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Mathematical Problems in Engineering
VolumeΒ 2012Β (2012), Article IDΒ 431576, 16 pages
http://dx.doi.org/10.1155/2012/431576
Research Article

Robust Reliable 𝐻∞ Control for Nonlinear Stochastic Markovian Jump Systems

1Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan University of Science and Technology, Wuhan 430081, China
2Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Received 29 November 2011; Accepted 2 April 2012

Academic Editor: WeihaiΒ Zhang

Copyright Β© 2012 Guici Chen and Yi Shen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The robust reliable 𝐻∞ control problem for a class of nonlinear stochastic Markovian jump systems (NSMJSs) is investigated. The system under consideration includes ItΓ΄-type stochastic disturbance, Markovian jumps, as well as sector-bounded nonlinearities and norm-bounded stochastic nonlinearities. Our aim is to design a controller such that, for possible actuator failures, the closed-loop stochastic Markovian jump system is exponential mean-square stable with convergence rate 𝛼 and disturbance attenuation 𝛾. Based on the Lyapunov stability theory and ItΓ΄ differential rule, together with LMIs techniques, a sufficient condition for stochastic systems is first established in Lemma 3. Then, using the lemma, the sufficient conditions of the solvability of the robust reliable 𝐻∞ controller for linear SMJSs and NSMJSs are given. Finally, a numerical example is exploited to show the usefulness of the derived results.

1. Introduction

In the past few decades, Markovian jump systems (MJSs) have been considerably studied since this kind of hybrid systems consists of a number of subsystems and a switch signal, which includes applications in safety-critical and high-integrity systems (e.g., aircraft, chemical plants, nuclear power station, robotic manipulator systems, large-scale flexible structures for space stations such as antenna, and solar arrays) typically systems, which may experience abrupt changes in their structure, see, for example, [1] and the references therein. And now, some results of stability and stabilization for ItΓ΄ type stochastic Markovian jump systems are also available in many papers, see, for example, [2–4] and the references therein.

The analysis and synthesis problems of Markovian jump systems (MJSs) or stochastic Markovian jump systems (SMJSs) have attracted plenty of attention from many researchers. Many important and remarkable achievements reasonable have obtained. If the control systems possess integrity against actuator and sensor failures, we called reliable control systems or fault-tolerant control systems [5]. Recently, the robust reliable control and filtering problems for time-delay systems or Markovian jump systems (MJSs) have attracted considerable attention, and several approaches have been developed, see, for example, [6–11] and the references therein. Via linear matrix inequalities (LMIs), the authors designed the robust reliable 𝐻∞ controller for uncertain nonlinear systems [6]. In [7], for admissible uncertainties as well as actuator failures occurring among a prespecified subset of actuators, Zhang et al. studied the reliable dissipative control of Markovian jump impulsive systems. The reliable 𝐻∞ control problem for discrete-time piecewise linear systems with infinite distributed delays have been investigated in [8]. Recently, the study of stochastic 𝐻∞ filtering for the systems governed by stochastic ItΓ΄-type equations has attracted a great deal of attention, and Zhang and Chen [9] firstly solved the nonlinear stochastic delay-free 𝐻∞ filtering problem by means of a stochastic bounded real lemma derived in [10]. The reliable 𝐻∞ filtering problems for discrete time-delay systems with randomly occurred nonlinearities [11] and discrete time-delay Markovian jump systems with partly unknown transition probabilities [12] also has been studied, respectively. The reliable control problem for a class of Markovian jump systems with interval time-varying delays and stochastic failure is studied in [13]. In recent years, the research begins to focusing on robust reliable control problems for stochastic systems or stochastic switched nonlinear systems, see, for example, [14–16] and the references therein.

However, all the aforementioned results are mainly focusing on the reliable control and filtering problems of discrete-time-delay systems and Markovian jump systems. Up to now, to the best of the authors’ knowledge, the robust reliable 𝐻∞ control problem for nonlinear stochastic Markovian jump systems (NSMJSs) has not been fully investigated, which is an open problem and gives the motivation of our present investigation. In this paper, our aim is to design a robust reliable 𝐻∞ controller for NSMJSs, such that the NSMJSs are globally mean exponential stable with convergence rate 𝛼and disturbance attenuation 𝛾.

1.1. Notations

Throughout this paper, for symmetric matrices 𝑋 and π‘Œ, the notation 𝑋β‰₯π‘Œ (resp., 𝑋>π‘Œ) means that the Matrix 𝑋-π‘Œ is positive semidefinite (respectively, positive definite). 𝐼is an identity matrix with appropriate dimensions; the subscript β€œπ‘‡β€ represents the Transposition. 𝐸(β‹…)denotes the expectation operator with respect to some probability measure 𝑃. β„’2[0,∞) is the space of square integrable vector functions over [0,∞); let (Ξ©,β„±,𝑃) be a complete probability space which is relative to an increasing family (ℱ𝑑)𝑑>0 of 𝜎 algebras (ℱ𝑑)𝑑>0βŠ‚β„±, where Ξ© is the samples space, β„± is 𝜎 algebra of subsets of the sample space, and 𝑃 is the probability measure on β„±.‖⋅‖𝐸2=‖𝐸(β‹…)β€–2, while β€–β‹…β€–2 stands for the usual β„’2[0,∞) norm, 𝑅𝑛 and π‘…π‘›Γ—π‘š denote the 𝑛 dimensional Euclidean space and the set of all π‘›Γ—π‘š real matrices, respectively. In this paper, we provide all spaces π•‚π‘˜,π‘˜β‰₯1 with the usual inner product βŸ¨β‹…,β‹…βŸ© and its corresponding 2-normβ€–β‹…β€–. Let 𝐿2(Ξ©,π•‚π‘˜) denote the space of square-integrable π•‚π‘˜-valued functions on the probability space (Ξ©,β„±,𝑃). For any 0<𝑇<∞, we write [0,𝑇] for the closure of the open interval (0,𝑇) in 𝑅 and denote by 𝐿𝑛2([0,𝑇];𝐿2(Ξ©,π•‚π‘˜)) the space of the nonanticipative stochastic processes 𝑦(β‹…)=(𝑦(β‹…))π‘‘βˆˆ[0,𝑇] with respect to (ℱ𝑑)π‘‘βˆˆ[0,𝑇] satisfying ‖𝑦(β‹…)β€–2𝐿𝑛2∫=𝐸(𝑇0‖𝑦(𝑑)β€–2dβˆ«π‘‘)=𝑇0𝐸(‖𝑦(𝑑)β€–2)d𝑑<∞. 𝑉(π‘₯(𝑑),𝑑,π‘Ÿ(𝑑)=𝑖)=𝑉(π‘₯(𝑑),𝑑,𝑖), 𝐴(π‘Ÿ(𝑑)=𝑖)=𝐴𝑖𝐡(π‘Ÿ(𝑑)=𝑖)=𝐡𝑖, 𝐴0(π‘Ÿ(𝑑)=𝑖)=𝐴0𝑖,𝐡0(π‘Ÿ(𝑑)=𝑖)=𝐡0𝑖,𝐢(π‘Ÿ(𝑑)=𝑖)=𝐢𝑖,𝐷(π‘Ÿ(𝑑)=𝑖)=𝐷𝑖.

2. Problem Formulation and Failure Model

In this paper, we mainly consider the following nonlinear stochastic Markovian jump systems (NSMJSs) with actuator failures: dξ€Ίπ‘₯(𝑑)=𝐴(π‘Ÿ(𝑑))π‘₯(𝑑)+𝐡(π‘Ÿ(𝑑))𝑒𝑓(𝑑,π‘Ÿ(𝑑))+𝐸(π‘Ÿ(𝑑))𝑣(𝑑)+𝑓(π‘Ÿ(𝑑),π‘₯(𝑑))d𝑑+𝐢(π‘Ÿ(𝑑))π‘₯(𝑑)+𝐷(π‘Ÿ(𝑑))𝑒𝑓(𝑑,π‘Ÿ(𝑑))+𝐻(π‘Ÿ(𝑑))𝑣(𝑑)+𝑔(π‘Ÿ(𝑑),π‘₯(𝑑))d𝑀π‘₯𝑑(𝑑),𝑧(𝑑)=𝐽(π‘Ÿ(𝑑))π‘₯(𝑑),0ξ€Έ=π‘₯0,(2.1) where π‘₯(𝑑)βˆˆπ‘…π‘› is the system state, 𝑒𝑓(𝑑)βˆˆπ‘…π‘™ is the control input of actuator fault, 𝑣(𝑑)βˆˆπ‘…π‘ž is the exogenous disturbance input of the systems which belong to β„’2[0,∞), 𝑧(𝑑)βˆˆπ‘…π‘Ÿ is the system control output, 𝑀(𝑑) is a zero mean real scalar Weiner processes on a probability space (Ξ©,β„±,𝑃) relative to an increase family (ℱ𝑑)𝑑>0 of 𝜎 algebras (ℱ𝑑)𝑑>0βŠ‚β„±. 𝐴𝑖,𝐡𝑖,𝐸𝑖,𝐢𝑖,𝐷𝑖,𝐹𝑖,𝐻𝑖,𝐽𝑖 are the known real constant matrices with appropriate dimensions. Morever, we assume that𝐸(dξ€·(𝑀(𝑑))=0,𝐸d𝑀(𝑑))2ξ€Έ=d𝑑.(2.2)

Let π‘Ÿ(𝑑),𝑑β‰₯0, be a right-continuous Markovian chain on the probability space taking values in a finite state space 𝑆=1,2,…,𝑁 with generator Ξ“=(πœ†π‘–π‘—)𝑁×𝑁 given by ξ‚»πœ†π‘ƒ{π‘Ÿ(𝑑+Ξ”)=π‘—βˆ£π‘Ÿ(𝑑)=𝑖}=𝑖𝑗Δ+π‘œ(Ξ”)if𝑖≠𝑗,1+πœ†π‘–π‘–Ξ”+π‘œ(Ξ”)if𝑖=𝑗,(2.3) where Ξ”>0. Here πœ†π‘–π‘—β‰₯0 is the transition rate from manner 𝑖 to manner 𝑗, if 𝑖≠𝑗 while πœ†π‘–π‘–βˆ‘=βˆ’π‘—β‰ π‘–πœ†π‘–π‘—. We assume that the Markovian chain π‘Ÿ(β‹…) is independent of the Wienner process 𝑀(β‹…). It is well known that almost every sample path of π‘Ÿ(𝑑) is a right-continuous step function with a finite number of simple jump in any finite subinterval of 𝑅+(∢=[0,+∞)).

𝑓(β‹…,β‹…)βˆΆπ‘†Γ—π‘…π‘›β†’π‘…π‘› is a unknown nonlinear function which describes the system nonlinearity satisfying the following sector-bounded conditions: 𝑓𝑖(π‘₯(𝑑))βˆ’π‘‡1𝑖π‘₯𝑇𝑓𝑖(π‘₯(𝑑))βˆ’π‘‡2𝑖π‘₯≀0,π‘–βˆˆπ‘†,(2.4)𝑔(β‹…,β‹…)βˆΆπ‘†Γ—π‘…π‘›β†’π‘…π‘› also is a unknown nonlinear function which describes the stochastic nonlinearity satisfying the following: 𝑔𝑇𝑖(π‘₯(𝑑))𝑔𝑖(π‘₯(𝑑))≀π‘₯𝑇𝐺𝑇𝑖𝐺𝑖π‘₯,π‘–βˆˆπ‘†,(2.5) where 𝑇1𝑖,𝑇2𝑖,𝐺𝑖 are known real constant matrices with approximate dimensions.

Remark 2.1. The nonlinearities 𝑓𝑖(π‘₯(𝑑)) are bounded by sectors, which belong to [𝐿1𝑖,𝐿2𝑖], and are very general that include the usual Lipschitz conditions as a special case which is considerable investigated and includes several other classes well studied nonlinear systems [17–19]. The nonlinearities 𝑔𝑖(π‘₯(𝑑)) satisfy the norm-bounded conditions.

When the actuator experiences failure, we use 𝑒𝑓(𝑑,π‘Ÿ(𝑑)) to describe the control signal form actuators. Consider the following actuator failure model with failure parameter 𝐹𝑖: 𝑒𝑓𝑖(𝑑)=𝐹𝑖𝑒𝑖(𝑑),(2.6) where 𝐹𝑖 is the actuator fault matrix with 𝐹𝑖𝑓=diag𝑖1,𝑓𝑖2,…,π‘“π‘–π‘šξ€Έ,0≀𝑓𝑖𝑗≀𝑓𝑖𝑗≀𝑓𝑖𝑗,𝑓𝑖𝑗β‰₯1,𝑗=1,2,…,π‘š.(2.7)

In which the variables 𝑓𝑖𝑗 quantify the failures of the actuators. 𝑓𝑖𝑗=0 means that 𝑗th actuator completely fails, and 𝑓𝑖𝑗=1 means that the 𝑗th actuator is normal.

Define the following: 𝐹0𝑖𝑓=diag0𝑖1,𝑓0𝑖2,…,𝑓0π‘–π‘šξ€Έ=𝐹𝑖+𝐹𝑖2,𝑓0𝑖𝑗=𝑓𝑖𝑗+𝑓𝑖𝑗2,(2.8)𝐹0𝑖𝑓=diag0𝑖1,𝑓0𝑖2𝑓,…,0π‘–π‘šξ‚=πΉπ‘–βˆ’πΉπ‘–2,𝑓0𝑖𝑗=π‘“π‘–π‘—βˆ’π‘“π‘–π‘—2,(2.9) and hence, the matrix 𝐹𝑖 can be rewritten as 𝐹𝑖=𝐹0𝑖+Δ𝑖=𝐹0π‘–ξ€·πœ‘+diag𝑖1,πœ‘π‘–2,…,πœ‘π‘–π‘šξ€Έ,||πœ‘π‘–π‘—||≀𝑓𝑖𝑗,𝑗=1,2,…,π‘š.(2.10)

In this paper, our aim is to design the controller 𝑒𝑖(𝑑)=𝐾𝑖π‘₯(𝑑),π‘–βˆˆπ‘†, such that the closed-loop systems satisfy the following conditions:(i)without the exogenous disturbance input (i.e., 𝑣(𝑑)=0), the closed-loop control systems (2.1) are globally exponentially stable with convergence rate 𝛼>0;(ii)with zero initial condition (i.e., π‘₯(𝑑0)=0) and nonzero exogenous disturbance input (i.e., 𝑣(𝑑)β‰ 0), the following inequality holds:‖𝑧‖𝐸2<𝛾‖𝑣‖2ξ‚΅i.eξ€œ.,𝑇0𝑧𝑇(𝑑)𝑧(𝑑)d𝑑≀𝛾2ξ€œπ‘‡0𝑣𝑇(𝑑)𝑣(𝑑)d𝑑.(2.11)

If the above two conditions hold, we also called the systems that are exponential mean-square stable with convergence rate 𝛼 and disturbance attenuation 𝛾.

3. Main Results

Lemma 3.1 (Schur complement lemma [20]). For a given matrix 𝑆=𝑆1𝑆3βˆ—π‘†2 with 𝑆𝑇1=𝑆1,𝑆𝑇2=𝑆2, the following conditions are equivalent:(1)𝑆<0,(2)𝑆2<0,𝑆1βˆ’π‘†3𝑆2βˆ’1𝑆𝑇3<0,(3)𝑆1<0,𝑆2βˆ’π‘†3𝑆1βˆ’1𝑆𝑇3<0.

Lemma 3.2 (see [21]). Let π‘₯βˆˆβ„π‘› and π‘¦βˆˆβ„π‘›. Then, for any positive scalar πœ€, we have π‘₯𝑇𝑦+𝑦𝑇π‘₯β‰€πœ€π‘₯𝑇π‘₯+πœ€βˆ’1𝑦𝑇𝑦.(3.1)

3.1. Robust Reliable 𝐻∞ for LSMJSs

To obtain our main results, we first consider the following linear stochastic Markovian jump systems (LSMJSs) without control input: dπ‘₯𝐴(𝑑)=𝑖π‘₯(𝑑)+𝐸𝑖𝑣(𝑑)d𝐢𝑑+𝑖π‘₯(𝑑)+𝐻𝑖𝑣(𝑑)d𝑀(𝑑),𝑧(𝑑)=𝐽𝑖π‘₯𝑑π‘₯(𝑑),0ξ€Έ=π‘₯0.(3.2)

Lemma 3.3. Suppose that 𝑃(𝑑,π‘Ÿ(𝑑))>0 is continuously differentiable, then the systems (3.2) are exponential mean-square stable with convergence rate 𝛼and disturbance attenuation𝛾 if and only if the following matrix functional inequalities hold: Ξžπ‘–βŽ›βŽœβŽœβŽπ‘€(𝑑)=𝑖(𝑑)+π½π‘‡π‘–π½π‘–π‘ƒπ‘–πΈπ‘–πΆπ‘‡π‘–βˆ—βˆ’π›Ύ2πΌπ»π‘‡π‘–βˆ—βˆ—βˆ’π‘ƒπ‘–βˆ’1⎞⎟⎟⎠(𝑑)<0,π‘–βˆˆπ‘†,(3.3) where 𝑀𝑖(𝑑)=𝐴𝑇𝑖𝑃𝑖(𝑑)+𝑃𝑖(𝑑)𝐴𝑖+Μ‡βˆ‘π‘ƒ(𝑑)+π‘—βˆˆπ‘†πœ†π‘–π‘—π‘ƒπ‘—(𝑑).

Proof. At first, let 𝑣(𝑑)=0, and defining the following Lyapunov function: 𝑉(π‘₯(𝑑),𝑑,𝑖)=𝑉(π‘₯(𝑑),𝑑,π‘Ÿ(𝑑)=𝑖)=π‘₯𝑇(𝑑)𝑃(𝑑,π‘Ÿ(𝑑)=𝑖)π‘₯(𝑑)=π‘₯𝑇(𝑑)𝑃𝑖(𝑑)π‘₯(𝑑).(3.4)
By ItΓ΄ formula, we get the following: ℒ𝑉(π‘₯(𝑑),𝑑,𝑖)=π‘₯𝑇𝑀(𝑑)𝑖(𝑑)+𝐢𝑇𝑖𝑃𝑖(𝑑)𝐢𝑖π‘₯(𝑑),(3.5) the matrix function inequalities (3.3) imply that ℒ𝑉(π‘₯(𝑑),𝑑,𝑖)<0, and let π‘Žπ‘–=πœ†max(βˆ’Ξžπ‘–(𝑑)), π‘Ž=maxπ‘–βˆˆπ‘†(π‘Žπ‘–), where πœ†max(β‹…) means the maximum eigenvalue of matrix (β‹…), and we have ℒ𝑉(π‘₯(𝑑),𝑑,𝑖)β‰€βˆ’π‘Žπ‘₯𝑇(𝑑)π‘₯(𝑑).(3.6)
Hence 𝑑𝑒𝛼𝑑𝑉(π‘₯(𝑑),𝑑,𝑖)=𝛼𝑒𝛼𝑑𝑉(π‘₯(𝑑),𝑑,𝑖)+𝑒𝛼𝑑𝑑𝑉(π‘₯(𝑑),𝑑,𝑖)≀(π‘π›Όβˆ’π‘Ž)𝑒𝛼𝑑(β€–π‘₯𝑑)β€–2+𝑒𝛼𝑑2π‘₯𝑇(𝑑)𝑃𝑖(𝑑)𝐢𝑖π‘₯(𝑑)𝑑𝑀(𝑑),(3.7) where 𝑏𝑖=sup𝑑β‰₯𝑑0{πœ†max(𝑃𝑖(𝑑))}, and 𝑏=maxπ‘–βˆˆπ‘†(𝑏𝑖). Integrating the both sides of above inequality from 𝑑0 to 𝑇and taking expectation, we obtain that 𝐸𝑒𝛼𝑇π‘₯𝑉(π‘₯(𝑇),𝑇,𝑖)βˆ’π‘‰0,𝑑0ξ€œ,𝑖≀(π‘π›Όβˆ’π‘Ž)𝐸𝑇𝑑0𝑒𝛼𝑠‖π‘₯(𝑠)β€–2d𝑠.(3.8)
Set 𝛼=π‘Ž/𝑏, and the following inequality is obtained: 𝑒𝛼𝑇minπ‘–βˆˆπ‘†πœ†min𝑃𝑖‖(𝑇)𝐸‖π‘₯(𝑇)2𝑒≀𝐸𝛼𝑇𝑉π‘₯(π‘₯(𝑇),𝑇,𝑖)≀𝐸𝑉0,𝑑0ξ€Έ,,𝑖(3.9) which implies that 𝐸‖‖π‘₯(𝑇)2ξ€·π‘₯≀𝐸𝑉0,𝑑0ξ€Έ1,𝑖minπ‘–βˆˆπ‘†πœ†min𝑃𝑖𝑒(𝑇)βˆ’π›Όπ‘‡.(3.10)
That is to say that the stochastic systems are globally exponentially stable with convergence rate 𝛼>0.
Then, considering the stochastic 𝐻∞ performance level for the resulting systems (3.2) with nonzero exogenous disturbance input (𝑣(𝑑)β‰ 0), for any𝑑>0, we define that ξ‚»ξ€œπ½(𝑑)=𝐸𝑑0𝑧𝑇(𝑠)𝑧(𝑠)βˆ’π›Ύ2𝑣𝑇(𝑠)𝑣(𝑠)d𝑠.(3.11)
By general ItΓ΄ formula, we get he following: ξ‚»ξ€œπ½(𝑑)=𝐸𝑑𝑑0𝑧𝑇(𝑠)𝑧(𝑠)βˆ’π›Ύ2𝑣𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(π‘₯(𝑠),𝑠,𝑖)dπ‘ ξ‚Όξ‚»ξ€œβˆ’πΈ(𝑉(π‘₯(𝑑),𝑑,𝑖))≀𝐸𝑑0𝑧𝑇(𝑠)𝑧(𝑠)βˆ’π›Ύ2𝑣𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(π‘₯(𝑠),𝑠,𝑖)dπ‘ ξ‚Όξ‚»ξ€œβ‰€πΈπ‘‘0πœ‚π‘‡(𝑠)Ω𝑖(𝑠)πœ‚(𝑠)d𝑠,(3.12) where πœ‚π‘‡(𝑑)=(π‘₯𝑇(𝑑)𝑣𝑇(𝑑)), Ω𝑖(𝑑)=𝑀𝑖(𝑑)+𝐽𝑇𝑖𝐽𝑖𝑃𝑖(𝑑)𝐸𝑖𝐸𝑇𝑖𝑃𝑖(𝑑)βˆ’π›Ύ2𝐼+𝐢𝑇𝑖𝐻𝑇𝑖𝑃𝑖(𝑑)𝐢𝑇𝑖𝐻𝑇𝑖𝑇 From (3.3) we know that Ξ©(𝑑)<0, which implies that 𝐽(𝑑)<0.(3.13)
Therefore, the inequality ‖𝑧‖𝐸2<𝛾‖𝑣‖2 holds. The proof is completed.

In the following time, we consider the following linear stochastic Markovian jump systems (LSMJSs) under the state feedback controller: dπ‘₯𝐴(𝑑)=𝑖+𝐡𝑖𝐹𝑖𝐾𝑖π‘₯(𝑑)+𝐸𝑖𝑣(𝑑)d𝐢𝑑+𝑖+𝐷𝑖𝐹𝑖𝐾𝑖π‘₯(𝑑)+𝐻𝑖𝑣(𝑑)d𝑀(𝑑),𝑧(𝑑)=𝐽𝑖π‘₯𝑑π‘₯(𝑑),0ξ€Έ=π‘₯0.(3.14)

Theorem 3.4. If there exist the positive matrices 𝑋𝑖>0, and the constant matrices π‘Œπ‘– with approximate dimensions, such that the following LMIs hold Ξ˜π‘–=βŽ›βŽœβŽœβŽΞ˜π‘–1πΈπ‘–Ξ˜π‘–2Ξ˜π‘–3βˆ—βˆ’π›Ύ2𝐼𝐻𝑇𝑖0βˆ—βˆ—βˆ’π‘‹π‘–0βˆ—βˆ—βˆ—Ξ˜π‘–4⎞⎟⎟⎠<0,π‘–βˆˆπ‘†,(3.15) where Ξ˜π‘–1=𝑋𝑖𝐴𝑇𝑖+𝐴𝑖𝑋𝑖+π΅π‘–πΉπ‘–π‘Œπ‘–+π‘Œπ‘‡π‘–πΉπ‘‡π‘–π΅π‘‡π‘–+πœ†π‘–π‘–π‘‹π‘–,Ξ˜π‘–2=𝑋𝑖𝐢𝑇𝑖+π‘Œπ‘‡π‘–πΉπ‘‡π‘–π·π‘‡π‘–, Ξ˜π‘–3=ξ€·βˆšπœ†π‘–1π‘‹π‘–β‹―βˆšπœ†π‘–,π‘–βˆ’1π‘‹π‘–βˆšπœ†π‘–,𝑖+1π‘‹π‘–β‹―βˆšπœ†π‘–π‘π‘‹π‘–π‘‹π‘–π½π‘‡π‘–ξ€Έ,Ξ˜π‘–4ξ€·=diagβˆ’π‘‹1,…,βˆ’π‘‹π‘–βˆ’1,βˆ’π‘‹π‘–+1,…,βˆ’π‘‹π‘ξ€Έ,,βˆ’πΌ(3.16) then the LSMJSs (3.14) are exponential mean-square stable with convergence rate 𝛼and disturbance attenuation 𝛾. In this case, the desired controllers are given as follows: 𝐾𝑖=π‘Œπ‘–π‘‹π‘–βˆ’1.(3.17)

Proof. Defining the following Lyapunov function: 𝑉(π‘₯(𝑑),𝑑,𝑖)=𝑉(π‘₯(𝑑),𝑑,π‘Ÿ(𝑑)=𝑖)=π‘₯𝑇(𝑑)𝑃𝑖π‘₯(𝑑).(3.18)
By Lemma 3.3, and similar to the proof of Lemma 3.3, we can get the following: ℒ𝑉(π‘₯(𝑑),𝑑,𝑖)β‰€πœ‚π‘‡(𝑑)Ξžπ‘–πœ‚(𝑑),(3.19) where Ξžπ‘–=𝑀𝑖𝑃𝑖𝐸𝑖𝐢𝑇𝑖+πΎπ‘‡π‘–πΉπ‘‡π‘–π·π‘‡π‘–βˆ—βˆ’π›Ύ2πΌπ»π‘‡π‘–βˆ—βˆ—βˆ’π‘ƒπ‘–βˆ’1ξƒͺ𝑀𝑖=(𝐴𝑖+𝐡𝑖𝐹𝑖𝐾𝑖)𝑇𝑃𝑖+𝑃𝑖(𝐴𝑖+π΅π‘–πΉπ‘–πΎπ‘–βˆ‘)+π‘—βˆˆπ‘†πœ†π‘–π‘—π‘ƒπ‘—.
Using Schur complement lemma together with contragredient transformation, we know that LMIs (3.15) imply that Ξžπ‘–<0. So we have ξ‚»ξ€œπ½(𝑑)=𝐸𝑑0𝑧𝑇(𝑠)𝑧(𝑠)βˆ’π›Ύ2𝑣𝑇(𝑠)𝑣(𝑠)dπ‘ ξ‚Όξ‚»ξ€œ=𝐸𝑑𝑑0𝑧𝑇(𝑠)𝑧(𝑠)βˆ’π›Ύ2𝑣𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(π‘₯(𝑠),𝑠,𝑖)dπ‘ ξ‚Όξ‚»ξ€œβˆ’πΈ(𝑉(π‘₯(𝑑),𝑑,𝑖))≀𝐸𝑑0𝑧𝑇(𝑠)𝑧(𝑠)βˆ’π›Ύ2𝑣𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(π‘₯(𝑠),𝑠,𝑖)d𝑠.<0.(3.20) Therefore, the inequality ‖𝑧‖𝐸2<𝛾‖𝑣‖2 holds. The proof is completed.

Theorem 3.5. If there exist the positive matrices 𝑋𝑖>0, the positive diagonal matrices 𝑅𝑖>0, and the constant matrices π‘Œπ‘– with approximate dimensions, such that the following LMIs hold: ξ‚Ξ˜π‘–=βŽ›βŽœβŽœβŽœβŽœβŽœβŽξ‚Ξ˜π‘–1πΈπ‘–ξ‚Ξ˜π‘–2Ξ˜π‘–3π΅π‘–π‘…π‘–π‘Œπ‘‡π‘–βˆ—βˆ’π›Ύ2𝐼𝐻𝑇𝑖000βˆ—βˆ—βˆ’π‘‹π‘–0𝐷𝑖𝑅𝑖0βˆ—βˆ—βˆ—Ξ˜π‘–400βˆ—βˆ—βˆ—βˆ—βˆ’π‘…π‘–0βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π‘…π‘–ξ‚πΉβˆ’2𝑖0⎞⎟⎟⎟⎟⎟⎠<0,π‘–βˆˆπ‘†,(3.21) where ξ‚Ξ˜π‘–1=𝑋𝑖𝐴𝑇𝑖+𝐴𝑖𝑋𝑖+𝐡𝑖𝐹𝑖0π‘Œπ‘–+π‘Œπ‘‡π‘–πΉπ‘‡π‘–0𝐡𝑇𝑖+πœ†π‘–π‘–π‘‹π‘–, ξ‚Ξ˜π‘–2=𝑋𝑖𝐢𝑇𝑖+π‘Œπ‘‡π‘–πΉπ‘‡π‘–0𝐷𝑇𝑖, Then the LSMJSs (3.14) are exponential mean-square stable with convergence rate 𝛼 and disturbance attenuation 𝛾. In this case, the desired controllers are given as follows: 𝐾𝑖=π‘Œπ‘–π‘‹π‘–βˆ’1.(3.22)

Proof. Noticing (2.10), we can see that Ξ˜π‘– in (3.15) can be rewritten as Ξ˜π‘–=Ξ˜π‘–0+𝐡𝑇𝑖0𝐷𝑇𝑖0ξ€»π‘‡Ξ”π‘–ξ€Ίπ‘Œπ‘–ξ€»+ξ€Ίπ‘Œ000𝑖000𝑇Δ𝑖𝐡𝑇𝑖0𝐷𝑇𝑖0ξ€»,(3.23) where Ξ˜π‘–0=βŽ›βŽœβŽœβŽξ‚Ξ˜π‘–1πΈπ‘–ξ‚Ξ˜π‘–2Ξ˜π‘–3βˆ—βˆ’π›Ύ2𝐼𝐻𝑇𝑖0βˆ—βˆ—βˆ’π‘‹π‘–0βˆ—βˆ—βˆ—Ξ˜π‘–4⎞⎟⎟⎠.
By Lemma 3.2, we have Ξ˜π‘–β‰€Ξ˜π‘–0+𝐡𝑇𝑖0𝐷𝑇𝑖0𝑇𝑅𝑖𝐡𝑇𝑖0𝐷𝑇𝑖0ξ€»+ξ€Ίπ‘Œπ‘–ξ€»000π‘‡π‘…π‘–βˆ’1𝐹20π‘–ξ€Ίπ‘Œπ‘–ξ€»,000(3.24) by Schur complement, we know that ξ‚Ξ˜π‘–<0 imply that Ξ˜π‘–<0. Therefore, we can know from Theorem 3.4 that the LSMJSs (3.14) are stabilizable with convergence rate 𝛼and disturbance attenuation𝛾. This completes the proof.

3.2. Robust Reliable 𝐻∞ for NSMJSs

In this section, we consider the following nonlinear stochastic Markovian jump systems (NSMJSs) under the state feedback controller: dπ‘₯𝐴(𝑑)=𝑖+𝐡𝑖𝐹𝑖𝐾𝑖π‘₯(𝑑)+𝐸𝑖𝑣(𝑑)+𝑓𝑖(π‘₯(𝑑))d𝑑+𝐢𝑖+𝐷𝑖𝐹𝑖𝐾𝑖π‘₯(𝑑)+𝐻𝑖𝑣(𝑑)+𝑔𝑖(ξ€»π‘₯(𝑑))d𝑀(𝑑),𝑧(𝑑)=𝐻𝑖π‘₯𝑑π‘₯(𝑑),0ξ€Έ=π‘₯0.(3.25)

Theorem 3.6. If there exist the positive matrices 𝑋𝑖>0, and the constant matrices π‘Œπ‘– with approximate dimensions, for the positive constant πœ€π‘– and the given scalar πœ†π‘–, such that the following LMIs hold: Ξ˜π‘–=βŽ›βŽœβŽœβŽœβŽœβŽœβŽΞ˜π‘–1πΈπ‘–πΌβˆ’πœ†π‘–π‘‹π‘–ξπ‘‡π‘–2Ξ˜π‘‡π‘–2Ξ˜π‘‡π‘–2Ξ˜π‘–3βˆ—βˆ’π›Ύ2𝐼0𝐻𝑇𝑖𝐻𝑇𝑖0βˆ—βˆ—βˆ’πœ†π‘–πΌ000βˆ—βˆ—βˆ—βˆ’π‘‹π‘–00βˆ—βˆ—βˆ—βˆ—βˆ’πœ€π‘–πΌ0βˆ—βˆ—βˆ—βˆ—βˆ—Ξ˜π‘–4⎞⎟⎟⎟⎟⎟⎠<0,π‘–βˆˆπ‘†,(3.26) where Ξ˜π‘–3=(πœ€π‘–πΊπ‘–πœ†π‘–π‘‹π‘–ξπ‘‡π‘–1Ξ˜π‘–3),Ξ˜π‘–4=diag(βˆ’πœ€π‘–πΌ,βˆ’πœ†π‘–ξπ‘‡π‘–1,Ξ˜π‘–4𝑇),𝑖1=(𝑇𝑇𝑖1𝑇𝑖2+𝑇𝑇𝑖2𝑇𝑖1)/2,Ti2=βˆ’(TTi1+TTi2)/2, then the NSMJSs (3.25) are exponential mean-square stable with convergence rate 𝛼and disturbance attenuation𝛾. In this case, the desired controllers are given as follows: 𝐾𝑖=π‘Œπ‘–π‘‹π‘–βˆ’1.(3.27)

Proof. Defining the following Lyapunov function: 𝑉(π‘₯(𝑑),𝑑,𝑖)=𝑉(π‘₯(𝑑),𝑑,π‘Ÿ(𝑑)=𝑖)=π‘₯𝑇(𝑑)𝑃𝑖π‘₯(𝑑),(3.28)
by ItΓ΄ formula, we get the following: ℒ𝑉(π‘₯(𝑑),𝑑,𝑖)=2π‘₯𝑇(𝑑)𝑃𝑖𝐴𝑖+𝐡𝑖𝐹𝑖𝐾𝑖π‘₯(𝑑)+𝐸𝑖𝑣(𝑑)+𝑓𝑖(ξ€»+π‘₯(𝑑))π‘—βˆˆπ‘†πœ†π‘–π‘—π‘₯𝑇(𝑑)𝑃𝑗+𝐢π‘₯(𝑑)𝑖+𝐷𝐡𝑖𝐹𝑖𝐾𝑖π‘₯(𝑑)+𝐻𝑖𝑣(𝑑)+𝑔𝑖(π‘₯(𝑑))𝑇×𝑃𝑖𝐢𝑖+𝐷𝐡𝑖𝐹𝑖𝐾𝑖π‘₯(𝑑)+𝐻𝑖𝑣(𝑑)+𝑔𝑖(π‘₯(𝑑))β‰€πœŽπ‘‡(𝑑)Ξ£π‘–πœŽ(𝑑)+π‘₯𝑇(𝑑)𝐺𝑇𝑖𝑃𝑖𝐺𝑖𝐢π‘₯(𝑑)+2𝑖+𝐷𝐡𝑖𝐹𝑖𝐾𝑖π‘₯(𝑑)+𝐻𝑖𝑣(𝑑)𝑇𝑃𝑖𝑔𝑖(π‘₯(𝑑)),(3.29) where πœŽπ‘‡(𝑑)=[π‘₯𝑇(𝑑),𝑣𝑇(𝑑),𝑓𝑇𝑖(π‘₯(𝑑))],Σ𝑖=𝑀𝑖𝑃𝑖𝐸𝑖𝑃𝑖𝐸𝑇𝑖𝑃𝑖𝑃00𝑇𝑖00ξ‚Ά+𝐢𝑇𝑖+𝐾𝑇𝑖𝐹𝑇𝑖𝐷𝑇𝑖𝐻𝑇𝑖0𝑃𝑖𝐢𝑇𝑖+𝐾𝑇𝑖𝐹𝑇𝑖𝐷𝑇𝑖𝐻𝑇𝑖0𝑇.
By Lemma 3.2, it follows that 2𝐢𝑖+𝐷𝐡𝑖𝐹𝑖𝐾𝑖π‘₯(𝑑)+𝐻𝑖𝑣(𝑑)𝑇𝑃𝑖𝑔𝑖(π‘₯(𝑑))β‰€πœŽπ‘‡ξƒ¬πΆ(𝑑)𝑇𝑖+𝐾𝑇𝑖𝐹𝑇𝑖𝐷𝑇𝑖𝐻𝑇𝑖0ξƒ­πœ€π‘–βˆ’1𝐼𝐢𝑇𝑖+𝐾𝑇𝑖𝐹𝑇𝑖𝐷𝑇𝑖𝐻𝑇𝑖0ξƒ­π‘‡πœŽ(𝑑)+π‘₯π‘‡ξ€·πœ€(𝑑)π‘–π‘ƒπ‘–πΊπ‘–ξ€Έπ‘‡πœ€π‘–βˆ’1πΌξ€·πœ€π‘–π‘ƒπ‘–πΊπ‘–ξ€Έπ‘₯(𝑑),(3.30) from (2.4) (𝑓𝑖(π‘₯(𝑑))βˆ’π‘‡1𝑖π‘₯)𝑇(𝑓𝑖(π‘₯(𝑑))βˆ’π‘‡2𝑖π‘₯)≀0,π‘–βˆˆπ‘† which are equivalent to 𝑓π‘₯(𝑑)(π‘₯(𝑑))𝑇𝑇𝑖1𝑇𝑖2𝑇𝑇𝑖2𝐼𝑓π‘₯(𝑑)(π‘₯(𝑑))≀0,π‘–βˆˆπ‘†.(3.31)
Considering the stochastic 𝐻∞ performance level for the resulting systems (3.25) with nonzero exogenous disturbance input (𝑣(𝑑)β‰ 0), for any𝑑>0, we define that ξ‚»ξ€œπ½(𝑑)=𝐸𝑑0𝑧𝑇(𝑠)𝑧(𝑠)βˆ’π›Ύ2𝑣𝑇(𝑠)𝑣(𝑠)d𝑠.(3.32)
By general ItΓ΄ formula, for a given positive scalar πœ†, we get the following: ξ‚»ξ€œπ½(𝑑)=𝐸𝑑𝑑0𝑧𝑇(𝑠)𝑧(𝑠)βˆ’π›Ύ2𝑣𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(π‘₯(𝑠),𝑠,𝑖)dπ‘ ξ‚Όξ‚»ξ€œβˆ’πΈ(𝑉(π‘₯(𝑑),𝑑,𝑖))≀𝐸𝑑0𝑧𝑇(𝑠)𝑧(𝑠)βˆ’π›Ύ2𝑣𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(π‘₯(𝑠),𝑠,𝑖)βˆ’πœ†π‘–ξ€·π‘“π‘–(π‘₯(𝑑))βˆ’π‘‡1𝑖π‘₯(𝑑)𝑇𝑓𝑖(π‘₯(𝑑))βˆ’π‘‡2𝑖π‘₯(𝑑)dπ‘ ξ‚Όξ‚»ξ€œβ‰€πΈπ‘‘0πœŽπ‘‡(𝑠)Ξ©π‘–πœŽ(𝑠)d𝑠,(3.33) where Ω𝑖=Σ𝑖+βŽ›βŽœβŽœβŽξ€·πœ€π‘–π‘ƒπ‘–πΊπ‘–ξ€Έπ‘‡πœ€π‘–βˆ’1πΌξ€·πœ€π‘–π‘ƒπ‘–πΊπ‘–ξ€Έ+𝐽𝑇𝑖𝐽𝑖000βˆ’π›Ύ2⎞⎟⎟⎠+𝐢𝐼0000𝑇𝑖+𝐾𝑇𝑖𝐹𝑇𝑖𝐷𝑇𝑖𝐻𝑇𝑖0ξƒ­πœ€π‘–βˆ’1𝐼𝐢𝑇𝑖+𝐾𝑇𝑖𝐹𝑇𝑖𝐷𝑇𝑖𝐻𝑇𝑖0𝑇+βŽ›βŽœβŽœβŽξπ‘‡βˆ’πœ†π‘–1𝑇0βˆ’πœ†π‘–2𝑇000βˆ’πœ†π‘‡π‘–2⎞⎟⎟⎠.0βˆ’πœ†πΌ(3.34)
By Schur complement lemma, we see that Ω𝑖<0 is equivalent to the following matrix inequalities: βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘€π‘–βˆ’πœ†π‘–ξπ‘‡π‘–1πΈπ‘–π‘ƒπ‘–βˆ’πœ†π‘–ξπ‘‡π‘–2π‘‹π‘–βˆ’1Ξ˜π‘‡π‘–2π‘‹π‘–βˆ’1Ξ˜π‘‡π‘–2πœ€π‘–π‘ƒπ‘–πΊπ‘–π½π‘‡π‘–βˆ—βˆ’π›Ύ2𝐼0𝐻𝑇𝑖𝐻𝑇𝑖00βˆ—βˆ—βˆ’πœ†π‘–πΌ0000βˆ—βˆ—βˆ—βˆ’π‘ƒπ‘–βˆ’1000βˆ—βˆ—βˆ—βˆ—βˆ’πœ€π‘–πΌ00βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’πœ€π‘–βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ πΌ0βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’πΌ<0,π‘–βˆˆπ‘†,(3.35) which is implied in LIMs (3.26). Hence 𝐽(𝑑)<0.
Therefore, the inequality ‖𝑧‖𝐸2<𝛾‖𝑣‖2 holds. The proof is completed.

Similar to the proof of Theorem 3.5, we can get the following theorem without proof immediately.

Theorem 3.7. If there exist the positive matrices 𝑋𝑖>0, and the constant matrices π‘Œπ‘– with approximate dimensions, for the positive constant πœ€π‘– and the given scalar πœ†π‘–, such that the following LMIs hold ξΞ˜π‘–=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽξ‚Ξ˜π‘–1πΈπ‘–πΌβˆ’πœ†π‘–π‘‹π‘–ξπ‘‡π‘–2ξ‚Ξ˜π‘‡π‘–2ξ‚Ξ˜π‘‡π‘–2π΅π‘–π‘…π‘–π‘Œπ‘‡π‘–Ξ˜π‘–3βˆ—βˆ’π›Ύ2𝐼0𝐻𝑇𝑖𝐻𝑇𝑖000βˆ—βˆ—βˆ’πœ†π‘–πΌ00000βˆ—βˆ—βˆ—βˆ’π‘‹π‘–0𝐷𝑖𝑅𝑖00βˆ—βˆ—βˆ—βˆ—βˆ’πœ€π‘–πΌ000βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π‘…π‘–00βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π‘…π‘–ξ‚πΉβˆ’2𝑖00βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—Ξ˜π‘–4⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠<0,π‘–βˆˆπ‘†.(3.36) then the NSMJSs (3.27) are exponential mean-square stable with convergence rate 𝛼and disturbance attenuation𝛾. In this case, the desired controllers are given as follows: 𝐾𝑖=π‘Œπ‘–π‘‹π‘–βˆ’1.(3.37)

4. Numerical Example with Simulation

In this section, we will give an example to show the usefulness of the derived results and the effectiveness of the proposed methods (Figure 1).

431576.fig.001
Figure 1: The Markovian chain π‘Ÿ(𝑑).

Consider linear SMJSs (3.14) with 𝑆={1,2}, and the system parameters are given as follows:𝐴1=ξƒͺ0.30.30.5βˆ’0.20βˆ’0.30.100.3,𝐴2=ξƒͺ,𝐢0.50.20.2βˆ’0.20βˆ’0.40.200.21=ξƒͺ0.50.20.100.2βˆ’0.10.3βˆ’0.1βˆ’0.3,𝐢2=ξƒͺ,𝐡0.20.10.30.1βˆ’0.30.500.1βˆ’0.51=diag(0.5,0.4,0.5),𝐡2𝐸=diag(0.5,0.4,0.5),1=𝐸2=(0.3,0.1,0.5)𝑇,𝐻2=𝐻1=(0.2,0.1,0.3)𝑇,𝐷2=𝐷1=diag(0.2,0.3,0.4),𝐽1𝐽=(0.3,0.2,0.6),2=(0.1,βˆ’0.1,0.4),𝛾=0.9.(4.1)

The actuator failure parameters are as follows:0.2≀𝑓𝑖1≀0.4,0.1≀𝑓𝑖2≀0.7,0.1≀𝑓𝑖3≀0.9,π‘–βˆˆπ‘†={1,2}.(4.2)

From (2.8) and (2.9), we have𝐹10=𝐹20𝐹=diag(0.3,0.4,0.5),10=𝐹20=diag(0.1,0.3,0.4).(4.3)

From Figure 2, we can see that the uncontrolled LSMJSs are not stable, according to Theorem 3.5. By using the LMI toolbox, the controller parameters can be calculated as follows:𝐾1=ξƒͺβˆ’56.2264βˆ’6.3843βˆ’67.8069βˆ’1.1129βˆ’8.9588βˆ’3.6802βˆ’0.97540.1795βˆ’3.2600,𝐾2=ξƒͺ.βˆ’41.78466.0578βˆ’200.8802βˆ’1.1365βˆ’7.5245βˆ’11.02090.1171βˆ’0.4055βˆ’0.7561(4.4)

431576.fig.002
Figure 2: The state curve of uncontrolled LSMJSs (3.14).

Figures 3 and 4 give the simulation results of the response for the closed-loop LSMJSs, which confirm that the closed-loop LSMJSs are exponential mean-square stable with convergence rate 𝛼 and disturbance attenuation 𝛾.

431576.fig.003
Figure 3: The state curve of closed-loop LSMJSs (3.14).
431576.fig.004
Figure 4: The curve of |𝑧(𝑑)|2βˆ’π›Ύ2|𝑣(𝑑)| for controlled LSMJSs (3.14).

5. Conclusions

In this paper, we have studied the robust reliable 𝐻∞ control problems for a class of NSMJSs. The system under study contains ItΓ΄-type stochastic disturbance, Markovian jumps, sector-bounded nonlinearities, and norm-bounded stochastic nonlinearities. Based on the Lyapunov stability theory and ItΓ΄ differential rule, sufficient condition which ensures exponential mean-square stable with convergence rate 𝛼and disturbance attenuation 𝛾 for SMJSs has been established in Lemma 3.3. By the lemma, together with the LMIs techniques, the sufficient conditions for the designation of the robust reliable 𝐻∞ controller of linear SMJSs and NSMJSs have been obtained in terms of LMIs. Finally, a numerical example has been given to show the usefulness of the derived results and the effectiveness of the proposed methods. It is possible to extend our main results to the NSMJSs with time delay by using delay-dependent techniques, which is one of the future research topics.

Acknowledgments

This work is supported by National Science Foundation of China (NSFC) under Grant 61104127, 60904060, and 61134012 Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Wuhan University of Science and Technology) under Grant Y201101.

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