Finite-time boundedness and L2-gain analysis for switched delay systems with norm-bounded disturbance

Applied Mathematics and Computation 217 (2011) 5982–5993

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Finite-time boundedness and L2-gain analysis for switched delay systems with norm-bounded disturbance Xiangze Lin a,⇑, Haibo Du b, Shihua Li b a b

College of Engineering, Nanjing Agricultural University, Nanjing 210031, PR China School of Automation, Southeast University, Key Laboratory of Measurement and Control of CSE, Ministry of Education, Nanjing 210096, PR China

a r t i c l e

i n f o

Keywords: Switched linear systems Time-delay Finite-time boundedness Multiple Lyapunov-like functions

a b s t r a c t Finite-time boundedness and finite-time weighted L2-gain for a class of switched delay systems with time-varying exogenous disturbances are studied. Based on the average dwell-time technique, sufficient conditions which guarantee the switched linear system with time-delay is finite-time bounded and has finite-time weighted L2-gain are given. These conditions are delay-dependent and are given in terms of linear matrix inequalities. Detail proofs are given by using multiple Lyapunov-like functions. An example is employed to verify the efficiency of the proposed method.  2010 Elsevier Inc. All rights reserved.

1. Introduction Switched systems, which consist of a family of subsystems described by differential or difference equations and a switching law that orchestrates switching between these subsystems, belong to a special class of hybrid systems. Recently, switched systems have received a great deal of attention, such as stability [1–10] and controllability and observability [11–13]. This is due to the fact that switched systems have numerous applications in mechanical control systems, automotive industry, traffic control, switching power converters, and many other fields. Time-delay is a common phenomenon and is unavoidable in engineering control design. Switched systems with time-delay have strong engineering background, such as power systems [14,15] and networked control systems [16,17]. However, due to the interaction among continuous-time dynamics, discrete-time dynamics and time-delay, the dynamics of switched systems with time-delay becomes more complex than switched systems without time-delay and time-delay systems without switching. Therefore, the study of switched systems with time-delay is very interesting and challenging. Recently, based on multiple Lyapunov-like functions method, many valuable results on such systems have been developed. In [18], by using the average dwell-time technique, Lyapunov stability and L2-gain were analyzed for a class of switched systems with timevarying delay. In [19], for a class of uncertain discrete-time switched systems with mode-dependent time delays, robust stability analysis and H1 control problem were discussed. In [20], by designing a class of state-based switching signals, the problem of stabilization for switched linear systems with mode-dependent time-varying delays was solved. Up to now, most of existing literature related to stability of switched systems focuses on Lyapunov asymptotic stability, which is defined over an infinite time interval. However, in practice, one is interested in not only system stability (usually in the sense of Lyapunov) but also a bound of system trajectories over a fixed short time, such as networked control systems [21,22] and network congestion control [23]. In addition, a system could be Lyapunov stable but completely useless because it possesses undesirable transient performances, such as the system with saturation elements in the control loop. To study the transient performances of a system, the concept of short time stability, i.e., finite-time stability, was introduced in [25]. ⇑ Corresponding author. E-mail address: [email protected] (X. Lin). 0096-3003/$ - see front matter  2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.12.032

X. Lin et al. / Applied Mathematics and Computation 217 (2011) 5982–5993

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Specifically, a system is said to be finite-time stable if, given a bound on the initial condition, its state remains within a prescribed bound in a fixed time interval. Note that finite-time stability and Lyapunov asymptotic stability are independent concepts: a system could be finite-time stable but not Lyapunov asymptotically stable, and vice versa [27]. In addition, it should be emphasized that practical stability also studies the boundedness of system trajectory. Practical stability means that the corresponding controller can drive the system to an arbitrarily neighborhood around the origin, which is also defined over an infinite time interval. Some early results on finite-time stability can be found in [24–26]. Recently, based on linear matrix inequality theory, many valuable results have been obtained for this type of stability [26–35]. In [27–29], the authors introduced the concept of finite-time boundedness which is an extension of finite-time stability, and presented some sufficient conditions for finitetime boundedness and stabilization of continuous-time systems or discrete-time systems. In [30], finite-time stabilization of linear time-varying systems has been studied. In [31–33], finite-time control problem for the impulsive systems was discussed. In [34], for a class of nonlinear quadratic systems, sufficient conditions for finite-time stability and stabilization of were also presented. For more analysis and synthesis results of finite control problem, the readers are referred to the literature [35,36] and the references therein. In addition, it should be pointed out that the authors of [37–40] have presented some results of finite-time stability for different systems, but finite-time stability in those systems which implies Lyapunov stability and finite-time convergence is different from that in this paper and [23–35]. So far, Lyapunov stability analysis for switched systems with time delay and finite-time stability for different systems have been extensively studied by many researchers. However, to the best of authors’ knowledge, there is no result available yet on finite-time stability of switched systems with time-delay. For the switched systems without time-delay, in [41], practical stability and finite-time stability were discussed. For ease of computation, in [42], based on linear matrix inequalities, finite-time stability and stabilization conditions were developed. Considering the wide application of switched systems with time-delay and the requirement for transient behavior in engineering fields, it motivates us to investigate finite-time stability and finite-time boundedness for a class of switched linear systems with time-delay. Our contributions are given as follows: (1) Definitions of finite-time boundedness and finite-time weighted L2-gain are extended to switched linear systems with time-delay. The system under consideration is subject to time-varying norm-bounded exogenous disturbance. (2) Sufficient conditions for finite-time boundedness and finite-time weighted L2-gain of switched linear systems with time-delay are given. The paper is organized as follows. In Section 2, some notations and problem formulations are presented. In Section 3, based on linear matrix inequalities, sufficient conditions which guarantee finite-time boundedness of switched linear systems with time-delay are given. Sufficient conditions which guarantee that system has finite-time weighted L2-gain are presented in Section 4. Finally, an example is presented to illustrate the efficiency of the proposed method in Section 5. Concluding remarks are given in Section 6. 2. Preliminaries and problem formulation In this paper, let P > 0 (P P 0, P < 0, P 6 0) denote a symmetric positive definite (positive-semidefinite, negative definite, negative-semidefinite) matrix P. For any symmetric matrix P, kmax(P) and kmin(P) denote the maximum and minimum eigenvalues of matrix P, respectively. The identity matrix of order n is denoted as In (or, simply, I if no confusion arises). Consider a switched linear systems with time-delay as follows

8 _ ¼ ArðtÞ xðtÞ þ BrðtÞ xðt  hÞ þ GrðtÞ xðtÞ; > < xðtÞ zðtÞ ¼ C rðtÞ xðtÞ þ DrðtÞ xðtÞ; t P 0; > : xðtÞ ¼ uðtÞ; t 2 ½h; 0;

ð1Þ

where x(t) 2 Rn is the state, z(t) 2 Rm is the control output, Ar(t), Br(t), Cr(t) and Dr(t) are constant real matrices, u(t) is a differentiable vector-valued initial function on [h, 0], h > 0 denotes the constant delay, x(t) is time-varying exogenous disturbance and satisfies Assumption 2, r(t) : [0, 1) ? M = {1, 2, . . . , m} is the switching signal which is a piecewise constant function depending on time t or state x(t), and m is the number of subsystems. Corresponding to the switching signal r(t), we have the following switching sequence:

fx0 ; ði0 ; t0 Þ; . . . ; ðik ; t k Þ; . . . ; jik 2 M; k ¼ 0; 1; . . .g; in which t0 is the initial time, x0 is the initial state and the ikth subsystem is activated when t 2 [tk, tk+1). Assumption 1. The state of switched linear system does not jump at switching instants, i.e., the trajectory x(t) is everywhere continuous. Switching signal r(t) has finite switching number in any finite interval time. R1 Assumption 2. The external disturbances x(t) is time-varying and satisfies the constraint 0 xT ðtÞxðtÞdt 6 d; d P 0. It should be pointed out that the assumption about the external disturbances x(t) in this paper is different from that of [27,31,32], where the external disturbances is constant.

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Definition 1 [43]. For any T P t P 0, let Nr(t, T) denote the switching number of r(t) over (t, T). If

Nr ðt; TÞ 6 N0 þ

T t

sa

holds for sa > 0 and an integer N0 P 0, then sa is called an average dwell-time. Without loss of generality, in this paper we choose N0 = 0, as [18]. In [27], the concept of finite-time boundedness was proposed to study the transient behavior of linear systems with external disturbances in a fixed time interval. In the sequel, let us extend the definition of finite-time boundedness to switched linear system with time-delay (1). Definition 2. Given four positive constants c1, c2, Tf, d with c1 < c2, d P 0, a positive definite matrix R, and a switching signal r(t). If

xðt 0 ÞT Rxðt0 Þ 6 c1 ) xðtÞT RxðtÞ < c2 ;

8t 2 ½0; T f ;

8xðtÞ :

Z

Tf

xT ðtÞxðtÞdt 6 d;

ð2Þ

0

where xðt0 ÞT Rxðt0 Þ ¼ suph6h60 fxðhÞT RxðhÞg, then system (1) is said to be finite-time bounded with respect to (c1, c2, Tf, d, R, r). If (2) holds for any switching signal r(t), system (1) is said to be uniformly finite-time bounded with respect to (c1, c2, Tf, R, d). Remark 1. The meaning of ‘‘uniformity’’ in Definition 2 is with respect to the switching signal, rather than the time, which is identical to that of [1,3]. Remark 2. Definition 2 means that once a switching signal is given, a switched system is finite-time bounded if, given a bound on the initial state and bounded constant disturbances, the state remains within the prescribed bound in the fixed time interval. Uniform finite-time boundedness requires that the finite-time bounded conditions holds for any switching signal. It should be remarked that the concepts of finite-time stability and finite-time boundedness are different from the concept of reachable set. The set of states that a dynamical system can attain under some given bounded inputs and starting from some given initial conditions is called to be reachable set. However, In most analysis about reachable set, it is assumed that system is asymptotically stable [44]. However, in the analysis of finite-time boundedness, the assumption of system asymptotic stability is unnecessary. For more detail discussions about the difference between two approaches can be found in Remark 4 of Ref. [27]. Recently, disturbance attenuation properties for switched systems have been widely studied. In [5], weighted L2-gain problem was introduced and some sufficient conditions were given. Then, in [18], weighted L2-gain analysis for a class switched linear systems with time-delay was discussed. Here, in this paper, we investigate weighted L2-gain in a fixed interval, i.e., finite-time weighted L2-gain. Definition 3. For Tf > 0, k P 0 and c > 0, system (1) is said to have finite-time weighted L2-gain, if under zero initial condition

u(t) = 0, "t 2 [h, 0], it holds that Z

Tf

eks zT ðsÞzðsÞds 6 c2

0

Z

Tf

xT ðsÞxðsÞds:

ð3Þ

0

3. Finite-time boundedness analysis In this section, we focus on finite-time boundedness of switched delay systems (1). First, consider a non-switched timedelay system with external disturbances



_ xðtÞ ¼ AxðtÞ þ Bxðt  hÞ þ GxðtÞ; xðtÞ ¼ uðtÞ;

t P t0 ;

t 2 ½t 0  h; t 0 ;

ð4Þ

where x(t) 2 Rn is the state, A, B and G are constant real matrices, u(t) is a differentiable vector-valued initial function on [t0  h, t0], h > 0 denotes the constant delay, x(t) satisfies Assumption 1. Choose a Lyapunov-like function as follows

VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ; where

e 1 xðtÞ; V 1 ðtÞ ¼ xT ðtÞ Q

V 2 ðtÞ ¼

Z

t

SxðhÞdh; xT ðhÞeaðthÞ e

th

e 1; e and a P 0; Q S are positive definite matrices to be determined. A lemma is given which will be useful in the subsequent analysis.

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e 1 > 0; Q > 0; e Lemma 1. Suppose that there exist matrices Q S > 0 and constants a P 0; b > 0 such that 2

0

e 1A þ e e1 þ Q e1 AT Q S  aQ B e1 @ BT Q

1

e 1B Q eah e S

e 2G Q

0

bQ 2

e2 G Q T

C A < 0:

0

ð5Þ

Then, along the trajectory of system (4),

VðtÞ < eaðtt0 Þ Vðt0 Þ þ b

Z

t

eaðtsÞ xT ðsÞQ 2 xðsÞds:

t0

Proof. Taking the derivative of V(t) with respect to t along the trajectory of system (4) yields

e 1 xðtÞ e 1 ½AxðtÞ þ Bxðt  hÞ þ GxðtÞ _ V_ 1 ðtÞ ¼ 2xT ðtÞ Q ¼ 2xT ðtÞ Q T e 1 AÞxðtÞ þ 2xT ðt  hÞBT Q e 1 GxðtÞ; e1 þ Q e 1 xðtÞ þ xT ðtÞGT Q e 1 xðtÞ þ xT ðtÞ Q ¼ xT ðtÞðA Q

ð6Þ

Sxðt  hÞ: V_ 2 ðtÞ ¼ aV 2 ðtÞ þ xT ðtÞe SxðtÞ  xT ðt  hÞeah e

ð7Þ

Then, it follows from (6) and (7) that

0

1T 0

e1 þ Q e1 e 1A þ e AT Q S  aQ C B T e _VðtÞ  aVðtÞ ¼ B @ xðt  hÞ A @ B Q1 e1 xðtÞ GT Q xðtÞ

e 1B Q S eah e 0

e 1G Q

10

xðtÞ

1

CB C 0 A@ xðt  hÞ A: xðtÞ 0

ð8Þ

Assuming condition (5) is satisfied, we obtain

_ VðtÞ  aVðtÞ < bxT ðtÞQ 2 xðtÞ:

ð9Þ

By calculation, we have

d at ðe VðtÞÞ < beat xT ðtÞQ 2 xðtÞ: dt

ð10Þ

Integrating (30) from t0 to t gives

VðtÞ < eaðtt0 Þ Vðt0 Þ þ b

Z

t

eaðtsÞ xT ðsÞQ 2 xðsÞds:

t0

Thus, the proof is completed. h Now, based on Lemma 1, let us discuss finite-time boundedness of switched systems (1). e 1;i ¼ R1=2 Q R1=2 ; e S i ¼ R1=2 Si R1=2 . Suppose that there exist matrices Q 1;i > 0; Q 2;i > 0; Si > 0 Theorem 1. For any i 2 M, let Q 1;i and constants ai P 0; bi > 0 such that

0 B B B @

e 1;i Ai þ e e 1;i þ Q e 1;i ATi Q S i  ai Q

e 1;i Bi Q

e 1;i Gi Q

e 1;i BTi Q

Si eai h e

0

e 1;i GTi Q

0

bi Q 2;i

ai h

ðk2 þ he

1 C C C < 0; A

k3 Þc1 þ k4 bd < c2 eai T f k1 :

ð11aÞ

ð11bÞ

If the average dwell-time of the switching signal r satisfies

sa > sa ¼

T f ln l h i ; ah lnðk1 c2 Þ  ln ðk2 þ he k3 Þc1 þ k4 bd  aT f

ð11cÞ

then switched systems (1) is finite-time bounded with respect to ðc1 ; c2 ; T f ; d; R; rÞ, where l P 1; Q 1;i 6 lQ 1;j ; Q 2;i 6 lQ 2;j ; Si 6 lSj ; 8i; j 2 M; a ¼ max8i2M ðai Þ; b ¼ max8i2M ðbi Þ; k1 ¼ min8i2M ðkmin ðQ 1;i ÞÞ; k2 ¼ max8i2M ðkmax ðQ 1;i ÞÞ; k3 ¼ max8i2M ðkmax ðSi ÞÞ; k4 ¼ max8i2M ðkmax ðQ 2;i ÞÞ. Proof. Choose a Lyapunov-like function as follows

VðtÞ ¼ V rðtÞ ðtÞ ¼ V 1;rðtÞ ðtÞ þ V 2;rðtÞ ðtÞ; where

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X. Lin et al. / Applied Mathematics and Computation 217 (2011) 5982–5993

e 1;rðtÞ xðtÞ; V 1;rðtÞ ðtÞ ¼ xT ðtÞ Q

V 2;rðtÞ ðtÞ ¼

Z

t

S rðtÞ xðhÞdh: xT ðhÞearðtÞ ðthÞ e

th

Step 1: When t 2 [tk, tk+1), by virtue of (11a) and Lemma 1, we have

VðtÞ ¼ V rðtÞ ðtÞ < earðtk Þ ðttk Þ V rðtk Þ ðt k Þ þ brðtk Þ

Z

t

tk

earðtk Þ ðtsÞ xT ðsÞQ 2;rðtk Þ xðsÞds:

ð12Þ

e i ¼ R1=2 Q R1=2 ; e Since l P 1, Q1,i 6 lQ1,j, Si 6 lSj, "i, j 2 M and Q S i ¼ R1=2 Si R1=2 , then i

e S i 6 le Sj ;

e 1;i 6 l Q e 1;j ; Q

8i; j 2 M:

Without loss of generality, assume that rðtk Þ ¼ i; rðt k Þ ¼ j at switching instant tk in the following, where rðtk Þ ¼ limv !0 rðtk þ v Þ. Noticing that xðtk Þ ¼ xðtk Þ from Assumption 1, one obtains

V rðtk Þ ðt k Þ 6 lV rðtk Þ ðtk Þ;

ð13Þ

where xðt  k Þ ¼ limv !0 xðt k þ v Þ. Since a = max"i2M(ai), b = max"i2M(bi), then it follows from (12) and (13) that

VðtÞ < eaðttk Þ lV rðtk Þ ðt k Þ þ b

Z

t

tk

eaðtsÞ xT ðsÞQ 2;rðtk Þ xðsÞds:

ð14Þ

Step 2: For any t 2 (0, Tf), let N be the switching number of r(t) over (0, Tf), which implies that Nr(0, t) 6 N. Using the iterative method in Step 1, we have

VðtÞ < eat lN V rð0Þ ð0Þ þ lN b þb

Z

t1

eaðtsÞ xT ðsÞQ 2;rð0Þ xðsÞds þ lN1 b

0

Z

t

tk

eaðtsÞ xT ðsÞQ 2;rðtk Þ xðsÞds ¼ eat lN V rð0Þ ð0Þ þ b

6 eaT f lN V rð0Þ ð0Þ þ eaT f lN bkmax ðQ 2;rðsÞ Þ

Z

Tf

Z

Z

t2 t1

t

eaðtsÞ xT ðsÞQ 2;rðt1 Þ xðsÞds þ   

eaðtsÞ lNr ðs;tÞ xT ðsÞQ 2;rðsÞ xðsÞds

0

xT ðsÞxðsÞds 6 eaT f lN V rð0Þ ð0Þ þ eaT f lN k4 bd:

ð15Þ

0 T

Noting that N 6 saf , then Tf

VðtÞ < eaT f lsa ðV rð0Þ ð0Þ þ k4 bdÞ:

ð16Þ

On the other hand,

Z

t

T T e 1;rðtÞ xðtÞ P kmin ðQ S rðtÞ xðhÞdh P xT ðtÞ Q xT ðhÞearðtÞ ðthÞ e ð17Þ 1;rðtÞ Þx ðtÞRxðtÞ ¼ k1 x ðtÞRxðtÞ; Z 0 ah e 1;rð0Þ xð0Þ þ xT ðhÞeharð0Þ e V rð0Þ ð0Þ ¼ xT ð0Þ Q S rð0Þ xðhÞdh 6 kmax ðQ 1;rð0Þ ÞxT ð0ÞRxð0Þ þ he kmax ðSrð0Þ Þ sup fxðhÞT RxðhÞg

e 1;rðtÞ xðtÞ þ VðtÞ ¼ xT ðtÞ Q

th

h6h60

h

  ah ah ah 6 kmax ðQ 1;rð0Þ Þ þ he kmax ðSrð0Þ Þ sup fxðhÞT RxðhÞg 6 ðk2 þ he k3 Þxðt 0 ÞT Rxðt0 Þ 6 ðk2 þ he k3 Þc1 :

ð18Þ

h6h60

Putting together (16)–(18), one obtains Tf

ah

VðtÞ eaT f lsa ðV rð0Þ ð0Þ þ k4 bdÞ ðk2 þ he k3 Þc1 þ k4 bd aT f sT f x ðtÞRxðtÞ 6 < 6 e l a: k1 k1 k1 T

ð19Þ

The following proof can be divided into two cases. Case 1: l = 1, which is a trivial case, from (11b),

xT ðtÞRxðtÞ < c2 eaT f eaT f ¼ c2 :

ð20Þ

Case 2: l > 1, from (11b),

h i ah lnðk1 c2 Þ  ln ðk2 þ he k3 Þc1 þ k4 bd  aT f > 0:

By virtue of (11c), we have

Tf

sa

h i ah lnðk1 c2 Þ  ln ðk2 þ he k3 Þc1 þ k4 bd  aT f <

ln l

ln ¼



aT

c 2 k1 e f ðk2 þheah k3 Þc1 þk4 bd

ln l

 :

ð21Þ

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Substituting (21) into (19) yields

xT ðtÞRxðtÞ <

! ah ðk2 þ he k3 Þc1 þ k4 bd aT f c2 k1 eaT f ¼ c2 : e ah k1 ðk2 þ he k3 Þc1 þ k4 bd

ð22Þ

Noticing that the trajectory of system (1) remains continuous at instant Tf, we conclude that (20) and (22) hold for all t 2 [0, Tf]. h Remark 3. The function V(t) in the proof procedure of Theorem 1 belongs to multiple Lyapunov-like functions. Unlike the classical Lyapunov function for switched systems in the case of asymptotical stability, there is no requirement of negative _ definiteness or negative semidefiniteness on VðtÞ. Actually, if the exogenous disturbance x(t) = 0 and we limit the constants _ ai < 0 ("i 2 M), then VðtÞ will be a negative definite function. For this case, we can obtain the system (1) is asymptotically stable on the infinite interval [0, +1) if the average dwell-time sa > ((ln l)/a). The detailed proof can be found in [18]. The advantages of multiple Lyapunov-like functions lie in their flexibility, because different Lyapunov-like functions can be constructed for different subsystems. However, there are some constraints on the switching signals such as condition (11c). Hence, Theorem 1 may not be suitable for the case of fast switching or stochastic switching. The following corollary will give some conditions which can guarantee switched systems (1) finite-time bounded under an arbitrary switching signal. Now, based on Theorem 1, some conditions which guarantee uniform finite-time boundedness of switched systems (1) are given. e 1 ¼ R1=2 Q R1=2 ; e Corollary 1. Let Q S ¼ R1=2 SR1=2 . Suppose that there exist matrices Q 1 > 0; Q 2 > 0; S > 0 and a constant 1 a P 0; b > 0 such that, for any i 2 M,

0 B B @ h

e1 þ Q e1 e 1 Ai þ e ATi Q S  aQ T e B Q1 i e1 GTi Q

ah

e 1 Bi Q S eah e

e 1 Gi Q

0

bQ 2

i

0

1 C C < 0; A

kmax ðQ 1 Þ þ he kmax ðSÞ c1 þ kmax ðQ 2 Þbd < c2 eaT f kmin ðQ 1 Þ;

ð23aÞ

ð23bÞ

then switched systems (1) is uniformly finite-time bounded with respect to ðc1 ; c2 ; T f ; R; dÞ. Proof. Choose a common Lyapunov-like function as follows

e 1 xðtÞ þ VðxðtÞÞ ¼ xT ðtÞ Q

Z

t

xT ðhÞeaðthÞ e SxðhÞc dh:

th

e 1;rðtÞ ; Q e e e Substituting Q h 2;rðtÞ ; S rðtÞ with Q 1 ; Q 2 ; S into the proof procedure of Theorem 1, it is easy to get the conclusion. 4. Finite-time weighted L2-gain analysis Having discussed finite-time boundedness of switched linear system with time delay, let us discuss the finite-time weighted L2-gain problem. First, consider a non-switched delay system

8 _ ¼ AxðtÞ þ Bxðt  hÞ þ GxðtÞ; > < xðtÞ zðtÞ ¼ CxðtÞ þ DxðtÞ; t P t0 ; > : xðtÞ ¼ uðtÞ; t 2 ½t 0  h; t 0 ;

ð24Þ

where x(t) 2 Rn is the state, z(t) 2 Rm is the control output, A, B, C, D and G are constant real matrices, u(t) is a differentiable vector-valued initial function on [t0  h, t0], h > 0 denotes the constant delay. Choose a Lyapunov-like function as follows

VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ; where

e 1 xðtÞ; V 1 ðtÞ ¼ xT ðtÞ Q

V 2 ðtÞ ¼

Z

t

xT ðhÞeaðthÞ e SxðhÞdh;

th

e 1; e and a P 0; Q S are positive definite matrices to be determined. e 1 > 0; e Lemma 2. Suppose that there exist matrices Q S > 0 and scalars a P 0; c > 0 such that

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0

e 1A þ e e1 þ Q e 1 þ CT C AT Q S  aQ B T e @ B Q1

e 1B Q eah e S

e2 þ D C G Q T

T

e 2G þ CT D Q 0 T

1 C A < 0:

ð25Þ

c2 I þ D D

0

Then, along the trajectory of system (24),

VðtÞ < eaðtt0 Þ Vðt 0 Þ þ

Z

t

eaðtsÞ ½c2 xT ðsÞxðsÞ  zT ðsÞzðsÞds:

t0

Proof. Taking derivative of V(t) with respect to t along the trajectory of system (24) yields

e 1 xðtÞ e 1 ½AxðtÞ þ Bxðt  hÞ þ GxðtÞ _ V_ 1 ðtÞ ¼ 2xT ðtÞ Q ¼ 2xT ðtÞ Q e1 þ Q e 1 xðtÞ þ xT ðtÞGT Q e 1 xðtÞ þ xT ðtÞ Q e 1 AÞxðtÞ þ 2xT ðt  hÞBT Q e 1 GxðtÞ; ¼ xT ðtÞðAT Q V_ 2 ðtÞ ¼ aV 2 ðtÞ þ xT ðtÞe SxðtÞ  xT ðt  hÞeah e Sxðt  hÞ:

ð26Þ ð27Þ

Then, it follows from (26) and (27) that

1T 0 T e 1A þ e e1 þ Q e1 xðtÞ A Q S  aQ B C B T e _ VðtÞ  aVðtÞ ¼ @ xðt  hÞ A @ B Q1 e1 xðtÞ GT Q 0

e 1B Q eah e S 0

10

1 xðtÞ CB C 0 A@ xðt  hÞ A: xðtÞ 0

e 1G Q

ð28Þ

Assuming condition (25) is satisfied, we obtain

0

1T 0 T xðtÞ C C B C B _ VðtÞ  aVðtÞ < @ xðt  hÞ A @ 0 xðtÞ DT C

1 xðtÞ C CB A@ xðt  hÞ A ¼ c2 xT ðtÞxðtÞ  zT ðtÞzðtÞ: 0 0 xðtÞ 0 c2 I þ DT D CT D

0

10

ð29Þ

By calculation, we have

  d at ðe VðtÞÞ < eat c2 xT ðtÞxðtÞ  zT ðtÞzðtÞ : dt

ð30Þ

Integrating (30) gives

VðtÞ < eaðtt0 Þ Vðt 0 Þ þ

Z

t

  eaðtsÞ c2 xT ðsÞxðsÞ  zT ðsÞzðsÞ ds:

t0

Thus, the proof is completed. h e 1;i ¼ R1=2 Q R1=2 ; e S i ¼ R1=2 Si R1=2 . Suppose that there exist matrices Q 1;i > 0; Q 2;i > 0; Si > 0 and Theorem 2. For any i 2 M, let Q 1;i constants ai P 0; c > 0 such that

0 B B B @

2

e 1;i þ Q e 1;i þ C T C i e 1;i Ai þ e ATi Q S i  ai Q i

e 1;i Bi Q

e 1;i Gi þ C T Di Q i

e 1;i BTi Q

eai h e Si

0

e 1;i GTi Q ai T f

c d < c2 e

þ

DTi C i

c2 I þ

0

DTi Di

1

C C C < 0; A

ð31aÞ

ð31bÞ

k1 :

If the average dwell-time of the switching signal r satisfies

sa > sa ¼ max



 T f ln l ln l ; ; lnðk1 c2 Þ  ln ðc2 dÞ  aT f a

ð31cÞ

then switched systems (1) is finite-time bounded with respect to ð0; c2 ; T f ; d; R; rÞ and has finite-time weighted L2 -gain, where l P 1; Q 1;i 6 lQ 1;j ; Q 2;i 6 lQ 2;j ; Si 6 lSj ; 8i; j 2 M; a ¼ max8i2M ðai Þ; k1 ¼ min8i2M ðkmin ðQ 1;i ÞÞ; k2 ¼ max8i2M ðkmax ðQ 1;i ÞÞ; k3 ¼ max8i2M ðkmax ðSi ÞÞ. Proof. Assuming condition (31a) is satisfied, then we have

0 B B @

e 1;i þ Q e 1;i e 1;i Ai þ e ATi Q S i  ai Q T e B Q 1;i i e 1;i GTi Q

e 1;i Bi Q Si eai h e 0

1

0 T C C C B i i þ @ 0 0 C A T 2 D c I i Ci

e 1;i Gi Q

0 0 0

C Ti Di

1

C 0 A < 0:

DTi Di

ð32Þ

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X. Lin et al. / Applied Mathematics and Computation 217 (2011) 5982–5993

Note that

0

C Ti C i

B B B 0 @

C Ti Di

0

0

1

C Ti

C B C C B C 0 C ¼ B 0 CðC i 0 Di Þ P 0; A @ A

0

DTi C i

1

0 DTi Di

ð33Þ

DTi

which implies that

0 B B B B @

e 1;i Ai þ e e 1;i þ Q e 1;i ATi Q S i  ai Q

e 1;i Bi Q

e 1;i BTi Q

S eai h e

e 1;i GTi Q

0

e 1;i Gi Q 0

i

c2 I

1 C C C < 0: C A

ð34Þ

From Theorem 1, conditions (34) and (31b) guarantee that system (1) is finite-time bounded with respect to (0, c2, Tf, d, R, r), where Q2,i = c2I, "i 2 M. In the following, we will prove that system (1) has finite-time weighted L2-gain. Let

VðtÞ ¼ V rðtÞ ðtÞ ¼ V 1;rðtÞ ðtÞ þ V 2;rðtÞ ðtÞ; where

e 1;rðtÞ xðtÞ; V 1;rðtÞ ðtÞ ¼ xT ðtÞ Q

Z

V 2;rðtÞ ðtÞ ¼

t

S rðtÞ xðhÞdh: xT ðhÞearðtÞ ðthÞ e

th

Step 1: When t 2 [tk, tk+1), by virtue of Eq. (31a) and Lemma 2, we have

VðtÞ ¼ V rðtÞ ðtÞ < earðtk Þ ðttk Þ V rðtk Þ ðtk Þ þ

Z

t

earðtk Þ ðtsÞ ½c2 xT ðsÞxðsÞ  zT ðsÞzðsÞds:

ð35Þ

tk

e i ¼ R1=2 Q R1=2 ; e Since l P 1, Q1,i 6 lQ1,j, Si 6 lSj, "i, j 2 M and Q S i ¼ R1=2 Si R1=2 , then i

e S i 6 le Sj;

e 1;i 6 l Q e 1;j ; Q

8i; j 2 M:

In what follows, without loss of generality, assume rðt k Þ ¼ i; rðt  at kÞ ¼ j rðtk Þ ¼ limv !0 rðtk þ v Þ. Noticing that xðtk Þ ¼ xðtk Þ from Assumption 1, one obtains

switching

instant

tk,

V rðtk Þ ðtk Þ 6 lV rðtk Þ ðtk Þ;

where

ð36Þ

where xðt k Þ ¼ limv !0 xðt k þ v Þ. Since a = max"i2M(ai), then it follows from (35) and (36) that

VðtÞ < eaðttk Þ lV rðtk Þ ðt k Þ þ

Z

t

eaðtsÞ ½c2 xT ðsÞxðsÞ  zT ðsÞzðsÞds:

ð37Þ

tk

Step 2: For any t 2 (0, Tf), let N be the switching number of r(t) over (0,Tf), which implies that Nr(0, t) 6 N. Using the iterative method in Step 1, we have

VðtÞ < eat lN V rð0Þ ð0Þ þ lN

Z

t1

Z

  eaðtsÞ c2 xT ðsÞxðsÞ  zT ðsÞzðsÞ ds þ lN1

0

þ  þ

Z

t

t2

  eaðtsÞ c2 xT ðsÞxðsÞ  zT ðsÞzðsÞ ds

t1

  eaðtsÞ c2 xT ðsÞxðsÞ  zT ðsÞzðsÞ ds ¼ eat lN V rð0Þ ð0Þ þ

tk

Z

t

  eaðtsÞ lNr ðs;tÞ c2 xT ðsÞxðsÞ  zT ðsÞzðsÞ ds:

ð38Þ

0

Under zero initial condition, (38) gives

0 6 VðtÞ <

Z

t

eaðtsÞ lNr ðs;tÞ ½c2 xT ðsÞxðsÞ  zT ðsÞzðsÞds;

ð39Þ

0

which implies that

Z

t

eaðtsÞ lNr ðs;tÞ zT ðsÞzðsÞds <

Z

0

0

eaðtsÞ lNr ðs;tÞ c2 xT ðsÞxðsÞds:

ð40Þ

0

Multiplying both sides of (40) by

Z

t

t

lNr ð0;tÞ yields

eaðtsÞ lNr ð0;sÞ zT ðsÞzðsÞds <

Z 0

t

eaðtsÞ lNr ð0;sÞ c2 xT ðsÞxðsÞds:

ð41Þ

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X. Lin et al. / Applied Mathematics and Computation 217 (2011) 5982–5993

Since sa P ln l/a, then 0 6 Nr(0, s) 6 s/sa 6 as/ln l. Substituting this inequality into (41) yields

Z

t

eat2as zT ðsÞzðsÞds <

Z

0

t

eaðtsÞ c2 xT ðsÞxðsÞds:

ð42Þ

0

Setting t = Tf, we obtain

Z

Tf

e2as zT ðsÞzðsÞds <

Z

0

Tf

eas c2 xT ðsÞxðsÞds 6 c2

Z

0

Tf

xT ðsÞxðsÞds:

ð43Þ

0

Let k = 2a. Thus, according to the Definition 3, the proof is completed. h Now, based on Theorem 2, some sufficient conditions which guarantee system (1) has finite-time weighted L2-gain under an arbitrary switching signal are given. e 1 ¼ R1=2 Q R1=2 ; e Corollary 2. Let Q S ¼ R1=2 SR1=2 . Suppose that there exist matrices Q 1 > 0; S > 0 and constants a P 0; c > 0 1 such that, for any i 2 M,

0 B B @

e 1 Ai þ e e1 þ Q e 1 þ CT Ci ATi Q S  aQ i T e B Q1

c2 d <

i

e 1 þ DT C i GTi Q i aT f c2 e kmin ðQ 1 Þ;

e 1 Bi Q eah e S 0

e 1 Gi þ C T Di Q i 0 2

c I þ

DTi Di

1 C C < 0; A

ð44aÞ ð44bÞ

then switched systems (1) is uniformly finite-time bounded with respect to ð0; c2 ; T f ; R; dÞ and has finite-time weighted L2 -gain. Proof. Choose a common Lyapunov-like function as follows

e 1 xðtÞ þ VðxðtÞÞ ¼ xT ðtÞ Q

Z

t

xT ðhÞeaðthÞ e SxðhÞdh:

th

e 1;rðtÞ ; e e 1; e Substituting Q S rðtÞ with Q S into the proof procedure of Theorem 2, it is easy to get the conclusion. h Remark 4. From a viewpoint of computation, it should be noted that the conditions in Theorems 1 and 2 and Corollaries 1 and 2 are not standard linear matrix inequalities (LMIs) conditions. However, once some values are fixed for ai, bi, these conditions, i.e., (11a), (23a), (31a), (44a), can be translated into LMIs conditions and thus solved involving Matlab’s LMI control toolbox. In addition, as in [27], the conditions (11b), (23b), (31b), (44b) can also be guaranteed by LMIs conditions once the values of ai, bi are fixed. Specifically, (1) The conditions (11b) and (23b) can be guaranteed by the following LMIs conditions, that is, for any i 2 M, there exist some positive numbers h1, h2, h3 and h4 such that

h1 I < Q 1;i < h2 I;

ð45aÞ

0 < Si < h3 I;

ð45bÞ

0 < Q 2;i < h4 I; ah

ðh2 þ he h3 Þc1 þ h4 bd < c2 e

ð45cÞ ai T f

h1 :

ð45dÞ

(2) The conditions (31b) and (44b) can be guaranteed by the following LMIs conditions, that is, for any i 2 M, there exists a positive number h such that

hI < Q 1;i ;

ð46aÞ

c2 d < c2 eai T f h:

ð46bÞ

Remark 5. Based on two kinds of Lyapunov-like functions, finite-time boundedness and L2-gain have been discussed under two kinds of switching conditions. In practical applications, the choosing order is given as follows: First, we choose the common Lyapunov-like function method since this function is more restrictive. If there exists a common Lyapunov-like function for all subsystems, then the finite-time boundedness and L2-gain can be guaranteed under an arbitrary switching. However, in many cases, it is difficult to find a common Lyapunov-like function. Second, we choose the multiple Lyapunov-like functions method. If there exist different Lyapunov-like functions for each subsystem, then the finite-time boundedness and L2-gain can be guaranteed if the switching is slow enough, i.e., there exists a lower bound for the average dwell time of switching signal.

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X. Lin et al. / Applied Mathematics and Computation 217 (2011) 5982–5993

5. Numerical examples and simulations Now, an example is employed to verify the proposed method in this paper. Example 1. Consider a switched linear system with time delay as follows

_ xðtÞ ¼ ArðtÞ xðtÞ þ BrðtÞ xðt  hÞ þ GrðtÞ xðtÞ 0

1

ð47Þ

0

1

0

1

0

1

1:7 1:7 0 1 1 0 1:5 1:7 0:1 1 0 0:1 with A1 ¼ @ 1:3 1 0:7 A; A2 ¼ @ 0:7 0 0:6 A; B1 ¼ @ 1:3 1 0:3 A; B2 ¼ @ 1:3 0:1 0:6 A; G1 ¼ G2 ¼ 0:7 1 0:6 1:7 0 1:7 0:7 1 0:6 1:5 0:1 1:8 0 1 0 1 0 1 0:7 1 0 0 0:03sinðtÞ A; h ¼ 0:2; xðtÞ ¼ @ 0 A; t 2 ½h; 0. @ 0 1 0 A; xðtÞ ¼ @ 0:02cosð2tÞ 0 0 0 1 0:015ðsinðt þ 1Þ þ cosðt  2ÞÞ The values of c1, c2, Tf, d and matrix R are given as follows:

c1 ¼ 0:5;

c2 ¼ 50;

T f ¼ 10;

R ¼ I;

d ¼ 0:01:

Solving (11a) and (45) for ai = 0.042, bi = 0.075, ("i 2 M) leads to feasible solutions

0

Q 1;1

Q 2;1

0:2188 B ¼ 1:0  10  @ 0:0008 0:0014 0 0:1197 B ¼ 1:0  107  @ 0:0076 10

0 1 0:1179 0:0223 0:0346 C B C 10 0:2477 0:0379 A; Q 1;2 ¼ 1:0  10  @ 0:0223 0:1746 0:0595 A; 0:0379 0:2078 0:0346 0:0595 0:1614 1 0 1 0:0076 0:0024 0:1013 0:0128 0:0079 C B C 0:1276 0:0000 A; Q 2;2 ¼ 1:0  107  @ 0:0128 0:0705 0:0224 A; 0:0008

0:0014

1

0:0024 0:0000 0:1187 1 0:2753 0:2163 0:0071 B C 10 S1 ¼ 1:0  10  @ 0:2163 0:2731 0:0662 A; 0

0:0071

0:0662

0:1449

0

0:0079

0:0105 B 10 S2 ¼ 1:0  10  @ 0:0725

0:0224 0:0725 0:0894

0:0493 0:0527

Fig. 1. The state trajectories of system (47) under switching signal r1 and r2.

0:1019 1 0:0493 C 0:0527 A: 0:1999

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X. Lin et al. / Applied Mathematics and Computation 217 (2011) 5982–5993

5 State trajectory x1 State trajectory x2

Trajectories

State trajectory x3

0

−5

0

1

2

3

4

5

6

7

8

9

10

Time(s) Fig. 2. Time histories of state trajectory of switched system (47) under switching signal r1(t).

15

Trajectories

10 5 0 −5

State trajectory x1 State trajectory x2

−10 −15

State trajectory x3

0

1

2

3

4

5

Time(s)

6

7

8

9

10

Fig. 3. Time histories of state trajectory of switched system (47) under switching signal r2(t).

According to (11c), one obtains

sa ¼ 3:7953: Then, by virtue of Theorem 1, for any switching signal r(t) with average dwell-time sa > sa , switched system (47) is finitetime bounded with respect to (0.5, 50, 10, 0.01, I, r). Fig. 1 shows the state trajectory over 0  10 s under a periodic switching signal r1(t) with interval time DT = 3.8 s. From Fig. 1, it is easy to see that system (47) is finite-time bounded. Fig. 2 shows the state trajectory of (47) under the switching signal r1(t). If the switching is too frequent, it is possible that the system is not finite-time bounded any more. For instance, simulation curves of system (47) under a periodic switching signal r2(t) with interval time DT = 1.3 s are shown in Fig. 1. The state trajectory of system (47) under the switching signal r2(t) is presented in Fig. 3.

6. Conclusion In this paper, finite-time boundedness and finite-time weighted L2-gain for a class of switched linear systems with timedelay have been investigated. Based on linear matrix techniques and multiple Lyapunov-like function method, some sufficient conditions which guarantee that the switched linear systems with time-delay is finite-time bounded and has finitetime weighted L2-gain have been provided respectively. A challenging and interesting future research topic is how to extend the results in this paper to uncertain switched systems with time-varying delay. Acknowledgements This work was supported by Natural Science Foundation of China (61074013), National 863 Project (2009AA01Z314), Specialized Research Fund for the Doctoral Program of Higher Education of China (20090092110022), The Program for Postgraduates Research Innovation in University of Jiangsu Province, and Youth Sci-Tech Innovation Fund, NJAU (KJ09029).

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References [1] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Systems Magazine 19 (5) (1999) 59–70. [2] R.A. Decarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of IEEE 88 (7) (2000) 1069–1082. [3] D. Liberzon, Switching in Systems and Control, Brikhauser, Boston, 2003. [4] Z. Sun, S.S. Ge, Switched Linear Systems: Control and Design, Springer-Verlag, New York, 2004. [5] G. Zhai, B. Hu, K. Yasuda, A. Michel, Disturbance attenuation properties of time-controlled switched systems, Journal of the Franklin Institute 338 (2001) 765–779. [6] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transaction on Automatic Control 43 (4) (1998) 475–482. [7] J.P. Hespanha, Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle, IEEE Transaction on Automatic Control 49 (4) (2004) 470–482. [8] J. Zhao, D.J. Hill, Passivity and stability of switched systems: a multiple storage function method, Systems & Control Letters 57 (2) (2008) 158–164. [9] X. Lin, S. Li, Output feedback stabilization of invariant sets for nonlinear switched systems, Acta Automatica Sinica 34 (7) (2008) 784–791. [10] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey on recent results, IEEE Transaction on Automatic Control 54 (2) (2009) 308–322. [11] A. Bemporad, G. Ferrari-Trecate, M. Morari, Observability and controllability of piecewise affine and hybrid systems, IEEE Transaction on Automatic Control 45 (10) (2000) 1864–1876. [12] Z. Sun, S.S. Ge, T.H. Lee, Controllability and reachability criteria for switched linear systems, Automatica 38 (5) (2002) 775–786. [13] Z. Ji, L. Wang, X. Guo, On controllability of switched linear systems, IEEE Transaction on Automatic Control 53 (3) (2008) 796–801. [14] C. Meyer, S. Schroder, R.W. De Doncker, Solid-state circuit breakers and current limiters for medium-voltage systems having distributed power systems, IEEE Transaction Power Electron 19 (5) (2004) 1333–1340. [15] J. Buisson, P.Y. Richard, H. Cormerais, On the stabilisation of switching electrical power converters, in: Proceedings of Hybrid Systems: Computation and Control, 2005, pp. 184–197. [16] H. Lin, G. Zhai, P.J. Antsaklis, Robust stability and disturbance attenuation analysis of a class of networked control systems, in: Proceedings of the IEEE Conference on Decision and Control, 2003, pp. 1182–1187. [17] X. Sun, G. Liu, W. Wang, D. Rees, Stability analysis for networked control systems based on average dwell time method, International Journal of Robust and Nonlinear Control 20 (2010) 1774–1784. [18] X. Sun, J. Zhao, D.J. Hill, Stability and L2 gain analysis for switched delay systems: a delay-dependent method, Automatica 42 (10) (2006) 1769–1774. [19] Y. Sun, L. Wang, G. Xie, Delay-dependent robust stability and H1 control for uncertain discrete-time switched systems with mode-dependent time delays, Applied Mathematics and Computation 187 (2) (2007) 1228–1237. [20] X. Liu, Stabilization of switched linear systems with mode-dependent time-varying delays, Applied Mathematics and Computation 216 (9) (2010) 2581–2586. [21] S. Mastellone, C.T. Abdallah, P. Dorato, Stability and finite time stability of discrete-time nonlinear networked control systems, in: Proceedings of American Control Conference, 2005, pp. 1239–1244. [22] F. Pan, X. Chen, L. Lin, Practical stability analysis of stochastic swarms, in: The 3rd International Conference on Innovative Computing Information and Control, 2008, pp. 32–36. [23] L. Wang, L. Cai, X. Liu, X. Shen, Practical stability and bounds of heterogeneous AIMA/RED system with time delay, in: IEEE International Conference on Communications, 2008, pp. 5558–5563. [24] G. Kamenkov, On stability of motion over a finite interval of time, Journal of Applied Mathematics and Mechanics 17 (5) (1953) 529–540 (in Russian). [25] P. Dorato, Short time stability in linear time-varying systems, in: Proceedings of the IRE International Convention Record Part 4, New York, 1961, pp. 83–87. [26] L. Weiss, E.F. Infante, Finite time stability under perturbing forces and on product spaces, IEEE Transaction on Automatic Control 12 (1) (1967) 54–59. [27] F. Amato, M. Ariola, P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances, Automatica 37 (9) (2001) 1459–1463. [28] F. Amato, M. Ariola, Finite-time control of discrete-time linear system, IEEE Transaction on Automatic Control 50 (5) (2005) 724–729. [29] F. Amato, M. Ariola, P. Dorato, Finite-time stabilization via dynamic output feedback, Automatica 42 (2) (2006) 337–342. [30] G. Garcia, S. Tarbouriech, J. Bernussou, Finite-time stabilization of linear time-varying continuous systems, IEEE Transactions on Automatic Control 54 (2) (2009) 364–369. [31] L. Liu, J. Sun, Finite-time stabilization of linear systems via impulsive control, International Journal of Control 81 (6) (2008) 905–909. [32] S. Zhao, J. Sun, L. Liu, Finite-time stability of linear time-varying singular systems with impulsive effects, International Journal of Control 81 (11) (2008) 1824–1829. [33] F. Amato, R. Ambrosino, A. Merola, C. Cosentino, Finite-time stability of linear time-varying systems with jumps, Automatica 45 (5) (2009) 1354–1358. [34] F. Amato, C. Cosentino, A. Merola, Sufficient conditions for finite-time stability and stabilization of nonlinear quadratic systems, IEEE Transactions on Automatic Control 55 (2) (2010) 430–434. [35] F. Amato, A. Merola, C. Cosentino, Finite-time stability of linear time-varying systems: analysis and controller design, IEEE Transactions on Automatic Control 55 (4) (2010) 1003–1008. [36] F. Amato, A. Merola, C. Cosentino, Finite-time control of discrete-time linear systems: analysis and design conditions, Automatica 46 (5) (2010) 919– 924. [37] L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, Systems & Control Letters 19 (4) (1992) 467–473. [38] S.P. Bhat, D.S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM Journal on Control and Optimization 38 (3) (2000) 751–766. [39] Y. Orlov, Finite time stability and robust control synthesis of uncertain switched systems, SIAM Journal of Control and Optimization 43 (4) (2005) 1253–1271. [40] S. Li, Y. Tian, Finite-time stability of cascaded time-varying systems, International Journal of Control 80 (4) (2007) 646–657. [41] X. Xu, G. Zhai, Practical stability and stabilization of hybrid and switched systems, IEEE Transaction on Automatic Control 50 (11) (2005) 1897–1903. [42] H. Du, X. Lin, S. Li, Finite-time stability and stabilization of switched linear systems, in: The 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, 2009, pp. 1938–1943. [43] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, in: Proceedings of IEEE Conference On Decision and Control, 1999, pp. 2655–2660. [44] J.E. Gayek, A survey of techniques for approximating reachable and controllable sets, in: Proceedings of the 30th IEEE Conference on Decision and Control 1991, pp. 1724–1729.