Novel LMI-based stability and stabilization analysis on impulsive switched system with time delays

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Journal of the Franklin Institute 349 (2012) 2650–2663 www.elsevier.com/locate/jfranklin

Novel LMI-based stability and stabilization analysis on impulsive switched system with time delays Bo Wanga,b, Peng Shic,d,n, Jun Wanga, Yongduan Songe a

School of Electrical and Information Engineering, Xihua University, Chengdu 610096, China School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, China c Department of Computing and Mathematical Sciences, University of Glamorgan, Pontypridd, CF37 1DL, United Kingdom d School of Engineering and Science, Victoria University, Melbourne, Vic 8001, Australia e School of Automation, Chongqing University, Chongqing 400044, China

b

Received 9 February 2012; received in revised form 19 April 2012; accepted 11 June 2012 Available online 19 June 2012

Abstract This paper is concerned with the problems of stability and stabilization of impulsive switched system with time delays. First, a novel Razumikhin function is constructed; then based on LMI approach and optimization techniques, we derive our theoretical result with good properties; subsequently, we extend our results to the case with perturbations. Finally, numerical simulations are provided to demonstrate the effectiveness of the proposed techniques. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Recently impulsive switched method gets popular and has successful applications in a wide variety of areas, such as chaotic systems [1], chemical reactor system [2], secure communication [3], neural networks [4], ecosystems system [5], disease control [6], finance systems [7], complex networks [8], and so on. Impulsive dynamical systems are considered as a type of hybrid systems consisting of three elements: a continuous differential equation, which governs the continuous evolution of the system between impulses; a difference

n Corresponding author at: School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, 5005, Australia. E-mail address: [email protected] (P. Shi).

0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.06.005

B. Wang et al. / Journal of the Franklin Institute 349 (2012) 2650–2663

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equation, which governs the way the system states are changed at impulse times; and an impulsive law for determining when the impulses occur. For stability and stabilization analysis of impulsive switched systems, piecewise Lyapunov functions are powerful approaches, usually can be classified into two types: piecewise Lyapunov–Krasovskii function and piecewise Lyapunov–Razumikhin function. Generally the former requires more strict conditions compared to the latter. In this paper we will employ Lyapunov–Krasovskii function method to study the stability and stabilization of impulsive switched system. Delay and disturbance are natural phenomena and exist in the engineering field widely. Up to now, many works have been done, see [9–25] and the references therein. Generally, impulsive switched system can be divided into two categories: non-delay system and delay system. For non-delay system, many investigations have been done. In [26], by using switched Lyapunov functions, the new general criteria of exponential stability and asymptotic stability with arbitrary and conditioned impulsive switching for a new class of hybrid impulsive and switching models have been established; in [27], the results on robust stability for this class of impulsive switched systems are obtained and sufficient conditions for the existence of a guaranteed cost control law are also given; in [28], the new fundamental properties are derived and several sufficient conditions are presented on the exponential stability and robust stabilization for singular impulsive closed-loop systems; in [29], through the receding horizon strategy, the problem on state feedback stabilization for a class of linear impulsive systems featuring arbitrarily-paced impulse times is investigated. However, for delay system, there are many problems need to be solved. For example, LMI method is a well-known way to study stability and stabilization of non-delay impulsive switched system. However, for impulsive switched system with delay, the LMI method will not be applicable unless some extra conditions are attached. For instance, some assumptions are imposed on system variable [30,31] and some limits are added on matrix parameter [32,33]. These not only can result in the decrease of the applied scope, but also may produce conservatism problems. Therefore, how to utilize the LMI method to analyze the stability and stabilization for delay impulsive switched system without extra constrains becomes a challenging topic. In this paper, a new Lyapunov–Razumikhin function will be constructed to solve this problem. Finally some numerical examples will be given to validate the correctness of the theoretic results given. Notations used in this paper are fairly standard. Let Rn be the n-dimensional Euclidean space, Rn  m the set of n  m real matrix, n the symmetric part in a matrix, I the identity matrix with appropriate dimensions, diagf  g the diagonal matrix. By A40 we mean that A is a real symmetric positive definitive matrix. Let Ln,h ¼ L([h,0], Rn) denote the Banach space of continuous functions mapping the interval [h, 0] into Rn with the topology of uniform convergence. 2. System description and preliminaries Consider the following class of impulsive switched time delay systems 8 _ > < xðtÞ ¼ AxðtÞ þ BxðthÞ tatk Dx ¼ Bk xðt Þ t ¼ tk > : xðt þ yÞ ¼ jðyÞ y 2 ½h 0 0

ð1Þ

B. Wang et al. / Journal of the Franklin Institute 349 (2012) 2650–2663

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where x(t)ARn is the state vector, hZ0 is the time-varying delay, j(y)ALn,h is the initial condition. System (1) can be represented as follow 8 _ ¼ yðtÞ xðtÞ > > > > < yðtÞ ¼ AxðtÞ þ BxðthÞ tatk ð2Þ Dx ¼ Bk xðt Þ t ¼ tk > > >   > : xðt0 þ yÞ ¼ jðyÞ y 2 h 0 Let us recall the following results and definitions which will be used throughout the paper. Lemma 1. [34]. For any constant matrices E, G and F with appropriate dimensions, FTFrkI, k is a positive scalar, then k 2xT EFGyrcxEE T x þ yT G T Gy c where c is a positive scalar, xARn and yARn.

ð3Þ

Lemma 2. [35]. For any positive definite matrix FARn  n, a positive scalarg, the vector function w:[0, g]-Rn such that the integrations concerned are well defined, then Z

g

T Z wðsÞds F

0

g

 Z wðsÞds rg

0

g

wT ðsÞFwðsÞds

ð4Þ

0

Definition 1. For differential dynamic system, the dynamics characteristics of state variable can be described as x(t)rrj0eat, r41, a is the system infimum of exponential power. 3. Main results In this section, we consider the problems of stability and stabilization of impulsive switched system based on LMI method and optimization technologies. Theorem 1. For a40, if there exist d41,ck ¼ lmax ½ðI þ Bk ÞT ðI þ Bk Þ, satisfy Inðck dÞ þ aDk r0,

k ¼ 1,2,   

ð5Þ

where parameter a is determined by the following optimization problem minfag

ð6Þ

P,P2 ,P3 ,Q

s.t.

2

S1,1

6 6 n S¼6 6 n 4 n

S1,2 S2,2

S1,3 S2,3

n

S3,3

n

n

3 S1,4 7 S2,4 7 7o0 S3,4 7 5 S4,4

B. Wang et al. / Journal of the Franklin Institute 349 (2012) 2650–2663

S1,1 ¼ P2 A þ AT P2 T þ l 2 eaP þ S1,2 ¼ PP2 þ AT P3 T þ

2653

lðm1Þ PþQ Dk m

l þ l=m 2

S2,2 ¼ P3 P3 T S1,3 ¼ P2 B S2,3 ¼ P3 B S3,3 ¼ Qeah ll=m S2,4 ¼ 2 S4,4 ¼ eI For ar0, if there exist d41,ck ¼ lmax ½ðI þ Bk ÞT ðI þ Bk Þ,b ¼ a, satisfy Inðck dÞbDk r0,

k ¼ 1,2, . . .

ð7Þ

where parameter b is determined by the following optimization problem maxfbg

ð8Þ

P,P2 ,P3 ,Q

s.t.

2

S1,1

6 6 n S¼6 6 n 4 n

S1,2

S1,3

S2,2

S2,3

n

S3,3

n

n

S1,4

3

7 S2,4 7 7o0 S3,4 7 5 S4,4

S1,1 ¼ P2 A þ AT P2 T þ l 2 e þ bP þ S1,2 ¼ PP2 þ AT P3 T þ

lðm1Þ PþQ Dk m

l þ l=m 2

S2,2 ¼ P3 P3 T S1,3 ¼ P2 B S2,3 ¼ P3 B S3,3 ¼ Qebh ll=m S2,4 ¼ 2 S4,4 ¼ eI Then, the impulse switched system (1) will be stable. Proof. For a40, when tA[tk, tkþ1), we chose the following Razumikhin Lyapunov functional V ðtÞ ¼ V1 ðtÞ þ V2 ðtÞ ¼ gðtÞxT ðtÞPxðtÞ þ

Z

t

xT ðsÞeaðstÞ QxðsÞds

th

where gðtÞ ¼

lðm1Þ l ðttk Þ þ , ðtkþ1 tk Þm m

t 2 ½tk ,tkþ1 Þ:

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Let tkþ1tk ¼ Dk, we have _ ¼ gðtÞ

lðm1Þ , Dk m

t 2 ½tk ,tkþ1 Þ

Then the derivative of V(t) along the trajectory of system (2) can be given by V_ ðtÞ ¼ V_ 1 ðtÞ þ V_ 2 ðtÞ lðm1Þ T x ðtÞPxðtÞ þ 2gðtÞxT ðtÞPyðtÞ þ 2ðxT ðtÞP2 þ yT ðtÞP3 ÞðyðtÞ ¼ Dk m þAxðtÞ þ BxðttÞÞ þ xT ðtÞQxðtÞxT ðthÞeah QxðthÞ þ aV2 ðtÞ Therefore, we have V_ ðtÞrxT Sx where T x ¼ ½ x ðtÞ

yT ðtÞ xT ðttÞ T

According to (7), we have V_ ðtÞraV ðtÞ Then ðV ðtÞeat Þ0 ¼ ðV_ ðtÞaV ðtÞÞeat r0 Integrating above inequality from tk to t, we get V ðtÞeat rV ðtk Þeatk Hence V ðtÞrV ðtk Þeaðttk Þ We can derive V1 ðtÞrV ðtk Þeaðttk Þ ¼ mV1 ðtk Þeaðttk Þ where m¼

V ðtk Þ lmax ðPÞ þ hlmax ðQÞ ¼ 41 V1 ðtk Þ lmin ðPÞ

At switching time t¼ tk, we have V1 ðtk Þ ¼ gðtk ÞxT ðtk ÞPxðtk Þr

l T  1  T  ck x ðtk Þck Pxðt gðt Þx ðtk Þck Pxðt V1 ðt kÞ¼ kÞ¼ kÞ m m k m

Then we can conclude V1 ðtÞrck V1 ðtk  Þeaðttk Þ rc1 c2    ck V1 ðt0 Þexpðaðtt0 ÞÞ ¼ ½c1 expðaD1 Þ½c2 expðaD2 Þ    ½ck expðaDk ÞV1 ðt0 Þexpðaðttk ÞÞ According to (5) lmin ðPÞjjxjj2 rV1 ðtÞrV1 ðt0 Þ

1 expðaðttk ÞÞ dk

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Considering V1 ðt0 þ Þrlmax ðPÞj0 2 , the following inequality is true jjxjjrG1 d k=2 where

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðaDkþ1 Þlmax ðPÞ j0 G1 ¼ lmin ðPÞ

Then the impulsive switched system (1) is stable. For ar0, when tA[tk, tkþ1), we chose the following Razumikhin Lyapunov functional Z t T xT ðsÞebðstÞ QxðsÞds V ðtÞ ¼ V1 ðtÞ þ V2 ðtÞ ¼ gðtÞx ðtÞPxðtÞ þ th

The derivative of V(t) along the trajectory of system (2) is given by V_ ðtÞ ¼ V_ 1 ðtÞ þ V_ 2 ðtÞ: where V_ ðtÞ ¼ V_ 1 ðtÞ þ V_ 2 ðtÞ lðm1Þ T x ðtÞPxðtÞ þ 2gðtÞxT ðtÞPyðtÞ þ 2ðxT ðtÞP2 ¼ Dk m þyT ðtÞP3 ÞðyðtÞ þ AxðtÞ þ BxðttÞÞ þ xT ðtÞQxðtÞxT ðthÞebh QxðthÞbV2 ðtÞ Therefore, we have V_ ðtÞrxT Sx where T T T x ¼ ½ x ðtÞ y ðtÞ x ðttÞ T

According to (7) V_ ðtÞrbV ðtÞ We have ðV ðtÞebt Þ0 ¼ ðV_ ðtÞ þ bV ðtÞÞebt r0 Integrating above inequality from tk to t, we get V ðtÞebt rV ðtk Þebtk Then V ðtÞrV ðtk Þebðttk Þ We have V1 ðtÞrV ðtk Þebðttk Þ ¼ mV1 ðtk Þebðttk Þ where m¼

V ðtk Þ llmax ðPÞ þ hlmax ðQÞ ¼ 41 V1 ðtk Þ llmin ðPÞ

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Like above, it can be concluded aðttk Þ V1 ðtÞrck V1 ðt rc1 c2    ck V1 ðt0 Þexpðbðtt0 ÞÞ k Þe

¼ ½c1 expðbD1 Þ½c2 expðbD2 Þ    ½ck expðbDk ÞV1 ðt0 Þexpðbðttk ÞÞ According to (8) lmin ðPÞjjxjj2 rV1 ðtÞrV1 ðtþ 0Þ

1 dk

Considering V1 ðt0 þ Þrlmax ðPÞj0 2 , the following inequality is hold jjxjjrG2 d k=2 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lmax ðPÞ j 41 G2 ¼ lmin ðPÞ 0 The above implies that the impulsive switched system (1) is stable, and the proof is completed.

4. Robust stability results In this section, we consider the following uncertain impulsive switched system 8 _ ¼ ðA þ DAÞxðtÞ þ ðB þ DBÞxðthÞ tatk xðtÞ > < Dx ¼ Bk xðt Þ t ¼ tk > : xðt þ yÞ ¼ jðyÞ y 2 ½ h 0  0

ð9Þ

where DAðtÞ ¼ E1 F1 ðtÞG1 DBðtÞ ¼ E2 F2 ðtÞG2 F1 T ðtÞF1 ðtÞrI F2 T ðtÞF2 ðtÞrI where E1, E2, G1, G2 are known constant real matrices with appropriate dimensions. Along the same line as that in the proof of Theorem 1, we have the following result. Theorem 2. For a40, if there exist d41,ck ¼ lmax ½ðI þ Bk ÞT ðI þ Bk Þ, satisfy Inðck dÞ þ aDk r0,

k ¼ 1,2, . . .

ð10Þ

where parameter a is determined by following optimization problem minfag P,P2 ,P3 ,Q,e1 ,e2

ð11Þ

B. Wang et al. / Journal of the Franklin Institute 349 (2012) 2650–2663

2 6 6 6 6 6 S¼6 6 6 6 4

n

S1,2 S2,2

S1,3 S2,3

S1,4 S2,4

S1,5 S2,5

n

n

S3,3

n

n

n

S3,4 S4,4

0 0

n

n

n

n

S5,5

n

n

n

n

n

S1,1

S1,1 ¼ P2 A þ AT P2 T þ l 2 eaP þ S1,2 ¼ PP2 þ AT P3 T þ

2657

3 S1,6 7 S2,6 7 7 0 7 7 7o0 0 7 7 0 7 5 S6,6

lðm1Þ P þ Qe1 G1 T G1 Dk m

l þ l=m 2

S2,2 ¼ P3 P3 T S1,3 ¼ P2 B S2,3 ¼ P3 B S3,3 ¼ Qeah þ e2 G2T G2 S2,4 ¼

ll=m 2

S4,4 ¼ eI S1,5 ¼ P2 E1 S2,5 ¼ P3 E1 S5,5 ¼ e1 I S1,6 ¼ P2 E2 S2,6 ¼ P3 E2 S6,6 ¼ e2 I For ar0, if there exist d41,b ¼ a,ck ¼ lmax ½ðI þ Bk ÞT ðI þ Bk Þ, satisfy Inðck dÞbDk r0,

k ¼ 1,2, . . .

ð12Þ

where parameter b is determined by the following optimization problem maxfbg

ð13Þ

P,P2 ,P3 ,Q,e1 ,e2

s.t.

2 6 6 6 6 6 S¼6 6 6 6 4

S1,1 n

S1,2 S2,2

S1,3 S2,3

S1,4 S2,4

S1,5 S2,5

n

n

S3,3

S3,4

0

n

n

n

S4,4

n

n

n

n

0 S5,5

n

n

n

n

n

S1,1 ¼ P2 A þ AT P2 T þ l 2 e þ bP þ S1,2 ¼ PP2 þ AT P3 T þ S2,2 ¼ P3 P3 T

l þ l=m 2

3 S1,6 7 S2,6 7 7 0 7 7 7o0 0 7 7 0 7 5 S6,6 lðm1Þ PþQ Dk m

2658

B. Wang et al. / Journal of the Franklin Institute 349 (2012) 2650–2663

S1,3 ¼ P2 B S2,3 ¼ P3 B S3,3 ¼ Qebh þ e2 G2T G2 S2,4 ¼

ll=m 2

S4,4 ¼ eI S1,5 ¼ P2 E1 S2,5 ¼ P3 E1 S5,5 ¼ e1 I S1,6 ¼ P2 E2 S2,6 ¼ P3 E2 S6,6 ¼ e2 I Then, the uncertain impulsive switched system (9) is stable. 5. Example and simulation In this section, we present some numerical examples to validate the effectiveness of the theoretical results derived. First we consider the impulsive switched system (1) with following parameters h ¼ 1,Dk ¼ D,Bk ¼ bI,m ¼ 1:2,l ¼ 1:0,d ¼ 2:0       1 0:5 0:4 0 0:2 0 A¼ ,B ¼ ,C ¼ 0 1 0:3 0:4 0:1 0:2 Now we study the relationship of some important parameters. Table 1 show the relationship between parameters a and D. Table 2 show the relationship of parameters a, b and D. Remark 1. From Table 1, we can see that the parameter a decreases with the augment of D, which means D has infection to the system dynamics. From Table 2, we can see that the stable minimal upper bound of parameter b decreases with the augment of D when a40, Table 1 The relationship between parameter a and D D

0.1

0.2

0.5

1.0

a

3.9630

3.1645

2.6961

2.5422

Table 2 The stable least upper bound of parameter b with different D and a. a\D

0.1

0.2

0.5

1.0

2.0 1.0 0.0 2.0 5.0

0.2185 0.2566 0.2929 0.3602 0.4493

0.1363 0.2185 0.2929 0.4211 0.5711

0.1658 0.0921 0.2929 0.5711 0.7974

0.9221 0.1658 0.2929 0.7399 0.9420

B. Wang et al. / Journal of the Franklin Institute 349 (2012) 2650–2663

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increases with the reduction of D when ao0, in addition, we can also see it decreases with the augment of a when D is given.

Then we carry on the numerical simulation to study the dynamics of impulsive switched system (1). The numerical simulation is with the initial state jðyÞ ¼ ½0:5; 0:5T , y 2 ð1,0Þ, parameters D ¼ 0.2, b¼ 0.5 and simulation step 0.001 second. Fig. 1 depicts the time response of state variable of system (1) without impulse switching. We can see the system is unstable for its state variable diverges with the lapse of time. Fig. 2 depicts the time response of state 200

x

1

100 0 −100 −1

0

1

2 t/s

3

4

5

0

1

2 t/s

3

4

5

200

x

2

150 100 50 0 −1

Fig. 1. Time response of state variable of the system (1) without impulse switching. 0.1 0

x1

−0.1 −0.2 −0.3 −0.4 −0.5 −1

0

1

2

3

4

5

3

4

5

t/s 0.5

x2

0.4 0.3 0.2 0.1 0 −1

0

1

2 t/s

Fig. 2. Time response of state variable of the system (1) with impulse switching.

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variable of system (1) with impulse switching. That means the unstable system can be stabilized through impulse switching based on Theorem 1. Next, we consider the uncertain impulsive switched system (9) with following parameters h ¼ 1,Dk ¼ D,Bk ¼ bI,m ¼ 1:2,l ¼ 1:0,d ¼ 2:0 

1 A¼ 0:5

  0:5 0:4 ,B ¼ 2 0:3

DAðtÞ ¼ E1 F1 ðtÞG1

8tZ0

DBðtÞ ¼ E2 F2 ðtÞG2

8tZ0

  0 0:2 ,C ¼ 0:4 0:1

G1 ¼ E1 ,G2 ¼ E2    0:3 0:1 0:3 E1 ¼ ,E2 ¼ 0:3 0:2 0:2

0 0:2

  0:1 0:9 ,F1 ¼ 0:3 0



   0 1 0 ,F2 ¼ 1 0 0:8

Table 3 The relationship between parameter a and D. D

0.1

0.2

0.5

1.0

a(without perturbation) a(with perturbation)

0.0110 0.2964

0.3684 0.0437

0.5770 0.2401

0.6455 0.3042

0.2

x1

0 −0.2 −0.4 −0.6 −1

0

1

2

3

4

5

3

4

5

t/s 0.6

x2

0.4 0.2 0 −0.2 −1

0

1

2 t/s

Fig. 3. Time response of state variable of the uncertain system (9) without impulse switching.

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0.1 0

x1

−0.1 −0.2 −0.3 −0.4 −0.5 −1

0

1

2

3

4

5

3

4

5

t/s 0.5

x2

0.4 0.3 0.2 0.1 0 −1

0

1

2 t/s

Fig. 4. Time response of state variable of the uncertain system (9) with impulse switching.

Remark 2. From Table 3, not only we can see D has infection to the dynamics of systems, but also the existence of perturbation will result in the decreasing of system stability. Finally, we carry on the numerical simulation to analyze the dynamics of the uncertain impulsive switched system (9). The numerical simulation is with jðyÞ ¼ ½0:5; 0:5T , y 2 ð1,0Þ, D ¼ 0.2, b ¼ 0.8 and simulation step 0.001 s. Fig. 3 depicts the time response of state variable of system (9) without impulse switching. We can see the system is stable, however spends 4 s converging to the zeros. Fig. 4 depicts the time response of state variable of system (9) with impulse switching. It can be seen the system converge to zeros within 1 s through impulse switching based on Theorem 2. 6. Conclusions For stability and stabilization analysis of impulsive switched system with time delay, LMI method is not mature compared to other methods so far, however still gets increasing attentions for its unmatchable solving and optimizing power. In this paper, a novel Razumikhin function has been constructed to make better use of LMI method to carry on the stability and stabilization analysis of delay impulsive switched system. Subsequent examples show the effectiveness and potential of the proposed techniques. Acknowledgements This research has been supported by key projects of Xihua University (Z1120946), National Key Basic Research Program, China (2012CB215202), the 111 Project (B12018) and the National Natural Science Foundation of China (61170030, 61174058).

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