Robust control analysis and design for discrete-time singular systems

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Pergamon

000S-1098(94)E0022-A

Automatica, Vol. 30, No. 11, pp. 1741-1750, 1994

Cowdght ~) 1994 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0005-109S/94 $7.00 + 0.00

Robust Control Analysis and Design for Discrete-time

**Singular Systems*t**

CHUN-HSIUNG FANGAt LI LEE§ and FAN-REN CHANGll

A new approach to robust control analysis and design for uncertain

discrete-time singular systems is proposed. Under the allowable structured

perturbations, the stability robustness, regularity preservation, and impulse

elimination are simultaneously guaranteed by the proposed method.

Key Words--Robust stability; robust control; structured perturbations; singular systems; regularity.

Abstract--In this paper, we propose a simple approach to analyse stability robustness of discrete-time singular systems under structured perturbations. The developed robustness criteria are then applied to solve robust regional pole- assignment problems of singular systems. A robust control design algorithm, via state feedback, is also given. The robust stability problem of singular systems is more complicated than that of regular systems. Not only stability robustness but system regularity and impulse elimination should be considered simultaneously. Since the results of robust control and analysis for singular systems is not available in the literature as much as other fields, the paper may be viewed as a complementary result in this field. Although only discrete-time case is discussed, several results can be directly applied to continuous-time systems as well.

1. NOTATION

THE FOLLOWINO notation will be used throughout the paper

p(M): spectral radius of M E C "×" IMIm: modulus matrix of M ~ C nx" IMl: determinant of matrix M E C "×" JZ]: magnitude of complex number z

D(-h,

f): a disk centered at point - h +jO with radius f

*Received 4 August 1992; revised 10 August 1993; received in final form 10 February 1994. The original version of this paper was presented at the 12th IFAC World Congress which was held in Sydney, Australia, during 18-23 July 1993. The Published Proceedings of this IFAC Meeting may be ordered from: Elsevier Science Limited, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, U.K. This paper was recommended for publication in revised form by Associate Editor Vladimir Ku~era under the direction of Editor Huibert Kwakernaak. Corresponding

author Dr L. Lee. Fax 886 7 5615137; E-mail

leeli@ee.nsysu.edu.tw.

t This work was supported by the National Science Council of Taiwan, R.O.C. under Grant No. NSC-82-0404-E- 151-025.

Department of Electrical Engineering, National Kaoh- siung Institute of Technology, Kaohsiung 807, Taiwan, R.O.C.

§ Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, R.O.C.

II Department of Electrical Engineering, National Taiwan University, Taipei 104, Taiwan, R.O.C.

1741

mij:

the (i, j)th element of matrix M

M>-N: mij, nijER

and

mij>-n~j

for i , j =

1, 2 , . . . , n

AAA: end of statement in quoted lemmas Q.E.D.: end of proof.

2. INTRODUCTION

In the past ten years, there has been a growing interest in the system-theoretic problems of singular systems due to the extensive applications of singular systems to large-scale systems, circuits, economics, polynomial matrices and other areas (Luenberger, 1977; Verghese

et al.,

1981; Lewis, 1986; Dai, 1989; Fang and Chang, 1991, 1992). Sometimes the system is called generalized state-space systems, or implicit systems, or descriptor systems or semistate systems (Lewis, 1986). Several important and fundamental results in standard state-space systems have been successfully extended to singular systems (Cobb, 1984; Bender and Laub, 1987; Shayman, 1987; Zhou

et al.,

1987; Kucera and Zagalak, 1988; Fang and Chang, 1990). Many excellent design methods for singular system control have also been well developed recently (Kucera and Zagalak, 1991; Paras- kevopoulos and Koumboulis, 1991). However, little effort has been devoted to studying the robust control problems of singular systems. In this paper, we will investigate such problems. Consider a discrete-time perturbed singular

where E, A e R " x -

perturbations. singular. We system

Ex(k +

1) =

Ax(k) + AAx(k), Ex(O) = Exo,

(1) and AA stands for the Here the matrix E may be will assume rank E - r -< n. In

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system (1), AA denotes an n × n time-invariant structured perturbation matrix. The perturbations in the various elements of the system matrix A are independent of one another (Sobel et al., 1989; Tesi and Vicino, 1990; Juang, 1991). In a practical situation, the perturbation matrix is not known exactly but the magnitude of the deviation which can be expected in the entries of A may be known. The perturbation considered in this paper is described as

IAA[m -< qH, (2)

where q 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.

It has been known that a singular system generally contains three kinds of modes: dynamical finite modes, dynamical infinite modes, and nondynamical infinite modes (Ver- ghese et al., 1981; Bender and Laub, 1987). The dynamical infinite modes can generate undesired impulsive behaviours. Hence, to eliminate or to avoid inducing dynamical infinite modes is a key work in singular system control (Wang et al., 1987). If we assume deg IzE - AI = r and AA = 0, the system (1) now has r dynamical finite modes, no dynamical infinite modes, and n - r nondynamical infinite modes. However, if the perturbation AA ~ 0, it would possibly introduce dynamical infinite modes into system (1) since it can change the degree of I z E - A - A , 4 1 . Furthermore, the perturbation AA can also possibly destroy the system regularity (i.e. I z E - A - AAI is identically zero). For example, let

E

E = [ ~ ~]' A = [ 0 4 5 ~i~]' L 4 = [ 0 . 2 ~]" Here d e g l z E - A r = l = r a n k E . It is easy to check deg IzE - A - AAI = 0 < rank EVE ~ 0 and IzE ~ A - AAI becomes identically zero for E = 0. This example indicates a fact that the robust control problems of singular systems must consider not only stability robustness but system regularity and impulse elimination. It should be noted that the latter two cases do not arise in the standard state-space systems. Therefore the robust control design problem of singular systems is more difficult than that of standard state-space systems.

This paper is organized as follows: Section 3 reviews some related results and states the problems concerned in this paper. The solutions to the problems are given in Section 4. Section 5

makes a brief conclusion. Some examples are also provided at the end of each section for illustrating our ideas.

3. RELATED LITERATURES AND PROBLEM STATEMENT

3.1. Review o f related results

In the literature, there are only a few papers dealing with the problems of robust control of singular systems. Mertzios (1984) developed a recursive formula to compute the approximated transfer function of perturbed singular systems. The sensitivity of poles and coefficients of transfer function to perturbations was also investigated. As the perturbations were given, Nichols (1986) used the condition number to measure the distance between nominal generalized eigenvalue and perturbed generalized eigenvalue. Based on the concept of eigenvalue sensitivity, Nichols proposed a nice algorithm to construct a state feedback gain such that the closed-loop system is insensitive to perturba- tions. However, since only first-order perturba- tion to eigenvalues is considered (other higher order perturbations are neglected), the perturbations allowed in her results must be small enough. Furthermore, her method can not be directly applied to calculate the upper bound of the perturbations under which the perturbed system remains stable. For checking the stability of perturbed systems, the perturbed ranges of all generalized eigenvalues must be computed. Syrmos and Lewis (1991) introduced a so-called chordal metric to replace the condition number used by Nichols and dealt with the problems in the same way as Nichols did. The problem of robust stability for singular systems with unstructured perturbations has been investigated by Qiu and Davison (1992). He and Ji (1992) proposed some sufficient conditions for practical stability of large scale systems with impulsive solutions. It seems that they are the first ones to discuss such a problem.

In this paper, from a different point of view, we propose a simple approach, which is an extension of Chou's (1990), to solve the robust control problem of singular systems. Since the matrix E in this paper is not restricted to the identity matrix, Chou's results may be viewed as a special case of this paper. By our approach, it is easy to obtain an upper bound of perturbations under which the perturbed system is guaranteed regular, impulse-free and asymptotically stable. The restriction that perturbations must be small is removed. To check the stability of perturbed systems, it is not necessary to

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Robust control for singular systems 1743 calculate the positions of all perturbed

eigenvalues.

4 . S O L U T I O N S

4.1. Robust stability analysis 3.2. Preliminaries and problem formulation

To state our problem, we need to review some preliminary results.

Lemma 1. The system Ex(k + 1) = Ax(k) is said to be asymptotically stable if and only if (Lewis, 1986):

all roots of IzE - AI = 0 lie inside

the disk D(0, 1). (3) AAA The statement of equation (3) can be replaced by

[ z E - A l # O f o r a l l z with I z l - 1. (4) Lemma 2. The following conditions are equiv- alent (Bender and Laub, 1987):

(a) the system Ex(k + 1 ) = A x ( k ) has no dyna- mical infinite modes;

(b) (zE - A) -1 is proper; (5a)

(c) deg IzE - AI = rank E. (5b)

AAA Note that it is not necessary to consider the dynamical infinite modes in standard state-space systems (E = I) since condition ( 5 b ) i s always true for such system.

Definition 1. For any two n × n real constant matrices E and A, the pair (E, A) is said to be regular, impulse-free (i.e. without dynamical infinite modes), and asymptotically stable if I z E - AI ~ O, ( z E - A) -1 is proper, and all roots of IzE - AI = 0 lie within the disk D(0, 1).

Three robust control problems are investigated in this paper. (1) Assume the pair (E, A) is regular, impulse-free, and asymptotically stable. What is the upper bound of perturbation AA such that the perturbed system (1) is still regular, impulse-free and asymptotically stable? (2) If the roots of I s E - A I = 0 lie inside a specified disk, what is the upper bound of AA such that the roots of I s E - A - A A I = 0 still lie inside the disk? This is called the robust root-clustering problem (Yedavalli, 1993). (3) How to design a robust controller for the perturbed system Ex(k + 1) = (A + AA)x(k) + (B + AB)u(k) by using the results developed in problems (1) and (2) is the third problem. In the above, u(k) is the input signal and AA and AB represent the uncertainties on matrices A and B, respectively.

Lemma 3. For any n × n constant matrices X, Y, Z with [Xlm <--Z, it is easy to obtain the following inequalities (Lancaster and Tism- enetsky, 1985):

(a) IXYIm <- IXlm IYJm <- Z Irlm (6) (b) IX + Vim <-IXlm + [Y[m <- Z + ]Yl,~ (7) (c) p(X) <- p(IXlm) <- p(Z) (8a) (d) p(Xg)<p([Xlm JgJm)<--p(Z IgJm) (8b) (e) p ( X + Y) <- pOX + Vim) <- p(IXlm + IYIm)

-< p(Z + IYim). (8c)

AAA The pair (E,A) in perturbed system (1) is assumed regular, impulse-free, and asymptotically stable. We define G p ( z ) - ( z E - A) -1 and denote by G(k) the pulse response sequence matrix of the multivariable system G,(z). One can rewrite Gp(z) as

Gp(z) -- (zE - A) -1 = ~ G(k)z-k. (9) k=0

Then we have the following key lemma.

Lemma 4. If the pair (E,A) is regular, asymptotically stable and impulse-free, and IAAlm <- qH, then

p((zE - A) -1 AA) -< p(qTH),

for all z with Izl--- 1 (10) where T is defined by

T =- ~ IG(k)lm. (11)

k=O

Proof. From Lemma 3 and equation (9), it is easy to check

p((zE - A) -t AA

-< p(I(zE - A) -1AAIm) for all z with Izl -> 1

~o

for all z with Izl > - 1 <- p IG(k)z-klm • qH for all z with IzL- > 1

\ k = 0 <--p q ~ [G(k)JmH for all k ~ O / = p(qTH). z with Izl ~ 1 (12) Q.E.D.

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The matrix T in equation (11) would be finite if the pair (E,A) is asymptotically stable and impulse-free. This fact can be verified in Remark 1 below.

Remark 1. By the Weierstrass decomposition (Dai, 1989; Lewis, 1986), we propose a simple method to evaluate the matrix T without performing the expansion of ( z E - A ) -~ re- quired in equation (9). Since the pair (E, A) is asymptotically stable and impulse-free, it can be transformed to the well-known Weierstrass form, i.e. there exist two constant nonsingular matrices U and V such that

U(zE - A ) V = [zL - Ar 0 ,

0 I~_,] (13)

where A, E R ~×~ and all eigenvalues of A, are within D(O, 1). Suppose U and V are decomposed as

-

-[U~] and V=[V~ Vb],

U = Us

where Ua E R "×", Ub E R ("-')×~, V~ E R "×~ and Vb E R "x("-°. Then it is easy to show that

G(k)z-* = Va(ZIr -- A,)-Iu~ + VbUb

k~O oo Va E - - k - 1 - k . , = fl'lr Z Ua + VbUb.(14a) k = l Therefore, we have Vb" Ub for k = 0 (14b) G ( k ) = [ V ~ A ~ _ I u °

for k = 1,

2 . . . . , oo.

The matrix T can be readily obtained from equation (14b). Since all eigenvalues of Ar are within unit disk D(0, 1), we can find a similarity transformation matrix S such that A, = SDrS -1,

where D, is a block diagonal Jordan matrix. For simplicity, assume here A~ e R s×5 and D, has the following form: A1 0 D , = 0 0 0 0 0 0 1~2 1 0 0 A2 1 0 0 A2 0 0 0 _ 0 0 , 1 A2 (15)

where A1 and A2 are eigenvalues of A, with

IA~I

< 1 and IA2I < 1. The following inequality is obvious:

T = ~ IG(k)lm k = 0

<-IVbUt, l,,, +

IValm ~ IA~-llm

tU.l,,,

k = l

<-IVbU~lm + IValm

ISlm ~

IDklm

k = 0

× IS-11m Igalm. (16)

Since I)til < 1 for i = 1, 2, it is not difficult to verify IDfl,,, -< ID~l~m 0 0 0 0 ]

J

P2 [A2[ p2 p3 p~ 0 P2 IA2I p2 p3 , 0 0 P2 I'X21

p~

0 0 0 P2 [A2[

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k = l k = l Pl IAI[ 0 = 0 0 0 1 1

where Pl - - - and P 2 - - - • Therefore,

1 -]All 1 -1'~21

all entries of matrix T must be finite if the pair (E, A) is asymptotically stable and impulse-free.

The following two lemmas will be used to prove Theorem 1.

Lemma 5. (Lancaster and Tismenetsky, 1985). For any n × n matrix Q, if p(Q) < 1 then

11 - QI ~ 0. AAA

Lemma 6. (Chen, 1984, Theorem 3-4). Let M(z)

be a square rational matrix and be decomposed uniquely as M(z) = Mp(z) + Msp(z), where

Mp(z) is a polynomial matrix and Msp(Z) is a strictly proper rational matrix. Then M-~(z) is proper if and only if M r ( z ) exists and is proper. AAA

Theorem 1. Let the pair (E,A) be regular, asymptotically stable and impulse-free. Assume the perturbation &A is bounded by Izkal,,, <-qH

for some q > 0. The perturbed system (1) is still regular, asymptotically stable and impulse-free if the following inequality:

1

q < (18)

p ( T ~ )

holds, where T is defined in equation (11).

Proof Since IzE - A ( is not identically zero (i.e. regular), by the determinant formula, we have

IzE - A - &AI

= pzE - A I . I I - (zE -A)-IAAI. (19) If inequality (18) holds, from Lemma 4, we have

p ( ( z E - A ) - l z S u 4 ) < l for all z with I z l - 1 . According to Lemma 5, the following inequality:

II - (zE - A ) - I A A I # 0 for all z with [z[-> 1

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holds. In view of equation (19), we have

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Robust control for singular systems 1745 show the impulse-free and regular properties. If

inequality (18) is satisfied, by Lemma 3, one can get

p(G(0)AA) <- p(q Ia(0)lm H)

<-p(qTH)< 1. (21) The nonsingularity of I - G ( 0 ) ~ A is ensured obviously from Lemma 5. The matrix ( z E -

A) -1 can be expressed as ( z E - A ) - 1 = G ( 0 ) +

Gsp(z), where Gsp(z) is the strictly proper part. We rewrite

(zE - A - AA ) -1

= [I - G(O)AA - Gsp(z)AA]-~(zE - A) -1. (22) By L e m m a 6, the nonsingularity of I - G(0)AA implies that [ I - G ( 0 ) A A - Gsp(z)AA] -~ is proper and so is ( z E - A - AA) -~. Therefore, by L e m m a 2, the perturbed system is impulse-free. If ( z E - A - A A ) -1 is proper, the system's regularity is also guaranteed (i.e. I z E - A - AA I

is not identically zero). Q.E.D.

Remark 2. From the above statements, we know that if the perturbed system is asymptotically stable and impulse-free, then the system regularity can be ensured as well. If E = / , from equations (11) and (14), then T = ~ IAg-llm.

k=l

The condition in equation (19) then coincides with Chou's results. Thus his results might be viewed as a special case of ours. In Chou's paper, he has showed many advantages of his approach. All his advantages are preserved in our approach to dealing with singular systems with highly structured uncertainties.

Remark 3. Many features of continuous-time singular systems and discrete-time singular systems are essentially different. For instance, stability of discrete-time systems can exclude the possibility of introducing the dynamical infinite modes, but this is not always true for the continuous-time case. The detailed derivation for continuous-time systems can be found in Fang and Lee (1"993).

Example 1. Consider the following system:

0 0 x(k + 1) 0 1

0.5 0 _121 _{+ AAx(k).}

The pair (E, A) is regular, asymptotically stable

and impulse-flee. By Weierstrass decomposition, it is easy to obtain U = 0 0 , V = 0 1 , 0 0 . 5 2 0 then we have and

ar:[7 00 ]

G(o) = o , 0.5

[0"5k-1

0 0 J

G ( k ) = 0 0 0.2 k-1 ,

2"05 k-1 0

0

for k = l , 2 . . .

00

T = 0 1.25 . 0.5 0

The following shows the allowable upper bounds on q for various cases of structured perturbations:

[10 ]

(a) H = 0 0 1

0 0 ' p(TH)=0"5;

(i.e. the perturbed singular system in this example remains regular, asymptotically stable and impulse-free if the perturbation is bounded

by IAmlm < 0.5H -0 (b) H = 0 0 -0 (c) H = 0 0

I-_

(d) H = / i

L v

(f) H = I ! 0 0- 0 0 0 1 0 0 0 0 , 1 0 1 , - - =0.5; 0 p(TH) 0 01 1 0 1 , =0.5; 1 o p ( T n )

1

1 1 1 1 1 , =0.129. 1 1 p(TH) 1 m - - O 0 ~ ' p ( T H ) 1 - - - 0 . 2 ; p(TH)

Note the upper bounds of perturbations found in cases (a)-(c) are exact with respect to their

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perturbation structures. An interesting case should be indicated as follows. In some cases, the perturbations do not affect the positions of eigenvalues (i.e. eigenvalue sensitivity to perturbations is zero) but can destroy the regularity. Assume the perturbation is AA = e •/-/1, where

/-/1 = 0 .

0 It is easy to check

JzE - A - AAI = (2 + E)(z - 0.5)(z - 0.2). The finite eigenvalues of nominal system, A1.2 = 0.5, 0.2, are not disturbed by the uncertainty E ~ ( - 2 , 2) for the perturbation structure //1. However, the regularity will vanish suddenly when E = - 2 . Using Theorem 1, we have

1/p(TH1)

= 2. This is the exact upper bound of

perturbations for the remaining system to be regular.

Remark 4. Consider a continuous-time singular system EYe(t) = Ax(t). If one is only interested in whether the finite eigenvalues of the continuous- time system lie inside the unit disk D(0, 1). Theorem 1 can be directly applied to check this matter without any modification. Although this theorem is developed for discrete-time systems, the derivation still holds if the variable z is replaced by s.

4.2. Robust root-clustering in a specified disk The dynamic response of a linear time- invariant system can be modified by means of placing the poles in predetermined locations. However, for systems with uncertain parameters, the exact placement of pole locations might be difficult to attain. Hence, the concept of pole placement within a specified region is a suitable and useful approach. Lately, many researchers have considered how to locate the closed-loop poles of a standard state-space system in a prescribed region to shape the dynamic response. However, the variation of pole locations due to parameter perturbations of the plant has not been discussed by them. Rachid (1989) and Fang (1993) have given some new methods to study the problem of root-clustering in a specified disk for standard state-space systems with linear time-invariant perturbations. The same problem for uncertain singular systems is still unsolved now.

In this section, we will apply the results derived in the previous section to solve the problem of robust root-clustering in a specified disk for uncertain singular systems. A sufficient condition is proposed to guarantee pole position robustness within a specified disk for singular

systems with structured uncertainties. Under the allowable highly structured perturbation, both stability robustness and certain performance robustness can thus be ensured. Consider two singular systems.

and

X~ :Ex(k + 1) = Ax(k) + AAx(k) (1) Z2 :Ex(k + 1) = A x ( k ) + &,gtx(k), (23) where ,~ ~ R n×n and A,~ denotes the associated structured perturbation.

Definition 2. For any two n × n real constant matrices X and Y, we say that all finite eigenvalues of the pair (X, Y) lie inside disk D ( - h , f ) if ( z X - y ) - i is proper and all roots of I z X - YI = 0 are within the disk D ( - h , f).

Lemma 7. Assume A and ,~ are related by

fi;t = A + hE. (24)

Then, all finite eigenvalues of the pair (E, A) lie inside disk D ( - h , f ) if and only if all finite eigenvalues of the pair ( E , / ] ) lie inside the unit disk D (0, 1).

Proof. This fact can be easily verified from the following identity:

f ( A E - ,4 ) = z E - A, (25)

where A = (1/f)(z + h). Q.E.D.

Remark 5. In comparison with the nominal parts of systems Y~ and Z2, all finite eigenvalues of the pair (E, A) lie inside disk D ( - h , f ) if and only if the pair (E, ,~) is asymptotically stable and impulse-free.

Assume all finite eigenvalues of the pair (E, ,~) are located within disk D(0, 1) and the perturbation A/~ can be bounded by

l&,4 Im -< q/?/, (26) where q is a positive real number and /z/ a nonnegative matrix. By the results of Theorem 1, we can readily obtain that all finite eigenvalues of the perturbed system Z2 lie inside disk D(0, 1) (i.e. asymptotically stable and impulse-free) if the following inequality:

1

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Robust control for singular systems 1747 is satisfied, where oo = ~ It~(k)lm, (28) k = 0 oo (zE-,~1-~ = ~ ~ ( k l z -k (29 / k = 0 and

1

.~ = ~ (A + hE). (30) The robust root-clustering theorem is stated as follows.

Theorem 2. Assume all finite eigenvalues of the pair (E, A) lie inside the disk D ( - h , f) and the perturbation AA is bounded by

IAA[m <- qH for some q > 0. (31) Then all finite eigenvalues of the perturbed system ~1 still lie inside disk D ( - h , f ) if the following inequality:

f

q < p(~'H) (32) holds, where 1" is defined in equation (28).

Proof. Let

ZkA = fA,~. (33)

Comparing equations (31) and (26), we have H =f/~. Then the inequalities in (32) and the inequalities in (27) are equivalent. If all finite eigenvalues of the pair (E,A) lie in disk

D ( - h , f ) , then from Lemma 7 all finite eigenvalues of the pair (E, A) are located within the unit circle D(0, 1). Define ~ and ~ as

- A + AA = A + fA,~ (34) and

-- ,~ + A,4. (35)

From Theorem 1, if inequality (32) [or inequality (27)] is satisfied, we can say that all finite eigenvalues of the pair (E, ~ ) are located inside disk D(0, 1). Since • and ~ are related by

f ~ = ~ + hE (36) it is easy to check by Lemma 7 that all finite eigenvalues of the pair ( E , ~ ) lie in disk

D ( - h , f ) . Thus all finite eigenvalues of the perturbed system Z1 are still located in disk

D ( - h , f), under the perturbation AA. Q.E.D.

Remark 6. The techniques of Theorem 2 with equations (27)-(30) can also be applied to solve the problems of root-clustering robustness inside a disk centred on the negative real axis for

continuous-time systems without any

modification. Example 2.

[i °°-

0

1 0

x(k + 1) 0 0.25 0 "1

1

0 -0.5 Jx(k)

-0.75 - 1 0 + aAx(k).

The pair (E, A) is impulse-free and its finite eigenvalues a r e / ~ 1 = -0.25 and A2 = -0.75 E

D ( - 0 . 5 , 0.5). If ,4 = (1/0.5)(A + 0.5E), it is easy to check that the finite eigenvalues of the pair ( E , A ) are hi = 1 and A2 = _1 e D(0, 1). Using equation (28), we obtain

2.6667 0 0.6667 ]

~ - 2 0 2.6667 1.

/

5.3333 1 1.33331

If the structured perturbation information is

then

[10 ]

H - - 0 1

0 0

0.5/p(~'H) = 0.1161. Hence if

I~AIm

< 0.1161H, not only is the perturbed system regular and impulse-free but all finite eigenvalues of the perturbed system are still located within D ( - 0 . 5 , 0.5). If the perturbation structure is changed to

E 1

H = 1

1

then 0.5/0(TH) = 0.0319.

11]

1 1 , 1 1

4.3. Robust control design

Based on the results derived in Section 4.1 and 4.2, we now propose a simple m e t h o d ' t o synthesize a controller for robust pole assignment of perturbed singular systems in a specified region. Let us consider a perturbed singular system

Ex(k + 1) = (A + AA)x(k) + (B + AB)u(k)

(37) where B ~ R "xm. We only assume the triple (E, A, B) is strongly controllable (Verghese et al., 1981). In equation (37), AA and AB are linear time-invariant perturbations which can be bounded by

IAAIm-<qlH1 and IABIm-<q2H2, (38) where qi, i = 1, 2 are both positive real numbers, and H;, i = 1, 2 are two nonnegative constant matrices. Here we assume the perturbation

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bounds ql and q2 and the structured perturbation information /-/1 and //2 are known in advance.

Let the static state feedback control be

u(k) = Fx(k). (39) Then the nominal closed-loop system is

Ex(k +

I)

= (A + BF)x(k)

(40)

and the perturbed closed-loop system

Ex(k + 1) = (A + BF)x(k) + (AA + ABF)x(k),

(41)

Therefore, the design problem to be considered is to determine the feedback gain F in (39), such that all finite poles of the nominal closed-loop system (40) lie inside a specified disk, and at the same time all finite poles of the perturbed closed-loop system (41) are also located within the same disk. Suppose the specified region for robust root-clustering is the interior of disk

D ( - h , f ) , then employing Theorem 2, we have the following result.

Theorem 3. All finite closed-loop poles of the perturbed system (41) will be located within the disk D ( - h , f), if AA and AB are bounded by (38), all finite eigenvalues of the pair (E, A +

BF) lie inside a disk D ( - h , f), and the following inequality: f (42) ql < p (~/.~) is satisfied, where o¢ = ~ It~(k)l., (43) k = 0 and =/-/1 + q2/-/2 IFI,,,. (44) ql In equation

(43), G(k)

is obtained by (zE _ ~ ) - 1 = ~ ~ ( k ) z - k , (45) k ~ 0 where ft = ~ (A + BF + hE). (46)

Proof. From equation (38) and Lemma 2, we have

IAA + ABEl., <--IAAI., + }ABI., IFl,,,

<-- qlH1 + q2H2 IFI.,

= ql/t. (47)

Therefore, according to Theorem 2, we have the

stated results. Q.E.D.

Now the following design algorithm, for a given set of perturbations, can be used to select the control gain F for which the system is pole-assignment robust.

Step 1. Specify the finite closed-loop eigenvalues

/~i ( i = 1, 2 . . . . , r) in the specified disk D ( - h , f).

Step 2. Use any pole-assignment technique of singular systems to design F (Wang et al., 1987; Fahmy and O'Reilley, 1989; Fang and Chang, 1990; Kucera and Zagalak, 1991; Paras- kevopoulos and Koumboulis, 1991), such that the nominal closed-loop system (40) has the specified A,-.

Step 3. Based on highly structured information (38) and on the designed F, check if the robust pole-assignment condition given in (42) is satisfied. If so, the design of robust controller for robust pole-assignment is finished. If not, since the different eigenstructure can help in satisfying the condition in (42), we shrink the nominal closed-loop eigenvalues closer to the centre of the specified disk and then go back to Step 2.

Remark 7. The inequality (42) indicates that the smaller the value p(TH), the more robust the controller becomes. It has been shown that (Lancaster and Tismenetsky, 1985)

p(T/4)---II

T/411-< IITII

rl/411,

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where I1"11 denotes any matrix norm. In some cases, one may want to choose the gain F to assign the nominal closed-loop poles to some predetermined positions in the specified disk. In MIMO systems, the gains F are not unique. Since the matrices T and /4 are functions of F, the gain F ma_y be selected so that the product of

IITII and IIHIf is as small as possible for obtaining robust stability. In general, if a gain F is chosen to make IITII decrease (increase), it will increase (decrease) II/-tll. The next example will display this interesting observation. The exact relation between the choice of F and these two matrix norms is still not clear.

Example 3. Consider a perturbed discrete-time singular system 1 0 0 0 0 1

i] 1-32-16 0]

x(k + 1) = 0 0 - 1 x(k) 0.8 2.4 0

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Robust control for singular systems 1749 which has finite eigenvalues at A1,2=2.1612,

-2.1612. Assume AA and &B are bounded by

IAAI,,, <--

qiH~

= 0.2. 0

1

and

IABim <--q2H2

= 0.01 •

Select a feedback gain

1 0

F = -1.2 - 2 '

then the nominal closed-loop system has finite eigenvalues at hi =0.2 +j0.5292 and A2 = 0 . 2 - j0.5292 E D(0, 0.8). From equation (43), we have

[ 2 . 3 1 9 1 0 2 . 6 3 8 3 ] 1"~ 1.3191 0 2.6383 . 9.6662 0.8 12.9324 Since

i0 6 il

/-) = Ha + q2 H2 Iflm = 0 ql 10.06 1.1 and 0.8 ql = 0.2 < p(~/~) = 0.2311

the feedback closed-loop system is impulse-free and all its finite eigenvalues would stay inside D(0,0.8) under the perturbations for such feedback gain.

If the feedback gain is selected as

and obtain

[:;I

o o]

F = 8 -2.4 0

the finite eigenvalues of the nominal closed-loop system a r e /~1 = 0.1 and A2 = 0 which are closer

to the centre of D(0, 0.8). Then we compute the corresponding T a n d / ~

I

1.1429 0 2.2857] T-~ 0 0 1 , 3.5429 0.8 7.0857 0 0 0.04 1.12 0.8 q 1 = 0 . 2 < - - =0.6520.

p(rn)

It tells us that the feedback closed-loop system can now tolerate perturbations larger than the former feedback system can. It is true for most systems.

In MIMO systems, many different feedback gains could be selected for the same pole- assignment locations. The following shows that the selection of feedback gains will affect the permissible perturbations. For assigning both eigenvalues to the origin, let

16 06 Ool

FI= -0.8 - 2 . 4 0.2 -1.4

be applied, respectively. Then using_Theorem 3, we find their corresponding T and H

~ 0 0 1 , T2 ~ 1.25 0 2.251,

3.2 0.8 1.6 4.2 0.8 2.6 J

i0 6 il I °: il

0 , 0

L0-04 1.12 L0-01 1.07

and upper bounds of associated ql

0.8 0.8

- - - - =0.7122 and - - =0.3314.

p( T~H1)

p(T2H2)

Hence the resultant permissible ranges of ql are quite different. Note that ll~ll~=5.6, IT2II~ = 7.6, II/~111o~ = 1.16, and 11/~211~ = 1.08.

Remark

8. It is noted that we may have to try many initial estimates of the nominal closed-loop eigenvalues before we find the desired feedback gain matrix for robust pole-assignment.

5 . C O N C L U S I O N S

The robust control analysis and design for discrete:time singular systems are studied in this paper. A sufficient condition for simultaneously checking stability robustness, regularity robustness, and impulse-free robustness of singular systems, under highly structured perturbation, is provided. Only simple computations are needed in this approach. Based on the criteria developed, we also provide a qualitative algorithm to design a robust controller for assigning poles inside a specified region. In the literature, little effort has been devoted to dealing with the problem of robust control analysis and design for singular systems. This paper may be viewed as complementary in this field. We also believe that the same problem can be solved by using the Lyapunov equation, as studied for standard state-space systems. By the proposed criterion, the computed upper bounds of perturbations may be exact for certain structures, but in some cases it may be too conservative. Therefore, to improve the

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criterion for obtaining a less conservative bound is also under investigation.

Acknowledgements---The authors would like to thank the

anonymous reviewers for their valuable comments and suggestions. The first author is grateful to Professor Jer-Guang Shieh in the Department of Electrical Engineer- ing at National Sun Yat-Sen University, Taiwan for his encouragement.

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