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Robust Stability and Stabilization for Singular Systems with State Delay and Parameter Uncertainty

Shengyuan Xu∗

Paul Van Dooren∗

Radu S ¸ tefan∗

James Lam†

Abstract This paper considers the problems of robust stability and stabilization for uncertain continuous singular systems with state delay. The parametric uncertainty is assumed to be norm-bounded. The purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular, impulse free and stable for all admissible uncertainties, while the purpose of robust stabilization is to design a state feedback control law such that the resulting closed-loop is robustly stable. These problems are solved via the notions of generalized quadratic stability and generalized quadratic stabilization, respectively. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are derived. A strict linear matrix inequality (LMI) design approach is developed. An explicit expression for the desired robust state feedback control law is also given. Finally, a numerical example is provided to demonstrate the application of the proposed method. Keywords: Continuous singular systems, state delay, parameter uncertainty, robust stability, robust stabilization, linear matrix inequality.

1

Introduction

Control of delay systems has been a topic of recurring interest over the past decades since time delays are often the main causes for instability and poor performance of systems and encountered in various engineering systems such as chemical processes, long transmission lines in pneumatic systems, and so on [8]. Recently, the problems of robust stability analysis and robust stabilization for uncertain delay systems have been studied. Like in the case of uncertain systems without delay, the method based on the concepts of quadratic stability and quadratic stabilizability has been shown to be effective in dealing with these problems in both continuous and discrete contexts [12, 19]. On the other hand, control of singular systems has been extensively studied in the past years due to the fact that singular systems better describe physical systems than regular ones. Singular systems are also referred to as descriptor systems, implicit systems, generalized state-space systems, differential-algebraic systems or semi-state systems [4, 11]. A great number of results based on the theory of regular systems (or state-space systems) have been extended to the area of singular systems [4, 11]. Recently, robust stability and robust stabilization for uncertain singular systems have been considered. The notions of quadratic stability ∗

Center for Systems Engineering and Applied Mechanics (CESAME), Université catholique de Louvain, Avenue G. Lemaître, 4, B—1348 Louvain-la-Neuve, Belgium. † Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong.

1

and quadratic stabilizability for regular systems have been extended [20, 22]. It should be pointed out that the robust stability problem for singular systems is much more complicated than that for regular systems because it requires to consider not only stability robustness, but also regularity and absence of impulses (for continuous singular systems) and causality (for discrete singular systems) at the same time [6, 7], and the latter two need not be considered in regular systems. Very recently, much attention has been paid to singular systems with time delay. For the discrete-time case, when structured uncertainty appears, some results on robust stability were given in [17] by using properties of modulus matrix. When unstructured uncertainty appears, the results on robust stability and robust stabilization were reported in [18], where an LMI design method was developed. For the continuous-time case, numerical methods for such systems were discussed in [1] and [3], while [23] studied the stability problem by analyzing the system’s characteristic equation and some frequency domain conditions for stability were given. It is worth pointing out that no parameter uncertainty was considered in [23]. To the best of our knowledge, when parameter uncertainty appears, there are no results on the problems of robust stability and stabilization for continuous singular delay systems in the literature. In this paper, we address the problems of robust stability and stabilization for uncertain continuous singular systems with state delay. The parameter uncertainties are time invariant and unknown but normbounded. The purpose of the robust stability problem is to develop conditions such that the uncertain singular system is regular, impulse free and stable for all admissible uncertainties. Following the same idea as in dealing with the robust stability problem for uncertain singular systems without delay [20, 22], we introduce the concept of generalized quadratic stability. It is shown that generalized quadratic stability implies robust stability. A necessary and sufficient condition for generalized quadratic stability is obtained in terms of a strict LMI. Similarly, the concept of generalized quadratic stabilization is proposed when dealing with the robust stabilization problem, the purpose of which is the design of memoryless state feedback control laws such that the resultant closed-loop system is regular, impulse free and stable for all admissible uncertainties. A strict LMI design approach is proposed and an explicit expression for the desired robust state feedback control law is given. It is worth pointing out that most LMI-type conditions for singular systems in the literature contain equality constraints [13, 21, 22], which will result in numerical problems when checking such non-strict LMI conditions since equality constraints are fragile and usually not met perfectly [15], therefore, the strict LMI design approach proposed in this paper is much more reliable in numerical computation. Notation. Throughout this paper, for real symmetric matrices X and Y, the notation X ≥ Y (respectively, X > Y ) means that the matrix X −Y is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimension, the superscript “T ” represents the transpose, kxk is the Euclidean norm of the vector x, while ρ(M) denotes the spectral radius of the matrix M.

2

Preliminaries and Problem Formulation

Consider a linear singular system with state delay and parameter uncertainties described by (Σ) :

E x(t) ˙ = (A + ∆A)x(t) + (Ad + ∆Ad )x(t − τ ) + (B + ∆B)u(t) x(t) = φ(t), t ∈ (−τ , 0]

(1) (2)

where x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input. The matrix E ∈ Rn×n may be singular, we shall assume that rank E = r ≤ n. A, Ad and B are known real constant matrices with appropriate dimensions. τ > 0 is a constant time delay, φ(t) is a compatible vector valued continuous function.. ∆A, ∆Ad and ∆B 2

are time-invariant matrices representing norm-bounded parameter uncertainties, and are assumed to be of the form: h i i h (3) ∆A ∆Ad ∆B = MF (σ) NA Nd NB

where M, NA , Nd and NB are known real constant matrices with appropriate dimensions. The uncertain matrix F (σ) satisfies (4) F (σ)F (σ)T ≤ I

and σ ∈ Θ, where Θ is a compact set in R. Furthermore, it is assumed that given any matrix F : F F T ≤ I, there exists a σ ∈ Θ such that F = F (σ). ∆A, ∆Ad and ∆B are said to be admissible if both (3) and (4) hold. Remark 1 It should be pointed out that the structure of the uncertainty with the form (3) and (4) has been used in other papers dealing with the problem of robust stabilization for regular and singular uncertain systems in both continuous and discrete time contexts; see e.g. [16, 22]. The nominal unforced singular delay system of (1) can be written as: E x(t) ˙ = Ax(t) + Ad x(t − τ )

(5)

Definition 1 [4, 11] (I) The pair (E, A) is said to be regular if det(sE − A) is not identically zero. (II) The pair (E, A) is said to be impulse free if deg(det(sE − A)) = rank E. The singular delay system (5) may have an impulsive solution, however, the regularity and the absence of impulses of the pair (E, A) ensure the existence and uniqueness of an impulse free solution to this system, which is shown in the following lemma. Lemma 1 Suppose the pair (E, A) is regular and impulse free, then the solution to (5) exists and is impulse free and unique on (0, ∞). Proof. Noting the regularity and the absence of impulses of the pair (E, A) and using the decomposition as in [4], the desired result follows immediately. ¤ In view of this, we introduce the following definition for singular delay system (5). Definition 2 (I) The singular delay system (5) is said to be regular and impulse free if the pair (E, A) is regular and impulse free. (II) The singular delay system (5) is said to be stable if for any ε > 0 there exists a scalar δ(ε) > 0 such that, for any compatible initial conditions φ(t) satisfying sup−τ ≤t≤0 kφ(t)k ≤ δ(ε), the solution x(t) of system (5) satisfies kx(t)k ≤ ε. Furthermore, x(t) → 0, t → ∞ Throughout this paper, we shall use the following notion of robust stability and robust stabilization for uncertain singular delay system (Σ). 3

Definition 3 The uncertain singular delay system (Σ) is said to be robustly stable if the system (Σ) with u(t) ≡ 0 is regular, impulse free and stable for all admissible uncertainties ∆A and ∆Ad . Definition 4 The uncertain singular delay system (Σ) is said to be robustly stabilizable if there exists a linear state feedback control law u(t) = Kx(t), K ∈ Rm×n such that the resultant closed-loop system is robustly stable in the sense of Definition 3. In this case, u(t) = Kx(t) is said to be a robust state feedback control law for system (Σ). The problem to be addressed in this paper is the development of conditions for robust stability and robust stabilizability for the uncertain singular delay system (Σ) given in (1) and (2).

3

Main Results

In this section, we give a solution to the robust stability analysis and robust stabilization problems formulated previously, by using a strict LMI approach. Firstly, we present the following result for singular delay system (5), which will play a key role in solving the above problems. Theorem 1 The singular delay system (5) is regular, impulse free and stable if there exist a matrix Q > 0 and a matrix P such that EP T = P E T

≥ 0

(6)

AP T + P AT + Ad P T Q−1 P ATd + Q < 0

(7)

For the proof of Theorem 1, we need the following results. Lemma 2 [13]The singular system E x(t) ˙ = Ax(t)

(8)

is regular, impulse free and stable if and only if there exists a matrix P such that EP T = P E T T

T

AP + P A

≥ 0 < 0

Lemma 3 Consider the function ϕ : R+ → R. If ϕ˙ is bounded on [0, ∞), that is, there exists a scalar α > 0 such that |ϕ(t)| ˙ ≤ α for all t ∈ [0, ∞), then ϕ is uniformly continuous on [0, ∞). Lemma 4 [9]Consider the function ϕ : R+ → R. If ϕ is uniformly continuous and lim ϕ(t) = 0

R∞ 0

ϕ(s)ds < ∞. Then

t→∞

Proof of Theorem 1. Suppose both (6) and (7) hold for Q > 0, then from (7) it is easy to see that AP T + P AT < 0

(9)

By Lemma 2, it follows from (6) and (9) that the pair (E, A) is regular and impulse free. Next, we shall show the stability of the singular delay system (5). To this end, we note that the regularity and the absence of impulses of the pair (E, A) implies that there exist two invertible matrices G and H ∈ Rn×n such that [4] " # " # I 0 A 0 r 1 ¯ := GEH = E , A¯ := GAH = (10) 0 0 0 In−r 4

where Ir ∈ Rr×r and In−r ∈ R(n−r)×(n−r) are identity matrices, A1 ∈ Rr×r . According to (10), " " " # # ¯11 P¯12 ¯ A P A d11 d12 ¯ := GQGT = Q11 A¯d := GAd H = , P¯ := GP H −T = , Q ¯T Ad21 Ad22 P¯21 P¯22 Q 12

let ¯ 12 Q ¯ 22 Q

#

(11)

Then, from (6) and (7) we have ¯T ≥ 0 ¯ P¯ T = P¯ E E ¯ −1 P¯ A¯Td + Q ¯ < 0 A¯P¯ T + P¯ A¯T + A¯d P¯ T Q

(12) (13)

By using a Schur complement argument, it follows from (13) that # " ¯ A¯d P¯ T A¯P¯ T + P¯ A¯T + Q 0. Define Z

−1 ζ 1 (t) + V (ζ t ) = ζ 1 (t) P¯11 T

t

¯ P¯ −T ζ(s)ds ζ(s)T P¯ −1 Q

t−τ

where ζ t = ζ(t + β),

β ∈ [−τ , 0]

Then, the time-derivative of V (ζ t ) along the solution of (21) and (22) is given by ¢ d ¡ ¯ ¯ P¯ −T ζ(t) − ζ(t − τ )T P¯ −1 Q ¯ P¯ −T ζ(t − τ ) ζ(t)T P¯ −1 Eζ(t) + ζ(t)T P¯ −1 Q dt ¯ ¯ P¯ −T ζ(t) + 2ζ(t)T P¯ −1 A¯d ζ(t − τ ) − ζ(t − τ )T P¯ −1 Q ¯ P¯ −T ζ(t − τ ) + ζ(t)T P¯ −1 Q = 2ζ(t)T P¯ −1 Aζ(t) ¢ ¡ ¯ −1 P¯ A¯T + Q ¯ P¯ −T ζ(t) ≤ ζ(t)T P¯ −1 A¯P¯ T + P¯ A¯T + A¯d P¯ T Q d

V˙ (ζ t ) =

It follows from this inequality and (13) that V˙ (ζ t ) < 0 and

Z t −1 T ¯ −1 ¯ ¯ P¯ −T ζ(s)ds − V (ζ 0 ) ζ(s)T P¯ −1 Q λ1 kζ 1 (t)k − V (ζ 0 ) ≤ ζ 1 (t) P11 ζ 1 (t) − V (ζ 0 ) ≤ ζ 1 (t) P11 ζ 1 (t) + t−τ Z t Z t Z t V˙ (ζ s )ds ≤ −λ2 kζ(s)k2 ds ≤ −λ2 kζ 1 (s)k2 ds < 0 (23) = 2

T

0

0

0

where ¡ −1 ¢ λ1 = λmin P¯11 > 0,

¢ £ ¡ ¤ ¯ −1 P¯ A¯Td + Q ¯ P¯ −T > 0 λ2 = −λmax P¯ −1 A¯P¯ T + P¯ A¯T + A¯d P¯ T Q

Taking into account (23), we can deduce that 2

λ1 kζ 1 (t)k + λ2

Z

t

0

kζ 1 (s)k2 ds ≤ V (ζ 0 )

Therefore

and

Z

0

where m1 =

t

kζ 1 (t)k2 ≤ m1

(24)

kζ 1 (s)k2 ds ≤ m2

(25)

1 V (ζ 0 ) > 0, λ1

m2 =

1 V (ζ 0 ) > 0 λ2

Thus, kζ 1 (t)k is bounded. Considering ° this° and (20), it can be deduced from (22) that kζ 2 (t)k is bounded, ° ° d kζ 1 (t)k2 is bounded too. By Lemma 3 and hence, it follows from (21) that °ζ˙ 1 (t)° is bounded, therefore, dt we have that kζ 1 (t)k2 is uniformly continuous. Therefore, noting (25) and using Lemma 4, we obtain lim kζ 1 (t)k = 0

t→∞

It then follows from (20) and (22) that lim kζ 2 (t)k = 0

t→∞

¤

Thus, (ΣD ) is stable. This completes the proof. 6

Remark 2 Theorem 1 provides a sufficient condition for the singular delay system (Σ) to be regular, impulse free and stable. When E = I, the singular delay system (Σ) reduces to a state-space delay system and it is easy to show that Theorem 1 coincides with Lemma 1 in [10]. Therefore, Theorem 1 can be viewed as an extension of existing results on state-space delay systems to singular delay systems. Furthermore, by comparing Theorem 1 with Lemma 2 in [13], we can regard Theorem 1 as an extension of existing results on singular systems without delay to singular delay systems. Following the same philosophy as in dealing with the problems of robust stability and robust stabilization for uncertain singular systems without delay [20, 22] and taking into account Theorem 1, we introduce the following definitions. Definition 5 The uncertain singular delay system (Σ) is said to be generalized quadratically stable if there exist matrices Q > 0 and P such that EP T = P E T ≥ 0

(A + ∆A) P T + P (A + ∆A)T + (Ad + ∆Ad ) P T Q−1 P (Ad + ∆Ad )T + Q < 0

(26) (27)

for all admissible uncertainties ∆A and ∆Ad . Definition 6 The uncertain singular delay system (Σ) is said to be generalized quadratically stabilizable if there exists a linear state feedback control law u(t) = Kx(t), K ∈ Rm×n , matrices Q > 0 and P such that EP T

= P ET ≥ 0

(AK + ∆AK ) P T + P (AK + ∆AK )T + (Ad + ∆Ad ) P T Q−1 P (Ad + ∆Ad )T + Q < 0

(28) (29)

for all admissible uncertainties ∆A, ∆Ad and ∆B, where AK = A + BK,

∆AK = ∆A + ∆BK.

(30)

The following lemma shows that generalized quadratic stability and generalized quadratic stabilization imply robust stability and robust stabilization, respectively. Lemma 5 Consider the uncertain singular delay system (Σ). If it is generalized quadratically stable, then it is robustly stable. If it is generalized quadratically stabilizable, then it is robustly stabilizable. Proof. From Theorem 1, the desired results follow immediately. ¤ In view of this, necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilizability for the uncertain singular delay system (Σ) are derived. In order to obtain these results, the following lemma is needed. Lemma 6 [14]Given matrices Ω, Γ and Ξ of appropriate dimensions and with Ω symmetrical, then Ω + ΓF (σ)Ξ + (ΓF (σ)Ξ)T < 0 for all F (σ) satisfying F (σ)F (σ)T ≤ I, if and only if there exists a scalar ² > 0 such that Ω + ²ΓΓT + ²−1 ΞT Ξ < 0 For simplicity we introduce the matrix Φ ∈ Rn×(n−r) satisfying EΦ = 0 and rank Φ = n − r. Now, we are in a position to give the quadratic stability result. 7

Theorem 2 The uncertain singular delay system (Σ) is generalized quadratically stable if and only if there exist a scalar ² > 0, matrices X > 0, Q > 0 and Y such that the following LMI holds:   ¡ ¢T ¡ ¢ ¡ ¢T ¡ ¢ A EX + Y ΦT + EX + Y ΦT AT + ²M M T + Q Ad EX + Y ΦT EX + Y ΦT NAT ¡ ¡ ¢ ¢   −Q EX + Y ΦT NdT  < 0 (31) EX + Y ΦT ATd  ¡ ¢T ¡ ¢T NA EX + Y ΦT Nd EX + Y ΦT −²I

Proof. (Sufficiency) Assume that there exist a scalar ² > 0, matrices X > 0, Q > 0 and Y satisfying (31). By setting P = EX + Y ΦT , it is easy to see that EP T = P E T ≥ 0

(32)

Observe that for any F (σ) satisfying (4) and any scalar ² > 0, " # " # # " i h h T P N M ∆AP T + P ∆AT ∆Ad P T T A T N PT + = F (σ) F (σ) MT P N A d P ∆ATd 0 0 P NdT " # " # i h T ²MM T 0 P N A T N PT ≤ + ²−1 P N A d 0 0 P NdT

0

i

Therefore, # (A + ∆A) P T + P (A + ∆A)T + Q (Ad + ∆Ad ) P T P (Ad + ∆Ad )T −Q # " " # h T AP T + P AT + ²MM T + Q Ad P T P N −1 A + ² ≤ NA P T P ATd −Q P NdT "

Nd P T

i

By using a Schur complement argument, it follows from this inequality and (31) that # " (A + ∆A) P T + P (A + ∆A)T + Q (Ad + ∆Ad ) P T < 0, P (Ad + ∆Ad )T −Q or, equivalently, (A + ∆A) P T + P (A + ∆A)T + (Ad + ∆Ad ) P T Q−1 P (Ad + ∆Ad )T + Q < 0 This inequality and (32) are precisely (26) and (27) in Definition 5. Hence, the uncertain singular delay system (Σ) is generalized quadratically stable. (Necessity) Assume that the uncertain singular delay system (Σ) is generalized quadratically stable. It follows from Definition 5 that there exist matrices Q > 0 and P such that (26) and (27) hold. Thus, for all F (σ) satisfying (3) and (4), the following inequality holds: # " (A + ∆A) P T + P (A + ∆A)T + Q (Ad + ∆Ad ) P T 0 and by replacing P into (33), the desired result follows immediately.

¤

The generalized quadratic stabilizability result is presented in the following theorem. Theorem 3 The uncertain singular delay system (Σ) is generalized quadratically stabilizable if and only if there exist a scalar ² > 0, matrices X > 0, Q > 0, Y and Z such that the following LMI holds:   W Ad Υ(X, Y )T Υ(X, Y )NAT + Z T NBT   (37) Υ(X, Y )ATd −Q Υ(X, Y )NdT  0 such that Υ(X, Y ) = Υ(X, Y ) + θI also satisfies (37) with Υ(X, Y) invertible. Applying the control law in (38) to the uncertain singular delay system (Σ), we obtain the resultant closed-loop system as E x(t) ˙ = Ac x(t) + (Ad + ∆Ad )x(t − τ ) where Ac = A + BZΥ(X, Y )−T + ∆A + ∆BZΥ(X, Y )−T Along a similar line as in the proof of the sufficiency part of Theorem 2, we can obtain Ac Υ(X, Y )T + Υ(X, Y )ATc + (Ad + ∆Ad ) Υ(X, Y )T Q−1 Υ(X, Y ) (Ad + ∆Ad )T + Q < 0 for all admissible uncertainties ∆A and ∆Ad . Thus, by Definition 6, we have that the uncertain singular delay system (Σ) is generalized quadratically stabilizable. (Necessity) Suppose the uncertain singular delay system (Σ) is generalized quadratically stabilizable, then by Definition 6 it follows that there exist a state feedback control law u(t) = Kx(t), K ∈ Rm×n , matrices Q > 0 and P such that (28) and (29) hold. Therefore, # " (AK + ∆AK ) P T + P (AK + ∆AK )T + Q (Ad + ∆Ad ) P T 0, matrices X > 0 and Y such that   (A + BK) Υ(X, Y )T + Υ(X, Y ) (A + BK)T + ²MM T + Q Ad Υ(X, Y )T Υ(X, Y ) (NA + NB K)T   Υ(X, Y )ATd −Q Υ(X, Y )NdT   T T (NA + NB K) Υ(X, Y ) Nd Υ(X, Y ) −²I Finally, by setting Z = KΥ(X, Y )T we obtain (37). This completes the proof.

¤

Remark 3 Theorem 3 presents a necessary and sufficient condition for generalized quadratic stabilizability. The desired robustly stabilizing state feedback for uncertain singular system (Σ) can be obtained by solving the strict LMI (37), which can be solved numerically very efficiently by using interior-point algorithm, and no tuning of parameters is involved [2]. It is worth pointing out that strict LMI (37) is expressed by using the system matrices of (Σ). The design procedure involves no decomposing of the system, which can get around certain numerical problems arising from decomposition of matrices and thus makes the design procedure relatively simple and reliable.

4

Numerical Example

In this section, we give an example to demonstrate the effectiveness of the proposed method. Consider the linear uncertain singular delay system (Σ) with parameters as follows:           0.5 1 1 1.5 0.5 1 −1 0 −1 1 1 0           E =  1 −1 1  , A =  −1 0 1  , Ad =  1 −1 0.5  , B =  1 0  , M =  0.2  0.1 1 1 0.5 0 1 0.3 0.5 −1 2 0 1 h i h i h i NA = 0.2 0.1 0.3 , Nd = 0.1 0.2 0.5 , NB = 0.1 0.1 10

In this example we assume that the time delay τ = 1.5. The purpose is the design of a state feedback control law such that, for all admissible uncertainties, the resultant closed-loop system is regular, impulse free and stable. To this end, we choose h i ΦT = −1 1 2 Using Matlab LMI Control Toolbox to solve the LMI (37), we obtain the    0.4272 −0.0076 0.2185 −0.0948 −0.1671    X =  −0.0948 0.5988 0.1269 0.1841  , Q =  −0.0076 0.2465 0.0071 −0.1671 0.1841 0.4025 " # −1.6809 −0.4028 −0.3668 Z = , ² = 0.6369 0.7616 1.2915 −0.0220

solution as follows:    0.2378 0.2465    0.0071  , Y =  −0.0710  −0.1259 0.4402

Therefore, by Theorem 3, a robustly stabilizing state feedback control law can be obtained as " # −2.4756 3.8449 −6.0927 u(t) = x(t) −1.9419 −30.7452 17.9458

5

Conclusions

The problems of robust stability and stabilization for uncertain continuous singular systems with state delay and parameter uncertainty have been studied. Based on the notions of generalized quadratic stability and generalized quadratic stabilization, these problems have been solved. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are presented in terms of a strict LMI, respectively. The proposed state feedback control law guarantees that the resultant closed-loop system is regular, impulse free as well as stable for all admissible uncertainties. Acknowledgements This paper presents research supported by the Belgium Programme on Inter-university Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture. The scientific responsibility rests with its authors. The third author has been partially supported by NATO.

References [1] U. Ascher and L. R. Petzold. The numerical solution of delay-differential-algebraic equations of retarded and neutral type. SIAM J. Numer. Anal., 32:1635—1657, 1995. [2] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in Systems and Control Theory. SIAM Studies in Applied Mathematics, Philadelphia, PA, 1994. [3] S. L. Campbell. Singular linear systems of differential equations with delays. Appl. Anal., 2:129—136, 1980. [4] L. Dai. Singular Control Systems. Berlin: Springer-Verlag, 1989. [5] M. C. de Oliveira, J. Bernussou, and J. C. Geromel. A new discrete-time robust stability condition. Systems Control Lett., 37:261—265, 1999. 11

[6] C.-H. Fang and F.-R. Chang. Analysis of stability robustness for generalized state-space systems with structured perturbations. Systems Control Lett., 21:109—114, 1993. [7] C.-H. Fang, L. Lee, and F.-R. Chang. Robust control analysis and design for discrete-time singular systems. Automatica, 30:1741—1750, 1994. [8] J. K. Hale. Theory of Functional Differential Equations. New York: Springer-Verlag, 1977. [9] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic. Nonlinear and Adaptive Control Design. New York: John Wiley Sons Inc., 1995. [10] J. H. Lee, S. W. Kim, and W. H. Kwon. Memoryless H∞ controllers for state delayed systems. IEEE Trans. Automat. Control, 39:159—162, 1994. [11] F. L. Lewis. A survey of linear singular systems. Circuits, Syst. Signal Processing, 5:3—36, 1986. [12] M. S. Mahmoud and N. F. Al-Muthairi. Quadratic stabilization of continuous time systems with statedelay and norm-bounded time-varying uncertainties. IEEE Trans. Automat. Control, 39:2135—2139, 1994. [13] I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda. H∞ control for descriptor systems: a matrix inequalities approach. Automatica, 33:669—673, 1997. [14] I. R. Petersen. A stabilization algorithm for a class of uncertain linear systems. Systems Control Lett., 8:351—357, 1987. [15] E. Uezato and M. Ikeda. Strict LMI conditions for stability, robust stabilization, and H∞ control of descriptor systems. In Proc. 38th IEEE Conf. Decision and Control, pages 4092—4097, Phoenix, Arizona, USA, December, 1999. [16] L. Xie and C. E. De Souza. Robust H∞ control for linear time-invariant systems with norm-bounded uncertainty in the input matrix. Syst. Contr. Lett., 14:389—396, 1990. [17] S. Xu and J. Lam. Robust stability analysis for uncertain discrete singular systems with state delay. Systems Control Lett. (under review). [18] S. Xu, J. Lam, and C. Yang. Robust H∞ control for discrete singular systems with state delay and parameter uncertainty. Dynamics of Continuous, Discrete and Impulsive Systems. (Revised). [19] S. Xu, J. Lam, and C. Yang. Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay. Systems Control Lett., 43:77—84, 2001. [20] S. Xu and C. Yang. An algebraic approach to the robust stability analysis and robust stabilization of uncertain singular systems. Int. J. Systems Sci., 31:55—61, 2000. [21] S. Xu and C. Yang. H∞ state feedback control for discrete singular systems. IEEE Trans. Automat. Control, 45(7):1405—1409, 2000. [22] S. Xu, C. Yang, Y. Niu, and J. Lam. Robust stabilization for uncertain discrete singular systems. Automatica, 37:769—774, 2001. [23] W. Zhu and L. Petzold. Asymptotic stability of linear delay differential-algebraic equations and numerical methods. Applied Numerical Mathematics, 24:247—264, 1997. 12