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Stabilization of switched linear systems with mode-dependent time-varying delays
Applied Mathematics and Computation 216 (2010) 2581–2586
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stabilization of switched linear systems with mode-dependent time-varying delays q Xiuhong Liu Department of Mathematics and Statistics, Shandong University of Finance, Jinan 250014, China
a r t i c l e
i n f o
a b s t r a c t In this paper, we investigate the problem of stabilization via state feedback and/or statebased switching for switched linear systems with mode-dependent time-varying delays. By using the multiple Lyapunov functional method, we establish sufficient conditions that guarantee the switched system is stabilizable via state feedback and/or switching under time-varying delays with appropriate upper bounds. The main results are presented in terms of linear matrix inequalities (LMIs) which generalize some known results and can be easily tested by using the Matlab’s LMI Tool-box. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Stabilization Switched linear delay system Multiple Lyapunov functions Linear matrix inequality
1. Introduction By a switched system we mean a hybrid dynamical system which consists of a family of continuous-time subsystems and a switching rule that orchestrates the switching between them. Generally speaking, the local behavior of the switched system is governed by the continuous dynamics of subsystems and the discrete dynamics of switching mechanisms determine the global performance of the system. In this way, switched systems provide a unifying formulation for conventional dynamic systems and complex systems. Many important engineering systems, such as computer networks, automatic highway systems, and power electronics, can be represented by switched systems [6,19,20]. In recent years, stability issues for switched linear systems of the form
_ xðtÞ ¼ Arðt;xÞ xðtÞ;
t 2 Rþ ¼ ½0; 1Þ
ð1Þ
have been extensively studied by many authors [1,3,4,7–18,21–24]. Here, the piecewise constant function rðt; xÞ : Rþ Rn ! N ¼ f1; 2; . . . ; Ng is the switching rule. So far, two main problems have been investigated in known literatures. One is to find conditions that guarantee asymptotic stability of (1) for arbitrary switching rule. The other is to identify those switching rules for which system (1) is asymptotically stable if it is not asymptotically stable for arbitrary switching. At present, one commonly used approach to solve the first problem is to find a common Lyapunov function for system (1). Finding a common Lyapunov function is still an open problem, even though several progresses have been done in [1,3,7–15]. In the context of the second problem, it is natural to distinguish between two situations. For the case when some or all of the individual subsystems are asymptotically stable, it is necessary to characterize, as completely as possible, the class of switching rules which preserve asymptotic stability (such switching rules clearly exist, for example, just let r ¼ i, where i is the index of some asymptotically stable subsystem). On the other hand, if all the individual subsystems are unstable, then a challenging task is to construct at least one stabilizing switching rule, which may actually be quite difficult or even impossible. There have been many references aimed at the problem of stability design for system (1), Wicks et al. [21,22] proved that the existence of a state-based switching rule stabilizing system (1) is implied by the existence of an asymptotically stable
q
This work was supported by the National Natural Science Foundation of China (10901093). E-mail address: [email protected]
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.101
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X. Liu / Applied Mathematics and Computation 216 (2010) 2581–2586
convex combination of the subsystem matrices. Feron [4] proved that this sufficient condition is also necessary for the case N ¼ 2. Zhai [24] derived parallel results for the discrete-time case. Li et al. [11] found that it is possible to find a stable convex combination for a class of switched systems. By using a multiple Lyapunov functional method, Wicks and DeCarlo [23] also proved that system (1) with N ¼ 2 is stable via the switching rule rðtÞ :¼ arg maxfV i ðxðtÞÞ : i ¼ 1; 2g if there exist b1 ; b2 > 0 and positive definite matrices P1 ; P 2 such that
AT1 P1 þ P1 A1 þ b1 ðP1 P2 Þ < 0 and
AT2 P2 þ P2 A2 þ b2 ðP2 P1 Þ < 0; where V i ðxðtÞÞ ¼ xT ðtÞPi xðtÞ. In this paper, we will consider the following switched linear system with mode-dependent time-varying delays
_ xðtÞ ¼ Arðt;xÞ xðtÞ þ Brðt;xÞ x t srðt;xÞ ðtÞ ;
t P 0;
ð2Þ
where x 2 Rn , the switching law rðt; xÞ : Rþ Rn ! N is a piecewise constant function that is continuous from the right, and si ðtÞ P 0; i 2 N, are time-varying delays satisfying appropriate conditions. It is well-known that time delays are frequently encountered in practical systems such as engineering, communications and biological systems and may introduce instability, oscillation, and poor performance. The problems of stability for delay systems have been of great importance and interest. However, it seems to us switched linear systems with delays receive less attention. Generally speaking, all the results for stability of non-switched linear delay systems can be easily extended to the stability of switched delay systems for arbitrary switching rule if we choose those Lyapunov functional candidates in the literature to be common Lyapunov functions for switched delay systems. However, the second problem, i.e., the construction of switching rules such that the switched delay system is asymptotically stable, is more challenging. To the best of our knowledge, little has been known about this problem. If we let all the delays in the switched linear delay system (2) be zero, system (2) reduces to the form of system (1). For P P system (1), we have known if there exists a Hurwitz stable convex combination Ni¼1 ai Ai with ai > 0 and Ni¼1 ai ¼ 1 (each Ai is not necessarily Hurwitz stable), then there exists at least one state-based switching rule such that system (1) is asymptotically stable [21,22]. It is reasonable to believe that there exists at least one switching rule such that the switched linear delay system (2) is asymptotically stable for suitable time delays if the corresponding system (1) has a Hurwitz stable convex combination. This idea has been proved in a recent paper [16] by using a common Lyapunov functional approach. However, such a common Lyapunov function may not exist (i.e., a stable convex combination may not exist), and the common Lyapunov functional method usually leads to conservativeness. In this paper, we will further study this problem by using a multiple Lyapunov functional method. The given results will extend some known ones in [23] to the switched delay system (2) without the restriction of N ¼ 2, and complement the results in [16]. Throughout this paper, the notation represents the elements below the main diagonal of a symmetric matrix. I denotes an identity matrix of appropriate dimension. AT means the transpose of the matrix A. We say X > Y if X Y is positive definite, where X and Y are symmetric matrices of same dimensions. k k refers to the Euclidean norm for vectors. This paper is organized as follows. Section 2 contains the problem formulation, Section 3 is the main results. Two numerical examples are presented in Section 4. The conclusion is given in Section 5. 2. Models and preliminaries In the sequel, we assume that the time-varying delays si ðtÞ; i 2 N, satisfy 0 6 si ðtÞ 6 h, where h > 0 is a constant. Here, we do not impose any restriction on the derivatives of delays si ðtÞ (e.g., 0 < s_ i ðtÞ < 1). Let C ¼ Cð½h; 0; Rn Þ be a Banach space of continuous functions mapping the interval ½h; 0 into Rn with the norm k/kh ¼ suph6h60 k/ðhÞk for / 2 C. We let xt 2 C be defined by xt ðhÞ ¼ xðt þ hÞ for h 6 h 6 0, then system (2) can be rewritten as a nonautonomous functional differential equation of the form
_ xðtÞ ¼ f ðt; xt Þ; þ
ð3Þ n
where f : R C ! R is defined by f ðt; xt Þ ¼ Ai xðtÞ þ Bi xðt si ðtÞÞ for rðt; xÞ ¼ i 2 N. Obviously, f ðt; xt Þ is piecewise continuous in t. For any initial function / 2 C and given switching rule, using the basic theory for functional differential equations in [5], it is easy to show that there is a unique solution xðtÞ of system (3) such that xðtÞ is continuous on ½h; 1Þ; xðtÞ ¼ /ðtÞ for t 2 ½h; 0 and xðtÞ satisfies (3) for t P 0. On the other hand, it is not difficult to prove that f takes Rþ (bounded sets of C) into bounded sets of Rn for any given switching rule. This makes it possible to apply Theorem 2.1 (a stability theorem of Lyapunov–Krasovskii type) in [5] to the switched system (2). When si ðtÞ 0, system (2) reduces to the following system without delays
e rðt;xÞ xðtÞ; _ xðtÞ ¼A
t P 0;
e rðt;xÞ ¼ Arðt;xÞ þ Brðt;xÞ . where A
ð4Þ
X. Liu / Applied Mathematics and Computation 216 (2010) 2581–2586
2583
In the following, we say system (2) or (4) is stabilizable via switching if there exists a switching rule rðt; xÞ under which system (2) or (4) is asymptotically stable. As we all know, system (4) is stabilizable via switching if the convex combination PN PN e i¼1 ai A i is Hurwitz [21,22], where ai > 0 and i¼1 ai ¼ 1. That is, there exist positive definite matrices P and Q such that N X
ai Ae Ti P þ P Ae i < Q < 0:
ð5Þ
i¼1
In a recent paper [16], the authors extended the main result in [21,22] to the switched linear delay system (2). It has been proved that system (2) is stabilizable via the switching rule S for all delays satisfying 0 6 si ðtÞ 6 h, where h > 0 can be obtained from the following feasible LMIs:
2
Q
6 4
PBi
hðAi þ Bi ÞT R
R
hBi R
R
T
3 7 5 < 0;
i 2 N;
ð6Þ
P > 0; R > 0 are matrices of appropriate dimensions to be determined, Q is defined by (5), and the switching rule S is constructed as follows: eT P þ PA e i Þx0 g (the minimum rule). Choose the initial mode rð0; x0 Þ ¼ arg mini2N fxT0 ð A i eT P þ PA e i Þx < xT Qxg. e i ¼ fx : xT ð A rðt; xÞ ¼ i as long as xðtÞ 2 X i e i , use the minimum rule to determine the next switch. If xðtÞ hits the boundary of X The result mentioned above was also extended to the the switched linear delay system with control input u 2 Rm :
_ xðtÞ ¼ Arðt;xÞ xðtÞ þ Brðt;xÞ xðt srðt;xÞ ðtÞÞ þ C rðt;xÞ uðtÞ:
ð7Þ
In this paper, we will further study the stabilizability of systems (2) and (7) by using a multiple Lyapunov functional approach that is different from the method used in [16]. We first need the following lemmas. Lemma 1 [2]. Let M; U; V be given matrices such that V > 0. Then
"
U
MT
M
V
# < 0 () U þ M T V 1 M < 0:
Lemma 2 [16]. For any 0 6 sðtÞ 6 h, any differentiable function xðtÞ and any constant matrix W > 0, we have the following inequality
½xðtÞ xðt sðtÞÞT W½xðtÞ xðt sðtÞÞ 6 h
Z
t
_ x_ T ðsÞW xðsÞds
tsðtÞ
for t P h.
3. Stabilization via switching Define the following multiple Lyapunov functions as
V i ðxðtÞÞ ¼ xT ðtÞP i xðtÞ þ h
Z
t
_ ðs t þ hÞx_ T ðsÞQ xðsÞds;
ð8Þ
th
where Pi > 0; i 2 N, and Q > 0. In the sequel, the following assumption will be supposed: (H1) There exist constants bi > 0, matrices P i > 0 and ki 2 N such that ki – i; ki – kj ; 8i; j 2 N, and
ðAi þ Bi ÞT Pi þ Pi ðAi þ Bi Þ þ bi ðPi Pki Þ ¼ W i < 0:
ð9Þ
Theorem 1. Assume that (H1) holds. Then there exists h > 0 such that system (2) is stabilizable via the switching rule rðtÞ :¼ arg maxfxT ðtÞPi xðtÞ : i 2 Ng, where h satisfies the following feasible LMIs:
2
W i
Pi Bi
hðAi þ Bi ÞT Q
Q
hBi Q Q
Ui ¼ 6 4
T
3 7 5 < 0;
and Q > 0 is a matrix to be determined.
i 2 N;
ð10Þ
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X. Liu / Applied Mathematics and Computation 216 (2010) 2581–2586
Proof. We first prove that (10) is always feasible if (9) holds. That is, if (9) holds, then there exist h > 0 and Q > 0 such that (10) holds for each i 2 N. In fact, by using Lemma 1, we can easily obtain that (10) is equivalent to
W i
P i Bi
Q
2
þ h ½Ai þ Bi Bi T Q ½Ai þ Bi Bi < 0; 1
where P i satisfy (9). Let Q ¼ h I. Then we have
W i
Pi Bi 1
h I
þ h½Ai þ Bi Bi T ½Ai þ Bi Bi < 0:
ð11Þ
Using Lemma 1 again, we can easily obtain that W i < 0 implies
W i
Pi Bi <0 1 h I 1
for sufficiently small h. Hence, (11) holds for sufficiently small h > 0 if (9) is valid. That is, there exist h > 0, and Q ¼ h I such that (10) holds. h Next, we prove system (2) is stabilizable via the switching rule rðtÞ :¼ arg maxfV i ðxðtÞÞ : i 2 Ng if (9) and (10) holds. Set wi ðtÞ ¼ xðt si ðtÞÞ xðtÞ. Then system (2) reduces to the following equivalent form:
_ xðtÞ ¼ ðAr þ Br ÞxðtÞ þ Br wr ðtÞ:
ð12Þ
Assume that the ith subsystem is activated at time t. Choose the Lyapunov functions defined by (8) and consider the righthand derivative of V along the solution of system (12). Under the switching rule rðtÞ :¼ arg maxfV i ðxðtÞÞ : i 2 Ng, we have 2 _ h Dþ V i ¼ xT ðtÞ½Pi ðAi þ Bi Þ þ ðAi þ Bi ÞT P i xðtÞ þ 2xT ðtÞPi wi ðtÞ þ h x_ T ðtÞQ xðtÞ
Z
t
_ x_ T ðsÞQ xðsÞds:
ð13Þ
th
By Lemma 2, (12) and (13), we get 2
Dþ V i 6 xT ðtÞ½P i ðAi þ Bi Þ þ ðAi þ Bi ÞT Pi xðtÞ þ 2xT ðtÞPi wi ðtÞ þ h gTi ðtÞ½Ai þ Bi Bi T Q ½Ai þ Bi Bi gi ðtÞ wTi ðtÞRi wi ðtÞ; where gTi ðtÞ ¼ xT ðtÞwTi ðtÞ . According to the definition of the switching rule rðtÞ and the assumption that the ith mode is activated at time t, we have xT ðtÞP i xðtÞ P xT ðtÞP j xðtÞ for j 2 N. From (9) we get
xT ðtÞ½Pi ðAi þ Bi Þ þ ðAi þ Bi ÞT PxðtÞ 6 xT ðtÞW i xðtÞ: Thus,
Dþ V i 6 gTi ðtÞXi gi ðtÞ; where
Xi ¼
W i
Pi Bi 1
h I
þ h½Ai þ Bi Bi T ½Ai þ Bi Bi ;
which is equivalent to (10) by Lemma 1. Therefore, there exists a positive constant ki such that
Dþ V i 6 ki kxðtÞk2 : On the other hand, based on the definition of the switching signal rðtÞ, we can easily get that xT ðtÞPi xðtÞ ¼ xT ðtÞP j xðtÞ if rðt Þ ¼ i and rðtÞ ¼ j (i.e., the ith subsystem is switched to the jth subsystem at time t). Hence V i ðxðtÞÞ ¼ V j ðxðtÞÞ. By using Theorem 2.1 in [5, Chapter 5] we have that system (2) is stabilizable via the switching rule rðtÞ :¼ arg maxfxT ðtÞP i xðtÞ : i 2 Ng. This completes the proof of Theorem 1. Remark 1. When si ðtÞ 0 and N ¼ 2, Theorem 1 reduces to the main result in [21]. If we design the switching rule as rðtÞ :¼ arg minfxT ðtÞPi xðtÞ : i 2 Ng, the following theorem is immediate. Theorem 2. Assume that there exist constants bi > 0, matrices P i > 0 and ki 2 N such that ki – i; ki – kj ; 8i; j 2 N, and
f i < 0: ðAi þ Bi ÞT Pi þ Pi ðAi þ Bi Þ þ bi ðPki Pi Þ ¼ W Then there exists h > 0 such that system (2) is stabilizable via the switching rule rðtÞ :¼ arg minfxT ðtÞP i xðtÞ : i 2 Ng, where h can f i ¼ W i. be obtained from (10) with W In the sequel, we extend Theorem 1 to the switched control system (7). We assume:
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X. Liu / Applied Mathematics and Computation 216 (2010) 2581–2586
(H2) there exist a series of state feedback matrices K i , positive constants bi , positive definite matrices P i and ki 2 N such that ki – i; ki – kj ; 8i; j 2 N, and
ðAi þ Bi þ C i K i ÞT Pi þ Pi ðAi þ Bi þ C i K i Þ þ bi ðPki Pi Þ ¼ U i < 0:
ð14Þ
Similar to the proof of Theorem 1, we can easily get the following theorem for system (7). Theorem 3. Assume that (H2) holds. Then for appropriate h > 0, system (7) is stabilizable via the state feedback u ¼ K r x and the switching rule rðtÞ :¼ arg minfxT ðtÞPi xðtÞ : i 2 Ng, where h > 0 can be obtained from
2
U i
6 4
P i Bi
hðAi þ Bi þ C i K i ÞT Q
Q
hBi Q
Q
T
3 7 5 < 0;
i 2 N;
ð15Þ
and Q > 0 is a matrix to be determined. 1 Remark 2. It is easy to see that (15) are not strict linear matrix inequalities. By multiplying diagfP1 ; Q 1 g on both sides i ;Q 1 1 T of (15), using Lemma 1 and letting Pi ¼ X i ; Q ¼ Y and X i K i ¼ Z i , we have that (15) is equivalent to
2
ei U
6 6 6 6 4
Bi Y
hX i ðAi þ Bi ÞT þ hZ i C Ti
Y
hYBi
0
Y
0
bi X ki
bi X i
T
3
7 7 7 < 0; 7 5
i 2 N;
ð16Þ
where
e i ¼ X i ðAi þ Bi ÞT þ ðAi þ Bi ÞX i þ Z i C T þ C i Z T b X i : U i i i For given bi > 0, by solving LMIs (16) we not only obtain an allowable upper bound h of time delays feedback matrices as K i ¼ Z Ti X 1 i .
si ðtÞ but also get the
4. Numerical examples In this section, we will present two numerical examples to illustrate the effectiveness of our theoretical results. Example 1. Consider system (2) with N ¼ 2 and system parameters as followings:
0:2 2 A2 ¼ ; 2 0 1 0 0:1 0 0:2 0 ; B2 ¼ : B1 ¼ 0 0 0 0
A1 ¼
0:1 1
;
Based on the analysis in [10, p. 41], we know that any convex combination of
e 1 ¼ A1 þ B1 ¼ 0 1 ; A 2 0
e 2 ¼ A2 þ B2 ¼ 0:5 1 A 1 2:8
are not Hurwitz stable. Thus, the result in [16] cannot be applied to this example. However, by choosing b1 ¼ 0:9 and b2 ¼ 1:4 and by using Theorem 1, we have that system (2) with the aforementioned parameters is stabilizable via the switching rule rðtÞ :¼ arg maxfV i ðxðtÞÞ : i 2 Ng for si ðtÞ 6 0:7441. The feasible solutions of (10) are as followings:
P1 ¼
0:6667
0:0676
0:0676
0:4179
;
P2 ¼
0:4347 0:1093 0:1093 0:6147
;
Q¼
0:0728
0
0
0:0001
:
Example 2. Consider system (2) with N ¼ 2 and system parameters as followings:
A1 ¼
0:3
1
;
0 0:5 0:2 1 ; A2 ¼ 0:3 0:5
; 0 0:2 0:1 0:2 B2 ¼ ; 0 0:3
B1 ¼
0:1
0:1
C1 ¼
0:3
; 1 0:5 C2 ¼ : 0:2
Choose b1 ¼ 0:5 and b2 ¼ 0:3. By solving LMIs (16), we find that system (7) with the aforementioned parameters is stabilizable via the state feedback u ¼ K i x and the switching rule rðtÞ :¼ arg minfV i ðxðtÞÞ : i 2 Ng for si ðtÞ 6 3:3311. The feedback matrices K i are as followings:
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X. Liu / Applied Mathematics and Computation 216 (2010) 2581–2586
K 1 ¼ ½ 0:6956 1:6685 ;
K 2 ¼ ½ 2:1990 1:2019 :
The feasible solutions of (16) are as followings:
X1 ¼
0:0262
0:0132
0:0132
0:0082
;
X2 ¼
0:0418
0:0123
0:0123
0:0067
;
Y¼
0:4087
0:0113
0:0113
0:0224
:
5. Conclusion In this paper, by employing a multiple Lyapunov functional approach we investigate the problem of stabilization via switching for the switched linear system with mode-dependent time-varying delays. We not only prove the switched linear delay system will be stabilizable via a designed switching rule for appropriate delays, but also establish some feasible linear matrix inequalities to get an allowable upper bound of delays. The results obtained in this paper generalize and complement some existing results in known literatures. Two numerical examples are worked out to illustrate the effectiveness of the theoretical results. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
T. Ando, Sets of matrices with a common Lyapunov solution, Archiv der Mathematik 77 (2001) 76–84. B. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. D. Cheng, L. Guo, J. Huang, On quadratic Lyapunov functions, IEEE Trans. Automat. Contr. 48 (2003) 885–890. E. Feron, Quadratic stabilizability of swithced systems via state and output feedback, Center for Intelligent Control Systems, Massachusetts Institute of Technology, Cambridge, MA, Tech. Rep. CICS P-468, February 1996. J. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. D. Jeon, M. Tomizuka, Learning hybrid force and position control of robot manipulators, IEEE Trans. Robotic Automat. 9 (1996) 423–431. H.K. Khalil, On the existence of positive diagonal P such that PA þ AT P < 0, IEEE Trans. Automat. Contr. 27 (1982) 181–184. D. Liberzon, J.P. Hespanha, A.S. Morse, Stability of switched systems: a Lie-algebraic condition, Syst. Contr. Lett. 37 (1999) 117–122. D. Liberzon, A.S. Morse, Basic problems on stability and design of switched systems, IEEE Contr. Syst. Mag. 34 (1999) 1914–1946. D. Liberzon, Control Using Logic and Switching, Part 1: Switching in Systems and Control, Handout Notes, CDC’01 Workshop. Z.G. Li, C.Y. Wen, Y.C. Soh, Stabilization of a class of switched systems via designing switching law, IEEE Trans. Automat. Contr. 46 (2001) 665–670. T. Mori, H. Kokame, An extension of a class of systems that have a common Lyapunov function, Math. Contr. Signal Syst. 10 (1997) 265–286. K.S. Narendra, J. Balakrishnan, A common Lyapunov function for stable LTI systems with com-muting A-matrices, IEEE Trans. Automat. Contr. 39 (1994) 2469–2471. T. Ooba, Y. Funahashi, Two conditions concerning common QLFs for linear systems, IEEE Trans. Automat. Contr. 42 (1997) 719–721. T. Ooba, Y. Funahashi, Stability robustness for linear state space models – a Lyapunov mapping approach, Syst. Contr. Lett. 29 (1997) 191–196. Y.G. Sun, L. Wang, G. Xie, Stabilization of switched linear systems with multiple time-varying delays, in: Proc. IEEE Conf. on Decision and Control, San Diego, CA, USA, December 13–15, 2006, pp. 4069–4074, . Y.G. Sun, L. Wang, G. Xie, Delay-dependent robust stability and stabilization for discrete-time switched systems with mode-dependent time-varying delays, Appl. Math. Comput. 180 (2006) 428–435. Y.G. Sun, L. Wang, G. Xie, Delay-dependent robust stability and H1 control for uncertain discrete-time switched systems with mode-dependent time delays, Appl. Math. Comput. 187 (2007) 1228–1237. P.P. Varaiya, Smart car on smart roads: problems of control, IEEE Trans. Automat. Contr. 38 (1993) 195–207. S.M. Williams, R.G. Hoft, Adaptive frequency domain control of PM switched power line conditioner, IEEE Trans. Power Electron. 6 (1991) 665–670. M.A. Wicks, P. Peleties, R.A. DeCarlo, Construction of piecewise Lyapunov functions for stabilizing switched systems, in: Proc. IEEE Conf. on Decision and Control, Lake Buena Vista, December 1994, pp. 3492–3497. M.A. Wicks, P. Peleties, R. DeCarlo, Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems, Eur. J. Contr. 4 (1998) 14–147. M.A. Wicks, R.A. DeCaro, Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems, in: Proc. 1997 American Control Conf., 1997, pp. 1709–1713. G. Zhai, Quadratic stabilizability of discrete-time switched systems via state and output feedback, in: Proc. IEEE Conf. on Decision and Control, Orlando, FL, December 2001, pp. 2165–2166.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stabilization of switched linear systems with mode-dependent time-varying delays q Xiuhong Liu Department of Mathematics and Statistics, Shandong University of Finance, Jinan 250014, China
a r t i c l e
i n f o
a b s t r a c t In this paper, we investigate the problem of stabilization via state feedback and/or statebased switching for switched linear systems with mode-dependent time-varying delays. By using the multiple Lyapunov functional method, we establish sufficient conditions that guarantee the switched system is stabilizable via state feedback and/or switching under time-varying delays with appropriate upper bounds. The main results are presented in terms of linear matrix inequalities (LMIs) which generalize some known results and can be easily tested by using the Matlab’s LMI Tool-box. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Stabilization Switched linear delay system Multiple Lyapunov functions Linear matrix inequality
1. Introduction By a switched system we mean a hybrid dynamical system which consists of a family of continuous-time subsystems and a switching rule that orchestrates the switching between them. Generally speaking, the local behavior of the switched system is governed by the continuous dynamics of subsystems and the discrete dynamics of switching mechanisms determine the global performance of the system. In this way, switched systems provide a unifying formulation for conventional dynamic systems and complex systems. Many important engineering systems, such as computer networks, automatic highway systems, and power electronics, can be represented by switched systems [6,19,20]. In recent years, stability issues for switched linear systems of the form
_ xðtÞ ¼ Arðt;xÞ xðtÞ;
t 2 Rþ ¼ ½0; 1Þ
ð1Þ
have been extensively studied by many authors [1,3,4,7–18,21–24]. Here, the piecewise constant function rðt; xÞ : Rþ Rn ! N ¼ f1; 2; . . . ; Ng is the switching rule. So far, two main problems have been investigated in known literatures. One is to find conditions that guarantee asymptotic stability of (1) for arbitrary switching rule. The other is to identify those switching rules for which system (1) is asymptotically stable if it is not asymptotically stable for arbitrary switching. At present, one commonly used approach to solve the first problem is to find a common Lyapunov function for system (1). Finding a common Lyapunov function is still an open problem, even though several progresses have been done in [1,3,7–15]. In the context of the second problem, it is natural to distinguish between two situations. For the case when some or all of the individual subsystems are asymptotically stable, it is necessary to characterize, as completely as possible, the class of switching rules which preserve asymptotic stability (such switching rules clearly exist, for example, just let r ¼ i, where i is the index of some asymptotically stable subsystem). On the other hand, if all the individual subsystems are unstable, then a challenging task is to construct at least one stabilizing switching rule, which may actually be quite difficult or even impossible. There have been many references aimed at the problem of stability design for system (1), Wicks et al. [21,22] proved that the existence of a state-based switching rule stabilizing system (1) is implied by the existence of an asymptotically stable
q
This work was supported by the National Natural Science Foundation of China (10901093). E-mail address: [email protected]
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.101
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X. Liu / Applied Mathematics and Computation 216 (2010) 2581–2586
convex combination of the subsystem matrices. Feron [4] proved that this sufficient condition is also necessary for the case N ¼ 2. Zhai [24] derived parallel results for the discrete-time case. Li et al. [11] found that it is possible to find a stable convex combination for a class of switched systems. By using a multiple Lyapunov functional method, Wicks and DeCarlo [23] also proved that system (1) with N ¼ 2 is stable via the switching rule rðtÞ :¼ arg maxfV i ðxðtÞÞ : i ¼ 1; 2g if there exist b1 ; b2 > 0 and positive definite matrices P1 ; P 2 such that
AT1 P1 þ P1 A1 þ b1 ðP1 P2 Þ < 0 and
AT2 P2 þ P2 A2 þ b2 ðP2 P1 Þ < 0; where V i ðxðtÞÞ ¼ xT ðtÞPi xðtÞ. In this paper, we will consider the following switched linear system with mode-dependent time-varying delays
_ xðtÞ ¼ Arðt;xÞ xðtÞ þ Brðt;xÞ x t srðt;xÞ ðtÞ ;
t P 0;
ð2Þ
where x 2 Rn , the switching law rðt; xÞ : Rþ Rn ! N is a piecewise constant function that is continuous from the right, and si ðtÞ P 0; i 2 N, are time-varying delays satisfying appropriate conditions. It is well-known that time delays are frequently encountered in practical systems such as engineering, communications and biological systems and may introduce instability, oscillation, and poor performance. The problems of stability for delay systems have been of great importance and interest. However, it seems to us switched linear systems with delays receive less attention. Generally speaking, all the results for stability of non-switched linear delay systems can be easily extended to the stability of switched delay systems for arbitrary switching rule if we choose those Lyapunov functional candidates in the literature to be common Lyapunov functions for switched delay systems. However, the second problem, i.e., the construction of switching rules such that the switched delay system is asymptotically stable, is more challenging. To the best of our knowledge, little has been known about this problem. If we let all the delays in the switched linear delay system (2) be zero, system (2) reduces to the form of system (1). For P P system (1), we have known if there exists a Hurwitz stable convex combination Ni¼1 ai Ai with ai > 0 and Ni¼1 ai ¼ 1 (each Ai is not necessarily Hurwitz stable), then there exists at least one state-based switching rule such that system (1) is asymptotically stable [21,22]. It is reasonable to believe that there exists at least one switching rule such that the switched linear delay system (2) is asymptotically stable for suitable time delays if the corresponding system (1) has a Hurwitz stable convex combination. This idea has been proved in a recent paper [16] by using a common Lyapunov functional approach. However, such a common Lyapunov function may not exist (i.e., a stable convex combination may not exist), and the common Lyapunov functional method usually leads to conservativeness. In this paper, we will further study this problem by using a multiple Lyapunov functional method. The given results will extend some known ones in [23] to the switched delay system (2) without the restriction of N ¼ 2, and complement the results in [16]. Throughout this paper, the notation represents the elements below the main diagonal of a symmetric matrix. I denotes an identity matrix of appropriate dimension. AT means the transpose of the matrix A. We say X > Y if X Y is positive definite, where X and Y are symmetric matrices of same dimensions. k k refers to the Euclidean norm for vectors. This paper is organized as follows. Section 2 contains the problem formulation, Section 3 is the main results. Two numerical examples are presented in Section 4. The conclusion is given in Section 5. 2. Models and preliminaries In the sequel, we assume that the time-varying delays si ðtÞ; i 2 N, satisfy 0 6 si ðtÞ 6 h, where h > 0 is a constant. Here, we do not impose any restriction on the derivatives of delays si ðtÞ (e.g., 0 < s_ i ðtÞ < 1). Let C ¼ Cð½h; 0; Rn Þ be a Banach space of continuous functions mapping the interval ½h; 0 into Rn with the norm k/kh ¼ suph6h60 k/ðhÞk for / 2 C. We let xt 2 C be defined by xt ðhÞ ¼ xðt þ hÞ for h 6 h 6 0, then system (2) can be rewritten as a nonautonomous functional differential equation of the form
_ xðtÞ ¼ f ðt; xt Þ; þ
ð3Þ n
where f : R C ! R is defined by f ðt; xt Þ ¼ Ai xðtÞ þ Bi xðt si ðtÞÞ for rðt; xÞ ¼ i 2 N. Obviously, f ðt; xt Þ is piecewise continuous in t. For any initial function / 2 C and given switching rule, using the basic theory for functional differential equations in [5], it is easy to show that there is a unique solution xðtÞ of system (3) such that xðtÞ is continuous on ½h; 1Þ; xðtÞ ¼ /ðtÞ for t 2 ½h; 0 and xðtÞ satisfies (3) for t P 0. On the other hand, it is not difficult to prove that f takes Rþ (bounded sets of C) into bounded sets of Rn for any given switching rule. This makes it possible to apply Theorem 2.1 (a stability theorem of Lyapunov–Krasovskii type) in [5] to the switched system (2). When si ðtÞ 0, system (2) reduces to the following system without delays
e rðt;xÞ xðtÞ; _ xðtÞ ¼A
t P 0;
e rðt;xÞ ¼ Arðt;xÞ þ Brðt;xÞ . where A
ð4Þ
X. Liu / Applied Mathematics and Computation 216 (2010) 2581–2586
2583
In the following, we say system (2) or (4) is stabilizable via switching if there exists a switching rule rðt; xÞ under which system (2) or (4) is asymptotically stable. As we all know, system (4) is stabilizable via switching if the convex combination PN PN e i¼1 ai A i is Hurwitz [21,22], where ai > 0 and i¼1 ai ¼ 1. That is, there exist positive definite matrices P and Q such that N X
ai Ae Ti P þ P Ae i < Q < 0:
ð5Þ
i¼1
In a recent paper [16], the authors extended the main result in [21,22] to the switched linear delay system (2). It has been proved that system (2) is stabilizable via the switching rule S for all delays satisfying 0 6 si ðtÞ 6 h, where h > 0 can be obtained from the following feasible LMIs:
2
Q
6 4
PBi
hðAi þ Bi ÞT R
R
hBi R
R
T
3 7 5 < 0;
i 2 N;
ð6Þ
P > 0; R > 0 are matrices of appropriate dimensions to be determined, Q is defined by (5), and the switching rule S is constructed as follows: eT P þ PA e i Þx0 g (the minimum rule). Choose the initial mode rð0; x0 Þ ¼ arg mini2N fxT0 ð A i eT P þ PA e i Þx < xT Qxg. e i ¼ fx : xT ð A rðt; xÞ ¼ i as long as xðtÞ 2 X i e i , use the minimum rule to determine the next switch. If xðtÞ hits the boundary of X The result mentioned above was also extended to the the switched linear delay system with control input u 2 Rm :
_ xðtÞ ¼ Arðt;xÞ xðtÞ þ Brðt;xÞ xðt srðt;xÞ ðtÞÞ þ C rðt;xÞ uðtÞ:
ð7Þ
In this paper, we will further study the stabilizability of systems (2) and (7) by using a multiple Lyapunov functional approach that is different from the method used in [16]. We first need the following lemmas. Lemma 1 [2]. Let M; U; V be given matrices such that V > 0. Then
"
U
MT
M
V
# < 0 () U þ M T V 1 M < 0:
Lemma 2 [16]. For any 0 6 sðtÞ 6 h, any differentiable function xðtÞ and any constant matrix W > 0, we have the following inequality
½xðtÞ xðt sðtÞÞT W½xðtÞ xðt sðtÞÞ 6 h
Z
t
_ x_ T ðsÞW xðsÞds
tsðtÞ
for t P h.
3. Stabilization via switching Define the following multiple Lyapunov functions as
V i ðxðtÞÞ ¼ xT ðtÞP i xðtÞ þ h
Z
t
_ ðs t þ hÞx_ T ðsÞQ xðsÞds;
ð8Þ
th
where Pi > 0; i 2 N, and Q > 0. In the sequel, the following assumption will be supposed: (H1) There exist constants bi > 0, matrices P i > 0 and ki 2 N such that ki – i; ki – kj ; 8i; j 2 N, and
ðAi þ Bi ÞT Pi þ Pi ðAi þ Bi Þ þ bi ðPi Pki Þ ¼ W i < 0:
ð9Þ
Theorem 1. Assume that (H1) holds. Then there exists h > 0 such that system (2) is stabilizable via the switching rule rðtÞ :¼ arg maxfxT ðtÞPi xðtÞ : i 2 Ng, where h satisfies the following feasible LMIs:
2
W i
Pi Bi
hðAi þ Bi ÞT Q
Q
hBi Q Q
Ui ¼ 6 4
T
3 7 5 < 0;
and Q > 0 is a matrix to be determined.
i 2 N;
ð10Þ
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X. Liu / Applied Mathematics and Computation 216 (2010) 2581–2586
Proof. We first prove that (10) is always feasible if (9) holds. That is, if (9) holds, then there exist h > 0 and Q > 0 such that (10) holds for each i 2 N. In fact, by using Lemma 1, we can easily obtain that (10) is equivalent to
W i
P i Bi
Q
2
þ h ½Ai þ Bi Bi T Q ½Ai þ Bi Bi < 0; 1
where P i satisfy (9). Let Q ¼ h I. Then we have
W i
Pi Bi 1
h I
þ h½Ai þ Bi Bi T ½Ai þ Bi Bi < 0:
ð11Þ
Using Lemma 1 again, we can easily obtain that W i < 0 implies
W i
Pi Bi <0 1 h I 1
for sufficiently small h. Hence, (11) holds for sufficiently small h > 0 if (9) is valid. That is, there exist h > 0, and Q ¼ h I such that (10) holds. h Next, we prove system (2) is stabilizable via the switching rule rðtÞ :¼ arg maxfV i ðxðtÞÞ : i 2 Ng if (9) and (10) holds. Set wi ðtÞ ¼ xðt si ðtÞÞ xðtÞ. Then system (2) reduces to the following equivalent form:
_ xðtÞ ¼ ðAr þ Br ÞxðtÞ þ Br wr ðtÞ:
ð12Þ
Assume that the ith subsystem is activated at time t. Choose the Lyapunov functions defined by (8) and consider the righthand derivative of V along the solution of system (12). Under the switching rule rðtÞ :¼ arg maxfV i ðxðtÞÞ : i 2 Ng, we have 2 _ h Dþ V i ¼ xT ðtÞ½Pi ðAi þ Bi Þ þ ðAi þ Bi ÞT P i xðtÞ þ 2xT ðtÞPi wi ðtÞ þ h x_ T ðtÞQ xðtÞ
Z
t
_ x_ T ðsÞQ xðsÞds:
ð13Þ
th
By Lemma 2, (12) and (13), we get 2
Dþ V i 6 xT ðtÞ½P i ðAi þ Bi Þ þ ðAi þ Bi ÞT Pi xðtÞ þ 2xT ðtÞPi wi ðtÞ þ h gTi ðtÞ½Ai þ Bi Bi T Q ½Ai þ Bi Bi gi ðtÞ wTi ðtÞRi wi ðtÞ; where gTi ðtÞ ¼ xT ðtÞwTi ðtÞ . According to the definition of the switching rule rðtÞ and the assumption that the ith mode is activated at time t, we have xT ðtÞP i xðtÞ P xT ðtÞP j xðtÞ for j 2 N. From (9) we get
xT ðtÞ½Pi ðAi þ Bi Þ þ ðAi þ Bi ÞT PxðtÞ 6 xT ðtÞW i xðtÞ: Thus,
Dþ V i 6 gTi ðtÞXi gi ðtÞ; where
Xi ¼
W i
Pi Bi 1
h I
þ h½Ai þ Bi Bi T ½Ai þ Bi Bi ;
which is equivalent to (10) by Lemma 1. Therefore, there exists a positive constant ki such that
Dþ V i 6 ki kxðtÞk2 : On the other hand, based on the definition of the switching signal rðtÞ, we can easily get that xT ðtÞPi xðtÞ ¼ xT ðtÞP j xðtÞ if rðt Þ ¼ i and rðtÞ ¼ j (i.e., the ith subsystem is switched to the jth subsystem at time t). Hence V i ðxðtÞÞ ¼ V j ðxðtÞÞ. By using Theorem 2.1 in [5, Chapter 5] we have that system (2) is stabilizable via the switching rule rðtÞ :¼ arg maxfxT ðtÞP i xðtÞ : i 2 Ng. This completes the proof of Theorem 1. Remark 1. When si ðtÞ 0 and N ¼ 2, Theorem 1 reduces to the main result in [21]. If we design the switching rule as rðtÞ :¼ arg minfxT ðtÞPi xðtÞ : i 2 Ng, the following theorem is immediate. Theorem 2. Assume that there exist constants bi > 0, matrices P i > 0 and ki 2 N such that ki – i; ki – kj ; 8i; j 2 N, and
f i < 0: ðAi þ Bi ÞT Pi þ Pi ðAi þ Bi Þ þ bi ðPki Pi Þ ¼ W Then there exists h > 0 such that system (2) is stabilizable via the switching rule rðtÞ :¼ arg minfxT ðtÞP i xðtÞ : i 2 Ng, where h can f i ¼ W i. be obtained from (10) with W In the sequel, we extend Theorem 1 to the switched control system (7). We assume:
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(H2) there exist a series of state feedback matrices K i , positive constants bi , positive definite matrices P i and ki 2 N such that ki – i; ki – kj ; 8i; j 2 N, and
ðAi þ Bi þ C i K i ÞT Pi þ Pi ðAi þ Bi þ C i K i Þ þ bi ðPki Pi Þ ¼ U i < 0:
ð14Þ
Similar to the proof of Theorem 1, we can easily get the following theorem for system (7). Theorem 3. Assume that (H2) holds. Then for appropriate h > 0, system (7) is stabilizable via the state feedback u ¼ K r x and the switching rule rðtÞ :¼ arg minfxT ðtÞPi xðtÞ : i 2 Ng, where h > 0 can be obtained from
2
U i
6 4
P i Bi
hðAi þ Bi þ C i K i ÞT Q
Q
hBi Q
Q
T
3 7 5 < 0;
i 2 N;
ð15Þ
and Q > 0 is a matrix to be determined. 1 Remark 2. It is easy to see that (15) are not strict linear matrix inequalities. By multiplying diagfP1 ; Q 1 g on both sides i ;Q 1 1 T of (15), using Lemma 1 and letting Pi ¼ X i ; Q ¼ Y and X i K i ¼ Z i , we have that (15) is equivalent to
2
ei U
6 6 6 6 4
Bi Y
hX i ðAi þ Bi ÞT þ hZ i C Ti
Y
hYBi
0
Y
0
bi X ki
bi X i
T
3
7 7 7 < 0; 7 5
i 2 N;
ð16Þ
where
e i ¼ X i ðAi þ Bi ÞT þ ðAi þ Bi ÞX i þ Z i C T þ C i Z T b X i : U i i i For given bi > 0, by solving LMIs (16) we not only obtain an allowable upper bound h of time delays feedback matrices as K i ¼ Z Ti X 1 i .
si ðtÞ but also get the
4. Numerical examples In this section, we will present two numerical examples to illustrate the effectiveness of our theoretical results. Example 1. Consider system (2) with N ¼ 2 and system parameters as followings:
0:2 2 A2 ¼ ; 2 0 1 0 0:1 0 0:2 0 ; B2 ¼ : B1 ¼ 0 0 0 0
A1 ¼
0:1 1
;
Based on the analysis in [10, p. 41], we know that any convex combination of
e 1 ¼ A1 þ B1 ¼ 0 1 ; A 2 0
e 2 ¼ A2 þ B2 ¼ 0:5 1 A 1 2:8
are not Hurwitz stable. Thus, the result in [16] cannot be applied to this example. However, by choosing b1 ¼ 0:9 and b2 ¼ 1:4 and by using Theorem 1, we have that system (2) with the aforementioned parameters is stabilizable via the switching rule rðtÞ :¼ arg maxfV i ðxðtÞÞ : i 2 Ng for si ðtÞ 6 0:7441. The feasible solutions of (10) are as followings:
P1 ¼
0:6667
0:0676
0:0676
0:4179
;
P2 ¼
0:4347 0:1093 0:1093 0:6147
;
Q¼
0:0728
0
0
0:0001
:
Example 2. Consider system (2) with N ¼ 2 and system parameters as followings:
A1 ¼
0:3
1
;
0 0:5 0:2 1 ; A2 ¼ 0:3 0:5
; 0 0:2 0:1 0:2 B2 ¼ ; 0 0:3
B1 ¼
0:1
0:1
C1 ¼
0:3
; 1 0:5 C2 ¼ : 0:2
Choose b1 ¼ 0:5 and b2 ¼ 0:3. By solving LMIs (16), we find that system (7) with the aforementioned parameters is stabilizable via the state feedback u ¼ K i x and the switching rule rðtÞ :¼ arg minfV i ðxðtÞÞ : i 2 Ng for si ðtÞ 6 3:3311. The feedback matrices K i are as followings:
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X. Liu / Applied Mathematics and Computation 216 (2010) 2581–2586
K 1 ¼ ½ 0:6956 1:6685 ;
K 2 ¼ ½ 2:1990 1:2019 :
The feasible solutions of (16) are as followings:
X1 ¼
0:0262
0:0132
0:0132
0:0082
;
X2 ¼
0:0418
0:0123
0:0123
0:0067
;
Y¼
0:4087
0:0113
0:0113
0:0224
:
5. Conclusion In this paper, by employing a multiple Lyapunov functional approach we investigate the problem of stabilization via switching for the switched linear system with mode-dependent time-varying delays. We not only prove the switched linear delay system will be stabilizable via a designed switching rule for appropriate delays, but also establish some feasible linear matrix inequalities to get an allowable upper bound of delays. The results obtained in this paper generalize and complement some existing results in known literatures. Two numerical examples are worked out to illustrate the effectiveness of the theoretical results. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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