprime method. Although some unweighted reduction methods and some stability weighted reduction methods do give reasonable results in some cases, their performances are very hard to predict. What is the worst is that the small approximation error for the
- 530. 19.5. Notes and References 513 reduction criterion may have little relevance to the closed-loop H1 performance. For example, Table 19.4 shows that the 7-th order weighted approximation error s using the Hankel/Optimization method is very small, however, the H1 performance is very far away from the desired level. Although the unweighted right coprime factor reduction method gives very good results for this example, one should not be led to conclude that the unweighted right coprime factor method will work well in general. If this is true, then one can easily conclude that the unweighted left coprime factor method will do equally well by considering a dual problem. Of course, Table 19.2 shows that this is not true because the unweighted left coprime factor method does not give good results. The only reason for the good performance of the right coprime factor method for this example is the special data structure of this example, i.e., the relationship between the B1 matrix and B2 matrix: h i B1 = pq2 B2 0 : The interested reader may want to explore this special structure further. ^ Order of K Lower bounds r 7 6 5 4 3 2 1 0 r+1 0.295 0.303 0.385 0.405 0.635 0.668 0.687 0.702 B 1.009 0.626 4.645 0.750 71.8 6.59 127.2 2.029 H/O 0.295 0.323 0.389 0.658 0.960 1 1 1 Table 19.3: PWRCF: and lower bounds 19.5 Notes and References The stability oriented controller reduction criterion is rst proposed by Enns 1984]. The weighted and unweighted coprime factor controller reduction methods are proposed by Liu and Anderson 1986, 1990], Liu, Anderson, and Ly 1990], Anderson and Liu 1989], and Anderson 1993]. For normalized H1 controller, Mustafa and Glover 1991] have proposed a controller reduction method with a prior performance bounds. Normalized coprime factors have been used in McFarlane and Glover 1990] for controller order reductions. Lenz, Khargonekar and Doyle 1987] have also proposed another H1 controller reduction method with guaranteed performance for a class of H1 problems. The main results presented in this chapter are based on the work of Goddard and Glover 1993,1994]. We should note that a satisfactory solution to the general frequency weighted L1 norm model reduction problem remains unavailable and this problem has a crucial implication toward controller reduction with preserving closed-loop H1 performance as its objective. The frequency weighted Hankel norm approximation is considered in Latham and Anderson 1986], Hung and Glover 1986], and Zhou 1993]. The L1 model reduction procedures discussed in this chapter are due to Zhou 1993].
- 531. CONTROLLER ORDER REDUCTION 514 ^ Order of K Lower bounds of s s u ^ Order of K Lower bounds of s s u r r+1 B H/O B H/O r r+1 B H/O B H/O 7 6 5 4 ;6 1:2 10;6 1:9 10;6 1:1 10 1:9 10;6 0.8421 0.5048 2.5439 0.5473 0.0001 0.2092 0.3182 0.3755 254.28 9.7018 910.01 21.444 185.9 30.85 305.3 15.38 3 2 1 0 9 10;6 6:23 10;5 1:66 10;4 2:145 10;4 11.791 1.3164 9.1461 1.5341 0.5403 0.7642 1 1 2600.2 365.45 3000.6 383.277 397.9 288.1 384.3 384.3 Table 19.4: SWRCF: s and the corresponding u
- 532. 20 Structure Fixed Controllers In this chapter we focus on the problem of designing optimal controllers with controller structures restricted for instance, the controller may be limited to be a state feedback or a constant output feedback or a xed order dynamic controller. We shall be interested in deriving some explicit necessary conditions that an optimal xed structure controller ought to satisfy. The fundamental idea is to formulate our optimal control problems as some constrained minimization problems. Then the rst-order Lagrange multiplier necessary conditions for optimality are applied to derive our optimal controller formulae. Readers should keep in mind that our purpose here is to introduce the method, not to try to solve as many problems as possible. Hence, we will try to be concise but clear. In section 20.1, we will review some Lagrange multiplier optimization methods. Then these tools will be used in section 20.2 to solve a xed order H2 optimal controller problem. 20.1 Lagrange Multiplier Method In this section, we consider the constrained minimization problem. The results quoted below are standard and can be found in any references given at the end of the chapter. Let f (x) := f (x1 x2 : : : xn ) 2 R be a real valued function de ned on a set S Rn . A point x0 2 Rn in S is said to be a (global) minimum point of f on S if f (x) f (x0 ) for all points x 2 S . A point x0 2 S is said to be a local minimum point of f on S if there is a neighborhood N of x0 such that f (x) f (x0 ) for all points x 2 N . 515
- 533. STRUCTURE FIXED CONTROLLERS 516 We will be particularly interested in the case where the set S is described by a set of functions, hi (x) = 0 i = 1 2 : : : m and m < n or equivalently h i H (x) := h1 (x) h2 (x) : : : hm (x) = 0: Hence we will focus on the local necessary (and su cient) conditions for the following problem: ( minimize f (x) (20:1) subject to H (x) = 0: We shall assume that f (x) and hi (x) are di erentiable and denote 2 6 6 rf (x) := 6 6 4 @f @x1 @f @x2 .. . @f @xn 3 7 7 7 7 5 2 i 6 6 rhm (x) = 6 6 4 h rH (x) := rh1 (x) rh2 (x) @h1 @x1 @h1 @x2 @h2 @x1 @h2 @x2 @hm @x1 @hm @x2 @h1 @xn @h2 @xn @hm @xn .. . .. . .. . 3 7 7: 7 7 5 De nition 20.1 A point x0 2 Rn satisfying the constraints H (x0 ) = 0 is said to be a regular point of the constraints if rH (x0 ) has full column rank (m) equivalently, let (x) := H (x)z for z 2 Rm , and then r (x0 ) = rH (x0 )z = 0 has the unique solution z = 0. Theorem 20.1 Suppose that x0 2 Rn is a local minimum of the f (x) subject to the constraints H (x) = 0 and suppose further that x0 is a regular point of the constraints. Then there exists a unique multiplier 2 6 6 =6 6 4 1 2 . . . m 3 7 7 2 Rm 7 7 5 such that, if we set F (x) = f (x) + H (x) , then rF (x0 ) = 0, i.e., rF (x0 ) = rf (x0 ) + rH (x0 ) = 0: In the case where the regular point conditions are either not satis ed or hard to verify, we have the following alternative.
- 534. 20.1. Lagrange Multiplier Method 517 Theorem 20.2 Suppose that x0 2 Rn is a local minimum of f (x) subject to the constraints H (x) = 0. Then there exists " 0 # 2 Rm+1 such that 0 rf (x0 ) + rH (x0 ) = 0. Remark 20.1 Although some second order necessary and su cient conditions for local minimality can be given, they are usually not very useful in our applications for the reason that they are very hard to verify except in some very special cases. Furthermore, even if some su cient conditions can be veri ed, it is still not clear whether the minima is global. It is a common practice in many applications that the rst-order necessary conditions are used to derive some necessary conditions for the existence of a minima. Then nd a solution (or solutions) from these necessary conditions and check if the solution(s) satis es our objectives regardless of the solution(s) being a global minima or not. ~ In most of the control applications, constraints are given by a symmetric matrix function, and in this case, we have the following lemma. Lemma 20.3 Let T (x) = T (x) 2 Rl l be a symmetric matrix function and let x0 2 Rn be such that T (x0 ) = 0. Then x0 is a regular point of the constraints T (x) = T (x) = 0 if, for P = P 2 Rl l , r Trace(T (x0 )P ) = 0 has the unique solution P = 0. Proof. Since T (x) = tij (x)] is a symmetric matrix, tij = tji and the e ective constraints for T (x) = 0 are given by the l(l + 1)=2 equations, tij (x) = 0 for i = 1 2 : : : l and i j l. By de nition, x0 is a regular point for the e ective constraints (tij (x) = 0 for i = 1 2 : : : l and i j l) if the following equation has the unique solution pij = 0 for i = 1 2 : : : l and i j l: (x0 ) := l X i=1 rtii pii + l l XX i=1 j =i+1 2rtij pji = 0: Now the result follows by de ning pij := pji and by noting that (x0 ) can be written as 0 1 l l XX (x0 ) = r @ with P = P . i=1 j =1 tij pji A = r Trace(T (x0 )P ) 2 Corollary 20.4 Suppose that x0 2 Rn is a local minimum of f (x) subject to the constraints T (x) = 0 where T (x) = T (x) 2 Rl l and suppose further that x0 is a regular
- 535. STRUCTURE FIXED CONTROLLERS 518 point of the constraints. Then there exists a unique multiplier P = P 2 Rl l such that if we set F (x) = f (x) + Trace(T (x)P ), then rF (x0 ) = 0, i.e., rF (x0 ) = rf (x0 ) + r Trace(T (x0 )P ) = 0: In general, in the case where a local minimal point x0 is not necessarily a regular point, we have the following corollary. Corollary 20.5 Suppose that x0 2 Rn is a local minimum of f (x) subject to the constraints T (x) = 0 where T (x) = T (x) 2 Rl l . Then there exist 0 6= ( 0 P ) 2 R Rl l with P = P such that 0 rf (x0 ) + r Trace(T (x0 )P ) = 0: Remark 20.2 We shall also note that the variable x 2 Rn may be more conveniently given in terms of a matrix X 2 Rk q , i.e., we have 2 6 x11 6 ... 6 6x 6 k1 6 6 x12 6 x = VecX := 6 .. 6 . 6 6 6 xk2 6 6 .. 6 . 4 Then is equivalent to 2 6 rF (x) := 6 4 2 6 @F (x) := 6 6 6 6 @X 4 xkq @F (x) @x11 .. . @F (x) @xkq 3 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 7 5 3 7=0 7 5 @F (x) @x11 @F (x) @x21 @F (x) @x12 @F (x) @x22 @F (x) @x1q @F (x) @x2q @xk1 @xk2 @xkq .. .. . . @F (x) @F (x) .. . @F (x) 3 7 7 7 = 0: 7 7 5 This later expression will be used throughout in the sequel. ~ As an example, let us consider the following H2 norm minimization with constant state feedback: the dynamic system is given by x = Ax + B1 w + B2 u _ z = C1 x + D12 u
- 536. 20.1. Lagrange Multiplier Method 519 and the feedback u = Fx is chosen so that A + B2 F is stable and J0 = kTzw k2 2 is minimized. For simplicity, we shall assume that D12 D12 = I and D12 C1 = 0. It is routine to verify that J0 = Trace(B1 B1 X ) where X = X 0 satis es T (X F ) := X (A + B2 F ) + (A + B2 F ) X + (C1 + D12 F ) (C1 + D12 F ) = 0: Hence the optimal control problem becomes a constrained minimization problem, and we can use the Lagrange multipliers method outlined above. (Note that in this case, " # VecX the variable x takes the form x = ). Let VecF J (X F ) := J0 + Trace(T (X F )P ) with P = P . We rst verify the regularity conditions: the equation r Trace(T (X F )P ) = 0 or, equivalently, # " @ Trace(T (X F )P ) # " P (A + B2 F ) + (A + B2 F )P = 0 @X @ Trace(T (X F )P ) = 2(B2 X + F )P @F has a unique solution P = 0 since A + B2 F is assumed to be stable at the minimum point. Hence regularity conditions are satis ed. Now the necessary condition for local optimum can be applied: @J (X F ) = B B + P (A + B F ) + (A + B F )P = 0 (20.2) 1 1 2 2 @X @J (X F ) = 2(B X + F )P = 0 (20.3) 2 @F @J (X F ) = X (A + B F ) + (A + B F ) X + (C + D F ) (C + D F ) = 0: (20.4) 2 2 1 12 1 12 @P It should be pointed out that, in general, we cannot conclude from equation (20.3) that B2 X + F = 0. Care must be exercised to arrive at such a conclusion. For example, if we assume that B1 is square and nonsingular, then we know that if F is such that A + B2 F is stable, then P > 0. Hence we have F = ;B 2 X and we substitute this relation into equation (20.4) and get the familiar Riccati equation: XA + A X ; XB2B2 X + C1 C1 = 0: This Riccati equation has a stabilizing solution if (A B2 ) is stabilizable and if (C1 A) has no unobservable modes on the imaginary axis. Indeed, in this case, the controller thus obtained is a global optimal control law.
- 537. STRUCTURE FIXED CONTROLLERS 520 20.2 Fixed Order Controllers In this section, we shall use the Lagrange multiplier method to derive some necessary conditions for a xed-order controller that minimizes an H2 performance. We shall again consider a standard system setup z y w G - u K where the system G is n-th order with a realization given by 2 A B B 3 2 6 C1 01 D12 7 : G(s) = 4 5 C2 D21 0 For simplicity, we shall assume that (i) (A B1 ) is stabilizable and (C1 A) is detectable (ii) (A B2 ) is stabilizable and (C2 A) is detectable h i h (iii) D12 C1 D12 = 0 I (iv) " # i " # B1 D = 0 . D21 21 I We shall be interested in the following xed order H2 optimal controller problem: given an integer nc n, nd an nc-th order controller " K = Ac Bc Cc 0 # that internally stabilizes the system G and minimizes the H2 norm of the transfer matrix Tzw . For technical reasons, we will further assume that the realization of the controller is minimal, i.e., (Ac Bc) is controllable and (Cc Ac ) is observable. Suppose a such controller exists, and then the closed loop transfer matrix Tzw can be written as 2 A BC B 3 " 2 1 ~ ~ # 6 BcC2 Ac c BcD21 7 =: A B Tzw = 4 5 ~ C 0 C1 D12 Cc 0
- 538. 20.2. Fixed Order Controllers ~ with A stable. Moreover, 521 ~~ ~ kTzw k2 = Trace(B B X ) 2 (20:5) ~ where X is the observability Gramian of Tzw : ~~ ~ ~ ~ ~ X A + A X + C C = 0: (20:6) Theorem 20.6 Suppose (Ac Bc Cc) is a controllable and observable triple and K = # " Ac Bc Cc 0 internally stabilizes the system G and minimizes the norm Tzw . Then ^ ^ there exist n n nonnegative de nite matrices X , Y , X , and Y such that Ac , Bc, and Cc are given by Ac = ;(A ; B2 B2 X ; Y C2 C2 ) Bc = ;Y C2 Cc = ;B2 X for some factorization (20.7) (20.8) (20.9) ^^ YX = M ; ; = Inc 1 and such that with := ; and ? := In ; the following with M positive-semisimple conditions are satis ed: 0 0 0 0 = = = = A X + XA ; XB2 B2 X + ? XB2 B2 X ? + C1 C1 AY + Y A ; Y C2 C2 Y + ? Y C2 C2 Y ? + B1 B1 ^ ^ (A ; Y C2 C2 ) X + X (A ; Y C2 C2 ) + XB2B2 X ; ? XB2B2 X ? ^ ^ (A ; B2 B2 X )Y + Y (A ; B2 B2 X ) + Y C2 C2 Y ; ? Y C2 C2 Y ? ^ ^ ^^ rank X = rank Y = rank Y X = nc : (20.10) (20.11) (20.12) (20.13) Proof. The problem can be viewed as a constrained minimization problem with the objective function given by equation (20.5) and with constraints given by equation (20.6). ~ ~ Let Y = Y 2 R(n+nc ) (n+nc ) and denote n o ~~ ~ ~ ~ ~ ~ J1 = Trace (X A + A X + C C )Y : Then @J1 = Y A + AY = 0 ~ ~ ~~ ~ @X ~ ~ has the unique solution Y = 0 since A is assumed to be stable. Hence the regularity conditions are satis ed and the Lagrange multiplier method can be applied. Form the Lagrange function J as ~~ ~ J := Trace(B B X ) + J1 1 A matrix M is called positive semisimple if it is similar to a positive de nite matrix.
- 539. STRUCTURE FIXED CONTROLLERS 522 ~ ~ and partition X and Y as " # " # ~ ~ X = X11 X12 Y = Y11 Y12 : X12 X22 Y12 Y22 The necessary conditions for (Ac Bc Cc ) to be a local minima are @J (20.14) @Ac = 2(X12 Y12 + X22 Y22 ) = 0 @J = 2(X B + X Y C + X Y C ) = 0 (20.15) 22 c 22 12 2 12 11 2 @Bc @J (20.16) @Cc = 2(B2 X11 Y12 + B2 X12 Y22 + Cc Y22 ) = 0 @J = X A + A X + C C = 0 ~~ ~ ~ ~ ~ (20.17) ~ @Y @J = Y A + AY + B B = 0: ~ ~ ~~ ~ ~ (20.18) ~ @X ~ ~ ~ It is clear that X 0 and Y 0 since A is stable. Equations (20.17) and (20.18) can be written as 0 0 0 0 0 0 = = = = = = X11 A + A X11 + X12 Bc C2 + C2 Bc X12 + C1 C1 X12 Ac + A X12 + X11 B2 Cc + C2 Bc X22 X22 Ac + Ac X22 + X12 B2 Cc + Cc B2 X12 + Cc Cc AY11 + Y11 A + B2 Cc Y12 + Y12 Cc B2 + B1 B1 AY12 + Y12 Ac + B2 Cc Y22 + Y11 C2 Bc Ac Y22 + Y22 Ac + Bc C2 Y12 + Y12 C2 Bc + Bc Bc : (20.19) (20.20) (20.21) (20.22) (20.23) (20.24) For clarity, we shall present the proof in three steps: ~ 1. X22 > 0 and Y22 > 0: We show rst that X22 > 0. Since X 0, we have X22 0 + X = X by Lemma 2.16. Hence equation (20.21) can be written as and X22 X22 12 12 + + 0 = X22 (Ac + X22 X12 B2 Cc ) + (Ac + X22 X12 B2 Cc ) X22 + Cc Cc : + Since (Cc Ac ) is observable by assumption, (Cc Ac + X22 X12 B2 Cc ) is also observable. Now it follows from the Lyapunov theorem (Lemma 3.18) that X22 > 0. Y22 > 0 follows by a similar argument. 2. formula for Ac , Bc , Cc , ;, and : given X22 > 0 and Y22 > 0, we de ne ; ; := ;X221X12 (20.25) ; := Y22 1 Y12 (20.26) := ; (20.27)
- 540. 20.2. Fixed Order Controllers ^ X ^ Y X Y 523 := := := := ; X12 X221X12 0 ; Y12 Y22 1 Y12 0 ; X11 ; X12 X221 X12 ;1 Y : Y11 ; Y12 Y22 12 (20.28) (20.29) (20.30) (20.31) Then it follows from equation (20.14) that ; =I ~ ~ and 2 = ; ; = ; = . It also follows from X 0 and Y and Y 0. Moreover, from equation (20.14), we have 0 that X 0 nc = rank X22 rank X12 nc : This implies that rank X12 = nc = rank Y12 and ^ ^ ^^ rank X = rank X12 = nc = rank Y = rank Y X: ^^ The product of Y X can be factored as ^^ Y X = (;Y12 X12 ); = Y22 X22 ; and 1 1 1 ; M := Y22 X22 = Y22=2 (Y22=2 X22 Y22=2 )Y22 1=2 is a positive semisimple matrix. Using these formulae in equations (20.15) and (20.16), we get ; Bc = ;(X221X12 Y11 + Y12 )C2 = (;Y11 ; (; )Y12 )C2 = ;(Y11 ; Y12 )C2 = ;Y C2 and ; Cc = ;B2 (X11 Y12 Y22 1 + X12 ) = ;B2 (X11 + X12 ;) = ;B2 X : The formula for Ac follows from (20.24)-; (20.23) and some algebra. ^ ^ 3. Equations for X , Y , X , and Y : Equations (20.12) and (20.13) follow by substituting Ac , Bc, and Cc into (20.20) ; and (20.23) and by using the fact ^ ^ ^ ^ that X = X and Y = Y . Finally, equations (20.10) and (20.11) follow from equations (20.19) and (20.22) with some tedious algebra. 2
- 541. STRUCTURE FIXED CONTROLLERS 524 Remark 20.3 It is interesting to note that if the full order controller nc = n is considered, then we have = I and ? = 0. In that case, equations (20.10) and (20.11) become standard H2 Riccati equations: 0 = A X + XA ; XB2 B2 X + C1 C1 0 = AY + Y A ; Y C2 C2 Y + B1 B1 Moreover, there exist unique stabilizing solutions X 0 and Y 0 to these two Riccati equations such that A ; B2 B2 X and A ; Y C2 C2 are stable. Using these facts, we get that equations (20.12) and (20.13) have unique solutions: Z1 ^ X = e(A;Y C2 C2 ) t XB2B2 Xe(A;Y C2 C2 )t dt 0 0 Z1 ^ Y = e(A;B2 B2 X )t Y C2 C2 Y e(A;B2 B2 X ) t dt 0: 0 ^ ^ It is a fact that X is nonsingular i (B2 X A ; Y C2 C2 ) is observable and that Y is nonsingular i (A ; B2 B2 X Y C2 ) is controllable, or equivalently i " # Ko := A ; B2 B2 X ; Y C2 C2 Y C2 ;B2 X 0 is controllable and observable. (Note that Ko is known to be the optimal H2 controller ^ ^ from Chapter 14). Furthermore, if X and Y are nonsingular, we can indeed nd ; and such that ; = In . In fact, in this case, ; and are both square and ; = ( );1 . Hence, we have " # " # ;1 C K = Ac Bc = ;(A ; B2 B2 X ; Y 1 2 C2 ); ;Y C2 C 0 ;B2 X ;; 0 " c # A ; B2 B2 X ; Y C2 C2 Y C2 = K = o ;B2 X 0 ^ ^ i.e., if X and Y are nonsingular or, equivalently, if optimal controller Ko is controllable and observable as we assumed, then Theorem 20.6 generates the optimal controller. However, in general, Ko is not necessarily minimal hence Theorem 20.6 will not be applicable. It is possible to derive some similar results to Theorem 20.6 without assuming the minimality of the optimal controller by using pseudo-inverse in the derivations, but that, in general, is much more complicated. An alternative solution to this dilemma would be simply by direct testing: if a given nc does not generate a controllable and observable controller, then lower nc and try again. ~ Remark 20.4 We should also note that although we have the necessary conditions for a reduced order optimal controller, it is generally hard to solve these coupled equations although some ad hoc homotopy algorithm might be used to nd a local minima. ~
- 542. 20.3. Notes and References 525 Remark 20.5 This method can also be used to derive the H1 results presented in the previous chapters. The interested reader should consult the references for details. It should be pointed out that this method su ers a severe de ciency: global results are hard to nd. This is due to (a) only rst order necessary conditions can be relatively easily derived (b) the controller order must be xed hence even if a xed-order optimal controller can be found, it may not be optimal over all stabilizing controllers. ~ 20.3 Notes and References Optimization using the Lagrange multiplier can be found in any standard optimization textbook. In particular, the book by Hestenes 1975] contains the nite dimensional case, and the one by Luenberger 1969] contains both nite and in nite dimensional cases. The Lagrange multiplier method has been used extensively by Hyland and Bernstein 1984], Bernstein and Haddard 1989], and Skelton 1988] in control applications. Theorem 20.6 was originally shown in Hyland and Bernstein 1984].
- 544. 21 Discrete Time Control In this chapter we discuss discrete time Riccati equations and some of their applications in discrete time control. A simpler form of a Riccati equation is the so-called Lyapunov equation. Hence we will start from the solutions of a discrete Lyapunov equation which are given in section 21.1. Section 21.2 presents the basic property of a Riccati equation solution as well as the necessary and su cient conditions for the existence of a solution to the LQR problem related Riccati equation. Various di erent characterizations of a bounded real transfer matrix are presented in section 21.3. The key is the relationship between the existence of a solution to a Riccati equation and the norm bound of a stable transfer matrix. Section 21.4 collects some useful matrix function factorizations and characterizations. In particular, state space criteria are stated (mostly without proof) for a transfer matrix to be inner, for the existence of coprime factorizations, innerouter factorizations, normalized coprime factorizations, and spectral factorizations. The discrete time H2 optimal control will be considered brie y in Section 21.5. Finally, the discrete time balanced model reduction is considered in section 21.6 and 21.7. 21.1 Discrete Lyapunov Equations Let A B , and Q be real matrices with appropriate dimensions, and consider the following linear equation: AXB ; X + Q = 0: (21:1) Lemma 21.1 The equation (21.1) has a unique solution if and only if i (A) j (B) 6= 1 for all i j . 527
- 545. DISCRETE TIME CONTROL 528 Proof. Analogous to the continuous case. 2 Remark 21.1 If i (A) j (B) = 1 for some i j , then the equation (21.1) has either no solution or more than one solution depending on the speci c data given. If B = A and Q = Q , then the equation is called the discrete Lyapunov equation. ~ The following results are analogous to the corresponding continuous time cases, so they will be stated without proof. Lemma 21.2 Let Q be a symmetric matrix and consider the following Lyapunov equation: AXA ; X + Q = 0 1. Suppose that A is stable, and then the following statements hold: P (a) X = 1 Ai Q(A )i and X 0 if Q 0. i=0 (b) if Q 0, then (Q A) is observable i X > 0. 2. Suppose that X is the solution of the Lyapunov equation then (a) j i (A)j 1 if X > 0 and Q 0. (b) A is stable if X 0, Q 0 and (Q A) is detectable. 21.2 Discrete Riccati Equations This section collects some basic results on the discrete Riccati equations. So the presentation of this section and the sections to follow will be very much like the corresponding sections in Chapter 13. Just as the continuous time Riccati equations play the essential roles in continuous H2 and H1 theories, the discrete time Riccati equations play the essential roles in discrete time H2 and H1 theories. Let a matrix S 2 R2n 2n be partitioned into four n n blocks as " " # S := S11 S12 S21 S22 # 0 ;I and let J = 2 R2n 2n then S is called simplectic if J ;1 S J = S ;1 . A I 0 simplectic matrix has no eigenvalues at the origin, and, furthermore, it is easy to see that if is an eigenvalue of a simplectic matrix S , then 1= , and 1= are also eigenvalues of S . Let A, Q, and G be real n n matrices with Q and G symmetric and A nonsingular. De ne a 2n 2n matrix: " # A + G(A );1 Q ;G(A );1 : S := ;(A );1 Q (A );1
- 546. 21.2. Discrete Riccati Equations 529 Then S is a simplectic matrix. Assume that S has no eigenvalues on the unit circle. Then it must have n eigenvalues in jz j < 1 and n in jz j > 1. Consider the two ndimensional spectral subspaces X; (S ) and X+ (S ): the former is the invariant subspace corresponding to eigenvalues in jz j < 1, and the latter corresponds to eigenvalues in jz j > 1. After nding a basis for X; (S ), stacking the basis vectors up to form a matrix, and partitioning the matrix, we get " X; (S ) = Im T1 T2 # where T1 T2 2 Rn n . If T1 is nonsingular or, equivalently, if the two subspaces " # X; (S ) Im 0 I are complementary, we can set X := T2T1;1 . Then X is uniquely determined by S , i.e., S 7! X is a function which will be denoted Ric thus, X = Ric(S ). As in the continuous time case, we make the following de nition. De nition 21.1 The domain of Ric, denoted by dom(Ric), consists of all (2n 2n) simplectic matrices S such that S has no eigenvalues on the unit circle and the two " # 0 subspaces X; (S ) and Im are complementary. I Theorem 21.3 Suppose S 2 dom(Ric) and X = Ric(S ). Then (a) X is unique and symmetric (b) I + XG is invertible and X satis es the algebraic Riccati equation A XA ; X ; A XG(I + XG);1 XA + Q = 0 (21:2) (c) A ; G(I + XG);1 XA = (I + GX );1 A is stable. Note that the discrete Riccati equation in (21.2) can also be written as A (I + XG);1 XA ; X + Q = 0: Remark 21.2 In the case that A is singular, all results presented in this chapter will still be true if the eigenvalue problem of S is replaced by the following generalized eigenvalue problem: " # " # I G ; A 0 0 A ;Q I and X; (S ) is taken to be the subspace spanned by the generalized principal vectors corresponding to those generalized eigenvalues in jz j < 1. Here the generalized principal
- 547. DISCRETE TIME CONTROL 530 vectors corresponding to a generalized eigenvalue fx1 : : : xk g satisfying " " # A 0 ; ;Q I " # are referred to the set of vectors " A 0 x = ;Q I 1 #! I G xi = 0 A # I G x 1 0 A # I G x i;1 i = 2 : : : k: 0 A " ~ See Dooren 1981] and Arnold and Laub 1984] for details. " Proof. (a): Since S 2 dom(Ric), 9 T = T1 T2 # such that " X; (S ) = Im T1 T2 and T1 is invertible. Let X := T2 T1;1, then " # " # # " # X; (S ) = Im T1 = Im I T1 = Im I : T2 X X Obviously, X is unique since " # " I = Im I Im X1 X2 # i X1 = X2 . Now let us show that X is symmetric. Since XT1 = T2 pre-multiply by T1 to get T1 XT1 = T1 T2: (21:3) We only need to show that T1 T2 is symmetric. Since X; (S ) is a stable invariant subspace, there is a stable n n matrix S; such that ST = TS;: (21:4) Pre-multiply by S; T J and note that S JS = J we get (S; T J )TS; = (S; T J )ST = (S; T )JST = T S JST = T JT S; (T JT )S; ; T JT = 0:
- 548. 21.2. Discrete Riccati Equations 531 This is a Lyapunov equation and S; is stable. Hence there is a unique solution T JT = 0 i.e., T2 T1 = T1 T2 : Thus we have X = X since T1 is nonsingular. (b): To show that X is the solution of the Riccati equation, we note that equation (21.4) can be written as S " h # " I = I X X Pre-multiply (21.5) by ;X I h i # T1S; T1;1: (21:5) to get i "I # ;X I S X = 0: Equivalently, we get ; XA + (I + XG)(A );1 (X ; Q) = 0: (21:6) We now show that I + XG is invertible. Suppose I + XG is not invertible, then 9 v 2 Rn such that v (I + XG) = 0: (21:7) Pre-multiply (21.6) by v , and we get v XA = 0. Since A is assumed to be invertible, it follows that v X = 0. This, in turn, implies v = 0 by (21.7). Hence I + XG is invertible and the equation (21.6) can be written as A (I + XG);1 XA ; X + Q = 0: This is the Riccati equation (21.2). h i (c): To show that X is the stabilizing solution, pre-multiply (21.5) by I 0 to get A + G(A );1 Q ; G(A );1 X = T1 S; T1;1: The above equation can be simpli ed by using Riccati equation to get (I + GX );1 A = T1 S; T1;1 which is stable. 2
- 549. DISCRETE TIME CONTROL 532 " # ~ ~ Remark 21.3 Let X+(S ) = Im T1 and suppose that T1 is nonsingular. Then the ~ T2 ~ ~~ ~ Riccati equation has an anti-stabilizing solution X = T2 T1;1 such that (I + GX );1 A is ~ antistable. Lemma 21.4 Suppose that G and Q are positive semi-de nite and that S has no eigen" # T values on unit circle. Let X; (S ) = Im 1 then T1 T2 0. T 2 " Proof. Let T = T1 T2 # and S; be such that ST = TS; (21:8) and S; has all eigenvalues inside of the unit disc. Let k Uk = TS; k = 0 1 : : :: Then Uk+1 = SUk with U0 = T . De ning V = Further de ne " # 0 I , we get T1 T2 = U0 V U0 . 0 0 Yk := ;Uk V Uk + U0 V U0 = ; = ; Now " k;1 X i=0 k;1 X i=0 #" (Ui+1 V Ui+1 ; Ui V Ui ) Ui (S V S ; V )Ui : #" # ;Q 0 I 0 S V S ; V = I ;Q 0 ;1 G(A );1 0 I 0 ;A Q I since G and Q are assumed to be positive semi-de nite. So Yk 0 for all k 0. Note that Uk ! 0 as k ! 1 since S; has all eigenvalues inside the unit disc. Therefore T1 T2 = limk!1 Yk 0. 2 Lemma 21.5 Suppose that G and Q are positive semi-de nite. Then S 2 dom(Ric) i (A G) is stabilizable and S has no eigenvalues on the unit circle.
- 550. 21.2. Discrete Riccati Equations 533 Proof. The necessary part is obvious. We now show that the stabilizability of (A G) and S having no eigenvalues on the unit circle are, in fact, su cient. To show this, we only need to show that T1 is nonsingular, i.e. Ker T1 = 0. First, it is claimed that KerT1 is S; -invariant. To prove this, let x 2 Ker T1 . Rewrite (21.8) as (A + G(A );1 Q)T1 ; G(A );1 T2 = T1 S; (21.9) ;1 QT1 + (A );1 T2 = T2 S; : ;(A ) (21.10) Substitute (21.10) into (21.9) to get AT1 ; GT2 S; = T1 S; (21:11) and pre-multiply the above equation by x S; T2 and post-multiply x to get ;x S; T2 GT2 S; x = x S; T2 T1S; x: According to Lemma 21.4 T2 T1 0, we have GT2 S; x = 0: This in turn implies that T1 S; x = 0 from (21.11). Hence Ker T1 is invariant under S; . Now to prove that T1 is nonsingular, suppose on the contrary that Ker T1 6= 0. Then S; jKer T1 has an eigenvalue, , and a corresponding eigenvector, x: S;x = x (21:12) j j < 1 0 6= x 2 Ker T1: Post-multiply (21.10) by x to get (A );1 T2 x = T2 S; x = T2 x: Now if T2 x 6= 0, then 1= is an eigenvalue of A and, furthermore, 0 = GT2 S; x = GT2 x, so GT2 x = 0, which is contradictory to the stabilizability assumption of (A G). Hence " # T we must have T2 x = 0. But this would imply 1 x = 0, which is impossible. 2 T2 Lemma 21.6 Let G and Q be positive semi-de nite matrices and " # A + G(A );1 Q ;G(A );1 : S= ;(A );1 Q (A );1 Then S has no eigenvalues on the unit circle i (A G) has no uncontrollable modes and (Q A) has no unobservable modes on the unit circle.
- 551. DISCRETE TIME CONTROL 534 Proof. (() Suppose, on the contrary, that S has an eigenvalue ej . Then " # for some " # S x = ej y " # x y x 6= 0, i.e., y (A + G(A );1 Q)x ; G(A ;(A );1 Qx + (A );1 y = ej x );1 y = ej y: Multiplying the second equation by G and adding it to the rst one give Ax ; ej Gy = ej x (21.13) j A y: ;Qx + y = e (21.14) Pre-multiplying equation (21.13) by e;j y and equation (21.14) by x yield e;j y Ax = y Gy + y x ;x Qx + x y = ej x A y: Thus It follows that ;y Gy ; x Qx = 0: yG = 0 Qx = 0: Substitute these relationships into (21.13) and (21.14) to get Ax = ej x ej A y = y: Since x and y cannot be zero simultaneously, ej is either an unobservable mode of (Q A) or an uncontrollable mode of (A G), a contradiction. ()): Suppose that S has no eigenvalue on the unit circle but ej is an unobservable mode of (Q A) and x is a corresponding eigenvector. Then it is easy to verify that " # " # 0 0 S x = ej x so ej is an eigenvalue of S , again a contradiction. The case for (A G) having uncontrollable mode on the unit circle can be proven similarly. 2
- 552. 21.2. Discrete Riccati Equations 535 Theorem 21.7 Let G and Q be positive semi-de nite matrices and " # A + G(A );1 Q ;G(A );1 : S= ;(A );1 Q (A );1 Then S 2 dom(Ric) i (A G) is stabilizable and (Q A) has no unobservable modes on the unit circle. Furthermore, X = Ric(S ) 0 if S 2 dom(Ric) and X > 0 if and only if (Q A) has no unobservable stable modes. Proof. Let Q = C C for some matrix C . The rst half of the theorem follows from Lemmas 21.5 and 21.6. Now rewrite the discrete Riccati equation as A (I + XG);1 X (I + GX );1 A ; X + A X (I + GX );2 GXA + C C = 0 (21:15) and note that by de nition (I + GX );1 A is stable and A X (I + GX );2 GXA + C C 0. Thus X 0 by Lyapunov theorem. To show that the kernel of X has the refereed property, suppose x 2 KerX , pre-multiply (21.15) by x , and post-multiply by x to get XAx = 0 Cx = 0: (21:16) This implies that KerX is an A-invariant subspace. If KerX 6= 0, then there is an 0 6= x 2 KerX , so Cx = 0, such that Ax = x. But for x 2 KerX , (I + GX );1 Ax = (I + GX );1 x = x, so j j < 1 since (I + GX );1A is stable. Thus is an unobservable stable mode of (Q A). On the other hand, suppose that j j < 1 is a stable unobservable mode of (Q A). Then there exists a x 2 C n such that Ax = x and Cx = 0 do the same pre- and postmultiplications on (21.15) as before to get j j2 x (XG + I );1 Xx ; x Xx = 0: This can be rewritten as x X 1=2 j j2 (I + X 1=2 GX 1=2 );1 ; I ]X 1=2 x = 0: Now X 0 G 0, and j j < 1 imply that j j2 (I + X 1=2GX 1=2 );1 ; I < 0. Hence Xx = 0, i.e., X is singular. 2 Lemma 21.8 Suppose that D has full column rank and let R = D D > 0 then the following statements are equivalent: " # A ; ej I B has full column rank for all 2 0 2 ]. (i) C D ;(I ; DR;1D )C A ; BR;1D C has no unobservable modes on the unit circle, (ii) or, equivalently, (D? C A ; BR;1 D C ) has no unobservable modes on the unit circle.
- 553. DISCRETE TIME CONTROL 536 ; Proof. Suppose that ej is an unobservable mode of (I ; DR;1D )C A ; BR;1D C then there is an x = 0 such that 6 (A ; BR;1 D C )x = ej x (I ; DR;1 D )Cx = 0 i.e., " But this implies that A ; ej I B C D #" " I 0 #" # x = 0: ;R;1D C I A ; ej I B C D 0 # (21:17) does not have full column rank. Conversely, suppose that (21.17) does not have full " # column rank for some then there exists " A ; ej I B C D Now let " # " #" # " # " Then u 6= 0 such that v and u = 0: v u = I 0 ;1D C I v ;R x = I 0 ;1D C I y R #" # x : y #" # u 6= 0 v (A ; BR;1 D C ; ej I )x + By = 0 (I ; DR;1D )Cx + Dy = 0: Pre-multiply (21.19) by D to get y = 0. Then we have (A ; BR;1 D C )x = ej x (I ; DR;1 D )Cx = 0 ; i.e., ej is an unobservable mode of (I ; DR;1 D )C A ; BR;1 D C . (21:18) (21:19) 2 Corollary 21.9 Suppose that D has full column rank and denote R = D D > 0. Let S have the form " # ;1 ;1 S = E + G(E;) Q ;G(E ;)1 1Q ;(E ) (E ) where E = A ; BR;1 D C , G = BR;1 B , Q = C (I ; DR;1D )C , and E is assumed " # A ; ej I B has to be invertible. Then S 2 dom(Ric) i (A B ) is stabilizable and C D full column rank for all 2 0 2 ]. Furthermore, X = Ric(S ) 0.
- 554. 21.3. Bounded Real Functions 537 Note that the Riccati equation corresponding to the simplectic matrix in Corollary 21.9 is E XE ; X ; E XG(I + XG);1 XE + Q = 0: This equation can also be written as A XA ; X ; (B XA + D C ) (D D + B XB );1 (B XA + D C ) + C C = 0: 21.3 Bounded Real Functions Let a real rational transfer matrix be given by " M (z ) = A B C D # where again A is assumed to be nonsingular and the realization is assumed to have no uncontrollable and no unobservable modes on the unit circle. Note again that all results hold for A singular case with the same modi cation as in the last section. De ne M (z ) := M T (z ;1 ). Then " ;1 ;(A );1 C M (z ) = (A ) ;1 B (A ) D ; B (A );1 C " # : # Lemma 21.10 Let M (z) = A B 2 RL1 and let S be a simplectic matrix de ned C 0 by " # A ;1 );1 S := A ; BB (;1 ) C C BB (A;1 : ;(A ) C C (A ) Then the following statements are equivalent: (i) kM (z )k1 < 1 (ii) S has no eigenvalues on the unit circle and kC (I ; A);1 B k < 1. Proof. It is easy to compute that 2 A ; BB (A );1C C ;BB (A );1 B 3 I ; M (z )M (z )];1 = 6 (A );1 C C (A );1 0 7: 4 5 ;1 C C ;B (A );1 I ;B (A ) It is claimed that I ; M (z )M (z )];1 has no uncontrollable and/or unobservable modes on the unit circle.
- 555. 538 DISCRETE TIME CONTROL To show " # that, suppose that = ej is an uncontrollable mode of I ; M (z )M (z )];1 . q Then 9 q = 1 2 C 2n such that q2 " # " # A ; BB (A );1 C C ;BB (A );1 = ej q q B = 0: q (A );1 C C (A );1 0 Hence q1 B = 0 and h i h i q1 A + q2 (A );1 C C q2 (A );1 = ej q1 q2 : There are two possibilities: 1. q2 6= 0. Then we have q2 (A );1 = ej q2 , i.e., Aq2 = e;j q2 . This implies e;j is an eigenvalue of A. This is a contradiction since M (z ) 2 RL1 . 2. q2 = 0. Then q1 A = ej q1 , which again implies that M (z ) has a mode on the unit circle if q1 6= 0, again a contradiction. Similar proof can be done for observability, hence the claim is true. Now note that " #" #" # ;I A ; BB (A );1 C C ;BB (A );1 ;I S= : I (A );1 C C (A );1 I Hence S does not have eigenvalues on the unit circle. It is clear that we have already proven that S has no eigenvalues on the unit circle i (I ; M M );1 2 RL1 . So it is su cient to show that kM (z )k1 < 1 , (I ; M M );1 2 RL1 andkM (1)k < 1: It is obvious that the right hand side is necessary. To show that it is also su cient, suppose kM (z )k1 1, then max (M (ej )) = 1 for some 2 0 2 ], since max (M (1)) < 1 and M (ej ) is continuous in . This implies that 1 is an eigenvalue of M (e;j )M (ej ), so I ; M (e;j )M (ej ) is singular. This contradicts to (I ; M M );1 2 RL1 . 2 In the above Lemma, we have assumed that the transfer matrix is strictly proper. We shall now see how to handle non-strictly proper case. For that purpose we shall focus our attention on the stable system, and we shall give an example below to show why this restriction is sometimes necessary for the technique to work. We rst note that H1 -norm of a stable system is de ned as kM (z )k1 = sup (M (z )) : jzj 1 Then it is clear that kM (z )k1 (M (1)) = kDk. Thus in particular if kM (z )k1 < 1 then I ; D D > 0. On the other hand, if a function is only known to be in RL1 , the above condition may not be true if the system is not stable.
- 556. 21.3. Bounded Real Functions 539 Example 21.1 Let 0 < < 1=2 and let # " M1 (z ) = z ; 1= = 1= 1 z 1 2 RL1 : Then kM1(z )k1 = 1; < 1, but 1 ; D D = 0. In general, if M 2 RL1 and kM k1 < 1, then I ; D D can be inde nite. 3 " A B C D # Lemma 21.11 Let M (z) = 2 RH1 . Then kM (z )k1 < 1 if and only if N (z ) 2 RH1 and kN (z )k1 < 1 where " # " ^# A + B (I ; D D);1 D C B (I ; D D);1=2 =: E B : N (z ) = ^ C 0 (I ; DD );1=2 C 0 Proof. This is exactly the analogy of Corollary 17.4. " # Theorem 21.12 Let M (z) = A B 2 RH1 and de ne C D E := A + B (I ; D D);1 D C G := ;B (I ; D D);1 B Q := C (I ; DD );1 C: Suppose that E is nonsingular and de ne a simplectic matrix as " # E + G(E );1 Q ;G(E );1 : S := ;(E );1 Q (E );1 Then the following statements are equivalent: (a) kM (z )k1 < 1 (b) S has no eigenvalues on the unit circle and kC (I ; A);1 B + Dk < 1 (c) 9 X 0 such that I ; D D ; B XB > 0 and E XE ; X ; E XG(I + XG);1 XE + Q = 0 and (I + GX );1 E is stable. Moreover, X > 0 if (C A) is observable. (d) 9 X > 0 such that I ; D D ; B XB > 0 and E XE ; X ; E XG(I + XG);1 XE + Q < 0 2
- 557. DISCRETE TIME CONTROL 540 (e) 9 X > 0 such that " A B C D #" (f) 9 T nonsingular such that " " (g) TAT ;1 TB CT ;1 D A B C D #! := X 0 0 I #! 1 In 0 = 2 < 1 with (" " 0 2 # #" # " # A B ; X 0 <0 C D 0 I T 0 0 I #" A B C D #" T 0 0 I #;1 <1 and : 12C 2 2 Cm p ) C (n+m) (n+p) (assuming that M (s) is a p m matrix and A is an n n matrix). Note that the Riccati equation in (c) can also be written as A XA ; X + (B XA + D C ) (I ; D D ; B XB );1 (B XA + D C ) + C C = 0: Proof. (a),(b) follows by Lemma 21.10 (a))(g) follows from Theorem 11.7. (g))(f) follows from Theorem 11.5. (f))(e) follows by letting X = T T . (e))(d) follows by Schur complementary formula. (d))(c) can be shown in the same way as in the proof of Theorem 13.11. (c))(a) We shall only give the proof for D = 0 case, the case D = 0 can be 6 transformed to the zero case by Lemma 21.11. Hence in the following we have E = A, G = ;BB , and Q = C C . Assuming (c) is satis ed by some X z := ej 0 and considering the obvious relation with (z ;1 I ; A )X (zI ; A) + (z ;1 I ; A )XA + A X (zI ; A) = X ; A XA = C C + A XB (I ; B XB );1 B XA:
- 558. 21.3. Bounded Real Functions 541 The last equality is obtained from substituting in Riccati equation. Now pre-multiply the above equation by B (z ;1 I ; A);1 and post-multiply by (zI ; A);1 B to get I ; M (z ;1 )M (z ) = W (z ;1 )W (z ) where # " A B W (z ) = ;1=2 B XA ;(I ; B XB )1=2 : (I ; B XB ) Suppose W (ej )v = 0 for some and v then ej is a zero of W (z ) if v 6= 0. However, all the zeros of W (z ) are given by the eigenvalues of A + B (I ; B XB );1 B XA = (I ; BB X );1 A that are all inside of the unit circle. Hence ej cannot be a zero of W . Therefore, we get I ; M (e;j )M (ej ) > 0 for all 2 0 2 ], i.e., kM k1 < 1. 2 The following more general results can be proven easily following the same procedure as in the proof of (c))(a). " # Corollary 21.13 Let M (z) = A B 2 RL1 and suppose 9 X = X such that C 0 A XA ; X + A XB (I ; B XB );1B XA + C C = 0: Then where I ; M (z ;1)M (z ) = W (z ;1 )(I ; B XB )W (z ) " # A B W (z ) = ;1 B XA ;I : (I ; B XB ) Moreover, the following statements hold: (1) if I ; B XB > 0 (< 0), then kM (z )k1 1 ( 1) (2) if I ; B XB > 0 (< 0) and j i f(I ; BB X );1 Agj 6= 1, then kM (z )k1 < 1 (> 1). Remark 21.4 As in the continuous time case, the equivalence between (a) and (b) in Theorem 21.12 can be used to compute the H1 norm of a discrete time transfer matrix. ~
- 559. DISCRETE TIME CONTROL 542 21.4 Matrix Factorizations 21.4.1 Inner Functions A transfer matrix N (z ) is called inner if N (z ) is stable and N (z )N (z ) = I for all z = ej . Note that if N has no poles at the origin, then N (ej ) = N (ej ). A transfer matrix is called outer if all its transmission zeros are stable (i.e., inside of the unit disc). " # Lemma 21.14 Let N = A B and suppose that X = X satis es C D A XA ; X + C C = 0: Then (a) D C + B XA = 0 implies N N = (D ; CA;1 B ) D (b) X 0, (A B ) controllable, and N N = (D ; CA;1 B ) D implies D C + B XA = 0. Proof. The results follow by noting the following equality: N (z )N (z ) = = 3 2 A 0 B 6 (A );1 C C (A );1 (A );1 C D 7 5 4 (D ; CA;1 B ) C ;B (A );1 (D ; CA;1 B ) D 2 3 A 0 B 6 6 7 XB + (A );1 C D 7 : 0 (A );1 4 5 DT C + B XA ;B (A );1 (D ; CA;1 B ) D 2 The following corollary is a special case of this lemma which gives the necessary and su cient conditions of a discrete inner transfer matrix with the state-space representation. Corollary 21.15 Suppose that N (z) = " A B C D # 2 RH1 is a controllable realiza- tion then N (z ) is inner if and only if there exists a matrix X = X (a) A XA ; X + C C = 0 (b) D C + B XA = 0 (c) (D ; CA;1 B ) D = D D + B XB = I . 0 such that
- 560. 21.4. Matrix Factorizations 543 The following alternative characterization of the inner transfer matrix is often useful and insightful. Corollary 21.16 Let N (s) = nonsingular such that " A B C D # 2 RH1 and assume that there exists a T " ;1 P = TAT;1 TB CT D # and P P = I: Then N (z ) is an inner. Furthermore, if the realization of N is minimal, then such T exists. Proof. Rewrite P P = I as " A B C D #" T T #" I # " Let X = T T , and then " # A B = T T : C D I # A XA ; X + C C A XB + C D = 0: B XA + D C B XB + D D ; I This is the desired equation, so N is inner. On the other hand, if the realization is minimal, then X > 0. This implies that T exists. 2 In a similar manner, Corollary 21.15 can be used to derive the state-space representation of the complementary inner factor (CIF). Lemma 21.17 Suppose that a p m (p > m) transfer matrix N (z) (minimal) is inner h i then there exists a p (p ; m) CIF N? 2 RH1 such that the matrix N N? is square and inner. A particular realization is " N? (z ) = A Y C Z where Y and Z satisfy # A XY + C Z = 0 B XY + D Z = 0 Z Z + Y XY = I: (21:20) (21:21) (21:22)
- 561. DISCRETE TIME CONTROL 544 Proof. Note that h i "A B Y # is inner. N N? = C D Z Now it is easy to prove the results by using the formula in Corollary 21.15. 2 21.4.2 Coprime Factorizations Recall " #that two transfer matrices M (z) N (z) 2 RH1 are said to be right coprime if M N is left invertible in RH1 i.e., 9 U V 2 RH1 such that U (z )N (z ) + V (z )M (z ) = I: The left coprime is de ned analogously. A plant G(z ) 2 RL1 is said to have double coprime factorization if 9 a right coprime factorization G = NM ;1 , a left coprime ~ ~ ~ ~ factorization G = M ;1 N , and U V U V 2 RH1 such that " #" # ~ V U M ;U = I: (21:23) ~ ~ ~ ;N M N V The state space formulae for discrete time transfer matrix coprime factorization are the same as for the continuous time. They are given by the following theorem. Theorem 21.18 Let G(z) = " A B C D # 2 RL1 be a stabilizable and detectable re- alization. Choose F and L such that A + BF and A + LC are both stable. Let ~ ~ ~ ~ U V U V N M N , and M be given as follows " # 2 A + BF M := 6 4 F N C + DF BZr Zr DZr " ~ # 2 A + BF LZl;1 U := 6 F 0 4 ~ V ;(C + DF ) Zl;1 " ~# " 3 7 5 3 7 5 # M := A + LC L B + LD ~ N Zl C Zl Zl D " # " # U := A + LC L ;(B + LD) V Zl;1 F 0 Zl;1 ~ ~ where Zr and Zl are any nonsingular matrices. Then G = NM ;1 = M ;1N are rcf and lcf, respectively, and (21.23) is satis ed.
- 562. 21.4. Matrix Factorizations 545 Some coprime factorizations are particularly interesting, for example, the coprime factorization with inner numerator. This factorization in the case of G(z ) 2 RH1 yields an inner-outer factorization. " # j B Theorem 21.19 Assume that (A B) is stabilizable, A ;Ce I D has full column rank for all 2 0 2 ], and D has full column rank. Then there exists a right coprime factorization G = NM ;1 such that N is inner. Furthermore, a particular realization is given by " # 2 A + BF M := 6 4 F N C + DF where BR;1=2 R;1=2 DR;1=2 3 7 5 R = D D + B XB F = ;R;1 (B XA + D C ) and X = X 0 is the unique stabilizing solution AD X (I + B (D D);1 B X );1AD ; X + C D? D? C = 0 where AD := A ; B (D D);1 D C . Using Lemma 21.17, the complementary inner factor of N in Theorem 21.19 can be obtained as follows " # A + BF Y N? = C + DF Z where Y and Z satisfy A XY + C Z = 0 B XY + D Z = 0 Z Z + Y XY = I: Note that Y and Z are only related to F implicitly through X . Remark 21.5 If G(z) 2 RH1 , then the denominator matrix M in Theorem 21.19 is an outer. Hence, the factorization G = N (M ;1) is an inner-outer factorization. ~ Suppose that the system G(z ) is not stable then a coprime factorization with inner denominator can also be obtained by solving a special Riccati equation.
- 563. 546 " # DISCRETE TIME CONTROL Theorem 21.20 Assume that G(z) := A B 2 RL1 and that (A B) is stabilizC D able. Then there exists a right coprime factorization G = NM ;1 such that M is inner if and only if G has no poles on the unit circle. A particular realization is " # 2 A + BF BR;1=2 3 M := 6 R;1=2 7 4 F 5 N ;1=2 C + DF DR where R = I + B XB F = ;R;1B XA and X = X 0 is the unique stabilizing solution to A X (I + BB X );1 A ; X = 0: Another special coprime factorization is called normalized coprime factorization which has found applications in many control problems such as model reduction controller design and gap metric characterization. Recall that a right coprime factorization of G = NM ;1 with N M 2 RH1 is called a normalized right coprime factorization if M M +N N =I " M i.e., if N # is an inner. ~ ~ Similarly, an lcf G = M ;1 N is called a normalized left coprime factorization if i ~ N is a co-inner. Then the following results follow in the same way as for the ~ M continuous time case. Theorem 21.21 Let a realization of G be given by h " G= A B C D # and de ne ~ R = I + D D > 0 R = I + DD > 0: (a) Suppose that (A B ) is stabilizable and that (C A) has no unobservable modes on the imaginary axis. Then there is a normalized right coprime factorization G = NM ;1 " # 2 A + BF BZ ;1=2 3 M := 6 Z ;1=2 7 2 RH1 5 4 F N ;1=2 C + DF DZ
- 564. 21.4. Matrix Factorizations where and X = X 547 Z = R + B XB F = ;Z ;1(B XA + D C ) 0 is the unique stabilizing solution ~ An X (I + BR;1 B X );1 An ; X + C R;1 C = 0 where An := A ; BR;1 D C . (b) Suppose that (C A) is detectable and that (A B ) has no uncontrollable modes on the imaginary axis. Then there is a normalized left coprime factorization G = ~ ~ M ;1N h i " A + LC L B + LD # ~ ~ M N := ~ ;1=2 ~ ;1=2 ~ ;1=2 Z C Z Z D where ~ ~ Z = R + CY C ~ L = ;(BD + AY C )Z ;1 and Y = Y 0 is the unique stabilizing solution ~ ~ ~ An Y (I + C R;1CY );1 An ; Y + BR;1 B = 0 ~ ~ where An := A ; BD R;1 C . " M (c) The controllability Gramian P and observability Gramian Q of N by # are given P = (I + Y X );1Y Q = X h ~ ~ ~ ~ while the controllability Gramian P and observability Gramian Q of M N are given by ~ P =Y i ~ Q = (I + XY );1 X: 21.4.3 Spectral Factorizations The following theorem gives a solution to a special class of spectral factorization problems. " # Theorem 21.22 Assume G(z) := A B 2 RH1 and > kG(z)k1. Then, there C D exists a transfer matrix M 2 RH1 such that M M = 2I ; G G and M ;1 2 RH1 . A particular realization of M is " # A B M (z ) = ;R1=2 F R1=2
- 565. DISCRETE TIME CONTROL 548 where RD = 2 I ; D D R = RD ; B XB F = (RD ; B XB );1 (B XA + D C ) and X = X 0 is the stabilizing solution of ; ; As X (I ; BRD1 B X );1 As ; X + C (I + DRD1 D )C = 0 where As := A + BR;1 D C . D Similar to the continuous time, we have the following theorem. # " Theorem 21.23 Let G(z) := A B 2 RH1 with D full row rank and G(ej )G (ej ) > C D 0 for all . Then, there exists a transfer matrix M 2 RH1 such that M M = GG . A particular realization of M is " M (z ) = A BW CW DW # where BW = APC + BD DW DW = DD CW = DW (DD );1 (C ; BW XA) and APA ; P + BB = 0 A XA ; X + (C ; BW XA) (DD );1 (C ; BW XA) = 0: 21.5 Discrete Time H2 Control Consider the system described by the block diagram z y G - K w u
- 566. 21.5. Discrete Time H2 Control 549 The realization of the transfer matrix G is taken to be of the form 2 A B B 3 " # 1 2 6 C1 D11 D12 7 =: A B : G(z ) = 4 5 C D C2 D21 0 The following assumptions are made: (A1) (A B2 ) is stabilizable and (C2 A) is detectable h i (A2) D12 is full column rank with D12 D? unitary and D21 is full row rank with " # D21 unitary ~ D? (A3) (A4) " " A ; ej I B2 C1 D12 A ; ej I B1 C2 D21 # # has full column rank for all 2 0 2 ] has full row rank for all 2 0 2 ]. The problem in this section is to nd an admissible controller K which minimizes kTzw k2 . Denote Let X2 0 and Y2 Ax := A ; B2 D12 C1 Ay := A ; B1 D21 C2 : 0 be the stabilizing solutions to the following Riccati equations: Ax (I + X2 B2 B2 );1 X2 Ax ; X2 + C1 D? D? C1 = 0 and ~ ~ Ay (I + Y2 C2 C2 );1 Y2 Ay ; Y2 + B1 D? D? B1 = 0: Note that the stabilizing solutions exist by the assumptions (A3) and (A4). Note also that if Ax and Ay are nonsingular, the solutions can be obtained through the following two simplectic matrices: " ;1 ( ;1 H2 := Ax + B2 B2 ;Ax ) C1 D?D? C1 ;B2 B2 (A1x ) ;(Ax ) 1 C1 D? D? C1 (Ax ); " # ;1 C ;1 ~ ~ J2 := Ay + C2;12 Ay ~ B1~D? D?B1 ;C2 C21Ay : ;Ay B1 D? D? B1 A; y #
- 567. DISCRETE TIME CONTROL 550 De ne Rb F2 F0 L2 L0 and I + B2 X2 B2 ;(I + B2 X2 B2 );1 (B2 X2 A + D12 C1 ) ;(I + B2 X2 B2 );1 (B2 X2 B1 + D12 D11 ) ;(AY2 C2 + B1 D21 )(I + C2 Y2 C2 );1 (F2 Y2 C2 + F0 D21 )(I + C2 Y2 C2 );1 := := := := := AF2 := A + B2 F2 C1F2 := C1 + D12 F2 AL2 := A + L2C2 B1L2 := B1 + L2 D21 ^ A2 := A + B2 F2 + L2C2 " # AF2 B1 + B2 F0 Gc (z ) := C1F2 D11 + D12 F0 " # B1L2 Gf (z ) := 1=2 AL2 1=2 (L0 D21 ; F0 ) : Rb (L0 C2 ; F2 ) Rb Theorem 21.24 The unique optimal controller is " ^ # A2 ; B2 L0 C2 ;(L2 ; B2 L0) : Kopt(z ) := F2 ; L0 C2 L0 Moreover, min kTzw k2 = kGc k2 + kGf k2 . 2 2 2 " # Remark 21.6 Note that for a discrete time transfer matrix G(z) = A B 2 RH2, C D its H2 norm can be computed as kG(z )k2 = TracefD D + B LoB g = TracefDD + CLc C g 2 where Lc and Lo are the controllability and observability Gramians ALcA ; Lc + BB = 0 A Lo A ; Lo + C C = 0: Using the above formula, we can compute min kTzw k2 by noting that X2 and Y2 satisfy 2 the equations AF2 X2 AF2 ; X2 + C1F2 C1F2 = 0 AL2 Y2 AL2 ; Y2 + B1L2 B1L2 = 0:
- 568. 21.5. Discrete Time H2 Control 551 For example, kGc k2 = Trace f(D11 + D12 F0 ) (D11 + D12 F0 ) + (B1 + D2 F0 )X2 (B1 + D2 F0 )g 2 and kGf k2 = Trace Rb f(L0 D21 ; F0 )(L0 D21 ; F0 ) + (L0 C2 ; F2 )Y2 (L0 C2 ; F2 ) g : 2 ~ Proof. Let x denote the states of the system G. Then the system can be written as x = Ax + B1 w + B2 u _ z = C1 x + D11 w + D12 u y = C2 x + D21 w: (21.24) (21.25) (21.26) De ne := u ; F2 x ; F0 w then the transfer function from w " z = AF2 B1 + B2 F0 B2 C1F2 D11 + D12 F0 D12 #" # w = G w + UR1=2 c b " where to z becomes # ; AF2 B2 Rb 1=2 : U (s) := ; C1F2 D12 Rb 1=2 It is easy to shown that U is an inner and that U Gc 2 RH? . Now denote the transfer 2 function from w to by T w . Then 1 Tzw = Gc + URb =2 T w and 2 1 kTzw k2 = kGc k2 + Rb =2 T w 2 kGc k2 2 2 2 for any given stabilizing controller K . Hence if the states (x) and the disturbance (w) are both available for feedback (i.e., full information control) and u = F2 x + F0 w, then T w = 0 and kTzw k2 = kGc k2 . Therefore, u = F2 x + F0 w is an optimal full information control law. Note that = T w w T w = F` (G K ) G y - K w u 2 A B B 3 1 2 G = 6 ;F2 ;F0 I 7 4 5 C2 D21 0
- 569. DISCRETE TIME CONTROL 552 Since A ; B2 (;F2 ) = A + B2 F2 is stable, from the relationship between an output estimation (OE) problem and a full control (FC) problem, all admissible controllers for G (hence for the output feedback problem) can be written as K = F` (Mt KFC ) 2 6 A + B2F2 Mt = 6 ;F2 6 4 C2 0 0 h I hI ;B2i h0 Ii 0 0 i3 7 7 7 5 where KFC is an internally stabilizing controller for the following system: h i3 2 A B1 I 0i7 6 h ^ 6 G = 6 ;F2 ;F0 h 0 I i 7 : 7 4 5 C2 D21 0 0 ^ Furthermore, T w = F`(G F` (Mt KFC )) = F` (G KFC ). Hence 1 ^ min kTzw k2 = kGc B1 k2 + K kRb =2F` (G KFC )k2 : 2 2 min 2 K FC Next de ne ~ G := = " 2 6 6 6 4 # 1 Rb =2 0 G ^ 0 I 2I 0 60 I 4 3 0 0 7 5 ;1=2 0 0 Rb h i3 A B1 I 0i7 h 1 1 ;Rb =2 F2 ;Rb =2F0 h 0 I i 7 7 5 D21 C2 0 0 and ~ KFC := " # I 0 KFC : 1 0 Rb =2 Then it is easy to see that 1 ^ ~ ~ Rb =2 F`(G KFC ) = F` (G KFC ) and 1 ^ ~ ~ min kRb =2F` (G KFC )k2 = min kF`(G KFC )k2 : ~ K KFC FC
- 570. 21.6. Discrete Balanced Model Reduction 553 " # L2 ~ ~ ~ A controller minimizing kF`(G KFC )k2 is KFC = 1=2 L since the transpose (or Rb 0 ~ KFC ) is a full information feedback problem considered at the beginning ~ dual) of F` (G of the proof. Hence we get ~ F` (G and " # L2 1=2 L ) = Gf Rb 0 min kTzw k2 = kGck2 + kGf k2 : 2 2 2 K Finally, the optimal output feedback controller is given by K = F` (Mt " # " # L2 ) = A + B2 F2 + L2 C2 ; B2 L0 C2 L2 ; B2 L0 : L0 ;F2 + L0 C2 L0 The proof of uniqueness is similar to the continuous time case, and hence omitted. 2 It should be noted that in contrast with the continuous time the full information optimal control problem in the discrete time is not a state feedback even when D11 = 0. The discrete time H1 control problem is much more involved and it is probably more e ective to obtain the discrete solution by using a bilinear transformation. 21.6 Discrete Balanced Model Reduction In this section, we will show another application of the LFT machinery in discrete balanced model reduction. We will show an elegant proof of the balanced truncation error bound. Consider a stable discrete time system G(z ) and assume that the transfer matrix has the following realization: # " G(s) = A B 2 RH1 : C D Let P and Q be two positive semi-de nite symmetric matrices such that APA ; P + BB 0 A QA ; Q + C C 0: Without loss of generality, we shall assume that P =Q= " 1 0 0 2 # (21:27) (21:28)
- 571. DISCRETE TIME CONTROL 554 with 1 = diag( 1 Is1 2 Is2 : : : r Isr ) 0 2 = diag( r+1 Isr+1 r+2 Isr+2 : : : n Isn ) 0 where si denotes the multiplicity of i . (Note that the singular values are not necessarily ordered.) Moreover, the realization for G(z ) is partitioned conformably with P and Q: 2A A B 3 11 12 6 A21 A22 B1 7 : G(s) = 4 2 5 C1 C2 D Theorem 21.25 Suppose 1 > 0. Then A11 is stable. Proof. We shall rst assume 2 > 0. From equation (21.28), we have A11 1 A11 ; 1 + A21 2 A21 + C1 C1 0: Assume that is an eigenvalue of A11 then there is an x = 0 such that 6 A11 x = x: (21:29) (21:30) Now pre-multiply x and post-multiply x to equation (21.29) to get (j j2 ; 1)x 1 x + x A21 2 A21 x + x C1 C1 x 0: It is clear that j j 1. However, if j j = 1, say = ej for some , we have A21 x = 0 C1 x = 0: These equations together with equation (21.30) imply that " A11 A12 A21 A22 #" # " # 0 0 x = ej x i.e., ej is an eigenvalue of A. This contradicts the stability assumption of A, so A11 is stable. Now assume that 2 is singular, and we will show that we can remove all those states corresponding to the zero singular values without changing the system stability. For that purpose, we assume 2 = 0. Then the inequality (21.27) can be written as " A11 1 A11 ; 1 + B1 B1 A11 1 A21 + B1 B2 A21 1 A11 + B2 B1 A21 1 A21 + B2 B2 # 0:
- 572. 21.6. Discrete Balanced Model Reduction This implies that and 555 A11 1 A11 ; 1 + B1 B1 0 (21:31) A21 1 A21 + B2 B2 0: (21:32) But inequality (21.32) implies that A21 = 0 Hence we have " B2 = 0: A = A11 A12 0 A22 # and the stability of A11 and A22 are ensured. Substitute this A matrix into the inequality (21.28), and we obtain A11 1 A11 ; 1 + C1 C1 0: The subsystem with A11 still satis es inequalities (21.27) and (21.28) with 1 > 0. This proves that we can assume without loss of generality that 2 > 0. 2 Remark 21.7 It is important to note that the realization for the truncated subsystem " Gr = A11 B1 C1 D # is still balanced in some sense1 since the system parameters satisfy the following equations: A11 1 A11 ; 1 + A12 2 A12 + B1 B1 0 A11 1 A11 ; 1 + A21 2 A21 + C1 C1 0: But these equations imply that hold. A11 1 A11 ; 1 + B1 B1 0 A11 1 A11 ; 1 + C1 C1 0 " # n X Theorem 21.26 Suppose Gr = A11 B1 . Then kG ; Gr k1 2 i. C1 D In particular, kGk1 kDk + 2 n X i=r+1 i. i=1 1 Balanced in the sense that the same inequalities as (21.27) and (21.28) are satis ed. ~
- 573. DISCRETE TIME CONTROL 556 Without loss of generality, we shall assume n = 1. We will prove that for 2 = n I = I , we have kG ; Gr k1 2 r = n ; 1: Then the theorem follows immediately by scaling and recursively applying this result since the reduced system Gr is still balanced. It will be seen that it is more convenient to set = 1=2 . The proof of the theorem 1 will follow from the following two lemmas and the bounded real lemma which establishes the relationship between the H1 norm of a transfer matrix and its realizations. (Note that in the following, a constant matrix X is said to be contractive or a contraction if kX k 1 and strictly contractive if kX k < 1). The lemma below shows that for any stable system there is a realization such that " # h i A A B is a contraction, and, similarly, there is another realization such that C is a contraction. Lemma 21.27 Suppose that a realization of the transfer matrix G satis es P = Q = diagf 2 I g then " ;1 # ;1 B1 A12 ;1A11 A22 A21 B2 and 2 A ;1 A 3 6 A21 ;1 A22 7 4 11 12 5 ;1 C1 C2 are contractive. Proof. Since P = " 2 0 0 I h h # satis es the inequality (21.27), A B i" P i 0 #" 0 I # A B P i.e, P ;1=2 AP 1=2 P ;1=2 B is a contraction. But h Hence i " P ;1=2 AP 1=2 P ;1=2 B = " ;1 A12 A22 ;1 A12 A22 ;1A11 A21 ;1 A11 A21 ;1 B1 B2 ;1 B1 B2 # #2 0 I 6I 0 4 3 0 0 7: 5 0 0 I
- 574. 21.6. Discrete Balanced Model Reduction 557 is also a contraction. The other part follows by a similar argument. " Lemma 21.28 Suppose that X = X11 X12 Z X22 # tive (strictly contractive). Then 2 6 M =6 4 " Y Z and Y = 11 Y21 Y22 1 p2 X11 X12 0 1 1 p2 Y11 p2 X22 Z 1 0 Y21 p2 Y22 is also contractive (strictly contractive). Proof. Dilate M to the following matrix: 2 0 6 p1 Y11 6 Md = 6 2 6 Y21 4 1 p2 Y11 1 p2 X11 Z 1 p2 Y22 X12 1 p2 X22 0 1 ; p2 X22 # 2 are contrac- 3 7 7 5 3 7: 0 7 7 1 p2 Y22 7 ; 5 1 p2 X11 0 ;Z Considering X and Y are contractive, we can easily verify that Md Md I , i.e, Md is a contraction. 2 We can now prove the theorem. Proof of Theorem 21.26. Note that " # 2 A11 0 B1 3 Gr = A11 B1 = 6 0 0 0 7 : 5 4 C1 D C1 0 D Hence 3 2 1 p2 B1 A11 0 0 0 7 6 0 0 0 6 0 0 7 7 6 1 (G ; G ) = 6 0 1 0 A11 A12 p2 B1 7 : 7 6 r 6 2 7 6 0 0 A21 A22 p12 B2 7 5 4 1 1 1 ; p2 C1 0 p2 C1 p2 C2 0 Now apply the similarity transformation 2 3 ; 0 0 6 7 6 7 T = 6 0 1 ;I 0 1 I 7 6 ; 0 ; 07 4 5 0 I 0 I
- 575. 558 DISCRETE TIME CONTROL to the realization of 1 (G ; Gr ) to get 2 3 2 A ;1 1 A 1 A12 0 0 11 2 12 2 6 1 A21 ;1 1 A22 1 A21 1 A22 1 B2 7 7 62 2 2 2 2 6 0 1 (G ; G ) = 6 1 ;1 A12 ;1 A11 1 ;1 A12 ;1 B1 7 : 7 r 7 6 1 ;1 2 1 2 2 6 A21 1 A21 1 A22 1 B2 7 A22 5 42 2 2 2 2 1 C2 1 C2 ;1 C1 0 0 2 2 It is easy to verify that as a constant matrix the right hand side of the above realization 1 for 2 (G ; Gr ) can be written as 20 0 I 03 2 3 6 0 p1 I 0 0 7 6 I 0 0 0 0 7 6 7 1 1 2 6 6 I 0 0 0 7 Md 6 0 p2 I 0 p2 I 0 7 7^ 6 7 (21:33) 6 1 60 pI 0 07 60 0 I 0 07 7 4 5 4 5 2 0 0 0 0 I 0 0 0 I where 2 0 1 ;1 B1 3 p2 ;1A12 ;1 A11 6 1 1 1 ;1 p2 A21 p2 B2 7 7: A 7 ^ 6 pA (21:34) Md = 6 2 21 ;1 p 22 6 A11 1 A12 0 0 7 5 4 2 1 p2 C2 0 0 C1 ;1 According to Theorem 21.12 (a) and (g), the theorem follows if we can show that 1 the realization for 2 (G ; Gr ) as a constant matrix is a contraction. However, this is ^ d is a contraction since both the right and left hand matrices in (21.33) guaranteed if M are contractive. ^ Finally, the contractiveness of Md follows from Lemmas 21.27 and 21.28 by identifying 1 1 Z = A22 X11 = p ;1 A12 Y11 = p A21 ;1 2 2 and " # " i " A11 ;1 p12 A12 # ;1B1 # h X12 = ;1 A11 : Y21 Y22 = 1 1 1 p2 A21 p2 B2 p2 C2 X22 C1 ;1 2 21.7 Model Reduction Using Coprime Factors In this section, we consider lower order controller design using coprime factor reduction. We shall only consider the special case where the normalized right coprime factors are used.
- 576. 21.7. Model Reduction Using Coprime Factors Suppose that a dynamic system is given by " G= A B C D 559 # and assume that the realization is stabilizable and detectable. Recall from Theorem 21.21 that there exists a normalized right coprime factorization G = NM ;1 such " # M is inner. that N Lemma 21.29 Let p (Y X ). i i = given by i= Proof. This is obvious since Then the Hankel singular values of " M N # are i p1 + 2 < 1: i i (Y X ) 2 i = i (PQ) = 1 + i (Y X ) and i (Y X ) 0. 2 It is known that there exists a transformation such that X and Y are balanced: " # 1 0 X=Y = = 0 2 with 1 = diag 1 Is1 : : : r Isr ] > 0. Now partitioning the system G and matrix F accordingly, 2A A B 3 6 A11 A12 B1 7 F = h F1 F2 i : G = 4 21 22 2 5 C1 C2 D Then the reduced coprime factors " # 2 A11 + B1F1 B1Z ;1=2 3 ^ M := 6 Z ;1=2 7 2 RH1 4 F1 5 ^ N DZ ;1=2 C1 + DF1 satisfy the following error bound. " # " ^ # Lemma 21.30 M ; M ^ N N 1 2 X i r+1 i =2 X i r+1 i p1 + 2 : i
- 577. DISCRETE TIME CONTROL 560 Proof. Analogous to the continuous time case. 2 Remark 21.8 It should be understood that the reduced model can be obtained by directly computing X and P and by obtaining a balanced model without solving the Riccati equation for Y . ~ This reduced coprime factors combined with the robust or H1 controller design methods can be used to design lower order controllers so that the system is robustly stable and some speci ed performance criteria are satis ed. We will leave the readers to explore the utility of this model reduction method. However, we would like to point out that unlike the continuous time case, the reduced coprime factors in discrete time may not be normalized. In fact, we can prove a more general result. Lemma 21.31 Let a realization for N (z) 2 RH1 be given by " N (z ) = A B C D # with A stable. Suppose that there exists an X = X " 0 such that # A XA ; X + C C A XB + C D = 0: B XA + D C B XB + D D ; I Then N is an inner, i.e., N N = I . Moreover, if the realization for N= " # 2 A11 A12 B1 3 A B =6 5 4 A21 A22 B2 7 C D C1 is also balanced with X= = and " 1 0 C2 D 0 2 # A A ; + BB = 0 with 1 > 0, then the truncated system " Nr = A11 B1 C1 D is stable and contractive, i.e., Nr Nr I. # (21:35)
- 578. 21.8. Notes and References 561 Proof. Pre-multiply equation (21.35) by U= "h I 0 i # 0 I 0 and post-multiply equation (21.35) by U to get " or " # A11 1 A11 ; 1 + C1 C1 + A21 2 A21 A11 1 B1 + C1 D + A21 2 B2 = 0 B1 1 A11 + D C1 + B2 2 A21 B1 1 B1 + D D ; I + B2 2 B2 # " A11 1 A11 ; 1 + C1 C1 A11 1 B1 + C1 D = ; A21 B1 1 A11 + D C1 B1 1 B1 + D D ; I B2 This gives " A11 B1 C1 D #" Now let T = 1=2 , and then 1 1 0 0 I " #" A11 B1 C1 D TA11 T ;1 TB1 C1 T ;1 D " # " # A21 i.e., kNr k1 1. It is clear that Nr is inner if B2 may not be true). ; 1 0 0 I # " A 2 B21 2 # # : 0: 1 # 2 = 0 (although the converse 2 21.8 Notes and References The results for the discrete Riccati equation are based mostly on the work of Kucera 1972] and Molinari 1975]. Numerical algorithms for solving discrete ARE with singular A matrix can be found in Arnold and Laub 1984], Dooren 1981], and references therein. The matrix factorizations are obtained by Chu 1988]. The normalized coprime factorizations are obtained by Meyer 1990] and Walker 1990]. The detailed treatment of discrete time H1 control can be found in Stoorvogel 1990], Limebeer, Green, and Walker 1989], and Iglesias and Glover 1991].
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- 598. Index Bode's gain and phase relation, 155 bounded real function, 353 discrete time, 532 bounded real lemma, 353 discrete time, 532 additive approximation, 157 additive uncertainty, 213, 219 adjoint operator, 94 admissible controller, 294 algebraic duality, 297 algebraic Riccati equation (ARE), 11, 168, 321 all solutions, 322 complementarity property, 327 discrete time, 523 maximal solution, 335 minimal solution, 335 solutions, 27 stability property, 327 stabilizing solution, 327 all-pass, 179, 357 dilation, 179 analytic function, 94 anticausal, 97 antistable, 97 Cauchy sequence, 90 Cauchy-Schwartz inequality, 92 causal function, 175 causality, 175 Cayley-Hamilton theorem, 21, 48, 51, 54 central solution H1 control, 411 matrix dilation problem, 41 classical control, 232 co-inner function, 357 compact operator, 178 complementary inner function, 358 complementary sensitivity, 128, 142 conjugate system, 66 contraction, 408 controllability, 45 Gramian, 71 matrix, 47 operator, 175 controllable canonical form, 60 controller parameterization, 293 controller reduction, 485 additive, 486 coprime factors, 489 H1 performance, 493 coprime factor uncertainty, 222, 467 coprime factorization, 123, 316, 358, 467 balanced model reduction additive, 157 discrete time, 548 error bounds, 161, 195 frequency-weighted, 164 multiplicative, 157 relative, 157 stability, 158 balanced realization, 69, 71, 157 Banach space, 90 basis, 17 Bezout identity, 123 Bode integral, 141 581
- 599. 582 discrete time, 539 normalized, 470 cyclic matrix, 21, 61 design limitation, 153 design model, 211 design tradeo , 141 destabilizing perturbation, 268 detectability, 45, 51 di erential game, 459 direct sum, 94 discrete algebraic Riccati equation, 523 discrete time, 523 coprime factorization, 539 H1 control, 548 H2 control, 543 inner-outer factorization, 540 Lyapunov equation, 523 model reduction, 553 normalized coprime factorization, 541 disturbance feedforward (DF), 10, 294, 298 H1 control, 423 H2 control, 386 D ; K iteration, 289 dom(Ric), 327 double coprime factorization, 123 dual system, 66 duality, 297 eigenvalue, 20 eigenvector, 20, 324 generalized, 21 lower rank generalized eigenvector, 21 entropy, 435 equivalence class, 305 equivalence relation, 305 equivalent control, 305 F` , 242 Fu , 242 INDEX feedback, 116 ltering H1 performance, 452 xed order controller, 516 Fourier transform, 96 frequency weighting, 135 frequency-weighted balanced realization, 164 Frobenius norm, 29 full control (FC), 10, 294, 298 H1 control, 422 H2 control, 387, 546 full information (FI), 10, 294, 297 H1 control, 416 H2 control, 385, 546 full rank, 18 column, 18 row, 18 gain, 155 gain margin, 232 gap metric, 484 generalized eigenvector, 26, 27, 324 generalized inverse, 35 generalized principal vector, 525 Gilbert's realization, 69 Gramian controllability, 71, 74 observability, 71, 74 graph metric, 484 graph topology, 484 Hamiltonian matrix, 321 Hankel norm, 173 Hankel norm approximation, 173, 188 Hankel operator, 173, 393 mixed Hankel-Toeplitz operator, 397 Hankel singular value, 75, 162, 173 Hardy spaces, 95 harmonic function, 142 Hermitian matrix, 21 H1 control, 405
- 600. INDEX discrete time, 548 loop shaping, 467 singular problem, 448 state feedback, 459 H1 ltering, 452 H1 optimal controller, 430, 437 H1 performance, 135, 138 H1 space, 89, 97 ; H1 space, 97 hidden modes, 77 Hilbert space, 91 homotopy algorithm, 520 H2 optimal control 365 discrete time, 543 H2 performance, 135 H2 space, 89, 95 H2 stability margin, 389 ? H2 space, 96 H2 (@ D ) space, 199 ? H2 (@ D ) space, 199 Hurwitz, 49 image, 18 induced norm, 28, 101 inertia, 179 inner function, 357 discrete time, 537 inner product, 92 inner-outer factorization, 143, 358 input sensitivity, 128 input-output stability, 406 integral control, 448 H1 control, 448 H2 control, 448 internal model principle, 448 internal stability, 119, 406 invariant subspace, 26, 322 invariant zero, 84, 333 inverse of a transfer function, 66 isometric isomorphism, 91, 174 Jordan canonical form, 20 583 Kalman canonical decomposition, 52 kernel, 18 Kronecker product, 25 Kronecker sum, 25 Lagrange multiplier, 511 left coprime factorization, 123 linear combination, 17 linear fractional transformation (LFT), 241 linear operator, 91 L1 space, 96 loop gain, 130 loop shaping, 132 H1 approach, 467 normalized coprime factorization, 475 loop transfer matrix (function), 128 LQG stability margin, 389 LQR problem, 367 LQR stability margin, 373 l2 (;1 1) space, 93 L2 space, 95 L2 (;1 1) space, 93 L2 (@ D ) space, 199 Lyapunov equation, 26, 70 discrete time, 523 main loop theorem, 275 matrix compression, 38 dilation, 38, 445 factorization, 343, 537 Hermitian, 21 inequality, 335 inertia, 179 inversion formulas, 22 norm, 27 square root of a, 36 maximum modulus theorem, 94 max-min problem, 400 McMillan degree, 81 McMillan form, 80
- 601. INDEX 584 minimal realization, 67, 73 minimax problem, 400 minimum entropy controller, 435 mixed Hankel-Toeplitz operator, 397 modal controllability, 51 modal observability, 51 model reduction additive, 157 multiplicative, 157 relative, 157 model uncertainty, 116, 211 , 263 lower bound, 273 synthesis, 288 upper bound, 273 multidimensional system, 249 multiplication operator, 98 multiplicative approximation, 157, 165 multiplicative uncertainty, 213, 220 Nehari's Theorem, 203 nominal performance (NP), 215 nominal stability (NS), 215 non-minimum phase zero, 141 norm, 27 Frobenius, 29 induced, 28 semi-norm, 28 normal rank, 79 normalized coprime factorization, 362 loop shaping, 475 null space, 18 Nyquist stability theorem, 123 observability, 45 Gramian, 71 operator, 175 observable canonical form, 61 observable mode, 51 observer, 62 observer-based controller, 62 operator extension, 91 restriction, 91 optimal Hankel norm approximation, 188 optimality of H1 controller, 430 optimization method, 511 orthogonal complement, 18, 94 orthogonal direct sum, 94 orthogonal matrix, 18 orthogonal projection theorem, 94 outer function, 358 output estimation (OE), 10, 294, 298 H1 control, 425 H2 control, 387 output injection, 298 H2 control, 386 output sensitivity, 128 Parrott's theorem, 40 Parseval's relations, 96 PBH (Popov-Belevitch-Hautus) tests, 51 performance limitation, 141 phase, 155 phase margin, 232 Plancherel theorem, 96 plant condition number, 230 Poisson integral, 141 pole, 77, 79 pole direction, 143 pole placement, 57 pole-zero cancelation, 77 positive real, 354 positive (semi-)de nite matrix, 36 power signal, 100 pseudo-inverse, 35 quadratic control, 365 quadratic performance, 365, 393 Rp (s) , 80 range, 18 realization, 67 balanced, 75
- 602. INDEX input normal, 77 minimal, 67 output normal, 77 Redhe er star-product, 259 regular point, 512 regulator problem, 365 relative approximation, 157, 165 return di erence, 128 RH1 space, 97 RH; space, 97 1 RH? space, 95 2 RH2 space, 96 Riccati equation, 321 Riccati operator, 327 right coprime factorization, 123 risk sensitive, 436 robust performance (RP), 215 H1 performance, 226, 275 H2 performance, 226 structured uncertainty, 279 robust stability (RS), 215 structured uncertainty, 278 robust stabilization, 467 roll-o rate, 144 Schmidt pair, 178 Schur complement, 23 self-adjoint operator, 94, 178 sensitivity function, 128, 142 bounds, 153 integral, 144 separation theory, 310 H1 control, 426 H2 control, 388 simplectic matrix, 525 singular H1 problem, 448 singular value, 32 singular value decomposition (SVD), 31 singular vector, 32 skewed performance speci cation, 230 small gain theorem, 215 Smith form, 79 span, 17 585 spectral factorization, 343 spectral radius, 20 spectral signal, 100 stability, 45, 49 internal, 119 stability margin LQG, H2 , 389 stabilizability, 45, 49 stabilization, 293 stabilizing controller, 293 stable invariant subspace, 27, 327, 525 star-product, 259 state feedback, 298 H1 control, 459 H2 control, 382 state space realization, 60 strictly positive real, 354 structured singular value, 263 lower bound, 273 upper bound, 273 structured uncertainty, 263 subharmonic function, 142 supremum norm, 89 Sylvester equation, 26 Sylvester's inequality, 19, 68 Toeplitz operator, 197, 395 trace, 19 tradeo , 141 transition matrix, 46 transmission zero, 81 uncertainty, 116, 211 state space, 252 unimodular matrix, 79 unitary matrix, 18 unstructured uncertainty, 211 weighted model reduction, 164, 500 weighting, 135 well-posedness, 117, 243 winding number, 409
- 603. 586 Youla parameterization, 303, 316, 454 zero, 77, 79 blocking, 81 direction, 142 invariant, 84 transmission, 81 INDEX