Finite-time H∞ control of switched systems with mode-dependent average dwell time

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Journal of the Franklin Institute 351 (2014) 1301–1315 www.elsevier.com/locate/jfranklin

Finite-time H1 control of switched systems with mode-dependent average dwell time Hao Liua,n, Xudong Zhaob a

Faculty of Aerospace Engineering, Shenyang Aerospace University, No. 37 Daoyi South Avenue, Daoyi Development District, Shenyang 110136, Liaoning Province, China b College of Information Science and Technology, Bohai University, Jinzhou 121013, China

Received 16 March 2013; received in revised form 18 October 2013; accepted 31 October 2013 Available online 7 November 2013

Abstract This paper is concerned with finite-time H 1 control problem for a class of switched linear systems by using a mode-dependent average dwell time (MDADT) method. The switching signal used in this paper is more general than the average dwell time (ADT), in which each mode has its own ADT. By combining the MDADT and Multiple Lyapunov Functions (MLFs) technologies, some sufficient conditions, which can guarantee that the corresponding closed-loop system is finite-time bounded with a prescribed H 1 performance, are derived for the switched systems. Moreover, a set of mode-dependent dynamic state feedback controllers are designed. Finally, two examples are given to verify the validity of the proposed approaches. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction A switched system is a dynamical system that consists of a finite number of subsystems and a logical rule that orchestrates switching between these subsystems [1]. Switched systems have been attracting considerable attention during the last decades, such as network control systems [2], flight control systems [3], DC/DC converters [4], chaos generators [5], to list a few. Recently, analysis and synthesis of switched systems have received much attention [6–10]. Moreover, the H 1 control problem or L2-gain analysis problem for switched systems with disturbance has become interesting issues [11–17], because the disturbance is commonly found in practical switched systems. n Corresponding author at: Faculty of Aerospace Engineering, Shenyang Aerospace University, Shenyang 110136, PR China. Tel.: +86 24 89728891. E-mail address: [email protected] (H. Liu).

0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.10.020

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On the other hand, most of the existing literature related to stability of switched systems focuses on Lyapunov asymptotic stability, which is defined over an infinite time interval. However, in many practical applications, the main concern is the behavior of the system over a fixed finite time interval. In this case, the concept of finite-time stability is proposed. More specifically, a system is said to be finite-time stable if, given a bound on the initial condition, its state remains within prescribed bounds in the fixed time interval [18]. Finite-time boundedness and stabilization problems for a class of switched linear systems with time-varying exogenous disturbances are studied in [19]. The problems of finite-time stability analysis and stabilization for switched nonlinear discrete-time systems are studied, and then the results are extended to the analysis of H 1 finite-time boundedness in [20]. In [21], the issues of finite-time bounded and finite-time weighted L2-gain for a class of switched delay systems are addressed. A dynamic output feedback controller is designed for the finite-time stabilization problem in [22]. As a class of switching signals, average dwell time switching signal means that the number of switches in a finite interval is bounded and the average time between consecutive switching is not less than a constant [23], which is more general than Dwell Time (DT) switching [24]. However, the property in the ADT switching that the average time interval between any two consecutive switching is at less τa , which is independent of the system mode, is probably still not anticipated [25]. In order to extend the existing studies on the switched systems with ADT by providing two mode-dependent parameters, therefore, a novel notion of mode-dependent average dwell time is proposed in [25,26], which will release the restriction of ADT. So far, there is no result available yet on finite-time H 1 control of switched systems with mode-dependent average dwell time, which will improve the disturbance attenuation performance. This motivates us for this study. The main contribution of this paper is that a novel notion of the MDADT switching is used to investigate the problem of finite-time H 1 control of switched linear systems. By using the MDADT switching, a new weighted finite-time H 1 performance condition of the underlying system is derived and a set of mode-dependent dynamic state feedback controllers are designed. The remainder of the paper is organized as follows. In Section 2, some definitions and problem formulations are presented. By using the MDADT switching approach, sufficient conditions that can guarantee finite-time bounded of switched linear systems with a prescribed H 1 performance are proposed in Section 3. Then, two examples are presented to illustrate the efficiency of the proposed method in Section 4. Conclusions are given in Section 5. Notations: The notations used in this paper are standard. Let R denote the field of real numbers. The notation P40 means that P is a real symmetric and positive definite; the superscript “T” stands for matrix transposition; Rn denotes the n-dimensional Euclidean space. λmin ðPÞ and λmax ðPÞ denote the minimum and maximum eigenvalues of matrix P, respectively.

2. Problem formulation Consider a class of switched linear systems given by x_ ðtÞ ¼ AsðtÞ xðtÞ þ BsðtÞ uðtÞ þ GsðtÞ ωðtÞ

ð1Þ

zðtÞ ¼ CsðtÞ xðtÞ þ H sðtÞ ωðtÞ

ð2Þ

H. Liu, X. Zhao / Journal of the Franklin Institute 351 (2014) 1301–1315

1303

where xðtÞA Rn is the state, uðtÞA Rm is the control input, zðtÞA Rq is the controlled output, and ωðtÞA Rr is the exogenous disturbance and satisfies Assumption 1. sðtÞ : ½0; 1ÞI ¼ f1; 2; …; Mg is the switching signal which is a piecewise constant function depending on time t or state x(t). Ap, Bp, Gp, Cp and Hp are constant real matrices for p A I . Assumption 1. For a given constant Tf, the exogenous disturbance ωðtÞ is time-varying and RT satisfies the constraint 0 f ωT ðtÞωðtÞ dt r d, d40. Now, the following definitions are presented for later development. Definition 1 (Zhao and Hill [17] Finite-Time Stable, FTS). Given a positive definite matrix R, three positive constants c1, c2, Tf , with c1 oc2 , and a switching signal s, the continuous-time switched linear system (1) with uðtÞ  0 and ωðtÞ  0 is said to be finite-time stable with respect to ðc1 ; c2 ; T f ; R; sÞ, if xT0 Rx0 oc1 ) xðtÞT RxðtÞoc2 , 8 t A ½0; T f . Definition 2 (Zhao and Hill [17] Finite-Time Bounded, FTB). Given a positive definite matrix R, three positive constants c1, c2, Tf , with c1 oc2 , and a switching signal s, the switched linear system (1) is said to be finite-time bounded with respect to ðc1 ; c2 ; d; T f ; R; sÞ, if xT0 Rx0 oc1 ) RT xðtÞT RxðtÞoc2 , 8 t A ½0; T f , 8 ωðtÞ : 0 f ωT ðtÞωðtÞ dt r d. Definition 3 (Finite-Time H 1 Performance). Given a positive definite matrix R, two positive constants c2 and Tf, and a switching signal s, the switched linear system (1) with uðtÞ  0 is said to have finite-time H 1 performance with respect to ð0; c2 ; d; T f ; γ; R; sÞ, if the system is finitetime bounded and the following inequality is satisfied: Z Tf Z Tf zT ðtÞzðtÞ dtoγ 2 ωT ðtÞωðtÞ dt ð3Þ 0

0

where γ40 is a prescribed scalar and ωðtÞ satisfies the Assumption 1. Definition 4 (Finite-Time H 1 Control). The switched linear system (1) is said to be finitetime stabilizable with H 1 disturbance attenuation level γ , if there exists a control input u(t), 8t A ½0; T f , such that: (i) The corresponding closed-loop system is finite-time bounded. (ii) Under the zero-initial condition, the controlled output z(t) satisfies inequality (3). Definition 5 (Zhao et al. [25]). For a switching signal sðtÞ and any T Z t Z 0, let N sp ðt; TÞ be the switching numbers that the pth subsystem is activated over the interval ½t; T and T p ðt; TÞ denote the total running time of the pth subsystem over the interval ½t; T, pA I . We say that sðtÞ has a mode-dependent average dwell time τap if there exist positive numbers N0p (we call N0p the mode-dependent chatter bounds here) and τap such that N sp ðt; T Þr N 0p þ

T p ðt; TÞ ; τap

8T Zt Z0

ð4Þ

Remark 1. As discussed in [25], Definition 5 constructs a different set of switching signals from that with the ADT property, that is, if there exist positive numbers τap , pA I such that a

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switching signal has the MDADT property, it only requires that the average time among the intervals associated with the pth subsystem is larger than τap . In this paper, the following mode-dependent dynamic state-feedback controllers with order nc are considered: ( x_ c ðtÞ ¼ Ac;sðtÞ xc ðtÞ þ Bc;sðtÞ xðtÞ; xc ð0Þ ¼ 0 ð5Þ uðtÞ ¼ C c;sðtÞ xc ðtÞ þ Dc;sðtÞ xðtÞ where Ac;s , Bc;s , Cc;s and Dc;s are matrices to be determined. Denoting " # " # Bc;sðtÞ Ac;sðtÞ xðtÞ x~ ðtÞ ¼ ; K c;sðtÞ Dc;sðtÞ C c;sðtÞ xc ðtÞ the problem can be cast as the search of a robust static state feedback control gain K c;s A Rðnc þmÞðnc þnÞ for the augmented switched systems x~_ ðtÞ ¼ ðA sðtÞ þ B sðtÞ K c;sðtÞ Þ~x ðtÞ þ G sðtÞ ωðtÞ

ð6Þ

zðtÞ ¼ C sðtÞ x~ ðtÞ þ H sðtÞ ωðtÞ

ð7Þ

where



A sðtÞ ¼

AsðtÞ

0

0

0



 ;

C sðtÞ ¼ ½C sðtÞ 0;

B sðtÞ ¼

0

BsðtÞ

I

0

 ;

 G sðtÞ ¼

GsðtÞ 0

 ;

H sðtÞ ¼ ½H sðtÞ :

The purpose of this paper is to design a mode-dependent dynamic state-feedback controller and a set of admissible switching signals with MDADT such that the closed-loop system is bounded and has finite-time H 1 performance. 3. Main results In this section, based on MDADT and MLFs, some sufficient conditions for the existence of mode-dependent dynamic state-feedback controllers which can ensure that the switched systems (1)–(2) are finite-time bounded with H 1 disturbance attenuation level γ are proposed. Theorem 1. For any p; qA f1; …; Mg; p a q, let X p ¼ R  1=2 X p R  1=2 and suppose there exist matrices X p 40, Qp ¼ QTp 40, Yp and constants γ p 40, μp 41, λp 40 such that " # T T Gp X p A p þ A p X p þ B p Y p þ Y Tp B p  λp X p o0 ð8Þ n  γ p Qp X q  μp X p o0 

 c1 γd ɛp c2 þ e o κ1 κ3 κ2

ð9Þ ð10Þ

H. Liu, X. Zhao / Journal of the Franklin Institute 351 (2014) 1301–1315

1305

If the MDADT switching signal sðtÞ satisfies τap 4τnap ¼

MT f ln μp lnðc2 =κ2 Þ lnðc1 =κ1 þ γd=κ 3 Þ ɛ p

ð11Þ

then there exists a set of mode-dependent dynamic state-feedback controllers such that the switched system (6) is finite-time boundedness with respect to ðc1 ; c2 ; d; T f ; R; sÞ, where κ1 ¼ minp A I ðλmin ðX p ÞÞ, κ2 ¼ maxp A I ðλmax ðX p ÞÞ, κ3 ¼ minp A I ðλmin ðQp ÞÞ, ɛp ¼ MT f λp þ ∑M p ¼ 1 N 0p ln μp . Moreover, the controller gains are given by K c;p ¼ Y p X p 1

ð12Þ

Proof. Choose the following Lyapunov function: V sðtÞ ð~x ðtÞÞ ¼ x~ T ðtÞPsðtÞ x~ ðtÞ

ð13Þ

1 where PsðtÞ ¼ X sðtÞ and X sðtÞ satisfy conditions (8)–(10). When t A ½t i ; t iþ1 Þ, the derivative of Vð~x ðtÞÞ along the trajectories of subsystem p yields " #" # x~ ðtÞ Υ p Pp G p T T V_p ð~x ðtÞÞ ¼ ½~x ðtÞ ω ðtÞ ωðtÞ n 0 T

T

T

where Υ p ¼ A p Pp þ Pp A p þ Pp B p K c;p þ K c;p B p Pp . Then, from Eq. (8), we have V_p ð~x ðtÞÞoλp V p ð~x ðtÞÞ þ γ p ωT ðtÞQp 1 ωðtÞ

ð14Þ

By integrating Eq. (14) for any t A ½t i ; t iþ1 Þ, it holds that Z t λsðti Þ ðt  t i Þ 1 V sðti Þ ð~x ðtÞÞ þ γ sðti Þ eλsðti Þ ðt  τÞ ωT ðτÞQsðt ωðτÞ dτ V sðtÞ ð~x ðtÞÞoe iÞ

ð15Þ

ti

On the other hand, we can obtain from Eqs. (9) and (13) that, 8 ðsðt i Þ ¼ p; sðt i Þ ¼ qÞA I  I , p a q, V sðti Þ ð~x ðt i ÞÞoμsðti Þ V sðti Þ ð~x ðt i ÞÞ This, together with Eq. (15), implies n  o    V sðtÞ ðx~ ðt ÞÞoexp λsðtN sð0;tÞ Þ t  t N sð0;tÞ V sðtN sð0;tÞ Þ x~ t N sð0;tÞ Z t n o 1 γ sðtN Þ exp λsðtN sð0;tÞ Þ ðt  τÞ ωT ðτÞQsðt þ N t N sð0;tÞ

sð0;tÞ

ð16Þ

sð0;tÞ

Þ ωð τ Þ

dτ

  n  o r μsðtN Þ exp λsðtN sð0;tÞ Þ t  t N sð0;tÞ V sðtN sð0;tÞ  1 Þ x~ t Nsð0;tÞ Z sð0;tÞ n o t 1 þ γ sðtN Þ exp λsðtN sð0;tÞ Þ ðt  τÞ ωT ðτÞQsðt Þ ωðτÞ dτ N t N sð0;tÞ

sð0;tÞ

sð0;tÞ

n    o r μsðtN Þ μsðtN  1 Þ exp λsðtN sð0;tÞ Þ t  t N sð0;tÞ þ λsðtN sð0;tÞ  1 Þ t N sð0;tÞ  t N sð0;tÞ  1 sð0;tÞ sð0;tÞ  n  o V sðtN sð0;tÞ  2 Þ x~ t Nsð0;tÞ  1 þ μsðtN Þ exp λsðtN sð0;tÞ Þ t  t N sð0;tÞ sð0;tÞ Z tN n  o sð0;tÞ 1  γ sðtN  1 Þ exp λsðtN sð0;tÞ  1 Þ t N sð0;tÞ  τ ωT ðτÞQsðt ωðτÞ dτ N  1Þ t N sð0;tÞ  1

sð0;tÞ

sð0;tÞ

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H. Liu, X. Zhao / Journal of the Franklin Institute 351 (2014) 1301–1315

Z þ r⋯

t

t N sð0;tÞ

γ sðtN

sð0;tÞ

Þ

n o 1 exp λsðtN sð0;tÞ Þ ðt  τÞ ωT ðτÞQsðt N

sð0;tÞ

Þ ωðτÞ

dτ

n   exp λsðtN sð0;tÞ Þ t  t N sð0;tÞ o   þλsðtN sð0;tÞ  1 Þ t N sð0;tÞ  t N sð0;tÞ  1 þ ⋯ þ λsðt1 Þ ðt 2  t 1 Þ V sð0Þ ðx~ ð0ÞÞ n   þμsðtN Þ μsðtN  1 Þ ⋯μsðt1 Þ exp λsðtN sð0;tÞ Þ t  t N sð0;tÞ sð0;tÞ sð0;tÞ o   þλsðtN sð0;tÞ  1 Þ t N sð0;tÞ  t N sð0;tÞ  1 þ ⋯ þ λsðt1 Þ ðt 2  t 1 Þ Z t1

 1  γ sðt0 Þ exp λsðt0 Þ ðt 1  τÞ ωT ðτÞQsðt ωðτÞ dτ þ ⋯ 0Þ t0 Z t n o 1 þ γ sðtN Þ exp λsðtN sð0;tÞ Þ ðt  τÞ ωT ðτÞQsðt Þ ωðτ Þ dτ N r μsðtN

t N sð0;tÞ

Þ μsðt N sð0;tÞ  1 Þ ⋯μsðt 1 Þ sð0;tÞ

sð0;tÞ

Ns  1

(

r ∏ μsðtjþ1 Þ exp

sð0;tÞ

Ns  1

∑ λstj ðt jþ1  t j Þ þ λstN

j¼0

j¼0

 Z 1 þγ maxλmax Qp pAI

M

r ∏

p¼1

(

μpN sp ð0;tÞ exp

t

M

∏

0 p¼1

( μpN sp ðτ;tÞ exp )

M

M

r ∏

p¼1

μNp sp ð0;T f Þ

( exp

p¼1

(

(

)

M

∑ λp T p ðτ; tÞ ωT ðτÞωðτÞ dτ

p¼1

)Z

M

∑ λp T p ð0; tÞ

p¼1

t

ωT ðτÞωðτÞ dτ

0

)

M

∑ λp T p ð0; T f Þ V sð0Þ ðx~ ð0ÞÞ

p¼1

 M þγd maxλmax Qp 1 ∏ μNp sp ð0;T f Þ exp pAI

ðt  t N sð0;tÞ Þ V sð0Þ ðx~ ð0ÞÞ

∑ λp T p ð0; tÞ V sð0Þ ðx~ ð0ÞÞ

p¼1

 M þγ maxλmax Qp 1 ∏ μpN sp ð0;tÞ exp pAI

sð0;tÞ

)

p¼1

)

M

)

∑ λp T p ð0; T f Þ

p¼1

)      ln μp γd r exp ∑ N 0p ln μp exp ∑ þ λp T p 0; T f V sð0Þ ðx~ ð0ÞÞ þ κ3 τap p¼1 p¼1 ( ) ( )    M M ln μp γd þ λp T f r exp ∑ N 0p ln μp exp ∑ V sð0Þ ðx~ ð0ÞÞ þ κ3 τap p¼1 p¼1 ( )  

  M ln μp γd þ λp r exp ∑ N 0p ln μp exp MT f max V sð0Þ ðx~ ð0ÞÞ þ pAI κ3 τap p¼1 M

(

(

M

where γ ¼ maxp A I ðγ p Þ. Considering X p ¼ R  1=2 X p R  1=2 , we have 1

1 V sð0Þ ðx~ ð0ÞÞ ¼ x~ T ð0ÞPsð0Þ x~ ð0Þ ¼ x~ T ð0ÞX sð0Þ x~ ð0Þ ¼ x~ T ð0ÞR1=2 X sð0Þ R1=2 x~ ð0Þ   1 1 r max λmax X sð0Þ x~ T ð0ÞR~x ð0Þr x~ T ð0ÞR~x ð0Þ pAI κ1

ð17Þ

H. Liu, X. Zhao / Journal of the Franklin Institute 351 (2014) 1301–1315

1307

1

1 V sðtÞ ðx~ ðt ÞÞ ¼ x~ T ðt ÞPsðtÞ x~ ðt Þ ¼ x~ T ðt ÞX sðtÞ x~ ðt Þ ¼ x~ T ðt ÞR1=2 X sðtÞ R1=2 x~ ðt Þ   1 1 Z min λmin X sðtÞ x~ T ðt ÞR~x ðt ÞZ x~ T ðt ÞR~x ðt Þ pAI κ2

ð18Þ

Assume that if condition (10) is satisfied, then ln c2 =κ 2  lnðc1 =κ1 þ γd=κ3 Þ ɛp 40, thus, 8p A I , we can obtain the following inequality from Eq. (11):     M ln μp c2 c1 γd MT f max þ λp oln  ln þ ð19Þ  ∑ N 0p ln μp pAI τap κ2 κ1 κ3 p¼1 From Eq. (11), by combining Eqs. (17)–(19), it gives x~ T ðt ÞR~x ðt Þoκ2 V sðtÞ ðx~ ðt ÞÞ (



ln μp r κ 2 exp ∑ N 0p ln μp þ MT f max þ λp pAI τap p¼1 M

 )

c1 γd þ κ1 κ3



r c2

ð20Þ

According to Definition 2, the switched linear system (6) is finite-time bounded with respect to ðc1 ; c2 ; d; T f ; R; sÞ. The proof is completed. □ Remark 2. It can be seen from Theorem 1 that the parameters μp and λp are mode-dependent, thus, the MDADT switching signal is more general than the ADT switching signal. It is also noted that if τa ¼ τap , 8 pA I we have readily known from Definition 5 that T p ðt; TÞ ; τa pAI

∑ N sp ðt; TÞr ∑ N 0p þ ∑

pAI

pAI

8T ZtZ0

Thus, there exist positive numbers N 0 ¼ ∑p A I N 0p and τa ¼ τap such that T t N s ðt; TÞ r N 0 þ ; 8T Zt Z0 τa In other words, a MDADT switching signal with bounded τnap also has bounded ADT τna  τnap , 8p A I in special case of λ  λp , μ  μp , 8p A I . In the case of finite-time stability for the switched linear system (6) with ωðtÞ  0, it is easy to obtain the sufficient conditions from Theorem 1. Corollary 1. For any p, q A f1; :::; Mg, p a q, let X p ¼ R  1=2 X p R  1=2 and suppose there exist matrices X p 40, Yp and constants μp 41, λp 40 such that T

T

X p A p þ A p X p þ B p Y p þ Y Tp B p  λp X p o0

ð21Þ

X q  μp X p o0

ð22Þ

c1 ɛ p c2 e  o0 κ1 κ2

ð23Þ

If the MDADT switching signal sðtÞ satisfies τap 4τnap ¼

MT f ln μp lnðc2 =κ2 Þ lnðc1 =κ1 Þ ɛp

ð24Þ

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then there exists a set of mode-dependent dynamic state-feedback controllers such that the switched linear system (6) with ωðtÞ  0 is finite-time stable with respect to ðc1 ; c2 ; T f ; R; sÞ, where ɛp ¼ MT f λp þ ∑M p ¼ 1 N 0p ln μp , κ 1 ¼ minp A I ðλmin ðX p ÞÞ, κ 2 ¼ maxp A I ðλmax ðX p ÞÞ. Moreover, the controller gains are given by Eq. (12). Based on the results in Theorem 1, sufficient conditions for both finite-time boundedness and H 1 performance are derived in the following theorem. Theorem 2. For any p, q A f1; …; Mg, p a q, let X p ¼ R  1=2 X p R  1=2 and suppose there exist matrices X p 40, Yp and constants μp 41, λp 40 such that 2 3 T T T X p A p þ A p X p þ B p Y p þ Y Tp B p  λp X p Gp Cp 6 7 T 7 6 ð25Þ n  γ 2 I H p 5o0 4 n n I X q  μp X p o0

ð26Þ

κ 2 γ 2 deɛp  c2 o0

ð27Þ

If the MDADT switching signal sðtÞ satisfies

MT f ln μp ln μp τap 4τnap ¼ max ; lnðc2 =κ 2 Þ lnðγ 2 dÞ ɛp λp

ð28Þ

then there exists a set of mode-dependent dynamic state-feedback controllers such that the corresponding closed-loop system is finite-time stabilizable with H 1 performance γ~ with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi respect to ð0; c2 ; T f ; d; γ~ ; R; sÞ, where γ~ ¼ maxp A I ðβp Þ, βp ¼ γ 2 αp expfMT f maxp A I λp g, M κ2 ¼ maxp A I ðλmax ðX p ÞÞ, ɛp ¼ MT f λp þ ∑M p ¼ 1 N 0p ln μp , αp ¼ expf∑p ¼ 1 N 0p ln μp g. Moreover, the controller gains are given by Eq. (12). Proof. Note that 2 T3 Cp 4 T 5½C p H p Z 0 Hp

ð29Þ

then, we can get from Eq. (25) that " T T X p A p þ A p X p þ B p Y p þ Y Tp B p  λp X p n

Gp  γ2I

# o0

ð30Þ

It can be easily concluded from Theorem 1 that conditions (26), (27) and (30) can guarantee that the switched system (6) is finite-time bounded with respect to ð0; c2 ; d; T f ; R; sÞ by setting Qp ¼ I and c1 ¼ 0. Choose Eq. (13) as the Lyapunov function. Then, from Eq. (25), we have V_p ð~x ðtÞÞoλp V p ð~x ðtÞÞ þ ΓðtÞ where ΓðtÞ ¼ γ 2 ωT ðtÞωðtÞ zT ðtÞzðtÞ.

ð31Þ

H. Liu, X. Zhao / Journal of the Franklin Institute 351 (2014) 1301–1315

1309

By integrating Eq. (31) for any t A ½t i ; t iþ1 Þ, it holds that Z t V sðtÞ ð~x ðtÞÞoeλsðti Þ ðt  ti Þ V sðti Þ ð~x ðtÞÞ þ eλsðti Þ ðt  τÞ ΓðτÞ dτ ti ( ) M

M

r ∏ μNp sp ð0;tÞ exp

∑ λp T p ð0; tÞ V sð0Þ ð~x ð0ÞÞ

p¼1

Z

t

þ

p¼1

(

M

∏ μpN sp ðτ;tÞ exp

0 p¼1

Z

t

¼

∑ λp T p ðτ; tÞ ΓðτÞ dτ

p¼1

(

M

∏ μNp sp ðτ;tÞ exp

0 p¼1

)

M

)

M

∑ λp T p ðτ; tÞ ΓðτÞ dτ

ð32Þ

p¼1

Next, under zero initial condition, we establish the weighted H 1 performance with the MDADT switching. By Eq. (32), we get ( ) Z t M M N sp ðτ;tÞ ∏ μp exp ∑ λp T p ðτ; tÞ zT ðτÞzðτÞ dτ 0 p¼1

oγ

p¼1

Z

t

M

∏

2

0 p¼1

μpN sp ðτ;tÞ

(

exp

)

M

∑ λp T p ðτ; tÞ ωT ðτÞωðτÞ dτ

ð33Þ

p¼1

Multiplying both side of Eq. (33) by expf  ∑M p ¼ 1 N sp ð0; tÞln μp g yields ( ) Z t M exp ∑ ½N sp ð0; τÞln μp þ λp T p ðτ; tÞ zT ðτÞzðτÞ dτ p¼1

0

Z

(

t

oγ 2

)

M

∑ ½N sp ð0; τÞln μp þ λp T p ðτ; tÞ ωT ðτÞωðτÞ dτ

exp

ð34Þ

p¼1

0

Due to N sp ð0; τÞr N 0p þ T p ð0; τÞ=τap and τap Z ln μp =λp , we obtain N sp ð0; τÞln μp r N 0p ln μp þ λp T p ð0; τÞ Thus, Z γ

(

t

2

p¼1

Z

oγ

)

M

∑ ½N sp ð0; τÞln μp þ λp T p ðτ; tÞ ωT ðτÞωðτÞ dτ

exp 0

(

t

2

exp 0

and Z

(

t

)

M

∑ ½N 0p ln μp þ λp T p ð0; tÞ ωT ðτÞωðτÞ dτ

ð36Þ

p¼1

)

M

∑ λp T p ðτ; tÞ zT ðτÞzðτÞ dτ

exp 0

ð35Þ

p¼1

Z

(

t

o

∑ ½N sp ð0; τÞln μp þ λp T p ðτ; tÞ zT ðτÞzðτÞ dτ

exp 0

oγ 2 exp

(

)

M

p¼1 M

)Z

∑ N 0p ln μp

p¼1

(

t

exp 0

M

)

∑ λp T p ð0; tÞ ωT ðτÞωðτÞ dτ

p¼1

ð37Þ

1310

H. Liu, X. Zhao / Journal of the Franklin Institute 351 (2014) 1301–1315

By setting t ¼ T f , we have Z Tf Z zT ðτÞzðτÞ dτoγ 2 αp 0

Z

)

M

∑ λp T p ð0; τÞ ωT ðτÞωðτÞ dτ

exp 0



(

Tf

p¼1 Tf

ωT ðτÞωðτÞ dτ oγ αp exp MT f maxðλp Þ pAI 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi!2 Z 2

Tf

γ 2 αp exp MT f maxðλp Þ

¼

pAI

Z ¼ β2p

Tf

Z

Tf

ωT ðτÞωðτÞ dτ r γ~ 2

0

ωT ðτÞωðτÞ dτ

0

ωT ðτÞωðτÞ dτ

ð38Þ

0

where αp ¼ expf∑M p ¼ 1 N 0p ln μp g, γ~ ¼ maxp A I ðβ p Þ. By Definition 4, we can concluded from Eq. (38) that the underlying system is finite-time bounded with H 1 performance γ~ in the sense of Definition 4 for any MDADT switching signal (28). The proof is completed. □ 4. Numerical examples and simulation In this section, we apply the proposed approaches to verify the validity of the results given above. Example 1. Consider the switched linear systems consisting of two subsystems described by x_ ðtÞ ¼ AsðtÞ xðtÞ þ BsðtÞ uðtÞ þ GsðtÞ ωðtÞ where



A1 ¼  B2 ¼

1

3

0:2

0:5

0:5

0:2

0:3

0:7



 ;

A2 ¼

;

G1 ¼





0:8

1

1

0:3

0:3 0:1

ð39Þ 

 ;

B1 ¼



 ;

G2 ¼

0:4 0:2

0:2

0:3

1

0:1

 ;

 ;

ωðtÞ ¼ 0:1 sin ðtÞ:

The corresponding parameters are given as follows: c1 ¼ 0:3; μ2 ¼ 1:02;

c2 ¼ 2;

T f ¼ 20;

λ1 ¼ λ2 ¼ 0:01;

R ¼ I; d ¼ 0:3;

γ 1 ¼ γ 2 ¼ 1;

μ1 ¼ 1:01;

N 01 ¼ N 02 ¼ 0:

Our purpose here is to design a set of mode-dependent dynamic state feedback controllers and find the admissible MDADT switching signals such that the resulting closed-loop system is finite-time bounded. In order to illustrate the merits of the proposed MDADT switching, we will present the design results of switching signals for the systems with the ADT switching for the sake of comparison. By choosing parameters appropriately and different design approaches, the results with different switching schemes are listed in Table 1. It can be seen from Table 1 that the minimal MDADT are reduced to τna1 ¼ 0:8501, n τa2 ¼ 1:6919, for given λ ¼ λ1 ¼ λ2 ¼ 0:01. One special case of the MDADT switching is

H. Liu, X. Zhao / Journal of the Franklin Institute 351 (2014) 1301–1315

1311

Table 1 Results for the system under different switching schemes. Switching schemes

ADT switching

MDADT switching

Criteria for switching design Switching signals

Theorem 1 in [19] τna ¼ 1:6919 ðμ ¼ 1:02; λ r0:01Þ

Theorem 1 in this paper τna1 ¼ 0:8501; τna2 ¼ 1:6919 ðλ1 ¼ λ2 ¼ 0:01Þ ðμ1 ¼ 1:01; μ2 ¼ 1:02Þ

Fig. 1. State response x(t).

τna ¼ τna1 ¼ τna2 ¼ 1:6919 by setting μ ¼ μ1 ¼ μ2 ¼ 1:02, which is the ADT switching, that is, the MDADT switching is more general than the ADT switching. Then, according to Theorem 1, under the following dynamic state feedback controllers (5) with order nc ¼ 3, where

1312

H. Liu, X. Zhao / Journal of the Franklin Institute 351 (2014) 1301–1315

and for any switching signal sðtÞ with MDADT τa1 ¼ 1, τa2 ¼ 2, the switched system (39) is finite-time bounded with respect to ð0:3; 2; 0:3; 20; I; sÞ. The simulation results are shown in Figs. 1–3. Applying the obtained mode-dependent dynamic state feedback controllers above, under the MDADT switching, we can get the state response for closed-loop system x(t) and dynamic state feedback controller xc(t), respectively, as shown in Figs. 1 and 2. It can be seen from Fig. 3 that xT ðtÞRxðtÞ52, 8 t A ½0; 20, i.e., the switched system (39) is finite-time bounded. Example 2. Consider the finite-time H 1 control problem for switched systems (39) and zðtÞ ¼ CsðtÞ xðtÞ þ H sðtÞ ωðtÞ

ð40Þ

Fig. 2. State response xc(t).

Fig. 3. The history of xT ðtÞRxðtÞ.

H. Liu, X. Zhao / Journal of the Franklin Institute 351 (2014) 1301–1315

where C1 ¼



0:3 0:4



 ;

C2 ¼

0:5  0:7

 ;

H 1 ¼  0:3;

1313

H 2 ¼ 0:2:

The corresponding parameters are chosen as c2 ¼ 1, λ1 ¼ λ2 ¼ 0:1. Other parameters are given in Example 1. Our purpose here is to design a set of mode-dependent dynamic state feedback controllers and find the admissible MDADT switching signals such that the resulting closed-loop system is finite-time bounded with a prescribed H 1 performance γ~ . Then, according to Theorem 2, under the following mode-dependent dynamic state feedback controllers (5) with order nc ¼ 3, where

Fig. 4. The controlled output z(t).

1314

H. Liu, X. Zhao / Journal of the Franklin Institute 351 (2014) 1301–1315

Fig. 5. The history of xT ðtÞRxðtÞ.

and for any switching signal sðtÞ with MDADT τa1 4τna1 ¼ 0:9520, τa2 4τna2 ¼ 1:8947, the switched systems (38) and (39) are finite-time bounded with H 1 performance γ~ ¼ 7:3891 with respect to ð0; 20; 0:3; 1; 7:3891; I; sÞ. The simulation results are shown in Figs. 4 and 5. The controlled output response of the systems (40) under the MDADT switching signal s with τa1 ¼ 1, τa2 ¼ 2 is given in Fig. 4. As shown in Fig. 3, xT ðtÞRxðtÞ is less than 1, 8 t A ½0; 20, therefore, the resulting closed-loop system is finite-time bounded with H 1 performance γ~ ¼ 7:3891. 5. Conclusions The finite-time H 1 control problem for a class of switched linear systems with modedependent ADT switching is investigated in this paper. The MDADT switching used above, which can improve the H 1 performance index potentially, is less strict than the ADT switching. Based on the MDADT approach, the weighted H 1 performance criterion for a class of switched linear systems is derived. Moreover, a set of mode-dependent dynamic state feedback controllers and the minimal MDADT for admissible switching signals are designed. Finally, two numerical examples are given to demonstrate the validity of the proposed results. Acknowledgement This work was partially supported by the National Natural Science Foundation of China (61203123), the Fundamental Research Funds for the Central Universities, China (11CX04044A), and the Shandong Provincial Natural Science Foundation, China (ZR2012FQ019).

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