![Problem 1.1
Stability and composition of transfer functions
G. Fernández-Anaya
Departamento de Ciencias Básicas
Universidad Iberoaméricana
Lomas de Santa Fe
01210 México D.F.
México
guillermo.fernandez@uia.mx
J. C. Martı́nez-Garcı́a
Departamento de Control Automático
CINVESTAV-IPN
A.P. 14-740
07300 México D.F.
México
martinez@ctrl.cinvestav.mx
1 INTRODUCTION
As far as the frequency-described continuous linear time-invariant systems
are concerned, the study of control-oriented properties (like stability) re-
sulting from the substitution of the complex Laplace variable s by rational
transfer functions have been little studied by the Automatic Control com-
munity. However, some interesting results have recently been published:
Concerning the study of the so-called uniform systems, i.e., LTI systems
consisting of identical components and amplifiers, it was established in [8]
a general criterion for robust stability for rational functions of the form
D(f(s)), where D(s) is a polynomial and f(s) is a rational transfer function.
By applying such a criterium, it gave a generalization of the celebrated
Kharitonov’s theorem [7], as well as some robust stability criteria under H∞-
uncertainty. The results given in [8] are based on the so-called H-domains.1
As far as robust stability of polynomial families is concerned, some Kharito-
1The H-domain of a function f (s) is defined to be the set of points h on the complex
plane for which the function f (s) − h has no zeros on the open right-half complex plane.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-23-320.jpg)
![4 PROBLEM 1.1
nov’s like results [7] are given in [9] (for a particular class of polynomials),
when interpreting substitutions as nonlinearly correlated perturbations on
the coefficients.
More recently, in [1], some results for proper and stable real rational SISO
functions and coprime factorizations were proved, by making substitutions
with α (s) = (as + b) / (cs + d), where a, b, c, and d are strictly positive real
numbers, and with ad − bc = 0. But these results are limited to the bilinear
transforms, which are very restricted.
In [4] is studied the preservation of properties linked to control problems (like
weighted nominal performance and robust stability) for Single-Input Single-
Output systems, when performing the substitution of the Laplace variable (in
transfer functions associated to the control problems) by strictly positive real
functions of zero relative degree. Some results concerning the preservation of
control-oriented properties in Multi-Input Multi-Output systems are given in
[5], while [6] deals with the preservation of solvability conditions in algebraic
Riccati equations linked to robust control problems.
Following our interest in substitutions we propose in section 22.2 three in-
teresting problems. The motivations concerning the proposed problems are
presented in section 22.3.
2 DESCRIPTION OF THE PROBLEMS
In this section we propose three closely related problems. The first one con-
cerns the characterization of a transfer function as a composition of transfer
functions. The second problem is a modified version of the first problem:
the characterization of a transfer function as the result of substituting the
Laplace variable in a transfer function by a strictly positive real transfer
function of zero relative degree. The third problem is in fact a conjecture
concerning the preservation of stability property in a given polynomial re-
sulting from the substitution of the coefficients in the given polynomial by
a polynomial with non-negative coefficients evaluated in the substituted co-
efficients.
Problem 1: Let a Single Input Single Output (SISO) transfer function G(s)
be given. Find transfer functions G0(s) and H(s) such that:
1. G (s) = G0 (H (s)) ;
2. H (s) preserves proper stable transfer functions under substitution of
the variable s by H (s), and:
3. The degree of the denominator of H(s) is the maximum with the prop-
erties 1 and 2.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-24-320.jpg)
![STABILITY AND COMPOSITION OF TRANSFER FUNCTIONS 5
Problem 2: Let a SISO transfer function G(s) be given. Find a transfer
function G0 (s) and a Strictly Positive Real transfer function of zero relative
degree (SPR0), say H(s), such that:
1. G(s) = G0 (H (s)) and:
2. The degree of the denominator of H(s) is the maximum with the prop-
erty 1.
Problem 3: (Conjecture) Given any stable polynomial:
ansn
+ an−1sn−1
+ · · · + a1s + a0
and given any polynomial q(s) with non-negative coefficients, then the poly-
nomial:
q(an)sn
+ q(an−1)sn−1
+ · · · + q(a1)s + q(a0)
is stable (see [3]).
3 MOTIVATIONS
Consider the closed-loop control scheme:
y (s) = G (s) u (s) + d (s) , u (s) = K (s) (r (s) − y (s)) ,
where: P (s) denotes the SISO plant; K (s) denotes a stabilizing controller;
u (s) denotes the control input; y (s) denotes the control input; d (s) denotes
the disturbance and r (s) denotes the reference input. We shall denote the
closed-loop transfer function from r (s) to y (s) as Fr (G (s) , K (s)) and the
closed-loop transfer function from d (s) to y (s) as Fd (G (s) , K (s)).
• Consider the closed-loop system Fr (G (s) , K (s)), and suppose that
the plant G(s) results from a particular substitution of the s Laplace
variable in a transfer function G0(s) by a transfer function H(s),
i.e., G(s) = G0(H(s)). It has been proved that a controller K0 (s)
which stabilizes the closed-loop system Fr (G0 (s) , K0 (s)) is such that
K0 (H (s)) stabilizes Fr (G (s) , K0 (H (s))) (see [2] and [8]). Thus, the
simplification of procedures for the synthesis of stabilizing controllers
(profiting from transfer function compositions) justifies problem 1.
• As far as problem 2 is concerned, consider the synthesis of a controller
K (s) stabilizing the closed-loop transfer function Fd (G (s) , K (s)),
and such that Fd (G (s) , K (s))∞ γ, for a fixed given γ 0. If we
known that G(s) = G0 (H (s)), being H (s) a SPR0 transfer function,
the solution of problem 2 would arise to the following procedure:
1. Find a controller K0(s) which stabilizes the closed-loop transfer
function Fd (G0 (s) , K0 (s)) and such that:
Fd (G0 (s) , K0 (s))∞ γ.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-25-320.jpg)
![6 PROBLEM 1.1
2. The composed controller K (s) = K0 (H (s)) stabilizes the closed-
loop system Fd (G (s) , K (s)) and:
Fd (G (s) , K (s))∞ γ
(see [2], [4], and [5]).
It is clear that condition 3 in the first problem, or condition 2 in
the second problem, can be relaxed to the following condition: the
degree of the denominator of H (s) is as high as be possible with
the appropriate conditions. With this new condition, the open
problems are a bit less difficult.
• Finally, problem 3 can be interpreted in terms of robustness under
positive polynomial perturbations in the coefficients of a stable transfer
function.
BIBLIOGRAPHY
[1] G. Fernández, S. Muñoz, R. A. Sánchez, and W. W. Mayol, “Simulta-
neous stabilization using evolutionary strategies,”Int. J. Contr., vol. 68,
no. 6, pp. 1417-1435, 1997.
[2] G. Fernández, “Preservation of SPR functions and stabilization by sub-
stitutions in SISO plants,”IEEE Transaction on Automatic Control, vol.
44, no. 11, pp. 2171-2174, 1999.
[3] G. Fernández and J. Alvarez, “On the preservation of stability in fam-
ilies of polynomials via substitutions,”Int. J. of Robust and Nonlinear
Control, vol. 10, no. 8, pp. 671-685, 2000.
[4] G. Fernández, J. C. Martı́nez-Garcı́a, and V. Kučera, “H∞-Robustness
Properties Preservation in SISO Systems when applying SPR Substitu-
tions,”Submitted to the International Journal of Automatic Control.
[5] G. Fernández and J. C. Martı́nez-Garcı́a, “MIMO Systems Properties
Preservation under SPR Substitutions,” International Symposium on the
Mathematical Theory of Networks and Systems (MTNS’2002), University
of Notre Dame, USA, August 12-16, 2002.
[6] G. Fernández, J. C. Martı́nez-Garcı́a, and D. Aguilar-George, “Preserva-
tion of solvability conditions in Riccati equations when applying SPR0
substitutions,” submitted to IEEE Transactions on Automatic Control,
2002.
[7] V. L. Kharitonov, “Asymptotic stability of families of systems of linear
differential equations, ”Differential’nye Uravneniya, vol. 14, pp. 2086-
2088, 1978.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-26-320.jpg)
![STABILITY AND COMPOSITION OF TRANSFER FUNCTIONS 7
[8] B. T. Polyak and Ya. Z. Tsypkin, “Stability and robust stability of uni-
form systems, ”Automation and Remote Contr., vol. 57, pp. 1606-1617,
1996.
[9] L. Wang, “Robust stability of a class of polynomial families under non-
linearly correlated perturbations,”System and Control Letters, vol. 30,
pp. 25-30, 1997.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-27-320.jpg)
![Problem 1.2
The realization problem for Herglotz-Nevanlinna
functions
Seppo Hassi
Department of Mathematics and Statistics
University of Vaasa
P.O. Box 700, 65101 Vaasa
Finland
sha@uwasa.fi
Henk de Snoo
Department of Mathematics
University of Groningen
P.O. Box 800, 9700 AV Groningen
Nederland
desnoo@math.rug.nl
Eduard Tsekanovskiı̆
Department of Mathematics
Niagara University, NY 14109
USA
tsekanov@niagara.edu
1 MOTIVATION AND HISTORY OF THE PROBLEM
Roughly speaking, realization theory concerns itself with identifying a given
holomorphic function as the transfer function of a system or as its linear frac-
tional transformation. Linear, conservative, time-invariant systems whose
main operator is bounded have been investigated thoroughly. However, many
realizations in different areas of mathematics including system theory, elec-
trical engineering, and scattering theory involve unbounded main operators,
and a complete theory is still lacking. The aim of the present proposal is
to outline the necessary steps needed to obtain a general realization theory
along the lines of M. S. Brodskiı̆ and M. S. Livšic [8], [9], [16], who have](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-28-320.jpg)
![THE REALIZATION PROBLEM FOR HERGLOTZ-NEVANLINNA FUNCTIONS 9
considered systems with a bounded main operator.
An operator-valued function V (z) acting on a Hilbert space E belongs to the
Herglotz-Nevanlinna class N, if outside R it is holomorphic, symmetric, i.e.,
V (z)∗
= V (z̄), and satisfies (Im z)(Im V (z)) ≥ 0. Here and in the following
it is assumed that the Hilbert space E is finite-dimensional. Each Herglotz-
Nevanlinna function V (z) has an integral representation of the form
V (z) = Q + Lz +
R
1
t − z
−
t
1 + t2
dΣ(t), (1)
where Q = Q∗
, L ≥ 0, and Σ(t) is a nondecreasing matrix-function on R with
R
dΣ(t)/(t2
+ 1) ∞. Conversely, each function of the form (1) belongs
to the class N. Of special importance (cf. [15]) are the class S of Stieltjes
functions
V (z) = γ +
∞
0
dΣ(t)
t − z
, (2)
where γ ≥ 0 and
∞
0
dΣ(t)/(t+1) ∞, and the class S−1
of inverse Stieltjes
functions
V (z) = α + βz +
∞
0
1
t − z
−
1
t
dΣ(t), (3)
where α ≤ 0, β ≥ 0, and
∞
0
dΣ(t)/(t2
+ 1) ∞.
2 SPECIAL REALIZATION PROBLEMS
One way to characterize Herglotz-Nevanlinna functions is to identify them
as (linear fractional transformations of) transfer functions:
V (z) = i[W(z) + I]−1
[W(z) − I]J, (4)
where J = J∗
= J−1
and W(z) is the transfer function of some general-
ized linear, stationary, conservative dynamical system (cf. [1], [3]). The
approach based on the use of Brodskiı̆-Livšic operator colligations Θ yields
to a simultaneous representation of the functions W(z) and V (z) in the form
WΘ(z) = I − 2iK∗
(T − zI)−1
KJ, (5)
VΘ(z) = K∗
(TR − zI)−1
K, (6)
where TR stands for the real part of T. The definitions and main results
associated with Brodskiı̆-Livšic type operator colligations in realization of
Herglotz-Nevanlinna functions are as follows, cf. [8], [9], [16].
Let T ∈ [H], i.e., T is a bounded linear mapping in a Hilbert space H, and
assume that Im T = (T −T∗
)/2i of T is represented as Im T = KJK∗
, where
K ∈ [E, H], and J ∈ [E] is self-adjoint and unitary. Then the array
Θ =
T K J
H E
(7)](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-29-320.jpg)
![10 PROBLEM 1.2
defines a Brodskiı̆-Livšic operator colligation, and the function WΘ(z) given
by (5) is the transfer function of Θ. In the case of the directing operator
J = I the system (7) is called a scattering system, in which case the main
operator T of the system Θ is dissipative: Im T ≥ 0. In system theory
WΘ(z) is interpreted as the transfer function of the conservative system
(i.e., Im T = KJK∗
) of the form (T −zI)x = KJϕ− and ϕ+ = ϕ− −2iK∗
x,
where ϕ− ∈ E is an input vector, ϕ+ ∈ E is an output vector, and x is
a state space vector in H, so that ϕ+ = WΘ(z)ϕ−. The system is said to
be minimal if the main operator T of Θ is completely non self-adjoint (i.e.,
there are no nontrivial invariant subspaces on which T induces self-adjoint
operators), cf. [8], [16]. A classical result due to Brodskiı̆ and Livšic [9]
states that the compactly supported Herglotz-Nevanlinna functions of the
form
b
a
dΣ(t)/(t − z) correspond to minimal systems Θ of the form (7) via
(4) with W(z) = WΘ(z) given by (5) and V (z) = VΘ(z) given by (6).
Next consider a linear, stationary, conservative dynamical system Θ of the
form
Θ =
A K J
H+ ⊂ H ⊂ H− E
. (8)
Here A ∈ [H+, H−], where H+ ⊂ H ⊂ H− is a rigged Hilbert space, A ⊃
T ⊃ A, A∗
⊃ T∗
⊃ A, A is a Hermitian operator in H, T is a non-Hermitian
operator in H, K ∈ [E, H−], J = J∗
= J−1
, and Im A = KJK∗
. In this case
Θ is said to be a Brodskiı̆-Livšc rigged operator colligation. The transfer
function of Θ in (8) and its linear fractional transform are given by
WΘ(z) = I − 2iK∗
(A − zI)−1
KJ, VΘ(z) = K∗
(AR − zI)−1
K. (9)
The functions V (z) in (1) which can be realized in the form (4), (9) with a
transfer function of a system Θ as in (8) have been characterized in [2], [5],
[6], [7], [18]. For the significance of rigged Hilbert spaces in system theory,
see [14], [16]. Systems (7) and (8) naturally appear in electrical engineering
and scattering theory [16].
3 GENERAL REALIZATION PROBLEMS
In the particular case of Stieltjes functions or of inverse Stieltjes functions
general realization results along the lines of [5], [6], [7] remain to be worked
out in detail, cf. [4], [10].
The systems (7) and (8) are not general enough for the realization of general
Herglotz-Nevanlinna functions in (1) without any conditions on Q = Q∗
and
L ≥ 0. However, a generalization of the Brodskiı̆-Livšic operator colligation
(7) leads to analogous realization results for Herglotz-Nevanlinna functions
V (z) of the form (1) whose spectral function is compactly supported: such
functions V (z) admit a realization via (4) with
W(z) = WΘ(z) = I − 2iK∗
(M − zF)−1
KJ,
V (z) = WΘ(z) = K∗
(MR − zF)−1
K,
(10)](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-30-320.jpg)
![THE REALIZATION PROBLEM FOR HERGLOTZ-NEVANLINNA FUNCTIONS 11
where M = MR + iKJK∗
, MR ∈ [H] is the real part of M, F is a finite-
dimensional orthogonal projector, and Θ is a generalized Brodskiı̆-Livšic
operator colligation of the form
Θ =
M F K J
H E
, (11)
see [11], [12], [13]. The basic open problems are:
Determine the class of linear, conservative, time-invariant dynamical sys-
tems (new type of operator colligations) such that an arbitrary matrix-valued
Herglotz-Nevanlinna function V (z) acting on E can be realized as a linear
fractional transformation (4) of the matrix-valued transfer function WΘ(z)
of some minimal system Θ from this class.
Find criteria for a given matrix-valued Stieltjes or inverse Stieltjes function
acting on E to be realized as a linear fractional transformation of the matrix-
valued transfer function of a minimal Brodskiı̆-Livšic type system Θ in (8)
with: (i) an accretive operator A, (ii) an α-sectorial operator A, or (iii) an
extremal operator A (accretive but not α-sectorial).
The same problem for the (compactly supported) matrix-valued Stieltjes or
inverse Stieltjes functions and the generalized Brodskiı̆-Livšic systems of the
form (11) with the main operator M and the finite-dimensional orthogonal
projector F.
There is a close connection to the so-called regular impedance conserva-
tive systems (where the coefficient of the derivative is invertible) that were
recently considered in [17] (see also [19]). It is shown that any function
D(s) with non-negative real part in the open right half-plane and for which
D(s)/s → 0 as s → ∞ has a realization with such an impedance conservative
system.
BIBLIOGRAPHY
[1] D. Alpay, A. Dijksma, J. Rovnyak, and H.S.V. de Snoo, “Schur func-
tions, operator colligations, and reproducing kernel Pontryagin spaces,”
Oper. Theory Adv. Appl., 96, Birkhäuser Verlag, Basel, 1997.
[2] Yu. M. Arlinskiı̆, “On the inverse problem of the theory of characteristic
functions of unbounded operator colligations”, Dopovidi Akad. Nauk
Ukrain. RSR, 2 (1976), 105–109 (Russian).
[3] D. Z. Arov, “Passive linear steady-state dynamical systems,” Sibirsk.
Mat. Zh., 20, no. 2, (1979), 211–228, 457 (Russian) [English transl.:
Siberian Math. J., 20 no. 2, (1979) 149–162].](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-31-320.jpg)
![12 PROBLEM 1.2
[4] S. V. Belyi, S. Hassi, H. S. V. de Snoo, and E. R. Tsekanovskiı̆,
“On the realization of inverse Stieltjes functions,” Proceedings
of the 15th International Symposium on Mathematical Theory of
Networks and Systems, Editors D. Gillian and J. Rosenthal,
University of Notre Dame, South Bend, Idiana, USA, 2002,
http://www.nd.edu/∼mtns/papers/20160 6.pdf
[5] S. V. Belyi and E. R. Tsekanovskiı̆, “Realization and factorization prob-
lems for J-contractive operator-valued functions in half-plane and sys-
tems with unbounded operators,” Systems and Networks: Mathemati-
cal Theory and Applications, Akademie Verlag, 2 (1994), 621–624.
[6] S. V. Belyi and E. R. Tsekanovskiı̆, “Realization theorems for operator-
valued R-functions,” Oper. Theory Adv. Appl., 98 (1997), 55–91.
[7] S. V. Belyi and E. R. Tsekanovskiı̆, “On classes of realizable operator-
valued R-functions,” Oper. Theory Adv. Appl., 115 (2000), 85–112.
[8] M. S. Brodskiı̆, “Triangular and Jordan representations of linear op-
erators,” Moscow, Nauka, 1969 (Russian) [English trans.: Vol. 32 of
Transl. Math. Monographs, Amer. Math. Soc., 1971].
[9] M. S. Brodskiı̆ and M. S. Livšic, “Spectral analysis of non-selfadjoint
operators and intermediate systems,” Uspekhi Mat. Nauk, 13 no. 1, 79,
(1958), 3–85 (Russian) [English trans.: Amer. Math. Soc. Transl., (2)
13 (1960), 265–346].
[10] I. Dovshenko and E. R.Tsekanovskiı̆, “Classes of Stieltjes operator-
functions and their conservative realizations,” Dokl. Akad. Nauk SSSR,
311 no. 1 (1990), 18–22.
[11] S. Hassi, H. S. V. de Snoo, and E. R. Tsekanovskiı̆, “An addendum
to the multiplication and factorization theorems of Brodskiı̆-Livšic-
Potapov,” Appl. Anal., 77 (2001), 125–133.
[12] S. Hassi, H. S. V. de Snoo, and E. R. Tsekanovskiı̆, “On commuta-
tive and noncommutative representations of matrix-valued Herglotz-
Nevanlinna functions,” Appl. Anal., 77 (2001), 135–147.
[13] S. Hassi, H. S. V. de Snoo, and E. R. Tsekanovskiı̆, “Realizations
of Herglotz-Nevanlinna functions via F-systems,” Oper. Theory: Adv.
Appl., 132 (2002), 183–198.
[14] J.W. Helton, “Systems with infinite-dimensional state space: the
Hilbert space approach,” Proc. IEEE, 64 (1976), no. 1, 145–160.
[15] I. S. Kac̆ and M. G. Kreı̆n, “The R-functions: Analytic functions map-
ping the upper half-plane into itself,” Supplement I to the Russian edi-
tion of F. V. Atkinson, Discrete and Continuous Boundary Problems,
Moscow, 1974 [English trans.: Amer. Math. Soc. Trans., (2) 103 (1974),
1–18].](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-32-320.jpg)
![THE REALIZATION PROBLEM FOR HERGLOTZ-NEVANLINNA FUNCTIONS 13
[16] M. S. Livšic, “Operators, Oscillations, Waves,” Moscow, Nauka, 1966
(Russian) [English trans.: Vol. 34 of Trans. Math. Monographs, Amer.
Math. Soc., 1973].
[17] O. J. Staffans, “Passive and conservative infinite-dimensional
impedance and scattering systems (from a personal point of view),” Pro-
ceedings of the 15th International Symposium on Mathematical Theory
of Networks and Systems, Ed., D. Gillian and J. Rosenthal, Univer-
sity of Notre Dame, South Bend, Indiana, USA, 2002, Plenary talk,
http://www.nd.edu/∼mtns
[18] E. R. Tsekanovskiı̆ and Yu. L. Shmul’yan, “The theory of biextensions
of operators in rigged Hilbert spaces: Unbounded operator colligations
and characteristic functions,” Uspekhi Mat. Nauk, 32 (1977), 69–124
(Russian) [English transl.: Russian Math. Surv., 32 (1977), 73–131].
[19] G. Weiss, “Transfer functions of regular linear systems. Part I: charac-
terizations of regularity”, Trans. Amer. Math. Soc., 342 (1994), 827–
854.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-33-320.jpg)
![Problem 1.3
Does any analytic contractive operator function on
the polydisk have a dissipative scattering nD
realization?
Dmitry S. Kalyuzhniy-Verbovetzky
Department of Mathematics
The Weizmann Institute of Science
Rehovot 76100
Israel
dmitryk@wisdom.weizmann.ac.il
1 DESCRIPTION OF THE PROBLEM
Let X, U, Y be finite-dimensional or infinite-dimensional separable Hilbert
spaces. Consider nD linear systems of the form
α :
x(t) =
n
k=1
(Akx(t − ek) + Bku(t − ek)),
y(t) =
n
k=1
(Ckx(t − ek) + Dku(t − ek)),
(t ∈ Zn
:
n
k=1
tk 0)
(1)
where ek := (0, . . . , 0, 1, 0, . . . , 0) ∈ Zn
(here unit is on the k-th place), for all
t ∈ Zn
such that
n
k=1 tk ≥ 0 one has x(t) ∈ X (the state space), u(t) ∈ U
(the input space), y(t) ∈ Y (the output space), Ak, Bk, Ck, Dk are bounded
linear operators, i.e., Ak ∈ L(X), Bk ∈ L(U, X), Ck ∈ L(X, Y), Dk ∈ L(U, Y)
for all k ∈ {1, . . . , n}. We use the notation α = (n; A, B, C, D; X, U, Y) for
such a system (here A := (A1, . . . , An), etc.). For T ∈ L(H1, H2)n
and
z ∈ Cn
denote zT :=
n
k=1 zkTk. Then the transfer function of α is
θα(z) = zD + zC(IX − zA)−1
zB.
Clearly, θα is analytic in some neighbourhood of z = 0 in Cn
. Let
Gk :=
Ak Bk
Ck Dk
∈ L(X ⊕ U, X ⊕ Y), k = 1, . . . , n.
We call α = (n; A, B, C, D; X, U, Y) a dissipative scattering nD system (see
[5, 6]) if for any ζ ∈ Tn
(the unit torus) ζG is a contractive operator, i.e.,](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-34-320.jpg)
![DISSIPATIVE SCATTERING ND REALIZATION 15
ζG ≤ 1. It is known [5] that the transfer function of a dissipative scatter-
ing nD system α = (n; A, B, C, D; X, U, Y) belongs to the subclass B0
n(U, Y)
of the class Bn(U, Y) of all analytic contractive L(U, Y)-valued functions on
the open unit polydisk Dn
, which is segregated by the condition of vanishing
of its functions at z = 0. The question whether the converse is true was
implicitly asked in [5] and still has not been answered. Thus, we pose the
following problem.
Problem: Either prove that an arbitrary θ ∈ B0
n(U, Y) can be realized
as the transfer function of a dissipative scattering nD system of the form
(1) with the input space U and the output space Y, or give an example
of a function θ ∈ B0
n(U, Y) (for some n ∈ N, and some finite-dimensional
or infinite-dimensional separable Hilbert spaces U, Y) that has no such a
realization.
2 MOTIVATION AND HISTORY OF THE PROBLEM
For n = 1 the theory of dissipative (or passive, in other terminology) scatter-
ing linear systems is well developed (see, e.g., [2, 3]) and related to various
problems of physics (in particular, scattering theory), stochastic processes,
control theory, operator theory, and 1D complex analysis. It is well known
(essentially, due to [8]) that the class of transfer functions of dissipative scat-
tering 1D systems of the form (1) with the input space U and the output
space Y coincides with B0
1(U, Y). Moreover, this class of transfer functions
remains the same when one is restricted within the important special case
of conservative scattering 1D systems, for which the system block matrix
G is unitary, i.e., G∗
G = IX⊕U, GG∗
= IX⊕Y. Let us note that in the
case n = 1 a system (1) can be rewritten in an equivalent form (without a
unit delay in output signal y) that is the standard form of a linear system,
then a transfer function does not necessarily vanish at z = 0, and the class
of transfer functions turns into the Schur class S(U, Y) = B1(U, Y). The
classes B0
1(U, Y) and B1(U, Y) are canonically isomorphic due to the relation
B0
1(U, Y) = zB1(U, Y).
In [1] an important subclass Sn(U, Y) in Bn(U, Y) was introduced. This
subclass consists of analytic L(U, Y)-valued functions on Dn
, say, θ(z) =
t∈Zn
+
θtzt
(here Zn
+ = {t ∈ Zn
: tk ≥ 0, k = 1, . . . , n}, zt
:=
n
k=1 ztk
k for
z ∈ Dn
, t ∈ Zn
+) such that for any n-tuple T = (T1, . . . , Tn) of commuting
contractions on some common separable Hilbert space H and any positive
r 1 one has θ(rT) ≤ 1, where θ(rT) = t∈Zn
+
θt ⊗ (rT)t
∈ L(U ⊗
H, Y ⊗ H), and (rT)t
:=
n
k=1(rTk)tk
. For n = 1 and n = 2 one has
Sn(U, Y) = Bn(U, Y). However, for any n 2 and any non-zero spaces U
and Y the class Sn(U, Y) is a proper subclass of Bn(U, Y). J. Agler in [1]
constructed a representation of an arbitrary function from Sn(U, Y), which
in a system-theoretical language was interpreted in [4] as follows: Sn(U, Y)](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-35-320.jpg)
![16 PROBLEM 1.3
coincides with the class of transfer functions of nD systems of Roesser type
with the input space U and the output space Y, and certain conservativity
condition imposed. The analogous result is valid for conservative systems of
the form (1). A system α = (n; A, B, C, D; X, U, Y) is called a conservative
scattering nD system if for any ζ ∈ Tn
the operator ζG is unitary. Clearly,
a conservative scattering system is a special case of a dissipative one. By [5],
the class of transfer functions of conservative scattering nD systems coincides
with the subclass S0
n(U, Y) in Sn(U, Y), which is segregated from the latter by
the condition of vanishing of its functions at z = 0. Since for n = 1 and n = 2
one has S0
n(U, Y) = B0
n(U, Y), this gives the whole class of transfer functions
of dissipative scattering nD systems of the form (1), and the solution to the
problem formulated above for these two cases.
In [6] the dilation theory for nD systems of the form (1) was developed.
It was proven that α = (n; A, B, C, D; X, U, Y) has a conservative dilation
if and only if the corresponding linear function LG(z) := zG belongs to
S0
n(X ⊕ U, X ⊕ Y). Systems that satisfy this criterion are called n-dissipative
scattering ones. In the cases n = 1 and n = 2 the subclass of n-dissipative
scattering systems coincides with the whole class of dissipative ones, and in
the case n 2 this subclass is proper. Since transfer functions of a system
and of its dilation coincide, the class of transfer functions of n-dissipative
scattering systems with the input space U and the output space Y is S0
n(U, Y).
According to [7], for any n 2 there exist p ∈ N, m ∈ N, operators Dk ∈
L(Cp
) and commuting contractions Tk ∈ L(Cm
), k = 1, . . . , n, such that
max
ζ∈Tn
n
k=1
zkDk = 1
n
k=1
Tk ⊗ Dk.
The system α = (n; 0, 0, 0, D; {0}, Cp
, Cp
) is a dissipative scattering one,
however not, n-dissipative. Its transfer function θα(z) = LG(z) = zD ∈
B0
n(Cp
, Cp
) S0
n(Cp
, Cp
).
Since for functions in B0
n(U, Y)S0
n(U, Y) the realization technique elaborated
in [1] and developed in [4] and [5] is not applicable, our problem is of current
interest.
BIBLIOGRAPHY
[1] J. Agler, “On the representation of certain holomorphic functions de-
fined on a polydisc,” Topics in Operator Theory: Ernst D. Hellinger
Memorial Volume (L. de Branges, I. Gohberg, and J. Rovnyak, Eds.),
Oper. Theory Adv. Appl. 48, pp. 47-66 (1990).
[2] D. Z. Arov, “Passive linear steady-state dynamic systems,” Sibirsk.
Math. Zh. 20 (2), 211-228 (1979), (Russian).
[3] J. A. Ball and N. Cohen, “De Branges-Rovnyak operator models and
systems theory: A survey,” Topics in Matrix and Operator Theory (H.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-36-320.jpg)
![DISSIPATIVE SCATTERING ND REALIZATION 17
Bart, I. Gohberg, and M.A. Kaashoek, eds.), Oper. Theory Adv. Appl.,
50, pp. 93-136 (1991).
[4] J. A. Ball and T. Trent, “Unitary colligations, reproducing kernel
hilbert spaces, and Nevanlinna-Pick interpolation in several variables,”
J. Funct. Anal. 157, pp. 1-61 (1998).
[5] D. S. Kalyuzhniy, “Multiparametric dissipative linear stationary dy-
namical scattering systems: Discrete case,” J. Operator Theory, 43 (2),
pp. 427-460 (2000).
[6] D. S. Kalyuzhniy, “Multiparametric dissipative linear stationary dy-
namical scattering systems: Discrete case, II: Existence of conservative
dilations,” Integr. Eq. Oper. Th., 36 (1), pp. 107-120 (2000).
[7] D. S. Kalyuzhniy, “On the von Neumann inequality for linear matrix
functions of several variables,” Mat. Zametki 64 (2), pp. 218-223 (1998),
(Russian); translated in Math. Notes 64 (2), pp. 186-189 (1998).
[8] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert
Spaces, North Holland, Amsterdam, 1970.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-37-320.jpg)
![PARTIAL DISTURBANCE DECOUPLING WITH STABILITY 19
Let a system (A, B, C, E) and an integer k ≥ 1 be given. Find necessary
and sufficient conditions for the existence of a static state feedback control
law u (t) = Fx (t)+Gd (t) , where F : X → U and G : D → U are linear maps
such as zeroing the first k Markov parameters of Tzd, the transfer function
between the disturbance and the controlled output, while insuring internal
stability, i.e.:
• C (A + BF)
l
(BG + E) ≡ 0, for i ∈ {0, 1, . . . , k − 1}, and
• σ (A + BF) ⊆ Cg,
where σ (A + BF) stands for the spectrum of A + BF and Cg stands
for the (good) stable part of the complex plane, e.g., the open left-half
complex plane (continuous-time case) or the open unit disk (discrete-
time case)
2 MOTIVATION
The literature contains a lot of contributions related to disturbance rejection
or attenuation. The early attempts were devoted to canceling the effect of the
disturbance on the controlled output, i.e., insuring Tzd ≡ 0. This problem
is usually referred to as the disturbance decoupling problem with internal
stability, noted as DDPS (see [11], [1]).
The solvability conditions for DDPS can be expressed as matching of infinite
and unstable (invariant) zeros of certain systems (see, for instance, [8]),
namely those of (A, B, C), i.e., (1) with d(t) ≡ 0, and those of (A, B E
,
C), i.e., (1) with d(t) considered as a control input. However, the rigid
solvability conditions for DDPS are hardly met in practical cases. This
is why alternative design procedures have been considered, such as almost
disturbance decoupling (see [10]) and optimal disturbance attenuation, i.e.,
minimization of a norm of Tzd (see, for instance, [12]).
The partial version of the problem, as defined in Section 1, offers another al-
ternative from the rigid design of DDPS. The partial disturbance decoupling
problem (PDDP) amounts to zeroing the first, say k, Markov parameters of
Tzd. It was initially introduced in [2] and later revisited in [5], without sta-
bility, [6, 7] with dynamic state feedback and stability, [4] with static state
feedback and stability (sufficient solvability conditions for the single-input
single-output case), [3] with dynamic measurement feedback, stability, and
H∞-norm bound. When no stability constraint is imposed, solvability con-
ditions of PDDP involve only a subset of the infinite structure of (A, B, C)
and (A, B E
, C), namely the orders which are less than or equal to
k − 1 (see details in [5]). For PDDPS (i.e., PDDP with internal stability),
the role played by the finite invariant zeros must be clarified to obtain the
necessary and sufficient conditions that we are looking for, and solve the
open problem.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-39-320.jpg)
![20 PROBLEM 1.4
Several extensions of this problem are also important:
• solve PDDPS while reducing the H∞-norm of Tzd;
• consider static measurement feedback in place of static state feedback.
BIBLIOGRAPHY
[1] G. Basile and G. Marro, Controlled and Conditioned Invariants in Linear
System Theory, Prentice-Hall, 1992.
[2] E. Emre and L. M. Silverman, “Partial model matching of linear sys-
tems,”IEEE Trans. Automat. Contr., vol. AC-25, no. 2, pp. 280-281,
1980.
[3] V. Eldem, H. Özbay, H. Selbuz, and K. Özcaldiran, “Partial disturbance
rejection with internal stability and H∞ norm bound, ”SIAM Journal
on Control and Optimization, vol. 36 , no. 1 , pp. 180-192, 1998.
[4] F. N. Koumboulis and V. Kučera, “Partial model matching via static
feedback (The multivariable case),”IEEE Trans. Automat. Contr., vol.
AC-44, no. 2, pp. 386-392, 1999.
[5] M. Malabre and J. C. Martı́nez-Garcı́a, “The partial disturbance re-
jection or partial model matching: Geometric and structural solutions,
”IEEE Trans. Automat. Contr., vol. AC-40, no. 2, pp. 356-360, 1995.
[6] V. Kučera, J. C. Martı́nez-Garcı́a, and M. Malabre, “Partial model
matching: Parametrization of solutions, ” Automatica, vol. 33, no. 5,
pp. 975-977, 1997.
[7] J. C. Martı́nez-Garcı́a, M. Malabre, and V. Kučera, “The partial model
matching problem with stability,”Systems and Control Letters, no. 24,
pp. 61-74, 1994.
[8] J. C. Martı́nez-Garcı́a, M. Malabre, J.-M. Dion, and C. Commault, “Con-
densed structural solutions to the disturbance rejection and decoupling
problems with stability,”International Journal of Control, vol. 72, No.
15, pp. 1392-1401, 1999.
[9] A. Saberi, P. Sannuti, A. A. Stoorvogel, and B. M. Chen, H2 Optimal
Control, Prentice-Hall, 1995.
[10] J. C. Willems, “Almost invariant subspaces: An approach to high gain
feedback design - part I: Almost controlled invariant subspaces,”IEEE
Trans. Automat. Contr., vol. AC-26, no.1, pp. 235-252, 1981.
[11] M. M. Wonham, Linear Multivariable Control: A Geometric Approach,
3rd ed., Springer Verlag, New York, 1985.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-40-320.jpg)
![PARTIAL DISTURBANCE DECOUPLING WITH STABILITY 21
[12] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control,
Upper Saddle River, NJ: Prentice-Hall, Inc., Simon Schuster, 1995.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-41-320.jpg)
![Problem 1.5
Is Monopoli’s model reference adaptive controller
correct?
A. S. Morse1
Center for Computational Vision and Control
Department of Electrical Engineering
Yale University, New Haven, CT 06520
USA
1 INTRODUCTION
In 1974 R. V. Monopoli published a paper [1] in which he posed the now
classical model reference adaptive control problem, proposed a solution and
presented arguments intended to establish the solution’s correctness. Sub-
sequent research [2] revealed a flaw in his proof, which placed in doubt the
correctness of the solution he proposed. Although provably correct solutions
to the model reference adaptive control problem now exist (see [3] and the
references therein), the problem of deciding whether or not Monopoli’s orig-
inal proposed solution is in fact correct remains unsolved. The aim of this
note is to review the formulation of the classical model reference adaptive
control problem, to describe Monopoli’s proposed solution, and to outline
what’s known at present about its correctness.
2 THE CLASSICAL MODEL REFERENCE ADAPTIVE
CONTROL PROBLEM
The classical model reference adaptive control problem is to develop a dy-
namical controller capable of causing the output y of an imprecisely modeled
SISO process P to approach and track the output yref of a prespecified ref-
erence model Mref with input r. The underlying assumption is that the
process model is known only to the extent that it is one of the members of
a pre-specified class M. In the classical problem M is taken to be the set of
1This research was supported by DARPA under its SEC program and by the NSF.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-42-320.jpg)
![IS MONOPOLI’S MODEL REFERENCE ADAPTIVE CONTROLLER CORRECT? 23
all SISO controllable, observable linear systems with strictly proper transfer
functions of the form g β(s)
α(s) where g is a nonzero constant called the high
frequency gain and α(s) and β(s) are monic, coprime polynomials. All g
have the same sign and each transfer function is minimum phase (i.e., each
β(s) is stable). All transfer functions are required to have the same relative
degree n̄ (i.e., deg α(s) − deg β(s) = n̄.) and each must have a McMillan
degree not exceeding some prespecified integer n (i.e., deg α(s) ≤ n). In the
sequel we are going to discuss a simplified version of the problem in which
all g = 1 and the reference model transfer function is of the form 1
(s+λ)n̄
where λ is a positive number. Thus Mref is a system of the form
ẏref = −λyref + c̄xref + ¯
dr ẋref = Āxref + b̄r (1)
where {Ā, b̄, c̄, ¯
d} is a controllable, observable realization of 1
(s+λ)(n̄−1) .
3 MONOPOLI’S PROPOSED SOLUTION
Monopoli’s proposed solution is based on a special representation of P that
involves picking any n-dimensional, single-input, controllable pair (A, b) with
A stable. It is possible to prove [1, 4] that the assumption that the process P
admits a model in M, implies the existence of a vector p∗
∈ IR2n
and initial
conditions z(0) and x̄(0), such that u and y exactly satisfy
ż =
A 0
0 A
z +
b
0
y +
0
b
u
˙
x̄ = Āx̄ + b̄(u − z
p∗
)
ẏ = −λy + c̄x̄ + ¯
d(u − z
p∗
)
Monopoli combined this model with that of Mref to obtain the direct control
model reference parameterization
ż =
A 0
0 A
z +
b
0
y +
0
b
u (2)
ẋ = Āx + b̄(u − z
p∗
− r) (3)
ėT = −λeT + c̄x + ¯
d(u − z
p∗
− r) (4)
Here eT is the tracking error
eT
∆
= y − yref (5)
and x
∆
= x̄ − xref . Note that it is possible to generate an asymptotically
correct estimate
z of z using a copy of (2) with
z replacing z. To keep
the exposition simple, we are going to ignore the exponentially decaying
estimation error
z − z and assume that z can be measured directly.
To solve the MRAC problem, Monopoli proposed a control law of the form
u = z
p + r (6)](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-43-320.jpg)
![IS MONOPOLI’S MODEL REFERENCE ADAPTIVE CONTROLLER CORRECT? 25
Tuner 1
From (8) and the differential equation for eT directly above it, it can be seen
that the estimation error2
e1
∆
=
e1 − eT (9)
satisfies the error equation
ė1 = −λe1 + φ
1(
p − p∗
) (10)
where
φ
1 = −(c̄H1 + ¯
dz
) (11)
Prompted by this, Monopoli proposed to tune
p1 using the pseudo-gradient
tuner
˙
p1 = −φ1e1 (12)
The motivation for considering this particular tuning law will become clear
shortly, if it is not already.
What is known about Version 1?
The overall model reference adaptive controller proposed by Monopoli thus
consists of the reference model (1), the control law (6), the multi-estimator
(2), (7), (8), the output estimation error (9) and the tuner (11), (12). The
open problem is to prove that this controller either solves the model reference
adaptive control problem or that it does not.
Much is known that is relevant to the problem. In the first place, note that
(1), (2) together with (5) - (11) define a parameter varying linear system
Σ1(
p) with input r, state (yref , xref , z, H1, w1,
e1, e1) and output e1. The
consequence of the assumption that every system in M is minimum phase is
that Σ1(
p) is detectable through e1 for every fixed value of
p [5]. Meanwhile
the form of (10) enables one to show by direct calculation, that the rate of
change of the partial Lyapunov function V
∆
= e2
1 +||
p−p∗
||2
along a solution
to (12) and the equations defining Σ1(
p), satisfies
V̇ = −2λe2
1 ≤ 0 (13)
From this it is evident that V is a bounded monotone nonincreasing function
and consequently that e1 and
p are bounded wherever they exist. Using and
the fact that Σ1(
p) is a linear parameter-varying system, it can be concluded
that solutions exist globally and that e1 and
p are bounded on [0, ∞). By
integrating (13) it can also be concluded that e1 has a finite L2
[0, ∞)-norm
and that ||e1||2
+||
p−p∗
||2
tends to a finite limit as t → ∞. Were it possible
to deduce from these properties that
p tended to a limit p̄, then it would
possible to establish correctness of the overall adaptive controller using the
detectability of Σ1(p̄).
2Monopoli called e1 an augmented error.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-45-320.jpg)
![26 PROBLEM 1.5
There are two very special cases for which correctness has been established.
The first is when the process models in M all have relative degree 1; that
is when n̄ = 1. See the references cited in [3] for more on this special case.
The second special case is when p∗
is taken to be of the form q∗
k where k
is a known vector and q∗
is a scalar; in this case
p
∆
=
qk where
q is a scalar
parameter tuned by the equation ˙
q = −k
φ1e1 [6].
Version 2
In the sequel we describe the multi-estimator and tuner that, together with
reference model (1) and control law (6), comprise the second version of them
adaptive controller relevant to the problem at hand.
Multi-Estimator 2
The second multi-estimator E2(
p), which is relevant to the problem under
consideration, is similar to E1(
p) but has the slight advantage of leading to
a tuner that is somewhat easier to analyze. To describe E2(
p), we need first
to define matrices
Ā2
∆
=
Ā 0
c̄ −λ
and b̄2
∆
=
b̄
¯
d
The form of E2(
p) is motivated by the readily verifiable fact that if H2 and
w2 are (n̄+1)×2n and (n̄+1)×1 signal matrices generated by the equations
Ḣ2 = Ā2H2 + b̄2z
and ẇ2 = Ā2w2 + b̄2(u − r) (14)
then w2 − H2p∗
is a solution to (3) - (4). In other words, x
eT
=
w2−H2p∗
+ where is an initial condition dependent time function decaying
to zero as fast as eĀ2t
. Again, for simplicity, we shall ignore . This means
that
eT = c̄2w2 − c̄2H2p∗
where c̄2 = 0 · · · 0 1
. Thus, in this case, a natural way to generate
an estimate
e2 of eT is by means of the equation
e2 = c̄2w2 − c̄2H2
p (15)
It is clear that the multi-estimator E2(
p) defined by (2), (14) and (15) has
the required property of delivering an asymptotically correct estimate
e2 of
eT if
p is fixed at p∗
.
Tuner 2
Note that in this case the estimation error
e2
∆
=
e2 − eT (16)](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-46-320.jpg)
![IS MONOPOLI’S MODEL REFERENCE ADAPTIVE CONTROLLER CORRECT? 27
satisfies the error equation
e2 = φ
2(
p2 − p∗
) (17)
where
φ
2 = −c̄2H2 (18)
Equation (17) suggests that one consider a pseudo-gradient tuner of the form
˙
p = −φ2e2 (19)
What is Known about Version 2?
The overall model reference adaptive controller in this case thus consists of
the reference model (1), the control law (6), the multi-estimator (2), (14),
(15), the output estimation error (16) and the tuner (18), (19). The open
problem is here to prove that this version of the controller either solves the
model reference adaptive control problem or that it does not.
Much is known about the problem. In the first place, (1), (2) together with
(5), (6) (14) - (18) define a parameter varying linear system Σ2(
p) with
input r, state (yref , xref , z, H2, w2) and output e2. The consequence of the
assumption that every system in M is minimum phase is that this Σ2(
p) is
detectable through e2 for every fixed value of
p [5]. Meanwhile the form of
(17) enables one to show by direct calculation that the rate of change of the
partial Lyapunov function V
∆
= ||
p − p∗
||2
along a solution to (19) and the
equations defining Σ2(
p), satisfies
V̇ = −2λe2
2 ≤ 0 (20)
It is evident that V is a bounded monotone nonincreasing function and
consequently that
p is bounded wherever they exist. From this and the fact
that Σ2(
p) is a linear parameter-varying system, it can be concluded that
solutions exist globally and that
p is bounded on [0, ∞). By integrating
(20) it can also be concluded that e2 has a finite L2
[0, ∞)-norm and that
||
p − p∗
||2
tends to a finite limit as t → ∞. Were it possible to deduce
from these properties that
p tended to a limit p̄ , then it would to establish
correctness using the detectability of Σ2(p̄).
There is one very special cases for which correctness has been established
[6]. This is when p∗
is taken to be of the form q∗
k where k is a known vector
and q∗
is a scalar; in this case
p
∆
=
qk where
q is a scalar parameter tuned
by the equation ˙
q = −k
φ2e2. The underlying reason why things go through
is because in this special case, the fact that ||
p − p∗
||2
and consequently
||
q − q∗
|| tend to a finite limits, means that
q tends to a finite limit as well.
4 THE ESSENCE OF THE PROBLEM
In this section we transcribe a stripped down version of the problem that
retains all the essential feature that need to be overcome in order to decide](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-47-320.jpg)
![28 PROBLEM 1.5
whether or not Monopoli’s controller is correct. We do this only for version
2 of the problem and only for the case when r = 0 and n̄ = 1. Thus, in
this case, we can take Ā2 = −λ and b̄2 = 1. Assuming the reference model
is initialized at 0, dropping the subscript 2 throughout, and writing φ
for
−H, the system to be analyzed reduces to
ż =
A 0
0 A
z +
b
0
(w + φ
p∗
) +
0
b
p
z (21)
φ̇ = −λφ − z (22)
ẇ = −λw +
p
z (23)
e = φ
(
p − p∗
) (24)
˙
p = −φe (25)
To recap, p∗
is unknown and constant but is such that the linear parameter-
varying system Σ(
p) defined by (21) to (24) is detectable through e for
each fixed value of
p. Solutions to the system (21) - (25) exist globally.
The parameter vector
p and integral square of e are bounded on [0, ∞) and
||
p − p∗
|| tends to a finite limit as t → ∞. The open problem here is to show
for every initialization of (21)-(25), that the state of Σ(
p) tends to 0 or that
it does not.
BIBLIOGRAPHY
[1] R. V. Monopoli, “Model reference adaptive control with an augmented
error,” IEEE Transactions on Automatic Control, pp. 474–484, October
1974.
[2] A. Feuer, B. R. Barmish, and A. S. Morse, “An unstable system as-
sociated with model reference adaptive control,” IEEE Transactions on
Automatic Control, 23:499–500, 1978.
[3] A. S. Morse, “Overcoming the obstacle of high relative degree,” European
Journal of Control, 2(1):29–35, 1996.
[4] K. J. Åström and B. Wittenmark, “On self-tuning regulators,” Automat-
ica, 9:185–199, 1973.
[5] A. S. Morse, “Towards a unified theory of parameter adaptive control -
Part 2: Certainty equivalence and implicit tuning,” IEEE Transactions
on Automatic Control, 37(1):15–29, January 1992.
[6] A. Feuer, Adaptive Control of Single Input Single Output Linear Systems,
Ph.D. thesis, Yale University, 1978.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-48-320.jpg)
![Problem 1.6
Model reduction of delay systems
Jonathan R. Partington
School of Mathematics
University of Leeds
Leeds, LS2 9JT
U.K.
J.R.Partington@leeds.ac.uk
1 DESCRIPTION OF THE PROBLEM
Our concern here is with stable single input single output delay systems,
and we shall restrict to the case when the system has a transfer function
of the form G(s) = e−sT
R(s), with T 0 and R rational, stable, and
strictly proper, thus bounded and analytic on the right half plane C+. It is
a fundamental problem in robust control design to approximate such systems
by finite-dimensional systems. Thus, for a fixed natural number n, we wish
to find a rational approximant Gn(s) of degree at most n in order to make
small the approximation error G − Gn, where . denotes an appropriate
norm. See [9] for some recent work on this subject.
Commonly used norms on a linear time-invariant system with impulse re-
sponse g ∈ L1
(0, ∞) and transfer function G ∈ H∞
(C+) are the H∞
norm G∞ = supRe s0 |G(s)|, the Lp
norms gp =
∞
0
|g(t)|p
dt
1/p
(1 ≤ p ∞), and the Hankel norm Γ, where Γ : L2
(0, ∞) → L2
(0, ∞) is
the Hankel operator defined by
(Γu)(t) =
∞
0
g(t + τ)u(τ) dτ.
These norms are related by
Γ ≤ G∞ ≤ g1 ≤ 2nΓ,
where the last inequality holds for systems of degree at most n.
Two particular approximation techniques for finite-dimensional systems are
well-established in the literature [14], and they can also be used for some
infinite-dimensional systems [5]:](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-49-320.jpg)
![30 PROBLEM 1.6
• Truncated balanced realizations, or, equivalently, output normal real-
izations [11, 13, 5];
• Optimal Hankel-norm approximants [1, 4, 5].
As we explain in the next section, these techniques are known to produce
H∞
-convergent sequences of approximants for many classes of delay systems
(systems of nuclear type). We are thus led to pose the following question:
Do the sequences of reduced order models produced by truncated balanced
realizations and optimal Hankel-norm approximations converge for all stable
delay systems?
2 MOTIVATION AND HISTORY OF THE PROBLEM
Balanced realizations were introduced in [11], and many properties of trun-
cations of such realizations were given in [13]. An H∞
error bound for the
reduced-order system produced by truncating a balanced realization was
given for finite-dimensional systems in [3, 4], and extended to infinite-di-
mensional systems in [5]. This commonly used bound is expressed in terms
of the sequence (σk)∞
k=1 of singular values of the Hankel operator Γ corre-
sponding to the original system G; in our case Γ is compact, and so σk → 0.
Provided that g ∈ L1
∩L2
and Γ is nuclear (i.e.,
∞
k=1 σk ∞) with distinct
singular values, then the inequality
G − Gb
n∞ ≤ 2(σn+1 + σn+2 + . . .)
holds for the degree-n balanced truncation Gb
n of G. The elementary lower
bound G − Gn ≥ σn+1 holds for any degree-n approximation to G.
Another numerically convenient approximation method is the optimal Han-
kel-norm technique [1, 4, 5], which involves finding a best rank-n Hankel
approximation ΓH
n to Γ, in the Hankel norm, so that Γ − ΓH
n = σn+1. In
this case the bound
G − GH
n − D0∞ ≤ σn+1 + σn+2 + . . .
is available for the corresponding transfer function GH
n with a suitable con-
stant D0. Again, we require the nuclearity of Γ for this to be meaningful.
3 AVAILABLE RESULTS
In the case of a delay system G(s) = e−sT
R(s) as specified above, it is known
that the Hankel singular values σk are asymptotic to A
T
πk
r
, where r is](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-50-320.jpg)
![MODEL REDUCTION OF DELAY SYSTEMS 31
the relative degree of R and |sr
R(s)| tends to the finite nonzero limit A as
|s| → ∞. Hence Γ is nuclear if and only if the relative degree of R is at least
2. (Equivalently, if and only if g is continuous.) We refer to [6, 7] for these
and more precise results.
Even for a very simple non-nuclear system such as G(s) = e−sT
s + 1, for which
kσk → T/π, no theoretical upper bound is known for the H∞
errors in
the rational approximants produced by truncated balanced realizations and
optimal Hankel-norm approximation, although numerical evidence suggests
that they should still tend to zero.
A related question is to find the best error bounds in L1
approximation of
a delay system. For example, a smoothing technique gives an L1
approx-
imation error O
ln n
n
for systems of relative degree r = 1 (see [8]), and
it is possible that the optimal Hankel norm might yield a similar rate of
convergence. (A lower bound of C/n for some constant C 0 follows easily
from the above discussion.)
One approach that may be useful in these analyses is to exploit Bonsall’s
theorem that a Hankel integral operator Γ is bounded if and only if it is
uniformly bounded on the set of all normalized L2
functions whose Laplace
transforms are rational of degree one [2, 12]. An explicit constant in Bon-
sall’s theorem is not known, and would be of great interest in its own right.
Another approach which may be relevant is that of Megretski [10], who
introduces maximal real part norms. Their interest stems from the inequality
G∞ ≥ Re G∞ ≥ Γ/2.
BIBLIOGRAPHY
[1] V. M. Adamjan, D. Z. Arov, and M. G. Kreı̆n, “Analytic properties of
Schmidt pairs for a Hankel operator and the generalized Schur–Takagi
problem,” Math. USSR Sbornik, 15:31–73, 1971.
[2] F. F. Bonsall, “Boundedness of Hankel matrices”, J. London Math. Soc.
(2), 29(2):289–300, 1984.
[3] D. Enns, Model Reduction for Control System Design, Ph.D. disserta-
tion, Stanford University, 1984.
[4] K. Glover, “All optimal Hankel-norm approximations of linear mul-
tivariable systems and their L∞
-error bounds, Internat. J. Control,
39(6):1115–1193, 1984.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-51-320.jpg)
![32 PROBLEM 1.6
[5] K. Glover, R. F. Curtain, and J. R. Partington, “Realisation and ap-
proximation of linear infinite-dimensional systems with error bounds,”
SIAM J. Control Optim., 26(4):863–898, 1988.
[6] K. Glover, J. Lam, and J. R. Partington, “Rational approximation of a
class of infinite-dimensional systems. I. Singular values of Hankel oper-
ators,” Math. Control Signals Systems, 3(4):325–344, 1990.
[7] K. Glover, J. Lam, and J. R. Partington,“Rational approximation of a
class of infinite-dimensional systems. II. Optimal convergence rates of
L∞ approximants,” Math. Control Signals Systems, 4(3):233–246, 1991.
[8] K. Glover and J. R. Partington, “Bounds on the achievable accuracy in
model reduction,” In: Modelling, Robustness and Sensitivity Reduction
in Control Systems (Groningen, 1986), pp. 95–118. Springer, Berlin,
1987.
[9] P. M. Mäkilä and J. R. Partington, “Shift operator induced approxi-
mations of delay systems,” SIAM J. Control Optim., 37(6):1897–1912,
1999.
[10] A. Megretski, “Model order reduction using maximal real part norms,”
Presented at CDC 2000, Sydney, 2000.
http://web.mit.edu/ameg/www/images/lund.ps.
[11] B. C. Moore, “Principal component analysis in linear systems: control-
lability, observability, and model reduction,” IEEE Trans. Automat.
Control, 26(1):17–32, 1981.
[12] J. R. Partington and G. Weiss, “Admissible observation operators for
the right-shift semigroup,” Math. Control Signals Systems, 13(3):179–
192, 2000.
[13] L. Pernebo and L. M. Silverman, “Model reduction via balanced state
space representations,” IEEE Trans. Automat. Control, 27(2):382–387,
1982.
[14] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control,
Upper Saddle River, NJ: Prentice Hall1996.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-52-320.jpg)
![Problem 1.7
Schur extremal problems
Lev Sakhnovich
Courant Institute of Mathematical Science
New York, NY 11223
USA
Lev.Sakhnovich@verizon.net
1 DESCRIPTION OF THE PROBLEM
In this paper we consider the well-known Schur problem the solution of which
satisfy in addition the extremal condition
w
(z)w(z) ≤ ρ2
min, |z| 1, (1)
where w(z) and ρmin are m × m matrices and ρmin 0. Here the matrix
ρmin is defined by a certain minimal-rank condition (see Definition 1). We
remark that the extremal Schur problem is a particular case. The general
case is considered in book [1] and paper [2]. Our approach to the extremal
problems does not coincide with the superoptimal approach [3],[4]. In paper
[2] we compare our approach to the extremal problems with the superoptimal
approach. Interpolation has found great applications in control theory [5],[6].
Schur Extremal Problem: The m×m matrices a0, a1, ..., an are given.
Describe the set of m×m matrix functions w(z) holomorphic in the circle
|z| 1 and satisfying the relation
w(z) = a0 + a1z + ... + anzn
+ ... (2)
and inequality (1.1).
A necessary condition of the solvability of the Schur extremal problem is the
inequality
R2
min − S ≥ 0, (3)
where the (n + 1)m×(n + 1)m matrices S and Rmin are defined by the
relations
S = CnC
n, Rmin = diag[ρmin, ρmin, ..., ρmin], (4)](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-53-320.jpg)
![34 PROBLEM 1.7
Cn =
a0 0 ... 0
a1 a0 ... 0
... ... ... ...
an an−1 ... a0
. (5)
Definition 1: We shall call the matrix ρ = ρmin 0 minimal if the following
two requirements are fulfilled:
1. The inequality
R2
min − S ≥ 0 (6)
holds.
2. If the m×m matrix ρ 0 is such that
R2
− S ≥ 0, (7)
then
rank(R2
min − S) ≤ rank(R2
− S), (8)
where R = diag[ρ, ρ, ..., ρ].
Remark 1: The existence of ρmin follows directly from definition 1.
Question 1: Is ρmin unique?
Remark 2: If m = 1 then ρmin is unique and ρ2
min = λmax, where λmax is
the largest eigenvalue of the matrix S.
Remark 3: Under some assumptions the uniqueness of ρmin is proved in
the case m 1, n = 1 (see [2],[7]).
If ρmin is known then the corresponding wmin(ξ) is a rational matrix func-
tion. This generalizes the well-known fact for the scalar case (see [7]).
Question 2: How to find ρmin?
In order to describe some results in this direction we write the matrix
S = CnC
n in the following block form
S11 S12
S21 S22
, (9)
where S22 is an m×m matrix.
Proposition 1: [1] If ρ = q 0 satisfies inequality (1.7) and the relation
q2
= S22 + S
12(Q2
− S11)−1
S12, (10)
where Q = diag[q, q, ..., q], then ρmin = q.
We shall apply the method of successive approximation when studying equa-
tion (1.10). We put q2
0 = S22, q2
k+1 = S22 +S
12(Q2
k − S11)−1
S12, where k≥0,
Qk = diag[qk, qk, ..., qk]. We suppose that
Q2
0 − S11 0. (11)
Theorem 1: [1] The sequence q2
0, q2
2, q2
4, ... monotonically increases and has
the limit m1. The sequence q2
1, q2
3, q2
5, ... monotonically decreases and has the
limit m2. The inequality m1≤m2 holds. If m1 = m2 then ρ2
min = q2
.
Question 3: Suppose relation (1.11) holds. Is there a case when m1=m2?
The answer is “no” if n = 1 (see [2],[8]).
Remark 4: In book [1] we give an example in which ρmin is constructed in
explicit form.](https://image.slidesharecdn.com/25977194-250513060943-251b2d62/85/Unsolved-Problems-In-Mathematical-Systems-And-Control-Theory-Course-Book-Vincent-D-Blondel-Editor-Alexandre-Megretski-Editor-54-320.jpg)
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- 5. Unsolved Problems in Mathematical Systems and Control Theory Edited by Vincent D. Blondel Alexandre Megretski PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD
- 6. iv Copyright c 2004 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540, USA In the United Kingdom: Princeton University Press, 3 Market Place, Wood- stock, Oxfordshire OX20 1SY, UK All rights reserved Library of Congress Cataloging-in-Publication Data Unsolved problems in mathematical systems and control theory Edited by Vincent D. Blondel, Alexandre Megretski. p. cm. Includes bibliographical references. ISBN 0-691-11748-9 (cl : alk. paper) 1. System analysis. 2. Control theory. I. Blondel, Vincent. II. Megretski, Alexandre. QA402.U535 2004 2003064802 003—dc22 The publisher would like to acknowledge the editors of this volume for pro- viding the camera-ready copy from which this book was printed. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
- 7. I have yet to see any problem, however complicated, which, when you looked at it in the right way, did not become still more compli- cated. Poul Anderson
- 8. This page intentionally left blank
- 9. Contents Preface xiii Associate Editors xv Website xvii PART 1. LINEAR SYSTEMS 1 Problem 1.1. Stability and composition of transfer functions Guillermo Fernández-Anaya, Juan Carlos Martı́nez-Garcı́a 3 Problem 1.2. The realization problem for Herglotz-Nevanlinna functions Seppo Hassi, Henk de Snoo, Eduard Tsekanovskiı̆ 8 Problem 1.3. Does any analytic contractive operator function on the polydisk have a dissipative scattering nD realization? Dmitry S. Kalyuzhniy-Verbovetzky 14 Problem 1.4. Partial disturbance decoupling with stability Juan Carlos Martı́nez-Garcı́a, Michel Malabre, Vladimir Kučera 18 Problem 1.5. Is Monopoli’s model reference adaptive controller correct? A. S. Morse 22 Problem 1.6. Model reduction of delay systems Jonathan R. Partington 29 Problem 1.7. Schur extremal problems Lev Sakhnovich 33 Problem 1.8. The elusive iff test for time-controllability of behaviors Amol J. Sasane 36
- 10. viii CONTENTS Problem 1.9. A Farkas lemma for behavioral inequalities A.A. (Tonny) ten Dam, J.W. (Hans) Nieuwenhuis 40 Problem 1.10. Regular feedback implementability of linear differential behaviors H. L. Trentelman 44 Problem 1.11. Riccati stability Erik I. Verriest 49 Problem 1.12. State and first order representations Jan C. Willems 54 Problem 1.13. Projection of state space realizations Antoine Vandendorpe, Paul Van Dooren 58 PART 2. STOCHASTIC SYSTEMS 65 Problem 2.1. On error of estimation and minimum of cost for wide band noise driven systems Agamirza E. Bashirov 67 Problem 2.2. On the stability of random matrices Giuseppe C. Calafiore, Fabrizio Dabbene 71 Problem 2.3. Aspects of Fisher geometry for stochastic linear systems Bernard Hanzon, Ralf Peeters 76 Problem 2.4. On the convergence of normal forms for analytic control systems Wei Kang, Arthur J. Krener 82 PART 3. NONLINEAR SYSTEMS 87 Problem 3.1. Minimum time control of the Kepler equation Jean-Baptiste Caillau, Joseph Gergaud, Joseph Noailles 89 Problem 3.2. Linearization of linearly controllable systems R. Devanathan 93 Problem 3.3. Bases for Lie algebras and a continuous CBH formula Matthias Kawski 97
- 11. CONTENTS ix Problem 3.4. An extended gradient conjecture Luis Carlos Martins Jr., Geraldo Nunes Silva 103 Problem 3.5. Optimal transaction costs from a Stackelberg perspective Geert Jan Olsder 107 Problem 3.6. Does cheap control solve a singular nonlinear quadratic problem? Yuri V. Orlov 111 Problem 3.7. Delta-Sigma modulator synthesis Anders Rantzer 114 Problem 3.8. Determining of various asymptotics of solutions of nonlinear time- optimal problems via right ideals in the moment algebra G. M. Sklyar, S. Yu. Ignatovich 117 Problem 3.9. Dynamics of principal and minor component flows U. Helmke, S. Yoshizawa, R. Evans, J.H. Manton, and I.M.Y. Mareels 122 PART 4. DISCRETE EVENT, HYBRID SYSTEMS 129 Problem 4.1. L2-induced gains of switched linear systems João P. Hespanha 131 Problem 4.2. The state partitioning problem of quantized systems Jan Lunze 134 Problem 4.3. Feedback control in flowshops S.P. Sethi and Q. Zhang 140 Problem 4.4. Decentralized control with communication between controllers Jan H. van Schuppen 144 PART 5. DISTRIBUTED PARAMETER SYSTEMS 151 Problem 5.1. Infinite dimensional backstepping for nonlinear parabolic PDEs Andras Balogh, Miroslav Krstic 153 Problem 5.2. The dynamical Lame system with boundary control: on the struc- ture of reachable sets M.I. Belishev 160
- 12. x CONTENTS Problem 5.3. Null-controllability of the heat equation in unbounded domains Sorin Micu, Enrique Zuazua 163 Problem 5.4. Is the conservative wave equation regular? George Weiss 169 Problem 5.5. Exact controllability of the semilinear wave equation Xu Zhang, Enrique Zuazua 173 Problem 5.6. Some control problems in electromagnetics and fluid dynamics Lorella Fatone, Maria Cristina Recchioni, Francesco Zirilli 179 PART 6. STABILITY, STABILIZATION 187 Problem 6.1. Copositive Lyapunov functions M. K. Çamlıbel, J. M. Schumacher 189 Problem 6.2. The strong stabilization problem for linear time-varying systems Avraham Feintuch 194 Problem 6.3. Robustness of transient behavior Diederich Hinrichsen, Elmar Plischke, Fabian Wirth 197 Problem 6.4. Lie algebras and stability of switched nonlinear systems Daniel Liberzon 203 Problem 6.5. Robust stability test for interval fractional order linear systems Ivo Petráš, YangQuan Chen, Blas M. Vinagre 208 Problem 6.6. Delay-independent and delay-dependent Aizerman problem Vladimir Răsvan 212 Problem 6.7. Open problems in control of linear discrete multidimensional sys- tems Li Xu, Zhiping Lin, Jiang-Qian Ying, Osami Saito, Yoshihisa Anazawa 221 Problem 6.8. An open problem in adaptative nonlinear control theory Leonid S. Zhiteckij 229 Problem 6.9. Generalized Lyapunov theory and its omega-transformable regions Sheng-Guo Wang 233
- 13. CONTENTS xi Problem 6.10. Smooth Lyapunov characterization of measurement to error sta- bility Brian P. Ingalls, Eduardo D. Sontag 239 PART 7. CONTROLLABILITY, OBSERVABILITY 245 Problem 7.1. Time for local controllability of a 1-D tank containing a fluid modeled by the shallow water equations Jean-Michel Coron 247 Problem 7.2. A Hautus test for infinite-dimensional systems Birgit Jacob, Hans Zwart 251 Problem 7.3. Three problems in the field of observability Philippe Jouan 256 Problem 7.4. Control of the KdV equation Lionel Rosier 260 PART 8. ROBUSTNESS, ROBUST CONTROL 265 Problem 8.1. H∞-norm approximation A.C. Antoulas, A. Astolfi 267 Problem 8.2. Noniterative computation of optimal value in H∞ control Ben M. Chen 271 Problem 8.3. Determining the least upper bound on the achievable delay margin Daniel E. Davison, Daniel E. Miller 276 Problem 8.4. Stable controller coefficient perturbation in floating point imple- mentation Jun Wu, Sheng Chen 280 PART 9. IDENTIFICATION, SIGNAL PROCESSING 285 Problem 9.1. A conjecture on Lyapunov equations and principal angles in sub- space identification Katrien De Cock, Bart De Moor 287
- 14. xii CONTENTS Problem 9.2. Stability of a nonlinear adaptive system for filtering and parameter estimation Masoud Karimi-Ghartemani, Alireza K. Ziarani 293 PART 10. ALGORITHMS, COMPUTATION 297 Problem 10.1. Root-clustering for multivariate polynomials and robust stability analysis Pierre-Alexandre Bliman 299 Problem 10.2. When is a pair of matrices stable? Vincent D. Blondel, Jacques Theys, John N. Tsitsiklis 304 Problem 10.3. Freeness of multiplicative matrix semigroups Vincent D. Blondel, Julien Cassaigne, Juhani Karhumäki 309 Problem 10.4. Vector-valued quadratic forms in control theory Francesco Bullo, Jorge Cortés, Andrew D. Lewis, Sonia Martı́nez 315 Problem 10.5. Nilpotent bases of distributions Henry G. Hermes, Matthias Kawski 321 Problem 10.6. What is the characteristic polynomial of a signal flow graph? Andrew D. Lewis 326 Problem 10.7. Open problems in randomized µ analysis Onur Toker 330
- 15. Preface Five years ago, a first volume of open problems in Mathematical Systems and Control Theory appeared.1 Some of the 53 problems that were published in this volume attracted considerable attention in the research community. The book in front of you contains a new collection of 63 open problems. The contents of both volumes show the evolution of the field in the half decade since the publication of the first volume. One noticeable feature is the shift toward a wider class of questions and more emphasis on issues driven by physical modeling. Early versions of some of the problems in this book have been presented at the Open Problem sessions of the Oberwolfach Tagung on Regelungstheorie, on February 27, 2002, and of the Conference on Mathematical Theory of Networks and Systems (MTNS) in Notre Dame, Indiana, on August 12, 2002. The editors thank the organizers of these meetings for their willingness to provide the problems this welcome exposure. Since the appearance of the first volume, open problems have continued to meet with large interest in the mathematical community. Undoubtedly, the most spectacular event in this arena was the announcement by the Clay Mathematics Institute2 of the Millennium Prize Problems whose solution will be rewarded by one million U.S. dollars each. Modesty and modesty of means have prevented the editors of the present volume from offering similar rewards toward the solution of the problems in this book. However, we trust that, notwithstanding this absence of a financial incentive, the intellectual challenge will stimulate many readers to attack the problems. The editors thank in the first place the researchers who have submitted the problems. We are also very thankful to the Princeton University Press, and in particular Vickie Kearn, for their willingness to publish this vol- ume. The full text of the problems, together with comments, additions, and solutions, will be posted on the book website at Princeton Univer- sity Press (link available from http://pup.princeton.edu/math/) and on http://www.inma.ucl.ac.be/∼blondel/op/. Readers are encouraged to submit contributions by following the instructions given on these websites. The editors, Louvain-la-Neuve, March 15, 2003. 1Vincent D. Blondel, Eduardo D. Sontag, M. Vidyasagar, and Jan C. Willems, Open Problems in Mathematical Systems and Control Theory, Springer Verlag, 1998. 2See http://www.claymath.org.
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- 17. Associate Editors Roger Brockett, Harvard University, USA Jean-Michel Coron, University of Paris (Orsay), France Roland Hildebrand, University of Louvain (Louvain-la-Neuve), Belgium Miroslav Krstic, University of California (San Diego), USA Anders Rantzer, Lund Institute of Technology, Sweden Joachim Rosenthal, University of Notre Dame, USA Eduardo Sontag, Rutgers University, USA M. Vidyasagar, Tata Consultancy Services, India Jan Willems, University of Leuven, Belgium
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- 19. Website The full text of the problems presented in this book, together with com- ments, additions and solutions, are freely available in electronic format from the book website at Princeton University Press: http://pup.princeton.edu/math/ and from an editor website: http://www.inma.ucl.ac.be/∼blondel/op/ Readers are encouraged to submit contributions by following the instruc- tions given on these websites.
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- 23. Problem 1.1 Stability and composition of transfer functions G. Fernández-Anaya Departamento de Ciencias Básicas Universidad Iberoaméricana Lomas de Santa Fe 01210 México D.F. México guillermo.fernandez@uia.mx J. C. Martı́nez-Garcı́a Departamento de Control Automático CINVESTAV-IPN A.P. 14-740 07300 México D.F. México martinez@ctrl.cinvestav.mx 1 INTRODUCTION As far as the frequency-described continuous linear time-invariant systems are concerned, the study of control-oriented properties (like stability) re- sulting from the substitution of the complex Laplace variable s by rational transfer functions have been little studied by the Automatic Control com- munity. However, some interesting results have recently been published: Concerning the study of the so-called uniform systems, i.e., LTI systems consisting of identical components and amplifiers, it was established in [8] a general criterion for robust stability for rational functions of the form D(f(s)), where D(s) is a polynomial and f(s) is a rational transfer function. By applying such a criterium, it gave a generalization of the celebrated Kharitonov’s theorem [7], as well as some robust stability criteria under H∞- uncertainty. The results given in [8] are based on the so-called H-domains.1 As far as robust stability of polynomial families is concerned, some Kharito- 1The H-domain of a function f (s) is defined to be the set of points h on the complex plane for which the function f (s) − h has no zeros on the open right-half complex plane.
- 24. 4 PROBLEM 1.1 nov’s like results [7] are given in [9] (for a particular class of polynomials), when interpreting substitutions as nonlinearly correlated perturbations on the coefficients. More recently, in [1], some results for proper and stable real rational SISO functions and coprime factorizations were proved, by making substitutions with α (s) = (as + b) / (cs + d), where a, b, c, and d are strictly positive real numbers, and with ad − bc = 0. But these results are limited to the bilinear transforms, which are very restricted. In [4] is studied the preservation of properties linked to control problems (like weighted nominal performance and robust stability) for Single-Input Single- Output systems, when performing the substitution of the Laplace variable (in transfer functions associated to the control problems) by strictly positive real functions of zero relative degree. Some results concerning the preservation of control-oriented properties in Multi-Input Multi-Output systems are given in [5], while [6] deals with the preservation of solvability conditions in algebraic Riccati equations linked to robust control problems. Following our interest in substitutions we propose in section 22.2 three in- teresting problems. The motivations concerning the proposed problems are presented in section 22.3. 2 DESCRIPTION OF THE PROBLEMS In this section we propose three closely related problems. The first one con- cerns the characterization of a transfer function as a composition of transfer functions. The second problem is a modified version of the first problem: the characterization of a transfer function as the result of substituting the Laplace variable in a transfer function by a strictly positive real transfer function of zero relative degree. The third problem is in fact a conjecture concerning the preservation of stability property in a given polynomial re- sulting from the substitution of the coefficients in the given polynomial by a polynomial with non-negative coefficients evaluated in the substituted co- efficients. Problem 1: Let a Single Input Single Output (SISO) transfer function G(s) be given. Find transfer functions G0(s) and H(s) such that: 1. G (s) = G0 (H (s)) ; 2. H (s) preserves proper stable transfer functions under substitution of the variable s by H (s), and: 3. The degree of the denominator of H(s) is the maximum with the prop- erties 1 and 2.
- 25. STABILITY AND COMPOSITION OF TRANSFER FUNCTIONS 5 Problem 2: Let a SISO transfer function G(s) be given. Find a transfer function G0 (s) and a Strictly Positive Real transfer function of zero relative degree (SPR0), say H(s), such that: 1. G(s) = G0 (H (s)) and: 2. The degree of the denominator of H(s) is the maximum with the prop- erty 1. Problem 3: (Conjecture) Given any stable polynomial: ansn + an−1sn−1 + · · · + a1s + a0 and given any polynomial q(s) with non-negative coefficients, then the poly- nomial: q(an)sn + q(an−1)sn−1 + · · · + q(a1)s + q(a0) is stable (see [3]). 3 MOTIVATIONS Consider the closed-loop control scheme: y (s) = G (s) u (s) + d (s) , u (s) = K (s) (r (s) − y (s)) , where: P (s) denotes the SISO plant; K (s) denotes a stabilizing controller; u (s) denotes the control input; y (s) denotes the control input; d (s) denotes the disturbance and r (s) denotes the reference input. We shall denote the closed-loop transfer function from r (s) to y (s) as Fr (G (s) , K (s)) and the closed-loop transfer function from d (s) to y (s) as Fd (G (s) , K (s)). • Consider the closed-loop system Fr (G (s) , K (s)), and suppose that the plant G(s) results from a particular substitution of the s Laplace variable in a transfer function G0(s) by a transfer function H(s), i.e., G(s) = G0(H(s)). It has been proved that a controller K0 (s) which stabilizes the closed-loop system Fr (G0 (s) , K0 (s)) is such that K0 (H (s)) stabilizes Fr (G (s) , K0 (H (s))) (see [2] and [8]). Thus, the simplification of procedures for the synthesis of stabilizing controllers (profiting from transfer function compositions) justifies problem 1. • As far as problem 2 is concerned, consider the synthesis of a controller K (s) stabilizing the closed-loop transfer function Fd (G (s) , K (s)), and such that Fd (G (s) , K (s))∞ γ, for a fixed given γ 0. If we known that G(s) = G0 (H (s)), being H (s) a SPR0 transfer function, the solution of problem 2 would arise to the following procedure: 1. Find a controller K0(s) which stabilizes the closed-loop transfer function Fd (G0 (s) , K0 (s)) and such that: Fd (G0 (s) , K0 (s))∞ γ.
- 26. 6 PROBLEM 1.1 2. The composed controller K (s) = K0 (H (s)) stabilizes the closed- loop system Fd (G (s) , K (s)) and: Fd (G (s) , K (s))∞ γ (see [2], [4], and [5]). It is clear that condition 3 in the first problem, or condition 2 in the second problem, can be relaxed to the following condition: the degree of the denominator of H (s) is as high as be possible with the appropriate conditions. With this new condition, the open problems are a bit less difficult. • Finally, problem 3 can be interpreted in terms of robustness under positive polynomial perturbations in the coefficients of a stable transfer function. BIBLIOGRAPHY [1] G. Fernández, S. Muñoz, R. A. Sánchez, and W. W. Mayol, “Simulta- neous stabilization using evolutionary strategies,”Int. J. Contr., vol. 68, no. 6, pp. 1417-1435, 1997. [2] G. Fernández, “Preservation of SPR functions and stabilization by sub- stitutions in SISO plants,”IEEE Transaction on Automatic Control, vol. 44, no. 11, pp. 2171-2174, 1999. [3] G. Fernández and J. Alvarez, “On the preservation of stability in fam- ilies of polynomials via substitutions,”Int. J. of Robust and Nonlinear Control, vol. 10, no. 8, pp. 671-685, 2000. [4] G. Fernández, J. C. Martı́nez-Garcı́a, and V. Kučera, “H∞-Robustness Properties Preservation in SISO Systems when applying SPR Substitu- tions,”Submitted to the International Journal of Automatic Control. [5] G. Fernández and J. C. Martı́nez-Garcı́a, “MIMO Systems Properties Preservation under SPR Substitutions,” International Symposium on the Mathematical Theory of Networks and Systems (MTNS’2002), University of Notre Dame, USA, August 12-16, 2002. [6] G. Fernández, J. C. Martı́nez-Garcı́a, and D. Aguilar-George, “Preserva- tion of solvability conditions in Riccati equations when applying SPR0 substitutions,” submitted to IEEE Transactions on Automatic Control, 2002. [7] V. L. Kharitonov, “Asymptotic stability of families of systems of linear differential equations, ”Differential’nye Uravneniya, vol. 14, pp. 2086- 2088, 1978.
- 27. STABILITY AND COMPOSITION OF TRANSFER FUNCTIONS 7 [8] B. T. Polyak and Ya. Z. Tsypkin, “Stability and robust stability of uni- form systems, ”Automation and Remote Contr., vol. 57, pp. 1606-1617, 1996. [9] L. Wang, “Robust stability of a class of polynomial families under non- linearly correlated perturbations,”System and Control Letters, vol. 30, pp. 25-30, 1997.
- 28. Problem 1.2 The realization problem for Herglotz-Nevanlinna functions Seppo Hassi Department of Mathematics and Statistics University of Vaasa P.O. Box 700, 65101 Vaasa Finland sha@uwasa.fi Henk de Snoo Department of Mathematics University of Groningen P.O. Box 800, 9700 AV Groningen Nederland desnoo@math.rug.nl Eduard Tsekanovskiı̆ Department of Mathematics Niagara University, NY 14109 USA tsekanov@niagara.edu 1 MOTIVATION AND HISTORY OF THE PROBLEM Roughly speaking, realization theory concerns itself with identifying a given holomorphic function as the transfer function of a system or as its linear frac- tional transformation. Linear, conservative, time-invariant systems whose main operator is bounded have been investigated thoroughly. However, many realizations in different areas of mathematics including system theory, elec- trical engineering, and scattering theory involve unbounded main operators, and a complete theory is still lacking. The aim of the present proposal is to outline the necessary steps needed to obtain a general realization theory along the lines of M. S. Brodskiı̆ and M. S. Livšic [8], [9], [16], who have
- 29. THE REALIZATION PROBLEM FOR HERGLOTZ-NEVANLINNA FUNCTIONS 9 considered systems with a bounded main operator. An operator-valued function V (z) acting on a Hilbert space E belongs to the Herglotz-Nevanlinna class N, if outside R it is holomorphic, symmetric, i.e., V (z)∗ = V (z̄), and satisfies (Im z)(Im V (z)) ≥ 0. Here and in the following it is assumed that the Hilbert space E is finite-dimensional. Each Herglotz- Nevanlinna function V (z) has an integral representation of the form V (z) = Q + Lz + R 1 t − z − t 1 + t2 dΣ(t), (1) where Q = Q∗ , L ≥ 0, and Σ(t) is a nondecreasing matrix-function on R with R dΣ(t)/(t2 + 1) ∞. Conversely, each function of the form (1) belongs to the class N. Of special importance (cf. [15]) are the class S of Stieltjes functions V (z) = γ + ∞ 0 dΣ(t) t − z , (2) where γ ≥ 0 and ∞ 0 dΣ(t)/(t+1) ∞, and the class S−1 of inverse Stieltjes functions V (z) = α + βz + ∞ 0 1 t − z − 1 t dΣ(t), (3) where α ≤ 0, β ≥ 0, and ∞ 0 dΣ(t)/(t2 + 1) ∞. 2 SPECIAL REALIZATION PROBLEMS One way to characterize Herglotz-Nevanlinna functions is to identify them as (linear fractional transformations of) transfer functions: V (z) = i[W(z) + I]−1 [W(z) − I]J, (4) where J = J∗ = J−1 and W(z) is the transfer function of some general- ized linear, stationary, conservative dynamical system (cf. [1], [3]). The approach based on the use of Brodskiı̆-Livšic operator colligations Θ yields to a simultaneous representation of the functions W(z) and V (z) in the form WΘ(z) = I − 2iK∗ (T − zI)−1 KJ, (5) VΘ(z) = K∗ (TR − zI)−1 K, (6) where TR stands for the real part of T. The definitions and main results associated with Brodskiı̆-Livšic type operator colligations in realization of Herglotz-Nevanlinna functions are as follows, cf. [8], [9], [16]. Let T ∈ [H], i.e., T is a bounded linear mapping in a Hilbert space H, and assume that Im T = (T −T∗ )/2i of T is represented as Im T = KJK∗ , where K ∈ [E, H], and J ∈ [E] is self-adjoint and unitary. Then the array Θ = T K J H E (7)
- 30. 10 PROBLEM 1.2 defines a Brodskiı̆-Livšic operator colligation, and the function WΘ(z) given by (5) is the transfer function of Θ. In the case of the directing operator J = I the system (7) is called a scattering system, in which case the main operator T of the system Θ is dissipative: Im T ≥ 0. In system theory WΘ(z) is interpreted as the transfer function of the conservative system (i.e., Im T = KJK∗ ) of the form (T −zI)x = KJϕ− and ϕ+ = ϕ− −2iK∗ x, where ϕ− ∈ E is an input vector, ϕ+ ∈ E is an output vector, and x is a state space vector in H, so that ϕ+ = WΘ(z)ϕ−. The system is said to be minimal if the main operator T of Θ is completely non self-adjoint (i.e., there are no nontrivial invariant subspaces on which T induces self-adjoint operators), cf. [8], [16]. A classical result due to Brodskiı̆ and Livšic [9] states that the compactly supported Herglotz-Nevanlinna functions of the form b a dΣ(t)/(t − z) correspond to minimal systems Θ of the form (7) via (4) with W(z) = WΘ(z) given by (5) and V (z) = VΘ(z) given by (6). Next consider a linear, stationary, conservative dynamical system Θ of the form Θ = A K J H+ ⊂ H ⊂ H− E . (8) Here A ∈ [H+, H−], where H+ ⊂ H ⊂ H− is a rigged Hilbert space, A ⊃ T ⊃ A, A∗ ⊃ T∗ ⊃ A, A is a Hermitian operator in H, T is a non-Hermitian operator in H, K ∈ [E, H−], J = J∗ = J−1 , and Im A = KJK∗ . In this case Θ is said to be a Brodskiı̆-Livšc rigged operator colligation. The transfer function of Θ in (8) and its linear fractional transform are given by WΘ(z) = I − 2iK∗ (A − zI)−1 KJ, VΘ(z) = K∗ (AR − zI)−1 K. (9) The functions V (z) in (1) which can be realized in the form (4), (9) with a transfer function of a system Θ as in (8) have been characterized in [2], [5], [6], [7], [18]. For the significance of rigged Hilbert spaces in system theory, see [14], [16]. Systems (7) and (8) naturally appear in electrical engineering and scattering theory [16]. 3 GENERAL REALIZATION PROBLEMS In the particular case of Stieltjes functions or of inverse Stieltjes functions general realization results along the lines of [5], [6], [7] remain to be worked out in detail, cf. [4], [10]. The systems (7) and (8) are not general enough for the realization of general Herglotz-Nevanlinna functions in (1) without any conditions on Q = Q∗ and L ≥ 0. However, a generalization of the Brodskiı̆-Livšic operator colligation (7) leads to analogous realization results for Herglotz-Nevanlinna functions V (z) of the form (1) whose spectral function is compactly supported: such functions V (z) admit a realization via (4) with W(z) = WΘ(z) = I − 2iK∗ (M − zF)−1 KJ, V (z) = WΘ(z) = K∗ (MR − zF)−1 K, (10)
- 31. THE REALIZATION PROBLEM FOR HERGLOTZ-NEVANLINNA FUNCTIONS 11 where M = MR + iKJK∗ , MR ∈ [H] is the real part of M, F is a finite- dimensional orthogonal projector, and Θ is a generalized Brodskiı̆-Livšic operator colligation of the form Θ = M F K J H E , (11) see [11], [12], [13]. The basic open problems are: Determine the class of linear, conservative, time-invariant dynamical sys- tems (new type of operator colligations) such that an arbitrary matrix-valued Herglotz-Nevanlinna function V (z) acting on E can be realized as a linear fractional transformation (4) of the matrix-valued transfer function WΘ(z) of some minimal system Θ from this class. Find criteria for a given matrix-valued Stieltjes or inverse Stieltjes function acting on E to be realized as a linear fractional transformation of the matrix- valued transfer function of a minimal Brodskiı̆-Livšic type system Θ in (8) with: (i) an accretive operator A, (ii) an α-sectorial operator A, or (iii) an extremal operator A (accretive but not α-sectorial). The same problem for the (compactly supported) matrix-valued Stieltjes or inverse Stieltjes functions and the generalized Brodskiı̆-Livšic systems of the form (11) with the main operator M and the finite-dimensional orthogonal projector F. There is a close connection to the so-called regular impedance conserva- tive systems (where the coefficient of the derivative is invertible) that were recently considered in [17] (see also [19]). It is shown that any function D(s) with non-negative real part in the open right half-plane and for which D(s)/s → 0 as s → ∞ has a realization with such an impedance conservative system. BIBLIOGRAPHY [1] D. Alpay, A. Dijksma, J. Rovnyak, and H.S.V. de Snoo, “Schur func- tions, operator colligations, and reproducing kernel Pontryagin spaces,” Oper. Theory Adv. Appl., 96, Birkhäuser Verlag, Basel, 1997. [2] Yu. M. Arlinskiı̆, “On the inverse problem of the theory of characteristic functions of unbounded operator colligations”, Dopovidi Akad. Nauk Ukrain. RSR, 2 (1976), 105–109 (Russian). [3] D. Z. Arov, “Passive linear steady-state dynamical systems,” Sibirsk. Mat. Zh., 20, no. 2, (1979), 211–228, 457 (Russian) [English transl.: Siberian Math. J., 20 no. 2, (1979) 149–162].
- 32. 12 PROBLEM 1.2 [4] S. V. Belyi, S. Hassi, H. S. V. de Snoo, and E. R. Tsekanovskiı̆, “On the realization of inverse Stieltjes functions,” Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems, Editors D. Gillian and J. Rosenthal, University of Notre Dame, South Bend, Idiana, USA, 2002, http://www.nd.edu/∼mtns/papers/20160 6.pdf [5] S. V. Belyi and E. R. Tsekanovskiı̆, “Realization and factorization prob- lems for J-contractive operator-valued functions in half-plane and sys- tems with unbounded operators,” Systems and Networks: Mathemati- cal Theory and Applications, Akademie Verlag, 2 (1994), 621–624. [6] S. V. Belyi and E. R. Tsekanovskiı̆, “Realization theorems for operator- valued R-functions,” Oper. Theory Adv. Appl., 98 (1997), 55–91. [7] S. V. Belyi and E. R. Tsekanovskiı̆, “On classes of realizable operator- valued R-functions,” Oper. Theory Adv. Appl., 115 (2000), 85–112. [8] M. S. Brodskiı̆, “Triangular and Jordan representations of linear op- erators,” Moscow, Nauka, 1969 (Russian) [English trans.: Vol. 32 of Transl. Math. Monographs, Amer. Math. Soc., 1971]. [9] M. S. Brodskiı̆ and M. S. Livšic, “Spectral analysis of non-selfadjoint operators and intermediate systems,” Uspekhi Mat. Nauk, 13 no. 1, 79, (1958), 3–85 (Russian) [English trans.: Amer. Math. Soc. Transl., (2) 13 (1960), 265–346]. [10] I. Dovshenko and E. R.Tsekanovskiı̆, “Classes of Stieltjes operator- functions and their conservative realizations,” Dokl. Akad. Nauk SSSR, 311 no. 1 (1990), 18–22. [11] S. Hassi, H. S. V. de Snoo, and E. R. Tsekanovskiı̆, “An addendum to the multiplication and factorization theorems of Brodskiı̆-Livšic- Potapov,” Appl. Anal., 77 (2001), 125–133. [12] S. Hassi, H. S. V. de Snoo, and E. R. Tsekanovskiı̆, “On commuta- tive and noncommutative representations of matrix-valued Herglotz- Nevanlinna functions,” Appl. Anal., 77 (2001), 135–147. [13] S. Hassi, H. S. V. de Snoo, and E. R. Tsekanovskiı̆, “Realizations of Herglotz-Nevanlinna functions via F-systems,” Oper. Theory: Adv. Appl., 132 (2002), 183–198. [14] J.W. Helton, “Systems with infinite-dimensional state space: the Hilbert space approach,” Proc. IEEE, 64 (1976), no. 1, 145–160. [15] I. S. Kac̆ and M. G. Kreı̆n, “The R-functions: Analytic functions map- ping the upper half-plane into itself,” Supplement I to the Russian edi- tion of F. V. Atkinson, Discrete and Continuous Boundary Problems, Moscow, 1974 [English trans.: Amer. Math. Soc. Trans., (2) 103 (1974), 1–18].
- 33. THE REALIZATION PROBLEM FOR HERGLOTZ-NEVANLINNA FUNCTIONS 13 [16] M. S. Livšic, “Operators, Oscillations, Waves,” Moscow, Nauka, 1966 (Russian) [English trans.: Vol. 34 of Trans. Math. Monographs, Amer. Math. Soc., 1973]. [17] O. J. Staffans, “Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view),” Pro- ceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems, Ed., D. Gillian and J. Rosenthal, Univer- sity of Notre Dame, South Bend, Indiana, USA, 2002, Plenary talk, http://www.nd.edu/∼mtns [18] E. R. Tsekanovskiı̆ and Yu. L. Shmul’yan, “The theory of biextensions of operators in rigged Hilbert spaces: Unbounded operator colligations and characteristic functions,” Uspekhi Mat. Nauk, 32 (1977), 69–124 (Russian) [English transl.: Russian Math. Surv., 32 (1977), 73–131]. [19] G. Weiss, “Transfer functions of regular linear systems. Part I: charac- terizations of regularity”, Trans. Amer. Math. Soc., 342 (1994), 827– 854.
- 34. Problem 1.3 Does any analytic contractive operator function on the polydisk have a dissipative scattering nD realization? Dmitry S. Kalyuzhniy-Verbovetzky Department of Mathematics The Weizmann Institute of Science Rehovot 76100 Israel dmitryk@wisdom.weizmann.ac.il 1 DESCRIPTION OF THE PROBLEM Let X, U, Y be finite-dimensional or infinite-dimensional separable Hilbert spaces. Consider nD linear systems of the form α : x(t) = n k=1 (Akx(t − ek) + Bku(t − ek)), y(t) = n k=1 (Ckx(t − ek) + Dku(t − ek)), (t ∈ Zn : n k=1 tk 0) (1) where ek := (0, . . . , 0, 1, 0, . . . , 0) ∈ Zn (here unit is on the k-th place), for all t ∈ Zn such that n k=1 tk ≥ 0 one has x(t) ∈ X (the state space), u(t) ∈ U (the input space), y(t) ∈ Y (the output space), Ak, Bk, Ck, Dk are bounded linear operators, i.e., Ak ∈ L(X), Bk ∈ L(U, X), Ck ∈ L(X, Y), Dk ∈ L(U, Y) for all k ∈ {1, . . . , n}. We use the notation α = (n; A, B, C, D; X, U, Y) for such a system (here A := (A1, . . . , An), etc.). For T ∈ L(H1, H2)n and z ∈ Cn denote zT := n k=1 zkTk. Then the transfer function of α is θα(z) = zD + zC(IX − zA)−1 zB. Clearly, θα is analytic in some neighbourhood of z = 0 in Cn . Let Gk := Ak Bk Ck Dk ∈ L(X ⊕ U, X ⊕ Y), k = 1, . . . , n. We call α = (n; A, B, C, D; X, U, Y) a dissipative scattering nD system (see [5, 6]) if for any ζ ∈ Tn (the unit torus) ζG is a contractive operator, i.e.,
- 35. DISSIPATIVE SCATTERING ND REALIZATION 15 ζG ≤ 1. It is known [5] that the transfer function of a dissipative scatter- ing nD system α = (n; A, B, C, D; X, U, Y) belongs to the subclass B0 n(U, Y) of the class Bn(U, Y) of all analytic contractive L(U, Y)-valued functions on the open unit polydisk Dn , which is segregated by the condition of vanishing of its functions at z = 0. The question whether the converse is true was implicitly asked in [5] and still has not been answered. Thus, we pose the following problem. Problem: Either prove that an arbitrary θ ∈ B0 n(U, Y) can be realized as the transfer function of a dissipative scattering nD system of the form (1) with the input space U and the output space Y, or give an example of a function θ ∈ B0 n(U, Y) (for some n ∈ N, and some finite-dimensional or infinite-dimensional separable Hilbert spaces U, Y) that has no such a realization. 2 MOTIVATION AND HISTORY OF THE PROBLEM For n = 1 the theory of dissipative (or passive, in other terminology) scatter- ing linear systems is well developed (see, e.g., [2, 3]) and related to various problems of physics (in particular, scattering theory), stochastic processes, control theory, operator theory, and 1D complex analysis. It is well known (essentially, due to [8]) that the class of transfer functions of dissipative scat- tering 1D systems of the form (1) with the input space U and the output space Y coincides with B0 1(U, Y). Moreover, this class of transfer functions remains the same when one is restricted within the important special case of conservative scattering 1D systems, for which the system block matrix G is unitary, i.e., G∗ G = IX⊕U, GG∗ = IX⊕Y. Let us note that in the case n = 1 a system (1) can be rewritten in an equivalent form (without a unit delay in output signal y) that is the standard form of a linear system, then a transfer function does not necessarily vanish at z = 0, and the class of transfer functions turns into the Schur class S(U, Y) = B1(U, Y). The classes B0 1(U, Y) and B1(U, Y) are canonically isomorphic due to the relation B0 1(U, Y) = zB1(U, Y). In [1] an important subclass Sn(U, Y) in Bn(U, Y) was introduced. This subclass consists of analytic L(U, Y)-valued functions on Dn , say, θ(z) = t∈Zn + θtzt (here Zn + = {t ∈ Zn : tk ≥ 0, k = 1, . . . , n}, zt := n k=1 ztk k for z ∈ Dn , t ∈ Zn +) such that for any n-tuple T = (T1, . . . , Tn) of commuting contractions on some common separable Hilbert space H and any positive r 1 one has θ(rT) ≤ 1, where θ(rT) = t∈Zn + θt ⊗ (rT)t ∈ L(U ⊗ H, Y ⊗ H), and (rT)t := n k=1(rTk)tk . For n = 1 and n = 2 one has Sn(U, Y) = Bn(U, Y). However, for any n 2 and any non-zero spaces U and Y the class Sn(U, Y) is a proper subclass of Bn(U, Y). J. Agler in [1] constructed a representation of an arbitrary function from Sn(U, Y), which in a system-theoretical language was interpreted in [4] as follows: Sn(U, Y)
- 36. 16 PROBLEM 1.3 coincides with the class of transfer functions of nD systems of Roesser type with the input space U and the output space Y, and certain conservativity condition imposed. The analogous result is valid for conservative systems of the form (1). A system α = (n; A, B, C, D; X, U, Y) is called a conservative scattering nD system if for any ζ ∈ Tn the operator ζG is unitary. Clearly, a conservative scattering system is a special case of a dissipative one. By [5], the class of transfer functions of conservative scattering nD systems coincides with the subclass S0 n(U, Y) in Sn(U, Y), which is segregated from the latter by the condition of vanishing of its functions at z = 0. Since for n = 1 and n = 2 one has S0 n(U, Y) = B0 n(U, Y), this gives the whole class of transfer functions of dissipative scattering nD systems of the form (1), and the solution to the problem formulated above for these two cases. In [6] the dilation theory for nD systems of the form (1) was developed. It was proven that α = (n; A, B, C, D; X, U, Y) has a conservative dilation if and only if the corresponding linear function LG(z) := zG belongs to S0 n(X ⊕ U, X ⊕ Y). Systems that satisfy this criterion are called n-dissipative scattering ones. In the cases n = 1 and n = 2 the subclass of n-dissipative scattering systems coincides with the whole class of dissipative ones, and in the case n 2 this subclass is proper. Since transfer functions of a system and of its dilation coincide, the class of transfer functions of n-dissipative scattering systems with the input space U and the output space Y is S0 n(U, Y). According to [7], for any n 2 there exist p ∈ N, m ∈ N, operators Dk ∈ L(Cp ) and commuting contractions Tk ∈ L(Cm ), k = 1, . . . , n, such that max ζ∈Tn n k=1 zkDk = 1 n k=1 Tk ⊗ Dk. The system α = (n; 0, 0, 0, D; {0}, Cp , Cp ) is a dissipative scattering one, however not, n-dissipative. Its transfer function θα(z) = LG(z) = zD ∈ B0 n(Cp , Cp ) S0 n(Cp , Cp ). Since for functions in B0 n(U, Y)S0 n(U, Y) the realization technique elaborated in [1] and developed in [4] and [5] is not applicable, our problem is of current interest. BIBLIOGRAPHY [1] J. Agler, “On the representation of certain holomorphic functions de- fined on a polydisc,” Topics in Operator Theory: Ernst D. Hellinger Memorial Volume (L. de Branges, I. Gohberg, and J. Rovnyak, Eds.), Oper. Theory Adv. Appl. 48, pp. 47-66 (1990). [2] D. Z. Arov, “Passive linear steady-state dynamic systems,” Sibirsk. Math. Zh. 20 (2), 211-228 (1979), (Russian). [3] J. A. Ball and N. Cohen, “De Branges-Rovnyak operator models and systems theory: A survey,” Topics in Matrix and Operator Theory (H.
- 37. DISSIPATIVE SCATTERING ND REALIZATION 17 Bart, I. Gohberg, and M.A. Kaashoek, eds.), Oper. Theory Adv. Appl., 50, pp. 93-136 (1991). [4] J. A. Ball and T. Trent, “Unitary colligations, reproducing kernel hilbert spaces, and Nevanlinna-Pick interpolation in several variables,” J. Funct. Anal. 157, pp. 1-61 (1998). [5] D. S. Kalyuzhniy, “Multiparametric dissipative linear stationary dy- namical scattering systems: Discrete case,” J. Operator Theory, 43 (2), pp. 427-460 (2000). [6] D. S. Kalyuzhniy, “Multiparametric dissipative linear stationary dy- namical scattering systems: Discrete case, II: Existence of conservative dilations,” Integr. Eq. Oper. Th., 36 (1), pp. 107-120 (2000). [7] D. S. Kalyuzhniy, “On the von Neumann inequality for linear matrix functions of several variables,” Mat. Zametki 64 (2), pp. 218-223 (1998), (Russian); translated in Math. Notes 64 (2), pp. 186-189 (1998). [8] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Spaces, North Holland, Amsterdam, 1970.
- 38. Problem 1.4 Partial disturbance decoupling with stability J. C. Martı́nez-Garcı́a Programa de Investigación en Matemáticas Aplicadas y Computación Instituto Mexicano del Petróleo Eje Central Lázaro Cárdenas No. 152 Col San Bartolo Atepehuacan, 07730 México D.F., México martinez@ctrl.cinvestav.mx M. Malabre Institut de Recherche en Communications et Cybernétique de Nantes CNRS-(Ecole Centrale-Université-Ecole des Mines) de Nantes 1 rue de la Noë, F-44321 Nantes Cedex 03, France Michel.Malabre@irccyn.ec-nantes.fr V. Kučera Faculty of Electrical Engineering Czech Technical University in Prague Technicka 2, 16627 Prague 6, Czech Republic kucera@fel.vcut.cz 1 DESCRIPTION OF THE PROBLEM Consider a linear time-invariant system (A, B, C, E) described by: σx (t) = Ax (t) + Bu (t) + Ed (t) , z (t) = Cx (t) , (1) where σ denotes either the derivation or the shift operator, depending on the continuous-time or discrete-time context; x (t) ∈ X Rn denotes the state; u (t) ∈ U Rm denotes the control input; z (t) ∈ Z Rm denotes the output, and d (t) ∈ D Rp denotes the disturbance. A : X → X, B : U → X, C : X → Z, and E : D → X denote linear maps represented by real constant matrices.
- 39. PARTIAL DISTURBANCE DECOUPLING WITH STABILITY 19 Let a system (A, B, C, E) and an integer k ≥ 1 be given. Find necessary and sufficient conditions for the existence of a static state feedback control law u (t) = Fx (t)+Gd (t) , where F : X → U and G : D → U are linear maps such as zeroing the first k Markov parameters of Tzd, the transfer function between the disturbance and the controlled output, while insuring internal stability, i.e.: • C (A + BF) l (BG + E) ≡ 0, for i ∈ {0, 1, . . . , k − 1}, and • σ (A + BF) ⊆ Cg, where σ (A + BF) stands for the spectrum of A + BF and Cg stands for the (good) stable part of the complex plane, e.g., the open left-half complex plane (continuous-time case) or the open unit disk (discrete- time case) 2 MOTIVATION The literature contains a lot of contributions related to disturbance rejection or attenuation. The early attempts were devoted to canceling the effect of the disturbance on the controlled output, i.e., insuring Tzd ≡ 0. This problem is usually referred to as the disturbance decoupling problem with internal stability, noted as DDPS (see [11], [1]). The solvability conditions for DDPS can be expressed as matching of infinite and unstable (invariant) zeros of certain systems (see, for instance, [8]), namely those of (A, B, C), i.e., (1) with d(t) ≡ 0, and those of (A, B E , C), i.e., (1) with d(t) considered as a control input. However, the rigid solvability conditions for DDPS are hardly met in practical cases. This is why alternative design procedures have been considered, such as almost disturbance decoupling (see [10]) and optimal disturbance attenuation, i.e., minimization of a norm of Tzd (see, for instance, [12]). The partial version of the problem, as defined in Section 1, offers another al- ternative from the rigid design of DDPS. The partial disturbance decoupling problem (PDDP) amounts to zeroing the first, say k, Markov parameters of Tzd. It was initially introduced in [2] and later revisited in [5], without sta- bility, [6, 7] with dynamic state feedback and stability, [4] with static state feedback and stability (sufficient solvability conditions for the single-input single-output case), [3] with dynamic measurement feedback, stability, and H∞-norm bound. When no stability constraint is imposed, solvability con- ditions of PDDP involve only a subset of the infinite structure of (A, B, C) and (A, B E , C), namely the orders which are less than or equal to k − 1 (see details in [5]). For PDDPS (i.e., PDDP with internal stability), the role played by the finite invariant zeros must be clarified to obtain the necessary and sufficient conditions that we are looking for, and solve the open problem.
- 40. 20 PROBLEM 1.4 Several extensions of this problem are also important: • solve PDDPS while reducing the H∞-norm of Tzd; • consider static measurement feedback in place of static state feedback. BIBLIOGRAPHY [1] G. Basile and G. Marro, Controlled and Conditioned Invariants in Linear System Theory, Prentice-Hall, 1992. [2] E. Emre and L. M. Silverman, “Partial model matching of linear sys- tems,”IEEE Trans. Automat. Contr., vol. AC-25, no. 2, pp. 280-281, 1980. [3] V. Eldem, H. Özbay, H. Selbuz, and K. Özcaldiran, “Partial disturbance rejection with internal stability and H∞ norm bound, ”SIAM Journal on Control and Optimization, vol. 36 , no. 1 , pp. 180-192, 1998. [4] F. N. Koumboulis and V. Kučera, “Partial model matching via static feedback (The multivariable case),”IEEE Trans. Automat. Contr., vol. AC-44, no. 2, pp. 386-392, 1999. [5] M. Malabre and J. C. Martı́nez-Garcı́a, “The partial disturbance re- jection or partial model matching: Geometric and structural solutions, ”IEEE Trans. Automat. Contr., vol. AC-40, no. 2, pp. 356-360, 1995. [6] V. Kučera, J. C. Martı́nez-Garcı́a, and M. Malabre, “Partial model matching: Parametrization of solutions, ” Automatica, vol. 33, no. 5, pp. 975-977, 1997. [7] J. C. Martı́nez-Garcı́a, M. Malabre, and V. Kučera, “The partial model matching problem with stability,”Systems and Control Letters, no. 24, pp. 61-74, 1994. [8] J. C. Martı́nez-Garcı́a, M. Malabre, J.-M. Dion, and C. Commault, “Con- densed structural solutions to the disturbance rejection and decoupling problems with stability,”International Journal of Control, vol. 72, No. 15, pp. 1392-1401, 1999. [9] A. Saberi, P. Sannuti, A. A. Stoorvogel, and B. M. Chen, H2 Optimal Control, Prentice-Hall, 1995. [10] J. C. Willems, “Almost invariant subspaces: An approach to high gain feedback design - part I: Almost controlled invariant subspaces,”IEEE Trans. Automat. Contr., vol. AC-26, no.1, pp. 235-252, 1981. [11] M. M. Wonham, Linear Multivariable Control: A Geometric Approach, 3rd ed., Springer Verlag, New York, 1985.
- 41. PARTIAL DISTURBANCE DECOUPLING WITH STABILITY 21 [12] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Upper Saddle River, NJ: Prentice-Hall, Inc., Simon Schuster, 1995.
- 42. Problem 1.5 Is Monopoli’s model reference adaptive controller correct? A. S. Morse1 Center for Computational Vision and Control Department of Electrical Engineering Yale University, New Haven, CT 06520 USA 1 INTRODUCTION In 1974 R. V. Monopoli published a paper [1] in which he posed the now classical model reference adaptive control problem, proposed a solution and presented arguments intended to establish the solution’s correctness. Sub- sequent research [2] revealed a flaw in his proof, which placed in doubt the correctness of the solution he proposed. Although provably correct solutions to the model reference adaptive control problem now exist (see [3] and the references therein), the problem of deciding whether or not Monopoli’s orig- inal proposed solution is in fact correct remains unsolved. The aim of this note is to review the formulation of the classical model reference adaptive control problem, to describe Monopoli’s proposed solution, and to outline what’s known at present about its correctness. 2 THE CLASSICAL MODEL REFERENCE ADAPTIVE CONTROL PROBLEM The classical model reference adaptive control problem is to develop a dy- namical controller capable of causing the output y of an imprecisely modeled SISO process P to approach and track the output yref of a prespecified ref- erence model Mref with input r. The underlying assumption is that the process model is known only to the extent that it is one of the members of a pre-specified class M. In the classical problem M is taken to be the set of 1This research was supported by DARPA under its SEC program and by the NSF.
- 43. IS MONOPOLI’S MODEL REFERENCE ADAPTIVE CONTROLLER CORRECT? 23 all SISO controllable, observable linear systems with strictly proper transfer functions of the form g β(s) α(s) where g is a nonzero constant called the high frequency gain and α(s) and β(s) are monic, coprime polynomials. All g have the same sign and each transfer function is minimum phase (i.e., each β(s) is stable). All transfer functions are required to have the same relative degree n̄ (i.e., deg α(s) − deg β(s) = n̄.) and each must have a McMillan degree not exceeding some prespecified integer n (i.e., deg α(s) ≤ n). In the sequel we are going to discuss a simplified version of the problem in which all g = 1 and the reference model transfer function is of the form 1 (s+λ)n̄ where λ is a positive number. Thus Mref is a system of the form ẏref = −λyref + c̄xref + ¯ dr ẋref = Āxref + b̄r (1) where {Ā, b̄, c̄, ¯ d} is a controllable, observable realization of 1 (s+λ)(n̄−1) . 3 MONOPOLI’S PROPOSED SOLUTION Monopoli’s proposed solution is based on a special representation of P that involves picking any n-dimensional, single-input, controllable pair (A, b) with A stable. It is possible to prove [1, 4] that the assumption that the process P admits a model in M, implies the existence of a vector p∗ ∈ IR2n and initial conditions z(0) and x̄(0), such that u and y exactly satisfy ż = A 0 0 A z + b 0 y + 0 b u ˙ x̄ = Āx̄ + b̄(u − z p∗ ) ẏ = −λy + c̄x̄ + ¯ d(u − z p∗ ) Monopoli combined this model with that of Mref to obtain the direct control model reference parameterization ż = A 0 0 A z + b 0 y + 0 b u (2) ẋ = Āx + b̄(u − z p∗ − r) (3) ėT = −λeT + c̄x + ¯ d(u − z p∗ − r) (4) Here eT is the tracking error eT ∆ = y − yref (5) and x ∆ = x̄ − xref . Note that it is possible to generate an asymptotically correct estimate z of z using a copy of (2) with z replacing z. To keep the exposition simple, we are going to ignore the exponentially decaying estimation error z − z and assume that z can be measured directly. To solve the MRAC problem, Monopoli proposed a control law of the form u = z p + r (6)
- 44. 24 PROBLEM 1.5 where p is a suitably defined estimate of p∗ . Motivation for this particular choice stems from the fact that if one knew p∗ and were thus able to use the control u = z p∗ + r instead of (6), then this would cause eT to tend to zero exponentially fast and tracking would therefore be achieved. Monopoli proposed to generate p using two subsystems that we will refer to here as a “multi-estimator” and a “tuner” respectively. A multi-estimator E( p) is a parameter-varying linear system with parameter p, whose inputs are u, y, and r and whose output is an estimate e of eT that would be asymptotically correct were p held fixed at p∗ . It turns out that there are two different but very similar types of multi-estimators that have the requisite properties. While Monopoli focused on just one, we will describe both since each is relevant to the present discussion. Both multi-estimators contain (2) as a subsystem. Version 1 There are two versions of the adaptive controller that are relevant to the problem at hand. In this section we describe the multi-estimator and tuner that, together with reference model (1) and control law (6), comprise the first version. Multi-Estimator 1 The form of the first multi-estimator E1( p) is suggested by the readily veri- fiable fact that if H1 and w1 are n̄ × 2n and n̄ × 1 signal matrices generated by the equations Ḣ1 = ĀH1 + b̄z and ẇ1 = Āw1 + b̄(u − r) (7) respectively, then w1 − H1p∗ is a solution to (3). In other words x = w1 − H1p∗ + where is an initial condition dependent time function decaying to zero as fast as eĀt . Again, for simplicity, we shall ignore . This means that (4) can be re-written as ėT = −λeT − (c̄H1 + ¯ dz )p∗ + c̄w1 + ¯ d(u − r) Thus a natural way to generate an estimate e1 of eT is by means of the equation ˙ e1 = −λ e1 − (c̄H1 + ¯ dz ) p + c̄w1 + ¯ d(u − r) (8) From this it clearly follows that the multi-estimator E1( p) defined by (2), (7) and (8) has the required property of delivering an asymptotically correct estimate e1 of eT if p is fixed at p∗ .
- 45. IS MONOPOLI’S MODEL REFERENCE ADAPTIVE CONTROLLER CORRECT? 25 Tuner 1 From (8) and the differential equation for eT directly above it, it can be seen that the estimation error2 e1 ∆ = e1 − eT (9) satisfies the error equation ė1 = −λe1 + φ 1( p − p∗ ) (10) where φ 1 = −(c̄H1 + ¯ dz ) (11) Prompted by this, Monopoli proposed to tune p1 using the pseudo-gradient tuner ˙ p1 = −φ1e1 (12) The motivation for considering this particular tuning law will become clear shortly, if it is not already. What is known about Version 1? The overall model reference adaptive controller proposed by Monopoli thus consists of the reference model (1), the control law (6), the multi-estimator (2), (7), (8), the output estimation error (9) and the tuner (11), (12). The open problem is to prove that this controller either solves the model reference adaptive control problem or that it does not. Much is known that is relevant to the problem. In the first place, note that (1), (2) together with (5) - (11) define a parameter varying linear system Σ1( p) with input r, state (yref , xref , z, H1, w1, e1, e1) and output e1. The consequence of the assumption that every system in M is minimum phase is that Σ1( p) is detectable through e1 for every fixed value of p [5]. Meanwhile the form of (10) enables one to show by direct calculation, that the rate of change of the partial Lyapunov function V ∆ = e2 1 +|| p−p∗ ||2 along a solution to (12) and the equations defining Σ1( p), satisfies V̇ = −2λe2 1 ≤ 0 (13) From this it is evident that V is a bounded monotone nonincreasing function and consequently that e1 and p are bounded wherever they exist. Using and the fact that Σ1( p) is a linear parameter-varying system, it can be concluded that solutions exist globally and that e1 and p are bounded on [0, ∞). By integrating (13) it can also be concluded that e1 has a finite L2 [0, ∞)-norm and that ||e1||2 +|| p−p∗ ||2 tends to a finite limit as t → ∞. Were it possible to deduce from these properties that p tended to a limit p̄, then it would possible to establish correctness of the overall adaptive controller using the detectability of Σ1(p̄). 2Monopoli called e1 an augmented error.
- 46. 26 PROBLEM 1.5 There are two very special cases for which correctness has been established. The first is when the process models in M all have relative degree 1; that is when n̄ = 1. See the references cited in [3] for more on this special case. The second special case is when p∗ is taken to be of the form q∗ k where k is a known vector and q∗ is a scalar; in this case p ∆ = qk where q is a scalar parameter tuned by the equation ˙ q = −k φ1e1 [6]. Version 2 In the sequel we describe the multi-estimator and tuner that, together with reference model (1) and control law (6), comprise the second version of them adaptive controller relevant to the problem at hand. Multi-Estimator 2 The second multi-estimator E2( p), which is relevant to the problem under consideration, is similar to E1( p) but has the slight advantage of leading to a tuner that is somewhat easier to analyze. To describe E2( p), we need first to define matrices Ā2 ∆ = Ā 0 c̄ −λ and b̄2 ∆ = b̄ ¯ d The form of E2( p) is motivated by the readily verifiable fact that if H2 and w2 are (n̄+1)×2n and (n̄+1)×1 signal matrices generated by the equations Ḣ2 = Ā2H2 + b̄2z and ẇ2 = Ā2w2 + b̄2(u − r) (14) then w2 − H2p∗ is a solution to (3) - (4). In other words, x eT = w2−H2p∗ + where is an initial condition dependent time function decaying to zero as fast as eĀ2t . Again, for simplicity, we shall ignore . This means that eT = c̄2w2 − c̄2H2p∗ where c̄2 = 0 · · · 0 1 . Thus, in this case, a natural way to generate an estimate e2 of eT is by means of the equation e2 = c̄2w2 − c̄2H2 p (15) It is clear that the multi-estimator E2( p) defined by (2), (14) and (15) has the required property of delivering an asymptotically correct estimate e2 of eT if p is fixed at p∗ . Tuner 2 Note that in this case the estimation error e2 ∆ = e2 − eT (16)
- 47. IS MONOPOLI’S MODEL REFERENCE ADAPTIVE CONTROLLER CORRECT? 27 satisfies the error equation e2 = φ 2( p2 − p∗ ) (17) where φ 2 = −c̄2H2 (18) Equation (17) suggests that one consider a pseudo-gradient tuner of the form ˙ p = −φ2e2 (19) What is Known about Version 2? The overall model reference adaptive controller in this case thus consists of the reference model (1), the control law (6), the multi-estimator (2), (14), (15), the output estimation error (16) and the tuner (18), (19). The open problem is here to prove that this version of the controller either solves the model reference adaptive control problem or that it does not. Much is known about the problem. In the first place, (1), (2) together with (5), (6) (14) - (18) define a parameter varying linear system Σ2( p) with input r, state (yref , xref , z, H2, w2) and output e2. The consequence of the assumption that every system in M is minimum phase is that this Σ2( p) is detectable through e2 for every fixed value of p [5]. Meanwhile the form of (17) enables one to show by direct calculation that the rate of change of the partial Lyapunov function V ∆ = || p − p∗ ||2 along a solution to (19) and the equations defining Σ2( p), satisfies V̇ = −2λe2 2 ≤ 0 (20) It is evident that V is a bounded monotone nonincreasing function and consequently that p is bounded wherever they exist. From this and the fact that Σ2( p) is a linear parameter-varying system, it can be concluded that solutions exist globally and that p is bounded on [0, ∞). By integrating (20) it can also be concluded that e2 has a finite L2 [0, ∞)-norm and that || p − p∗ ||2 tends to a finite limit as t → ∞. Were it possible to deduce from these properties that p tended to a limit p̄ , then it would to establish correctness using the detectability of Σ2(p̄). There is one very special cases for which correctness has been established [6]. This is when p∗ is taken to be of the form q∗ k where k is a known vector and q∗ is a scalar; in this case p ∆ = qk where q is a scalar parameter tuned by the equation ˙ q = −k φ2e2. The underlying reason why things go through is because in this special case, the fact that || p − p∗ ||2 and consequently || q − q∗ || tend to a finite limits, means that q tends to a finite limit as well. 4 THE ESSENCE OF THE PROBLEM In this section we transcribe a stripped down version of the problem that retains all the essential feature that need to be overcome in order to decide
- 48. 28 PROBLEM 1.5 whether or not Monopoli’s controller is correct. We do this only for version 2 of the problem and only for the case when r = 0 and n̄ = 1. Thus, in this case, we can take Ā2 = −λ and b̄2 = 1. Assuming the reference model is initialized at 0, dropping the subscript 2 throughout, and writing φ for −H, the system to be analyzed reduces to ż = A 0 0 A z + b 0 (w + φ p∗ ) + 0 b p z (21) φ̇ = −λφ − z (22) ẇ = −λw + p z (23) e = φ ( p − p∗ ) (24) ˙ p = −φe (25) To recap, p∗ is unknown and constant but is such that the linear parameter- varying system Σ( p) defined by (21) to (24) is detectable through e for each fixed value of p. Solutions to the system (21) - (25) exist globally. The parameter vector p and integral square of e are bounded on [0, ∞) and || p − p∗ || tends to a finite limit as t → ∞. The open problem here is to show for every initialization of (21)-(25), that the state of Σ( p) tends to 0 or that it does not. BIBLIOGRAPHY [1] R. V. Monopoli, “Model reference adaptive control with an augmented error,” IEEE Transactions on Automatic Control, pp. 474–484, October 1974. [2] A. Feuer, B. R. Barmish, and A. S. Morse, “An unstable system as- sociated with model reference adaptive control,” IEEE Transactions on Automatic Control, 23:499–500, 1978. [3] A. S. Morse, “Overcoming the obstacle of high relative degree,” European Journal of Control, 2(1):29–35, 1996. [4] K. J. Åström and B. Wittenmark, “On self-tuning regulators,” Automat- ica, 9:185–199, 1973. [5] A. S. Morse, “Towards a unified theory of parameter adaptive control - Part 2: Certainty equivalence and implicit tuning,” IEEE Transactions on Automatic Control, 37(1):15–29, January 1992. [6] A. Feuer, Adaptive Control of Single Input Single Output Linear Systems, Ph.D. thesis, Yale University, 1978.
- 49. Problem 1.6 Model reduction of delay systems Jonathan R. Partington School of Mathematics University of Leeds Leeds, LS2 9JT U.K. J.R.Partington@leeds.ac.uk 1 DESCRIPTION OF THE PROBLEM Our concern here is with stable single input single output delay systems, and we shall restrict to the case when the system has a transfer function of the form G(s) = e−sT R(s), with T 0 and R rational, stable, and strictly proper, thus bounded and analytic on the right half plane C+. It is a fundamental problem in robust control design to approximate such systems by finite-dimensional systems. Thus, for a fixed natural number n, we wish to find a rational approximant Gn(s) of degree at most n in order to make small the approximation error G − Gn, where . denotes an appropriate norm. See [9] for some recent work on this subject. Commonly used norms on a linear time-invariant system with impulse re- sponse g ∈ L1 (0, ∞) and transfer function G ∈ H∞ (C+) are the H∞ norm G∞ = supRe s0 |G(s)|, the Lp norms gp = ∞ 0 |g(t)|p dt 1/p (1 ≤ p ∞), and the Hankel norm Γ, where Γ : L2 (0, ∞) → L2 (0, ∞) is the Hankel operator defined by (Γu)(t) = ∞ 0 g(t + τ)u(τ) dτ. These norms are related by Γ ≤ G∞ ≤ g1 ≤ 2nΓ, where the last inequality holds for systems of degree at most n. Two particular approximation techniques for finite-dimensional systems are well-established in the literature [14], and they can also be used for some infinite-dimensional systems [5]:
- 50. 30 PROBLEM 1.6 • Truncated balanced realizations, or, equivalently, output normal real- izations [11, 13, 5]; • Optimal Hankel-norm approximants [1, 4, 5]. As we explain in the next section, these techniques are known to produce H∞ -convergent sequences of approximants for many classes of delay systems (systems of nuclear type). We are thus led to pose the following question: Do the sequences of reduced order models produced by truncated balanced realizations and optimal Hankel-norm approximations converge for all stable delay systems? 2 MOTIVATION AND HISTORY OF THE PROBLEM Balanced realizations were introduced in [11], and many properties of trun- cations of such realizations were given in [13]. An H∞ error bound for the reduced-order system produced by truncating a balanced realization was given for finite-dimensional systems in [3, 4], and extended to infinite-di- mensional systems in [5]. This commonly used bound is expressed in terms of the sequence (σk)∞ k=1 of singular values of the Hankel operator Γ corre- sponding to the original system G; in our case Γ is compact, and so σk → 0. Provided that g ∈ L1 ∩L2 and Γ is nuclear (i.e., ∞ k=1 σk ∞) with distinct singular values, then the inequality G − Gb n∞ ≤ 2(σn+1 + σn+2 + . . .) holds for the degree-n balanced truncation Gb n of G. The elementary lower bound G − Gn ≥ σn+1 holds for any degree-n approximation to G. Another numerically convenient approximation method is the optimal Han- kel-norm technique [1, 4, 5], which involves finding a best rank-n Hankel approximation ΓH n to Γ, in the Hankel norm, so that Γ − ΓH n = σn+1. In this case the bound G − GH n − D0∞ ≤ σn+1 + σn+2 + . . . is available for the corresponding transfer function GH n with a suitable con- stant D0. Again, we require the nuclearity of Γ for this to be meaningful. 3 AVAILABLE RESULTS In the case of a delay system G(s) = e−sT R(s) as specified above, it is known that the Hankel singular values σk are asymptotic to A T πk r , where r is
- 51. MODEL REDUCTION OF DELAY SYSTEMS 31 the relative degree of R and |sr R(s)| tends to the finite nonzero limit A as |s| → ∞. Hence Γ is nuclear if and only if the relative degree of R is at least 2. (Equivalently, if and only if g is continuous.) We refer to [6, 7] for these and more precise results. Even for a very simple non-nuclear system such as G(s) = e−sT s + 1, for which kσk → T/π, no theoretical upper bound is known for the H∞ errors in the rational approximants produced by truncated balanced realizations and optimal Hankel-norm approximation, although numerical evidence suggests that they should still tend to zero. A related question is to find the best error bounds in L1 approximation of a delay system. For example, a smoothing technique gives an L1 approx- imation error O ln n n for systems of relative degree r = 1 (see [8]), and it is possible that the optimal Hankel norm might yield a similar rate of convergence. (A lower bound of C/n for some constant C 0 follows easily from the above discussion.) One approach that may be useful in these analyses is to exploit Bonsall’s theorem that a Hankel integral operator Γ is bounded if and only if it is uniformly bounded on the set of all normalized L2 functions whose Laplace transforms are rational of degree one [2, 12]. An explicit constant in Bon- sall’s theorem is not known, and would be of great interest in its own right. Another approach which may be relevant is that of Megretski [10], who introduces maximal real part norms. Their interest stems from the inequality G∞ ≥ Re G∞ ≥ Γ/2. BIBLIOGRAPHY [1] V. M. Adamjan, D. Z. Arov, and M. G. Kreı̆n, “Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur–Takagi problem,” Math. USSR Sbornik, 15:31–73, 1971. [2] F. F. Bonsall, “Boundedness of Hankel matrices”, J. London Math. Soc. (2), 29(2):289–300, 1984. [3] D. Enns, Model Reduction for Control System Design, Ph.D. disserta- tion, Stanford University, 1984. [4] K. Glover, “All optimal Hankel-norm approximations of linear mul- tivariable systems and their L∞ -error bounds, Internat. J. Control, 39(6):1115–1193, 1984.
- 52. 32 PROBLEM 1.6 [5] K. Glover, R. F. Curtain, and J. R. Partington, “Realisation and ap- proximation of linear infinite-dimensional systems with error bounds,” SIAM J. Control Optim., 26(4):863–898, 1988. [6] K. Glover, J. Lam, and J. R. Partington, “Rational approximation of a class of infinite-dimensional systems. I. Singular values of Hankel oper- ators,” Math. Control Signals Systems, 3(4):325–344, 1990. [7] K. Glover, J. Lam, and J. R. Partington,“Rational approximation of a class of infinite-dimensional systems. II. Optimal convergence rates of L∞ approximants,” Math. Control Signals Systems, 4(3):233–246, 1991. [8] K. Glover and J. R. Partington, “Bounds on the achievable accuracy in model reduction,” In: Modelling, Robustness and Sensitivity Reduction in Control Systems (Groningen, 1986), pp. 95–118. Springer, Berlin, 1987. [9] P. M. Mäkilä and J. R. Partington, “Shift operator induced approxi- mations of delay systems,” SIAM J. Control Optim., 37(6):1897–1912, 1999. [10] A. Megretski, “Model order reduction using maximal real part norms,” Presented at CDC 2000, Sydney, 2000. http://web.mit.edu/ameg/www/images/lund.ps. [11] B. C. Moore, “Principal component analysis in linear systems: control- lability, observability, and model reduction,” IEEE Trans. Automat. Control, 26(1):17–32, 1981. [12] J. R. Partington and G. Weiss, “Admissible observation operators for the right-shift semigroup,” Math. Control Signals Systems, 13(3):179– 192, 2000. [13] L. Pernebo and L. M. Silverman, “Model reduction via balanced state space representations,” IEEE Trans. Automat. Control, 27(2):382–387, 1982. [14] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Upper Saddle River, NJ: Prentice Hall1996.
- 53. Problem 1.7 Schur extremal problems Lev Sakhnovich Courant Institute of Mathematical Science New York, NY 11223 USA Lev.Sakhnovich@verizon.net 1 DESCRIPTION OF THE PROBLEM In this paper we consider the well-known Schur problem the solution of which satisfy in addition the extremal condition w (z)w(z) ≤ ρ2 min, |z| 1, (1) where w(z) and ρmin are m × m matrices and ρmin 0. Here the matrix ρmin is defined by a certain minimal-rank condition (see Definition 1). We remark that the extremal Schur problem is a particular case. The general case is considered in book [1] and paper [2]. Our approach to the extremal problems does not coincide with the superoptimal approach [3],[4]. In paper [2] we compare our approach to the extremal problems with the superoptimal approach. Interpolation has found great applications in control theory [5],[6]. Schur Extremal Problem: The m×m matrices a0, a1, ..., an are given. Describe the set of m×m matrix functions w(z) holomorphic in the circle |z| 1 and satisfying the relation w(z) = a0 + a1z + ... + anzn + ... (2) and inequality (1.1). A necessary condition of the solvability of the Schur extremal problem is the inequality R2 min − S ≥ 0, (3) where the (n + 1)m×(n + 1)m matrices S and Rmin are defined by the relations S = CnC n, Rmin = diag[ρmin, ρmin, ..., ρmin], (4)
- 54. 34 PROBLEM 1.7 Cn = a0 0 ... 0 a1 a0 ... 0 ... ... ... ... an an−1 ... a0 . (5) Definition 1: We shall call the matrix ρ = ρmin 0 minimal if the following two requirements are fulfilled: 1. The inequality R2 min − S ≥ 0 (6) holds. 2. If the m×m matrix ρ 0 is such that R2 − S ≥ 0, (7) then rank(R2 min − S) ≤ rank(R2 − S), (8) where R = diag[ρ, ρ, ..., ρ]. Remark 1: The existence of ρmin follows directly from definition 1. Question 1: Is ρmin unique? Remark 2: If m = 1 then ρmin is unique and ρ2 min = λmax, where λmax is the largest eigenvalue of the matrix S. Remark 3: Under some assumptions the uniqueness of ρmin is proved in the case m 1, n = 1 (see [2],[7]). If ρmin is known then the corresponding wmin(ξ) is a rational matrix func- tion. This generalizes the well-known fact for the scalar case (see [7]). Question 2: How to find ρmin? In order to describe some results in this direction we write the matrix S = CnC n in the following block form S11 S12 S21 S22 , (9) where S22 is an m×m matrix. Proposition 1: [1] If ρ = q 0 satisfies inequality (1.7) and the relation q2 = S22 + S 12(Q2 − S11)−1 S12, (10) where Q = diag[q, q, ..., q], then ρmin = q. We shall apply the method of successive approximation when studying equa- tion (1.10). We put q2 0 = S22, q2 k+1 = S22 +S 12(Q2 k − S11)−1 S12, where k≥0, Qk = diag[qk, qk, ..., qk]. We suppose that Q2 0 − S11 0. (11) Theorem 1: [1] The sequence q2 0, q2 2, q2 4, ... monotonically increases and has the limit m1. The sequence q2 1, q2 3, q2 5, ... monotonically decreases and has the limit m2. The inequality m1≤m2 holds. If m1 = m2 then ρ2 min = q2 . Question 3: Suppose relation (1.11) holds. Is there a case when m1=m2? The answer is “no” if n = 1 (see [2],[8]). Remark 4: In book [1] we give an example in which ρmin is constructed in explicit form.
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- 57. CHAPTER V Spanish Rule Returns When they reoccupied Florida in 1784, the Spaniards had changed but little during their twenty-year absence from the scene. With their return St. Augustine reverted to its former status as an isolated military post, heavily dependent upon outside sources for its supplies and financial support. Agriculture was neglected and brush soon covered the plantation fields, which the English and their slaves had cleared. Indians again roamed at will through the countryside. On the heels of the departing English they burned Bella Vista, the beautiful country estate of Lieutenant-Governor Moultrie, located a few miles south of St. Augustine in the community now bearing his name. The population of the capital, which had overflowed into new districts just before the English left, shrank to a fraction of its former size. Only a few score English remained to take the required oath of allegiance to the Spanish Crown. A relatively small number of St. Augustine’s former Spanish residents, or Floridanos, uprooted in 1763, returned from Cuba to claim their former homes. The Minorcan group, including a few Greeks and Italians, made up the major portion of St. Augustine’s civilian inhabitants. Vacant houses stared blankly along the narrow streets. Some with flat roofs and outside kitchens were relics of the first Spanish period. Others had been remodelled after the English taste with glass window panes, gabled roofs, and chimneys. St. Peter’s Church, in which the English had worshipped, remained unoccupied and soon became a ruin.
- 58. 53 Although a Spanish possession, St. Augustine acquired from time to time interesting residents of other nationalities. Juan McQueen, a close friend of George Washington, Thomas Jefferson, and Lafayette, came to the city in 1791 to escape embarrassing debts, and held official positions under the Spanish regime until death closed his colorful career in 1807. John Leslie, the famous English trader, also lived here after the Revolutionary War. The firm of Panton, Leslie and Company enjoyed a monopoly in trading with the Indians of Florida, and supplied St. Augustine with many of its needs on liberal credit. Ruins of the Fish mansion on Anastasia, or Fish’s Island, from a pencil sketch made by the Rev. Henry J. Morton in 1867. Philip Fatio, a Swiss, owned a large plantation on the St. Johns River in a section now known as Switzerland. He maintained a store and residence at St. Augustine, and had other extensive land holdings. Among the Minorcan group was an Estevan Benet, one of whose descendants was Stephen Vincent Benet, the noted writer. Jesse Fish lived across the bay on what is now called Fish’s Island with his many slaves and famous orange grove, from which he shipped fruit and juice to England. He was sent to St. Augustine as a youth by a trading firm during the first Spanish period, won the
- 59. 54 confidence of the Spaniards, and remained as custodian of some of their property through the English regime. The old patriarch still occupied his coquina mansion across the bay when the Spaniards returned. Father Pedro Camps, Padre of the Minorcan group, followed them to St. Augustine from New Smyrna in 1777, and continued as their beloved spiritual leader until his death in 1790. Also prominent in the city’s religious life was Father Michael O’Reilly, an Irish priest, who came with Governor Zéspedes in 1784 and remained active until removed by death in 1812. Life in St. Augustine followed a distinctive pattern, due to its isolation and lack of frequent communication with other cities. It was Spanish in language, dress, customs, and for the most part in architecture and population. Some of its officials and planters owned slaves, fine horses, and lived comfortably if not elaborately. They enjoyed leisure time for gambling, cock fighting, and to lounge through the long summers in a cool patio or at a congenial tavern. The populace was characteristically lazy and did little more than necessary to keep body and soul together. As in other Spanish colonies, the siesta, or after- dinner nap, was routine. During the mid-day heat streets were deserted and nothing stirred as if under the spell of an enchanter’s wand.
- 60. Old print of Plaza showing Cathedral and Constitution monument. One of the chief additions made to the city during its second Spanish period was the construction of a graceful new Parish Church. The building was begun in 1791, dedicated in 1797, and later consecrated as a Cathedral. Damaged by fire in 1887, it was restored the following year with the addition of the present clock tower. The Spaniards also commenced a new Treasury building, which was never completed due to lack of funds. Its mute walls remained standing until after the Civil War. For a time the Spanish government offered grants of land in East Florida on liberal terms to attract settlers. Hardy pioneers from the adjacent South poured in, who secretly wanted to overthrow Spanish rule. Fearing this influence, Spain closed the territory to further settlement by Americans in 1804. The story of East Florida and its capital from 1800 on is one of increasing difficulties, caused by the course of events in Europe and friction with neighboring southern states. Spain’s wealth and power were rapidly declining. One after another her American colonies
- 61. 55 sought and won their independence. In the southeastern United States sentiment for the possession of Florida was fanned by Indian raids and the loss of slaves across the border, which Spanish officials seemed to do little to control. In 1812, to assuage popular clamor, the Spanish Cortés adopted a more liberal constitution, and decreed that monuments be erected to commemorate it. At St. Augustine a coquina shaft was raised that still graces its Plaza, but scarcely had it been dedicated when the constitution was revoked, and the monuments were ordered dismantled. Here only the tablets were removed and later replaced. The North Florida Republic When the war of 1812 broke out between England and the United States, it was feared that England, then allied with Spain, might seize the Floridas as a base for military operations. The Congress authorized President Madison to appoint two agents, who were to endeavor to secure the temporary cession of East and West Florida to the United States. In the event this failed, steps were to be taken to forcibly occupy the provinces, should England threaten to seize them. President Madison appointed old General Matthews as his agent to East Florida. He was a Revolutionary War veteran and a former governor of Georgia. With promises of liberal grants of land, Matthews encouraged the planters along the northern borders of East Florida to set up an independent republic. The plan was to then turn over the territory it occupied to the United States. After seizing Fernandina these Patriots, as they were termed, advanced on St. Augustine with a small detachment of regular troops, occupied Fort Mosa on its northern outskirts, and called upon the Spanish governor to surrender. He sent a gunboat up the river to dislodge them, but they continued to camp in the vicinity for several months. St.
- 62. 56 Augustine was cut off from supplies and the surrounding country plundered by Indians and outlaws. The unfinished Spanish Treasury on St. George Street, from a sketch made in 1867. Present Old Spanish Treasury, shown in the background, still stands. Loud Spanish and English protests caused President Madison to recall his agents and repudiate their actions.
- 63. Streets such as this once were gay with costumed revelers. A Bit of Spain In a Narrative of a Voyage to the Spanish Main, published in 1819, an Englishman gives the following description of St. Augustine’s residents during this period: “The women are deservedly celebrated for their charm, their lovely black eyes have a vast deal of expression, their complexions a clear brunette; much attention is paid to the arrangement of their hair; at Mass they are always well dressed in black silk basquinas with the little mantilla over their heads; the men in their military costumes.”
- 64. 57 The same traveler later returned to St. Augustine by land, and found the city in a gay mood despite its difficulties. “I had arrived at the season of general relaxation, on the eve of the Carnival, which is celebrated with much gaiety in all Catholic countries. Masks, dominoes, harlequins, punchinelloes, and a variety of grotesque disguises, on horseback, in carts, gigs, and on foot paraded the streets with guitars, violins, and other instruments; and in the evening the houses were opened to receive masks, and balls were given in every direction.” Ceded to the United States After the War of 1812 there was still friction between Spanish Florida and the United States. Bands of Indians and escaped slaves occupied choice lands of the Florida interior, fortified the navigable rivers, and made occasional raids across the border. The Spanish garrison was not large enough to control lawless elements. In 1817 Fernandina and Amelia Island were taken over by MacGregor, an English soldier of fortune, later occupied by the pirate Autry, and became a den of outlaws and smugglers. United States troops were sent to dislodge them and restore law and order. General Andrew Jackson led an expedition into north central and west Florida in 1818 to punish the Indians, and after destroying their strongholds occupied Pensacola. England and Spain vehemently protested these violations of Spanish territory. Negotiations for the purchase of Florida were reopened. During February of 1819 a treaty was concluded whereby Spain finally ceded Florida to the United States, which appropriated up to five million dollars to pay the claims of Americans arising from the recent depredations. Spain ratified the treaty in 1820. On July 10, 1821, Colonel Robert Butler and a small detachment of United States troops received possession of East Florida and Castillo
- 65. 58 de San Marcos from José Coppinger, the last of the Spanish governors. After the Spanish flag was lowered, leaving the stars and stripes flying over the fortress, Spanish troops marched out between lines of American soldiers and they mutually saluted. The Spaniards then boarded American transports waiting to convey them to Cuba, one of the few remaining possessions of Spain’s great colonial empire in America. The Llambias House, a picturesque St. Augustine home dating back to the first Spanish period.
- 66. CHAPTER VI Under the United States St. Augustine was at last a part of the United States. Most of its Spanish residents bid the narrow streets farewell. The Minorcans, now firmly domiciled here, made up the major portion of the town’s population. Many by this time had risen to positions of influence in its affairs. Officials of the new regime found St. Augustine a rather dilapidated old town, devoid of progress and ambition. Due to the poverty that had marked the closing years of the second Spanish period, public and private buildings were badly run down, some almost in ruins. Soon after the change of flags, speculators and promoters flocked to the city, and were quartered in some of the deserted houses. In the fall of 1821 an epidemic of dreaded yellow fever carried off many of the newcomers. A new cemetery was opened up near the City Gates to receive the victims, a few of whom may have been of Huguenot descent. It became known as the Huguenot, or Protestant cemetery. In spite of its unkempt condition, St. Augustine possessed a certain mellow charm. At times the scent of orange blossoms hung heavy in the air and could be noticed by passing ships at sea. Along the narrow streets latticed gates led into cool courtyards and secluded gardens. There was no industry or commerce to disturb the serenity of the scene. St. Augustine’s shallow inlet, which preserved it from its enemies, also prevented it from becoming a place of bustling trade. Visitors Begin to Arrive
- 67. 59 Although difficult to reach by sea because of its treacherous bar, and by land over a road that was little more than a trail, a few adventurous travelers began to visit this quaint old city, which the United States had recently acquired. They were chiefly invalids and tubercular victims, for whom the mild winter climate was considered beneficial. Ralph Waldo Emerson, who was later to become the noted New England poet and philosopher, visited St. Augustine in 1827, at the age of 23, suffering from what he termed a “stricture of the chest.” During his ten weeks’ stay he recorded in his journal and letters his impressions of the city as he then saw it. “St. Augustine is the oldest town of Europeans in North America,” he observed, “full of ruins, chimneyless houses, lazy people, horse- keeping intolerably dear, and bad milk from swamp grass, as all their hay comes from the North.”
- 68. Napoleon Achille Murat, one of St. Augustine’s early visitors. But it restored his health and later he was inspired to comment: “The air and sky of this ancient, fortified, dilapidated sandbank of a town are delicious. It is a queer place. There are eleven or twelve hundred people and these are invalids, public officials, and Spaniards, or rather Minorcans.” While here Emerson met another distinguished visitor of the time, Prince Napoleon Achille Murat, son of the King of Naples, and
- 69. 60 nephew of the great Napoleon. Murat came to Florida in 1824, purchased an estate south of St. Augustine, and was a frequent visitor to the city, living here for a time during the Seminole War. He later settled on a plantation near Tallahassee. St. Augustine began to prosper in a small way from its increasing number of visitors and winter residents. The Freeze of 1835 The growing of oranges was an important industry in St. Augustine and its vicinity at this time. Many of its residents derived their principal income from the sale of the golden fruit, which was shipped by sloop to northern cities. The town was described by visitors as being virtually bowered in groves, and on each side of the Plaza were two rows of handsome orange trees, planted by Governor Grant during the English occupation. During February of 1835 a biting cold of extended duration swept down out of the northwest. At nearby Jacksonville the thermometer dropped to eight degrees, and ice formed on the St. Johns River. St. Augustine’s beautiful orange groves were killed to the ground, sweeping away the main source of livelihood for many of its people. Only the bare trunks and branches remained, making the city look bleak and desolate. Some of the trees sprouted from their damaged roots; others were planted, and in a few decades St. Augustine’s orange groves were again the subject of admiring comment on the part of visitors. But during the winter of 1894-95 another freeze destroyed them. The citrus industry moved farther south and was not again revived on a commercial scale in St. Augustine or its immediate vicinity.
- 70. 61 Osceola, colorful leader of the Seminoles. From a portrait by George Catlin, painted during the chief’s imprisonment at Fort Moultrie, S. C. The Seminole War The Seminole War followed closely on the heels of the disastrous freeze of 1835. Shortly after New Year’s day of 1836 St. Augustine
- 71. learned of the massacre of Major Dade and his command of 110 men. They were ambushed by Seminoles while enroute from Fort Brooke (Tampa) to Fort King (Ocala). On the same day, December 28, 1835, General Wiley Thompson, the Indian agent at Fort King, and another officer were killed. Soon plantations in the vicinity of St. Augustine were attacked and burned, and refugees arrived with gory tales of Indian atrocities. The February 27, 1836, issue of Niles Register carried the following item: “The whole country south of St. Augustine has been laid waste during the past week, and not a building of any value left standing. There is not a single house remaining between this city and Cape Florida, a distance of 250 miles.” When this occurred the original Indian tribes of Florida encountered by the early Spaniards had completely disappeared. Some had been wiped out during the long period of border conflict with the English. Others had succumbed to epidemics of disease. By the early 1800’s the principal Indians found in Florida were called Seminoles, and were a combination of several tribal remnants from Georgia and Alabama. Under United States rule the Seminoles were first restricted to a more limited area by the Treaty of Moultrie in 1823. But as settlers continued to pour in, a demand arose for their complete removal from Florida to reservations in the West, which the younger Seminole leaders were determined to resist. The effort to force their removal to western reservations resulted in conflict that dragged on for seven years, from 1835 to 1842. Officer after officer was sent to Florida to take command of operations against the Indians, including General Winfield Scott of subsequent Mexican War fame; and General Zachary Taylor, later to become President of the United States. But roving bands of Seminoles continued to strike and vanish into the dense swamps and little known woodlands.
- 72. 62 In 1837 two prominent Seminole leaders, Osceola and Coacoochee, with seventy of their warriors, were seized by General Hernandez under orders from General Jesup at a point a few miles south of St. Augustine. The Indians had come in under a white flag for a parley with United States officers. The captives were brought to St. Augustine and imprisoned in the Castillo, from which Coacoochee and twenty companions managed to escape. Osceola died soon after transfer to Fort Moultrie, Charleston. During May of 1840 a party of actors enroute from Picolata to St. Augustine were attacked by Indians, and near the same point two St. Augustine residents were murdered. “It is useless to complain,” stated a news item of the day. “The fact remains that we have been pent up in this little city for the last four years and a half by a few worthless outlaws. Our friends and neighbors, one after another, have been hastened to the mansions of the dead, and he who is foolhardy enough to venture beyond the gates may be the next victim.” But St. Augustine as usual managed to be gay. A young lieutenant, William Tecumseh Sherman of later Civil War fame, was stationed at Picolata and frequently rode into St. Augustine for diversion. In one of his letters home he wrote under date of February 15, 1842: “The inhabitants (of St. Augustine) still preserve the old ceremonies and festivities of old Spain. Balls, masquerades, etc., are celebrated during the gay season of the Carnival (just over), and the most religious observance of Lent in public, whilst in private they can not refrain from dancing and merry making. Indeed, I never saw anything like it—dancing, dancing, and nothing but dancing, but not such as you see in the North. Such ease and grace as I never before beheld.” Dr. Motte, a young military surgeon, made a similar observation in his journal: “The St. Augustine ladies certainly danced more gracefully, and kept better time, than any of my fair country women I ever saw
- 73. 63 in northern cities. It was really delightful to see the beautiful Minorcan girls moving through their intricate waltz to the music of violin and tambourine.” Finally most of the Seminoles were killed or surrendered for transfer to reservations in the West. A few were allowed to remain deep in the Everglades. There were probably less than 5,000 Indians in Florida at the outset, yet the war involved the enlistment of 20,000 men, an estimated cost of thirty million dollars, and 1,500 United States casualties. St. Augustine somewhat reluctantly saw the war come to an end. The presence of officers and troops had enlivened its social life, and poured government funds into the city. A Peaceful Interlude The end of the Seminole War made Florida safe again for travelers. William Cullen Bryant, the popular poet and author, paid St. Augustine a visit in 1843 and wrote articles about the city that were widely read. He noted that gabled roofs were rapidly replacing the flat roofs of the first Spanish period, and that some “modern” wooden buildings had been constructed. More than half the inhabitants still spoke the Minorcan, or Mahonese language. Another visitor of 1843 was Henry B. Whipple, later a prominent Episcopal Bishop. He found masquerading still a popular pastime in the city. Masking began during the Christmas holidays and continued until Lent. Small groups of people dressed in various disguises spent the evenings going from house to house, acting out their parts and furnishing their own music with guitar and violin. Whipple wrote that St. Augustine was still full of old ruins, and that “he liked to wander through the narrow streets and gaze upon these
- 74. monitors of time, which whispered that the hands that built them were long since mouldering in the grave.”
- 75. St. George Street as it looked in the 1870’s. In 1845 Florida became the twenty-seventh state admitted to the Union. Tallahassee had been selected as its territorial capital in 1824, being a compromise between St. Augustine and Pensacola, both of which were difficult to reach from most of the state.
- 76. 64 General Edmund Kirby Smith. During the Civil War St. Augustine lived on, enlivened during the winter by an influx of visitors, and drowsing undisturbed through the long summers until aroused by another conflict—the Civil War.
- 77. 65 Slaves played a relatively minor role in its economy, as compared with the rest of the state. Although a few plantations in the immediate vicinity employed slave labor, they were chiefly used as domestic servants and were generally well treated. There was considerable Union sentiment in the city due to its number of northern-born residents. Edmund Kirby-Smith, who had played in St. Augustine’s streets as a boy, became one of the leading Confederate Generals. His father came to the city in 1822 as Judge of the Superior Court and died here in 1846. His mother continued to occupy their home on what is now Aviles Street. During January of 1861 she wrote her son: “Our hearts are steeped in sadness and anxiety. Forebodings of evil yet to come depress us. We are threatened with the greatest calamity that can befall a nation. Civil war stares us in the face.” In the same letter she tells of how the news of Florida’s secession from the Union was received at St. Augustine: “Our state has seceded, and it was announced here by the firing of cannon and musketry, and much shooting. A large flag made by the ladies is waving on the square. By order of the Governor of this State, the Fort, Barracks, and Federal property were taken possession of. Cannon are mounted on the ramparts of the Fort to defend it if any attempt should be made to retake it.” Soon the shouting ceased and war became a stark reality with its heartaches, poverty, and privation. Many young men from St. Augustine went into the Confederate armies. The majority of its northern-born residents returned to the North to live for the duration of the war. The flow of visitors to the city ceased. During March of 1862 a Union blockading squadron appeared off the inlet, and an officer came ashore with a white flag to demand the city’s surrender. During the night its small Confederate garrison withdrew. Next morning St. Augustine was occupied by Union forces and held by them during the remainder of the conflict. Before the Federal troops landed the women of the city cut down the
- 78. flag pole in the Plaza so that the Union standard could not be raised where their Confederate banner had waved. Travelers complained bitterly of the service on the Picolata stage line, here shown bogged down enroute to St. Augustine. From a sketch made in 1867. Tourist Industry Resumed When the Civil War came to an end in 1865, St. Augustine was three centuries old. As the effects of the war and the reconstruction period wore away, the entertainment of winter residents and visitors was resumed. The city was still exceptionally quaint and foreign in appearance.
- 79. A visitor of 1869 found the Florida House, one of the city’s three small hotels, crowded with guests and wrote: “The number of strangers here greatly exceeded our expectations, and thronged in every street and public place. The fashionable belle of Newport and Saratoga, the pale, thoughtful clergyman of New England, were at all points encountered.” The city badly needed better hotels and travel facilities. Visitors then had to come up the St. Johns River by steamer to Picolata, and from there a horse-drawn stage jolted them for eighteen miles over a
- 80. 66 miserable road to the San Sebastian River, where a flatboat ferried the carriage across the river to the city’s outskirts. By 1871 travelers could go up the St. Johns River by steamer to Tocoi Landing, and there take a mule-drawn car over a crude railroad that ran fifteen miles east through the wilderness to St. Augustine. It was called the St. Johns Railway and a few years later installed two wood-burning locomotives. The San Marco, St. Augustine’s first great resort hotel, was opened in 1886, and burned to the ground in 1897. Its Isolation Broken The bonds of isolation and inaccessibility, which had retarded St. Augustine’s growth yet preserved its Old World character, were gradually being removed. Some signs of this awakening were
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