Robust Reliable 𝐻_∞ Control for Nonlinear Stochastic Markovian Jump Systems

Abstract

The robust reliable 𝐻_∞ control problem for a class of nonlinear stochastic Markovian jump systems (NSMJSs) is investigated. The system under consideration includes Itô-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 Itô 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 Itô 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 Itô-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 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the Matrix 𝑋-𝑌 is positive semidefinite (respectively, positive definite). 𝐼is an identity matrix with appropriate dimensions; the subscript “𝑇” represents the Transposition. 𝐸(⋅)denotes the expectation operator with respect to some probability measure 𝑃. ℒ₂[0,∞) is the space of square integrable vector functions over [0,∞); let (Ω,ℱ,𝑃) be a complete probability space which is relative to an increasing family (ℱ_𝑡)_𝑡>0 of 𝜎 algebras (ℱ_𝑡)_𝑡>0⊂ℱ, where Ω is the samples space, ℱ is 𝜎 algebra of subsets of the sample space, and 𝑃 is the probability measure on ℱ.‖⋅‖_𝐸2=‖𝐸(⋅)‖₂, while ‖⋅‖₂ stands for the usual ℒ₂[0,∞) norm, 𝑅^𝑛 and 𝑅^𝑛×𝑚 denote the 𝑛 dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices, respectively. In this paper, we provide all spaces 𝕂^𝑘,𝑘≥1 with the usual inner product ⟨⋅,⋅⟩ and its corresponding 2-norm‖⋅‖. Let 𝐿²(Ω,𝕂^𝑘) denote the space of square-integrable 𝕂^𝑘-valued functions on the probability space (Ω,ℱ,𝑃). For any 0<𝑇<∞, we write [0,𝑇] for the closure of the open interval (0,𝑇) in 𝑅 and denote by 𝐿^𝑛2([0,𝑇];𝐿²(Ω,𝕂^𝑘)) the space of the nonanticipative stochastic processes 𝑦(⋅)=(𝑦(⋅))_{𝑡∈[0,𝑇]} with respect to (ℱ_𝑡)_{𝑡∈[0,𝑇]} satisfying ‖𝑦(⋅)‖^2𝐿_𝑛2=𝐸(∫^𝑇0‖𝑦(𝑡)‖²d𝑡)=∫^𝑇0𝐸(‖𝑦(𝑡)‖²)d𝑡<∞. 𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)=𝑖)=𝑉(𝑥(𝑡),𝑡,𝑖), 𝐴(𝑟(𝑡)=𝑖)=𝐴_𝑖𝐵(𝑟(𝑡)=𝑖)=𝐵_𝑖, 𝐴₀(𝑟(𝑡)=𝑖)=𝐴_0𝑖,𝐵₀(𝑟(𝑡)=𝑖)=𝐵_0𝑖,𝐶(𝑟(𝑡)=𝑖)=𝐶_𝑖,𝐷(𝑟(𝑡)=𝑖)=𝐷_𝑖.

2. Problem Formulation and Failure Model

In this paper, we mainly consider the following nonlinear stochastic Markovian jump systems (NSMJSs) with actuator failures: d𝑥(𝑡)=+𝐴𝐶(𝑟((𝑟𝑡)()𝑡)𝑥)(𝑥𝑡)(𝑡+)+𝐵𝐷(𝑟((𝑟𝑡)()𝑡)𝑢)^𝑓𝑢(_𝑓𝑡,𝑟(𝑡))+𝐸(𝑟(𝑡))𝑣(𝑡)+𝑓(𝑟(𝑡),𝑥(𝑡))d𝑡𝑥𝑧𝑡(𝑡)=𝐽(𝑟(𝑡))𝑥(𝑡),₀(𝑡,𝑟(𝑡))+𝐻(𝑟(𝑡))𝑣(𝑡)+𝑔(𝑟(𝑡),𝑥(𝑡))d𝑤(𝑡),=𝑥₀,(2.1) where 𝑥(𝑡)∈𝑅^𝑛 is the system state, 𝑢^𝑓(𝑡)∈𝑅^𝑙 is the control input of actuator fault, 𝑣(𝑡)∈𝑅^𝑞 is the exogenous disturbance input of the systems which belong to ℒ₂[0,∞), 𝑧(𝑡)∈𝑅^𝑟 is the system control output, 𝑤(𝑡) is a zero mean real scalar Weiner processes on a probability space (Ω,ℱ,𝑃) relative to an increase family (ℱ_𝑡)_𝑡>0 of 𝜎 algebras (ℱ_𝑡)_𝑡>0⊂ℱ. 𝐴_𝑖,𝐵_𝑖,𝐸_𝑖,𝐶_𝑖,𝐷_𝑖,𝐹_𝑖,𝐻_𝑖,𝐽_𝑖 are the known real constant matrices with appropriate dimensions. Morever, we assume that𝐸(d𝑤(𝑡))=0,𝐸(d𝑤(𝑡))²=d𝑡.(2.2)

Let 𝑟(𝑡),𝑡≥0, be a right-continuous Markovian chain on the probability space taking values in a finite state space 𝑆=1,2,…,𝑁 with generator Γ=(𝜆_𝑖𝑗)_𝑁×𝑁 given by 𝑃{𝑟(𝑡+Δ)=𝑗∣𝑟(𝑡)=𝑖}=𝜆^𝑖𝑗1+Δ𝜆+_𝑖𝑖𝑜(Δ)if𝑖≠𝑗,Δ+𝑜(Δ)if𝑖=𝑗,(2.3) where Δ>0. Here 𝜆_𝑖𝑗≥0 is the transition rate from manner 𝑖 to manner 𝑗, if 𝑖≠𝑗 while 𝜆_𝑖𝑖=−∑_𝑗≠𝑖𝜆_𝑖𝑗. We assume that the Markovian chain 𝑟(⋅) is independent of the Wienner process 𝑤(⋅). It is well known that almost every sample path of 𝑟(𝑡) is a right-continuous step function with a finite number of simple jump in any finite subinterval of 𝑅₊(∶=[0,+∞)).

𝑓(⋅,⋅)∶𝑆×𝑅^𝑛→𝑅^𝑛 is a unknown nonlinear function which describes the system nonlinearity satisfying the following sector-bounded conditions: 𝑓_𝑖(𝑥(𝑡))−𝑇_1𝑖𝑥^𝑇𝑓_𝑖(𝑥(𝑡))−𝑇_2𝑖𝑥≤0,𝑖∈𝑆,(2.4)𝑔(⋅,⋅)∶𝑆×𝑅^𝑛→𝑅^𝑛 also is a unknown nonlinear function which describes the stochastic nonlinearity satisfying the following: 𝑔^𝑇𝑖(𝑥(𝑡))𝑔_𝑖(𝑥(𝑡))≤𝑥^𝑇𝐺^𝑇𝑖𝐺_𝑖𝑥,𝑖∈𝑆,(2.5) where 𝑇_1𝑖,𝑇_2𝑖,𝐺_𝑖 are known real constant matrices with approximate dimensions.

Remark 2.1. The nonlinearities 𝑓_𝑖(𝑥(𝑡)) are bounded by sectors, which belong to [𝐿_1𝑖,𝐿_2𝑖], 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 𝑔_𝑖(𝑥(𝑡)) satisfy the norm-bounded conditions.

When the actuator experiences failure, we use 𝑢^𝑓(𝑡,𝑟(𝑡)) to describe the control signal form actuators. Consider the following actuator failure model with failure parameter 𝐹_𝑖: 𝑢^𝑓𝑖(𝑡)=𝐹_𝑖𝑢_𝑖(𝑡),(2.6) where 𝐹_𝑖 is the actuator fault matrix with 𝐹_𝑖=diag𝑓_𝑖1,𝑓_𝑖2,…,𝑓_𝑖𝑚,0≤𝑓_𝑖𝑗≤𝑓_𝑖𝑗≤𝑓_𝑖𝑗,𝑓_𝑖𝑗≥1,𝑗=1,2,…,𝑚.(2.7)

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

Define the following: 𝐹_0𝑖=diag𝑓_0𝑖1,𝑓_0𝑖2,…,𝑓_0𝑖𝑚=𝐹_𝑖+𝐹_𝑖2,𝑓^0𝑖𝑗=𝑓^𝑖𝑗+𝑓_𝑖𝑗2,(2.8)𝐹_0𝑖=diag𝑓_0𝑖1,𝑓_0𝑖2,…,𝑓_0𝑖𝑚=𝐹_𝑖−𝐹_𝑖2,𝑓^0𝑖𝑗=𝑓_𝑖𝑗−𝑓_𝑖𝑗2,(2.9) and hence, the matrix 𝐹_𝑖 can be rewritten as 𝐹_𝑖=𝐹_0𝑖+Δ_𝑖=𝐹_0𝑖+diag𝜑_𝑖1,𝜑_𝑖2,…,𝜑_𝑖𝑚,||𝜑_𝑖𝑗||≤𝑓_𝑖𝑗,𝑗=1,2,…,𝑚.(2.10)

In this paper, our aim is to design the controller 𝑢_𝑖(𝑡)=𝐾_𝑖𝑥(𝑡),𝑖∈𝑆, such that the closed-loop systems satisfy the following conditions:(i)without the exogenous disturbance input (i.e., 𝑣(𝑡)=0), the closed-loop control systems (2.1) are globally exponentially stable with convergence rate 𝛼>0;(ii)with zero initial condition (i.e., 𝑥(𝑡₀)=0) and nonzero exogenous disturbance input (i.e., 𝑣(𝑡)≠0), the following inequality holds:‖𝑧‖_𝐸2<𝛾‖𝑣‖₂i.e.,^𝑇0𝑧^𝑇(𝑡)𝑧(𝑡)d𝑡≤𝛾²^𝑇0𝑣^𝑇(𝑡)𝑣(𝑡)d𝑡.(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=^𝑆1𝑆3_∗𝑆2 with 𝑆^𝑇1=𝑆₁,𝑆^𝑇2=𝑆₂, the following conditions are equivalent:(1)𝑆<0,(2)𝑆₂<0,𝑆₁−𝑆₃𝑆²⁻¹𝑆^𝑇3<0,(3)𝑆₁<0,𝑆₂−𝑆₃𝑆¹⁻¹𝑆^𝑇3<0.

Lemma 3.2 (see [21]). Let 𝑥∈ℝ^𝑛 and 𝑦∈ℝ^𝑛. Then, for any positive scalar 𝜀, we have 𝑥^𝑇𝑦+𝑦^𝑇𝑥≤𝜀𝑥^𝑇𝑥+𝜀⁻¹𝑦^𝑇𝑦.(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: d𝑥(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐸_𝑖𝑣𝑧((𝑡𝑡))=d𝑡𝐽+_𝑖𝐶_𝑖𝑥(𝑡)+𝐻_𝑖𝑣(𝑡)d𝑤(𝑡),𝑥𝑡𝑥(𝑡),₀=𝑥₀.(3.2)

Lemma 3.3. Suppose that 𝑃(𝑡,𝑟(𝑡))>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: Ξ_𝑖(𝑡)=⎛⎜⎜⎝𝑀^𝑖(𝑡)+𝐽^𝑇𝑖𝐽_𝑖𝑃_𝑖𝐸_𝑖𝐶^𝑇𝑖∗−𝛾²𝐼𝐻^𝑇𝑖∗∗−𝑃^𝑖−1(𝑡)⎞⎟⎟⎠<0,𝑖∈𝑆,(3.3) where 𝑀_𝑖(𝑡)=𝐴^𝑇𝑖𝑃_𝑖(𝑡)+𝑃_𝑖(𝑡)𝐴_𝑖+𝑃̇(𝑡)+∑_𝑗∈𝑆𝜆_𝑖𝑗𝑃_𝑗(𝑡).

Proof. At first, let 𝑣(𝑡)=0, and defining the following Lyapunov function: 𝑉(𝑥(𝑡),𝑡,𝑖)=𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)=𝑖)=𝑥^𝑇(𝑡)𝑃(𝑡,𝑟(𝑡)=𝑖)𝑥(𝑡)=𝑥^𝑇(𝑡)𝑃_𝑖(𝑡)𝑥(𝑡).(3.4)
By Itô formula, we get the following: ℒ𝑉(𝑥(𝑡),𝑡,𝑖)=𝑥^𝑇(𝑡)𝑀_𝑖(𝑡)+𝐶^𝑇𝑖𝑃_𝑖(𝑡)𝐶_𝑖𝑥(𝑡),(3.5) the matrix function inequalities (3.3) imply that ℒ𝑉(𝑥(𝑡),𝑡,𝑖)<0, and let 𝑎_𝑖=𝜆_max(−Ξ_𝑖(𝑡)), 𝑎=max_𝑖∈𝑆(𝑎_𝑖), where 𝜆_max(⋅) means the maximum eigenvalue of matrix (⋅), and we have ℒ𝑉(𝑥(𝑡),𝑡,𝑖)≤−𝑎𝑥^𝑇(𝑡)𝑥(𝑡).(3.6)
Hence 𝑑𝑒^𝛼𝑡𝑉(𝑥(𝑡),𝑡,𝑖)=≤𝛼(𝑏𝑒𝛼^𝛼𝑡𝑉−(𝑎𝑥)𝑒(𝑡)_𝛼𝑡,𝑡,𝑖)+𝑒^𝛼𝑡𝑑𝑉(𝑥(𝑡),𝑡,𝑖)‖𝑥(𝑡)‖²+𝑒^𝛼𝑡2𝑥^𝑇(𝑡)𝑃_𝑖(𝑡)𝐶_𝑖𝑥(𝑡)𝑑𝑤(𝑡),(3.7) where 𝑏_𝑖=sup_𝑡≥𝑡0{𝜆_max(𝑃_𝑖(𝑡))}, and 𝑏=max_𝑖∈𝑆(𝑏_𝑖). Integrating the both sides of above inequality from 𝑡₀ to 𝑇and taking expectation, we obtain that 𝐸𝑒^𝛼𝑇𝑉(𝑥(𝑇),𝑇,𝑖)−𝑉𝑥₀,𝑡₀,𝑖≤(𝑏𝛼−𝑎)𝐸^𝑇𝑡₀𝑒^𝛼𝑠‖𝑥(𝑠)‖²d𝑠.(3.8)
Set 𝛼=𝑎/𝑏, and the following inequality is obtained: 𝑒^𝛼𝑇min_𝑖∈𝑆𝜆_min𝑃_𝑖(𝑇)𝐸‖𝑥(𝑇)‖²≤𝐸𝑒^𝛼𝑇𝑉(𝑥(𝑇),𝑇,𝑖)≤𝐸𝑉𝑥₀,𝑡₀,𝑖,(3.9) which implies that 𝐸‖𝑥(𝑇)‖²≤𝐸𝑉𝑥₀,𝑡₀,𝑖1min_𝑖∈𝑆𝜆_min𝑃_𝑖(𝑇)𝑒^−𝛼𝑇.(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(𝑡)≠0), for any𝑡>0, we define that 𝐽(𝑡)=𝐸^𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)d𝑠.(3.11)
By general Itô formula, we get he following: 𝐽(𝑡)=≤𝐸𝐸^𝑡𝑡_0𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(𝑥(𝑠),𝑠,𝑖)d𝑠−𝐸(𝑉(𝑥(𝑡),𝑡,𝑖))𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(𝑥(𝑠),𝑠,𝑖)d𝑠≤𝐸^𝑡0𝜂^𝑇(𝑠)Ω_𝑖(𝑠)𝜂(𝑠)d𝑠,(3.12) where 𝜂^𝑇(𝑡)=(𝑥^𝑇(𝑡)𝑣^𝑇(𝑡)), Ω_𝑖(𝑡)=^{𝑀𝑖(𝑡)+𝐽𝑇𝑖𝐽𝑖𝑃𝑖(𝑡)𝐸𝑖}_{𝐸𝑇𝑖𝑃𝑖(𝑡)−𝛾2𝐼}+^𝐶𝑇𝑖_𝐻^𝑇𝑖𝑃^𝑖(𝑡)^𝐶𝑇𝑖_𝐻𝑇𝑖^𝑇 From (3.3) we know that Ω(𝑡)<0, which implies that 𝐽(𝑡)<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: d𝑥(𝑡)=𝐴_𝑖+𝐵_𝑖𝐹_𝑖𝐾_𝑖𝑥(𝑡)+𝐸_𝑖𝑣𝑧((𝑡𝑡))=d𝑡𝐽+_𝑖𝐶_𝑖+𝐷_𝑖𝐹_𝑖𝐾_𝑖𝑥(𝑡)+𝐻_𝑖𝑣(𝑡)d𝑤(𝑡),𝑥𝑡𝑥(𝑡),₀=𝑥₀.(3.14)

Theorem 3.4. If there exist the positive matrices 𝑋_𝑖>0, and the constant matrices 𝑌_𝑖 with approximate dimensions, such that the following LMIs hold Θ_𝑖=⎛⎜⎜⎝Θ^𝑖1𝐸_𝑖Θ_𝑖2Θ_𝑖3∗∗−𝛾∗²𝐼−𝐻𝑋^𝑇𝑖_𝑖0∗∗∗Θ0_𝑖4⎞⎟⎟⎠<0,𝑖∈𝑆,(3.15) where Θ_𝑖1=𝑋_𝑖𝐴^𝑇𝑖+𝐴_𝑖𝑋_𝑖+𝐵_𝑖𝐹_𝑖𝑌_𝑖+𝑌^𝑇𝑖𝐹^𝑇𝑖𝐵^𝑇𝑖+𝜆_𝑖𝑖𝑋_𝑖,Θ_𝑖2=𝑋_𝑖𝐶^𝑇𝑖+𝑌^𝑇𝑖𝐹^𝑇𝑖𝐷^𝑇𝑖, Θ_𝑖3=Θ√_𝑖4𝜆_𝑖1𝑋_𝑖⋯√𝜆_{𝑖,𝑖−1}𝑋_𝑖√𝜆_𝑖,𝑖+1𝑋_𝑖⋯√𝜆_𝑖𝑁𝑋_𝑖𝑋_𝑖𝐽^𝑇𝑖,=diag−𝑋₁,…,−𝑋_𝑖−1,−𝑋_𝑖+1,…,−𝑋_𝑁,−𝐼,(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: 𝐾_𝑖=𝑌_𝑖𝑋^𝑖−1.(3.17)

Proof. Defining the following Lyapunov function: 𝑉(𝑥(𝑡),𝑡,𝑖)=𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)=𝑖)=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡).(3.18)
By Lemma 3.3, and similar to the proof of Lemma 3.3, we can get the following: ℒ𝑉(𝑥(𝑡),𝑡,𝑖)≤𝜂^𝑇(𝑡)Ξ_𝑖𝜂(𝑡),(3.19) where Ξ_𝑖=^{𝑀𝑖𝑃𝑖𝐸𝑖𝐶𝑇𝑖+𝐾𝑇𝑖𝐹𝑇𝑖𝐷𝑇𝑖∗−𝛾2𝐼𝐻𝑇𝑖}_{∗∗−𝑃𝑖−1}𝑀^𝑖=(𝐴_𝑖+𝐵_𝑖𝐹_𝑖𝐾_𝑖)^𝑇𝑃_𝑖+𝑃_𝑖(𝐴_𝑖+𝐵_𝑖𝐹_𝑖𝐾_𝑖)+∑_𝑗∈𝑆𝜆_𝑖𝑗𝑃_𝑗.
Using Schur complement lemma together with contragredient transformation, we know that LMIs (3.15) imply that Ξ_𝑖<0. So we have 𝐽(𝑡)==𝐸𝐸^𝑡0_𝑡𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)d𝑠≤𝐸_𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(𝑥(𝑠),𝑠,𝑖)d𝑠−𝐸(𝑉(𝑥(𝑡),𝑡,𝑖))𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(𝑥(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 𝑋_𝑖>0, the positive diagonal matrices 𝑅_𝑖>0, and the constant matrices 𝑌_𝑖 with approximate dimensions, such that the following LMIs hold: Θ_𝑖=⎛⎜⎜⎜⎜⎜⎝Θ^𝑖1𝐸_𝑖Θ_𝑖2Θ_𝑖3𝐵_𝑖𝑅_𝑖𝑌^𝑇𝑖∗∗−𝛾∗²𝐼−𝐻𝑋^𝑇𝑖_𝑖000∗∗∗Θ0_𝑖4𝐷_𝑖𝑅_𝑖0∗∗∗∗−0𝑅_𝑖0∗∗∗∗∗−𝑅0_𝑖𝐹^−2𝑖0⎞⎟⎟⎟⎟⎟⎠<0,𝑖∈𝑆,(3.21) where Θ_𝑖1=𝑋_𝑖𝐴^𝑇𝑖+𝐴_𝑖𝑋_𝑖+𝐵_𝑖𝐹_𝑖0𝑌_𝑖+𝑌^𝑇𝑖𝐹^𝑇𝑖0𝐵^𝑇𝑖+𝜆_𝑖𝑖𝑋_𝑖, Θ_𝑖2=𝑋_𝑖𝐶^𝑇𝑖+𝑌^𝑇𝑖𝐹^𝑇𝑖0𝐷^𝑇𝑖, 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: 𝐾_𝑖=𝑌_𝑖𝑋^𝑖−1.(3.22)

Proof. Noticing (2.10), we can see that Θ_𝑖 in (3.15) can be rewritten as Θ_𝑖=Θ_𝑖0+𝐵^𝑇𝑖0𝐷^𝑇𝑖0^𝑇Δ_𝑖𝑌^𝑖000+𝑌_𝑖000^𝑇Δ_𝑖𝐵^𝑇𝑖0𝐷^𝑇𝑖0,(3.23) where Θ_𝑖0=⎛⎜⎜⎝Θ^{𝑖1𝐸𝑖}Θ_{𝑖2Θ𝑖3∗−𝛾2𝐼𝐻𝑇𝑖0∗∗−𝑋𝑖0∗∗∗Θ𝑖4}⎞⎟⎟⎠.
By Lemma 3.2, we have Θ_𝑖≤Θ_𝑖0+𝐵^𝑇𝑖0𝐷^𝑇𝑖0^𝑇𝑅_𝑖𝐵^𝑇𝑖0𝐷^𝑇𝑖0+𝑌_𝑖000^𝑇𝑅^𝑖−1𝐹^20𝑖𝑌^𝑖000,(3.24) by Schur complement, we know that Θ_𝑖<0 imply that Θ_𝑖<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: d𝑥(𝑡)=+𝐴𝐶_𝑖+_𝑖𝐵_𝑖𝐹_𝑖𝐾_𝑖𝑥(𝑡)+𝐸_𝑖𝑣(𝑡)+𝑓_𝑖(𝑥(𝑡))d𝑡𝑧(𝑡)=𝐻_𝑖+𝐷_𝑖𝐹_𝑖𝐾_𝑖𝑥(𝑡)+𝐻_𝑖𝑣(𝑡)+𝑔_𝑖(𝑥(𝑡))d𝑤(𝑡),𝑥𝑡𝑥(𝑡),₀=𝑥₀.(3.25)

Theorem 3.6. If there exist the positive matrices 𝑋_𝑖>0, and the constant matrices 𝑌_𝑖 with approximate dimensions, for the positive constant 𝜀_𝑖 and the given scalar 𝜆_𝑖, such that the following LMIs hold: Θ_𝑖=⎛⎜⎜⎜⎜⎜⎝Θ^𝑖1𝐸_𝑖𝐼−𝜆_𝑖𝑋_𝑖𝑇_𝑖2Θ^𝑇𝑖2Θ^𝑇𝑖2Θ_𝑖3∗∗−𝛾∗²𝐼−𝜆0_𝑖𝐻^𝑇𝑖𝐻^𝑇𝑖0∗∗∗𝐼−0𝑋_𝑖00∗∗∗∗−𝜀0_𝑖0∗∗∗∗∗𝐼0Θ_𝑖4⎞⎟⎟⎟⎟⎟⎠<0,𝑖∈𝑆,(3.26) where Θ_𝑖3=(𝜀_𝑖𝐺_𝑖𝜆_𝑖𝑋_𝑖𝑇_𝑖1Θ_𝑖3),Θ_𝑖4=diag(−𝜀_𝑖𝐼,−𝜆_𝑖𝑇_𝑖1,Θ_𝑖4),𝑇_𝑖1=(𝑇^𝑇𝑖1𝑇_𝑖2+𝑇^𝑇𝑖2𝑇_𝑖1)/2,T_i2=−(T^Ti₁+T^Ti₂)/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: 𝐾_𝑖=𝑌_𝑖𝑋^𝑖−1.(3.27)

Proof. Defining the following Lyapunov function: 𝑉(𝑥(𝑡),𝑡,𝑖)=𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)=𝑖)=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡),(3.28)
by Itô formula, we get the following: ℒ𝑉(𝑥(𝑡),𝑡,𝑖)=+2𝑥^𝑇𝐶(𝑡_𝑖)𝑃_𝑖𝐴_𝑖+𝐵_𝑖𝐹_𝑖𝐾_𝑖𝑥(𝑡)+𝐸_𝑖𝑣(𝑡)+𝑓_𝑖(𝑥(𝑡))+_𝑗∈𝑆𝜆_𝑖𝑗𝑥^𝑇(𝑡)𝑃_𝑗𝑥(𝑡)+𝐷𝐵_𝑖𝐹_𝑖𝐾_𝑖𝑥(𝑡)+𝐻_𝑖𝑣(𝑡)+𝑔_𝑖(𝑥(𝑡))^𝑇≤×𝜎_𝑇𝑃_𝑖𝐶_𝑖+𝐷𝐵_𝑖𝐹_𝑖𝐾_𝑖𝑥(𝑡)+𝐻_𝑖𝑣(𝑡)+𝑔_𝑖(𝑥(𝑡))(𝑡)Σ_𝑖𝜎(𝑡)+𝑥^𝑇(𝑡)𝐺^𝑇𝑖𝑃_𝑖𝐺_𝑖𝑥(𝑡)+2𝐶_𝑖+𝐷𝐵_𝑖𝐹_𝑖𝐾_𝑖𝑥(𝑡)+𝐻_𝑖𝑣(𝑡)^𝑇𝑃_𝑖𝑔_𝑖(𝑥(𝑡)),(3.29) where 𝜎^𝑇(𝑡)=[𝑥^𝑇(𝑡),𝑣^𝑇(𝑡),𝑓^𝑇𝑖(𝑥(𝑡))],Σ_𝑖=^{𝑀𝑖𝑃𝑖𝐸𝑖𝑃𝑖𝐸𝑇𝑖𝑃}_𝑖^𝑃00_𝑇𝑖00+^{𝐶𝑇𝑖+𝐾𝑇𝑖𝐹𝑇𝑖𝐷𝑇𝑖𝐻𝑇𝑖}₀𝑃^𝑖^{𝐶𝑇𝑖+𝐾𝑇𝑖𝐹𝑇𝑖𝐷𝑇𝑖𝐻𝑇𝑖}₀^𝑇.
By Lemma 3.2, it follows that 2≤𝐶𝜎_𝑖𝑇+𝐷𝐵_𝑖𝐹_𝑖𝐾_𝑖𝑥(𝑡)+𝐻_𝑖𝑣(𝑡)^𝑇𝑃_𝑖𝑔_𝑖(𝑥(𝑡))(𝑡)𝐶^𝑇𝑖+𝐾^𝑇𝑖𝐹^𝑇𝑖𝐷^𝑇𝑖𝐻^𝑇𝑖0𝜀^𝑖−1𝐼𝐶^𝑇𝑖+𝐾^𝑇𝑖𝐹^𝑇𝑖𝐷^𝑇𝑖𝐻^𝑇𝑖0^𝑇𝜎(𝑡)+𝑥^𝑇(𝑡)𝜀_𝑖𝑃_𝑖𝐺_𝑖^𝑇𝜀^𝑖−1𝐼𝜀_𝑖𝑃_𝑖𝐺_𝑖𝑥(𝑡),(3.30) from (2.4) (𝑓_𝑖(𝑥(𝑡))−𝑇_1𝑖𝑥)^𝑇(𝑓_𝑖(𝑥(𝑡))−𝑇_2𝑖𝑥)≤0,𝑖∈𝑆 which are equivalent to 𝑓𝑥(𝑥(𝑡()𝑡))^𝑇𝑇^𝑖1𝑇_𝑖2𝑇^𝑇𝑖2𝐼𝑓𝑥(𝑥(𝑡()𝑡))≤0,𝑖∈𝑆.(3.31)
Considering the stochastic 𝐻_∞ performance level for the resulting systems (3.25) with nonzero exogenous disturbance input (𝑣(𝑡)≠0), for any𝑡>0, we define that 𝐽(𝑡)=𝐸^𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)d𝑠.(3.32)
By general Itô formula, for a given positive scalar 𝜆, we get the following: 𝐽=(𝑡𝐸)_𝑡𝑡0≤𝐸_𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(𝑥(𝑠),𝑠,𝑖)d𝑠−𝐸(𝑉(𝑥(𝑡),𝑡,𝑖))≤𝐸_𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)+ℒ𝑉(𝑥(𝑠),𝑠,𝑖)−𝜆_𝑖𝑓_𝑖(𝑥(𝑡))−𝑇_1𝑖𝑥(𝑡)^𝑇𝑓_𝑖(𝑥(𝑡))−𝑇_2𝑖𝑥(𝑡)d𝑠𝜎^𝑇(𝑠)Ω_𝑖𝜎(𝑠)d𝑠,(3.33) where Ω_𝑖=Σ_𝑖+⎛⎜⎜⎝𝜀^𝑖𝑃_𝑖𝐺_𝑖^𝑇𝜀^𝑖−1𝐼0𝜀_𝑖𝑃_𝑖𝐺_𝑖+𝐽^𝑇𝑖𝐽_𝑖−𝛾0₂0+𝐶00𝐼0_𝑇𝑖0⎞⎟⎟⎠+𝐾^𝑇𝑖𝐹^𝑇𝑖𝐷^𝑇𝑖𝐻^𝑇𝑖0𝜀^𝑖−1𝐼𝐶^𝑇𝑖+𝐾^𝑇𝑖𝐹^𝑇𝑖𝐷^𝑇𝑖𝐻^𝑇𝑖0^𝑇+⎛⎜⎜⎝−𝜆𝑇^𝑖10−𝜆𝑇_𝑖2−𝜆0𝑇_𝑇𝑖2000−𝜆𝐼⎞⎟⎟⎠.(3.34)
By Schur complement lemma, we see that Ω_𝑖<0 is equivalent to the following matrix inequalities: ⎛⎜⎜⎜⎜⎜⎜⎝𝑀_𝑖−𝜆_𝑖𝑇_𝑖1𝐸_𝑖𝑃_𝑖−𝜆_𝑖𝑇_𝑖2𝑋^𝑖−1Θ^𝑇𝑖2𝑋^𝑖−1Θ^𝑇𝑖2𝜀_𝑖𝑃_𝑖𝐺_𝑖𝐽^𝑇𝑖∗∗−𝛾∗²𝐼−𝜆0_𝑖𝐻^𝑇𝑖𝐻^𝑇𝑖00∗∗∗𝐼−𝑃0_𝑖−1000∗∗∗∗−𝜀0_𝑖00∗∗∗∗∗𝐼−𝜀0_𝑖0∗∗∗∗∗∗𝐼−0𝐼⎞⎟⎟⎟⎟⎟⎟⎠<0,𝑖∈𝑆,(3.35) which is implied in LIMs (3.26). Hence 𝐽(𝑡)<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 𝑋_𝑖>0, and the constant matrices 𝑌_𝑖 with approximate dimensions, for the positive constant 𝜀_𝑖 and the given scalar 𝜆_𝑖, such that the following LMIs hold Θ_𝑖=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝Θ^𝑖1𝐸_𝑖𝐼−𝜆_𝑖𝑋_𝑖𝑇_𝑖2Θ^𝑇𝑖2Θ^𝑇𝑖2𝐵_𝑖𝑅_𝑖𝑌^𝑇𝑖Θ_𝑖3∗∗−𝛾∗²𝐼−𝜆0_𝑖𝐻^𝑇𝑖𝐻^𝑇𝑖000∗∗∗𝐼−0𝑋_𝑖0000∗∗∗∗−𝜀0_𝑖𝐷_𝑖𝑅_𝑖00∗∗∗∗∗𝐼−0𝑅_𝑖00∗∗∗∗∗∗−𝑅0_𝑖0∗∗∗∗∗∗∗𝐹^−2𝑖00Θ_𝑖4⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠<0,𝑖∈𝑆.(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: 𝐾_𝑖=𝑌_𝑖𝑋^𝑖−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).

Figure 1 

The Markovian chain 𝑟(𝑡).

Consider linear SMJSs (3.14) with 𝑆={1,2}, and the system parameters are given as follows:𝐴𝐶₁=−000..31.2000.3−000..53.3,𝐴₂=−000..52.2000.2−000..22₁.4,𝐵=000..53−000..22.1−−000.1..31,𝐶₂=000..21−000..11.3−000..35.5₁,𝐸=diag(0.5,0.4,0.5),𝐵₂=diag(0.5,0.4,0.5),₁𝐷=₂𝐸₂=(0.3,0.1,0.5)^𝑇,𝐻₂=𝐻₁=(0.2,0.1,0.3)^𝑇,=𝐷₁=𝐽diag(0.2,0.3,0.4),𝐽₁=(0.3,0.2,0.6),₂=(0.1,−0.1,0.4),𝛾=0.9.(4.1)

The actuator failure parameters are as follows:0.2≤𝑓_𝑖1≤0.4,0.1≤𝑓_𝑖2≤0.7,0.1≤𝑓_𝑖3≤0.9,𝑖∈𝑆={1,2}.(4.2)

From (2.8) and (2.9), we have𝐹₁₀=𝐹₂₀=diag(0.3,0.4,0.5),𝐹₁₀=𝐹₂₀=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=−−−5106...192172256944−−086.1..93758984583−−−6337..62.8860006209,𝐾₂=−−041.11..11737861456−−670.0..54520745855−−−21001.0.70.5826801092.(4.4)

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.

Figure 3 

The state curve of closed-loop LSMJSs (3.14).

Figure 4 

The curve of |𝑧(𝑡)|2−𝛾2|𝑣(𝑡)| 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 Itô-type stochastic disturbance, Markovian jumps, sector-bounded nonlinearities, and norm-bounded stochastic nonlinearities. Based on the Lyapunov stability theory and Itô 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.