Sliding mode control for stochastic Markovian jumping systems subject to successive packet losses

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Journal of the Franklin Institute 351 (2014) 2169–2184 www.elsevier.com/locate/jfranklin

Sliding mode control for stochastic Markovian jumping systems subject to successive packet losses Bei Chen, Yugang Niun, Yuanyuan Zou Key Laboratory of Advanced Control and Optimization for Chemical Processes, East China University of Science and Technology, Ministry of Education, Shanghai 200237, PR China Received 5 April 2012; received in revised form 8 October 2012; accepted 9 October 2012 Available online 23 October 2012

Abstract In this work, the problem of sliding mode control is considered for a class of Markovian jumping systems. The packet dropout may happen when the state information is transmitted from the sensor to the controller. By means of an estimator for lost signals, an integral-like sliding function is constructed. And then, a sliding mode controller involving in dropout probability is designed such that the effect of packet losses can be effectively attenuated. Besides, the analysis on both the stability of sliding mode dynamics and the reachability of sliding surface are made. Finally, the numerical simulation results are given. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Markovian jumping systems (MJSs) have received considerable attention in the past decades, since many real-world systems with abrupt variations in their structures can be effectively represented by MJSs, where the abrupt variations may happen due to random failures or repairs of components, changing of subsystem interconnections, abrupt variations in the operating point, etc. More importantly, different from the traditional single controller, the control strategy designed for MJSs is based on the idea of switching to improve the performance of closed-loop system. Up to now, many results on the stability and stabilization of MJSs have been obtained, see [1–5] and the reference therein.

n

Corresponding author. E-mail address: [email protected] (Y. Niu).

0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.10.004

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Sliding mode control (SMC) has been recognized as an effective robust control approach due to its excellent advantage of strong robustness against model uncertainties, parameter variations, and external disturbances [6,7]. Over the past few decades, a variety of SMC methods have been developed for various types of dynamical systems [8–10]. More recently, the application of SMC has been also extended to MJSs in [11–15]. Among them, the SMC problem of MJSs was first considered by Shi et al. [11]. By using a transformation, a set of linear sliding surfaces and a reaching motion controller were designed in [11], respectively. Niu et al. [12] further extended SMC method to Itoˆ stochastic systems with Markovian switching, and both the reachability and the stability of sliding mode dynamics were established. More recently, the problem of SMC for MJSs with unmeasured states were also considered, respectively, by Wu et al. [13] and Zhang et al. [15]. These above works have shown the effectiveness of SMC method for MJSs. Nevertheless, it is worthy noting that all of the aforementioned works were considered under the assumption that system signals could be successfully transmitted to the controller or actuator, i.e., there did not happen packet losses. Actually, this case might only be true for those traditional point-to-point controlled systems. As is well known, with the rapid advances in communication network, more and more system information is transmitted via communication networks. The insertion of communication networks in feedback control loops not only brings great advantages, e.g., low cost, reduced weight and power, etc., but also yields some detrimental phenomena. Among them, the data packet dropout, termed as data missing [16,17], is a potential source of instability and poor performance for the controlled systems. Hence, the problem of packet dropout has recently received increasing attention, and many results have been reported in, to mention a few [18–25] and the reference therein. However, to the authors’ best knowledge, there exist little work reported on the design problem of SMC for MJSs subject to packet losses. Especially, due to the complexity of the structure of MJSs with stochastic perturbation, these existing methods on SMC without packet dropout cannot be trivially extended to such class of systems. Motivated by the above discussions, the problem of SMC is investigated in this work for a class of MJSs with stochastic perturbation. It is assumed that the transmitted information may be lost, and the probability distribution of packet dropout obeys Bernoulli process. In this work, an estimation method is proposed to cope with the packet losses, based on which an integral-like sliding surface is chosen, and a dropout-probabilitydependent SMC law is designed. Moreover, by means of the stochastic Lyapunov method dependent on sliding variable and system state, the analysis on the reachability is made, simultaneously, with the stability of system states, and some sufficient conditions are derived. It should be pointed out that the main contribution in this paper is to provide a design method of sliding mode control for Itoˆ stochastic Markovian jumping systems subject to successive packet losses. In fact, it is difficult to investigate this issue due to the effect of both the Markovian switching and the data missing. Thus, some existing works cannot be simply extended to the systems under consideration in this work. Firstly, in the design of sliding mode control, what is the connection among different sliding functions under Markovian switching for SMC systems? Remark 2 in this work gives a deeper investigation on the problem. Secondly, it is highly desirable to synthesize a SMC law so as to ensure the attraction of the switching surface when the switching surface changes from one to another under Markovian switching. Thirdly, the effect of packet losses must be considered in

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design of sliding mode control, since the existence of data packet dropout may make the controlled system unstable or poor performance. Finally, it is worthy of noting that the case in this work is different from that in [26], which was synthesized under the assumption that there did not exist successive packet dropouts. Notation. Throughout this paper, J  J denotes the Euclidean norm of a vector (sum of absolute values) or its induced matrix norm. For a real matrix, M40 means that M is symmetric and positive definite, and I is used to represent an identity matrix of appropriate dimensions. In symmetric block matrices, n is used as an ellipsis for terms that are induced by symmetry. ðO,F ,PÞ is a probability space with O the sample space, and F the s-algebra of subsets of the sample space, and P is the probability measure. Matrices, if not explicitly stated, are assumed to have compatible dimensions. 2. Problem formulation Given the probability space ðO,F ,PÞ, consider the discrete-time MJSs with stochastic perturbation: xðk þ 1Þ ¼ Aðrk ÞxðkÞ þ Bðrk Þuðk,rk Þ þ f ðxðkÞ,kÞÞ þ ðCðrk Þ þ DCðrk ÞÞxðkÞwðkÞ, n

ð1Þ

m

where xðkÞ 2 R is the system state, uðk,rk Þ 2 R is the control input, w(k) is a scalar Wiener process on a probability space ðO,F ,PÞ relative to an increasing family ðF k Þk2N of s-algebra F k  F generated by ðwðkÞÞk2N with N the set of natural numbers, and is assumed to satisfy EfwðkÞg ¼ 0, EfwðkÞ2 g ¼ 1, and EfwðiÞwðjÞg ¼ 0 for iaj. The unknown nonlinear function f ðxðkÞ,kÞ is the external disturbance with known constant bound. The Markov chain frk ,kZ0g, taking values in a finite set S9f1,2, . . . ,Ng, and governs the switching among the different system modes, whose mode transition probabilities are given as Pfrkþ1 ¼ jjrk ¼ ig ¼ pij , ð2Þ PN where pij 40, 8i,j 2 S, and j ¼ 1 pij ¼ 1, and the transition probability matrix is as follows: 2 3 p11 p12    p1N 6 p21 p22    p2N 7 6 7 P¼6 ð3Þ 7: ^ & ^ 5 4 ^ pN1 pN2    pNN For each rk ¼ i 2 S, the system matrices, Aðrk Þ, Bðrk Þ, Cðrk Þ, and DCðrk Þ, of the ith mode are denoted by Ai ,Bi ,Ci and DCi , respectively. Then, the system (1) can be rewritten as xðk þ 1Þ ¼ Ai xðkÞ þ Bi ðuðk,iÞ þ f ðxðkÞ,kÞÞ þ ðCi þ DCi ÞxðkÞwðkÞ:

ð4Þ

Here, Ai, Bi, and Ci are real and known. Without loss of generality, it is assumed that the pair ðAi ,Bi Þ is controllable and the input matrix Bi has full column rank. The unknown matrix DCi represents uncertainty satisfying DCi ¼ Ei Fi ðkÞHi , where Ei and Hi are known real constant matrices, and Fi(k) is an unknown time-varying matrix satisfying FiT ðkÞFi ðkÞrI,8t. In this work, it is assumed that the system states may be lost when transmitted from sensor to the controller, and the probability distribution of the packet dropout obeys Bernoulli process y 2 R as follows: Pfy ¼ 1g ¼ y,

Pfy ¼ 0g ¼ 1y,

ð5Þ

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where 0ryo1 represents the probability that any data packet will be lost. Moreover, it is assumed that the packet dropouts may happen successively. In order to compensate the lost packet, the following method will be utilized in this work: ( xðkÞ if data packet is received, ð6Þ xs ðkÞ ¼ xs ðk1Þ if data packet is lost, which may be further rewritten as xs ðkÞ ¼ ð1yÞxðkÞ þ yxs ðk1Þ,

ð7Þ

which is termed as compensator in this work. Remark 1. It is worthy of noting that the case in this work is different from that in [26], which was synthesized under the assumption that the dropout of only single packet may happen, i.e., there did not exist successive packet dropouts. Besides, due to the structure characteristic of the MJSs, the approach proposed in [26] cannot be directly utilized to deal with the case in this work. Now, the objective of this work is to design a SMC law for the MJSs (4) with stochastic perturbation such that the resultant closed-loop system is exponentially mean-square stable despite packet losses and external disturbance. To this end, a sliding surface will firstly be chosen and the stability of ideal sliding motion will be analyzed. Furthermore, the desired SMC law will be designed such that both the reachability and the stability of sliding mode dynamics can be ensured. 3. Ideal sliding motion By taking the probability of packet losses into account, an integral-like sliding function is constructed as follows: sðk,iÞ ¼ ð1yÞGi xðkÞ þ yGi Ai xs ðk2Þ,

ð8Þ

where y is the dropout probability in Eq. (5) and the matrix Gi will be chosen later to ensure that Gi Bi is nonsingular. It can be shown from the matrix P theory that the nonsingularity of Gi Bi can be ensured by choosing Gi ¼ BTi P i with P i ¼ N j ¼ 1 pij Pj , since Bi is assumed to be of full column rank. The matrices Pj 40 (j 2 S) will be determined in Theorem 1 later. Remark 2. It is seen from Eq. (8) that the sliding function sðk,iÞ (i 2 S) depends on a set of specified matrices Pj (j 2 S), the solutions of some coupled LMIs concerning the transition probability of modes. This implies that the effect of Markovian switching from one mode to another is reflected in sðk,iÞ just via the matrices Pj . Hence, in the resultant SMC systems with Markovian switching, the connections among sliding surfaces corresponding to every mode are also established via matrices Pj. From Eqs. (4) and (8), we have sðk þ 1,iÞ ¼ ð1yÞGi Ai xðkÞ þ ð1yÞGi Bi uðk,iÞ þ ð1yÞGi Bi f ðxðkÞ,kÞ þyGi Ai xs ðk1Þ þ ð1yÞGi ðCi þ DCi ÞxðkÞwðkÞ:

ð9Þ

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It is noted that the ideal sliding motion satisfies sðk þ 1,iÞ ¼ sðk,iÞ ¼ 0,

for kZkn ,

ð10Þ

where kn is a positive constant. That is, after the state trajectory attains the sliding surface sðk,iÞ ¼ 0 at the step kn , it will stay there in the subsequence time. Thus, the equivalent control law for the ideal sliding motion may be obtained from Eqs. (4), (9) and (10) that ueq ðkÞ ¼ ðGi Bi Þ1 Gi Ai xðkÞf ðxðkÞ,kÞfðGi Bi Þ1 Gi Ai xs ðk1Þ ðGi Bi Þ1 Gi ðCi þ DCi ÞxðkÞwðkÞ,

ð11Þ

with f ¼ y=ð1yÞ, which substituted into Eq. (4) may yield the ideal sliding mode dynamics in the sliding surface sðk,iÞ ¼ 0 as follows: xðk þ 1Þ ¼ Ai xðkÞBi ðGi Bi Þ1 Gi Ai xðkÞBi ðGi Bi Þ1 Gi ðCi þ DCi ÞxðkÞwðkÞ fBi ðGi Bi Þ1 Gi Ai xs ðk1Þ þ ðCi þ DCi ÞxðkÞwðkÞ ¼ ½Ai Bi ðGi Bi Þ1 Gi Ai xðkÞfBi ðGi Bi Þ1 Gi Ai xs ðk1Þ þ½IBi ðGi Bi Þ1 Gi ðCi þ DCi ÞxðkÞwðkÞ:

ð12Þ

Define ZðkÞ ¼ ½xT ðkÞ xTs ðk1ÞT . Then, it follows from Eqs. (7) and (12) that the augment dynamic system is given as ~ i ÞZðkÞ þ C i ZðkÞwðtÞ, Zðk þ 1Þ ¼ ðA i þ A where

"

Ai ¼ " ~i¼ A

Ai Bi ðGi Bi Þ1 Gi Ai

fBi ðGi Bi Þ1 Gi Ai

ð1yÞI

yI

0

0

ðyyÞI

ðyyÞI

"

# ,

ð13Þ # ,

½IBi ðGi Bi Þ1 Gi ðCi þ DCi Þ Ci ¼ 0

# 0 : 0

The system (13) is a stochastic parameter system dependent on stochastic variable y. In order to investigate the stochastic stability of system (13), the following definition and lemmas will be introduced. Definition 1. For the stochastic system (4), if there exist constants a40 and t 2 ð0,1Þ such that: EfJZðkÞJ2 gratk EfJZð0ÞJ2 g, where ZðkÞ denotes the solution of stochastic systems with initial state Zð0Þ, then, the stochastic system (4) is said to be exponentially mean-square stable. Lemma 1 (Xie and de Souza [27]). For any real vectors a, b and matrix X 40 of compatible dimensions: aT b þ bT araT Xa þ bT X 1 b:

Lemma 2 (Petersen et al. [28]). Let A, E, H, and F(t) be real matrices of appropriate dimensions with F(t) satisfying F T ðtÞF ðtÞrI. Then, for any real matrices Q ¼ QT , we have: Q þ EF ðtÞH þ H T F T ðtÞE T o0

8F ðtÞ s:t: F ðtÞT F ðtÞrI,

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if and only if there exists some scalar e40 such that Q þ eEE T þ e1 H T Ho0: Lemma 3 (Yang et al. [29]). Let V ðZðkÞÞ be a Lyapunov function. If there exist real scalars lZ0, m40, n40, and 0oco1 such that mJZðkÞJ2 rV ðZðkÞÞrnJZðkÞJ2 and EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞrlcV ðZðkÞÞ then the sequence ZðkÞ satisfies: n l : EfJZðkÞJ2 gr JZð0ÞJ2 ð1cÞk þ m mc In the following theorem, the condition of stability on the ideal sliding motion (12) is derived. Theorem 1. For the system (4) subject to packet dropout (12), if there exist symmetric matrices Pi 40, Qi 40, and scalar ei 40 satisfying the following LMI: 3 2 n n n n n n n Pi þ ei HiT Hi 7 6 0 Qi n n n n n n 7 6 7 6 6 2P i Ai 0 2P i n n n n n 7 7 6 7 6 2P i Ci 0 0 2P i n n n n 7 6 7o0: ð14Þ 6 6 J1i J2i 0 0 J3i n n n 7 7 6 7 6 2Gi Ci 0 0 0 0 2Gi Bi n n 7 6 7 6 6 J4i J5i 0 0 0 0 J6i n 7 5 4 0 0 0 2EiT P i 0 2EiT P i Bi 0 2ei I P PN T 2 T T T T T with P i ¼ N j ¼ 1 pij Pj , Q i ¼ j ¼ 1 pij Qj , a 9ð1yÞy, J1i ¼ ½2Ai Gi 0 , J2i ¼ ½0 fAi Gi  , T T J3i ¼ diagf2Gi Bi ,Gi Bi g, J4i ¼ ½ð1yÞQ i Q i  , J5i ¼ ½yQ i Q i  , J6i ¼ diagfQ i ,a2 Q i g, then the augment dynamic system is mean-square asymptotically stable, i.e., the sliding mode dynamics (12) in sðk,iÞ ¼ 0 with Gi ¼ BTi P i is mean-square asymptotically stable. Proof. Choose the Lyapunov function candidate for sliding motion (12) as V ðZðkÞ,iÞ ¼ xT ðkÞPi xðkÞ þ xTs ðk1ÞQi xs ðk1Þ:

ð15Þ

Remark 3. It is noted that the effect of packet losses has been considered in the choice of Lyapunov functional (15), which includes x(k) and xs ðk1Þ, where xs ðk1Þ is the compensation when the data packet is lost at time k. We have from Eqs. (7) and (12) that EfV ðZðk þ 1Þ,iÞjZðkÞgV ðZðkÞ,iÞ ¼ EfxT ðk þ 1ÞP i xðk þ 1ÞjZðkÞg þ EfxTs ðkÞQ i xs ðkÞjZðkÞgV ðZðkÞ,iÞ:

ð16Þ

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Note that EfðyyÞg ¼ 0, EfðyyÞ2 g ¼ ð1yÞy9a2 ,EfwðkÞg ¼ 0,EfwðkÞ2 g ¼ 1, EfwðiÞwðjÞg ¼ 0 for iaj. By Lemma 1, we obtain

2175

and

EfxT ðk þ 1ÞP i xðk þ 1ÞjZðkÞg ¼ Ef½ðAi Bi ðGi Bi Þ1 Gi Ai ÞxðkÞfBi ðGi Bi Þ1 Gi Ai xs ðk1Þ þðIBi ðGi Bi Þ1 Gi ÞðCi þ DCi ÞxðkÞwðkÞT P i ½ðAi Bi ðGi Bi Þ1 Gi Ai ÞxðkÞfBi ðGi Bi Þ1 Gi Ai xs ðk1Þ þðIBi ðGi Bi Þ1 Gi ÞðCi þ DCi ÞxðkÞwðkÞg ¼ ½ðAi Bi ðGi Bi Þ1 Gi Ai ÞxðkÞfBi ðGi Bi Þ1 Gi Ai xs ðk1ÞT P i ½ðAi Bi ðGi Bi Þ1 Gi Ai ÞxðkÞfBi ðGi Bi Þ1 Gi Ai xs ðk1Þ þððIBi ðGi Bi Þ1 Gi ÞðCi þ DCi ÞxðkÞÞT P i ðIBi ðGi Bi Þ1 Gi ÞðCi þ DCi ÞxðkÞ rxT ðkÞð2ATi P i Ai þ 2ATi GiT ðGi Bi Þ1 Gi Ai ÞxðkÞ þf2 xTs ðk1ÞATi GiT ðGi Bi Þ1 Gi Ai xs ðk1Þ 2fxT ðkÞðAi Bi ðGi Bi Þ1 Gi Ai ÞT P i Bi ðGi Bi Þ1 Gi Ai xs ðk1Þ þxT ðkÞð2ðCi þ DCi ÞT P i ðCi þ DCi Þ þ 2ðCi þ DCi ÞT GiT ðGi Bi Þ1 Gi ðCi þ DCi ÞÞxðkÞ ð17Þ and EfxTs ðkÞQ i xs ðkÞjZðkÞg ¼ Efðð1yÞxðkÞ þ yxs ðk1ÞÞT Q i ðð1yÞxðkÞ þ yxs ðk1ÞÞg ¼ ðð1yÞxðkÞ þ yxs ðk1ÞÞT Q i ðð1yÞxðkÞ þ yxs ðk1ÞÞ þEfððyyÞxðkÞ þ ðyyÞxs ðk1ÞÞT Q i ððyyÞxðkÞ þ ðyyÞxs ðk1ÞÞg rxT ðkÞðð1yÞ2 Q i ÞxðkÞ þ 2ð1yÞyxT ðkÞQ i xs ðk1Þ 2

þy xs ðk1ÞT Q i xs ðk1Þ þ a2 xT ðkÞQ i xðkÞ þa2 xTs ðk1ÞQ i xs ðk1Þa2 xT ðkÞQ i xs ðk1Þa2 xTs ðk1ÞQ i xðkÞ:

ð18Þ

Then, it follows from (16)–(18) that: EfV ðZðk þ 1Þ,iÞjZðkÞgV ðZðkÞ,iÞ rxT ðkÞ½2ATi P i Ai þ 2ATi GiT ðGi Bi Þ1 Gi Ai þ 2ðCi þ DCi ÞT P i ðCi þ DCi Þ þ2ðCi þ DCi ÞT GiT ðGi Bi Þ1 Gi ðCi þ DCi Þ þ ð1yÞ2 Q i þ a2 Q i Pi ÞxðkÞ 2

þxTs ðk1Þðf2 ATi GiT ðGi Bi Þ1 Gi Ai þ y Q i þ a2 Q i Qi Þxs ðk1Þ ¼ ZT ðkÞYi ZðkÞ, ð19Þ with " Yi ¼ ½L1i L2i 

T

P1 1i ½L1i

L2i  þ

Pi

0

0

Qi

#

and L1i ¼ ½2ATi P i 2ðCi þ DCi ÞT P i 2ATi GiT 0 2ðCi þ DCi ÞT GiT ð1yÞQ i Q i T , L2i ¼ ½0 0 0 fATi GiT 0 yQ i Q i T , P1i ¼ diagf2P i ,2P i ,2ðGi Bi Þ,ðGi Bi Þ,2ðGi Bi Þ,Q i ,a2 Q i g:

ð20Þ

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By Schur’s complement, Yi o0 is equivalent to the inequality: 2 3 Pi n n 6 0 Qi n 7 4 5o0: L1i L2i P1i Furthermore, it can be obtained by Lemma 2 that: 2 3 2 3 Pi n n n n Pi 6 0 6 Qi n 7 Qi n 7 4 5¼4 0 5 þ Mi Fi ðtÞNi þ NiT FiT ðtÞMiT L1i L2i P1i L3i L2i P1i 2 3 n n Pi 6 T Qi n 7 r4 0 5 þ ei NiT Ni þ e1 i Mi Mi , L3i L2i P1i

ð21Þ

ð22Þ

with L3i ¼ ½2ATi P i 2CiT P i 2ATi GiT 0 2CiT GiT ð1yÞQ i Q i T , pffiffiffi pffiffiffi Mi ¼ ½0 0 0 2EiT P i 0 0 2EiT P i Bi 0 0T , Ni ¼ ½Hi 0 0 0 0 0 0 0 0: Then, by Schur’s complement, one can readily conclude from Eqs. (20)–(22) that the sliding mode dynamics (12) is mean-square asymptotically stable if there exist Pi 40, Qi 40, and scalar ei 40 satisfy LMI (14). & In the following section, we shall further analyze the reachability of the sliding surface. 4. Sliding mode control and reachability For expression (9), define DðkÞ ¼ ð1yÞGi Bi f ðxðkÞ,kÞ ¼ ½d1 ðkÞ d2 ðkÞ . . . dm ðkÞT . Since the nonlinear function vector f ðxðkÞ,kÞ is assumed to be bounded, there exist known constants d i , d i (i ¼ 1,2, . . . ,m) satisfying d i rdi ðkÞrd i . Then, let dio ¼

di þ di , 2

dis ¼

d i d i 2

ði ¼ 1,2, . . . ,mÞ,

ð23Þ

and Do ¼ ½d1o d2o . . . dmo T , Ds ¼ diagfd1s ,d2s , . . . ,dms g. It should be pointed out that these bounds, d i and d i , may be time-varying or dependent on state x(k). To simplify, the variables in these bounds are omitted in the derivation later. By means of the information from compensator (7) and the bounds in Eq. (23), the desired SMC law is designed as follows: uðkÞ ¼ 

1 ðGi Bi Þ1 ½Gi Ai xs ðkÞ þ Do þ Ds sgnðss ðk,iÞÞ, 1y

where ss ðk,iÞ is the sliding variable sðk,iÞ in Eq. (8) with x(k) replaced by xs(k). From Eqs. (4), (8) and (24), we have sðk þ 1,iÞ ¼ ð1yÞGi Ai xðkÞ þ ð1yÞGi Bi uðk,iÞ þ ð1yÞGi Bi f ðxðkÞ,kÞ þyGi Ai xs ðk1Þ þ ð1yÞGi ðCi þ DCi ÞxðkÞwðkÞ ¼ ðyyÞGi Ai xðkÞ þ ðyyÞGi Ai xs ðk1Þ þ DðkÞDo Ds sgnðss ðk,iÞÞ

ð24Þ

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þð1yÞGi ðCi þ DCi ÞxðkÞwðkÞ:

ð25Þ

Theorem 2. For the system (4) subject to packet dropout (5), if there exist symmetric matrices Pi 40, Qi 40, and scalar ei 40,0ogi o1 satisfying the following LMI: 3 2 Pi þ ei HiT Hi n n n n n n n n 6 0 Qi n n n n n n n 7 7 6 7 6 6 K1i K2i K3i n n n n n n 7 7 6 6 P i Ci 0 0 P i n n n n n 7 7 6 7 6 7o0, 6 K K 0 0 K n n n n 4i 5i 6i 7 6 6 ð1yÞG C 0 0 0 0 I n n n 7 i i 7 6 7 6 7 6 ð1yÞQ i yQ 0 0 0 0 Q n n i i 7 6 6 2 Qi Q i 0 0 0 0 0 a Q i n 7 5 4 0 0 0 EiT P i 0 EiT P i Bi 0 0 ei I ð26Þ with f ¼ y=ð1yÞ, f ¼ ð4 þ fÞ , f^ ¼ ð2f þ 2f Þ , K1i ¼ ½0 fATi P i ^ T G T T , K3i ¼ diagfg I,P i ,Gi Bi ,Gi Bi g, K4i ¼ ½2aAT GT 0T , ½0 0 0 fA i i i i i 1=2

2 1=2

fATi GiT

T

0 , K2i ¼ K5i ¼ ½0 2aATi GiT T , K6i ¼ diagf2I,2Ig, then the SMC law (24) can ensure that the state trajectories are driven (with mean square) into a band of the sliding surface specified by Eq. (8) with Gi ¼ BTi P i . Proof. It follows from Eqs. (4) and (24) that the closed-loop control system is given as     y1 y 1 Bi ðGi Bi Þ Gi Ai xðkÞ þ xðk þ 1,iÞ ¼ I þ Bi ðGi Bi Þ1 Gi Ai xs ðk1Þ 1y 1y 1 1 Bi ðGi Bi Þ ½DðkÞDo Ds sgnðss ðkÞÞ þ ðCi þ DCi ÞxðkÞwðkÞ: ð27Þ þ 1y Define xðkÞ ¼ ½xT ðkÞ xTs ðk1Þ sT ðk,iÞT , and choose Lyapunov function candidate as V ðxðkÞ,iÞ ¼ xT ðkÞPi xðkÞ þ xTs ðk1ÞQi xs ðk1Þ þ sT ðk,iÞsðk,iÞ:

ð28Þ

Then it follows from Eqs. (7), (25), and (27) and Lemma 1 that EfV ðxðk þ 1Þ,iÞjxðkÞgV ðxðkÞ,iÞ

¼ EfxT ðk þ 1ÞP i xðk þ 1ÞjxðkÞg þ EfxTs ðkÞQ i xs ðkÞjxðkÞg þEfsT ðk þ 1,iÞsðk þ 1,iÞjxðkÞgV ðxðkÞ,iÞ rxT ðkÞðð4 þ fÞATi P i Ai þ ð4 þ fÞATi GiT ðGi Bi Þ1 Gi Ai þðCi þ DCi ÞT P i ðCi þ DCi ÞÞxðkÞ þ ð2f þ 2f2 ÞxTs ðk1ÞATi GiT ðGi Bi Þ1 Gi Ai xs ðk1Þ   1 2 þ3 ½DðkÞDo Ds sgnðss ðkÞÞT ðGi Bi Þ1 ½DðkÞDo Ds sgnðss ðkÞÞ 1y 2

þxT ðkÞðð1yÞ2 Q i þ a2 Q i ÞxðkÞ þ xTs ðk1Þðy Q i þ a2 Q i Þxs ðk1Þ þxT ðkÞð2a2 ATi GiT Gi Ai þ ð1yÞ2 ðCi þ DCi ÞT GiT Gi ðCi þ DCi ÞÞxðkÞ þxTs ðk1Þð2a2 ATi GiT Gi Ai Þxs ðk1Þ

B. Chen et al. / Journal of the Franklin Institute 351 (2014) 2169–2184

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þ½DðkÞDo Ds sgnðss ðkÞÞT ½DðkÞDo Ds sgnðss ðkÞÞV ðxðkÞ,iÞ:

ð29Þ

Besides, it is noted from the expression (23) that JDðkÞDo Ds sgnðsp ðkÞÞJr2JDs J:

ð30Þ

Thus, we have EfV ðxðk þ 1ÞÞjxðkÞgV ðxðkÞ,iÞ rxT ðkÞðð4 þ fÞATi P i Ai þ ð4 þ fÞATi GiT ðGi Bi Þ1 Gi Ai þðCi þ DCi ÞT P i ðCi þ DCi Þ þ ð1yÞ2 Q i þ a2 Q i þ2a2 ATi GiT Gi Ai þ ð1yÞ2 ðCi þ DCi ÞT GiT Gi ðCi þ DCi ÞPi ÞxðkÞ 2

þxTs ðk1Þðð2f þ 2f2 ÞATi GiT ðGi Bi Þ1 Gi Ai þ y Q i þa2 Q i þ 2a2 ATi GiT Gi Ai Qi Þxs ðk1ÞsT ðk,iÞsðk,iÞ þ bJDs J2 , 1

2

with b ¼ 4 þ 12JðGi Bi Þ J=ð1yÞ , for b we obtain

1=2

JDs Jrð1gi Þ

1=2

Jsðk,iÞJ, where 0ogi o1,

EfV ðxðk þ 1ÞÞjxðkÞgV ðxðkÞÞrxT ðkÞSi xðkÞ, with

2

Pi

6 0 Si ¼ ½K7i K8i 0T P1 2i ½K7i K8i 0 þ 4 0

ð31Þ

ð32Þ n

Qi 0

n

3

n 7 5

ð33Þ

gi I

and K7i ¼ ½f i ATi P i f i ATi GiT 0 ðCi þ DCi ÞT P i 2aATi GiT 0 ð1yÞðCi þ DCi ÞT GiT ð1yÞQ i Q i T ,

K8i ¼ ½0 0 f^ i ATi GiT 0 0 2aATi GiT 0 yQ i Q i T , P2i ¼ diagfP i ,ðGi Bi Þ,ðGi Bi Þ,P i ,2I,2I,I,Q i ,a2 Q i g: According to Lemma 2 and Schur’s complement, one can readily conclude that the matrices inequality Si o0 can be ensured by LMI (26). Furthermore, we obtain EfV ðxðk þ 1ÞÞjxðkÞgV ðxðkÞÞrxT ðkÞSi xðkÞrlmin ðSi ÞxðkÞT xðkÞomxðkÞT xðkÞ,

ð34Þ where 0omominflmin ðSi Þ,dg,

d :¼ maxflmax ðPi Þ, lmax ðQi Þ,1g:

Thus, it follows from Eq. (34) that m EfV ðxðk þ 1ÞÞjxðkÞgV ðxðkÞÞomxðkÞT xðkÞo V ðxðkÞÞ :¼ cV ðxðkÞÞ, d

ð35Þ

with 0oco1. Therefore, it is obtained from Lemma 3 and Definition 1 that for Jsðk,iÞJZb1=2 =ð1gÞ1=2 JDs J, the closed-loop system (27) is exponentially mean-square

B. Chen et al. / Journal of the Franklin Institute 351 (2014) 2169–2184

2179

stable. This means that the state trajectories will be driven (with mean square) into the band of the sliding surface specified by Eq. (8). & Remark 4. The condition Jsðk,iÞJZb1=2 =ð1gÞ1=2 JDs J can be easily obtained when the system state is not on the sliding surface sðk,iÞ ¼ 0 at the beginning. Otherwise, as a special case, the system state may stay on the sliding surface sðk,iÞ ¼ 0, which is just the ideal sliding motion discussed in Section 3. Remark 5. By Schur’s complement, one can readily find that LMIs (14) in Theorem 1 are implied by LMIs (26) in Theorem 2. Thus, If there exist matrices Pi 40, Qi 40, and scalar ei 40,0ogi o1 satisfying LMIs (26), the state trajectories will be driven (with mean square) into the band of the sliding surface specified by Eq. (8). Meanwhile, the ideal sliding motion in sðk,iÞ ¼ 0 is asymptotically stable. 5. Simulation example Consider the stochastic system (1) with two modes and parameters as follows: 2 3 2 3 2 3 0:05 0:1 0:2 1:2 2:5 0:3 0:1 0:3 6 7 6 7 6 7 3:0 5, C1 ¼ 4 0:2 0:1 0:1 5, A1 ¼ 4 0:1 0:1 0:1 5, B1 ¼ 4 1:5 0 0:1 0:2 5:8 3:5 0:3 0:4 0:2 E1 ¼ ½0:1 0:1 0:1T , H1 ¼ ½0:1 0:1 0:1, 2 3 2 0:1 0:3 0:2 1:0 6 7 6 A2 ¼ 4 0:2 0:1 0:2 5, B2 ¼ 4 1:1 0:1 0:2 0:2 3:4 T

E2 ¼ ½0:1 0:1 0:1 ,

3 2:0 7 1:5 5, 2:7

2

0:2 6 C2 ¼ 4 0:2 0:2

0:1 0:1 0:3

3 0:3 7 0:1 5, 0:4

H2 ¼ ½0:2 0:2 0:2:

The transition probability matrix (3) and the dropout probability (5) are given by   0:3 0:7 P¼ , y ¼ 0:2: 0:6 0:4 The solutions of LMI (26) are given as 2 3 0:0497 0:0248 0:0316 6 7 0:0115 5, P1 ¼ 4 0:0248 0:0318

2

0:0301

0:0143 0:0099

3

6 7 P2 ¼ 4 0:0143 0:0374 0:0052 5, 0:0316 0:0115 0:0381 0:0099 0:0052 0:0384 2 3 2 3 0:0039 0:0004 0:0007 0:0040 0:0002 0:0007 6 7 6 7 0:0098 0:0040 5, Q2 ¼ 4 0:0002 0:0083 0:0026 5, Q1 ¼ 4 0:0004 0:0007 0:0040 0:0044 0:0007 0:0026 0:0045 e1 ¼ 0:0277, e2 ¼ 0:0066, g1 ¼ g2 ¼ 0:5:

Thus, the desired sliding function (8) and the SMC law (24) can be obtained as follows: For mode 1     0:0016 0:0009 0:0010 0:0718 0:0296 0:0777 xs ðk2Þ, xðtÞ þ sðk,1Þ ¼ 0:0024 0:0009 0:0005 0:0804 0:0555 0:0621

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2180

 uðk,1Þ ¼



0:0082

0:0044

0:0052

0:0120 11:2065 12:0569 þdiagf0:0116 0:0585gsgnðss ðk,1ÞÞÞ:

0:0047

0:0024

12:1372

11:2065

 xs ðkÞ

For mode 2: 0:0409

0:0076

0:0081

  0:0039 0:0048 xðtÞ þ 0:0008 0:0023 0:0387

0:0662

0:0016 0:0027

2 x1 x2 x3

1.5 1 0.5 0 −0.5 −1 system mode

sðk,2Þ ¼

0:0064

−1.5 −2

3 2 1 0 0

Modes evolution r1 (k)

10 time (sec)

−2.5 −3

0

5

10 time (sec)

20

15

20

Fig. 1. Trajectories of state xðkÞ for r1 ðkÞ. 0.1 s1 s2

0

system mode



−0.1

−0.2

0

5

3 2 1 0

Modes evolution r1 (k)

0

10

10 time (sec)

15

time (sec)

Fig. 2. Switching function sðkÞ for r1 ðkÞ.

20

20

 xs ðk2Þ,

B. Chen et al. / Journal of the Franklin Institute 351 (2014) 2169–2184

 uðk,2Þ ¼

6:6876



0:0193

0:0241

0:0079

0:0038 0:0115 6:6876 14:1934 þdiagf0:0055 0:0132gsgnðss ðk,2ÞÞÞ:

0:0136

6:7653

2181

 xs ðkÞ

In this simulation, the uncertain parameter and external disturbance are given as F1 ðkÞ ¼ 0:1ðcosðkÞ þ sinðkÞÞ,F2 ðkÞ ¼ 0:1 sinðkÞ þ cosðkÞ,f1 ðxðkÞ,kÞ ¼ ½0:05 sinðx1 ðkÞ þ x2 ðkÞÞ 0:2 sinðx2 ðkÞx3 ðkÞÞ,f2 ðxðkÞ,kÞ ¼ ½0:1 sinððx1 ðkÞÞ þsinðx2 ðkÞÞÞ0:1 sinðx2 ðkÞx3 ðkÞÞ. To prevent

0.4 u1 u2

0.3 0.2 0.1 0

system mode

−0.1 −0.2 −0.3

3 2 1 0

Modes evolution r1 (k)

0

10 time (sec)

−0.4 −0.5

0

5

10

20

15

20

time (sec)

Fig. 3. Control signal uðtÞ for r1 ðkÞ.

2 x1 x2 x3

1.5 1 0.5 0 −0.5 system mode

−1 −1.5 −2

3 2 1 0

Modes evolution r2 (k)

0

−2.5 −3

0

5

10

10 time (sec)

15

time (sec)

Fig. 4. Trajectories of state xðkÞ for r2 ðkÞ.

20

20

2182

B. Chen et al. / Journal of the Franklin Institute 351 (2014) 2169–2184 0.05 s1 s2

0

−0.05

system mode

−0.1

−0.15

−0.2

0

3 2 1 0

Modes evolution r2 (k)

0

5

10 time (sec)

10

20

15

20

time (sec)

Fig. 5. Switching function sðkÞ for r2 ðkÞ. 0.4 u1 u2

0.3 0.2 0.1 0

system mode

−0.1 −0.2 −0.3

3 2 1 0

Modes evolution r2 (k)

0

−0.4 −0.5

0

5

10

10 time (sec)

15

20

20

time (sec)

Fig. 6. Control signal uðtÞ for r2 ðkÞ.

the control signals from chattering, we replace sgnðss ðk,iÞÞ with ss ðk,iÞ=ðss ðk,iÞ þ 0:1Þ in simulation. Giving two different modes variations r1 ðkÞ and r2 ðkÞ generated randomly, we get the simulation results as shown in Figs. 1–6 for the giving initial state xð0Þ ¼ ½10:51T . Among them, Figs. 1 and 4 are the state response of the closed-loop system, the sliding function is given in Figs. 2 and 5, and 3 and 6 show the SMC input. It is seen that the proposed sliding mode controller can effectively cope with the effect of Markovian switching and packet losses, and ensure the exponentially mean-square stable of the overall closed-loop system. Although there are some notable variations in the curves, which may

B. Chen et al. / Journal of the Franklin Institute 351 (2014) 2169–2184

2183

be caused by parameter uncertainties and exogenous disturbance, these undesirable effects are effectively attenuated. 6. Conclusions This paper has investigated the problem of SMC for uncertain MJSs with stochastic perturbation subject to successive packet losses. By using an estimation method on packet losses, both sliding surface and SMC law, dropout-probability-dependent, were synthesized, and the stability of the controlled system are ensured despite stochastic perturbation and packet losses. Moreover, the connections among various sliding surfaces have been considered, which is quite important for Markovian jumping systems. The main results obtained in this paper could be easily extended to more general nonlinear stochastic systems and networked-based control systems. Acknowledgment This work was supported in part by NNSF from China (61273073, 61074041, and 61004062).

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