![6 Will-be-set-by-IN-TECH
Example 4. Given the polynomial p(s) = s5 + 1s4 + 2s3 + 2s2 + 1s + 1 by means of criterion 2
determine the number of roots in the right half of the complex plane. Applying this criterion
we get as following.
σ1 1 2 1
σ2 1 2 1
σ3 0 0
Table 7. Example 4. A row equal zero.
The table (7) generated, it shows the third row equal zero. Then obtaining the derivative of the
polynomial “corresponding” to the immediately superiory row p(s) = s4 + 2s2 + 1 is obtained
p(s) = 4s3 + 4s. Now the coefficients of this polynomial replace the zeros of the third row and
the procedure continues, obtaining in this way the follow arrangement.
σ1 = + 1 2 1
σ2 = + 1 2 1
σ3 = + 4 4
σ4 = + 4 4
σ5 = +
σ6 = + 4
Table 8. Example 4. Solution to the problem of a row equal zero.
We can see in table (8) that there is no sign change in sigma column, then there are not roots
in the right half complex plane.
3. Linear time invariant systems with parametric uncertainty
3.1 Parametric uncertainty
All phisical systems are dependent on parameters qi and in the physical world does not know
the value of the parameters, only know the lower q−
i and upper q+
i bounds of each parameter,
so that q−
i ≤ qi ≤ q+
i , this expression is also written as qi ∈ [q−
i , q+
i ].
For example if we have several electrical resistances with color code of 1,000 ohm, if one
measures one of them, the measurement can be: 938, 1,024, or a value close to 1,000 ohm but
it is rather difficult that it is exactly 1,000 ohm. By means of tolerance code can be deduced
that the resistance will be greater than 900 and less than 1,100 ohm. Another example is
the mass of a commercial aircraft, it can fly with few passengers and little baggage or with
with many passengers and much baggage, then the mass of the plane is not known until the
last passenger to be registered, but not when the plane was designed, however the plane is
designed to fly from a minimum mass to a maximum mass.
The set of parameters involved in a system makes a Parametric Vector q = [q1, q2, · · · , q]T,
q ∈ and the set of all the possible parameter vectors that may exist makes a Parametric
Uncertainty Box Q = { q = [q1, q2, · · · , q]T
qi ∈ [q−
i , q+
i ] ∀i}. In the case of qi 0 ∀i then
Q = { q = [q1, q2, · · · , q]T
qi 0, qi ∈ [q−
i , q+
i ] ∀i } and Q is contained in a positive convex
cone P, Q ⊂ P ⊂ .
For the study of cases involving parametric uncertainty is necessary to define the minimum
and maximum vertices of the parametric uncertainty box, so the minimum vmin and
8 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-24-320.jpg)
![8 Will-be-set-by-IN-TECH
4. Brief description of sign decomposition
In different areas of sciences the fundamental problem can be mapped to a problem of robust
positivity of multivariable polynomic functions. For example the no singularity of a matrix
can be analyzed by mean of the robust positivity of its determinant, so it is very useful to have
a mathematical tool that solves the problem of robust positivity of multivariable polynomic
functions. Practically there are three tools for this purpose: Interval Arithmetic (Moor, 1966);
Bernstein Polynomials (Zettler, et all 1998) and Sign Decomposition ((Elizondo, 1999)) whose
complete version is developed in (Elizondo, 1999) and its partial versions are presented in
(Elizondo, 2000; 2001A;B; 2002A;B), for simplicity only will be mentioned (Elizondo, 1999).
Interval arithmetic is very difficult to use because it requires much more calculations than
other methods. When robust positivity is analyzed in a very simple function, Bernstein
polynomials have advantages over sign decomposition, but when the function is not simple,
sign decomposition has advantages over Bernstein polynomials (Graziano et al., 2004). There
are several works using sign decomposition instead of Bernstein polynomials, some of them
are: (Bhattacharyya et al., 2009; Guerrero, 2006; Keel et al., 2008; 2009; Keel, 2011; Knap et al.,
2010; 2011)
4.1 Definition of sign decomposition
The following is a brief description of the more relevant results of Sign Decomposition
(Elizondo, 1999). By means of this tool it is possible to determine, in necessary and sufficient
conditions, the robust positivity of a multivariable polynomic function depending on
parameters, employing extreme points analysis.
Since mathematically exist the possibility that a parameter q̂i has negative value , then this tool
begins by a “coordinates transformation” from ˆ
qi to qi such that the new parameters will be
positive qi 0, then an uncertainty box Q = { q = [q1, q2, · · · , q]T
qi 0, qi ∈ [q−
i , q+
i ] } is
makes, in other words, Q is in a positive convex cone P, Q ⊂ P ⊂ with minimum vmin and
maximum vmax Euclidean vertices. The transformation is very easy as shown in the equation
(1)
qi = q−
i +
q̂i − q̂−
i
q̂+
i − q̂−
i
(q+
i − q−
i ) (1)
From here on we will assume that if necessary, the transformation was made and work with
parameters qi 0. Under this consideration will continue with the rest of this topic.
Definition 4. (Elizondo, 1999) Let f : → be a continuous function and let Q ⊂ P ⊂
be a box. It is said that f (q) has Sign Decomposition in Q if there exist two bounded continuous
nondecreasing and nonnegative functions fn(·) ≥ 0, fp(·) ≥ 0, such that f (q) = fp(q) − fn(q)
∀ q ∈ Q. In this way there are defined the Positive Part fp(q) and Negative Part fn(q) of the
function.
Negative Part is only a name since Negative Part and Positive Part are nonnegative.
4.2 ( fn, fp) representation
Is obvious that for the general case, fn(·) and fp(·) are independent functions then they make
a basis in 2 with graphical representation in the ( fn(·), fp(·)) plane in accordance with figure
( 1).
10 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-26-320.jpg)
![Parametric Robust Stability 9
If we take a particular vector q ∈ Q and evaluated the fn(q) and fp(q) parts, we obtain the
coordinates ( fn(q), fp(q)) of the function in the ( fn, fp) plane. The 45o line is the set of points
where the function is equal zero because fp(q) = fn(q) so f (q) = fp(q) − fn(q) = 0 . If a point
is above the 45o line means that fp(q) fn(q) then f (q) 0. If a point is below the 45o line
means that fp(q) fn(q) then f (q) 0.
Fig. 1. ( fn, fp) plane
It should be noted that independently of the number of parameters in which the function
depends on, the function will always be represented in 2 via ( fn(q), fp(q)). For example,
the function f (q) = 4 − q2 + q1q3 + 8q2
1q2 − 9q1q2
2q3
3 such that q ∈ Q ⊂ P ⊂ 3,
Q = { q = [q1, q2, q3]T
qi ∈ [0, 1] }. The function has sign decomposition because it is
decomposed in two bounded continuous nondecreasing and nonnegative functions fp(q) =
4 + q1q3 + 8q2
1q2, fn(q) = q2 + 9q1q2
2q3
3 and f (q) = fp(q) − fn(q). The figure ( 2) was obtained
by plotting a hundred lines blue color, (one hundred fifty points per line) of variable q3 holding
(q1, q2) constant uniformly distributed in different positions. The process was repeated
varying q2 in green color and finally varying q1 in red color. According to the position shown
in the graph of the function with respect to the 45o line, it appears that the function is robustly
positive. But it must be demonstrated mathematically.
Fig. 2. Function in ( fn, fp) plane
11
Parametric Robust Stability](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-27-320.jpg)
![10 Will-be-set-by-IN-TECH
Some preliminary properties of the continuous functions f (q), g(q), h(q) with sign
decomposition in Q and for all u(q) nondecreasing function in Q, are proved in (Elizondo,
1999) as so facts, lemmas and theorems. This properties are employed on the following
theorems.
a) ( fn(q) + u(q), fp(q) + u(q)) is a ( fn, fp) representation of the function f (q) ∀q ∈ Q; b)
the representation ( fn(q) + u(q), fp(q) + u(q)) of the function is reduced to its minimum
expression: ( fp(q), fn(q)); c) f (q) + g(q); d) f (q) − g(q) and e) f (q)g(q) are functions with
sign decomposition in Q; f) if f (q) = g(q) + h(q), then the positive and negative parts of
f (q) − g(q) are reduced to their minimum expressions, as follows: f (q) − g(q) = ( f (q) −
g(q))p − ( f (q) − g(q))n, ( f (q) − g(q))n = fn(q) − gn(q), ( f (q) − g(q))p = fp(q) − gp(q).
4.3 The rectangle theorem
Since negative part and positive part are bounded continuous nondecreasing functions, then
the following inequalities ( 2) are fulfilled.
fn(νmin) ≤ fn(q) ≤ fn(νmax)
fp(νmin) ≤ fp(q) ≤ fp(νmax)
(2)
This means that a function f (q) with sign decomposition, evaluated at any vector q ∈ Q,
its negative part is contained in a segment and also the positive part is contained in another
segment. So, on ( fn, fp) plane the function is contained in a rectangle as expressed by the
following theorem according to figure ( 3 ).
Theorem 5. (Elizondo, 1999) Rectangle Theorem. Let f : → be a continuous function
with sign decomposition in a box Q ⊂ P ⊂ with minimum and maximum Euclidean vertices
vmin, vmax, then: a) f (q) is lower and upper bounded by fp(vmin) − fn(vmax) and fp(vmax) −
fn(vmin) respectively; b) The graphical representation of the function f (q), ∀q ∈ Q in ( fn, fp)
plane is contained in the rectangle with vertices ( fn(vmin), fp(vmin)), ( fn(vmax), fp(vmax)),
( fn(vmin), fp(vmax)) and ( fn(vmax), fp(vmin)); c) if the lower right vertex ( fn(vmax), fp(vmin))
is over the 45o line then f (q) 0 ∀q ∈ Q; d) if the upper left vertex ( fn(vmin), fp(vmax)) is below
the 45o line then f (q) 0 ∀q ∈ Q. In accordance with figure ( 3 ).
The above result seems to be very useful, we can say that the rectangle is the “house” where
the multivariable function lives in 2. We can know the robust positivity of a function
analyzing only one point. It is important to note that this is only sufficient conditions, the
lower right vertex can be below the 45o line and the function could be robustly positive or not
be. But if the lower right vertex is above the 45o line then the function is robustly positive.
For example, the function f (q) = 4 − q2 + q1q3 + 8q2
1q2 − 9q3
3q1q2
2 such that q ∈ Q ⊂ P ⊂ 3,
Q = { q = [q1, q2, q3]T
qi ∈ [0, 1] }, has sign decomposition, its minimum and maximum
Euclidean vertices are νmin = [0, 0]T, νmax = [1, 1]T, their positive and negative psrtes are:
fp(q) = 4 + q1q3 + 8q2
1q2, fn(q) = q2 + 9q3
3q1q2
2. Then the lower bound is fp(vmin) − fn(vmax),
fp(vmin) = 4 + (0)(0) + 8(0)(0) = 4, fn(vmax) = 1 + 9(1)(1)(1) = 10, the lower bound is
4 − 10 = −9. The function could be robustly positive, but for now we do not know, It is
necessary see more signs of decomposition items.
Remark 6. Should be noted three important concepts:
The graph of the function does not fills the whole rectangle, but it is contained in.
12 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-28-320.jpg)
![Parametric Robust Stability 11
The graph of the function always touches the rectangle in lower left vertice and upper right vertice.
The graph of the function is not necessarily convex.
Fig. 3. Rectangle theorem
4.4 The polygon theorem
For the purpose of improving the results shown up to this point, the following proposition
is necessary. In some cases it is necessary to analyze the function in a Γ box contained in Q,
Γ ⊂ Q. The Γ box has Euclidean Vertices μmin and μmax. So, a vector in Γ is expressed as so as
q = μmin + δ, where δ is a vector in Γ, with origins in μmin.
Proposition 7. (Elizondo, 1999) Let f : → be a continuous function in Q ⊂ P ⊂ , let
Γj ⊂ Q be a box with its vertices set {μi} with minimum and maximum Euclidean vertices μmin,
μmax, let Δ = {δ | δi ∈ [0, δmax
i ], δmax
i = μmax
i − μmin
i } ⊂ P ⊂ be a box with its vertices set
{δi} with minimum and maximum Euclidean vertices 0, δmax = μmax − μmin, and let q ∈ Γj a vector
such that q = μmin + δ where δ ∈ Δ. Then the function f (q) is expressed by its: linear, nonlinear and
independent parts, in its minimum expression for all q ∈ Γj.
f (q) = f min + fL(δ) + fN(δ) | δ ∈ Δ∀q ∈ Γj
f min Indepent Part = f (μmin)
fL(δ) Linear Part = ∇ f (q)|μmin · δ ∀δ ∈ Δ
fN(δ) Nonlinear Part = f (μmin + δ) − f min − fL(δ) ∀δ ∈ Δ
∇ f (q)|μmin · δ =
∂ f (q)
q1
μmin
δ1 +
∂ f (q)
q2
μmin
δ2 + · · · +
∂ f (q)
q
μmin
δ
Must be noted that f min = f (μmin). On other hand, it is clear that we can use the concepts of
positive part and negative part in the above proposition, So, fp(q) − fn(q) = f min
p − f min
n +
fLp(δ) − fLn(δ) + fNp(δ) − fNn(δ) obtaining the following equations (3) where the relation
between δ and q can be appreciated in the figure (4).
fp(q) = f min
p + fLp(δ) + fNp(δ)
fn(q) = f min
n + fLn(δ) + fNn(δ)
(3)
13
Parametric Robust Stability](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-29-320.jpg)
![Parametric Robust Stability 13
fNp(δ) = fp(q) − f min
p − fLp(δ)
fNn(δ) = fn(q) − f min
n − fLn(δ)
fLp(δ) = ∇ fp(q)
μmin · δ
fLn(δ) = ∇ fn(q)|μmin · δ
(4)
As an illustration of this theme, by means of rectangle theorem and polygon, we will analyze
the lower bound of a function in a gama box. Consider the function corresponding to
the figure ( 2), f (q) = 4 − q2 + q1q3 + 8q2
1q2 − 9q3
3q1q2
2 such that q ∈ Q ⊂ P ⊂ 3,
Q = { q = [q1, q2, q3]T
qi ∈ [0, 1] }. Suppose that the function is analyzed into a gamma
box Γ ⊂ Q, with Euclidean vertices μmin = [0.2 0.2 0.2 ]T and μmax = [0.85 0.85 0.85 ]T.
In accordance with the Rectangle Theorem (3) the lower bound is fp(vmin) − fn(vmax) =
−0.1403. Applying the Polygon Theorem (8) the lower bound is f min + fLmin − fNn(δmax),
so it is necessary to obtain each of these expressions, the results are as follows: f min =
f (μmin) = 3.9034, fL min = −0.4457, fNn(δmax) = 3.3825. The last value is obtained of
ecuations (4), thus the lower bound is 0.0752. By means of the Rectangle Theorem is obtained
f (q) −0.1403 ∀q ∈ Γ, following the Polygon Theorem is obtained f (q) 0.0752 ∀q ∈ Γ, so
the function is robustly positive in the Γ box.
4.5 The box partition theorem
By means of Rectangle Theorem (3) and Polygon Theorem (8) are obtained sufficient
conditions of robust positivity, so to obtain necessary and sufficient conditions is necessary
to obtain new results.
When it is not possible to know whether the function is positive or not in Q = [q−
1 , q+
1 ]
×[q−
2 , q+
2 ] × · · · × [q−
, q+
]. In this case it is possible to divide each variable [q−
i , q+
i ] in k parts,
generating k new intervals: [q−
i , q1
i ], [q1
i , q2
i ], · · · , [q
j
i, q
j+1
i ], · · · [qk−1
i , q+
i ], let [γ−
i , γ+
i ] be a
k new interval, giving cause to the generation of k new boxes Γi = [γ−
1 , γ+
1 ] × [γ−
2 , γ+
2 ] ×
· · · × [γ−
, γ+
] with μmin, μmax ∈ Γi minimum and maximum Euclidean vertices of Γi and
Q =
i
Γi. Through these concepts, the following theorem is obtained.
Theorem 9. Box Partition Theorem (Elizondo, 1999). Let f : → be a continuous function
with sign decomposition in Q such that Q ⊂ P ⊂ is a box with minimum and maximum Euclidean
vertices vmin, vmax. Then the function f (q) is positive (negative) in Q if and only if a Γ boxes set exists,
such that Q =
j
Γj and Lower Bound ≥ c 0 for each Γj box (Upper Bound ≤ c 0 for each one
Γj box).
This theorem can be applied in two ways, one of them we call “ Analytical Partition” and
the other one “Constant Partition”. In analytical partition, the box where the function has a
negative lower bound is subdivided iteratively. In the case of the function is robustly positive
is also obtained information about where the function is close to losing positivity. By means
of constant partition is only obtained information on whether the function is robustly positive
or not.
To illustrate both procedures, we analyze the robust positivity of the function (Elizondo, 1999)
f (q) =
4 + q1 + 8q2
1q2
−
q2 + 9q1q2
2
, such that Q = { q = [q1, q2]T
qi ∈ [0, 1] ∀i } . The
robust positivity is analized by means of the rectangle theorem because it is more easier to
15
Parametric Robust Stability](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-31-320.jpg)
![18 Will-be-set-by-IN-TECH
Lemma 12. (Elizondo, 1999) Let M(q) be a (n × n) matrix with elements mi,j(q) ∈ with
representation (αi,j(q) , βi,j(q) ). Then the (α, β) representation of the determinant of the matrix
M(q) is detα(M(q)), detβ(M(q)) . In accordance with definition (11)
4.7.2 Linear, nonlinear and independent parts of the determinant
When the positivity of the determinant of a matrix with elements mi,j(q) is analyzed via sign
decomposition, it is normally necessary to use the box partition and polygon theorems. Then,
the independent, linear and nonlinear parts of the determinant need to be obtained. These are
obtained in the following theorem.
Theorem 13. (Elizondo, 1999) (Sign Decomposition of the Determinant Theorem) Let q ∈ Γ ⊆ Q |
q = μmin + δ be according to the proposition (7 ). Let M(q) ∈ n×n be a matrix with elements mi,j(q)
with sign decomposition in Q with representation (αmin
i,j + αi,j,L(δ) + αi,j,N(δ), βmin
i,j + βi,j,L(δ) +
βi,j,N(δ)), then the (α, β) representation of the determinant of the matrix M(q) is as follows:
α(q) = αmin
+ αL(δ) + αN(δ),
β(q) = βmin
+ βL(δ) + βN(δ)
αmin
= detα αmin
i,j , βmin
= det βmin
i,j
αL(q) =
k=n
∑
k=1
detα Φ(k) αmin
i,j + [I − Φ(k)] αi,j,L(δ)
βL(q) =
k=n
∑
k=1
det Φ(k) βmin
i,j + [I − Φ(k)] βi,j,L(δ)
Φ(k) = ϕi,j(k)
|
ϕ1,1(k) = |sign(1 − k)|
ϕ2,2(k) = |sign(2 − k)|
.
.
.
ϕn,n(k) = |sign(n − k)|
ϕi,j(k) = 0 ∀i = j
αN(δ) = α(q) − αmin
− αL(δ), βN(δ) = β(q) − βmin
− βL(δ)
4.7.3 Example
(Elizondo, 1999; 2001A). The Frazer and Duncan Theorem is presented in (Ackermann et al.,
1993) in the boundary crossing version as follows. Let P(s, Q) = {p(s, q) | q ∈ Q ⊂ P ⊂
} be a family of polynomials of invariant degree with parametric uncertainty and real
continuous coefficients, then the family P(s, Q) is robust stable if and only if: 1) a stable
polynomial p(s, q̂) ∈ P(s, Q) exists, 2) det (H(q)) = 0 for all q ∈ Q.
(Ackermann et al., 1993) Given the family of invariant degree polynomials with parametric
uncertainty described by: p(s, q) = c0 + c1s + c2s2 + c3s3 + c4s4, with real continuous
coefficients: c0(q) = 3, c1(q) = 2, c2(q) = 0.25 + 2q1 + 2q2, c3(q) = 0.5(q1 + q2), c4(q) = q1q2,
such that qi ∈ [1, 5]. Determine the robust stability of the family by means of the Frazer
and Duncan theorem applying in graphical way the sign decomposition of the determinant
theorem (13).
20 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-36-320.jpg)
![Parametric Robust Stability 19
The Hurwitz matrix H(q) is obtained, it is proved that the polynomial p(s, q̂) is stable for
q̂ = [1 1]T and that the determinant of the Hurwitz matrix H(q̂) is positive. Having the first
condition of the Frazer and Duncan theorem satisfied, and proving that the determinant is
robust positive in Q, the second condition of the Frazer and Duncan theorem will be satisfied
too.
H(q) =
⎡
⎢
⎢
⎣
c3(q) c1(q) 0 0
c4(q) c2(q) c0(q) 0
0 c3(q) c1(q) 0
0 c4(q) c2(q) c0(q)
⎤
⎥
⎥
⎦
The robust positivity of the determinant problem is solved by means of: the box partition
theorem 9, the polygon theorem 8 in (α, β) representation and the sign decomposition of
the determinant theorem (13). Taking the partition in 9 equal parts in each one of the two
variables qi and applying sign decomposition in constant partition way, the function values
in minimum and maximum vertices “×” and lower bound “+” are plotted for each Γi box, as
it appears in the figure (10). All lower bound marks “+” are above the alpha axis, then all of
bounds are positive, therefore the determinant of the Hurwitz matrix H(q) is robust positive
implying that the polynomials family is robust stable.
Fig. 10. Positivity of the determinant
5. A solution for the parametric robust stability problem
5.1 Problem identification
In control area, the robust stability of LTI systems with parametric uncertainty problem has
been studied in different interesting ways. The problem can be divided in two parts. One of
them is that it is not possible to be obtained roots of a polynomial by analytical means for the
general case. The second is that we have now a family of polynomials to study instead of a
single polynomial.
Since to obtain roots of polynomials for the general case is a difficult problem. Then the
extraction of roots of polynomials went mapped firstly to a “position” of roots problem in the
complex plane, Routh never tried to extract the roots, his work begun studying the position of
the roots. This problem was subjected to a second mapping, it was transferred to mathematical
problems of smaller level for example to a positivity problem, as it is the case of: Routh,
Hurwitz, Lienard-Chipart and Elizondo-González 2001 criterions.
21
Parametric Robust Stability](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-37-320.jpg)
![20 Will-be-set-by-IN-TECH
The objective in this chapter is to study the stability of a family of polynomials with invariant
degree (the reder can see poles and zeros canellation cases) and real continuous coefficients
dependent on parameters with uncertainty. The essence of the problem is that we have now
a set of roots in the the complexes plane, and for stability condition all of them must be in
the left half of the complex plane for asymptotic stability. How to obtain that the set of roots
remains in the left side of the complex plane?
A well known solution is: a) the family P(s, Q) has at least one element p(s, q∗) stable and
b) | H(q)|= 0 ∀q ∈ Q. The explanation is because the determinant of a Hurwitz matrix is
zero when the polynomial has roots in the imaginary axis, so if there is a q∗ ∈ Q vector such
that p(s, q∗) is stable then its roots are at the left half of the complex plane. On other hand,
if a vector q slides into Q starting from q∗ implies that the coefficients ci(q) will change in
continuous way and the roots of p(s, q∗) will slides too on the complex plane. But if | H(q)|=
0 ∀q ∈ Q, it means that does not exist a vector q for which p(s, q) has roots in the imaginary
axis, implying that the displacement of the roots never cross the imaginary axis. This solution
is very difficult to use because to test the robust positivity of a determinant in the general case
is a very difficult problem (Ackermann et al., 1993)(page 93).
Another solution was through the subsetting test, the idea worked well in convex families
as interval (Kharitonov, 1978) and affine (Bartlett et al., 1988), but it was not in nonconvex
families as multilinear and polynomic.
Then it can be concluded that the solution for robust stability of LTI systems with parametric
uncertainty problem for the general case: interval, affine, multilinear, polinomic, cannot be
sustained in convexity properties nor subsetting test.
5.2 A proposed solution
In (Elizondo, 1999) it was developed a solution for the general case of robust stability of LTI
systems with parametric uncertainty without concerning the convexity of the families, the
solution consists of two parts.
A part of the solution was the development of a stability criterion, operating with
multivariable polynomic functions in parametric uncertainty case, simpler than Hurwitz
and Lienard-Chipart criterions (Elizondo et al., 2005). The mentioned criterion is similar to
criterion (Elizondo, 2001B) but without the σ column, therefore it does not determine the
number of unstables roots, it only determines whether the polynomial is stable or not. The
amount of mathematical operations required in this criterion is equal to the one of (Elizondo,
2001B) but they are much less that the required ones in Hurwitz and Lienard-Chipart
criterions (Elizondo et al., 2005).
The other part of the solution was the development of a mathematical tool capable of solving
robust positivity problems of multivariable polynomic functions in necessary and sufficient
conditions by means of extreme point analysis.The mathematical tool developed in (Elizondo,
1999) was Sign Decomposition.
Then, the solution proposed for robust stability in LTI systems with parametric uncertainty in
the general case is supported in two results: the stability criterion for LTI systems (Elizondo,
2001B) and sign decomposition (Elizondo, 1999). Given a polynomial p(s, q) = cn(q)sn +
cn−1(q)sn−1 + · · · + c0(q) with real coefficients, where q ∈ Q ⊂ P, Q = {[q1 q2 · · · q ]T|qi ∈
[0, 1] ∀i}. The procedure easier to use is by means of the partition box theorem (9) in the
modality “Constant Partition”, its application could be of the following way.
a) Take the equations of the coefficients ci(q) and decompose them into positive and negative
parts cip(q) and cin(q). In symbolic way.
22 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-38-320.jpg)
![Parametric Robust Stability 21
b) By means of the positive and negative parts, to obtain the components in alpha and beta
representation. αi = cip(q) + cin(q), βi = cip(q) − cin(q).
c) To make a table in accordance to the criterion (2).
d) By means of the rectangle theorem (5) or polygon theorem (8), to analyze the robust
positivity in Q of the coefficients cn(q) and cn−1(q). In case of negative bound in a coefficient,
include its graph in the following software.
e) To make a software to develop the table in accordance to the partition box theorem and to
graph the wished ei,1 element.
Remark 14. The sigma column in the criterion (2) is not necessary calculate for robust stability
5.3 Example
Given a LTI system with parametric uncertainty Q = {[q1 q2 q3]T|qi ∈ [0, 1] ∀i}, its
characteristic polynomial of invariant degree is p(s, q) = c4(q)s4 + c3(q)s3 + +c2(q)s2 +
+c1(q)s + c0(q). To analyze the robust stability of the system.
a) Positive and negative parts cpi(q) and cni(q).
c0(q) = 2 + q1q2q3
3 − q2q3
c1(q) = 5 + q1q3
2 − q2q3
c2(q) = 10 + 4q1q3 − q1q2
2 − q3
2
c3(q) = 5 + q2
2 − q1q2
2
c4(q) = 3 + q1q3
2 − q2q3
c0p(q) = 2 + q1q2q3
3
c1p(q) = 5 + q1q3
2
c2p(q) = 10 + 4q1q3
c3p(q) = 5 + q2
2
c4p(q) = 3 + q1q3
2
c0n(q) = q2q3
c1n(q) = q2q3
c2n(q) = q1q2
2 + q3
2
c3n(q) = q1q2
2
c4n(q) = q2q3
b) The alpha and beta representation of the coefficients is as follows.
αi = cpi(q) + cni(q),
α0 = cp0(q) + cn0(q)
α1 = cp1(q) + cn1(q)
α2 = cp2(q) + cn2(q)
α3 = cp3(q) + cn3(q)
α4 = cp4(q) + cn4(q)
βi = cpi(q) − cni(q)
β0 = cp0(q) − cn0(q)
β1 = cp1(q) − cn1(q)
β2 = cp2(q) − cn2(q)
β3 = cp3(q) − cn3(q)
β4 = cp4(q) − cn4(q)
c) To make a table in accordance to the criterion (2).
σ1 (α4, β4) (α2, β2) (α0, β0)
σ2 (α3, β3) (α1, β1)
σ3 α3,1 = cα3cα2 + cα4cα1, β3,1 = cβ3cβ2 − cβ4cβ1 α3,2 = cα3cα0, β3,2 = cβ3cβ0
σ4 α4,1 = α3,1cα1 + cα3α3,2, β4,1 = β3,1cβ1 − cβ3β3,2
σ5 Check robust positivity of β4,1 and β3,2
d) The lower bound of c4(q) and c3(q) are as follows.
For c4(q) is LB c4 = c4p
[0 0 0]T
− c4n
[1 1 1]T
= 3 + (0)(0)3 − (1)(1) = 2.
For c3(q) is LB c3 = c3p
[0 0 0]T
− c3n
[1 1 1]T
= 5 + (0)2 − (1)(1)2 = 4.
23
Parametric Robust Stability](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-39-320.jpg)
![10 Will-be-set-by-IN-TECH
that the null space of the projection (I − RPR
2) is a subset of R2A2. Conversely, let Z ∈ A2,
then
(I − RPR
2 )R2Z = R2Z − R2PZ = R2Z − R2Z = 0 (34)
hence R2Z belongs to the null space of (I − RPR
2 ), and (30) holds.
The operator Ξ has the following explicit form
Ξ = (I − R2PR
2 )T1 (35)
which leads to the explicit solution
μo = (I − R2PR
2)T1 (36)
The expression generalizes the solution of the mixed sensitivity problem in the LTI case (25;
43; 46) to the LTV case. This result also applies to solve the robustness problem of feedback
systems in the gap metric (38) in the TV case as outlined in (11; 21; 33), since the latter was
shown in (11) to be equivalent to a special version of the mixed sensitivity problem (20).
5. Conclusion
The optimal disturbance rejection and the mixed sensitivity problems for LTV systems involve
solving shortest distance minimization problems posed in different spaces of bounded linear
operators. LTV causal and stable systems form a nest algebras, this allows the commutant
lifting theorem for nest algebras to be applied and solve both problems in term of abstract
TV Hankel and a TV version generalization of Hankel-Topelitz operators under fairly weak
assumptions. Future work includes investigation of numerical solutions based on finite
dimensional approximations, and computation of the corresponding controllers.
Acknowledgement: This work was partially supported by the National Science Foundation
under NSF Award No. CMMI-0825921.
6. References
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[3] Davidson K.R. Nest Algebras, Longman Scientific Technical, UK, 1988.
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[6] G. Zames and A.K. El-Sakkary, Unstable Systems and Feedback: The Gap Metric, in
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[7] S.Q. Zhu, M.L.J. Hautus and C. Praagman, Sufficient Conditions for Robust BIBO
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1988.
36 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-52-320.jpg)
![Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 11
[8] Khammash M., Dahleh M. Time-varying control and the robust performance of systems
with structured norm-bounded perturbations, Proceedings of the IEEE Conference on
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[10] A. Ichikawa, Quadratic Games and H∞-Type Problems for Time-Varying Systems, Int. J.
Control, vol. 54, pp. 1249-1271, 1991.
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[12] Feintuch A., Francis B.A. Uniformly Optimal Control of Linear Time-Varying Systems,
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[13] Feintuch A., Francis B.A. Uniformly Optimal Control of Linear Feedback Systems,
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[15] Shamma J.S. Robust stability with time-varying structured uncertainty, Proceedings of the
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[21] S. M. Djouadi, Commutant Lifting for Linear Time-Varying Systems, American Control
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J. Control and Optimization, vol. 25, (1987) 815-844.
[23] Francis B.A. A Course in H∞ Control Theory, Springer-Verlag, 1987.
[24] Zhou K., Doyle J.C., Glover K. Robust and Optimal Control, Prentice Hall, 1996.
[25] M. Verma and E. Jonckheere, L∞-Compensation with Mixed Sensitivity as a Broadband
Matching Problem, Systems Control Letters, vol. 4, pp. 125-129, 1984.
[26] Power S. Commutators with the Triangular Projection and Hankel Forms on Nest
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[27] Power S.C. Factorization in Analytic Operator Algebras, J. Func. Anal., vol. 67, (1986)
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[28] S.M. Djouadi and C.D. Charalambous, On Optimal Performance for Linear-Time
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[29] Feintuch A., Saeks R. System Theory: A Hilbert Space Approach, Academic Press, N.Y.,
1982.
[30] Peters M.A. and Iglesias P.A., Minimum Entropy Control for Time-varying Systems,
Boston, Birkhäuser, 1997.
37
Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-53-320.jpg)
![12 Will-be-set-by-IN-TECH
[31] T.T. Georgiou and M.C. Smith, Optimal Robustness in the Gap Metric, IEEE Trans. on
Automatic Control, vol. 35, No. 6, pp. 673-686, 1990.
[32] Georgiou T., On the Computation of teh Gap Metric, Systems and Control Letters, vol.
11, pp. 253-257, 1988.
[33] Foias C., Georgiou T. and Smith M.C., Robust Stability of Feedback Systems: A
Geometric Approach Using The Gap Metric, SIAM J. Control and Optimization, vol.
31, No.6, pp. 1518-1537, 1993.
[34] Foias C., Georgiou T. and Smith M.C., Geometric Techniques for Robust Stabilization of
Linear Time-Varying Systems, Proc. of he 29th CDC, pp. 2868-2873, December 1990.
[35] B. Sz. Nagy and C. Foias, Dilatation des commutants des commutateurs d’operateurs,
C.R. Academie Sci. paris, Serie A, 266, pp. 493-495, 1968.
[36] B. Sz. Nagy, C. Foias, H. Bercovici, and L. Kerchy, Harmonic Analysis of Operators on
Hilbert Space, Springer, 2010.
[37] Ravi R., Nagpal K.M. and Khargonekar P.P., H∞ Control of Linear Time-Varying
Systems: A State Space Approach, SIAM J. Control and Optimization, vol. 29, No.6,
pp. 1394-1413, 1991.
[38] T.T. Georgiou and M.C. Smith, Robust Stabilization in the Gap Metric: Controller Design
for Distributed Plants, IEEE Trans. on Automatic Control, vol. 37, No. 8, pp. 1133-1143,
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[39] Dale W.N. and Smith M.C., Stabilizability and Existence of System Representation for
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1538-1557, 1993.
[40] Dullerud G.E. and Lall S., A new approach for analysis and synthesis of time-varying
systems IEEE Trans. on Automatic Control, Vol.: 44 , Issue: 8 , pp. 1486 - 1497, 1999.
[41] C. Pirie and G.E. Dullerud, Robust Controller Synthesis for Uncerttain Time-Varying
Systems. SIAM J. Control Optimization., Vol. 40, No. 4, pp. 1312Ű1331, 2002.
[42] J.C. Doyle, B.A. Francis and A.R. Tannenbaum, Feedback Control Theory, Macmillan,
NY, 1990.
[43] C. Foias, H. Ozbay and A.R. Tannenbaum, Robust Control of Infinite Dimensional
Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1996.
[44] C. Foias, A.E. Frazho, I. Gohberg, and M.A. Kaashoek, Metric constrained interpolation,
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Birkhäuser Verlag, Basel, 1998
[45] Zhou K. and Doyle J.C., Essentials of Robust Control, Prentice Hall, 1998.
[46] Djouadi S.M. and Birdwell J.D., On the Optimal Two-Block H∞ Problem, Proceedings of
the American Control Conference, pp. 4289 - 4294, June 2005.
38 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-54-320.jpg)
![A. Machinist.
Q. Were you present during the disturbance last July?
A. Yes, I was.
Q. State what came under your observation?
A. I was there on Twenty-eighth street, on the side of the hill, when
the troops were coming up, and I waited there until they came
pretty well up the track, then I came down the hill, and I looked for
my boy.
Q. What time was this?
A. On Saturday afternoon, when I seen the troops come up—and
they came in regular—I think it was four deep. They came up and
stopped and halted. They turned around in open order, formed in a
hollow square, and I turned around and came away with the young
man, and I took him off the track, and started him down towards
Penn street. I turned around and looked for my boy, and I could not
see him. Walked in towards the cars, and stood by the side of the
sand-house—the cars extended up a little ways beyond the sand-
house—and I got right in to the end of the car. Then came orders to
charge bayonets. I turned to get back, and the crowd was behind,
and I could not get back. I got a bayonet right in behind here.
[Indicating.] Just at that time Pearson stood in about that direction.
Q. Who do you mean by Pearson?
A. General Pearson, or Pierson, or whatever you call him. He was
looking in the direction towards the watch-box. There was some
stones being thrown over there—it would fly all to pieces, it
appeared to be like clay. There was only one stone I could
distinguish, about that large, [indicating,] the shape of an oyster
shell. It came from there. Then there was a couple of old shoes—I
didn't see anybody struck with them. Pearson turned around, when
he was standing there he was looking about this—he turned around](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-68-320.jpg)
![[Witness illustrates on paper, the situation of the troops during the
firing.]
Q. Just say where the first fire commenced in that hollow square, so
that the reporter can take it down.
A. It was near the corner, on the side of the track next to the hill.
Q. Near the corner of the square of troops, next to the hill?
A. There was no square there. It was round.
Q. It came from the right, next towards the hill?
A. Next towards the hill.
Q. How many shots were fired at first?
A. They shot like a little fellow would throw a lot of shooting crackers
out.
Q. How long after you heard the command to fire did this shooting
commence?
A. It was right by those other fellows standing over on this side—
right by them. I guess there was four or five of them repeated it. I
took them to be officers.
Q. Dressed differently?
A. Yes, sir.
By Senator Reyburn:
Q. Were they in front of their men when they gave the order?
A. They were in this hollow square.
Q. Did they turn round to fire?](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-70-320.jpg)
![A. Yes, sir.
Q. How close was he to the men he gave the order to?
A. He was standing looking towards the switch box, where these
were coming from. Right in front there were some men had a
bayonet in another, and he wanted to get it away. He wheeled round
this way and gave the orders for these men to fire, and then walked
right down the track, but wherever he went to I could not say.
Q. Was he standing close to where the men commenced firing when
he gave the order to those men? You say he wheeled around; those
he gave the command to must have been behind him?
A. Here is where he stood. [Illustrating.] He wheeled around to the
officers to the rear, and they passed the command to the officers in
front. They repeated the order to fire.
By Mr. Means:
Q. Did the officers who repeated that command repeat it in a
distinct, loud tone of voice?
A. Yes; they repeated it distinctly.
Q. How many of them?
A. There were some three or four, I could not say exactly. I am
positive there was three or four, if not more—positive of three.
Q. And then the firing was done—it was not a volley of musketry—it
was just done at random, was it?
A. The first squad that fired there, I don't believe there was more
than about half a second between them, and the squads, as soon as
this squad fired on this side first, then these other fellows here fired,
[indicating,] and the crowd broke away and run down. I laid there.
They wheeled right down over me, and fired down Twenty-eighth
street.](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-72-320.jpg)
![Q. Did you see the operation?
A. I saw them pull the cock back—that part that turns back—and put
in a cartridge. I was lying right underneath them when they were
doing it. The parties in front fired the last shot down Twenty-eighth
street. I don't know who they were. They were men with black
plumes in their hats.
Q. You say the parties next to Twenty-eighth street were the last to
fire?
A. They wheeled right round——
Q. That was the party that fired first?
A. No; the party that fired first was over here. [Illustrating.]
Q. The party facing down Liberty street?
A. Yes; they fired. There was some that was against a car that could
not do anything.
Q. Did they fire before the front line fired?
A. They fired about the same time. There was a car stood in here
when they fired, so that you could do nothing. Some of them fired
up like.
Q. Fired in the air, did they?
A. They appeared to fire in the air.
Q. How did these men of this side? You say they turned and fired
the other way?
A. No, sir; those men in front wheeled round this way, and fired
down Twenty-eighth. Fired up first and then——
Q. Where did they deliver their first fire—the men on that north side
of the track?](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-75-320.jpg)
![A. Order your men to fire.
Q. That was the exact language?
A. That is the very words.
Q. You are positive you heard the words, order your men?
A. Right in that way: Order your men to fire.
Q. Were any of the soldiers struck?
A. I did not see anybody struck. There was a man carried away.
They say he got sun struck. I seen him carried away. I didn't see
anybody hit.
Q. Did you hear any pistol shots, or any shots fired from the crowd,
or in the crowd, before the firing of the soldiery?
A. No, sir.
Q. Did you hear any?
A. One appeared to be like a cap—it was right in that corner.
[Indicating.]
Q. In the crowd?
A. That was the first shot I heard fired. That soldier fired it. He didn't
fire straight out. His gun went off up in that way.
Q. You heard a noise like a cap before the firing of the troops?
A. I couldn't tell exactly where that come from. It appeared to be
round the watch-box.
Q. About the switch-box?
A. Yes, sir.
Q. That is, the watch-box on the corner of the street, and the
railroad toward the hill?](https://image.slidesharecdn.com/2057408-250604150424-04047ab9/85/Recent-Advs-In-Robust-Ctl-Theory-Applns-In-Robotics-Electromech-A-Mueller-77-320.jpg)
Recent Advs In Robust Ctl Theory Applns In Robotics Electromech A Mueller
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- 5. RECENT ADVANCES IN ROBUST CONTROL – THEORY AND APPLICATIONS IN ROBOTICS AND ELECTROMECHANICS Edited by Andreas Mueller
- 6. Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics Edited by Andreas Mueller Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Sandra Bakic Technical Editor Teodora Smiljanic Cover Designer Jan Hyrat Image Copyright PeterPhoto123, 2011. Used under license from Shutterstock.com First published November, 2011 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics, Edited by Andreas Mueller p. cm. ISBN 978-953-307-421-4
- 7. free online editions of InTech Books and Journals can be found at www.intechopen.com
- 9. Contents Preface IX Part 1 Theoretical Aspects of Robust Control 1 Chapter 1 Parametric Robust Stability 3 César Elizondo-González Chapter 2 Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 27 Seddik M. Djouadi Chapter 3 A Sum of Squares Optimization Approach to Robust Control of Bilinear Systems 39 Eitaku Nobuyama, Takahiko Aoyagi and Yasushi Kami Chapter 4 Spatially Sampled Robust Repetitive Control 55 Cheng-Lun Chen and George T.-C. Chiu Chapter 5 An Iterative Approach to the Fixed-Order Robust H Control Problem Using a Sequence of Infeasible Controllers 87 Yasushi Kami and Eitaku Nobuyama Chapter 6 Optimizing the Tracking Performance in Robust Control Systems 103 Hossein Oloomi and Bahram Shafai Part 2 Robust Control of Robotic Systems 115 Chapter 7 Robust Adaptive Position/Force Control of Mobile Manipulators 117 Tatsuo Narikiyo and Michihiro Kawanishi Chapter 8 Positioning Control of One Link Arm with Parametric Uncertainty Using the QFT Method 139 Takayuki Kuwashima, Jun Imai and Masami Konishi
- 10. VI Contents Chapter 9 Robust Visual Servoing of Robot Manipulators Based on Passivity 149 A. Luis Rodríguez and Yu Tang Chapter 10 Modelling of Bound Estimation Laws and Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 163 Recep Burkan Chapter 11 Missile Cooperative Engagement Formation Configuration Control Method 179 Changzhu Wei, Yi Shen, Xiaoxiao Ma, Naigang Cui and Jifeng Guo Chapter 12 Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy 207 Andreas Mueller Part 3 Robust Control of Electromechanical Systems 227 Chapter 13 Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 229 Huangfu Yigeng, S. Laghrouche, Liu Weiguo and A. Miraoui Chapter 14 Sliding Controller of Switched Reluctance Motor 263 Ahmed Tahour and Abdel Ghani Aissaoui Chapter 15 Robust Control of Sensorless AC Drives Based on Adaptive Identification 277 Birou M.T. Iulian Chapter 16 A Robust Decoupling Estimator to Identify Electrical Parameters for Three-Phase Permanent Magnet Synchronous Motors 303 Paolo Mercorelli Chapter 17 LMI Robust Control of PWM Converters: An Output-Feedback Approach 317 Carlos Olalla, Abdelali El Aroudi, Ramon Leyva and Isabelle Queinnec Chapter 18 Analysis, Dimensioning and Robust Control of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains 343 Andrea Tilli, Lorenzo Marconi and Christian Conficoni Chapter 19 Passivity Based Control for Permanent-Magnet Synchronous Motors 371 Achour Abdelyazid
- 13. Preface The vital condition for the practical applicability of any control system is that its basic characteristics are invariant to variations of its constituent parameters and to external disturbances, at least in some well-defined range. This is considered as Robustness, and the design of control systems with prescribed performance and stability limits is the subject of Robust Control. The aim of this two-volume book `Robust Control' is to provide a selective overview of recent developments in the theory and application of robust control. The book is a compilation of 39 contributions by recognized experts in the broad field of robust control. Since robust control is a diverse field of research it is difficult to provide an exhaustive and at the same time balanced coverage of this topic. The chapters were selected so to equally account for recent developments in the theory as well as different application areas. Volume I comprises 19 chapters covering selected problems in the theory of robust control and its application to robotic and electromechanical systems. The first part of this volume consists of six chapters addressing specific theoretical issues. Chapter 1 deals with the robustness of control systems to parametric uncertainties of linear time invariant (LTI) systems. Many practical problems can be classified as time delay systems. The time delay may in general be time dependent, which must be accounted for by the controller in order to ensure stability. This robustness problem is addressed in chapter 2. Often robust control problems are formulated as L2 minimization problem. This approach fails for bilinear systems, however. Such systems are treated in section 3, where sum of squares formulation is used. The principle of repetitive control, a variant of internal model control, is applied to spatially sampled systems in chapter 4, where attention is paid to non-linear saturation effects. In chapter 5 the problem of robust H controller design for time invariant systems with polytopic uncertainties is addresses and an iterative design method is proposed.
- 14. X Preface Chapter 6 addresses the selection of optimal weights in mixed sensitivity H1 design. In particular the selection problem for tracking sinusoidal reference signals is studied and a procedure for the weight selection is proposed. The weight parameters are related to the tracking error specification via a functional approximation. The second part of volume I is dedicated to robust control of robotic and mobile manipulators that are inherently nonlinear. In chapter 7 an adaptive hybrid position-force control scheme is proposed for mobile platforms exhibiting kinematic as well as dynamic uncertainties. While the method is developed for the particular case of a mobile manipulator it represents a general method potentially applicable to general non-holonomically constrained mechanisms. The classical quantitative feedback theory (QFT) is applied to the control of a robotic arm in chapter 8 where the arm is assumed to possess uncertainties in its dynamical parameters. This case study demonstrates the application of the QFT. Visual servoing, being an established method for the control of robotic systems, is used in section 9 for motion control of a robotic arm with three degrees of freedom. In this method the dynamics of the controlled system is represented in the image space. It is crucial that fundamental properties of the dynamics model in joint space are inherited by that in image space. This allows designing a robust visual servoing control scheme assuming certain bounds on the variation of the robot parameters as shown in the chapter. Since any robust control method assumes certain bounds of the model parameters it is crucial to provide bounds on the uncertainties for the particular control problem. This is pursued in chapter 10 for the robotic manipulators. A bound estimation is derived in terms of the function that gives rise to a particular solution of the defining condition. Hence the main result of this chapter is a family of bound estimations. Three different particular solutions and the corresponding Lyapunov-stable control laws are presented. Chapter 11 addresses the formation control of cooperating missiles. Optimal control is applied to establish and control formations. Besides the robustness of the actual model-based controller of a robotic manipulator the robustness of the underlying model is crucial. This problem is addressed in chapter 12 for the control of parallel manipulators with actuation redundancy. It is pointed out that the problem of input-singularities, that are eliminated in the plant by the actuation redundancy, remains for the dynamics model. A globally valid dynamics formulation is proposed that does not suffer from this problem. The third part of this volume is reserved for applications of robust control to electromechanical problems. Chapter 13 presents a robust variant of a higher order sliding mode controller used for the control of permanent magnet synchronous
- 15. Preface XI motors. The concept of higher order sliding mode control is briefly recalled before the case study is discussed in detail. Chapter 14 presents the design of a sliding mode controller for switched reluctance motors. The robustness of the controller is shown. Chapter 15 addresses the robust control of sensorless AC drives using H control design. Adaptivity is introduced to account for the system uncertainties and for identification. The identification of electrical parameters of a three-phase permanent magnet synchronous machine is approached in section 16 by means of a robust estimation strategy. Chapter 17 considers the effect of uncertainties in pulse with modulation (PVM) DC- DC converters. A robust controller is proposed subject to linear matrix inequality (LMI) constraints. The compensation of harmonic currents in electrical mains by means of active filtering is addressed in chapter 18. A nonlinear robust controller is designed and tested. Passivity is a crucial property of non-linear control systems. This is discussed in chapter 19 for the control of permanent magnet synchronous drives. Given the wide spectrum covered by this monograph the editor and the authors are confident that the two volumes of 'Robust Control' will be a valuable and stimulating reference for researchers from different disciplines. Andreas Mueller Chair of Mechanics and Robotics University Duisburg-Essen Germany
- 17. Part 1 Theoretical Aspects of Robust Control
- 19. 0 Parametric Robust Stability César Elizondo-González Facultad de Ingeniería Mecánica y Eléctrica Universidad Autónoma de Nuevo León México 1. Introduction Robust stability of LTI systems with parametric uncertainty is a very interesting topic to study, industrial world is contained in parametric uncertainty. In industrial reality, there is not a particular system to analyze, there is a family of systems to be analyzed because the values of physical parameters are not known, we know only the lower and upper bounds of each parameter involved in the process, this is known as Parametric Uncertainty (Ackermann et al., 1993; Barmish, 1994; Bhattacharyya et al., 1995). The set of parameters involved in a system makes a Parametric Vector, the set of all vectors that can exists such that each parameter is kept within its lower and upper bounds is called a Parametric Uncertainty Box. The system we are studying is now composed of an infinite number of systems, each system corresponds to a parameter vector contained in the parametric uncertainty box. So as to test the stability of the LTI system with parametric uncertainty we have to prove that all the infinite number of systems are stable, this is called Parametric Robust Stability. The parametric robust stability problem is considerably more complicated than determine the stability of an LTI system with fixed parameters. The stability of a LTI system can be analyzed in different ways, this chapter will be analyzed by means of its characteristic polynomial, in the case of parametric uncertainty now exists a set with an infinite number of characteristic polynomials, this is known as a Family of Polynomials, and we have to test the stability of the whole family. The parametric robust stability problem in LTI systems with parametric uncertainty is solved in this chapter by means of two tools, the first is a recent stability criterion for LTI systems (Elizondo, 2001B) and the second is the mathematical tool “Sign Decomposition” (Elizondo, 1999). The recent stability criterion maps the prametric robust stability problem to a robust positivity problem of multivariable polynomic functions, sign decomposition solves this problem in necessary and sufficient conditions. By means of the recent stability criterion (Elizondo, 2001B) is possible to analyze the characteristic polynomial and determine the number of unstable roots on the right side in the complex plane. This criterion is similar to the Routh criterion although without using the traditional division of the Routh criterion. This small difference makes a big advantage when it is analized the robust stability in LTI systems with parametric uncertainty, the elements of the first column of the table (Elizondo, 2001B) they are multivariable polynomic functions and these must be positive for stability conditions. Robust positivity of a multivariable polynomial function is more easier to prove that in the case of quotients of this class of functions, therefore, the recent criterion (Elizondo, 2001B) is easier to use than Routh criterion. There are other 1
- 20. 2 Will-be-set-by-IN-TECH criterions whose its elements are multivariable polynomic functions, such as the Hurwitz criterion and Lienard-Chipart criterion (Gantmacher, 1990), but both use a huge amount of mathematical operations in comparison with the recently stablished stability criterion Elizondo et al. (2005). When industrial cases are analyzed, the difference of mathematical operations is paramount, if the recently stability criterion takes several hours to determine the robust stability, the other criterions take several days. For these reasons the recently stability criterion is used in this chapter instead of other criterions. Sign Decomposition (Elizondo, 1999) also called by some authors as Sign definite Decomposition is a mathematical tool able to determine in necessary and sufficient conditions the robust positivity of multivariable polynomic functions by means of extreme points analysis. Sign Decomposition begun as incipient orthogonal ideas of the author in his PhD research. It was not easy to develop this tool as thus it happens in orthogonal works with respect to the contemporary research line, the orthogonal ideas normally are not well seen. This is a very difficult situation on any research work, there may be many opinions, but we must accept that the world keeps working by the aligned but it changes by the orthogonals. In LTI systems with parametric uncertainty applications, the multivariable polynomic functions to be analyzed depend on bounded physical parameters and some bounds could be negative. So sign decomposition begins with a coordinates transformation from the physical parameters to a set of mathematical parameters such that all the vectors of the new parameters are contained in a positive convex cone; in other words, all the new parameters are non-negatives. In this way, the multivariable polynomic function is made by non-decreasing terms, some of them are preceded by a positive sign and some by a negative sign. Grouping all the positive terms and grouping all the negative terms, then factorizing the negative sign and defining a “positive part” and a “negative part” of the function we obtain two non-decreasing functions. Now the function can be expressed as the positive part minus the negative part. It is obvious that both parts are independent functions, so they can be taken as a basis in with a graphical representation using two axis, the axis of the negative part and the axis of the positive part. Now, suppose that we have a particular vector contained in the parametric uncertainty box , then evaluating the negative part and the positive part a point on the “negative part, positive part plane” is obtained, this point represents the function evaluated in the particular vector in . The forty five degree line crossing at the origin on the “negative part, positive part plane” represents the set of functions with zero value, a point above this line represents a function with positive value and a point below this line represents a function with negative value. The decomposition of the function in its negative and positive parts may look very simple and non-transcendent but taking into acount that the negative and positive parts are made by the addition of non-decreasing terms, then the negative and positive parts are nondecreasing functions in a vector space, this implies that the positive part and the negative part are bonded. So, geometrically, any point representing the function evaluated at any parameter vector is contained in a rectangle on the “negative part, positive part plane” and if the lowest right vertex is above the forty five degree line then the function is robust positive, obtaining in this way the basis of the “rectangle theorem”. By means of this theorem upper and lower bounds of the multivariable polynomic function in the parametric uncertainty box are obtained. Sign decomposition contains a set of definitions, propositions, facts, lemmas, theorems and corollaries, sign decomposition can be applied to several disciplines; in the case of LTI systems with parametric uncertainty, this mathematical tool can be applied to robust controllability, 4 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 21. Parametric Robust Stability 3 obsevability or stability analysis. In this chapter sign decomposition is applied to parametric robust stability. In this chapter the following topics are studied: recent stability criterion, linear time invariant systems with parametric uncertainty, brief description of sign decomposition and finally a solution for the parametric robust stability problem. All demonstrations of the criterions, theorems, corollaries, lemmas, etc, will be omitted because they are results previously published. 2. A recent stability criterion for LTI systems The study of stability of the LTI systems begun approximately one and a half century ago with three important criterions: Hermite in 1856 (Ackermann et al., 1993), 1854 (Bhattacharyya et al., 1995); Routh in 1875 (Ackermann et al., 1993), 1877 (Gantmacher, 1990) and Hurwitz in 1895 (Gantmacher, 1990). Routh, using Sturm’s theorem and Cauchy Index theory of a real rational function, set up a theorem to determine the number k of roots of polynomial with real coefficients on the right half plane of the complex numbers. Theorem 1. (Routh) (Gantmacher, 1990) The number of roots of the real polynomial p(s) = c0 + c1s + c2s2 + · · · + cnsn in the right half of the complex plane is equal to the number of variations of sign in the first column of the Routh’s table with coefficients: ai,j = (ai−1,1ai−2,j+1 − ai−2,1ai−1,j+1)/ai−1,1 ∀i ≥ 3, ai,j = cn+1−i−2(j−1) ∀i ≤ 2 There are several results related to the Routh criterion, for example (Fuller, 1977; Meinsma, 1995), but they are not appropriate to use in the parametric uncertainty case and they use more mathematical calculations than the Routh criterion. In this chapter a recent criterion, an arrange similar to the Rouht table, it is presented. The stability in this recent criterion depends on the positivity of a sign column. The recent criterion has two advantages: 1) the numerical operations are reduced with respect to above mentioned criterions; 2) the coefficients are multivariable polynomic functions in the case of parametric uncertainty and robust positivity is easier to test than Routh criterion. The criterion is as described below. Theorem 2. (Elizondo, 2001B) Given a polynomial p(s) = c0 + c1s + c2s2 + · · · + cn−1sn−1 + cnsn with real coefficients, the number of roots on the right half of the complex plane is equal to the number of variations of sign in the sign σ column on the follow arrange. σ1 cn cn−2 cn−4 · · · σ2 cn−1 cn−3 cn−5 · · · σ3 e3,1 e3,2 · · · . . . . . . . . . ei,j = (ei−1,1ei−2,j+1 − ei−2,1ei−1,j+1), ∀3 ≤ i ≤ n + 1 ei,j = cn+1−i−2(j−1) ∀i ≤ 2 σi = Sign(ei,1) ∀i ≤ 2, σi = Sign(ei,1) (i+1−m)/2 ∏ j=1 Sign(em+2(j−1),1) ∀i ≥ 3 The procedure for calculating the elements (ei,j) is similar to the Routh table but without using the division. On the other hand, the calculation of an element σi is more easier than it looks mathematical expression. We can get the sign σi, multiplying the sign of the 5 Parametric Robust Stability
- 22. 4 Will-be-set-by-IN-TECH element (ei,1) by the sign of the immediate superior element (ei−1,1) and then jumping in pairs. For example σ6 = Sign(e6,1)Sign(e5,1)Sign(e3,1)Sign(e1,1). Also σ1 = Sign(cn) and σ2 = Sign(cn−2). So also it is not necessary to calculate the last element (en+1,1), only its sign is necessary to calculate. Each row of (ei,j) elements is obtained by means of (ei−1,j) and (ei−2,j) elements previously calculated and in Hurwitz criterion a principal minor is not calculated from previous, then the Elizondo-González criterion is more advantageous than Hurwitz criterion as shown in table (1) Remark 3. a) Given the relation of the above criterion with the Routh criterion, the cases in that one element ei,j is equal to cero or all the elements of a row are cero, they are treated as so as it is done in the Routh criterion. b) The last element en+1,1 is not necessary to calculate, but it is necessary to obtain only its sign Mathematical operations in polynomials n degree grado Hurwitz C. Elizondo n × + o − × + o − 3 4 1 2 1 4 9 2 5 2 5 66 18 9 4 6 193 45 14 6 7 780 145 20 9 Table 1. A comparison of stability criterions. 2.1 Examples Example 1. Given the polynomial p(s) = s5 + 2s4 + 1s3 + 5s2 + 2s + 2 by means of criterion 2 determine the number of roots in the right half of the complex plane and compare the results with the Routh criterion. Applying 2 criterion we obtain the left table. As an example of the procedure to obtain the elements ei,j and σi, we have: e3,1 = 2 × 1 − 1 × 5, e3,2 = 2 × 2 − 1 × 2, σ6 = Sign(+) × Sign(−56) × Sign(−3) × Sign(1), σ5 = Sign(−56) × Sign(−19) × Sign(2). Elizondo-González 2001 σ1 = + 1 1 2 σ2 = + 2 5 2 σ3 = − −3 2 σ4 = + −19 −6 σ5 = + −56 σ6 = + + Routh 1 1 2 2 5 2 −1.5 1 6.3333 2 1.4737 + Table 2. Example 1. Comparison of stability criterions. The left arrengment shows two sign changes in σ column so the polynomial has two roots on the right half of the complex plane. By means of Routh criterion is obtained the right table, it shows too two sign changes in the first column which is the same previous result. An interesting observation (see table (2)) is that the left table presents a minus sign in the third row of the σ column and the right table presents a minus sign in the same third row but in the first column. 6 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 23. Parametric Robust Stability 5 Example 2. Given the polynomial p(s) = s5 + 2s4 + 2s3 + 2s2 + s + 3 by means of criterion 2 determine the number of roots in the right half of the complex plane and compare the results with the Routh criterion. Elizondo-González 2001 σ1 = + 1 2 1 σ2 = + 2 2 3 σ3 = + 2 −1 σ4 = + 6 6 σ5 = − −18 σ6 = + − Routh 1 2 1 2 2 3 1 −0.5 3 3 −1.5 + Table 3. Example 2. Comparison of stability criterions. It is easy to see by means of two criterions that the polynomial has two roots on the right half of the complex plane in accordance to the table (3). Example 3. Given the polynomial p(s) = s5 + 1s4 + 2s3 + 2s2 + 2s + 1 by means of criterion 2 determine the number of roots in the right half of the complex plane. When we try to make the table by means of Elizondo-González 2001 criterion or Routh criterion, it is truncated because e3,1 = 0 σ1 1 2 2 σ2 1 2 1 σ3 0 1 Table 4. Example 3. Presence of a zero in the first column of elements. Since the element e3,1 is equal zero (see table (4)) then this element is replaced by by an 0, thus obtaining the following arrangement. σ1 1 2 2 σ2 1 2 1 σ3 1 σ4 2 − 1 σ5 2 − 1 − 2 σ6 (2 − 1 − 2) Table 5. Example 3. Solution of the problem of zero in the first column. Applying the limit → 0 in table (5) is obtained the table (6). σ1 = + 1 2 2 σ2 = + 1 2 1 σ3 = + 1 σ4 = − −1 σ5 = + −1 σ6 = + − Table 6. Example 3. Final result to the solution of the problem of zero in the first column. From the table (6) is easy to see that the polynomial has two roots on the right half of the complex plane. 7 Parametric Robust Stability
- 24. 6 Will-be-set-by-IN-TECH Example 4. Given the polynomial p(s) = s5 + 1s4 + 2s3 + 2s2 + 1s + 1 by means of criterion 2 determine the number of roots in the right half of the complex plane. Applying this criterion we get as following. σ1 1 2 1 σ2 1 2 1 σ3 0 0 Table 7. Example 4. A row equal zero. The table (7) generated, it shows the third row equal zero. Then obtaining the derivative of the polynomial “corresponding” to the immediately superiory row p(s) = s4 + 2s2 + 1 is obtained p(s) = 4s3 + 4s. Now the coefficients of this polynomial replace the zeros of the third row and the procedure continues, obtaining in this way the follow arrangement. σ1 = + 1 2 1 σ2 = + 1 2 1 σ3 = + 4 4 σ4 = + 4 4 σ5 = + σ6 = + 4 Table 8. Example 4. Solution to the problem of a row equal zero. We can see in table (8) that there is no sign change in sigma column, then there are not roots in the right half complex plane. 3. Linear time invariant systems with parametric uncertainty 3.1 Parametric uncertainty All phisical systems are dependent on parameters qi and in the physical world does not know the value of the parameters, only know the lower q− i and upper q+ i bounds of each parameter, so that q− i ≤ qi ≤ q+ i , this expression is also written as qi ∈ [q− i , q+ i ]. For example if we have several electrical resistances with color code of 1,000 ohm, if one measures one of them, the measurement can be: 938, 1,024, or a value close to 1,000 ohm but it is rather difficult that it is exactly 1,000 ohm. By means of tolerance code can be deduced that the resistance will be greater than 900 and less than 1,100 ohm. Another example is the mass of a commercial aircraft, it can fly with few passengers and little baggage or with with many passengers and much baggage, then the mass of the plane is not known until the last passenger to be registered, but not when the plane was designed, however the plane is designed to fly from a minimum mass to a maximum mass. The set of parameters involved in a system makes a Parametric Vector q = [q1, q2, · · · , q]T, q ∈ and the set of all the possible parameter vectors that may exist makes a Parametric Uncertainty Box Q = { q = [q1, q2, · · · , q]T qi ∈ [q− i , q+ i ] ∀i}. In the case of qi 0 ∀i then Q = { q = [q1, q2, · · · , q]T qi 0, qi ∈ [q− i , q+ i ] ∀i } and Q is contained in a positive convex cone P, Q ⊂ P ⊂ . For the study of cases involving parametric uncertainty is necessary to define the minimum and maximum vertices of the parametric uncertainty box, so the minimum vmin and 8 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 25. Parametric Robust Stability 7 maximum vmax Euclidean vertices of Q are defined as so as vmin 2 = min q∈Q q 2, vmax 2 = max q∈Q q 2. 3.2 Parametric robust stability in LTI systems In the LTI systems with parametric uncertainty, the characteristic polynomial has coefficients dependent on physical parameters, p(s, q) = c0(q) + c1(q)s + c2(q)s2 + · · · cn(q)sn; so Routh criterion is very difficult to use because it is necessary to test the robust positivity of rational functions dependent on physical parameters. By means of Hurwitz criterion is possible to solve the problem of parametric robust stability by means of robust positivity of principal minors of a matrix dependent on physical parameters, this procedure uses a lot of mathematical calculations. The robust positivity of rational function dependent on physical parameters can be considered as so as a very much difficult problem since only the robust positive test of multivariable polynomic function is very difficult problem (Ackermann et al., 1993) (page 93). So the parametric robust stability problem in LTI systems with parametric uncertainty in the general case is not an easy problem to solve, however in this chapter is presented a solution. The characteristic polynomials are classified according to its coefficient of maximum complexity; from the simplest structure coefficient to the most complex are: Interval, Affine, Multilinear and Polynomic. For example, the coefficients: ci(q) = qi, ci(q) = 2q1 + 3q2 + 5q3 + q4, ci(q) = 5q1q2 + 2q2q4 + 5q3 + q4, ci(q) = 2q3 1q2 + 2q2 2q5 4 + q3, correspond to classification: Interval, Affine, Multilinear and Polynomic respectively. The number of polynomials p(s, q) that can exist is infinite since the number of vectors that exist is infinite, the collection of all polynomials that exist is a Family of Polynomials P(s, Q) = {p(s, q)|q ∈ Q}. The families of polynomials interval and afin are convex sets and these families have subsetting test. This concept, subsetting test, means that a family of polynomials is robustly stable if and only if all polynomials contained in the subsetting test are stable. Kharitonov in (Kharitonov, 1978), by means of his theorem demonstrates that a family of interval polynomials is robust stable if and only if a set of four polynomials are stable. In (Bartlett et al., 1988) by means of their edge theorem, demonstrated that a family of afin polynomials is robustly stable if and only if all the polynomials corresponding to the edges of the parametric uncertainty box are stable. The multilinear an polynomic families are not convex set and they do not have subsetting test. So parametric robust stability of these families can not be resolved by tools based on convexity. In (Elizondo, 1999) was presented a solution for parametric robust stability of any kind of family: Interval, Affine, Multilinear or Polynomic. The solution is based on sign decomposition, and by means of this tool can also solve the problem of robust controllability or robust observability. 3.3 Robust stability mapped to robust positivity The parametric robust stability problem of LTI systems can be mapped to a problem of robust positivity of polynomial functions for at least three ways. The first two are: the Hurwitz and Lienard-Chipart criterions, the other is the recently stability criterion (2). By Hurwitz or Lienard-Chipart criterions can do the mapping but as explained these require making a lot of mathematical calculations. The criterion (2) requires much less mathematical calculations that the criterions mentioned as was shown in table (1), (Elizondo et al., 2005) 9 Parametric Robust Stability
- 26. 8 Will-be-set-by-IN-TECH 4. Brief description of sign decomposition In different areas of sciences the fundamental problem can be mapped to a problem of robust positivity of multivariable polynomic functions. For example the no singularity of a matrix can be analyzed by mean of the robust positivity of its determinant, so it is very useful to have a mathematical tool that solves the problem of robust positivity of multivariable polynomic functions. Practically there are three tools for this purpose: Interval Arithmetic (Moor, 1966); Bernstein Polynomials (Zettler, et all 1998) and Sign Decomposition ((Elizondo, 1999)) whose complete version is developed in (Elizondo, 1999) and its partial versions are presented in (Elizondo, 2000; 2001A;B; 2002A;B), for simplicity only will be mentioned (Elizondo, 1999). Interval arithmetic is very difficult to use because it requires much more calculations than other methods. When robust positivity is analyzed in a very simple function, Bernstein polynomials have advantages over sign decomposition, but when the function is not simple, sign decomposition has advantages over Bernstein polynomials (Graziano et al., 2004). There are several works using sign decomposition instead of Bernstein polynomials, some of them are: (Bhattacharyya et al., 2009; Guerrero, 2006; Keel et al., 2008; 2009; Keel, 2011; Knap et al., 2010; 2011) 4.1 Definition of sign decomposition The following is a brief description of the more relevant results of Sign Decomposition (Elizondo, 1999). By means of this tool it is possible to determine, in necessary and sufficient conditions, the robust positivity of a multivariable polynomic function depending on parameters, employing extreme points analysis. Since mathematically exist the possibility that a parameter q̂i has negative value , then this tool begins by a “coordinates transformation” from ˆ qi to qi such that the new parameters will be positive qi 0, then an uncertainty box Q = { q = [q1, q2, · · · , q]T qi 0, qi ∈ [q− i , q+ i ] } is makes, in other words, Q is in a positive convex cone P, Q ⊂ P ⊂ with minimum vmin and maximum vmax Euclidean vertices. The transformation is very easy as shown in the equation (1) qi = q− i + q̂i − q̂− i q̂+ i − q̂− i (q+ i − q− i ) (1) From here on we will assume that if necessary, the transformation was made and work with parameters qi 0. Under this consideration will continue with the rest of this topic. Definition 4. (Elizondo, 1999) Let f : → be a continuous function and let Q ⊂ P ⊂ be a box. It is said that f (q) has Sign Decomposition in Q if there exist two bounded continuous nondecreasing and nonnegative functions fn(·) ≥ 0, fp(·) ≥ 0, such that f (q) = fp(q) − fn(q) ∀ q ∈ Q. In this way there are defined the Positive Part fp(q) and Negative Part fn(q) of the function. Negative Part is only a name since Negative Part and Positive Part are nonnegative. 4.2 ( fn, fp) representation Is obvious that for the general case, fn(·) and fp(·) are independent functions then they make a basis in 2 with graphical representation in the ( fn(·), fp(·)) plane in accordance with figure ( 1). 10 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 27. Parametric Robust Stability 9 If we take a particular vector q ∈ Q and evaluated the fn(q) and fp(q) parts, we obtain the coordinates ( fn(q), fp(q)) of the function in the ( fn, fp) plane. The 45o line is the set of points where the function is equal zero because fp(q) = fn(q) so f (q) = fp(q) − fn(q) = 0 . If a point is above the 45o line means that fp(q) fn(q) then f (q) 0. If a point is below the 45o line means that fp(q) fn(q) then f (q) 0. Fig. 1. ( fn, fp) plane It should be noted that independently of the number of parameters in which the function depends on, the function will always be represented in 2 via ( fn(q), fp(q)). For example, the function f (q) = 4 − q2 + q1q3 + 8q2 1q2 − 9q1q2 2q3 3 such that q ∈ Q ⊂ P ⊂ 3, Q = { q = [q1, q2, q3]T qi ∈ [0, 1] }. The function has sign decomposition because it is decomposed in two bounded continuous nondecreasing and nonnegative functions fp(q) = 4 + q1q3 + 8q2 1q2, fn(q) = q2 + 9q1q2 2q3 3 and f (q) = fp(q) − fn(q). The figure ( 2) was obtained by plotting a hundred lines blue color, (one hundred fifty points per line) of variable q3 holding (q1, q2) constant uniformly distributed in different positions. The process was repeated varying q2 in green color and finally varying q1 in red color. According to the position shown in the graph of the function with respect to the 45o line, it appears that the function is robustly positive. But it must be demonstrated mathematically. Fig. 2. Function in ( fn, fp) plane 11 Parametric Robust Stability
- 28. 10 Will-be-set-by-IN-TECH Some preliminary properties of the continuous functions f (q), g(q), h(q) with sign decomposition in Q and for all u(q) nondecreasing function in Q, are proved in (Elizondo, 1999) as so facts, lemmas and theorems. This properties are employed on the following theorems. a) ( fn(q) + u(q), fp(q) + u(q)) is a ( fn, fp) representation of the function f (q) ∀q ∈ Q; b) the representation ( fn(q) + u(q), fp(q) + u(q)) of the function is reduced to its minimum expression: ( fp(q), fn(q)); c) f (q) + g(q); d) f (q) − g(q) and e) f (q)g(q) are functions with sign decomposition in Q; f) if f (q) = g(q) + h(q), then the positive and negative parts of f (q) − g(q) are reduced to their minimum expressions, as follows: f (q) − g(q) = ( f (q) − g(q))p − ( f (q) − g(q))n, ( f (q) − g(q))n = fn(q) − gn(q), ( f (q) − g(q))p = fp(q) − gp(q). 4.3 The rectangle theorem Since negative part and positive part are bounded continuous nondecreasing functions, then the following inequalities ( 2) are fulfilled. fn(νmin) ≤ fn(q) ≤ fn(νmax) fp(νmin) ≤ fp(q) ≤ fp(νmax) (2) This means that a function f (q) with sign decomposition, evaluated at any vector q ∈ Q, its negative part is contained in a segment and also the positive part is contained in another segment. So, on ( fn, fp) plane the function is contained in a rectangle as expressed by the following theorem according to figure ( 3 ). Theorem 5. (Elizondo, 1999) Rectangle Theorem. Let f : → be a continuous function with sign decomposition in a box Q ⊂ P ⊂ with minimum and maximum Euclidean vertices vmin, vmax, then: a) f (q) is lower and upper bounded by fp(vmin) − fn(vmax) and fp(vmax) − fn(vmin) respectively; b) The graphical representation of the function f (q), ∀q ∈ Q in ( fn, fp) plane is contained in the rectangle with vertices ( fn(vmin), fp(vmin)), ( fn(vmax), fp(vmax)), ( fn(vmin), fp(vmax)) and ( fn(vmax), fp(vmin)); c) if the lower right vertex ( fn(vmax), fp(vmin)) is over the 45o line then f (q) 0 ∀q ∈ Q; d) if the upper left vertex ( fn(vmin), fp(vmax)) is below the 45o line then f (q) 0 ∀q ∈ Q. In accordance with figure ( 3 ). The above result seems to be very useful, we can say that the rectangle is the “house” where the multivariable function lives in 2. We can know the robust positivity of a function analyzing only one point. It is important to note that this is only sufficient conditions, the lower right vertex can be below the 45o line and the function could be robustly positive or not be. But if the lower right vertex is above the 45o line then the function is robustly positive. For example, the function f (q) = 4 − q2 + q1q3 + 8q2 1q2 − 9q3 3q1q2 2 such that q ∈ Q ⊂ P ⊂ 3, Q = { q = [q1, q2, q3]T qi ∈ [0, 1] }, has sign decomposition, its minimum and maximum Euclidean vertices are νmin = [0, 0]T, νmax = [1, 1]T, their positive and negative psrtes are: fp(q) = 4 + q1q3 + 8q2 1q2, fn(q) = q2 + 9q3 3q1q2 2. Then the lower bound is fp(vmin) − fn(vmax), fp(vmin) = 4 + (0)(0) + 8(0)(0) = 4, fn(vmax) = 1 + 9(1)(1)(1) = 10, the lower bound is 4 − 10 = −9. The function could be robustly positive, but for now we do not know, It is necessary see more signs of decomposition items. Remark 6. Should be noted three important concepts: The graph of the function does not fills the whole rectangle, but it is contained in. 12 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 29. Parametric Robust Stability 11 The graph of the function always touches the rectangle in lower left vertice and upper right vertice. The graph of the function is not necessarily convex. Fig. 3. Rectangle theorem 4.4 The polygon theorem For the purpose of improving the results shown up to this point, the following proposition is necessary. In some cases it is necessary to analyze the function in a Γ box contained in Q, Γ ⊂ Q. The Γ box has Euclidean Vertices μmin and μmax. So, a vector in Γ is expressed as so as q = μmin + δ, where δ is a vector in Γ, with origins in μmin. Proposition 7. (Elizondo, 1999) Let f : → be a continuous function in Q ⊂ P ⊂ , let Γj ⊂ Q be a box with its vertices set {μi} with minimum and maximum Euclidean vertices μmin, μmax, let Δ = {δ | δi ∈ [0, δmax i ], δmax i = μmax i − μmin i } ⊂ P ⊂ be a box with its vertices set {δi} with minimum and maximum Euclidean vertices 0, δmax = μmax − μmin, and let q ∈ Γj a vector such that q = μmin + δ where δ ∈ Δ. Then the function f (q) is expressed by its: linear, nonlinear and independent parts, in its minimum expression for all q ∈ Γj. f (q) = f min + fL(δ) + fN(δ) | δ ∈ Δ∀q ∈ Γj f min Indepent Part = f (μmin) fL(δ) Linear Part = ∇ f (q)|μmin · δ ∀δ ∈ Δ fN(δ) Nonlinear Part = f (μmin + δ) − f min − fL(δ) ∀δ ∈ Δ ∇ f (q)|μmin · δ = ∂ f (q) q1 μmin δ1 + ∂ f (q) q2 μmin δ2 + · · · + ∂ f (q) q μmin δ Must be noted that f min = f (μmin). On other hand, it is clear that we can use the concepts of positive part and negative part in the above proposition, So, fp(q) − fn(q) = f min p − f min n + fLp(δ) − fLn(δ) + fNp(δ) − fNn(δ) obtaining the following equations (3) where the relation between δ and q can be appreciated in the figure (4). fp(q) = f min p + fLp(δ) + fNp(δ) fn(q) = f min n + fLn(δ) + fNn(δ) (3) 13 Parametric Robust Stability
- 30. 12 Will-be-set-by-IN-TECH Fig. 4. Gamma box Theorem 8. Polygon Theorem (Elizondo, 1999). Let f : → be a continuous function with sign decomposition in Q, let q, δ, Γj and Δ in accordance with the proposition (7). Then, a) the lower and upper bounds of the function f (q) are: Lower Bound = f min + fL min − fNn(δmax) and Upper Bound = f min + fL max + fNp(δmax) ∀q ∈ Q, b) the bounds of incise ”a”, are contained in the interval defined by the bounds of the rectangle theorem 3. fp(μmin) − fn(μmax) ≤ Lower Bound ≤ Upper Bound ≤ fp(μmax) − fn(μmin), c) The graphical representation of the function f (q) ∀q ∈ Γ in the ( fn, fp) plane is contained in the polygon defined by the intersection of the rectangle of the rectangle theorem (5) and the space between the two 45o lines separated from the origin by the Lower Bound and Upper Bound in accordance with figure (5). Fig. 5. Bounding of the function The symbolic expression of the nonlinear part used in the above theorem is not necessary to obtain, because we will use only its numerical value. So, from the equations (3), the nonlinear parts are obtained as so as equations ( 4). 14 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 31. Parametric Robust Stability 13 fNp(δ) = fp(q) − f min p − fLp(δ) fNn(δ) = fn(q) − f min n − fLn(δ) fLp(δ) = ∇ fp(q) μmin · δ fLn(δ) = ∇ fn(q)|μmin · δ (4) As an illustration of this theme, by means of rectangle theorem and polygon, we will analyze the lower bound of a function in a gama box. Consider the function corresponding to the figure ( 2), f (q) = 4 − q2 + q1q3 + 8q2 1q2 − 9q3 3q1q2 2 such that q ∈ Q ⊂ P ⊂ 3, Q = { q = [q1, q2, q3]T qi ∈ [0, 1] }. Suppose that the function is analyzed into a gamma box Γ ⊂ Q, with Euclidean vertices μmin = [0.2 0.2 0.2 ]T and μmax = [0.85 0.85 0.85 ]T. In accordance with the Rectangle Theorem (3) the lower bound is fp(vmin) − fn(vmax) = −0.1403. Applying the Polygon Theorem (8) the lower bound is f min + fLmin − fNn(δmax), so it is necessary to obtain each of these expressions, the results are as follows: f min = f (μmin) = 3.9034, fL min = −0.4457, fNn(δmax) = 3.3825. The last value is obtained of ecuations (4), thus the lower bound is 0.0752. By means of the Rectangle Theorem is obtained f (q) −0.1403 ∀q ∈ Γ, following the Polygon Theorem is obtained f (q) 0.0752 ∀q ∈ Γ, so the function is robustly positive in the Γ box. 4.5 The box partition theorem By means of Rectangle Theorem (3) and Polygon Theorem (8) are obtained sufficient conditions of robust positivity, so to obtain necessary and sufficient conditions is necessary to obtain new results. When it is not possible to know whether the function is positive or not in Q = [q− 1 , q+ 1 ] ×[q− 2 , q+ 2 ] × · · · × [q− , q+ ]. In this case it is possible to divide each variable [q− i , q+ i ] in k parts, generating k new intervals: [q− i , q1 i ], [q1 i , q2 i ], · · · , [q j i, q j+1 i ], · · · [qk−1 i , q+ i ], let [γ− i , γ+ i ] be a k new interval, giving cause to the generation of k new boxes Γi = [γ− 1 , γ+ 1 ] × [γ− 2 , γ+ 2 ] × · · · × [γ− , γ+ ] with μmin, μmax ∈ Γi minimum and maximum Euclidean vertices of Γi and Q = i Γi. Through these concepts, the following theorem is obtained. Theorem 9. Box Partition Theorem (Elizondo, 1999). Let f : → be a continuous function with sign decomposition in Q such that Q ⊂ P ⊂ is a box with minimum and maximum Euclidean vertices vmin, vmax. Then the function f (q) is positive (negative) in Q if and only if a Γ boxes set exists, such that Q = j Γj and Lower Bound ≥ c 0 for each Γj box (Upper Bound ≤ c 0 for each one Γj box). This theorem can be applied in two ways, one of them we call “ Analytical Partition” and the other one “Constant Partition”. In analytical partition, the box where the function has a negative lower bound is subdivided iteratively. In the case of the function is robustly positive is also obtained information about where the function is close to losing positivity. By means of constant partition is only obtained information on whether the function is robustly positive or not. To illustrate both procedures, we analyze the robust positivity of the function (Elizondo, 1999) f (q) = 4 + q1 + 8q2 1q2 − q2 + 9q1q2 2 , such that Q = { q = [q1, q2]T qi ∈ [0, 1] ∀i } . The robust positivity is analized by means of the rectangle theorem because it is more easier to 15 Parametric Robust Stability
- 32. 14 Will-be-set-by-IN-TECH apply, although it must be said that the bounds of the polygon theorem are better than the rectangle theorem. Analytical Partition (Elizondo, 1999). In the subfigure 1 of figure (6) shows that the function is robustly positive in boxes Γ1 and Γ3 but not in the boxes Γ2 and Γ4. So it is necessary apply iteratively the partition box to the boxes where the function is not robust positive, in this way is obtained the subfigure 2 of figure (6). Since there is a set of boxes such that Q = j Γj| f (q) 0 ∀Γj, then the function is robustly positive in Q. The graphs were made to show the procedure in visual way, but for more than two dimensions, using software we can get the coordinates and dimensions of sub boxes where the function is close to losing positivity. (a) Subfigure 1 (b) Subfigure 2 Fig. 6. Partition box Constant Partition (Elizondo, 1999). In this procedure the domaine of each one of the parameters is divide in k equal parts (not necessarily equal), in this way, it is generated a boxes set of k sub boxes Γi such that Q = j Γj. The robust positivity of each Γi box can be analyzed by a computer program so that the computer give us the final result about the robust positivity of the function. Another way is through a software which plot a × (blue) mark in the ( fn, fp) plane in each ( fn(μmin), fp(μmin)) and ( fn(μmax), fp(μmax)) coordinates corresponding to the minimum and maximum vertices of each Γi box, and plot too a + (red) mark corresponding to the lower bound of each Γi box, as can be appreciated in figure (7) that it was obtained with k = 13. If a × (blue) mark is below the 45o line, means that there is at least one vector for which the function is negative and therefore the function is not robustly positive. If all the × (blue) marks are above the 45o line, and a + (red) mark is below the 45o line means that it is necessary to increase the k number of partitions up to all the + (red) and × (blue) marks are above the 45o line. If this is achieved then the function is robustly positive, as shown in figure (7). In the figure (7) we can see that it is difficult to see that all + (red) marks are above the 45o line, then with purpose to resolve this difficulty is proposed the following representation. 16 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 33. Parametric Robust Stability 15 Fig. 7. Function in ( fn, fp) plane 4.6 (α, β) Representation In some cases as so as figure (7) it is not easy to determine in graphic way whether a point close to the 45o line is over this line or not. So in (Elizondo, 1999) the (α, β) representation was developed, α(q) = fp(q) + fn(q), β(q) = fp(q) − fn(q) , it is similar to rotated 45o the axis with respect to ( fn, fp) representation implying some graphical and algebraic advantages over the negative and positive representation. Definition 10. (Elizondo, 1999) Let fn(q) and fp(q) be the negative and positive parts of a continuous function f (q) with sign decomposition in Q. Let T be the linear transformation described below such that T −1 exists, then it is called a representation of the function f (q), in (α, β) coordinates, to the linear transformation (α(q), β(q)) = T( fn(q), fp(q)) and the inverse transformation of an (α(q), β(q)) representation is a ( fn(q), fp(q)) representation of the function f (q). T = 1 1 −1 1 T−1 = 1 2 1 −1 1 1 α(q) β(q) = T fn(q) fp(q) fn(q) fp(q) = T−1 α(q) β(q) α(q) = fp(q) + fn(q) fp(q) = 1 2 (α(q) + β(q)) β(q) = fp(q) − fn(q) fn(q) = 1 2 (α(q) − β(q)) With the purpose to show the advantages of the (α, β) representation, by means of the rectangle theorem we analyze the same function in the previous subsection f (q) = 4 + q1 + 8q2 1q2 − q2 + 9q1q2 2 applying k = 13. We can see in the figure (8) beta axis scale is positive implying that all the bounds are positives and consequently the function is robustly positive. The function f (q) = 4 − q2 + q1q3 + 8q2 1q2 − 9q3 3q1q2 2 corresponding to the figure (2) is shown in the figure (9) in (α, β) representation. We can see that beta axis scale is positive implying the function is robustly positive. The original idea to develop the representation (α , β) (Elizondo, 1999) was to solve a visual geometric problem, but this representation has interesting algebraic properties on continuous functions f (q), g(q), h(q) with sign decomposition in Q and for all u(q) nondecreasing function in Q, (Elizondo, 1999) as the following: 17 Parametric Robust Stability
- 34. 16 Will-be-set-by-IN-TECH Fig. 8. Function in (α, β) representation Fig. 9. Function in (α, β) representation a) α(q) is a non-decreasing and non-negative function in Q; b) α(q) ≥ β(q); c) β(q) = f (q) ∀ f (q), ∀q ∈ Q; d) the (α(q) + u(q), β(q) + u(q)) is a α, β representation of f (q); e) the (α(q) + u(q), β(q)) representation is reduced to its minimum expression (α(q), β(q)); f) Addition f (q) + g(q) : α(q) = αf (q) + αg(q), β(q) = βf (q) + βg(q); g) Subtraction f (q) − g(q) : α(q) = αf (q) + αg(q), β(q) = βf (q) − βg(q); h) Product f (q)g(q), α(q) = αf (q)αg(q), β(q) = βf (q)βg(q); i) the (α, β) representation of −g(q) is as follows: (αg(q), −βg(q)); j) if f (q) = g(q) + h(q) then the alpha an beta parts of f (q) − g(q) are reduced to its minimum expression as follows α(q) = αf (q) − αg(q), β(q) = βf (q) − βg(q). Computationally the (α, β) representation is better than ( fn, fp) because if the computer does not generate the negative scale in the β axis it is implying that all “marks” are positives. 18 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 35. Parametric Robust Stability 17 This is an usful and inetresting property, but above all properties there are three outstanding properties, it would be very useful if they were fulfilled in complex numbers, they are as follows: Addition f (q) + g(q) α(q) = αf (q) + αg(q) β(q) = βf (q) + βg(q) Subtraction f (q) − g(q) α(q) = αf (q) + αg(q) β(q) = βf (q) − βg(q) Product f (q)g(q) α(q) = αf (q)αg(q) β(q) = βf (q)βg(q) (5) Most be noted that the alpha componet of subtraction is correct with α(q) = αf (q) + αg(q), it is an “addition” of alphas. It is also important to highlight the simplicity with which made the addition, subtraction and product in alpha beta representation. 4.7 Sign decomposition of the determinant Sign decomposition of the determinant was developed in (Elizondo, 1999) and it was presented an application in (Elizondo, 2001A; 2002B), by simplicity only will mention (Elizondo, 1999). In parametric robust stability is not very useful the sign decomposition of the determinant, but it is a part of sign decomposition. We can analyze robust stability by means of the Hurwitz criterion means the robust positivity of determinants, but it is so much easier by means of criterion (2), see table (1). Taking account that the reader could work in other areas where the nonsingularity of a matrix dependent in parameters is important, then sign decomposition of the determinant is included in this chapter. 4.7.1 The (α, β) representation of the determinant In order to achieve the procedure to determine the robust positivity in necessary and sufficient conditions of a determinant with real coefficients depending on parameters qi, the following fact is presented. By means of the (α, β) properties (5) is obtained the following fact, in the development of the determinant appears the alpha part and beta part, as shown in the following fact. Fact 1. (Elizondo, 1999) Let M(q) be a (2 × 2) matrix with elements mi,j(q) ∈ with representation (αi,j(q) , βi,j(q)). Then the (α, β) representation of the determinant of the matrix M(q) is: (det(M(q)))α = (α1,1(q)α2,2(q) + α2,1(q)α1,2(q)) (det(M(q)))β = (β1,1(q)β2,2(q) − β2,1(q)β1,2(q)). Definition 11. (Elizondo, 1999) Let M(q) = mi,j(q) be a matrix with elements mi,j(q) ∈ with (αi,j(q) , βi,j(q)) representation. Then the matrix Mα(q) = αi,j(q) will be called the alpha part of the matrix M(q), and the determinant detα(M(q)) = |M(q)|α = |Mα(q)|α will be called the alpha part of the determinant |M(q)| , which is symilar to the usual determinant changing all the subtractions by additions including the sign rule of Cramer. In a similar way, the matrix Mβ(q) = βi,j(q) will be called the beta part of the matrix M(q), and the determinant detβ(M(q)) = |M(q)|β = Mβ(q) will be called the beta part of the determinant |M(q)| . Most be noted that: a) βi,j(q) = mi,j(q), then, Mβ(q) = M(q) and detβ(M(q)) = det(M(q)), b) In accordance with the above fact, for a (2 × 2) matrix, the (α, β) representation of the determinant of the matrix M(q) is detα(M(q)), detβ(M(q)) . In the following lemma a generalization of the last expression for a (n × n) matrix is stablished. 19 Parametric Robust Stability
- 36. 18 Will-be-set-by-IN-TECH Lemma 12. (Elizondo, 1999) Let M(q) be a (n × n) matrix with elements mi,j(q) ∈ with representation (αi,j(q) , βi,j(q) ). Then the (α, β) representation of the determinant of the matrix M(q) is detα(M(q)), detβ(M(q)) . In accordance with definition (11) 4.7.2 Linear, nonlinear and independent parts of the determinant When the positivity of the determinant of a matrix with elements mi,j(q) is analyzed via sign decomposition, it is normally necessary to use the box partition and polygon theorems. Then, the independent, linear and nonlinear parts of the determinant need to be obtained. These are obtained in the following theorem. Theorem 13. (Elizondo, 1999) (Sign Decomposition of the Determinant Theorem) Let q ∈ Γ ⊆ Q | q = μmin + δ be according to the proposition (7 ). Let M(q) ∈ n×n be a matrix with elements mi,j(q) with sign decomposition in Q with representation (αmin i,j + αi,j,L(δ) + αi,j,N(δ), βmin i,j + βi,j,L(δ) + βi,j,N(δ)), then the (α, β) representation of the determinant of the matrix M(q) is as follows: α(q) = αmin + αL(δ) + αN(δ), β(q) = βmin + βL(δ) + βN(δ) αmin = detα αmin i,j , βmin = det βmin i,j αL(q) = k=n ∑ k=1 detα Φ(k) αmin i,j + [I − Φ(k)] αi,j,L(δ) βL(q) = k=n ∑ k=1 det Φ(k) βmin i,j + [I − Φ(k)] βi,j,L(δ) Φ(k) = ϕi,j(k) | ϕ1,1(k) = |sign(1 − k)| ϕ2,2(k) = |sign(2 − k)| . . . ϕn,n(k) = |sign(n − k)| ϕi,j(k) = 0 ∀i = j αN(δ) = α(q) − αmin − αL(δ), βN(δ) = β(q) − βmin − βL(δ) 4.7.3 Example (Elizondo, 1999; 2001A). The Frazer and Duncan Theorem is presented in (Ackermann et al., 1993) in the boundary crossing version as follows. Let P(s, Q) = {p(s, q) | q ∈ Q ⊂ P ⊂ } be a family of polynomials of invariant degree with parametric uncertainty and real continuous coefficients, then the family P(s, Q) is robust stable if and only if: 1) a stable polynomial p(s, q̂) ∈ P(s, Q) exists, 2) det (H(q)) = 0 for all q ∈ Q. (Ackermann et al., 1993) Given the family of invariant degree polynomials with parametric uncertainty described by: p(s, q) = c0 + c1s + c2s2 + c3s3 + c4s4, with real continuous coefficients: c0(q) = 3, c1(q) = 2, c2(q) = 0.25 + 2q1 + 2q2, c3(q) = 0.5(q1 + q2), c4(q) = q1q2, such that qi ∈ [1, 5]. Determine the robust stability of the family by means of the Frazer and Duncan theorem applying in graphical way the sign decomposition of the determinant theorem (13). 20 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 37. Parametric Robust Stability 19 The Hurwitz matrix H(q) is obtained, it is proved that the polynomial p(s, q̂) is stable for q̂ = [1 1]T and that the determinant of the Hurwitz matrix H(q̂) is positive. Having the first condition of the Frazer and Duncan theorem satisfied, and proving that the determinant is robust positive in Q, the second condition of the Frazer and Duncan theorem will be satisfied too. H(q) = ⎡ ⎢ ⎢ ⎣ c3(q) c1(q) 0 0 c4(q) c2(q) c0(q) 0 0 c3(q) c1(q) 0 0 c4(q) c2(q) c0(q) ⎤ ⎥ ⎥ ⎦ The robust positivity of the determinant problem is solved by means of: the box partition theorem 9, the polygon theorem 8 in (α, β) representation and the sign decomposition of the determinant theorem (13). Taking the partition in 9 equal parts in each one of the two variables qi and applying sign decomposition in constant partition way, the function values in minimum and maximum vertices “×” and lower bound “+” are plotted for each Γi box, as it appears in the figure (10). All lower bound marks “+” are above the alpha axis, then all of bounds are positive, therefore the determinant of the Hurwitz matrix H(q) is robust positive implying that the polynomials family is robust stable. Fig. 10. Positivity of the determinant 5. A solution for the parametric robust stability problem 5.1 Problem identification In control area, the robust stability of LTI systems with parametric uncertainty problem has been studied in different interesting ways. The problem can be divided in two parts. One of them is that it is not possible to be obtained roots of a polynomial by analytical means for the general case. The second is that we have now a family of polynomials to study instead of a single polynomial. Since to obtain roots of polynomials for the general case is a difficult problem. Then the extraction of roots of polynomials went mapped firstly to a “position” of roots problem in the complex plane, Routh never tried to extract the roots, his work begun studying the position of the roots. This problem was subjected to a second mapping, it was transferred to mathematical problems of smaller level for example to a positivity problem, as it is the case of: Routh, Hurwitz, Lienard-Chipart and Elizondo-González 2001 criterions. 21 Parametric Robust Stability
- 38. 20 Will-be-set-by-IN-TECH The objective in this chapter is to study the stability of a family of polynomials with invariant degree (the reder can see poles and zeros canellation cases) and real continuous coefficients dependent on parameters with uncertainty. The essence of the problem is that we have now a set of roots in the the complexes plane, and for stability condition all of them must be in the left half of the complex plane for asymptotic stability. How to obtain that the set of roots remains in the left side of the complex plane? A well known solution is: a) the family P(s, Q) has at least one element p(s, q∗) stable and b) | H(q)|= 0 ∀q ∈ Q. The explanation is because the determinant of a Hurwitz matrix is zero when the polynomial has roots in the imaginary axis, so if there is a q∗ ∈ Q vector such that p(s, q∗) is stable then its roots are at the left half of the complex plane. On other hand, if a vector q slides into Q starting from q∗ implies that the coefficients ci(q) will change in continuous way and the roots of p(s, q∗) will slides too on the complex plane. But if | H(q)|= 0 ∀q ∈ Q, it means that does not exist a vector q for which p(s, q) has roots in the imaginary axis, implying that the displacement of the roots never cross the imaginary axis. This solution is very difficult to use because to test the robust positivity of a determinant in the general case is a very difficult problem (Ackermann et al., 1993)(page 93). Another solution was through the subsetting test, the idea worked well in convex families as interval (Kharitonov, 1978) and affine (Bartlett et al., 1988), but it was not in nonconvex families as multilinear and polynomic. Then it can be concluded that the solution for robust stability of LTI systems with parametric uncertainty problem for the general case: interval, affine, multilinear, polinomic, cannot be sustained in convexity properties nor subsetting test. 5.2 A proposed solution In (Elizondo, 1999) it was developed a solution for the general case of robust stability of LTI systems with parametric uncertainty without concerning the convexity of the families, the solution consists of two parts. A part of the solution was the development of a stability criterion, operating with multivariable polynomic functions in parametric uncertainty case, simpler than Hurwitz and Lienard-Chipart criterions (Elizondo et al., 2005). The mentioned criterion is similar to criterion (Elizondo, 2001B) but without the σ column, therefore it does not determine the number of unstables roots, it only determines whether the polynomial is stable or not. The amount of mathematical operations required in this criterion is equal to the one of (Elizondo, 2001B) but they are much less that the required ones in Hurwitz and Lienard-Chipart criterions (Elizondo et al., 2005). The other part of the solution was the development of a mathematical tool capable of solving robust positivity problems of multivariable polynomic functions in necessary and sufficient conditions by means of extreme point analysis.The mathematical tool developed in (Elizondo, 1999) was Sign Decomposition. Then, the solution proposed for robust stability in LTI systems with parametric uncertainty in the general case is supported in two results: the stability criterion for LTI systems (Elizondo, 2001B) and sign decomposition (Elizondo, 1999). Given a polynomial p(s, q) = cn(q)sn + cn−1(q)sn−1 + · · · + c0(q) with real coefficients, where q ∈ Q ⊂ P, Q = {[q1 q2 · · · q ]T|qi ∈ [0, 1] ∀i}. The procedure easier to use is by means of the partition box theorem (9) in the modality “Constant Partition”, its application could be of the following way. a) Take the equations of the coefficients ci(q) and decompose them into positive and negative parts cip(q) and cin(q). In symbolic way. 22 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 39. Parametric Robust Stability 21 b) By means of the positive and negative parts, to obtain the components in alpha and beta representation. αi = cip(q) + cin(q), βi = cip(q) − cin(q). c) To make a table in accordance to the criterion (2). d) By means of the rectangle theorem (5) or polygon theorem (8), to analyze the robust positivity in Q of the coefficients cn(q) and cn−1(q). In case of negative bound in a coefficient, include its graph in the following software. e) To make a software to develop the table in accordance to the partition box theorem and to graph the wished ei,1 element. Remark 14. The sigma column in the criterion (2) is not necessary calculate for robust stability 5.3 Example Given a LTI system with parametric uncertainty Q = {[q1 q2 q3]T|qi ∈ [0, 1] ∀i}, its characteristic polynomial of invariant degree is p(s, q) = c4(q)s4 + c3(q)s3 + +c2(q)s2 + +c1(q)s + c0(q). To analyze the robust stability of the system. a) Positive and negative parts cpi(q) and cni(q). c0(q) = 2 + q1q2q3 3 − q2q3 c1(q) = 5 + q1q3 2 − q2q3 c2(q) = 10 + 4q1q3 − q1q2 2 − q3 2 c3(q) = 5 + q2 2 − q1q2 2 c4(q) = 3 + q1q3 2 − q2q3 c0p(q) = 2 + q1q2q3 3 c1p(q) = 5 + q1q3 2 c2p(q) = 10 + 4q1q3 c3p(q) = 5 + q2 2 c4p(q) = 3 + q1q3 2 c0n(q) = q2q3 c1n(q) = q2q3 c2n(q) = q1q2 2 + q3 2 c3n(q) = q1q2 2 c4n(q) = q2q3 b) The alpha and beta representation of the coefficients is as follows. αi = cpi(q) + cni(q), α0 = cp0(q) + cn0(q) α1 = cp1(q) + cn1(q) α2 = cp2(q) + cn2(q) α3 = cp3(q) + cn3(q) α4 = cp4(q) + cn4(q) βi = cpi(q) − cni(q) β0 = cp0(q) − cn0(q) β1 = cp1(q) − cn1(q) β2 = cp2(q) − cn2(q) β3 = cp3(q) − cn3(q) β4 = cp4(q) − cn4(q) c) To make a table in accordance to the criterion (2). σ1 (α4, β4) (α2, β2) (α0, β0) σ2 (α3, β3) (α1, β1) σ3 α3,1 = cα3cα2 + cα4cα1, β3,1 = cβ3cβ2 − cβ4cβ1 α3,2 = cα3cα0, β3,2 = cβ3cβ0 σ4 α4,1 = α3,1cα1 + cα3α3,2, β4,1 = β3,1cβ1 − cβ3β3,2 σ5 Check robust positivity of β4,1 and β3,2 d) The lower bound of c4(q) and c3(q) are as follows. For c4(q) is LB c4 = c4p [0 0 0]T − c4n [1 1 1]T = 3 + (0)(0)3 − (1)(1) = 2. For c3(q) is LB c3 = c3p [0 0 0]T − c3n [1 1 1]T = 5 + (0)2 − (1)(1)2 = 4. 23 Parametric Robust Stability
- 40. 22 Will-be-set-by-IN-TECH Then c4(q) and c3(q) are robustly positives in Q e) By means of software applying 8 partitions the graphs e3,1, e3,2, e4,1 were obtained as following. Fig. 11. Element e31 in (α, β) representation Fig. 12. Element e32 in (α, β) representation 24 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 41. Parametric Robust Stability 23 Fig. 13. Element e41 in (α, β) representation Since c4(q), c3(q), e31(q), e32(q), e41(q) are robustly positive, then the system is robustly stable. 6. References Ackermann, J. Bartlett, A. (1993). Robust Control Systems with Uncertain Physical Parameters, Springer, ISBN 978-0387198439. Barmish, B.R. (1990). New Tools for Robustness of Linear Systems, Prentice Hall, ISBN 978-0023060557. Bartlett, A.C.; Hollot, C.V. Lin, H. (1988). Root locations of an entire polytope of polynomials: It suffices to check the edges. Mathematics of Control Signals Systems, Vol. 1, No. 1, 61-71, DOI: 10.1007/BF02551236. Bhattacharyya, S.P.; Chapellat, H. Keel, L.H. (1995). Robust Control the Parametric Approach, Prentice Hall, ISBN 0-13-781576-X, NJ, USA. Bhattacharyya, S.P.;Keel, L.H. Datta, A. (2009). Linear Control Theory: Structure, Robustness and Optimization, CRC Press,ISBN 978-0-8493-4063-5, Boca Raton. Elizondo-González, C. (1999). Estabilidad y Controlabidad Robusta de Sistemas Lineales con Incertidumbre Multilineal. Programa Doctoral de la Facultad de Ingeniería Mecánica y Eléctrica de la Universidad Autónoma de Nuevo León. Elizondo-González, C. (2000). Necessary and Sufficient Conditions for Robust Positivity of Polynomic Functions Via Sign Decomposition, 3 rd IFAC Symposium on Robust Control Design ROCOND 2000, pp. 14-17, ISBN-13:9780080432496 , Prague Czech Republic, April, 2000. Elizondo-González C. (2001). Robust Positivity of the Determinant Via Sign Decomposition, The 5th World Multi-Conference on Systemics Cybernetics and Informatics SCI 2001, pp. 14-17, ISBN 980-07-7545-5, Orlando, Florida, USA, July, 2001. Elizondo-González C. (2001). New Stability Criterion on Space Coefficients, Conferences on Decision and Control IEEE, SBN 0-7803-7063-5, Orlando, Florida, USA. Diciembre, 2001. 25 Parametric Robust Stability
- 42. 24 Will-be-set-by-IN-TECH Elizondo-González, C. (2002). An Application of Recent Reslts on Parametric Robust Stability, X Congreso Latinoamericano de Control Autmático, Guadalajara Jalisco, 2002. Elizondo-González, C. (2002). An Applicaton of Sign Decomposition of the Determinant on Parametric Robust Stability, X Congreso Latinoamericano de Control Autmático, Guadalajara Jalisco, 2002. Elizondo-González, C. Alcorta-García, E. (2005). Análisis de cotas de raíces de polinomios característicos y nuevo criterio de estabilidad, Congreso Nacional 2005 de la Asociación de México de Control Automático, ISBN 970-32-2974-3 Cuernavaca, Morelos, México, Octubre, 2005. Fuller, A.T. (1977). On Redundance in Stability Criteria, Internatinal Journal Control, Vol. 26, No. 2, pp. 207-224. Gantmacher, F.R. (1990). The Theory of Matrices, American Mathematical Society, ISBN-10: 0821813935, ISBN-13: 978-0821813935. Guerrero, J.; Romero, G.; Mendez, A.; Dominguez, R.; Panduro, M. Perez, I. (2006). Lecture Notes in Control and Information Sciences, Robust Absolute Stability Using Polynomial Positivity and Sign Decomposition, Vol. 341, No. 1, pp. 423-430, 2006. ISSN: 0170-8643. Graziano-Torres, R.; Elizondo-González, C. (2010). Herramientas para el Análisis de Estabilidad Robusta de Sistemas LTI con Incertidumbre Paramétrica, Congreso Nacional 2004 de la Asociación de México de Control Automático, ISBN: 970-32-2137-8 , México D.F., Octubre, 2004. Keel, L.H. Bhattacharyya, S.P. (2008). Fixed Order Multivariable Controller Synthesis: A New Algorithm, Proceedings of the 47th Conference on Decision and Control, Cancun, Mexico, December, 2008. Keel, L.H. Bhattacharyya, S.P. (2009). Fixed Order Multivariable Discrete-Time Control, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shangai, P.R. China, December, 2009. Keel, L.H. Bhattacharyya, S.P. (2011). Robust Stability via Sign-Definite Decomposition, Journal of IEEE Transactions on Automatic Control, Vol. 56, No. 140 − 155, ISSN: 0018-9286, Jan -2011. Kharitonov, V. (1978). On a Generalization of a Stability Criterion. Seria Fizico-matematicheskaia, Vol.1, pp. 53-57, Izvestiia Akademii Nauk Kazakhskoi SSR. Knap, M.J.; Keel, L.H. Bhattacharyya, S.P. (2010). Robust stability of complex systems with applications to performance attainment problems, American Control Conference., Marriot Waterfront, Baltimore, MD, USA, July, 2010. Knap, M.J.; Keel, L.H. Bhattacharyya, S.P. (2010). Robust Hurwitz stability via sign-definite decomposition , Linear Algebra and its Applications., Volume 434, Issue 7, pp 1663-1676, 1 April 2011. Meinsma G. (1995). Elementary Proof of the Routh-Hurwitz Test Systems Control Letters 25, pp. 237-242. Moore, Ramon E.(1966). Robust Control the Parametric Approach, Prentice Hall, NJ, USA. Zettler M.; Garloff J. (1998). Robustness Analysis of Polynomials with Polynomial Parameter Dependency Using Bernstein Expansion, IEEE Transactions on Automatic Control vol. 43 pages 1017-1049, 1998. 26 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 43. 0 Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach Seddik M. Djouadi Electrical Engineering Computer Science Department, University of Tennessee, Knoxville, TN 37996-2100 USA 1. Introduction There have been numerous attempts in the literature to generalize results in robust control theory (42; 45) to linear time-varying (LTV) systems (for e.g. (10–13; 30; 33; 37; 39; 40) and references therein). In (12)(13) and (11) the authors studied the optimal weighted sensitivity minimization problem, the two-block problem, and the model-matching problem for LTV systems using inner-outer factorization for positive operators. Abstract solutions involving the computation of induced operators norms of operators are obtained. However, there is no clear indication on how to compute optimal linear LTV controllers. In (40) the authors rely on state space techniques which lead to algorithms based on infinite dimensional operator inequalities which are difficult to solve. These methods lead to suboptimal controllers and are restricted to finite dimensional systems. An extension of these results to uncertain systems is reported in (41) relying on uniform stability concepts. In (9) both the sensitivity minimization problem in the presence of plant uncertainty, and robust stability for LTV systems in the ∞ induced norm is considered. However, their methods could not be extended to the case of systems operating on finite energy signals. In (37) the standard problem of H∞ control theory for finite-dimensional LTV continuous-time plants is considered. It is shown that a solution to this problem exists if and only if a pair of matrix Riccati differential equations admits positive semidefinite stabilizing solutions. State-space formulae for one solution to the problem are also given. The gap metric was introduced to study stability robustness of feedback systems. It induces the weakest topology in which feedback stability is robust (6; 7; 31; 32; 38). Extensions of the gap to time-varying systems have been proposed in (33; 34) where a geometric framework was developed. Several results on the gap metric and the gap topology were established, in particular, the concept of a graphable subspace was introduced. In (21) the problem of robust stabilization for LTV systems subject to time-varying normalized coprime factor uncertainty is considered. Operator theoretic results which generalize similar results known to hold for linear time-invariant (infinite-dimensional) systems are developed. In particular, a tight upper bound for the maximal achievable stability margin under TV normalized coprime factor uncertainty in terms of the norm of an operator with a time-varying Hankel structure is computed. Analysis of time-varying control strategies for optimal disturbance rejection for known time-invariant plants has been studied in (2; 16). A robust version of these problems was 2
- 44. 2 Will-be-set-by-IN-TECH considered in (8; 15) in different induced norm topologies. All these references showed that for time-invariant nominal plants and weighting functions, time-varying control laws offer no advantage over time-invariant ones. In this paper, we are interested in optimal disturbance rejection for (possibly infinite-dimensional, i.e., systems with an infinite number of states) LTV systems. These systems have been used as models in computational linear algebra and in a variety of computational and communication networks (17). This allows variable number of states which is predominant in networks which can switch on or off certain parts of the system (17), and infinite number of states as in distributed parameter systems. Using inner-outer factorizations as defined in (3; 11) with respect of the nest algebra of lower triangular (causal) bounded linear operators defined on 2 we show that the problem reduces to a distance minimization between a special operator and the nest algebra. The inner-outer factorization used here holds under weaker assumptions than (12; 13), and in fact, as pointed in ((3) p. 180), is different from the factorization for positive operators used there. The optimal disturbance attenuation for LTV systems has been addressed using Banach space duality theory in (20; 28). Its robust version which deals with plant uncertainty is addressed in (4; 5; 19) using also duality theory ideas. Furthermore, using the commutant lifting theorem for nest algebras the optimum is shown to be equal to the norm of a compact time-varying Hankel operator defined on the space of causal Hilbert-Schmidt operators. The latter is the “natural” analogous to the Hankel operator used in the LTI case. An operator identity to compute the optimal TV Youla parameter is also provided. The results are generalized to the mixed sensitivity problem for TV systems as well, where it is shown that the optimum is equal to the operator induced of a TV mixed Hankel-Toeplitz operator generalizing analogous results known to hold in the linear time-invariant (LTI) case (22; 38; 43). Our approach is purely input-output and does not use any state space realization, therefore the results derived here apply to infinite dimensional LTV systems, i.e., TV systems with an infinite number of state variables (33). Although the theory is developed for causal stable system, it can be extended in a straightforward fashion to the unstable case using coprime factorization techniques for LTV systems discussed in (11; 13). The rest of the chapter is organized as follows. Section 2 the commutant lifting theorem for nest algebras is introduced. In section 3 the optimal disturbance rejection problem is formulated and solved in terms of a TV Hankel operator. A Generalization to the TV mixed sensitivity problem is carried out in section 4. Section 5 contains some concluding remarks. Definitions and notation • B(E, F) denotes the space of bounded linear operators from a Banach space E to a Banach space F, endowed with the operator norm A := sup x∈E, x≤1 Ax, A ∈ B(E, F) • 2 denotes the usual Hilbert space of square summable sequences with the standard norm x2 2 := ∞ ∑ j=0 |xj|2 , x := x0, x1, x2, · · · ∈ 2 28 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 45. Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 3 • Pk the usual truncation operator for some integer k, which sets all outputs after time k to zero. • An operator A ∈ B(E, F) is said to be causal if it satisfies the operator equation: Pk APk = Pk A, ∀k positive integers • tr(·) denotes the trace of its argument. The subscript “c” denotes the restriction of a subspace of operators to its intersection with causal (see (11; 29) for the definition) operators. “⊕” denotes for the direct sum of two spaces. “” stands for the adjoint of an operator. 2. The commutant lifting theorem The commutant lifting theorem has been proposed by Sz.Nagy and Foias (35; 36). It has been used successfully to solve several interpolation problems including H∞ control problems for linear time invariant (LTI) systems (31; 32; 43; 44). In this chapter, we rely on a time-varying version of the commutant lifting theorem which corresponds to nest or triangular algebras. Following (3; 18) a nest N of a Hilbert space Ȟ is a family of closed subspaces of Ȟ ordered by inclusion. The triangular or nest algebra T (N ) is the set of all operators T such that TN ⊆ N for every element N in N . A representation of T (N ) is an algebra homomorphism h from T (N ) into the algebra B(H) of bounded linear operators on a Hilbert space H. A representation is contractive if h(A) ≤ A, for all A ∈ T (N ). It is weak continuous if h(Ai) converges to zero in the weak topology of B(H) whenever the net {Ai} converges to zero in the weak topology of B(Ȟ). The representation h is said to be unital if h(IȞ) = IH, where IȞ is the identity operator on Ȟ, and IH the identity operator on H. The Sz. Nagy Theorem asserts that any such a representation h has a B(Ȟ)-dilation, that is, there exists a Hilbert space K containing H, and a positive representation H of B(Ȟ) such that PHH(A) |H= h(A), where PH is the orthogonal projection from K into H (3; 18). We now state the commutant lifting theorem for nest algebras from (3; 18) (see also references therein). Theorem 1. (3; 18) Let h : T (N ) −→ B(H) h : T (N ) −→ B(H ) be two unital weak continuous contractive representations with B(Ȟ)-dilations H : B(Ȟ) −→ B(K) H : B(Ȟ) −→ B(K ) respectively. Assume that X : H −→ H is a linear operator with X ≤ 1, such that Xh(A) = h (A)X for all A ∈ T (N ), that is, X intertwines h and h . Then there exists an operator Y : K −→ K such that i) Y ≤ 1. ii) Y intertwines H and H , that is, YH(A) = H (A)Y for all A ∈ B(Ȟ). 29 Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach
- 46. 4 Will-be-set-by-IN-TECH iii) Y dilates X, that is, Y : M −→ M , and PH Y |M= XPH |M, where H = M N is the orthogonal representation of H as the orthogonal difference of invariant subspaces for H |T (N ), and similarly for H . In the next section the optimal disturbance rejection problem is formulated and solved using this Theorem in terms of a TV Hankel operator. 3. Time-varying optimal disturbance rejection problem In this chapter, we first consider the problem of optimizing performance for causal linear time varying systems by considering the standard block diagram for the optimal disturbance attenuation problem represented in Fig. 1, where u represents the control inputs, y the measured outputs, z is the controlled output, w the exogenous perturbations. P denotes a causal stable linear time varying plant, and K denotes a time varying controller. The P K w u z y Fig. 1. Block Diagram for Disturbance Rejection closed-loop transmission from w to z is denoted by Tzw. Using the standard TV Youla parametrization of all stabilizing controllers the closed loop operator Tzw can be written as (2; 11; 16), Tzw = T1 − T2QT3 (1) where T1, T2 and T3 are stable causal LTV operators, that is, T1, T2 and T3 ∈ Bc(2, 2). Here it is assumed without loss of generality that P is stable, the Youla parameter Q := K(I + PK)−1 is then an operator belonging to Bc(2, 2), and is related in a one-to-one onto fashion to the controller K (29). Note that Q is allowed to be time-varying. If P is unstable it suffices to use the coprime factorization techniques in (11; 39) which lead to similar results. The magnitude of the signals w and z is measured in the 2-norm. The performance index which quantifies 30 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 47. Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 5 optimal disturbance rejection can be written in the following form (20) μ := inf {Tzw : K being robustly stabilizing linear time − varying controller} = inf Q∈Bc(2,2) T1 − T2QT3 (2) The performance index (2) will be transformed into a shortest distance minimization between a certain bounded linear operator and a subspace to be specified shortly. In order to do so, following (11) define a nest N as a family of closed subspaces of the Hilbert space 2 containing {0} and 2 which is closed under intersection and closed span. Let Qn := I − Pn, for n = −1, 0, 1, · · · , where P−1 := 0 and P∞ := I. Then Qn is a projection, and we can associate to it the following nest N := {Qn2, n = −1, 0, 1, · · · }. In this case the triangular or nest algebra T (N ) is the set of all operators T such that TN ⊆ N for every element N in N . That is T (N ) = {A ∈ B(2 , 2 ) : PnA(I − Pn) = 0, ∀ n} = {A ∈ B(2 , 2 ) : (I − Qn)AQn = 0, ∀ n} (3) Note that the Banach space Bc(2, 2) is identical to the nest algebra T (N ). For N belonging to the nest N , N has the form Qn2 for some n. Define N− = {N ∈ N : N N} (4) N+ = {N ∈ N : N N} (5) where N N means N ⊂ N, and N N means N ⊃ N. The subspaces N N− are called the atoms of N . Since in our case the atoms of N span 2, then N is said to be atomic (3). The early days of H∞ control theory saw solutions based on the so-called inner-outer factorizations of functions belonging to the Hardy spaces H2 and H∞, and their corresponding matrix valued counterparts for multi-input multi-output (MIMO) systems (22; 23). Generalizations in the context of nest algebras have been proposed in (1; 3) as follows: An operator A in T (N ) is called outer if the range projection P(RA), RA being the range of A and P the orthogonal projection onto RA, commutes with N and AN is dense in N ∩ RA for every N ∈ N . A partial isometry U is called inner in T (N ) if UU commutes with N (1; 3; 11). In our case, A ∈ T (N ) = Bc(2, 2) is outer if P commutes with each Qn and AQn2 is dense in Qn2 ∩ A2. U ∈ Bc(2, 2) is inner if U is a partial isometry and UU commutes with every Qn. Applying these notions to the time-varying operator T2 ∈ Bc(2, 2), we get T2 = T2iT2o, where T2i and T2o are inner outer operators in Bc(2, 2), respectively. Similarly, the operator T3 can be factored as T3 = T3oT3i where T3i ∈ Bc(2, 2) is inner, T3o ∈ Bc(2, 2) is outer. The performance index μ in (2) can then be written as μ = inf Q∈Bc(2,2) T1 − T2iT2oQT3oT3i (6) Following the classical H∞ control theory (22; 23; 45),we assume (A1) that T2o and T3o are invertible both in Bc(2, 2). Assumption (A1) can be relaxed by assuming instead that the outer operators T2o and T3o are bounded below (see Lemma (1) p. 220). Assumption (A1) guarantees that the map Q −→ T2oBc(2, 2)T3o is bijective. Under this assumption T2i becomes an isometry and T3i 31 Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach
- 48. 6 Will-be-set-by-IN-TECH a co-isometry in which case T 2iT2i = I and T3iT 3i = I. The operators T2o and T3o can be absorbed into the Youla operator Q, and expression (6) is then equivalent to μ = inf Q∈Bc(2,2) T 2iT1T 3i − Q (7) Expression (7) is the distance from the operator T 2iT1T 3i ∈ B(2, 2) to the nest algebra Bc(2, 2). It is the shortest distance from the bounded linear operator T 2iT1T 3i to the space of causal bounded linear operators Bc(2, 2), which is a subspace of B(2, 2). In the sequel, the commutant lifting theorem is used to solve the minimization (7) in terms of a time varying version of Hankel operators. First, let C2 denote the special class of compact operators on 2 called the Hilbert-Schmidt or Schatten 2-class (3; 14) under the norm, A2 := tr(A A) 1 2 (8) Note that C2 is a Hilbert space under the inner product (3) (A, B) = tr(B A), ∀ A, B ∈ C2 (9) Define the space A2 := C2 ∩ Bc(2 , 2 ) (10) Then A2 is the space of causal Hilbert-Schmidt operators. This space can be viewed as the TV counterpart of the standard Hardy space H2 in the standard H∞ theory. Define the orthogonal projection P of C2 onto A2. P is the lower triangular truncation, and is analogous to the standard positive Riesz projection (for functions on the unit circle) for the LTI case. Following (27) an operator X in B(2, 2) determines a Hankel operator HX on A2 if HX A = (I − P)XA, for A ∈ A2 (11) We shall show that the shortest distance μ is equal to the norm of a particular LTV Hankel operator using the time varying version of the commutant lifting theorem in Theorem 1, thus generalizing a similar result in the LTI setting. let HB be the Hankel operator (I − P)BP associated with the symbol B := T 2iT1T 3ci. The Hankel operator HB belongs to the Banach space of bounded linear operators on C2, namely, B(C2, C2). We have then the following Theorem which relates the optimal distance minimization μ to the induced norm of the Hankel operator HT 2i T1 T 3ci . Theorem 2. Under assumptions (A1) the following holds: μ = HT 2i T1 T 3ci = (I − P)T 2iT1T 3ciP (12) Proof. Following (3; 18) let H1 = A2 and H2 = C2 A2 the orthogonal complement of A2 in C2. Define the representations h and h of A2 by h(A) = RA |H1 , A ∈ Bc(2 , 2 ) (13) h (A) = (I − P)RA|H2 , A ∈ Bc(2 , 2 ) (14) 32 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 49. Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 7 where now RA denotes the right multiplication associated to the operator A defined on the specified Hilbert space, i.e., RAB = BA, B ∈ A2. The representation h(·) and h (·) have dilations H = H given by H(A) = H (A) = RA on C2, A ∈ Bc(2 , 2 ) (15) (16) Let M := Bc(2, 2), N = {0}, M := C2, N := A2, and H1 = M N, H2 = M N are orthogonal representations of H1 and H2 of invariant subspaces under H|Bc(2,2), that is, RABc(2, 2) ⊂ Bc(2, 2). Now we have to show that the operator HT 2iT1 T 3ci intertwines h and h , that is, if B := T 2iT1T 3ci, then h (A)HB = HBh(A) holds for for all A ∈ Bc(2, 2), h (A)HB = (I − P)RA(I − P)B |A2 = (I − P)RAB |A2 (17) = (I − P)BRA |A2 = (I − P)BPRA |A2 = HBh(A) (18) Applying the Commutant Lifting Theorem for representations of nest algebras implies that HB has a dilation H̃B that intertwines H and H , i.e., H̃BH(A) = H (A)H̃B, ∀A ∈ B(2, 2). By Lemma 4.4. in (18) H̃B is a left multiplication operator acting from A2 into C2 A2. That is, H̃B = LK for some K ∈ B(2, 2), with LK = K = H̃X = HB by Lemma 4.5. (18). By Lemma 4.3. (18) K = B − Q, ∃Q ∈ Bc(2, 2) with K = HB as required. By Theorem 2.1. (26) the Hankel operator HB is a compact operator if and only if B belongs to the space Bc(2, 2) + K, where K is the space of compact operators on the Hilbert space 2. A basic property of compact operators on Hilbert spaces is that they have maximizing vectors, that is, there exists at least one operator Ao ∈ A2, Ao2 = 1 such that HB achieves its induced norm at Ao. That is, HB Ao 2 = HBAo 2 = HB (19) We can then deduce from (7) and (12) an operator identity for the minimizer, that is, the optimal TV Youla parameter Qo as follows Qo Ao = T 2iT1T 3ci Ao − HT 2iT1 T 3ci Ao where the unknown is Qo. In the next section the mixed sensitivity problems for LTV systems is formulated and solved using the commutant lifting theorem. 4. The time-varying mixed sensitivity problem The mixed sensitivity problem for stable plants (42; 45) involves the sensitivity operator T1 := W 0 , the complementary sensitivity operator T2 = W V P and T3 := I which are all assumed to belong to Bc(2, 2 × 2), and is given by the minimization (13; 20; 46) μo = inf Q∈Bc(2,2) W 0 − W V PQ (20) where · stands for the operator norm in B(2, 2 × 2) given by B = sup x2≤1, x∈2 B1x2 2 + B2x2 2 1 2 , B = B1 B2 (21) 33 Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach
- 50. 8 Will-be-set-by-IN-TECH The optimization problem (20) can be expressed as a shortest distance problem from the operator T1 to the subspace S = T2P Bc(2, 2) of B(2, 2 × 2). To ensure closedness of S, we assume that WW + VV 0, i.e., WW + VV as an operator acting on 2 is a positive operator. In this case, there exists an outer spectral factorization Λ1 ∈ Bc(2, 2), invertible in Bc(2, 2) such that Λ 1Λ1 = WW + VV (1; 11). Consequently, Λ1P as a bounded linear operator in Bc(2, 2) has an inner-outer factorization U1G, where U1 is inner and G an outer operator defined on 2 (3). Next we assume (A2) G is invertible, so U1 is unitary, and the operator G and its inverse G−1 ∈ Bc(2, 2). The assumption (A2) is satisfied when, for e.g., the outer factor of the plant is invertible. Let R := T2Λ−1 1 U1, assumption (A2) implies that the operator RR ∈ B(2, 2) has a bounded inverse, this ensures closedness of S. It follows from Corollary 2 (1), that the self-adjoint operator RR has a spectral factorization of the form: RR = ΛΛ, where Λ, Λ−1 ∈ Bc(2, 2). Define the operator R2 := RΛ−1, then R 2 R2 = I, and S has the equivalent representation, S = R2Bc(2, 2). After absorbing Λ into the free parameter Q, the optimization problem (20) is then equivalent to: μo = inf Q∈Bc(2,2) T1 − R2Q (22) The minimization problem (22) gives the optimal mixed sensitivity with respect to controller design (as represented by Q). It is solved in terms of a projection of a multiplication operator. If the minimization (22) is achieved by a particular Qo, we call it optimal. Theorem 3. Introduce the orthogonal projection Π as follows Π : A2 ⊕ A2 −→ (A2 ⊕ A2) R2A2 Under assumptions (A2) the following holds: μo = ΠT1 (23) Proof. Denote by S := (A2 ⊕ A2) R2A2. That is, S is the orthogonal complement of the subspace R2A2 in A2 ⊕ A2, and define the operator Ξ : A2 −→ S Ξ := ΠT1 (24) We shall show with the help of the commutant lifting theorem that μo = Ξ (25) To see this we need, as before, a representation of Bc(2, 2), that is, an algebra homomorphism, say, h(·) (respectively h (·)), from Bc(2, 2), into the algebra B(A2, A2) (respectively Bc(S, S)), of bounded linear operators from A2 into A2 respectively from S into S . Define the representations h and h by h : Bc(2 , 2 ) −→ B(A2, A2), h : Bc(2 , 2 ) −→ Bc(S, S) (26) h(A) := RA, A ∈ Bc(2 , 2 ), h (A) := ΠRA, A ∈ Bc(2 , 2 ) 34 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 51. Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 9 where now RA denotes the right multiplication associated to the operator A defined on the specified Hilbert space. By the Sz. Nagy dilation Theorem there exist dilations H (respectively H ) for h (respectively h ) given by H(A) = RA on A2 for A ∈ Bc(2 , 2 ) (27) H (A) = RA on A2 ⊕ A2 for A ∈ Bc(2 , 2 ) (28) The spaces A2 and S can be written as orthogonal differences of subspaces invariant under H and H , respectively, as A2 = A2 {0}, S = A2 ⊕ A2 R2A2 (29) Now we have to show that the operator Ξ intertwines h and h , that is, h (A)Ξ = Ξh(A) for all A ∈ Bc(2, 2), h (A)Ξ = ΠRAΠT1 |A2 = ΠRAΠT1 |A2 = ΠRAT1 |A2 = ΠT1RA |A2 = Ξh(A) Applying the commutant lifting theorem for representations of nest algebras implies that Ξ has a dilation Ξ that intertwines H and H , i.e., Ξ H(A) = H (A)Ξ , ∀A ∈ B(2, 2). By Lemma 4.4. in (18) Ξ is a left multiplication operator acting from A2 into A2 ⊕ A2, and causal. That is, Ξ = LK for some K ∈ Bc(A2, A2 ⊕ A2), with K = Ξ = Ξ. Then Ξ = ΠT1 = ΠK, which implies that Π(T1 − K) = 0. Hence, (T1 − K) f ∈ R2A2, for all f ∈ A2. That is, (T1 − K) f = R2g, ∃g ∈ A2, which can be written as R 2(T1 − K) f = g ∈ A2. In particular, R 2(T1 − K) f ∈ Bc(2, 2), for all f ∈ Bc(2, 2) of finite rank. By Theorem 3.10 (3) there is a sequence Fn of finite rank contractions in Bc(2, 2) which converges to the identity operator in the strong *-topology. By an approximation argument it follows that R 2 (T1 − K) ∈ Bc(2, 2). Letting Q := R 2(T1 − K) we have g = Q f . We conclude that T1 − K = R2Q, that is, T1 − R2Q = K, with K = Ξ, and the Theorem is proved. The orthogonal projection Π can be computed as Π = I − R2PR 2 (30) where I is the identity operator on A2 ⊕ A2, R 2 is the adjoint operator of R2. To see that (30) holds note that for any Y ∈ A2 ⊕ A2, we have (I − RPR 2 )2 Y = (I − RPR 2 )(I − RPR 2)Y (31) = (I − R2PR 2 − R2PR 2 + R2PR 2PR 2 )Y (32) but R 2 R2 = I and P2 = P, therefore (I − RPR 2 )2 Y = (I − RPR 2)Y (33) This shows that (I − RPR 2) is a projection. The adjoint of (I − RPR 2), (I − RPR 2), is clearly equal to (I − RPR 2 ) showing that it is an orthogonal projection. Now we need to show that the null space of (I − RPR 2 ) is R2A2. Let Z ∈ A2 ⊕ A2 such that (I − RPR 2 )Z = 0, so Z = R2PR 2 Z. But R 2 Z ∈ C2, then PR 2 Z ∈ A2, implying that Z ∈ R2A2. We have showed 35 Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach
- 52. 10 Will-be-set-by-IN-TECH that the null space of the projection (I − RPR 2) is a subset of R2A2. Conversely, let Z ∈ A2, then (I − RPR 2 )R2Z = R2Z − R2PZ = R2Z − R2Z = 0 (34) hence R2Z belongs to the null space of (I − RPR 2 ), and (30) holds. The operator Ξ has the following explicit form Ξ = (I − R2PR 2 )T1 (35) which leads to the explicit solution μo = (I − R2PR 2)T1 (36) The expression generalizes the solution of the mixed sensitivity problem in the LTI case (25; 43; 46) to the LTV case. This result also applies to solve the robustness problem of feedback systems in the gap metric (38) in the TV case as outlined in (11; 21; 33), since the latter was shown in (11) to be equivalent to a special version of the mixed sensitivity problem (20). 5. Conclusion The optimal disturbance rejection and the mixed sensitivity problems for LTV systems involve solving shortest distance minimization problems posed in different spaces of bounded linear operators. LTV causal and stable systems form a nest algebras, this allows the commutant lifting theorem for nest algebras to be applied and solve both problems in term of abstract TV Hankel and a TV version generalization of Hankel-Topelitz operators under fairly weak assumptions. Future work includes investigation of numerical solutions based on finite dimensional approximations, and computation of the corresponding controllers. Acknowledgement: This work was partially supported by the National Science Foundation under NSF Award No. CMMI-0825921. 6. References [1] Arveson W. Interpolation problems in nest algebras, Journal of Functional Analysis, 4 (1975) pp. 67-71. [2] Chapellat H., Dahleh M. Analysis of time-varying control strategies for optimal disturbance rejection and robustness, IEEE Transactions on Automatic Control, 37 (1992) 1734-1746. [3] Davidson K.R. Nest Algebras, Longman Scientific Technical, UK, 1988. [4] Djouadi S.M. Optimal Robust Disturbance Attenuation for Continuous Time-Varying Systems, Proceedings of CDC, vol. 4, (1998) 3819-3824. 1181-1193. [5] Djouadi S.M. Optimal Robust Disturbance Attenuation for Continuous Time-Varying Systems, International journal of robust and non-linear control, vol. 13, (2003) 1181-1193. [6] G. Zames and A.K. El-Sakkary, Unstable Systems and Feedback: The Gap Metric, in Proc. of the Allerton Conference, pp. 380-385, 1980. [7] S.Q. Zhu, M.L.J. Hautus and C. Praagman, Sufficient Conditions for Robust BIBO Stabilisation: Given by the Gap Metric, Systems and Control Letters, vol. 11, pp. 53-59, 1988. 36 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 53. Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 11 [8] Khammash M., Dahleh M. Time-varying control and the robust performance of systems with structured norm-bounded perturbations, Proceedings of the IEEE Conference on Decision and Control, Brighton, UK, 1991. [9] Khammash M., J.B. Pearson J.B. Performance robustness of discrete-time systems with structured uncertainty, IEEE Transactions on Automatic Control, 36 (1991) 398-412. [10] A. Ichikawa, Quadratic Games and H∞-Type Problems for Time-Varying Systems, Int. J. Control, vol. 54, pp. 1249-1271, 1991. [11] Feintuch A. Robust Control Theory in Hilbert Space, Springer-Verlag, vol. 130, 1998. [12] Feintuch A., Francis B.A. Uniformly Optimal Control of Linear Time-Varying Systems, Systems Control Letters, vol. 5, (1984) 67-71. [13] Feintuch A., Francis B.A. Uniformly Optimal Control of Linear Feedback Systems, Systems Control Letters, vol. 21, (1985) 563-574. [14] Schatten R. Norm Ideals of Completely Continuous Operators, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1960. [15] Shamma J.S. Robust stability with time-varying structured uncertainty, Proceedings of the IEEE Conference on Decision and Control, 3163-3168, 1992 . [16] Shamma J.S., Dahleh M.A. Time-varying versus time-invariant compensation for rejection of persistent bounded disturbances and robust stabilization, IEEE Transactions on automatic Control, 36 (1991) 838-847. [17] P. DeWilde A-J. Van der Veen, Time-Varying Systems and Computations, Springer-Verlag, 1998. [18] J.A. Ball, Commutant Lifting and Interpolation: The Time Varying Case, Intergral Equat. and Operator Theory, vol. 25, pp. 377-405, 1996. [19] S.M. Djouadi and C.D. Charalambous, Time-Varying Optimal Disturbance Minimization in Presence of Plant Uncertainty, SIAM Journal on Optimization and Control, Vol. 48, No. 5, pp. 3354-3367, 2010 [20] S.M. Djouadi, Disturbance rejection and robustness for LTV Systems, Proceedings of the American Control Conference, pp. 3648-2653-6080, June 14-26, 2006. [21] S. M. Djouadi, Commutant Lifting for Linear Time-Varying Systems, American Control Conference Proceedings, St. Louis, MO, June 10-12, 2009, pp. 4067-4072. [22] Francis B.A., Doyle J.C. Linear Control Theory with an H∞ Optimality Criterion, SIAM J. Control and Optimization, vol. 25, (1987) 815-844. [23] Francis B.A. A Course in H∞ Control Theory, Springer-Verlag, 1987. [24] Zhou K., Doyle J.C., Glover K. Robust and Optimal Control, Prentice Hall, 1996. [25] M. Verma and E. Jonckheere, L∞-Compensation with Mixed Sensitivity as a Broadband Matching Problem, Systems Control Letters, vol. 4, pp. 125-129, 1984. [26] Power S. Commutators with the Triangular Projection and Hankel Forms on Nest Algebras, J. London Math. Soc., vol. 2, (32), (1985) 272-282. [27] Power S.C. Factorization in Analytic Operator Algebras, J. Func. Anal., vol. 67, (1986) 413-432. [28] S.M. Djouadi and C.D. Charalambous, On Optimal Performance for Linear-Time Varying Systems, Proc. of the IEEE 43th Conference on Decision and Control, Paradise Island, Bahamas, pp. 875-880, December 14-17, 2004. [29] Feintuch A., Saeks R. System Theory: A Hilbert Space Approach, Academic Press, N.Y., 1982. [30] Peters M.A. and Iglesias P.A., Minimum Entropy Control for Time-varying Systems, Boston, Birkhäuser, 1997. 37 Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach
- 54. 12 Will-be-set-by-IN-TECH [31] T.T. Georgiou and M.C. Smith, Optimal Robustness in the Gap Metric, IEEE Trans. on Automatic Control, vol. 35, No. 6, pp. 673-686, 1990. [32] Georgiou T., On the Computation of teh Gap Metric, Systems and Control Letters, vol. 11, pp. 253-257, 1988. [33] Foias C., Georgiou T. and Smith M.C., Robust Stability of Feedback Systems: A Geometric Approach Using The Gap Metric, SIAM J. Control and Optimization, vol. 31, No.6, pp. 1518-1537, 1993. [34] Foias C., Georgiou T. and Smith M.C., Geometric Techniques for Robust Stabilization of Linear Time-Varying Systems, Proc. of he 29th CDC, pp. 2868-2873, December 1990. [35] B. Sz. Nagy and C. Foias, Dilatation des commutants des commutateurs d’operateurs, C.R. Academie Sci. paris, Serie A, 266, pp. 493-495, 1968. [36] B. Sz. Nagy, C. Foias, H. Bercovici, and L. Kerchy, Harmonic Analysis of Operators on Hilbert Space, Springer, 2010. [37] Ravi R., Nagpal K.M. and Khargonekar P.P., H∞ Control of Linear Time-Varying Systems: A State Space Approach, SIAM J. Control and Optimization, vol. 29, No.6, pp. 1394-1413, 1991. [38] T.T. Georgiou and M.C. Smith, Robust Stabilization in the Gap Metric: Controller Design for Distributed Plants, IEEE Trans. on Automatic Control, vol. 37, No. 8, pp. 1133-1143, 1992. [39] Dale W.N. and Smith M.C., Stabilizability and Existence of System Representation for Discrete-Time-Varying Systems, SIAM J. Control and Optimization, vol. 31, No.6, pp. 1538-1557, 1993. [40] Dullerud G.E. and Lall S., A new approach for analysis and synthesis of time-varying systems IEEE Trans. on Automatic Control, Vol.: 44 , Issue: 8 , pp. 1486 - 1497, 1999. [41] C. Pirie and G.E. Dullerud, Robust Controller Synthesis for Uncerttain Time-Varying Systems. SIAM J. Control Optimization., Vol. 40, No. 4, pp. 1312Ű1331, 2002. [42] J.C. Doyle, B.A. Francis and A.R. Tannenbaum, Feedback Control Theory, Macmillan, NY, 1990. [43] C. Foias, H. Ozbay and A.R. Tannenbaum, Robust Control of Infinite Dimensional Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1996. [44] C. Foias, A.E. Frazho, I. Gohberg, and M.A. Kaashoek, Metric constrained interpolation, commutant lifting and systems. Operator Theory: Advances and Applications, 100. Birkhäuser Verlag, Basel, 1998 [45] Zhou K. and Doyle J.C., Essentials of Robust Control, Prentice Hall, 1998. [46] Djouadi S.M. and Birdwell J.D., On the Optimal Two-Block H∞ Problem, Proceedings of the American Control Conference, pp. 4289 - 4294, June 2005. 38 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics
- 55. Another Random Scribd Document with Unrelated Content
- 56. there, and found General Brown and Colonel Howard with a portion of the command in a shed surrounded by cars. I asked Colonel Howard—he was the first officer I saw—asked him what he was doing there. He said he was there by orders, and asked me the state of affairs in the city. I told him the condition in the neighborhood of the Union depot. There was a large mob upon the street—from the Union depot to Twenty-eighth street was crowded with a mob, and they were hooting, yelling, and threatening what they would do. I heard them shouting, they would break into the armories and gun stores—that was the common threat that was used by the mob on the street. That afternoon, Colonel Howard said to me, that he would see General Brown, who was up stairs in the transfer office. General Brown came down. He asked me the condition. I made to him the same statement I had made to Colonel Howard, and I advised him to take another position than that—he had first asked me what my advice would have been. I told him that I thought he might get a better place for the men than that. I then came down to the city, to the armory, and found the mob there who had broken in and taken all the arms that had been left in the armory, with the exception of those that had been concealed. They got the arms and left. By Senator Reyburn: Q. Where is your armory? A. It is on Market street. It is in the city property, over one of the market-houses. All of the regiments are quartered there in the city property. I suppose I had been there fifteen or twenty minutes when the balance of the officers, and the men of the regiment, came down into the armory. There was nothing more done that night. I was on the streets, I suppose, until twelve o'clock, until an alarm from the box at Twenty-eighth street sounded for fire, and, in company with several other persons, we started on the hill above the hospital, or near the hospital, not quite so far out, and there saw
- 57. what I took to be the first car that was burned that had been set on fire. It was running down near the round-house. I remained on the hill a good portion of that evening—it was then morning. Q. Sunday morning? A. Sunday morning; yes, sir. Sunday morning I came home and laid down awhile, and got breakfast, and started to the city to see if anything had been done to gather the battalion together. I saw none of the officers on the street at all, I believe, with the exception of the assistant adjutant general, Colonel Moore, of our division, and one officer of General Pearson's staff. There was nothing done that day until afternoon. A citizen came up to the house—I had gone home. He said to me, says he, The citizens are going to put this thing down. They would like to get your arms. I said, No; if the citizens will back us up, we will find a gun for every man. We have got all ours. All we want is ammunition and backing. I sent out and I had as many men of my company who lived near me come to a room in my own house, and I there ordered those men to go out, and order the balance of the company to take their accoutrements, which they had so thrown off, and come to the mayor's office. I did not want to go to my own armory. In fact, I was afraid to go there, from the fact that it was a place that could not have been defended at all. It could easily be set fire to. Went to the mayor, and asked him if I could have one of the rooms in which to assemble men. He said, Certainly. He was glad I had come. I sent one of my officers out—my first lieutenant, Mr. Brown—out to our colonel's house. I sent my lieutenant out to notify Colonel Howard what I had done—I had ordered the company to assemble at the mayor's office—and asking if he would procure us more ammunition than what I had. He came in—General Brown came in—and General Brown gave me an order on Major Buffington, and Mayor McCarthy furnished us with a wagon in which to go out and get ammunition. Major Buffington returned word that they had scarcely sufficient ammunition to give to their own guard. I stated the case to Mayor McCarthy, and Mayor
- 58. McCarthy, about eleven o'clock, started out, and, in the course of a half or three quarters of an hour, he came back and handed me two hundred and forty rounds which he had got. He asked me if I would remain with him during the night, and put down any disturbance. I said I would, subject to the orders of my superior officers. On Monday morning, about half past eight or nine o'clock, I had left the men go to get something to eat, and the mayor said to me, I don't want you to let the men go away from here unless under orders, and I will see that they receive provisions; that I should take my men around to a saloon near there; they would be attended to there. Shortly after breakfast he notified me there was a boat load of miners coming down on the packet from Elizabeth. He had received information that they had come down, and expected a pretty rough time, and asked me to go with them. I assembled the men, and General Brown went down with us. There was a squad of police. As I recollect the line of march, there was a squad of police in front. There was my company, and then there was a company of citizens, armed with shot-guns, rifles, and carbines, under the command of General Negley. General Negley and Mayor McCarthy and General Joe Brown were with us. We went down to Smithfield street, and we learned that the men, in place of coming down on the packet as far as its regular landing, had got off about half a mile above the landing, and come down that way; I suppose, so as to get in the city without any trouble. I threw my company across Grant street, and blocked up the passage. Ordered the men to load, and I saw then, while standing in front of the command, General Negley and Mayor McCarthy and others, making addresses to this band of miners—they were all reputed to be miners—I do not know whether they were or not. The crowd was dispersed. We marched down to Water street. There were no shots fired—no disturbance of any kind. They seemed to be pacified by the remarks made by the officers. We then went back to city hall, to the mayor's office, and were quartered there until the afternoon, when I received orders to report to my colonel. I reported to him on First avenue, and after supper we were sent to
- 59. our armory again. During the night, between ten and eleven o'clock, I was ordered out again to support another detachment of the police. It seems that a party of roughs from Cumberland had taken a train, and taken possession of it, and the mayor was notified, and he sent down a detail of police, and we were sent down to support the police. The police had men under arrest before we got there, and the next day we escorted these men and the police over to Allegheny, to take the cars for Claremont. That was about all the trouble—all the duty we really did, with the exception of some ordinary patrolling—marching around. There was nothing of any importance. By Senator Yutzy: Q. You came from the transfer station to your armory in the city— this was on Saturday night? A. This was on Saturday night. Q. What time in the night? A. I should judge it would be about eleven o'clock. Q. Bring your command down to the armory? A. No, sir. Q. Did you leave your command there? A. Yes, sir. Q. And never went back to the regiment after that, that night? A. That night. No, sir. The regiment was disbanded, as I understood it. Q. You understood that next day?
- 60. A. Yes, sir; well, I knew of the fact within half an hour afterwards as I stated. The balance of the officers came there while I was still in the armory. Q. What was your object in going to the armory? A. My object in going there was to save any property that could possibly be saved. We had considerable property there of ours independent of the State's. Q. Were you in uniform when you came down to the armory? A. No, sir. Q. Citizen's dress? A. Plain clothes. Q. Were you in citizen's dress during the time you were in command? A. No, sir. Q. You changed your uniform? A. It would be impossible, in my own opinion, for any man to have gone through that mob in uniform. Q. You rallied your men on Sunday and went to the mayor's office— did you remain there during all of Sunday? A. All of Sunday evening and Sunday night and up until Monday, until we went with the mayor to stop the progress of these miners. Q. Where was the balance of your regiment? A. I do not know, sir. I know they were assembled on Monday—I do this from hearsay, which, of course, is not evidence—I know that there were two of the companies in Allegheny who were doing duty of one kind or other over there.
- 61. By Senator Reyburn: Q. There was no organization of the regiment—you received no orders from the colonel? A. No, sir. Q. You were acting independently? A. Yes; I was acting independently in support of the mayor. Thomas Graham, sworn: By Senator Reyburn: Q. Give your full name and address? A. Thomas C. Graham. Q. Where do you reside? A. I reside in the Fifteenth ward of Pittsburgh. Q. What is your occupation? A. I am janitor of the city hall. Q. Have you any information concerning the riots of July, the subject which we are investigating? A. I was present at Twenty-eighth street, at the side of the hill, at the time the firing was done. Q. Be good enough to state, then, what you know about it? A. I was on my way home about a quarter past four o'clock, I think, to the best of my knowledge, and General Brown was in the car with me; the car was pretty well crowded; we were standing up. When
- 62. we arrived at Twenty-eighth street he said to me, Come along up and see the military; come up along. I said I didn't care about going up, and I didn't think that any one outside of the railroad employés had any business there; but he said he was not going to stop, and rather insisted, and I got off the car, and went up there. We crossed over the track, and away from the crowd altogether, on the south side of the track, beside of the hill, and we walked up that way, leading along the public road—it is used as a public thoroughfare, to a certain extent—and there halted. I thought by going up there we could have a good view of everything that was going on, and get out of danger. I found out my mistake afterwards, though. When the military came up I was standing right about seventy-five or one hundred feet from the tracks, on the side of the hill. Was elevated a considerable distance, and had a good view over all that was going on. I kept moving my head, or rather my eyes rolling, from one point to the other, taking all in that was going on, and when the troops came up and cleared the track on both sides, the battery came up and across, through Twenty-eighth street; they could not get through, but part of them came to support the battery; came up, and when they got up to the crowd—there is a watch-box stationed on the side of the hill, by the side of the hospital gate, and there was a crowd of half grown boys congregated around that box. I seen, as soon as they came up by the boys, they stooped down and picked up stones, there was like to be trouble, but I concluded I would be safe where I was, and remained there. As soon as the troops came up to Twenty-eighth street crossing, the boys commenced to throw at the troops, and some missiles were thrown from the corner of the hospital grounds. Then the firing began, and continued for quite a length of time. I stayed where I was, until I saw two men fall, one of them as close as to that window, the other one further down, towards the track. There was a ravine in the rear of where I was standing, and I made the remark to a gentleman standing by me that it appeared to be getting very warm here, we had better get out of this, and I leaped right into the ravine, and
- 63. there remained until the firing was over, and then came down and went home. I didn't come out of my house down the street, as I live eight squares further from there out. I didn't come in till that night. That was about what I saw of the occurrence. Q. How many people were engaged in this throwing? A. Well, as I said, the starting point of the throwing came from the side of the watch-box—the watch-box of the man who tends switch. There was about a dozen of them around there, and that was where the throwing commenced. Q. Did you see any of the troops struck with stones? A. No; I didn't see anyone struck particularly, but I would consider it would be impossible most to throw into that crowd without striking some one. I didn't see any missiles. Q. Were you close enough to hear any command given by the officers? A. I was about seventy-five or one hundred feet when the throwing begun; I heard the word fire very distinctly. Q. Where did it come from? A. It appeared to come from the head of the column. Q. Was it in the crowd? A. I should say, that it came from the head of the military column— there was not more than a space of, I suppose, thirty feet, and it might have been a little more, it could not have been much more than that from the head of the column to Twenty-eighth street, where the crowd was. Q. From the head of the column? A. Yes, sir.
- 64. Q. Which column do you mean? A. I am not a military man. Q. You mean the company marched up? A. No, sir; the company that marched up was then as close as I am to the other side of the table, with their arms at a charge. Q. Close to what? A. To the crowd at Twenty-eighth street. Q. Trying to press the crowd back? A. Trying to press the crowd or make their way through them—was not thrusting or anything of that kind. Q. You don't know who gave this command, or was it an exclamation you heard in the crowd? A. I am under the impression that it was a command or a military order, but I would not swear who it was that gave it. I could not do that. By Senator Yutzy: Q. Did it appear to be in front of the command—the head of the column where the command came from to fire? A. I was standing immediately opposite the head of the column, and the sound of that command appeared to come directly opposite to me, down at the head of the column. Q. Did you see any officers in advance of that command? A. When the military marched up? Q. At that time, when you heard this command, were any officers in advance of the column?
- 65. A. I seen several officers. The most of the officers were strangers. I don't know who they were. I could see they were officers, by their uniform. Q. Were they in front of the men? A. They were in front of them at the head of the column. The troops were formed in a hollow square. They marched up, and then got the command front. They marched to the side of the hill, to clear the track of any that might be there, and those who were standing on the track got up on the side of the hill. There were very few on that side. The rear rank got the command, To the rear, open order, march. Q. You heard these commands? A. Yes; very distinctly. They got the command to about, and then they marched to the north side, which left an interval of two or three tracks clear—formed a hollow square. Q. This company marched down between these two lines? A. This company appeared to be at the head of the column, and they marched through up the lines. There was a portion of them came away to let them in, and they marched up. Q. Was there a company marched up between those two lines, after the open order? A. The company appeared to come from the head of the column. Q. Wheeled out from the head of the column? A. I don't think—I am not positive, but I am under the impression it was separated from the column. I would not swear positively. They were dressed in blue. Q. How did they march up—company front—in line of battle?
- 66. A. They marched up, I think, in sections of about four—I am not positive about that. Q. Until they reached—— A. Nearly to Twenty-eighth street, then they halted. Q. And saw them open order, and one rank faced about, and they took positions on two sides of the track, which left a place between? A. Yes, sir. Q. Then was there a company marched up between these two? A. I am not positive, but I think this company was taken from the head, or a portion of it—it was not a full company, it was what you might term a squad—there was not, I suppose, over twenty-five. Q. It was taken from the head of the column? A. Yes, sir. Q. Was there a company marched up between those two ranks? A. I think a battery. I think this company, as I said, was taken from the head of the column, and marched up to support the battery, in order to get through the crowd at Twenty-eighth street. Q. Did this company turn around and face the crowd—how did they face? This crowd, you said, they marched up and tried to press the crowd back, and they wheeled out from the column and marched up against the crowd, or did they go down between the two lines, and march up against the crowd? A. I stated that they appeared to be reserved for the purpose of supporting that battery, as they were not brought from the rear in front rank at all. By Senator Reyburn:
- 67. Q. Did you see the sheriff and his posse? A. I did. Q. Where were they? A. They were at the head, coming up—the head of the column. Q. All of them? A. All of them. I recognized Sheriff Fife, Mr. Pitcairn, superintendent of the Pennsylvania railroad, and General Pearson at the head. When they came up, the sheriff attempted to say something to the crowd, and there was such jeering and hallooing, it was impossible to be heard from where I was standing. Q. You are positive about hearing this command to fire—was not this jeering—— A. That jeering and hooting was not at that particular time. There appeared to be a little confusion when that portion of the company came up, the jeering only began during the speaking of the sheriff. I don't know whether he was reading the riot act or not. I don't know what he was doing; of course I could not hear it. E. F. A. Hastings, sworn: By Senator Reyburn: Q. What is your full name? A. E. F. A., not quite the whole alphabet. Q. Where do you reside, Mr. Hastings? A. I live between Twenty-second and Twenty-third now. Q. What is your occupation?
- 68. A. Machinist. Q. Were you present during the disturbance last July? A. Yes, I was. Q. State what came under your observation? A. I was there on Twenty-eighth street, on the side of the hill, when the troops were coming up, and I waited there until they came pretty well up the track, then I came down the hill, and I looked for my boy. Q. What time was this? A. On Saturday afternoon, when I seen the troops come up—and they came in regular—I think it was four deep. They came up and stopped and halted. They turned around in open order, formed in a hollow square, and I turned around and came away with the young man, and I took him off the track, and started him down towards Penn street. I turned around and looked for my boy, and I could not see him. Walked in towards the cars, and stood by the side of the sand-house—the cars extended up a little ways beyond the sand- house—and I got right in to the end of the car. Then came orders to charge bayonets. I turned to get back, and the crowd was behind, and I could not get back. I got a bayonet right in behind here. [Indicating.] Just at that time Pearson stood in about that direction. Q. Who do you mean by Pearson? A. General Pearson, or Pierson, or whatever you call him. He was looking in the direction towards the watch-box. There was some stones being thrown over there—it would fly all to pieces, it appeared to be like clay. There was only one stone I could distinguish, about that large, [indicating,] the shape of an oyster shell. It came from there. Then there was a couple of old shoes—I didn't see anybody struck with them. Pearson turned around, when he was standing there he was looking about this—he turned around
- 69. towards the men, and his officer standing here—I don't know whether they belonged to Pittsburgh, or where they belonged, and I don't know whether they were officers. Monkey jackets it was, I think. He says, Order your men to fire. He repeated the word fire louder than he did the others, and turned around, and walked right down the track after that. I did not see them commence firing, and I dropped right down. They fired on that corner, on the side of the hill first—these men in front with the black plumes in their hat. I don't know one from the other. They wheeled round, and fired down Twenty-eighth street. They walked over the top of me. I laid there. I don't know where Pearson or any of the rest went after that. I got up, and helped this man in front of me, that was killed—I helped him back. Q. What man do you mean? A. Some say it was Dearmot, I don't know his name. Q. Killed by the cars? A. No; killed by the firing. A gun was right up against his breast when he was shot. Q. Where did this fire come from—those men standing by the cars? A. The military all around that hollow-square, except the lower end. I laid there; I was right underneath them; could see the whole thing that was going on. Some of the men fired right up in the air. I don't know whether they belonged to Philadelphia or not. By Senator Yutzy: Q. From what part of the line of this hollow-square did the firing commence? A. I will show you in just about a minute.
- 70. [Witness illustrates on paper, the situation of the troops during the firing.] Q. Just say where the first fire commenced in that hollow square, so that the reporter can take it down. A. It was near the corner, on the side of the track next to the hill. Q. Near the corner of the square of troops, next to the hill? A. There was no square there. It was round. Q. It came from the right, next towards the hill? A. Next towards the hill. Q. How many shots were fired at first? A. They shot like a little fellow would throw a lot of shooting crackers out. Q. How long after you heard the command to fire did this shooting commence? A. It was right by those other fellows standing over on this side— right by them. I guess there was four or five of them repeated it. I took them to be officers. Q. Dressed differently? A. Yes, sir. By Senator Reyburn: Q. Were they in front of their men when they gave the order? A. They were in this hollow square. Q. Did they turn round to fire?
- 71. A. They were facing the crowd to fire. Pearson turned and gave these men the command. Q. It was General Pearson that gave the command? A. Yes, sir. Q. You are positive it was General Pearson? A. I am positive. I was standing close enough to hear him. Q. Did he give the command to fire, or was he cautioning the men? A. He gave the command to fire, and repeated the word fire louder than he did all the others. By Senator Yutzy: Q. Were there any other officers in that hollow square, that you knew, besides General Pearson? A. None to my knowledge, that I knew. There were some men that I knew, coming up ahead of them. Mr. Pitcairn and Mr. Watt came up ahead of them. I know some of them, beside, but don't know their names. Q. Do you know Colonel Brown or Moore? A. I don't know him by name. Q. Did General Pearson give this command to an officer standing close by him? A. Yes; called them officers. Q. You saw him when he gave the command? A. The men that had swords, I think. Q. You saw General Pearson when he gave this order?
- 72. A. Yes, sir. Q. How close was he to the men he gave the order to? A. He was standing looking towards the switch box, where these were coming from. Right in front there were some men had a bayonet in another, and he wanted to get it away. He wheeled round this way and gave the orders for these men to fire, and then walked right down the track, but wherever he went to I could not say. Q. Was he standing close to where the men commenced firing when he gave the order to those men? You say he wheeled around; those he gave the command to must have been behind him? A. Here is where he stood. [Illustrating.] He wheeled around to the officers to the rear, and they passed the command to the officers in front. They repeated the order to fire. By Mr. Means: Q. Did the officers who repeated that command repeat it in a distinct, loud tone of voice? A. Yes; they repeated it distinctly. Q. How many of them? A. There were some three or four, I could not say exactly. I am positive there was three or four, if not more—positive of three. Q. And then the firing was done—it was not a volley of musketry—it was just done at random, was it? A. The first squad that fired there, I don't believe there was more than about half a second between them, and the squads, as soon as this squad fired on this side first, then these other fellows here fired, [indicating,] and the crowd broke away and run down. I laid there. They wheeled right down over me, and fired down Twenty-eighth street.
- 73. By Senator Reyburn: Q. Did the crowd scatter when they fired? A. The crowd scattered. By Mr. Means: Q. How many of the soldiers fired at that time—at the first command? A. The first fire? Q. When General Pearson gave the command to fire, then his officers under him repeated the command. Now how many soldiers fired when these officers repeated that command? A. It looked about like a platoon—about twenty-five, I suppose, if not more. By Senator Reyburn: Q. Were the crowd resisting these men? What were they doing when the soldiers came up? A. They were on Twenty-eighth street crossing—the railroad crosses Twenty-eighth street—they were on there. They were talking and hollering. Some man called Pearson—that was the man I took away —called Pearson a son of a bitch, and these men took Pearson's part. Says he: Don't call Al a son of a bitch; he is a friend of mine! I thought there was going to be a fight between them, too, and a man called me over and told me to get him away. I took him away with me across the track towards the round-house, and he started on down. There was nobody in front of me at all, and I had hardly got in there until the crowd was right at my heels. By Mr. Means:
- 74. Q. You say there was about twenty-five men fired? How many men fired after that? A. They appeared to be firing in squads all over the line. Q. How long did this firing last? A. I don't suppose it lasted more than about two and a half or three minutes—could not have lasted any longer, I think. Q. What were the officers doing? A. Some of them went down the track flying—running over the other side. Q. They were hunting their quarters? A. They were hunting their quarters. By Mr. Dewees: Q. Did you hear the order to load? A. No, sir; they were all loaded before they got there. They could not have loaded in that time. After the first volley was fired, then I seen them loading—those men in front. Q. Don't you know who gave the order to load? A. I didn't hear anybody give the order to load. Q. They loaded without orders? A. I suppose so, after the first fire. By Senator Reyburn: Q. Did you see them load afterwards? A. Yes; breech loaders.
- 75. Q. Did you see the operation? A. I saw them pull the cock back—that part that turns back—and put in a cartridge. I was lying right underneath them when they were doing it. The parties in front fired the last shot down Twenty-eighth street. I don't know who they were. They were men with black plumes in their hats. Q. You say the parties next to Twenty-eighth street were the last to fire? A. They wheeled right round—— Q. That was the party that fired first? A. No; the party that fired first was over here. [Illustrating.] Q. The party facing down Liberty street? A. Yes; they fired. There was some that was against a car that could not do anything. Q. Did they fire before the front line fired? A. They fired about the same time. There was a car stood in here when they fired, so that you could do nothing. Some of them fired up like. Q. Fired in the air, did they? A. They appeared to fire in the air. Q. How did these men of this side? You say they turned and fired the other way? A. No, sir; those men in front wheeled round this way, and fired down Twenty-eighth. Fired up first and then—— Q. Where did they deliver their first fire—the men on that north side of the track?
- 76. A. They fired some of them right up square—down below the car. I could not see on account of the cars. By Senator Yutzy: Q. You are positive that the firing did not come from that side of the line next towards Liberty street first? A. I am positive it did not, for the first firing commenced along the side of the hill—that part I saw. By Mr. Dewees: Q. How far was General Pearson out from the military when this command to fire was given? A. He was in the hollow square. Q. How far from the men? A. It would appear to be in the center, about Twenty-eighth street. You know this line went round on Twenty-eighth, and covered over part of Twenty-eighth street towards the hill. Q. Did you see the fire during the night? A. I was down on Penn street when the fire started. I do not know anything about that. I went up there to see it, and stood on Liberty street, watching the fire. Q. Did you see the troops come out of the round-house? A. No, sir; I did not. I was not there at that time. By Senator Yutzy: Q. You said General Pearson—he repeated the word fire louder than the balance of his order. Are you right positive what the balance of the order was?
- 77. A. Order your men to fire. Q. That was the exact language? A. That is the very words. Q. You are positive you heard the words, order your men? A. Right in that way: Order your men to fire. Q. Were any of the soldiers struck? A. I did not see anybody struck. There was a man carried away. They say he got sun struck. I seen him carried away. I didn't see anybody hit. Q. Did you hear any pistol shots, or any shots fired from the crowd, or in the crowd, before the firing of the soldiery? A. No, sir. Q. Did you hear any? A. One appeared to be like a cap—it was right in that corner. [Indicating.] Q. In the crowd? A. That was the first shot I heard fired. That soldier fired it. He didn't fire straight out. His gun went off up in that way. Q. You heard a noise like a cap before the firing of the troops? A. I couldn't tell exactly where that come from. It appeared to be round the watch-box. Q. About the switch-box? A. Yes, sir. Q. That is, the watch-box on the corner of the street, and the railroad toward the hill?
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