# Robust ReliableH∞ Control for Nonlinear Stochastic Markovian Jump Systems

Authors: Guici Chen; Yi Shen
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The robust reliableH∞ 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.


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## Body

## 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 reliableH∞ 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 matricesX 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 (ℱt)t>0 of σ algebras (ℱt)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, Rn and Rn×m denote the n dimensional Euclidean space and the set of all n×m real matrices, respectively. In this paper, we provide all spaces 𝕂k,k≥1 with the usual inner product 〈·,·〉 and its corresponding 2-norm∥·∥. Let L2(Ω,𝕂k) denote the space of square-integrable 𝕂k-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 L2n([0,T];L2(Ω,𝕂k)) the space of the nonanticipative stochastic processes y(·)=(y(·))t∈[0,T] with respect to (ℱt)t∈[0,T] satisfying ∥y(·)∥L2n2=E(∫0T∥y(t)∥2dt)=∫0TE(∥y(t)∥2)dt<∞. V(x(t),t,r(t)=i)=V(x(t),t,i), A(r(t)=i)=AiB(r(t)=i)=Bi, A0(r(t)=i)=A0i,B0(r(t)=i)=B0i,C(r(t)=i)=Ci,D(r(t)=i)=Di.


## 1.1. Notations

Throughout this paper, for symmetric matricesX 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 (ℱt)t>0 of σ algebras (ℱt)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, Rn and Rn×m denote the n dimensional Euclidean space and the set of all n×m real matrices, respectively. In this paper, we provide all spaces 𝕂k,k≥1 with the usual inner product 〈·,·〉 and its corresponding 2-norm∥·∥. Let L2(Ω,𝕂k) denote the space of square-integrable 𝕂k-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 L2n([0,T];L2(Ω,𝕂k)) the space of the nonanticipative stochastic processes y(·)=(y(·))t∈[0,T] with respect to (ℱt)t∈[0,T] satisfying ∥y(·)∥L2n2=E(∫0T∥y(t)∥2dt)=∫0TE(∥y(t)∥2)dt<∞. V(x(t),t,r(t)=i)=V(x(t),t,i), A(r(t)=i)=AiB(r(t)=i)=Bi, A0(r(t)=i)=A0i,B0(r(t)=i)=B0i,C(r(t)=i)=Ci,D(r(t)=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:(2.1)dx(t)=[A(r(t))x(t)+B(r(t))uf(t,r(t))+E(r(t))v(t)+f(r(t),x(t))]dt+[C(r(t))x(t)+D(r(t))uf(t,r(t))+H(r(t))v(t)+g(r(t),x(t))]dw(t),z(t)=J(r(t))x(t),x(t0)=x0,
where x(t)∈Rn is the system state, uf(t)∈Rl is the control input of actuator fault, v(t)∈Rq is the exogenous disturbance input of the systems which belong to ℒ2[0,∞), z(t)∈Rr 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 (ℱt)t>0 of σ algebras (ℱt)t>0⊂ℱ. Ai,Bi,Ei,Ci,Di,Fi,Hi,Ji are the known real constant matrices with appropriate dimensions. Morever, we assume that(2.2)E(dw(t))=0,E((dw(t))2)=dt.Letr(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=(λij)N×N given by
(2.3)P{r(t+Δ)=j∣r(t)=i}={λijΔ+o(Δ)ifi≠j,1+λiiΔ+o(Δ)ifi=j,
where Δ>0. Here λij≥0 is the transition rate from manner i to manner j, if i≠j while λii=-∑j≠iλij. 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×Rn→Rn is a unknown nonlinear function which describes the system nonlinearity satisfying the following sector-bounded conditions:
(2.4)(fi(x(t))-T1ix)T(fi(x(t))-T2ix)≤0,i∈S,g(·,·):S×Rn→Rn also is a unknown nonlinear function which describes the stochastic nonlinearity satisfying the following:
(2.5)giT(x(t))gi(x(t))≤xTGiTGix,i∈S,
where T1i,T2i,Gi are known real constant matrices with approximate dimensions.Remark 2.1.
The nonlinearitiesfi(x(t)) 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(x(t)) satisfy the norm-bounded conditions.When the actuator experiences failure, we useuf(t,r(t)) to describe the control signal form actuators. Consider the following actuator failure model with failure parameter Fi:
(2.6)uif(t)=Fiui(t),
where Fi is the actuator fault matrix with
(2.7)Fi=diag⁡(fi1,fi2,…,fim),0≤f̲ij≤fij≤f̅ij,f̅ij≥1,j=1,2,…,m.In which the variablesfij 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:(2.8)F0i=diag⁡(f0i1,f0i2,…,f0im)=F̅i+F̲i2,f0ij=f̲ij+f̅ij2,(2.9)F̃0i=diag⁡(f̃0i1,f̃0i2,…,f̃0im)=F̅i-F̲i2,f0ij=f̅ij-f̲ij2,
and hence, the matrix Fi can be rewritten as
(2.10)Fi=F0i+Δi=F0i+diag⁡(φi1,φi2,…,φim),|φij|≤f̃ij,j=1,2,…,m.In this paper, our aim is to design the controllerui(t)=Kix(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:(2.11)‖z‖E2<γ‖v‖2(i.e.,∫0TzT(t)z(t)dt≤γ2∫0TvT(t)v(t)dt).If the above two conditions hold, we also called the systems that are exponential mean-square stable with convergence rateα and disturbance attenuation γ.


## 3. Main Results

Lemma 3.1 (Schur complement lemma [20]).
For a given matrixS=(S1S3*S2) with S1T=S1,S2T=S2, the following conditions are equivalent:(1)
S<0,(2)
S2<0,S1-S3S2-1S3T<0,(3)
S1<0,S2-S3S1-1S3T<0.Lemma 3.2 (see [21]).
Letx∈ℝn and y∈ℝn. Then, for any positive scalar ε, we have
(3.1)xTy+yTx≤εxTx+ε-1yTy.


### 3.1. Robust ReliableH∞ for LSMJSs

To obtain our main results, we first consider the following linear stochastic Markovian jump systems (LSMJSs) without control input:(3.2)dx(t)=[Aix(t)+Eiv(t)]dt+[Cix(t)+Hiv(t)]dw(t),z(t)=Jix(t),x(t0)=x0.Lemma 3.3.
Suppose thatP(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:
(3.3)Ξi(t)=(Mi(t)+JiTJiPiEiCiT*-γ2IHiT**-Pi-1(t))<0,i∈S,
where Mi(t)=AiTPi(t)+Pi(t)Ai+Ṗ(t)+∑j∈SλijPj(t).Proof.
At first, letv(t)=0, and defining the following Lyapunov function:
(3.4)V(x(t),t,i)=V(x(t),t,r(t)=i)=xT(t)P(t,r(t)=i)x(t)=xT(t)Pi(t)x(t).
By Itô formula, we get the following:(3.5)LV(x(t),t,i)=xT(t)(Mi(t)+CiTPi(t)Ci)x(t),
the matrix function inequalities (3.3) imply that ℒV(x(t),t,i)<0, and let ai=λmax(-Ξi(t)), a=maxi∈S(ai), where λmax(·) means the maximum eigenvalue of matrix (·), and we have
(3.6)LV(x(t),t,i)≤-axT(t)x(t).
Hence(3.7)d[eαtV(x(t),t,i)]=αeαtV(x(t),t,i)+eαtdV(x(t),t,i)≤(bα-a)eαt‖x(t)‖2+eαt2xT(t)Pi(t)Cix(t)dw(t),
where bi=supt≥t0{λmax(Pi(t))}, and b=maxi∈S(bi). Integrating the both sides of above inequality from t0 to Tand taking expectation, we obtain that
(3.8)EeαT[V(x(T),T,i)-V(x0,t0,i)]≤(bα-a)E∫t0Teαs‖x(s)‖2ds.
Setα=a/b, and the following inequality is obtained:
(3.9)eαTmin⁡i∈Sλmin⁡(Pi(T))E‖x(T)‖2≤E[eαTV(x(T),T,i)]≤EV(x0,t0,i),
which implies that
(3.10)E‖x(T)‖2≤EV(x0,t0,i)1min⁡i∈Sλmin⁡(Pi(T))e-αT.
That is to say that the stochastic systems are globally exponentially stable with convergence rateα>0.
Then, considering the stochasticH∞ performance level for the resulting systems (3.2) with nonzero exogenous disturbance input (v(t)≠0), for anyt>0, we define that
(3.11)J(t)=E{∫0t[zT(s)z(s)-γ2vT(s)v(s)]ds}.
By general Itô formula, we get he following:(3.12)J(t)=E{∫t0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)]ds}-E(V(x(t),t,i))≤E{∫0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)]ds}≤E{∫0tηT(s)Ωi(s)η(s)ds},
where ηT(t)=(xT(t)vT(t)), Ωi(t)=(Mi(t)+JiTJiPi(t)EiEiTPi(t)-γ2I)+(CiTHiT)Pi(t)(CiTHiT)T From (3.3) we know that Ω(t)<0, which implies that
(3.13)J(t)<0.
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:(3.14)dx(t)=[(Ai+BiFiKi)x(t)+Eiv(t)]dt+[(Ci+DiFiKi)x(t)+Hiv(t)]dw(t),z(t)=Jix(t),x(t0)=x0.Theorem 3.4.
If there exist the positive matricesXi>0, and the constant matrices Yi with approximate dimensions, such that the following LMIs hold
(3.15)Θi=(Θi1EiΘi2Θi3*-γ2IHiT0**-Xi0***Θi4)<0,i∈S,
where Θi1=XiAiT+AiXi+BiFiYi+YiTFiTBiT+λiiXi,Θi2=XiCiT+YiTFiTDiT,
(3.16)Θi3=(λi1Xi⋯λi,i-1Xiλi,i+1Xi⋯λiNXiXiJiT),Θi4=diag⁡(-X1,…,-Xi-1,-Xi+1,…,-XN,-I),
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:
(3.17)Ki=YiXi-1.Proof.
Defining the following Lyapunov function:(3.18)V(x(t),t,i)=V(x(t),t,r(t)=i)=xT(t)Pix(t).
By Lemma3.3, and similar to the proof of Lemma 3.3, we can get the following:
(3.19)LV(x(t),t,i)≤ηT(t)Ξiη(t),
where Ξi=(MiPiEiCiT+KiTFiTDiT*-γ2IHiT**-Pi-1)Mi=(Ai+BiFiKi)TPi+Pi(Ai+BiFiKi)+∑j∈SλijPj.
Using Schur complement lemma together with contragredient transformation, we know that LMIs (3.15) imply that Ξi<0. So we have
(3.20)J(t)=E{∫0t[zT(s)z(s)-γ2vT(s)v(s)]ds}=E{∫t0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)]ds}-E(V(x(t),t,i))≤E{∫0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)]ds}<0..
Therefore, the inequality ∥z∥E2<γ∥v∥2 holds. The proof is completed.Theorem 3.5.
If there exist the positive matricesXi>0, the positive diagonal matrices Ri>0, and the constant matrices Yi with approximate dimensions, such that the following LMIs hold:
(3.21)Θ̃i=(Θ̃i1EiΘ̃i2Θi3BiRiYiT*-γ2IHiT000**-Xi0DiRi0***Θi400****-Ri0*****-RiF̃i0-2)<0,i∈S,
where Θ̃i1=XiAiT+AiXi+BiFi0Yi+YiTFi0TBiT+λiiXi, Θ̃i2=XiCiT+YiTFi0TDiT, 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:
(3.22)Ki=YiXi-1.Proof.
Noticing (2.10), we can see that Θi in (3.15) can be rewritten as
(3.23)Θi=Θi0+[BiT0DiT0]TΔi[Yi000]+[Yi000]TΔi[BiT0DiT0],
where Θi0=(Θ̃i1EiΘ̃i2Θi3*-γ2IHiT0**-Xi0***Θi4).
By Lemma3.2, we have
(3.24)Θi≤Θi0+[BiT0DiT0]TRi[BiT0DiT0]+[Yi000]TRi-1F0i2[Yi000],
by Schur complement, we know that Θ̃i<0 imply that Θi<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 ReliableH∞ for NSMJSs

In this section, we consider the following nonlinear stochastic Markovian jump systems (NSMJSs) under the state feedback controller:(3.25)dx(t)=[(Ai+BiFiKi)x(t)+Eiv(t)+fi(x(t))]dt+[(Ci+DiFiKi)x(t)+Hiv(t)+gi(x(t))]dw(t),z(t)=Hix(t),x(t0)=x0.Theorem 3.6.
If there exist the positive matricesXi>0, and the constant matrices Yi with approximate dimensions, for the positive constant εi and the given scalar λi, such that the following LMIs hold:
(3.26)Θ̅i=(Θi1EiI-λiXiT̂i2Θi2TΘi2TΘ̅i3*-γ2I0HiTHiT0**-λiI000***-Xi00****-εiI0*****Θ̅i4)<0,i∈S,
where Θ̅i3=(εiGiλiXiT̂i1Θi3),Θ̅i4=diag(-εiI,-λiT̂i1,Θi4),T̂i1=(Ti1TTi2+Ti2TTi1)/2,T̂i2=-(Ti1T+Ti2T)/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:
(3.27)Ki=YiXi-1.Proof.
Defining the following Lyapunov function:(3.28)V(x(t),t,i)=V(x(t),t,r(t)=i)=xT(t)Pix(t),
by Itô formula, we get the following:(3.29)LV(x(t),t,i)=2xT(t)Pi[(Ai+BiFiKi)x(t)+Eiv(t)+fi(x(t))]+∑j∈SλijxT(t)Pjx(t)+[(Ci+DBiFiKi)x(t)+Hiv(t)+gi(x(t))]T×Pi[(Ci+DBiFiKi)x(t)+Hiv(t)+gi(x(t))]≤σT(t)Σiσ(t)+xT(t)GiTPiGix(t)+2[(Ci+DBiFiKi)x(t)+Hiv(t)]TPigi(x(t)),
where σT(t)=[xT(t),vT(t),fiT(x(t))],Σi=(MiPiEiPiEiTPi00PiT00)+[CiT+KiTFiTDiTHiT0]Pi[CiT+KiTFiTDiTHiT0]T.
By Lemma3.2, it follows that
(3.30)2[(Ci+DBiFiKi)x(t)+Hiv(t)]TPigi(x(t))≤σT(t)[CiT+KiTFiTDiTHiT0]εi-1I[CiT+KiTFiTDiTHiT0]Tσ(t)+xT(t)(εiPiGi)Tεi-1I(εiPiGi)x(t),
from (2.4) (fi(x(t))-T1ix)T(fi(x(t))-T2ix)≤0,i∈S which are equivalent to
(3.31)[x(t)f(x(t))]T[T̂i1T̂i2Ti2TI][x(t)f(x(t))]≤0,i∈S.
Considering the stochasticH∞ performance level for the resulting systems (3.25) with nonzero exogenous disturbance input (v(t)≠0), for anyt>0, we define that
(3.32)J(t)=E{∫0t[zT(s)z(s)-γ2vT(s)v(s)]ds}.
By general Itô formula, for a given positive scalarλ, we get the following:
(3.33)J(t)=E{∫t0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)]ds}-E(V(x(t),t,i))≤E{∫0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)-λi(fi(x(t))-T1ix(t))T(fi(x(t))-T2ix(t))]ds}≤E{∫0tσT(s)Ω̅iσ(s)ds},
where
(3.34)Ω̅i=Σi+((εiPiGi)Tεi-1I(εiPiGi)+JiTJi000-γ2I0000)+[CiT+KiTFiTDiTHiT0]εi-1I[CiT+KiTFiTDiTHiT0]T+(-λT̂i10-λT̂i2000-λT̂i2T0-λI).
By Schur complement lemma, we see thatΩ̅i<0 is equivalent to the following matrix inequalities:
(3.35)(Mi-λiT̂i1EiPi-λiT̂i2Xi-1Θi2TXi-1Θi2TεiPiGiJiT*-γ2I0HiTHiT00**-λiI0000***-Pi-1000****-εiI00*****-εiI0******-I)<0,i∈S,
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 Theorem3.5, we can get the following theorem without proof immediately.Theorem 3.7.
If there exist the positive matricesXi>0, and the constant matrices Yi with approximate dimensions, for the positive constant εi and the given scalar λi, such that the following LMIs hold
(3.36)Θ̂i=(Θ̃i1EiI-λiXiT̂i2Θ̃i2TΘ̃i2TBiRiYiTΘ̅i3*-γ2I0HiTHiT000**-λiI00000***-Xi0DiRi00****-εiI000*****-Ri00******-RiF̃i0-20*******Θ̅i4)<0,i∈S.
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:
(3.37)Ki=YiXi-1.


## 3.1. Robust ReliableH∞ for LSMJSs

To obtain our main results, we first consider the following linear stochastic Markovian jump systems (LSMJSs) without control input:(3.2)dx(t)=[Aix(t)+Eiv(t)]dt+[Cix(t)+Hiv(t)]dw(t),z(t)=Jix(t),x(t0)=x0.Lemma 3.3.
Suppose thatP(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:
(3.3)Ξi(t)=(Mi(t)+JiTJiPiEiCiT*-γ2IHiT**-Pi-1(t))<0,i∈S,
where Mi(t)=AiTPi(t)+Pi(t)Ai+Ṗ(t)+∑j∈SλijPj(t).Proof.
At first, letv(t)=0, and defining the following Lyapunov function:
(3.4)V(x(t),t,i)=V(x(t),t,r(t)=i)=xT(t)P(t,r(t)=i)x(t)=xT(t)Pi(t)x(t).
By Itô formula, we get the following:(3.5)LV(x(t),t,i)=xT(t)(Mi(t)+CiTPi(t)Ci)x(t),
the matrix function inequalities (3.3) imply that ℒV(x(t),t,i)<0, and let ai=λmax(-Ξi(t)), a=maxi∈S(ai), where λmax(·) means the maximum eigenvalue of matrix (·), and we have
(3.6)LV(x(t),t,i)≤-axT(t)x(t).
Hence(3.7)d[eαtV(x(t),t,i)]=αeαtV(x(t),t,i)+eαtdV(x(t),t,i)≤(bα-a)eαt‖x(t)‖2+eαt2xT(t)Pi(t)Cix(t)dw(t),
where bi=supt≥t0{λmax(Pi(t))}, and b=maxi∈S(bi). Integrating the both sides of above inequality from t0 to Tand taking expectation, we obtain that
(3.8)EeαT[V(x(T),T,i)-V(x0,t0,i)]≤(bα-a)E∫t0Teαs‖x(s)‖2ds.
Setα=a/b, and the following inequality is obtained:
(3.9)eαTmin⁡i∈Sλmin⁡(Pi(T))E‖x(T)‖2≤E[eαTV(x(T),T,i)]≤EV(x0,t0,i),
which implies that
(3.10)E‖x(T)‖2≤EV(x0,t0,i)1min⁡i∈Sλmin⁡(Pi(T))e-αT.
That is to say that the stochastic systems are globally exponentially stable with convergence rateα>0.
Then, considering the stochasticH∞ performance level for the resulting systems (3.2) with nonzero exogenous disturbance input (v(t)≠0), for anyt>0, we define that
(3.11)J(t)=E{∫0t[zT(s)z(s)-γ2vT(s)v(s)]ds}.
By general Itô formula, we get he following:(3.12)J(t)=E{∫t0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)]ds}-E(V(x(t),t,i))≤E{∫0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)]ds}≤E{∫0tηT(s)Ωi(s)η(s)ds},
where ηT(t)=(xT(t)vT(t)), Ωi(t)=(Mi(t)+JiTJiPi(t)EiEiTPi(t)-γ2I)+(CiTHiT)Pi(t)(CiTHiT)T From (3.3) we know that Ω(t)<0, which implies that
(3.13)J(t)<0.
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:(3.14)dx(t)=[(Ai+BiFiKi)x(t)+Eiv(t)]dt+[(Ci+DiFiKi)x(t)+Hiv(t)]dw(t),z(t)=Jix(t),x(t0)=x0.Theorem 3.4.
If there exist the positive matricesXi>0, and the constant matrices Yi with approximate dimensions, such that the following LMIs hold
(3.15)Θi=(Θi1EiΘi2Θi3*-γ2IHiT0**-Xi0***Θi4)<0,i∈S,
where Θi1=XiAiT+AiXi+BiFiYi+YiTFiTBiT+λiiXi,Θi2=XiCiT+YiTFiTDiT,
(3.16)Θi3=(λi1Xi⋯λi,i-1Xiλi,i+1Xi⋯λiNXiXiJiT),Θi4=diag⁡(-X1,…,-Xi-1,-Xi+1,…,-XN,-I),
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:
(3.17)Ki=YiXi-1.Proof.
Defining the following Lyapunov function:(3.18)V(x(t),t,i)=V(x(t),t,r(t)=i)=xT(t)Pix(t).
By Lemma3.3, and similar to the proof of Lemma 3.3, we can get the following:
(3.19)LV(x(t),t,i)≤ηT(t)Ξiη(t),
where Ξi=(MiPiEiCiT+KiTFiTDiT*-γ2IHiT**-Pi-1)Mi=(Ai+BiFiKi)TPi+Pi(Ai+BiFiKi)+∑j∈SλijPj.
Using Schur complement lemma together with contragredient transformation, we know that LMIs (3.15) imply that Ξi<0. So we have
(3.20)J(t)=E{∫0t[zT(s)z(s)-γ2vT(s)v(s)]ds}=E{∫t0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)]ds}-E(V(x(t),t,i))≤E{∫0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)]ds}<0..
Therefore, the inequality ∥z∥E2<γ∥v∥2 holds. The proof is completed.Theorem 3.5.
If there exist the positive matricesXi>0, the positive diagonal matrices Ri>0, and the constant matrices Yi with approximate dimensions, such that the following LMIs hold:
(3.21)Θ̃i=(Θ̃i1EiΘ̃i2Θi3BiRiYiT*-γ2IHiT000**-Xi0DiRi0***Θi400****-Ri0*****-RiF̃i0-2)<0,i∈S,
where Θ̃i1=XiAiT+AiXi+BiFi0Yi+YiTFi0TBiT+λiiXi, Θ̃i2=XiCiT+YiTFi0TDiT, 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:
(3.22)Ki=YiXi-1.Proof.
Noticing (2.10), we can see that Θi in (3.15) can be rewritten as
(3.23)Θi=Θi0+[BiT0DiT0]TΔi[Yi000]+[Yi000]TΔi[BiT0DiT0],
where Θi0=(Θ̃i1EiΘ̃i2Θi3*-γ2IHiT0**-Xi0***Θi4).
By Lemma3.2, we have
(3.24)Θi≤Θi0+[BiT0DiT0]TRi[BiT0DiT0]+[Yi000]TRi-1F0i2[Yi000],
by Schur complement, we know that Θ̃i<0 imply that Θi<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 ReliableH∞ for NSMJSs

In this section, we consider the following nonlinear stochastic Markovian jump systems (NSMJSs) under the state feedback controller:(3.25)dx(t)=[(Ai+BiFiKi)x(t)+Eiv(t)+fi(x(t))]dt+[(Ci+DiFiKi)x(t)+Hiv(t)+gi(x(t))]dw(t),z(t)=Hix(t),x(t0)=x0.Theorem 3.6.
If there exist the positive matricesXi>0, and the constant matrices Yi with approximate dimensions, for the positive constant εi and the given scalar λi, such that the following LMIs hold:
(3.26)Θ̅i=(Θi1EiI-λiXiT̂i2Θi2TΘi2TΘ̅i3*-γ2I0HiTHiT0**-λiI000***-Xi00****-εiI0*****Θ̅i4)<0,i∈S,
where Θ̅i3=(εiGiλiXiT̂i1Θi3),Θ̅i4=diag(-εiI,-λiT̂i1,Θi4),T̂i1=(Ti1TTi2+Ti2TTi1)/2,T̂i2=-(Ti1T+Ti2T)/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:
(3.27)Ki=YiXi-1.Proof.
Defining the following Lyapunov function:(3.28)V(x(t),t,i)=V(x(t),t,r(t)=i)=xT(t)Pix(t),
by Itô formula, we get the following:(3.29)LV(x(t),t,i)=2xT(t)Pi[(Ai+BiFiKi)x(t)+Eiv(t)+fi(x(t))]+∑j∈SλijxT(t)Pjx(t)+[(Ci+DBiFiKi)x(t)+Hiv(t)+gi(x(t))]T×Pi[(Ci+DBiFiKi)x(t)+Hiv(t)+gi(x(t))]≤σT(t)Σiσ(t)+xT(t)GiTPiGix(t)+2[(Ci+DBiFiKi)x(t)+Hiv(t)]TPigi(x(t)),
where σT(t)=[xT(t),vT(t),fiT(x(t))],Σi=(MiPiEiPiEiTPi00PiT00)+[CiT+KiTFiTDiTHiT0]Pi[CiT+KiTFiTDiTHiT0]T.
By Lemma3.2, it follows that
(3.30)2[(Ci+DBiFiKi)x(t)+Hiv(t)]TPigi(x(t))≤σT(t)[CiT+KiTFiTDiTHiT0]εi-1I[CiT+KiTFiTDiTHiT0]Tσ(t)+xT(t)(εiPiGi)Tεi-1I(εiPiGi)x(t),
from (2.4) (fi(x(t))-T1ix)T(fi(x(t))-T2ix)≤0,i∈S which are equivalent to
(3.31)[x(t)f(x(t))]T[T̂i1T̂i2Ti2TI][x(t)f(x(t))]≤0,i∈S.
Considering the stochasticH∞ performance level for the resulting systems (3.25) with nonzero exogenous disturbance input (v(t)≠0), for anyt>0, we define that
(3.32)J(t)=E{∫0t[zT(s)z(s)-γ2vT(s)v(s)]ds}.
By general Itô formula, for a given positive scalarλ, we get the following:
(3.33)J(t)=E{∫t0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)]ds}-E(V(x(t),t,i))≤E{∫0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s,i)-λi(fi(x(t))-T1ix(t))T(fi(x(t))-T2ix(t))]ds}≤E{∫0tσT(s)Ω̅iσ(s)ds},
where
(3.34)Ω̅i=Σi+((εiPiGi)Tεi-1I(εiPiGi)+JiTJi000-γ2I0000)+[CiT+KiTFiTDiTHiT0]εi-1I[CiT+KiTFiTDiTHiT0]T+(-λT̂i10-λT̂i2000-λT̂i2T0-λI).
By Schur complement lemma, we see thatΩ̅i<0 is equivalent to the following matrix inequalities:
(3.35)(Mi-λiT̂i1EiPi-λiT̂i2Xi-1Θi2TXi-1Θi2TεiPiGiJiT*-γ2I0HiTHiT00**-λiI0000***-Pi-1000****-εiI00*****-εiI0******-I)<0,i∈S,
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 Theorem3.5, we can get the following theorem without proof immediately.Theorem 3.7.
If there exist the positive matricesXi>0, and the constant matrices Yi with approximate dimensions, for the positive constant εi and the given scalar λi, such that the following LMIs hold
(3.36)Θ̂i=(Θ̃i1EiI-λiXiT̂i2Θ̃i2TΘ̃i2TBiRiYiTΘ̅i3*-γ2I0HiTHiT000**-λiI00000***-Xi0DiRi00****-εiI000*****-Ri00******-RiF̃i0-20*******Θ̅i4)<0,i∈S.
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:
(3.37)Ki=YiXi-1.


## 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 (Figure1).Figure 1
The Markovian chainr(t).Consider linear SMJSs (3.14) with S={1,2}, and the system parameters are given as follows:(4.1)A1=(0.30.30.5-0.20-0.30.100.3),A2=(0.50.20.2-0.20-0.40.200.2),C1=(0.50.20.100.2-0.10.3-0.1-0.3),C2=(0.20.10.30.1-0.30.500.1-0.5),B1=diag⁡(0.5,0.4,0.5),B2=diag⁡(0.5,0.4,0.5),E1=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),J1=(0.3,0.2,0.6),J2=(0.1,-0.1,0.4),γ=0.9.The actuator failure parameters are as follows:(4.2)0.2≤fi1≤0.4,0.1≤fi2≤0.7,0.1≤fi3≤0.9,i∈S={1,2}.From (2.8) and (2.9), we have(4.3)F10=F20=diag⁡(0.3,0.4,0.5),F̃10=F̃20=diag⁡(0.1,0.3,0.4).From Figure2, 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:(4.4)K1=(-56.2264-6.3843-67.8069-1.1129-8.9588-3.6802-0.97540.1795-3.2600),K2=(-41.78466.0578-200.8802-1.1365-7.5245-11.02090.1171-0.4055-0.7561).Figure 2
The state curve of uncontrolled LSMJSs (3.14).Figures3 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 reliableH∞ 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.



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*Source: 431576-2012-05-31.xml*