Analysis of stability robustness for generalized state-space systems with structured perturbations

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Analysis of stability robustness for

generalized state-space systems with

structured perturbations*

Chun-Hsiung Fang

Department of Electrical Engineering. National Kaohsiung Insti- tute of Technology, 415 Chien-Kung Road. Kaohsiung 807. Taiwan, ROC

Fan-Ren Chang

Department of Electrical Engineering, National Taiwan Univer- sity, Taipei 107. Taiwan. ROC

Received 15 August 1992 Revised 19 December 1992

Abstract: A new approach is proposed to analyze the stability robustness of generalized state-space systems with structured perturbations. The presented method is computationally simple to use and can easily be calculated by computer. As far as we are aware, this paper seems to be the first one to solve the robust stability problems for generalized state-space systems with structured uncertainties. The robust stability problem of generalized state-space systems is more complicated than that of regular state-space systems because it needs consideration of not only stability robustness but also system regularity and impulse elimination. Th~ latter two ones need not be considered in regular state-space systems.

Keywords: Robust stability; generalized state-space systems; generalized eigenvalues; structured perturbations

1. Notations and introduction

o(M)

] M i r a

IMI

It,~l

T > N Re(s) : spectral radius of M e C " ×" : modulus matrix of M e C" ×" : determinant of matrix M e C " ×" :the (i,j)th element of T e ~ m×" : [ t i j ] >_ [ n j f o r i = 1 , 2 , . . . , n , j = 1,

2,...:~n

:real part of a complex number s Correspondence to: C.-H. Fang, Department of Electrical Engineering, National Kaohsiung Institute of Technology, 415 Chien-Kung Road, Kaohsiung 807, Taiwan, ROC.

* This work was supported by the National Science Council of Taiwan, ROC under Grant No. NSC-81-0404-E-151-513.

In recent years, there has been a growing interest in the system-theoretic problems of generalized state-space systems (or singular systems, or descriptor systems) due to the extensive applications of generalized state-space systems in large-scale systems, circuits, economics, control theory, and other areas [5,9, 10, 17]. Several important and fundamental results, except about robustness property, in regular systems have been successfully extended to generalized state-space systems [1, 6- 8,12, 18, 20]. In this paper we consider a continuous-time perturbed generalized state-space system described by E~(t) = Ax(t) + AAx(t), Ex(O)= Exo, (1) where E, A e ~ "×" and AA stands for the perturbations. Here the matrix E may be singular. It usually represents system's structure. We will assume rank E - r < n. Generally, system's perturbation can be characterized by using structured perturbations or unstructured perturbations [2, 3, 13-16]. Struc- tured perturbations are those for which bounds on the individual elements of the perturbation matrix are known (or derived), whereas unstructured perturbations are those for which only a norm bound on the perturbation matrix is known (or derived). In system (1), AA denotes an n x n time-invariant structured perturbation matrix. The perturbations in the various elements of the system matrix are independent of one another. In a practical situ- ation, the perturbation matrix is not known exactly and only the magnitude of the deviation that can be expected in the entries of A may be known. Suppose the perturbation can be bounded by

I Aa Ira -~ kH, (2)

where k is a real positive number and H is a constant nonnegative matrix. The constant matrix H represents the highly structured information for the additive perturbation matrix AA. In this paper, the nominal system (E, A) is assumed to be

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asymptotically stable and impulse-free, and the perturbation AE on matrix E will not be considered. The cases that nominal singular system is not impulsive-free and AE exists are discussed in Remark 2.11.

If E = I or E is nonsingular, system (1) becomes the well-known perturbed regular state-space system. The robust control problems of regular state- space systems under structured uncertainties have been solved by many authors 1,2,3,14-16, 19]. However, when E is singular, the problems of stability robustness are still open and unsolved. The stability robustness analysis for generalized state- space systems with structured uncertainties has not been dealt with. In this paper, we extend Chou's C3] results for the generalized system (1) with structured uncertainties. Since E is not restricted to the identity matrix, the results of 13] may be viewed as a special case of ours.

It is well known that the system (1)contains three kinds of mode:s: finite dynamical modes, infinite dynamical modes, and infinite nondynamical modes 1,1, 17]. The infinite dynamical modes can generate undesired impulse behaviors. Hence, to eliminate or to avoid inducing infinite dynamical modes is a key work in generalized state-space system control 1,8, 12, 18]. If we assume deg IsE - AI = r and AA = 0, system (1) now has r finite dynamical modes, none of infinite dynamical modes, and n - r infinite nondynamical modes, However, if the perturbation AA # 0, it would possibly introduce dynamical infinite modes into system (1) since it can change the degree of l sE - A - AA [. Furthermore, the perturbation AA can also possibly destroy the system's regularity (i.e. I s E - A - AAI is identically zero). For example, let

0,1 A A I = 0.2 4

o,]

[: o,]

0 ' AA2 = - 0.2 " Here d e g l s E - A l = l = r a n k E . It is easy to check deglsE - A - AAal = 0 < rank E (i.e. with infinite dynamical modes) and I sE - A - AAtI is identically zero (i.e. without regularity). Hence, the robust stability analysis of uncertain generalized state-space systems should consider not only stability robustness but also system regularity and impulse-free behaviors. The latter two problems do not arise in regular state-space systems.

2. Robust stability analysis

Lemma 2.1. (Bender and Laub 1,1] and Verghese et al. [17]). The response of EYe(t) = Ax(t) is said to be asymptotically stable and impulse-free if and only if the following two conditions are satisfied:

deg J sE - A [ = rank E = r (3)

and all roots of

I s E - A I = 0 (4)

have negative real parts.

Equation (4) in Lemma 2.1 can be rewritten as I s E - A [ # 0 for all R e ( s ) > 0 . (5) Note that when E = I or E is nonsingular, equation (3) is always satisfied.

Definition 2.2. For any real constant matrices X, y ~ , x , , the pair (X, Y) is said to be asymp- totically stable and impulse-free if d e o [ s X - YI = r and all roots of I s X - Y] = 0 have negative real parts.

Lemma 2.3. (Ortega [11]). For any n × n constant complex matrices X, Y, Z with I XIm < Z, we have {Xglm ~ IXlm[ Ylm <- Zl Ylm, (6)

I x + VIm-~ I xlra + I Ylm ~ Z + I rim, (7)

p(X) < Pit X I m ) <- P(Z), (8a)

p ( X V ) <_

p(IXl= I rim) _< p(Zl Ylm),

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p ( X + Y) < p ( l X + Ylm)

< P(I X Im + I Y[m) < P(Z -F I YIm). (8c) For the perturbed system (1), assume that the pair (E, A) is asymptotically stable and impulse- free, one can expand (sE - A ) - t as

(sE - A) -1 = Gsp(S) + J , (9)

where Gsv(S) is a strictly proper rational matrix which is analytic in right-half s-plane and d is a constant matrix. Denote by G(t) the impulse response of Gsp(S) and define

I;

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The matrix T would be finite since Gsp(S) is stable and strictly proper. A simple method to evaluate the matrix T is given in Remark 2.5.

Lemma 2.4. I f the pair (E,A) is asymptotically

stable and impulse-free, then

[(sE - A ) - ~Im < T + I J[m for all Re(s) >_ 0. (11) Proof. Taking Laplace transform of G(t), we have

f;

Gsp(S) = G(t)e-" dt. (12)

By Lemma 2.3 and equation (9), it is easy to check

I(sE - A) -1 [m ~ IGsp(S)[m -I-- l Jim

for all Re(s) > 0 for all Re(s) > 0

_< I G(t)e-"lmdt + l Jim

IG(t)lmdt + l Jim ~,for all Re(s) > 0

= T + l Jim. [] (13)

Remark 2.5. We propose a simple method to evaluate G(t) by the Weierstrass decomposition [5, 17]. Assume that the pair (E, A) is asymptotically stable and impulse-free. It can be trans- formed to the well-known Weierstrass form, i.e. there exist two constant nonsingular matrices

U and V such that

U ( s E - A ) V = [ s l ' - I._,0 1 ' (14) where A,¢9l '×'. Suppose U and Vare decomposed as

U = Ub '

where UoegV ×", Ub~91 ~"-'~×', V ~ "×', and Vb ~ ~ " × ~"-'L Then it is easy to show that

Gsp(S)-" V a ( s l r - A r ) - l U a , (16a)

J = Vb" Ub. (16b)

From equation (16a), we have

If:

]

Gsp(S) -" V a eA"e - ' d t U,,. (17)

A comparison with (12) gives

G(t) = Vae~4"tUa . (18)

Lemma 2.6. For a matrix Q ~ C "×", if p(Q) < 1,

then 11 -T- Q I ~ O.

Theorem 2.7. Assume the perturbation AA can be

bounded by I AA I m <<- kH. The perturbed system (1) is

asymptotically stable and impulse-free if the pair ( E, A) is asymptotically stable and impulse-free, and the following inequality is satisfied:

1

k < (19)

p ( T H + IJlmH)

Proof. By the determinant formula, we have

IsE - A - AAI = lsE - AI " II

- ( s E - A ) - i AA I. (20) From Lemma 2.3 and equation (13), the following inequalities are obvious,

p ( ( s E - A)- IAA)

< p(l(sE - A) -x ImkH) for all Re(s) > 0 <_ p ( k l J I m n + kTH) for all Re(s) >_ 0. (21)

If equation (19) is satisfied, then p ( ( s E - A ) - I A A ) < 1 for all Re(s)>0. According to Lemma 2.6, we have

I I - ( s E - A ) - t A A I ~ O for all Re(s) > 0. (22) Hence, we obtain [ s E - A - A A I ~ 0 for all

Re(s) _> 0 from equation (20). Using equations (9) and (20), it is easy to verify that

I s E - A - AAI

= I sE - A I I I - JAA - Gso(s)AA I. (23) Since G,p(S) is a stable and strictly proper rational matrix, the constant term of l I - JAA - G,p(s)AA [ will be decided by (I - JAA). I f ( / - J A A ) is non- singular, we can obtain, from equation (23), d e g l s E - A - A A I - d e g ( l s E - A I I I - JAAI).

(24) By Lemma 2.3, it is easy to show

p(JAA) <_ p ( k l J l m H ) ~ p ( k Y n + klJlmH). (25)

If inequality (19) is satisfied, p ( J A A ) < 1, and (I - JAA) is nonsingular. Then from equation (24),

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we have deg ]

sE

- A - AA } = deg

I sE - A [ = r.

Therefore, the perturbed system (1) is asymptotically stable and impulse-free. Furthermore, if inequality (19) holds, equations (20) and (24) can guarantee the regularity of the perturbed system. []

Remark 2.8. From the above statements, we know that the regularity can be guaranteed if the perturbed generalized state-space system is asymptotically stable and impulse-free.

R e m a r k 2.9. If E = I, then J = 0. The sufficient condition in equation (19) coincides with the results in [3]. Thus, Chou's results might be viewed as a special case of ours.

Remark 2.10. One can use the criterion developed in this paper to design a robust controller for an uncertain generalized state-space system. Assume that the given perturbed system is

EYe(t)=

Ax(t) + AAx(t) + Bu(t),

where the triple

(E,A,B)

is strongly controllable [17]. We can apply a state feedback

u(t)= Fx(t)

such that the pair

(E,A + BF)

is asymptotically stable and impulse- free [8, 18]. Then one may use the inequality (19) to check whether the designed feedback gain F is robust under the perturbation AA.

R e m a r k 2.11. If the nominal system is not impulse- free, it can be theoretically decomposed by nonsingular matrtix transformation as

U ( s E - A ) V = [ s I ' - Ap

0 sJ +0

I, _p ] , (26) where p < rank E,

ApCR p×p, J

is a nilpotent Jordan matrix, and both U and V are nonsingular constant matrices. In this case, an

"arbitrarily small"

matrix

As

of the form

A~,t (27)

can be constructed such that deg

I sJ + l,,_p +

As4] = rank(J) (i.e. d e g J s E - A + A~J = rank E) and all roots of isJ

+ l,,_p + As41= 0

lie in the left-half plane. By the As, we can rewrite the perturbed singular system as

E~(t) = Ax(t) + AAx(t)

= (A - A s ) x ( t ) + (As + A A ) x ( t )

= ,,ix(t) + a / i x ( t ) , ₍₂₈₎

w h e r e / i = A - As and A/i = AA +

As.

Now the new nominal system (E,/i) is asymptotically stable and impulse-free and the A/i may be viewed as a new structured perturbation on matrix/i. Note that here A, is a

"selected arbitrarily small"

matrix. In many applications, the matrix E is a

structure

information

matrix rather than a

parameter

matrix, i.e. the elements of E contain only structure information regarding the problem considered. Hence, they are not subject to variations. In the case when the matrix E does contain uncertain parameters and when the perturbation is of the form

V(AE)V =

aE2]

AE4J'

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then we have, from equation (14),

Is(E + A E ) - AI = I U - ' I " I U-I ]' Is(It + aEt)

-

A,I" I sAE4 +

I,_,[. (30) In equation (30) we assume that the nominal system is stable and impulse-free. If

Is(I, +

AEt) - A,J is not identically zero. one can find an

"arbitrarily

small"

matrix AE, such that

[sAE,

+ I,_, [ = 0 has at least one root in the right-half plane. Hence, in this case the stability of (E, A) has zero tolerance to AE4 on matrix E. The influence of AEt to system stability can be checked by using

Is(l, +

A E t ) -

A,I = s'l(l, +

A E , ) - hA, l, (31) where s = I/A, and A, is a stable matrix. A, is nonsingular implies that I(I, + A E t ) - AA,[ = 0 always has r finite roots for arbitrary AEt. If all r roots of I(l, + A E t ) - AA, I - - 0 have negative real parts for perturbation AEI, we may say that all r roots of [ s ( l , + A E t ) - A,[ = 0 also have negative real parts for perturbation AEt. If we define /~ =- At, A - I,, and AA m AEt, we may use the method proposed in Theorem 2.7 to check whether all r roots of [1, + AEt - ~.A,[ = 0 lie in left-half plane.

Recently, Qiu and Davison [13] have dealt with problem same as those considered in this paper. However, the perturbation AA on matrix A they investigated is unstructured (measured by spectral norm). They also assumed that the perturbation AE on matrix E is zero and the nominal system (E, A) is asymptotically stable and impulse-free. We would like to refer to [13] for a discussion of the assump- tions made by these authors.

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3. Illustrative examples

then we have Example 3.1. Let the nominal system be

i o0] [oO0]

E = 1 0 , A = - ~ 0 ,

0 0 0 s

where ~ is a real positive number. Assume that the perturbation is bounded by

oo]

IAAlm_<~" 1 1 - k ' H ,

1 1

where k -- ~. It is easy to get

0Ii 0 il

T = -1 , J = 0 .

0 0

We then have p ( T H + I J I m H ) = 2/~. Since

1

k = ~ :

p ( T H + I J l m H ) =

2'

we cannot make any conclusion for this perturbation according to the sufficient condition proposed in Theorem 2.7. In fact, if we let AA = ~-H, it is easy to check d e g l s E - A l = 2 > d e g l s E - A

- A A I - 1 . Obviously, this perturbation intro- duces a dynamical impulsive mode to the nominal system.

- 8

A , = 0 and

[oOi]

-- 1 ' l Jim-

0

0.5 G(t) =

e 00]

0 0 e - ' , T = O 0 . 2e - s ' 0 0 ¼ 0

Given the structured perturbation information the allowable upper bounds on k are as follows:

t:00]

i f H = 0 0 ,

0 0 p ( T H + IJImH) = 8

(i.e. the perturbed generalized state-space system in this example is asymptotically stable and impulse-free if the perturbation is bounded by

I AAIm

~-~

8" H.

In fact, the constant 8 is the exact upper bound for the structured perturbation).

I 0 0 ]

1

ifH = 0 1 , =

0 0 p ( T H + IJImH) 2.

The constant 2 is the exact upper bound for the structured perturbation. Another structured perturbations bounds are calculated in the following: Example 3.2. Consider the following system:

oo]

_{o o}

1 0

4 0 0 2 ]

= 0 - x ( t ) + A A x ( t ) , - 1 0

where the pair ( E , A ) is asymptotically stable and impulse-free. By Weierstrass decomposition, it is easy to obtain U = 1 , V = 1 0 0

[

0 0 ]

1

if H = 0 0 , 0 I p ( T H + l J l m H ) ' - ° ° '

[oo]

if H = 1 0 , 0 1

Ol]

if H = 1 0 , 0 0

o,]

i f H = 0 0 , 0 l p ( T H + [J[mH) p ( T H + IJImH) p ( T H + [J]m H ) = 1.414, = 2 , = 2.6667,

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if H = 0 , 1 p ( T H + [J[mH)

,1]

if H = 1 1 , 1 1 o(TH + IJ[rnH) = 1 and = 0.533. 4. C o n c l u s i o n s

A new and simple approach is proposed to analyze the stability robustness for generalized state- space systems with structured perturbations. The robust stability problem of generalized state-space system is more complicated than that of regular state-space system because not only stability robustness but also regularity preservation and impulse elimination should be simultaneously considered. Our method only requires evaluation of a time-invariant stable transition matrix. So it is computationally simple and can easily be calculated by a computer. To the best of authors' knowledge, the paper seems to be the first one to solve the stability robustness problem for generalized state-space system with structured perturbations. Research is ongoing in order to extend the proposed robustness criteria to the design problem of feedback generalized state-space systems. Exten- sion of the results in this paper to nonsquare generalized state-space systems is a topic for future research.

Acknowledgment

The authors gratefully acknowledge the re- viewers for their s u g g e s t i o n s a n d c o m m e n t s . R e f e r e n c e s

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