# Observer-Based Sliding Mode Control for Stochastic Nonlinear Markovian Jump Systems

Authors: Xiaohan Yin; Quanxin Zhu
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

We consider a class of stochastic nonlinear Markovian jump systems (MJSs) with partly unknown transition rate and time-varying delays. The system under consideration is subject to the mode uncertainties and nonlinear term and disturbance term which are unknown. The main mission is to design the observer-based sliding mode controller for such a complex system. An observer is first constructed, and then we design an integral sliding mode surface such that the MJSs satisfy the reaching condition. The sliding mode control law ensures the stochastic stability of the closed-loop system. Finally, an example is given to illustrate the proposed results.


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## Body

## 1. Introduction

The sliding mode control (SMC) has attractive features to keep systems insensitive to the parameter uncertainties and external disturbances on the sliding mode plane [1]. Many practical dynamics, for example, manufacturing systems, chemical process systems, computer controlled systems and communication systems, solar receiver control, and power systems, experience abrupt random changes in their structure. These changes are usually caused by random failure or repairs of the components, changing in subsystems interconnections, sudden environmental changes, and so forth. Such systems can be modeled as hybrid systems. One special class of hybrid systems is named Markovian jump systems (MJSs), whose system mode is governed by a Markov process.On the other hand, stability is one of the best topics in dynamic systems and a large number of results have appeared in the literature (e.g., see [2–28] and references therein). For MJSs, considerable attention has been recently paid to the problems of stability and stabilization [29–33], H∞ control and filtering [34–36], optimal tracking problem [37], etc. The robust stability and stabilization of uncertain stochastic systems with time-varying delays are investigated by using the linear matrix inequality approach (LMI) [38]. Huang and Mao [39] investigated the stabilization of stochastic linear systems by delayed state feedback controller. In [40], the method of Lyapunov functional is employed to study p-moment stability of nonlinear stochastic systems with impulsive and Markovian switching.The SMC for stochastic systems has received an increasing attention. For example, see [41–51]. K. Chang and W. Chang [52] developd a SMC method to guarantee the robust state covariance assignment for perturbed stochastic multivariable systems via variable structure control. In [53], robust observer design for Ito^’s stochastic time-delay systems was studied via sliding mode control and the sufficient conditions for the asymptotic stability (in probability) of the sliding motion were derived. Niu [54] dealed with the connection among the designed sliding surface corresponding to every mode for MJSs. However, there has been few results reported on the SMC of stochastic systems with Markovian switching. The existence of uncertainties, time-varying delays, Markovian switching with partly unknown transition probability will make the problem more complex and challenging.Motivated by the above discussion, in this paper we investigate the problem of SMC problem for a class of stochastic nonlinear Markovian jump systems (MJSs) with partly unknown transition probabilities and time-varying delay. Firstly, an appropriate integral sliding mode surface is constructed. Secondly, the observer-based sliding mode controller is designed to guarantee the stochastic stability of the closed-loop system. Finally, a numerical example is provided to show the effectiveness of the proposed scheme.The organization of this paper is given as follows. The system description and some preliminaries are given in Section2. Section 3 presents the design of an integral sliding mode surface function and designs a sliding mode controller. Section 4 provides a numerical example to certify the feasibility of the mentioned method, and we conclude this paper in Section 5.Notations. ( Ω , F , P ) is a probability space, where Ω denotes the sample space, F represents the σ-algebra of subsets of the sample space and P denotes the probability measure. Xi>0 means that Xi is symmetric positive definite. I denotes the identity matrix. · refers to the Euclidean norm of a vector.


## 2. Preliminaries

In this section, we consider a class of stochastic nonlinear Markovian jump systems (MJSs) with partly unknown transition rates and time-varying delays on a complete probability space(Ω,F,P):(1)dxt=Aγt+△Aγtxt+A1γt+△A1γtxt-τt+Bγtut+fxt,t+dtdt+Cγtxtdωt,yt=Dγtxt,xt=φt,t∈-τ,0,wherex(t)∈Rn represents the state vector, u(t)∈Rm represents control input, fxt,t∈Rm is the nonlinear function, and d(t)∈Rm is the system disturbance. y(t)∈Rp denotes the measured output. A(γt)∈Rn×n, A1(γt)∈Rn×n, B(γt)∈Rn×m, C(γt)∈Rn×n, and D(γt)∈Rp×n are the constant matrices and △A(γt)∈Rn×n  △A1γt(t)∈Rn×n are the system uncertainties. ω(t) is a standard Brownian motion defined on the complete probability spare (Ω,F,P). τ(t) is time-varying delay and supposed to satisfy(2)0≤τt≤τ<∞,whereτ is known real constant scalar and φ(t) is a continuous vector-valued initial function. {γt,t≥0} represent the system mode, which take values in a given state space S={1,2,…,N}. Let Π=(πij)N×N(i,j=1,2,…,N) represent the transition rate matrix. The mode transition probabilities can be denoted as follows:(3)Prγt+△t=j∣γt=i=πij△t+∘△t,i≠j,1+πii△t+∘△t,i=j,where△t>0 and lim△t→0∘(△t)/△t=0, πij satisfies πij>0 with i≠j, and πii=-∑j=1,j≠iNπij for each mode i. In addition, we assume that the probabilities of the Markovain jump system are partly unknown; for example, if there are four operation modes in system (1), the matrix of transition rates may be described as(4)Π=?π12???π22??π31??π34???π44,where “?” denotes the unknown elements. We representS=Ski∪Suki with Ski={j:πijisknown} and Suki={j:πijisunknown]. We denote πki≜∑j∈Skiπij. We can rewrite MJSs for notational convenience when the system mode is i:(5)dxt=Ait+△Aitxt+A1it+△A1itxt-τt+Biut+fx,t+dtdt+Cixtdωt,yt=Dixt,xt=φt,t∈-τ,0,whereA(γt)=Ai, B(γt)=Bi, Cγt=CiandD(γt)=Di, △Aγt=△Ai, and △A1(γt)=△A1i.Throughout this paper, we assume that the following conditions are satisfied.Assumption 1.
△ A i and △A1i satisfy the following form:(6)△Ai=GiFi,tEi,△A1i=G1iF1i,tE1i,
whereDi, Ei, D1i, and E1i are the constant matrices with appropriate dimensions and F(i,t), F1(i,t) are unknown time-varying matrix function satisfying(7)FTi,tFi,t≤I,F1Ti,tF1i,t≤I,i∈S.Assumption 2.
The external disturbance vectord(t) is an unknown function which satisfies(8)dt≤d,
whered is an unknown scalar.Assumption 3.
B i has full column rank and the pair (Ai,Bi) is controllable.Assumption 4.
The nonlinear functionf(x,t) satisfies(9)fx,t≤ρixt,i∈S,
whereρi>0 is a constant scalar.Obviously, under Assumptions1–4, it follows from [55] that there exists a unique global solution x(t) for any initial data φ(t), t∈[-τ,0].For any givenV(x^,e,i)∈C12(Rn×Rn×S;Rn), we define an operator LV from Rn×Rn×S to R by(10)LVx^,e,i=Vx^x^,e,i+Vex^,e,iAγt+△Aγtxt+A1γt+△A1γtxt-τt+Bγtut+fxt,t+dt+12trCTγtxTtxtCγt+∑j=1NπijVx^,e,i.Next, we will gave a lemma.Lemma 5 (see [43]).
For any positive constantε, X∈Rn, and Y∈Rn, the following inequality holds:(11)XTY+YTX≤εYTY+1εXTX.
Finally, we give the definition of stochastic stability to end this section.Definition 6 (see [56]).
The stochastic system (1) is said to be stochastically stable when u(t)=0, if the condition satisfies(12)E∫0∞xt2dt∣x0,r0<+∞,
wherex0∈Rn and r0 is the initial mode.


## 3. Main Results

We first design a state observer to estimate unmeasured state components. Then we construct an integral sliding mode surface which is based on the state observer as the first step of SMC design. The state observer we synthesize is as follows:(13)dx^t=Aix^t+A1ix^t-τt+Biut+Biuet+Liy-Dix^tdt+Cix^tdωt,wherex^(t)∈Rn represents the estimation of x(t), Li∈Rn×p, ∀i∈S is the observer gain matrix, and ue(t) is chosen to reduce the effect of unknown nonlinear function f(t) and unknown external disturbance d(t).Remark 7.
We can notice that the state observer not only contains state observationAix^(t) without time delays but also contains state observation A1ix^(t-τ(t)) with time delays; this can be a simplification of the controller.The solutionx^(t) of the state observer can be expressed as follows:(14)x^t=φ0+∫0tAix^s+A1ix^s-τs+Bius+Biues+Liy-Dix^sds+∫0tCix^sdωs.Lete(t) be the estimation error:(15)et=xt-x^t,and then we can obtain the following estimated dynamic:(16)det=Ai+△Aiet+A1i+△A1iet-τt+△Aix^t+△A1ix^t-τt+Bifx,t+dt-uet-LiDietdt+Cietdωt.Obviously, it follows from (16) that de(t) depends on x^(t). The integral sliding mode surface is constructed as follows:(17)sx^t,i=BiTXix^t-∫0tBiTXiAi+BiKix^ldl,∀i∈S,whereKi satisfies that Ai+BiKi is Hurwitz and Xi∈Rn×n is a constant matrix such that BiTXiBi is nonsingular and BiTXiCi=0. Combining (14) and (17), we have(18)sx^t,i=BiTXiφ0+∫0tBiTXiAix^s+A1ix^s-τs+Bius+Biues+Liyt-Dix^sds-∫0tBiTXiAi+BiKix^sds.According to SMC theory, when the system trajectory reaches the sliding surface, it follows thatsx^t,i=0 and s˙x^t,i=0.Remark 8.
The linear termBiTXix^(t) is continuous in the integral sliding mode surface (17), the integral term ∫0tBiTXi(Ai+BiKi)x^(t) is continuous, and then the integral sliding mode surface (17) is continuous when the MJS states switch.Remark 9.
It follows from (17) that the integral sliding mode surface s(x^(t),i) depends on a specific set of matrices Xi, i∈S, which means that switching effect from one state to another of the Markovian jump system is reflected on s(x^(t),i) by the matrix Xi. Thus, in the sliding mode control system accompanied by Markovian jump systems, the sliding mode surfaces establish contact with each other through Xi.By (18), we have(19)s˙it=BiTXiAix^t+A1ix^t-τt+Biut+Biuet+Liy-Dix^t-BiTXiAi+BiKix^t=BiTXiA1ix^t-τt+Biut+Biuet+LiDiet-BiKix^t.Based on the state observerx^(t), we design a new sliding mode control (SMC) law as follows:(20)uit=-ststuet+Kix^t+BiTXiBi-1BiTXiA1ix^t-τt+BiTXiBi-1BiTXiLiDiet+β,whereβ>0.The control termue(t) is designed as(21)uet=ρixt+d+γ1sgn⁡BiTXiet,whereγ1>0.Theorem 10.
Assume that Assumptions1–4 hold and the integral sliding mode controller is given by (18). Then, the reachabilty of the sliding mode surface s(x^(t),i)=0 can be satisfied under the action of the SMC law (20).Proof.
Taking(22)Vaxt=12sTx^t,iBiTXiBi-1sx^t,i,
then it follows from (2.14) that(23)LVaxt=sTx^t,iBiTXiBi-1s˙x^t,i=sTx^t,iBiTXiBi-1BiTXiA1ix^t-τt+Biut+Biuet+LiDiet-BiKix^t=sTx^t,iut+sTx^t,iuet-Kix^t+BiTXiBi-1BiTXiA1ix^t-τt+BiTXiBi-1BiTXiLiDiet≤sTx^t,iut+sx^t,iuet+Kix^t+BiTXiBi-1BiTXiA1ix^t-τt+BiTXiBi-1BiTXiLiDiet.
Substitutingu(t) into LVa(x(t)), we get(24)LVaxt≤-βsx^t,i<0.
Thus, the reachabilty of the sliding mode surfaces(x^,i)=0 can be satisfied.The equivalent control law in the sliding mode can be obtained by solvings˙(t)=0 as(25)ueq=Kix^t-uet-BiTXiBi-1BiTXiA1ix^t-τt-BiTXiBi-1BiTXiLiDiet.Substitutingueq(t) into dx^(t), then the state observer can be written as(26)dx^t=Ai+BiKix^t+A1i-BiBiTXiBi-1BiTXiA1ix^t-τt+I-BiBiTXiBi-1BiTXiLiDietdt+Cietdωt.The stability of system (1) can be determined by analyzing the stability of the closed-loop system which consists of (16) and (26).The following theorem gives sufficient conditions for the stability of the closed-loop system, which is expressed in terms of linear matrix inequalities (LMIs).Theorem 11.
Considering system (16) and (26) with Assumptions 1–4, the sliding mode surface function is given by (18). If there exist positive definite symmetric matrix Xi∈Rn×n, matrices H1>0 and H2>0, matrix Pi, matrix Qi, and positive scalars ε1i, ε2i, ε3i, and ε4i satisfy the following LMIs and the inequalities:(27)Ξ1iΞ2i∗Ξ3i<0,∀i∈Ski,(28)△1i△2i∗△3t<0,∀i∈Suki,(29)∑j=1NπijPi≤H1,∑j=1NπijQi≤H2,(30)A1iTA1i+ε2iE1iTE1i-Pi<0,∀i∈S,(31)ε4iE1iTE1i+ψ2Tψ2-Qi<0,∀i∈S,(32)XiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCTC+Pi+Xj≤0,∀j∈Suki,j≠i,(33)XiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCTC+Pi+Xi≥0,∀j∈Suki,j=i,(34)XiAi+BiKi+Ai+BiKiTXi+12λmaxCTC+Qi+XiXi+Xj≤0,∀j∈Suki,j≠i,(35)XiAi+BiKi+Ai+BiKiTXi+12λmaxCTC+Qi+XiXi+Xi≥0,∀j∈Suki,j=i,
where(36)Ξ1i=Ξ110ψ1TXi∗Ξ220∗∗-I,(37)Ξ2i=00000LiDiDiTXiD1iTXiDiTXiD1iTXi00000,(38)Ξ3i=diag⁡-I,-ε1iI,-ε2iI,-ε3iI,-ε4iI,(39)Ξ11=1+πkiXiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCTC+Pi+ε1iEiTEi+∑j∈SkiπijXj,(40)Ξ22=1+πkiXiAi+BiKi+Ai+BiKiTXi+12λmaxCTC+Qi+XiXi+ε3iEiTEi+∑j∈SkiπijXj,(41)△1i=△110ψ1TXi∗△220∗∗-I,(42)△2i=00000LiDiDiTXiD1iTXiDiTXiD1iTXi00000,(43)△3i=diag⁡-I,-ε1iI,-ε2iI,-ε3iI,-ε4iI,(44)△11=1+∑j∈Ski,j≠iπijXiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCTC+Pi+ε1iEiTEi+∑j∈Ski,j≠iπijXj,(45)△22=1+∑j∈Ski,j≠iπijXiAi+BiKi+Ai+BiKiTXi+12λmaxCTC+Qi+XiXi+ε3iEiTEi+∑j∈Ski,j≠iπijXj,(46)ψ1=I-BiBiTXiBi-1BiTXi,(47)ψ2=A1i-BiBiTXiBi-1BiTXiA1i,
and then the overall closed-loop system composed of (16) and (26) is stochastically stable.Proof.
Take(48)Vx^,e,i=eTtXiet+x^TtXix^t+∫t-τtteTsPiesds+∫t-τttx^TsQix^sds+∫-τ0∫t+θteTsH1esdθds+∫-τ0∫t+θtx^TsH2x^sdθds
Then,(49)LVx^,e,i=2eTtXiAi+△Aiet+A1i+△A1iet-τt+△Aix^t+△A1ix^t-τt+Bifx,t+dt-uet-LiDiet+12traceCietTCiet+2x^TtXiAi+BiKix^t+A1i-BiBiTXiBi-1BiTXiA1ix^t-τt+I-BiBiTXiBi-1BiTXiLiDiet+12traceCix^tTCix^t+eTtPiet-eTt-τtPiet-τt+x^TtQix^t-x^t-τtQix^t-τt+x^Tt∑j=1NπijXjx^t+eTt∑j=1NπijXjet+∑j=1Nπij∫t-τtteTsPiesds+∫t-τttx^TsQix^sds+τeTtH1et-∫t-τteTθH1eθdθ+τx^TtH2x^t-∫t-τtx^TθH2x^θdθ.
According to (29),(50)∑j=1Nπij∫t-τtteTsPiesds+∫t-τttx^TsQix^sds-∫t-τteTθH1eθdθ-∫t-τtx^TθH2x^θdθ≤0,
Using Assumption1 and Lemma 5, we have(51)2eTtXi△Aiet=2eTtXiDiFi,tEiet≤ε1i-1eTtXiDiDiTXiet+ε1ieTtEiTEiet,(52)2eTtXi△A1iet-τt=2eTtXiD1iF1i,tE1iet-τt≤ε2i-1eTtXiD1iD1iTXiet+ε2ieTt-τtE1iTE1iet-τt,(53)2eTtXi△Aix^t=2eTtXiDiFi,tEix^t≤ε3i-1eTtXiDiDiTXiet+ε3ix^TtEiTEix^t,(54)2eTtXi△A1ix^t-τt=2eTtXiD1iF1i,tE1ix^t-τt≤ε4i-1eTtXiD1iD1iTXiet+ε4ix^Tt-τtE1iTE1ix^t-τt,(55)2dTtXiBifx,t+dt-uet≤-2γ1BiTXiet<0,(56)2eTtXiA1iet-τt≤eTtXiXiet+eTt-τtA1iTA1iet-τt,(57)2x^TtXiψ1LiDiet≤x^TtXiψ1ψ1TXix^t++eTtDiTLiTLiDiet,(58)2x^TtXiψ2x^t-τt≤x^TtXiXix^t+x^Tt-τtψ2Tψ2x^t-τt,(59)12traceCietTCiet≤12λmaxCiTCieTtet,(60)12traceCix^tTCix^t≤12λmaxCiTCix^Ttx^t,
whereλmax(CiTCi) is the largest eigenvalue of CiTCi.
Letξ(t)≜e(t),et-τt,x^t,x^t-τtT, and then LV(x^,e,i)≤ξT(t)Λξ(t),
where(61)Λ=Λ110∗Λ22,(62)Λ11=Φ10∗Φ2,Λ22=Φ30∗Φ4,(63)Φ1=XiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCiTCi+Pi+∑j=1NπijXj+XiXi+DiTLiTLiDi+ε1i-1XiDiDiTXi+ε1iEiTEi+ε2i-1XiD1iD1iTXi+ε3i-1XiDiDiTXi+ε4i-1XiD1iD1iTXi+τH1,(64)Φ2=A1iTA1i+ε2iE1iTE1i-Pi,(65)Φ3=XiAi+BiKi+Ai+BiKiTXi+12λmaxCiTCi+Qi+XiXi+ε3iEiTEi+∑j=1NπijXj+Xiψ1ψ1TXi+τH2,(66)Φ4=ε4iE1iTE1i+ψ2Tψ2-Qi.
Fori∈Ski, the term XiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+1/2λmax(CiTCi)+Pi+τH1+∑j=1NπijXj can be rewritten as follows:(67)Θi≜XiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCiTCi+Pi+τH1+∑j=1NπijXj=1+∑j∈SkiπijXiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCiTCi+Pi+τH1+∑j∈SukiXiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCiTCi+Pi+τH1+∑j∈SkiπijXj+∑j∈SukiπijXj=1+πkiXiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCiTCi+Pi+τH1+∑j∈SkiπijXj+∑j∈SukiπijXiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCiTCi+Pi+τH1+Xj.
Then it follows from (28) and (32) that Θi<0. For ∀j∈Suki, i∉Ski,(68)Θi=1+∑j∈Ski,j≠iπijXiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCiTCi+Pi+τH1+πiiXiAi+AiTXi+XiLiDi+DiTLiTXi+12λmaxCiTCi+Pi+τH1+XiXi+Xi+∑j∈Suki,j≠iπijXiAi+AiTXi+XiLiDi+DiTLiTXi+XiXi+12λmaxCiTCi+Pi+τH1+Xj+∑j∈Ski,j≠iπijXj.
Then, combining (29), (32), and (33), we obtain Θi<0.
Similarly, fori∈Ski, the term Xi(Ai+BiKi)+Ai+BiKiTXi+1/2λmax(CiTCi)+Qi+τH2+XiXi+∑j=1NπijXj can be rewritten as follows:(69)Ωi≜XiAi+BiKi+Ai+BiKiTXi+12λmaxCiTCi+Qi+τH2+XiXi+∑j=1NπijXj=1+πkiXiAi+BiKi+Ai+BiKiTXi+12λmaxCiTCi+Qi+τH2+XiXi+∑j∈SkiπijXj+∑j∈SukiπijXiAi+BiKi+Ai+BiKiTXi+12λmaxCiTCi+Qi+τH2+XiXi+Xj.
According to (28) and (34), Ωi<0. For ∀j∈Suki,  i∉Ski,(70)Ωi=1+∑j∈Ski,j≠iπijXiAi+BiKi+Ai+BiKiTXi+12λmaxCiTCi+Qi+τH2+XiXi+πiiXiAi+BiKi+Ai+BiKiTXi+12λmaxCiTCi+Qi+τH2+XiXi+Xi+∑j∈Suki,j≠iπijXiAi+BiKi+Ai+BiKiTXi+12λmaxCiTCi+Qi+τH2+XiXi+Xj+∑j∈Ski,j≠iπijXj.
Then, combining (29), (34), and (35), we get Ωi<0.
Hence, we obtainΦ1<0, Φ2<0, Φ3<0, and Φ4<0 and Λ11<0 and Λ22<0, which implies Λ<0. Thus, we have(71)LVx^,e,i<0,i∈S,
which implies that the overall closed-loop system composed of (27) and (35) is stochastically stable.Remark 12.
Our result extends and improves those given in the previous literature. For example, ifCi=0, then the system (1) is reduced to that in [36]; if τ=0, then the system (1) is reduced to that in [52]. Moreover, the transition rates in this paper are partially unknown, which is quite different from those in [36, 52].Remark 13.
When analyzing the stability of the stochastic system with Markovian switching and time-varying delay, Theorem11 considers both state observer system and error system, which can conform the reality better and get better stability. When we construct the controller, because the system contains a Markovain stochastic term, we cannot directly apply the equivalent control method, so we construct the integral sliding surface and get our controller through the reachability of the integral sliding surface.


## 4. An Example

Let us consider system (1) above with the following matrix parameters:(72)A1=-1.8761.50.20-3.27-2.302.92-2.08,A2=-1.8761.50.20-4.27-2.302.92-2.08,A3=-1.8761.50.20-5.27-2.302.92-2.08,A4=-1.8761.50.20-6.27-2.302.92-2.08,B1=0.10.2-0.4,B2=0.20.1-0.1,B3=0.10.5-0.2,B4=10.2-0.5,D1=0.5-0.20.1-0.20.1-0.50.20.20,D2=0.5-0.20.1-0.20.1-0.150.20.20,D3=0.5-0.20.1-0.20.1-0.150.20.20,D4=0.5-0.20.1-0.20.1-0.250.20.20,C1=-10.6-2.420.1-0.50.12.20.5,C2=0.20.20.1-0.10.1-0.500.1-0.5,C3=10.8100.5-0.60.30.4-0.5,C4=-0.20.20.300.50.10.10.40.5,G1=0.20.20.200.20.1000.3,G2=0.20.20.200.20.1000.3,G3=0.20.20.200.20.1000.3,G4=0.20.20.200.20.1000.5,E1=0.20000.200.20.21,E2=0.20000.200.20.21,E3=0.10000.200.20.21,E4=0.10.1000.200.20.21,X1=7.5-4.7-4.1-4.79.5-0.8-4.10.856.7,X2=11.9-5.8-5.7-5.812.4-2.6-5.7-2.653.5,X3=13.8-6.1-6.4-6.114.2-2.1-6.4-2.150.3,X4=14.5-7.2-7.6-7.213.7-3.2-7.6-3.251.4,G11=0.10.10.100.10.2000.3,G12=0.10.10.100.10.2000.3,G13=0.10.10.100.10.2000.3,G14=0.10.10.100.10.2004,E11=0.10000.100.10.11,E12=0.10000.100.10.11,E13=0.10000.100.10.11,E14=0.10000.100.20.21.The transition matrixΠ is given as follows:(73)Π=?2.2???-1.91.7?1.2?-1.8?2.6??-2.8,where “?” represents the unknown element. We select the matricesK1, K2, K3, and K4 as follows:(74)K1=-7.544-6.6561.1216,K2=-12.296-4.808-0.194,K3=-9.568-3.5990.424,K4=-2.8651-2.051-0.251.Letτ(t)=0.5sin⁡t+0.5, τ=1, and ρi=0.5, i=1,2,3,4. The nonlinear function f(x,t), external disturbance d(t), and time-varying matrix function F(x,t) are given, respectively, as follows:(75)fx,t=0.6+0.2sin⁡20tx1t,dt=0.5+0.5sin⁡2t,F1,t=F2,t=F3,t=F4,t=0.02sin⁡2t.The initial conditions are given asx0=2.5-1.50.9T and x^0=0.5-0.50.9T. Select the constant matrix F=0.4680.963-1.915T. The adaptive laws are governed by (20); by solving the conditions (28)-(35), the observer is obtained as follows: (76)L1=4.6982-5.20201.82756.5907-7.02141.4326-2.5591-1.6181-1.3803,L2=6.1547-6.40961.24207.2397-2.20576.3323-1.1957-3.1032-3.1239,L3=1.7779-5.05171.38266.0450-2.00257.70804.0434-6.7840-2.0256,L4=5.5767-4.04522.27689.0970-3.03677.0766-2.2871-4.6401-1.5033.Then, the integral sliding mode surface in (18) for the 4 operating modes is synthesized as(77)st=1.451.11-23.25x^t-∫0t-75.6480-133.688356.9395x^ldl,i=1,2.370.34-6.75x^t-∫0t-18.9923-23.294713.5025x^ldl,i=2,-0.396.91-11.75x^t-∫0t-54.4374-92.062510.9138x^ldl,i=3,16.86-2.86-33.94x^t-∫0t-126.9169-124.094872.1974x^ldl,i=4.Selectingγ1=β=1, then we get(78)uet=0.5xt+d^t+1sgn⁡BiTXiet,ut=-ststuet+-7.544-6.6561.1216x^t+9.6035.0565.9285x^t-τt+4.24354.24355.2456et+1,i=1,-ststuet+-12.296-4.808-0.194x^t+2.9685.9372.968x^t-τt+3.3353.3963.631et+1,i=2,-ststuet+-9.568-3.5990.424x^t+9.5033.0371.793x^t-τt+4.2655.24355.6456et+1,i=3,-ststuet+-2.8651-2.051-0.251x^t+6.6035.0262.9285x^t-τt+2.24353.22255.2756et+1,i=1,Finally, we use Figures1–5 to show our results. In fact, Figure 1 shows the shows the trajectory of the Markov jumping signal with state 1,2,3. Figures 2–4 show state trajectory x(t), state estimation trajectory x^(t), and state estimation error trajectory e(t), respectively with state 1,2,3. Figure 5 shows the sliding mode surface function s(t).Figure 1
Trajectory ofr(t).Figure 2
State trajectoriesx(t).Figure 3
State estimation trajectoriesx^(t).Figure 4
State estimation error trajectories ofe(t).Figure 5
Sliding mode surface functions(t).


## 5. Conclusion

Because systems are inevitably influenced by stochastic factors, nonlinear factors, and time delay in the actual situation, discussing reachablity and stability of sliding mode control (SMC) in the context of stochastic system with time delay has more practical significance. This paper has investigated the SMC problem for stochastic nonlinear Markovian jump systems (MJSs) with partly unknown transition rates and time-varying delays. A SMC law has been synthesized such that the sliding mode surface can satisfy the reachable conditions. Sufficient conditions for the stability of the closed-loop system composed of the state observer system and error system have been proposed in terms of LMIs. A simulation result has been given to illustrate the feasibility of the results. Furture work will consider the SMC problem with Markov and impulse for nonlinear SDEs.



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*Source: 5036917-2019-01-01.xml*