Robust Stabilizability of Switched Linear Time-Delay Systems with Polytopic Uncertainties

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1 Proceedings of the 17th World Congress The International Federation of Automatic Control Robust Stabilizability of Switched Linear Time-Delay Systems with Polytopic Uncertainties Yijing Wang Zhenxian Yao Zhiqiang Zuo Huimin Zhao Guoshan Zhang School of Electrical Engineering & Automation, Tianjin University, Tianjin, P. R. China 372. School of Electrical Engineering & Automation, Tianjin University, Tianjin, P. R. China 372. Corresponding Author: School of Electrical Engineering & Automation, Tianjin University, Tianjin, P. R. China 372 ( General Courses Department, Academy of Military Transportation, Tianjin, P. R. China School of Electrical Engineering & Automation, Tianjin University, Tianjin, P. R. China 372. Abstract: This paper is devoted to robust stability analysis via state feedback of linear systems with state delay that are composed of polytopic uncertain subsystems. By state feedback, we mean that the switchings among subsystems are dependent on system states. For continuous-time switched linear systems, we show that if there exists a common positive definite matrix for stability of all convex combinations of the extreme points which belong to different subsystem matrices, then the switched system is robustly stabilizable via state feedback. The stabilityconditions ofbothdelay-independentdelay-dependentare analyzed using Lyapunov- Razumikhin functional approach. Furthermore, we propose the switching rules by using the obtained common positive definite matrix. Keywords: Switched systems; Robust Stabilizability; Lyapunov-Razumikhin function; Switching rule; State feedback. 1. INTRODUCTION In recentyears, the study of switched systems has received growing attention. By a switched system, we mean a hybrid dynamical system that is composed of a family of continuous-time or discrete-time subsystems a rule orchestrating the switching between the subsystems (see Branicky 1998 Ye et al 1998). The stard model for such systems is given in Branicky et al These systems arise as models for phenomena which cannot be described byexclusivelycontinuous or exclusivelydiscrete processes. Examples include the control of manufacturing systems (Pepyne et al 2, Song et al 2), communication networks, traffic control (Horowitz et al 2, Livadas et al 2, Varaiva 1993), chemical processing (Engell 2) automotive engine control aircraft control (Antsaklis 2). In the last two decades, there has been increasinginterest in stability analysis control design for switched systems (for example Branicky 1998,Ye et al 1998,Liberzon et al 1999). A survey of basic problems in stability design of switched systems has been proposed recently in Liberzonet al The motivation for studying switched This work was supported by National Natural Science Foundation of China under Grant 65412, , systems is from the fact that many practical systems are inherently multi-modelin the sense that severaldynamical subsystems are required to describe their behavior which may depend on various environmental factors, that the methods of intelligent control design are based on the idea of switching between different controllers. In Zhai et al 23, the authors only consider the quadratic stabilizability via state feedback for both continuous-time discrete-time switched linear uncertain systems without time-delay. Systems with time-delay constitute basic mathematical models of real phenomena, for instance, in circuit theory, economics mechanics. The analysis synthesis of controllers for switched systems with time-delay have been attracting increasingly more attention. Some results have been obtained. In Lu et al 26, the asymptotic stability of switched systems whose subsystems are time delay systems is studied via LMI (linear matrix inequalities) approach; in Sehjeong et al 26, they consider a switching system composed of a finite number of linear delay differential equations. There are a few existing results concerning quadratic stabilization of switched linear systems that are composedof severalunstable linear timeinvariantsubsystems. However, modeluncertaintyis often encountered in control systems. At the knowledge of the /8/$2. 28 IFAC / KR

2 authors, there is little research focused on dealing with such systems. Motivated by the above results, here we will consider switched linear systems with both polytopic uncertainties time-delay. Specifically, we use Lyapunov- Razumikhin function approach to deal with time-delay. Notations: The following notations are used throughout this paper. R is the set of real numbers R n, R m n are sets of real vectors with dimension n real matrices with dimension m n, respectively. The notation X Y (respectively, X > Y ), where X Y are symmetric matrices, means that the matrix X Y is positive semidefinite (respectively, positive definite). I denote the identity matrix zero matrix with compatible dimensions. denotes the symmetry elements in the symmetric matrices, that is X Y = Z X Y Y T Z 2. PROBLEM STATEMENT In this section, we consider the linear time-delay system with polytopic uncertainties ẋ(t) = A σ(x,t) x(t) A dσ(x,t) x(t τ). (1) where x(t) R n is the state, σ(x, t) is a switching rule defined by σ(x, t) : R n R {1, 2}, R denotes nonnegative real numbers. τ is assumed to be a constant time delay. Therefore, the switched system is composed of two continuous-time subsystems with timedelay S 1 : ẋ(t) = A 1 x(t) A d1 x(t τ). (2) S 2 : ẋ(t) = A 2 x(t) A d2 x(t τ). (3) Here, we assume that both S 1 S 2 are uncertain systems of polytopic type described as N i N i A i = µ ij A ij A di = µ ij A dij. (4) j=1 j=1 where µ i = (µ i1, µ i2,, µ ini ) belongs to µ N i i : µ ij = 1, µ ij j=1 (5) Here i {1, 2} denotes the i th subsystem. And A ij, A dij, j = 1, 2,, N i are constant matrices denoting the extreme points of the polytope A i, A di, N i is the number of the extreme points. The following lemma will be useful in the proof of our main results. Lemma 1. (see Cao et al 1998) For any x, y R n a matrix W > with compatible dimensions, the following inequality holds 2x T y x T Wx y T W 1 y (6) 3. MAIN RESULTS In this section, we will first give a delay-independent robust stability condition for system (1). Then a delaydependent one will be derived. 3.1 DELAY-INDEPENDENT ANALYSIS If S 1 or S 2 is robustly stable, we can always activate the stable subsystem so that the entire switched system is robustly stable. Therefore, to make the switching problem nontrivial, we make the following assumptions. Assumption 1: Both S 1 S 2 are delay-independent robustly unstable. Now, we need the definition of robust stabilizability via state feedback for the switched system (1). Definition 1: The system (1) is said to be robustly stabilizable via state feedback if there exist positive-definite function V (x) = x T Px, a positive number ǫ a switching rule σ(x, t) depending on x such that d dt V (x) < ǫxt x (7) holds for all trajectories of the system (1). Our aim in this section is to find a state feedback (statedependent switching rule) σ(x, t) such that the switched system (1) is robustly stable. We state prove the following main result. Theorem 2. The switched system (1) is delay-independent robustly stabilizable via state feedback if there exist constant scalars λ ij s (i = 1, 2; j = 1, 2,, N i ) satisfying λ ij 1 P >, W > such that P W 1 (8) λ ij A 1i (1 λ ij )A 2j T P Pλ ij A 1i (1 λ ij )A 2j Pλ ij A d1i WA T d 1i (1 λ ij )A d2j WA T d 2j P P < (9) holds for all i = 1, 2; j = 1, 2,, N i. Proof. For the benefit of notation simplicity, we only give the proof in the case of N 1 = N 2 = 2. The extension from N 1 = N 2 = 2 to general case is very obvious. From (9), we know that there always exists a positive scalar ǫ such that λ ij A 1i (1 λ ij )A 2j T P Pλ ij A 1i (1 λ ij )A 2j Pλ ij A d1i WA T d 1i (1) (1 λ ij )A d2j WA T d 2j P P < ǫi Then, for any x, we obtain x T λ 11 A 11 (1 λ 11 )A 21 T Px x T Pλ 11 A 11 (1 λ 11 )A 21 x x T Pλ 11 A d11 WA T d 11 (1 λ 11 )A d21 WA T d 21 Px x T Px < ǫx T x x T λ 12 A 11 (1 λ 12 )A 22 T Px x T Pλ 12 A 11 (1 λ 12 )A 22 x x T Pλ 12 A d11 WA T d 11 (1 λ 12 )A d22 WA T d 22 Px x T Px < ǫx T x x T λ 21 A 12 (1 λ 21 )A 21 T Px x T Pλ 21 A 12 (1 λ 21 )A 21 x x T Pλ 21 A d12 WA T d 12 (1 λ 21 )A d21 WA T d 21 Px x T Px < ǫx T x (11) (12) (13) x T λ 22 A 12 (1 λ 22 )A 22 T Px x T Pλ 22 A 12 (1 λ 22 )A 22 x x T Pλ 22 A d12 WA T d 12 (14) (1 λ 22 )A d22 WA T d 22 Px x T Px < ǫx T x which can be rewritten as λ 11 x T Ξ 11 x (1 λ 11 )x T Ξ 21 x < ǫx T x (15) λ 12 x T Ξ 11 x (1 λ 12 )x T Ξ 22 x < ǫx T x (16) 7649

3 λ 21 x T Ξ 12 x (1 λ 21 )x T Ξ 21 x < ǫx T x (17) λ 22 x T Ξ 12 x (1 λ 22 )x T Ξ 22 x < ǫx T x (18) Here Ξ ij := A T ij P PA ij PA dij WA T d ij P P From the above inequalities, it is obvious to verify that either x T Ξ 11 x < ǫx T x x T Ξ 12 x < ǫx T x (19) or x T Ξ 21 x < ǫx T x x T Ξ 22 x < ǫx T x (2) is true. For example, if (19) is not true with x T Ξ 11 x > ǫx T x, then we get x T Ξ 21 x < ǫx T x from (15) x T Ξ 22 x < ǫx T x from (16). The same is true for other cases. Now, we define the switching rule as σ(x, t) {i x T Ξ ij x < ǫx T x, j = 1, 2} (21) Then, based on the above discussion, we get x T Ξ σ1 x < ǫx T x x T Ξ σ2 x < ǫx T x (22) Thus it follows that x T Ξ σ x < ǫx T x (23) since A σ is a linear convex combination of A σ1 A σ2, A dσ is a linear convex combination of A dσ1 A dσ2. Given P >, here we choose the Lyapunov function as V (x(t)) = x T (t)px(t). The derivative of V (x(t)) along the solution of (1) is V (x(t)) = 2x T (t)pa σ x(t) 2x T (t)pa dσ x(t τ) (24) From Lemma 1, it follows that V (x(t)) x T (t)(a T σ P PA σ PA dσ WA T d σ P)x(t) x T (t τ)w 1 x(t τ) (25) From (8), we get V (x(t)) x T (t)(a T σ P PA σ PA dσ WA T d σ P)x(t) V (x(t τ)) (26) By Razumikhin Theorem, to prove (1) is robustly stable, it suffices to show that there exist an η > 1 a δ > such that V (x(t)) δx T (t)x(t) if V (x(t θ)) < ηv (x(t)), θ τ, In the remainder of the proof, we will construct such η δ show that they satisfy the above condition. From (23), we can see that there exists a δ 1 > such that A T σ P PA σ PA dσ WA T d σ P (1 2δ 1 )P < εi Let η = 1 δ 1. Now suppose that V (x(t θ)) < ηv (x(t)), θ τ,. Then from (26), it follows that V (x(t)) x T (t)(a T σ P PA σ PA dσ WA T d σ P ηp)x(t) < (δ 1 ǫ)x T (t)x(t) = δx T (t)x(t) (27) is true for all trajectories of (1). Thus the switched system is robustly stable. Remark 3. Multiplying P 1 for both sides of inequalities (8) (9) letting Q = P 1, we see that the matrix inequalities (8) (9) are equivalent to the following matrix inequalities Q W (28) Qλ ij A 1i (1 λ ij )A 2j T λ ij A 1i (1 λ ij )A 2j Q λ ij A d1i WA T d 1i (1 λ ij )A d2j WA T d 2j Q < (29) If the values of λ ij, i, j = 1, 2 are fixed, with respect to parameters Q W,then inequalities (28) (29) can be converted to linear matrix inequalities which can be solved efficiently using the interior-point method. Remark 4. It is easy to extend Theorem 2 to the case of more than two subsystems. But the number of conditions to be checked grows very fast when the number of subsystems the extreme points of each A i are large. 3.2 DELAY-DEPENDENT ANALYSIS Here to make the switching problem nontrivial, we also make the following assumption. Assumption 2: Both S 1 S 2 are delay-dependent robustly unstable. Theorem 5. The switched system (1) is delay-dependent robustly stabilizable via state feedback if there exist constant scalars λ ij s(i = 1, 2; j = 1, 2,, N i) satisfying λ ij 1, τ > P >, P 1 >, P 2 > such that A T ij P 1 1 A ij P (3) A T d ij P 1 2 A dij P (31) λ ij  1i (1 λ ij )Â2j T P Pλ ij  1i (1 λ ij )  2j τpλ ij A d1i (P 1 P 2 )A T d 1i (1 λ ij )A d2j (P 1 P 2 )A T d 2j P 2τP < (32) holds for all i = 1, 2; j = 1, 2,, N i. Here Âij = A ij A dij Proof. For the benefit of notation simplicity, here we also give the proof in the case of N 1 = N 2 = 2. From (32), we know that there always exists a positive scalar ǫ such that λ ij  1i (1 λ ij )Â2j T P Pλ ij  1i (1 λ ij )  2j τpλ ij A d1i (P 1 P 2 )A T d 1i (1 λ ij )A d2j (33) (P 1 P 2 )A T d 2j P 2τP < ǫi Then, for any x, we have x T λ 11  11 (1 λ 11 )Â21 T Px x T Pλ 11  11 (1 λ 11 )Â21x τx T Pλ 11 A d11 (P 1 P 2 )A T d 11 (1 λ 11 )A d21 (P 1 P 2 )A T d 21 Px 2τx T Px < ǫx T x (34) x T λ 12  11 (1 λ 12 )Â22 T Px x T Pλ 12  11 (1 λ 12 )Â22x τx T Pλ 12 A d11 (P 1 P 2 )A T d 11 (1 λ 12 )A d22 (P 1 P 2 )A T d 22 Px 2τx T Px < ǫx T x (35) x T λ 21  12 (1 λ 21 )Â21 T Px x T Pλ 21  12 (1 λ 21 )Â21x τx T Pλ 21 A d12 (P 1 P 2 )A T d 12 (1 λ 21 )A d21 (P 1 P 2 )A T d 21 Px 2τx T Px < ǫx T x (36) x T λ 22  12 (1 λ 22 )Â22 T Px x T Pλ 22  12 (1 λ 22 )Â22x τx T Pλ 22 A d12 (P 1 P 2 )A T d 12 (1 λ 22 )A d22 (P 1 P 2 )A T d 22 Px 2τx T Px < ǫx T x (37) 765

4 which can be rewritten as λ 11 x T Ω 11 x (1 λ 11 )x T Ω 21 x < ǫx T x (38) λ 12 x T Ω 11 x (1 λ 12 )x T Ω 22 x < ǫx T x (39) λ 21 x T Ω 12 x (1 λ 21 )x T Ω 21 x < ǫx T x (4) λ 22 x T Ω 12 x (1 λ 22 )x T Ω 22 x < ǫx T x (41) Here Ω ij = ÂT ij P PÂij τpa dij (P 1 P 2 )A T d ij P 2τP It is very easy to verify from the above inequalities that either or x T Ω 11 x < ǫx T x x T Ω 12 x < ǫx T x (42) x T Ω 21 x < ǫx T x x T Ω 22 x < ǫx T x (43) is true. For example, if (42) is not true with x T Ω 11 x > ǫx T x,then we get x T Ω 21 x < ǫx T x from (38) x T Ω 22 x < ǫx T x from (39). The same is true for other cases. Now, we define the switching rule as σ(x, t) {i x T Ω ij x < ǫx T x, j = 1, 2} (44) Then, based on the above discussion, we get x T Ω σ1 x < ǫx T x x T Ω σ2 x < ǫx T x (45) And thus x T Ω σ x < ǫx T x (46) since A σ is a linear convex combination of A σ1 A σ2, A dσ is a linear convex combination of A dσ1 A dσ2. Given P >, here we choose definite Lyapunov function V (x(t)) = x T (t)px(t). Since x(t) is continuously differentiable for t, using the Leibniz-Newton formula, one can write x(t τ) = x(t) = x(t) t t τ t t τ ẋ(s)ds A σ x(s) A dσ x(s τ)ds for t τ. Thus the system (1) can be written as ẋ(t) = (A σ A dσ )x(t) t A dσ A σ x(s) A dσ x(s τ)ds t τ (47) (48) The derivative of V (x(t)) along the solution of (1) is V (x(t)) = 2x T (t)p(a σ A dσ )x(t) t 2x T (t)pa dσ A σ x(s) A dσ x(s τ)ds (49) t τ Now we translate the interval t τ, t of x(t) to τ,. And from the Lemma 1, we get V (x(t)) 2x T (t)p(a σ A dσ )x(t) τx T (t)pa dσ (P 1 P 2 )A T d σ Px(t) τ τ x T (t s)a T σ P 1 1 A σ x(t s)ds It follows from (3) (31) that x T (t τ s)a T d σ P 1 2 A dσ x(t τ s)ds x T (t)a T σ P 1 1 A σ x(t) x T (t)px(t) x T (t)a T d σ P2 1 A dσ x(t) x T (t)px(t) (5) (51) Let Âσ = A σ A dσ. Hence V (x(t)) 2x T (t)pâσx(t) τx T (t)pa dσ (P 1 P 2 )A T d σ Px(t) τ V (x(t s))ds τ V (x(t τ s))ds (52) By Razumikhin Theorem, to guarantee the asymptotic stability of system, it is suffices to find two scalars η > 1 δ > such that V x(t) < δv (x(t)) if V (x(t θ)) < ηv (x(t)), θ 2τ, From (46), we know that there exists a δ 1 > such that x T ÂT σ P PÂσ τpa dσ (P 1 P 2 )A T d σ P 2τ(1 2δ 1 )Px < ǫx T x Let η = 1δ 1. Suppose that V (x(tθ)) < ηv (x(t)), θ 2τ,. Then from (52), we get V (x(t)) 2x T (t)pâσx(t) 2τηx T (t)px(t) τx T (t)pa dσ (P 1 P 2 )A T d σ Px(t) = x T (t)(ât σ P PÂσ τpa dσ (P 1 P 2 )A T d σ P 2τηP)x(t) < 2τδ 1 x T (t)px(t) (53) Thus the switched system is robustly stable. Remark 6. If we multiple P 1 by the left right to the inequalities (3) (32) let Q = P 1, we can easily see that the matrix inequalities (3) (32) are equivalent to the followingmatrixinequalities usingschurcomplement. Q QA T ij (54) P 1 Q QA T dij (55) P 2 Qλ ij  1i (1 λ ij )Â2j T λ ij  1i (1 λ ij )  2j Q τλ ij A d1i (P 1 P 2 )A T d 1i (1 λ ij )A d2j (P 1 P 2 )A T d 2j 2τQ < (56) Remark 7. The above three conditions involves bilinear matrix inequalities (BMI) with respect to λ ij s. Although there is LMI Toolbox in Matlab which is convenient for us for solving LMI problem, it is not easy to solve BMI problem up till. We can use the branch bound methods suggested or the homotopy-based algorithm to deal with the BMI problem. 4. NUMERICAL EXAMPLE Example 1. (Delay-independent case) Consider the system (1) composed of two subsystems where A 11 = A 4 12 = 4 (57) A d11 = A 1 1 d12 = A 21 = 1 1 A d21 = A 22 =.1 1 (58) 1.1 A d22 =

5 Both A 11, A d11 A 12, A d12 are delay-independent robustly unstable. Similarly, both A 21, A d21 A 22, A d22 are delay-independent robustly unstable. Therefore, both S 1 S 2 are robustly unstable. Now, we let λ 11 = λ 12 = λ 21 = λ 22 =.5, in order to satisfy the conditions in Theorem 2, by using MATLAB tools we get P = W = Here we let Γ ij := λ ij A 1i (1 λ ij )A 2j T P Pλ ij A 1i (1 λ ij )A 2j Pλ ij A d1i WA T d 1i (1 λ ij )A d2j WA T d 2j P P Then we obtained Γ 11 = Γ 21 = Γ 12 = Γ 22 = We can see they satisfy the conditions of (8) (9). Therefore, this switched system is robustly stablilizable via state feedback. Now, we will investigate the system state trajectory using two specific subsystems as follows A 1 =.5A 11.5A 12 = 4. (59) 1..5 A d1 =.5A d11.5a d12 = A 2 =.4A 21.6A 22 =.6 1. (6) 1..6 A d2 =.4A d21.6a d22 = which belong to S 1 S 2 are both unstable. Here, we suppose that the initial state is x = 2 5 T. Then, we get x T (A T 11P PA 11 PA d11 WA T d 11 P P)x = x T (A T 12P PA 12 PA d12 WA T d 12 P P)x = x T (AT 21 P PA 21 PA d21 WA T d 21 P P)x = x T (AT 22 P PA 22 PA d22 WA T d 22 P P)x = we choose σ(x, ) = 1 according to the switching rule (21). If we suppose the initial state is x = 2 4 T. Then, we get x T (AT 11 P PA 11 PA d11 WA T d 11 P P)x = x T (A T 12P PA 12 PA d12 WA T d 12 P P)x = x T (A T 21P PA 21 PA d21 WA T d 21 P P)x = x T (AT 22 P PA 22 PA d22 WA T d 22 P P)x = we choose σ(x, ) = 2 according to the switching rule (21). In the same way, we use the switching rule (21) to choose the subsystem mode for every time instant evolve the system forth on. Although both A 1 A 2 are unstable, but we see the system state converge to zero under the switching rule we proposed in (21) from Figure Fig. 1. The state of switched system with τ =.1 Example 2. (Delay-dependentcase) Considerthe switched system (1) composed of two subsystems where A 11 = A = (61) A d11 = A.3 d12 = A 21 = 2.6 A d21 = A 22 =.1 2 (62).6.1 A d22 =.2.5 Both A 11, A d11 A 12, A d12 are delay-dependent robustly unstable. Similarly, both A 21, A d21 A 22, A d22 are delay-dependent robustly unstable. Therefore, both S 1 S 2 are robustly unstable. Now, we let λ 11 = λ 12 = λ 21 = λ 22 =.4, τ =.172 in order to satisfy the conditions in Theorem 5, by using MATLAB tools we get P 1 = P = P = Here we let Σ ij := λ ij  1i (1 λ ij )Â2j T P Pλ ij  1i (1 λ ij )Â2j τpλ ij A d1i (P 1 P 2 )A T d 1i (1 λ ij )A d2j (P 1 P 2 )A T d 2j P 2τP Then we obtained Σ 11 = Σ 21 = Σ 12 = Σ 22 = We can see they satisfy the conditions of (3) to (32). Therefore, this switched system is robustly stablilizable via state feedback. Now, we will investigate the system state trajectory using two specific subsystems as follows 1..8 A 1 =.2A 11.8A 12 = (63).8.8 A d1 =.2A d11.8a d12 =

6 Fig. 2. The state of the switched system with τ = A 2 =.4A 21.6A 22 =.6 2. (64).6.6 A d2 =.4A d21.6a d22 =.2.5 which belong to S 1 S 2 are both unstable. Here, we suppose that the initial state is x =.6 4 T. Then, we get x T Υ 11 x = x T Υ 12 x = x T Υ 21x = x T Υ 22x = Υ ij := ÂT ijp PÂij τpa dij (P 1 P 2 )A T d ij P 2τP we choose σ(x, ) = 1 according to the switching rule (44). If we suppose that the initial state is x = 1 4 T. Then, we get x T Υ 11 x = x T Υ 12 x = x T Υ 21x = x T Υ 22x = we choose σ(x, ) = 2 according to the switching rule (44). In the same way, we use the switching rule (44) to choose the subsystem mode for every time instant evolve the system forth on. Although both A 1 A 2 are unstable, but we see the system state converge to zero under the switching rule we proposed in (44) from Figure CONCLUSION In this paper, we have considered robust stabilizability via state feedback for continuous-time system that are composed of polytopic uncertain subsystems. We have shown that if there exists common positive definite matrices for stability of all convex combinations of the extreme points which belong to different subsystem matrices, then the switched system is robustly stabilizable via state feedback. We have analyzed the stability conditions for the case of delay-independent delay-dependent also given the switching rules σ. Numerical examples show the effectiveness of the proposed method. M.S. Branicky. Multiple lyapunov functions others analysis tools for switched hybrid systems. IEEE Trans. Autom. Contr, vol.43, no.4, pages , Apr M. Branicky, V. Borkar, S. Mitter. A unified framework for hybrid control: Modal optimal control theory. IEEE Trans. Autom. Contr, vol.43, no.1, pages 31 45, Jan Y.-Y. Cao, Y.-X. Sun, C. Cheng. Delay dependent robust stabilization of uncertain systems with multiple state delays. IEEE Trans. Autom. Control, vol.43, pages , S. Engell, S. Kowalewski, C. Schulz, O. Strusberg. Continuous discrete interactions in chemical processing plants. Proc IEEE, vol.88, no.7, pages , Jul 2. R. Horowitz, P. Varaiya. Control design of an automated highway system. Proc IEEE, vol.88, no.7, pages , Jul 2. Lu Jian-ning, ZHAO Guang-zhou. Stability analysis based on LMI for switched systems with time delay. Journal of Southern Yangtze University (Natural Science Edition), vol.5, no.2, Apr 26. Sehjeong Kim, Sue Ann Campbell, Xinzhi Liu. Stability of a class of linear switching system with time delay. IEEE Transactions on circuits systems, vol.53, no.2, pages , Feb 26. D. Liberzon, A.S. Morse. Basic problems in stability analysis of switched systems. IEEE Control Systems Magazine, vol.19, no.5, pages 59 7, Oct C. Livadas, J. Lygeros, N. A. Lynch. High-level modeling analysis of the traffic alert collision avoidance system (TCAS). Proc IEEE, vol.88, no.7, pages , Jul 2. D. Pepyne, C. Cassaras. Optimal control of hybrid systems in manufacturing. Proc. IEEE, vol.88, no.7, pages , Jul 2. M. Song, T. Tran, N. Xi. Integration of task scheduling, action planning, control in robotic manufacturing systems. Proc IEEE, vol.88, no.7, pages , Jul 2. P. Varaiva. Smart cars on smart roads: Problems of control. IEEE Trans. Autom. Contr, vol.38, no.2, pages , Feb H. Ye, A.N. Michel, L. Hou. Stability analysis of switched systems. Presented at the Conf. Decision Control, Tampa, Florida, G. Zhai, H. Lin, P. J. Antsaklis. Quadratic stabilizability of switched linear systems with polytopic uncertainties. Int. J. Control, vol.76, no.7, pages , 23. REFERENCES P. Antsaklis. Special issue on hybrid systems: Theory applications Abriefintroductiontothe theory applications of hybrid systems. Proc IEEE, vol.88, no.7, pages , Jul