# H

Authors: Lijun Gao; Yuqiang Wu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper addresses the problem ofH∞ control for a class of uncertain stochastic systems with Markovian switching and time-varying delays. The system under consideration is subject to time-varying norm-bounded parameter uncertainties and an unknown nonlinear function in the state. An integral sliding surface corresponding to every mode is first constructed, and the given sliding mode controller concerning the transition rates of modes can deal with the effect of Markovian switching. The synthesized sliding mode control law ensures the reachability of the sliding surface for corresponding subsystems and the global stochastic stability of the sliding mode dynamics. A simulation example is presented to illustrate the proposed method.


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## Body

## 1. Introduction

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. During the past decades, due to their potential applications in manufacturing systems and communications, considerable attention has been paid to the problems of stability and stabilization [1–5], H∞ control and filtering [6–8], optimal tracking problem [9], and so on. The robust stability and stabilization of uncertain stochastic systems with time-varying delays are investigated by using the linear matrix inequality approach (LMI) [10]. Some stability criteria are obtained for a class of bilinear continuous time-delay uncertain systems with Markovian jump parameter [11]. In [12], a robust H∞ controller is designed for uncertain systems with state delay. Huang and Mao [13] investigated the stabilization of stochastic linear systems by delayed state feedback controller. In [14], 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. K. Chang and W. Chang [15] developd an SMC method to guarantee the robust state covariance assignment for perturbed stochastic multivariable systems via variable structure control. In [16], robust observer design for Ito stochastic time-delay systems has been studied via sliding mode control and the sufficient conditions for the asymptotic stability (in probability) of the sliding motion are derived. Niu et al. [17] paid some efforts to coping with the connection among the designed sliding surface corresponding to every mode for MJSs. However, there are little results reported on the SMC of stochastic systems with Markovian switching. The existence of uncertainties, time-varying delays, Markovian switching, and bilinear perturbations will make the problem more complex and challenging.This paper studies the robustH∞  control problem for uncertain stochastic systems with time-varying delays and Markovian switching. The systems under consideration may contain time-varying parameter uncertainties and bilinear stochastic perturbations, nonlinearities, and external disturbance. An integral-type sliding surface is constructed. Due to the existence of Markovian switching, in the design of sliding surface, a set of specified matrices are given to establish the connections among sliding surfaces corresponding to every mode. SMC law is synthesized to guarantee that the trajectories can be driven onto the specified sliding surfaces for each mode in finite time and the dynamics along the sliding surface for each mode is stochastically stable.


## 2. Preliminaries

Let(Ω,F,P) be a probability space with Ω the sample space, F the σ-algebra of subsets of the sample space, and P the probability measure. ∥·∥ and ∥·∥1 denote the Euclidean norm and 1-norm of a vector or its induced matrix norm, respectively. If a∈Rn, we have that ∥a∥<∥a∥1. ∥·∥2 stands for the usual L2[0,∞] norm. For a real matrix M, M>0 means that M is symmetric positive definite. I is the identity matrix with compatible dimension.The Markov process{rt,t≥0} represents the switching between the different modes taking values in a finite state space S={1,2,…,N} with generator π=(πij)N×N given by
(1)Pr{rt+Δ=j∣rt=i}={πijΔ+ο(Δ),ifi≠j1+πiiΔ+ο(Δ),ifi=j,whereπij is the transition rate from mode i to j and satisfies the following relations:
(2)πij≥0,πii=-∑j≠iπij,andο(Δ) is such that limΔ→0ο(Δ)/Δ=0.Consider the following stochastic system with Markovian switching and time-varying delay:(3)dx(t)=[(A(rt)+A~(rt))x(t)dx(t)=V+(Ad(rt)+A~d(rt))x(t-τ(t))dx(t)=V+B(rt)(u(t)+f(t,x(t),rt))+Bv(rt)v(t)(A(rt)+A~(rt))]dtdx(t)=+D(rt)[(C(rt)+C~(rt))x(t)dx(t)=DDDDD+(Cd(rt)+C~d(rt))x(t-τ(t))]dω(t),z(t)=(E(rt)+E~(rt))x(t)+F(rt)v(t),x(t)=φ(t),t∈[-d,0],wherex(t)∈Rn is the system state, u(t)∈Rm is the control input, z(t)∈Rp is the controlled output, v(t)∈Rq is the exogenous disturbance input belonging to L2[0,∞]∩L∞[0,∞], and ω(t) is a one-dimensional Brownian motion defined on the probability space (Ω,F,P). A(rt), Ad(rt), B(rt), Bv(rt), C(rt), Cd(rt), D(rt), E(rt), and F(rt) are known real constant matrices with appropriate dimensions. In general, it is assumed that the pair (A(rt),B(rt)) is controllable and the input matrix B(rt) has full column rank. τ(t) is the time-varying delay satisfying
(4)0<τ(t)≤d<∞,τ˙(t)≤h<1,whered and h are known real constant scalars, and φ(t) is a continuous vector-valued initial function. A~(rt), A~d(rt), C~(rt), C~d(rt), and E~(rt) are parameter uncertainties, and unknown function f(t,x(t),rt) is an unknown nonlinear uncertainties.For eachrt=i∈S, for the sake of convenience, denote A(rt)=Ai and A~(rt)=A~i(t). Other matrices are defined as above. Then system (3) becomes
(5)dx(t)=[(Ai+A~i(t))x(t)+(Adi+A~di(t))x(t-τ(t))+Bi(u(t)+f(t,x(t),i)),+Bviv(t)(Adi+A~di(t))]dt+Di[(Ci+C~i(t))x(t),+(Cdi+C~di(t))x(t-τ(t))]dω(t),(6)z(t)=(Ei+E~i(t))x(t)+Fiv(t),(7)x(t)=φ(t),t∈[-d,0].Forrt=i, the admissible uncertainties are assumed to be norm bounded and can be described as
(8)[A~i(t)A~di(t)E~i(t)]=N1iF1i(t)[HaiHadiHei],[C~i(t)C~di(t)]=N2iF2i(t)[HciHcdi],(9)∥f(t,x(t),i)∥≤ηi∥x(t)∥,whereηi>0 is a constant scalar, Hai, Hadi, Hci, Hcdi, Hei, N1i, and N2i are known real constant matrices, and F1i(t) and F2i(t) are unknown matrix functions satisfying
(10)F1iT(t)F1i(t)≤I,F2iT(t)F2i(t)≤I,∀t.In the sequel, some concepts and lemmas about the stability of stochastic systems are used.Definition 1.
System (5)–(7) with u(t)=0 is stochastically stable for v(t)=0, if there exists a constant matrix G>0 such that the following inequality holds for any pair of initial conditions (x0,r0)(11)E[∫0∞xT(t)x(t)dt∣x0,σ(0)=r0]≤x0TGx0.Definition 2.
For given scalarγ>0, the system (5)–(7) is said to stochastically stabilizable and satisfy ∥Tzv(s)∥∞≤γ if the closed-loop system is stochastically stable for v(t)=0 and for all mismatched uncertainties,
(12)∥z(t)∥E2<γ∥v(t)∥E2,
holds,  where∥z(t)∥E2=(E{∫0t|z(t)|2dt})1/2.Definition 3.
LetC2,1(Rn×[t0-r,∞]×S;R+) denote the family of all nonnegative functions V(x,t,i) that are continuously twice differentiable in x and once differentiable with respect to t. For each V(x,t,i)∈C2,1(Rn×[t0-r,∞]×S;R+), define an infinitesimal operator LV from Rn×[t0-r,∞]×S to R+ as follows:
(13)LV(x,t,i)=LV(x,t,i)LV(x,t,i)=limΔ→01Δ[E{V(x(t+Δ),rt+Δ,t+Δ)∣LV(x,t,i)=DDDx(t)=x,rt=i,t(x(t+Δ),rt+Δ,t+Δ)}-V(x,i,t)(x(t+Δ),rt+Δ,t+Δ)].
The following matrix inequalities will be essential for the proofs in Section3.Lemma 4 (see [18]).
LetQ, H, F, and G be real matrices of appropriate dimensions with FTF≤I, then, for any real matrix Q=QT, there exists scalar ε>0, and one has the following:
(14)Q+HF(t)G+GTFT(t)HT<0,∀F(t)s.t.FT(t)F(t)≤I,
if and only if there exists some scalarε>0 such that
(15)Q+εHHT+ε-1GTG<0.Lemma 5 (see [18]).
For any real vectorsa,b and matrix X>0 of compatible dimensions
(16)aTb+bTa≤aTXa+bTX-1b.


## 3. Main Results

This section constructs an SMC lawu(t) for system (5)–(7) such that the resultant closed-loop system is stochastically stable despite uncertainties, exogenous disturbance, time delay, and Markovian switching.


### 3.1. Integral-Type Sliding Surface

As the first step of SMC design, the integral-type sliding surface is constructed as follows:(17)s(x(t),i)=BiTXix(t)s(x(t),i)=-∫0tBiTXi(Ai+BiKi)x(τ)dτ,for eachrt=i∈S. In (17), the matrix Ki∈Rm×n is chosen such that the matrix Ai+BiKi is Hurwitz. In particular, the matrix Xi∈Rn×n is to be designed so that BiTXiBi is nonsingular and BiTXiDi=0. Under the assumption that Bi is full rank, it can be easily shown that the nonsingularity of BiTXiBi can be guaranteed if Xi is symmetric positive definite, that is, Xi>0. In addition, the condition BiTXiDi=0 will be incorporated in Theorem 6.The solutionx(t) of the stochastic system (5)–(7) is expressed as follows:
(18)x(t)=φ(0)x(t)=+∫0t[(Ai+A~i(s))x(s)+(Adi+A~di(s))x(s-τ(s))x(t)=VVVV+Bi(u(s)+f(s,x(s),i))+Bviv(s)(Ai+A~i(s))]dsx(t)=+∫0tDi[(Ci+C~i(s))x(s)x(t)=+DiDi+(Cdi+C~di(s))x(s-τ(s))]dω(s).Under the condition thatBiTXiDi=0, it can be obtained from (17) and (18) that
(19)s(x(t),i)=BiTXiφ(0)+∫0tBiTXi[(Ai+A~i(s))x(s)DDDDDDD+(Adi+A~di(s))x(s-τ(s))DDDDDDD+Bi(u(s)+f(s,x(s),i))+Bviv(s)(Ai+A~i(s))]ds.According to the sliding mode theory, we haves(x(t),i)=0 and s˙(x(t),i)=0. The equivalent control law in the sliding mode can be obtained by solving s˙(x(t),i)=0 as
(20)ueq(t)=Kix(t)-(BiTXiBi)-1BiTXi×[A~i(t)x(t)+(Adi+A~di(t))x(t-τ(t))+Bviv(t)(Adi+A~di(t))]-f(t,x(t),i).Substituting (20) into (5), the sliding mode dynamics in s(x(t),i)=0 can be written as
(21)dx(t)={[Ai+BiKi+A~i(t)-Bi(BiTXiBi)-1BiTXiA~i(t)]x(t)+[Adi+A~di(t)-Bi(BiTXiBi)-1×BiTXi(Adi+A~di(t))]x(t-τ(t))+[Bviv(t)-Bi(BiTXiBi)-1BiTXiBviv(t)]}dt+Di[(Ci+C~i(t))x(t)+(Cdi+C~di(t))x(t-τ(t))]dω(t).Now, the stochastic stability of the sliding mode dynamics described by (21) will be analyzed.


### 3.2. Stochastic Stability of Sliding Surface

Theorem 6.
Consider the stochastic delay system (5)–(7) with assumptions (4), (8)–(10), and the sliding mode surface (17). For a given scalar γ>0, if there exist matrices Xi>0 and Q>0, scalars ε1i>0, ε2i>0, ε3i>0, and ε4i>0 (i=1,2,…,N) satisfy the following LMIs: (22)[Π1i******Π5iTΠ2i*****BviTXi0Π3i****XiDiCiXiDiCdi0-Xi***N2iTDiTXiDiCiN2iTDiTXiDiCdi00Π4i**Ei0Fi00-I*000000-Xi0XiAdiT000002BiTXi000000BiTXi000000N1iTXi000000N1iTXi00000000000N1iTXi0000000N1iTXi0000000********************************************************-Xi*******0-BiTXiBi******00-BiTXiBi*****000-ε1iI****0000-ε2iI***00000-ε4iI**000000-ε4iI*N1iTXi000000-ε4iI],(23)BiTXiDi=0,
with(24)Π1i=Xi(Ai+BiKi)+(Ai+BiKi)TXiΠ1i=+Q+ε1iHaiTHai+ε3iHciTHci+ε4iHaiTHaiΠ1i=+ε4iHeiTHei+∑j=1NπijXj,Π2i=-(1-h)Q+ε2iHadiTHadi+ε3iHciTHci+ε4iHadiTHadi,Π3i=BviTXiBvi-γ2I,Π4i=-ε3iI+N2iTDiTXiDiN2i,Π5i=XiAdi+ε3iHciTHcdi,
and then the sliding motion (21) is stochastically stable and satisfies H∞ performance for all v(t)∈L2[0,∞).Proof.
Consider the following Lyapunov function:(25)V(x(t),i)=xT(t)Xix(t)+∫t-τ(t)txT(s)Qx(s)ds.
By Definition2, the stochastic stability for systems (21) is firstly established with v(t)=0. From Definition 3, the infinitesimal operator LV(x(t),i) for (21) with v(t)=0 is obtained for each i∈S(26)LV(x(t),i)=2xT(t)Xi[(BiTXiBi)-1Ai+BiKi+A~i(t)-Bi(BiTXiBi)-1BiTXiA~i(t)]x(t)+2xT(t)Xi×[(BiTXiBi)-1Adi+A~di(t)-Bi(BiTXiBi)-1BiTXi(Adi+A~di(t))]x(t-τ(t))+[(Ci+C~i(t))x(t)+(Cdi+C~di(t))x(t-τ(t))]T×DiTXiDi×[(Ci+C~i(t))x(t)+(Cdi+C~di(t))x(t-τ(t))]+∑j=1NπijxT(t)Xjx(t)+xT(t)Qx(t)-(1-τ˙(t))xT(t-τ(t))Qx(t-τ(t)).
Using Lemma5, one has
(27)2xT(t)XiA~i(t)x(t)≤ε1i-1xT(t)XiN1iN1iTXix(t)+ε1ixT(t)HaiTHaix(t),2xT(t)XiA~di(t)x(t-τ(t))≤ε2i-1xT(t)XiN1iN1iTXix(t)+ε2ixT(t-τ(t))HadiTHadix(t-τ(t)),-2xT(t)XiBi(BiTXiBi)-1BiTXiA~i(t)x(t)≤xT(t)XiBi(BiTXiBi)-1BiTXix(t)+xT(t)A~iT(t)XiA~i(t)x(t),-2xT(t)XiBi(BiTXiBi)-1BiTXi(Adi+A~di(t))x(t-τ(t))≤xT(t)XiBi(BiTXiBi)-1BiTXix(t)+xT(t-τ(t)(Adi+A~di(t))T×Xi(Adi+A~di(t))x(t-τ(t)).
From (22), we can obtain ε3iI-N2iTDiTXiDiN2i>0. According to assumption (8), it has
(28)[DiC-i+DiN2iF2i(t)H-ci]TXi[DiC-i+DiN2iF2i(t)H-ci]≤C-iTDiTXiDiC-i+C-iTDiTXiDiN2i×(ε3iI-N2iTDiTXiDiN2i)-1N2iTDiTXiDiC-i+ε3iH-ciTH-ci,
whereC-i=[CiCdi], H-ci=[HciHcdi]. Define Φ=[xT(t),xT(t-τ(t))]T. Then it follows from (26)–(28) that
(29)LV(x(t),i)≤ΦTΓiΦ,
in which(30)Γi=[Ξ1iXiAdiAdiTXiΞ2i]+C-iTDiTXiDiC-i+C-iTDiTXiDiN2i(ε3iI-N2iTDiTXiDiN2i)-1×N2iTDiTXiDiC-i+ε3iH-ciTH-ci,
with(31)Ξ1i=Xi(Ai+BiKi)+(Ai+BiKi)TXi+Q+ε1iHaiTHaiΞ1i=+ε1i-1XiN1iN1iTXi+ε2i-1XiN1iN1iTXi+A~iT(t)XiA~i(t)Ξ1i=+2XiBi(BiTXiBi)-1BiTXi,Ξ2i=-(1-h)Q+ε2iHadiTHadiΞ2i=+(Adi+A~di(t))TXi(Adi+A~di(t)).
So, ifΓi<0, for any Φ≠0, we have
(32)LV(x(t),i)<0.
According to Definition1, the sliding mode dynamics (21) is stochastically stable.
By Schur’s complement,Γi<0 is equivalent to the following LMI: (33)[Ξ1i*********Π5iTΞ2i********XiDiCiXiDiCdi-Xi******E2iTDiTXiDiCiE2iTDiTXiDiCdi0Π3i*****0000-Xi****0XiAdi000-Xi***2BiTXi00000-BiTXiBi**N1iTXi000000-ε1iI*N1iTXi0000000-ε2iI]+GF(t)H+HTFT(t)GT<0,
where(34)Ξ1i*=Xi(Ai+BiKi)+(Ai+BiKi)TXiΞ1i*=+Q+ε1iHaiTHai+ε3iHciTHci+∑j=1NπijXj,Ξ2i*=-(1-h)Q+ε2iHadiTHadi+ε3iHcdiTHcdi,G=[Hai000000000Hadi0000000]T,F(t)=[F1iT(t)00F2iT(t)],H=[0000N1iTXi000000000N1iTXi000].
According to Lemma4, there exists positive scalar ε4i  such that the following LMI holds:(35)[Ξ3i**********Π5iTΞ4i*********XiDiCiXiDiCdi-Xi********E2iTDiTXiDiCiE2iTDiTXiDiCdi0Π3i*******0000-Xi******0XiAdi000-Xi*****2BiTXi00000-BiTXiBi****N1iTXi000000-ε1iI***N1iTXi0000000-ε2iI**0000N1iTXi0000-ε4iI*00000N1iTXi0000-ε4iI]<0,
with(36)Ξ3i=Xi(Ai+BiKi)+(Ai+BiKi)TXi+Q+ε1iHaiTHaiΞ3i=+ε3iHciTHci+ε4iHaiTHai+∑j=1NπijXj,Ξ4i=-(1-h)Q+ε2iHadiTHadi+ε3iHcdiTHcdi+ε4iHadiTHadi.
Moreover, by Schur’s complement lemma, the above inequality is implied by (22).
Then, we will prove that the stochastic system (21) with (6) satisfies
(37)∥z(t)∥E2<γ∥v(t)∥2,
for all nonzerov(t)∈L2[0,∞).
Choose the same Lyapunov function as (25). And the generator LV(x(t),i) with v(t)≠0 can be calculated as follows:
(38)LV(x(t),i)=2xT(t)Xi[(BiTXiBi)-1Ai+BiKi+A~i(t)-Bi(BiTXiBi)-1BiTXiA~i(t)]x(t)+2xT(t)Xi[(BiTXiBi)-1Adi+A~di(t)-Bi(BiTXiBi)-1BiTXi(Adi+A~di(t))]×x(t-τ(t))+2xT(t)Xi×[I-Bi(BiTXiBi)-1BiTXi]Bviv(t)+[(Ci+C~i(t))x(t)+(Cdi+C~di(t))x(t-τ(t))]T×DiTXiDi×[(Ci+C~i(t))x(t)+(Cdi+C~di(t))x(t-τ(t))]+∑j=1NπijxT(t)Xjx(t)+xT(t)Qx(t)-(1-τ˙(t))xT(t-τ(t))Qx(t-τ(t)).
Let(39)J(t)≜E{∫t0t[zT(s)z(s)-γ2vT(s)v(s)]ds}.
So, under zero initial condition, we haveV(x(t0),i)=0 for x(t0)=0. From Dynkin’s formula, one has
(40)E{V(x(t),t)}=E{∫t0tLV(x(s),s)ds}.
Then, similar to the (26)–(29), we can obtain
(41)J(t)=E{∫t0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s)]ds}J(t)=-E{V(x(t),t)}J(t)≤E{∫0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s)]ds}J(t)=E{∫0t[xT(s)xT(s-τ(s))vT(s)]×Ωi[xT(s)xT(s-τ(s))vT(s)]Tds∫0t[xT(s)xT(s-τ(s))vT(s)]},
where(42)Ωi=[Θ1iΘ3iXiBvi+(Ei+E~i)TFiΘ3iTΘ2i0BviTXi+FiT(Ei+E~i)0BviTXiBvi+FiTFi-γ2I],
with(43)Θ1i=Ξ1i+XiBi(BiTXiBi)-1BiTXiΘ1i=+(Ei+E~i)T(Ei+E~i)+CiTDiTXiDiCiΘ1i=+CiTDiTXiDiN2iΘ1i×(ε3iI-N2iTDiTXiDiN2i)-1N2iTDiTXiDiCi,Θ2i=Ξ2i+CdiTDiTXiDiCdi+CdiTDiTXiDiN2iΘ2i×(ε3iI-N2iTDiTXiDiN2i)-1N2iTDiTXiDiCdi,Θ3i=XiAdi+CiTDiTXiDiN2iΘ3i=×(ε3iI-N2iTDiTXiDiN2i)-1Θ3i×N2iTDiTXiDiCdi,
andΞ1i and Ξ2i are defined as in (30).
It is observed thatΩi<0 from (22) and Schur’s complement. Hence, by (39), inequality ∥z(t)∥E2<γ∥v(t)∥2 holds for all nonzero v(t)∈L2[0,∞). The proof is complete.Remark 7.
Notice that the condition in Theorem6 is not a convex set due to the matrix equality constraints in (23). A simple algorithm is given to solve the feasibility problem. The equality condition BiTXiDi=0 with Xi satisfying inequality (22) can be equivalently converted to tr[(BiTXiDi)TBiTXiDi]=0.
Consider the following matrix inequality forα>0:
(44)(BiTXiDi)TBiTXiDi≤αI,fori∈S.
By Schur’s complement, one has(45)[-αIBiTXiDiDiTXiBi-I]≤0,fori∈S.
Now, the problem is changed to a problem of finding a global solution of the following minimization problem:(46)minα,subjectto(22)and(45).
It can be seen that if the global infimumα of (46) equals to zero, the corresponding solutions will satisfy the LMIs (22) and (23). So, the sliding control problem is solvable.


### 3.3. SMC Law Synthesis

Now, we will synthesize a SMC law to ensure that the trajectories of system (5)–(7) can be driven to reach and keep the predefined surface s(x(t),i)=0 from the initial time.Theorem 8.
For the uncertain stochastic delay system (5)–(7) with Markovian jumping and assumptions (4) and (8)–(10), the sliding surface is designed as (17) where Xi, (i∈S) are the solutions of LMI (22)-(23). Then the trajectories of system can be driven to the sliding surface s(x(t),i)=0 in finite time (with probability one) by employing the following SMC law:
(47)ui(t)=Kix(t)-μis(x(t),i)ui(t)=-(BiTXiBi)-1BiTXiAdix(t-τ(t))ui(t)=-12∑j=1Nπij(BjTXjBj)-1s(x(t),i)ui(t)=-ρi(t)sgn(s(x(t),i)),
where(48)ρi(t)=ηi∥x(t)∥+∥(BiTXiBi)-1BiTXiN1i∥ρi(t)×(∥Haix(t)∥+∥Hadix(t-τ(t))∥)ρi(t)=+∥(BiTXiBi)-1BiTXiBvi∥∥v(t)∥,
withμi are a small positive constant for each i∈S.Proof.
By (19) and (47), and by observing that BiTXiDi=0, it follows that
(49)s˙(x(t),i)=BiTXiA~i(t)x(t)+BiTXiA~di(t)x(t-τ(t))s˙(x(t),i)=-μiBiTXiBis(x(t),i)+BiTXiBviv(t)s˙(x(t),i)=-12∑j=1NπijBiTXiBi(BjTXjBj)-1s(x(t),i)s˙(x(t),i)=-ρi(t)BiTXiBisgn(s(x(t),i))s˙(x(t),i)=+BiTXiBif(t,x(t),i).
Choose the following Lyapunov function candidate:(50)V(t)=12sT(t)(BiTXiBi)-1s(t).
Utilizing expression (49) yields
(51)V˙(t)=sT(x(t),i)(BiTXiBi)-1s˙(x(t),i)V˙(t)=+12∑j=1NπijsT(x(t),i)(BjTXjBj)-1s(x(t),i)V˙(t)=sT(x(t),i)(BiTXiBi)-1BiTXiA~i(t)x(t)V˙(t)=+sT(x(t),i)(BiTXiBi)-1BiTXiA~di(t)x(t-τ(t))V˙(t)=-μisT(x(t),i)s(x(t),i)V˙(t)=-ρi(t)sT(x(t),i)sgn(s(x(t),i))V˙(t)=+sT(x(t),i)f(t,x(t),i)V˙(t)=+sT(x(t),i)(BiTXiBi)-1BiTXiBviv(t)V˙(t)≤∥s(x(t),i)∥∥(BiTXiBi)-1BiTXiN1i∥V˙(t)×(∥Haix(t)∥+∥Hadix(t-τ(t))∥)V˙(t)=-ρi(t)∥s(x(t),i)∥1+∥s(x(t),i)∥×∥(BiTXiBi)-1BiTXiBvi∥∥v(t)∥-μi∥s(x(t),i)∥2V˙(t)=+ηi∥s(x(t),i)∥∥x(t)∥.
Noting that∥s(x(t),i)∥1≥∥s(x(t),i)∥ and substituting (48) into (51), one has
(52)V˙(t)≤-μi∥s(x(t),i)∥2<0,for∥s(x(t),i)∥≠0.
This means that the trajectories of system (5)–(7) can be driven to remain in the sliding surface from the initial time.Remark 9.
The use of sign functionsgn(s(x(t),i)) may cause chattering behavior, which is undesired in practical engineering background [19]. To avoid the shortcomings, the function tanh(s(x(t),i)) is introduced to approximate function sgn(s(x(t),i)) [20].From SMC (47)-(48), it is seen that the bound ηi of f(t,x(t),i) is needed to synthesize the SMC law. It is well known that a practical engineering system may lose the exact knowledge of the bound due to the noise pollution and measurement error. Then, an adaptive SMC law is further presented for the case when the bound ηi is unknown.Theorem 10.
For the uncertain stochastic delay systems (5)–(7) with Markovian jumping and assumptions (4) and (8)–(10), the sliding surface is designed as (17) where Xi, (i∈S) are the solutions of LMI (22)-(23). Then the trajectories of system can be driven to the sliding surface s(x(t),i)=0 in finite time (with probability one) by employing the following SMC law:
(53)ui(t)=Kix(t)-μis(x(t),i)ui(t)=-(BiTXiBi)-1BiTXiAdix(t-τ(t))ui(t)=-12∑j=1Nπij(BjTXjBj)-1s(x(t),i)ui(t)=-ρ^i(t)sgn(s(x(t),i)),
where(54)ρ^i(t)=η^i∥x(t)∥+∥(BiTXiBi)-1BiTXiN1i∥ρ^i(t)×(∥Haix(t)∥+∥Hadix(t-τ(t))∥)ρ^i(t)×+∥(BiTXiBi)-1BiTXiBvi∥∥v(t)∥,
and adaptive law as(55)η^˙i=βi∥s(t)∥∥x(t)∥,
withμi, βi as small positive constants for each i∈S.Proof.
Choose the following Lyapunov function:(56)V(t)=12sT(t)(BiTXiBi)-1s(t)+12βi-1η~i2(t),
whereη~i(t)=η^i(t)-ηi. In line with the proof of Theorem 8, it can easily obtained that
(57)V˙(t)≤∥s(x(t),i)∥∥(BiTXiBi)-1BiTXiN1i∥×(∥Haix(t)∥+∥Hadix(t-τ(t))∥)-ρ^i(t)∥s(x(t),i)∥1V˙(t)≤+∥s(x(t),i)∥∥(BiTXiBi)-1BiTXiBvi∥∥v(t)∥V˙(t)≤-μi∥s(x(t),i)∥2+ηi∥s(x(t),i)∥∥x(t)∥V˙(t)≤+η~i(t)∥s(x(t),i)∥∥x(t)∥.
Noting that∥s(x(t),i)∥1≥∥s(x(t),i)∥ and substituting (54) into (57), one has
(58)V˙(t)≤-μi∥s(x(t),i)∥2<0,for∥s(x(t),i)∥≠0.
This means that the trajectories of system (5)–(7) can be driven to remain in the sliding surface from the initial time. This completes the proof.


## 3.1. Integral-Type Sliding Surface

As the first step of SMC design, the integral-type sliding surface is constructed as follows:(17)s(x(t),i)=BiTXix(t)s(x(t),i)=-∫0tBiTXi(Ai+BiKi)x(τ)dτ,for eachrt=i∈S. In (17), the matrix Ki∈Rm×n is chosen such that the matrix Ai+BiKi is Hurwitz. In particular, the matrix Xi∈Rn×n is to be designed so that BiTXiBi is nonsingular and BiTXiDi=0. Under the assumption that Bi is full rank, it can be easily shown that the nonsingularity of BiTXiBi can be guaranteed if Xi is symmetric positive definite, that is, Xi>0. In addition, the condition BiTXiDi=0 will be incorporated in Theorem 6.The solutionx(t) of the stochastic system (5)–(7) is expressed as follows:
(18)x(t)=φ(0)x(t)=+∫0t[(Ai+A~i(s))x(s)+(Adi+A~di(s))x(s-τ(s))x(t)=VVVV+Bi(u(s)+f(s,x(s),i))+Bviv(s)(Ai+A~i(s))]dsx(t)=+∫0tDi[(Ci+C~i(s))x(s)x(t)=+DiDi+(Cdi+C~di(s))x(s-τ(s))]dω(s).Under the condition thatBiTXiDi=0, it can be obtained from (17) and (18) that
(19)s(x(t),i)=BiTXiφ(0)+∫0tBiTXi[(Ai+A~i(s))x(s)DDDDDDD+(Adi+A~di(s))x(s-τ(s))DDDDDDD+Bi(u(s)+f(s,x(s),i))+Bviv(s)(Ai+A~i(s))]ds.According to the sliding mode theory, we haves(x(t),i)=0 and s˙(x(t),i)=0. The equivalent control law in the sliding mode can be obtained by solving s˙(x(t),i)=0 as
(20)ueq(t)=Kix(t)-(BiTXiBi)-1BiTXi×[A~i(t)x(t)+(Adi+A~di(t))x(t-τ(t))+Bviv(t)(Adi+A~di(t))]-f(t,x(t),i).Substituting (20) into (5), the sliding mode dynamics in s(x(t),i)=0 can be written as
(21)dx(t)={[Ai+BiKi+A~i(t)-Bi(BiTXiBi)-1BiTXiA~i(t)]x(t)+[Adi+A~di(t)-Bi(BiTXiBi)-1×BiTXi(Adi+A~di(t))]x(t-τ(t))+[Bviv(t)-Bi(BiTXiBi)-1BiTXiBviv(t)]}dt+Di[(Ci+C~i(t))x(t)+(Cdi+C~di(t))x(t-τ(t))]dω(t).Now, the stochastic stability of the sliding mode dynamics described by (21) will be analyzed.


## 3.2. Stochastic Stability of Sliding Surface

Theorem 6.
Consider the stochastic delay system (5)–(7) with assumptions (4), (8)–(10), and the sliding mode surface (17). For a given scalar γ>0, if there exist matrices Xi>0 and Q>0, scalars ε1i>0, ε2i>0, ε3i>0, and ε4i>0 (i=1,2,…,N) satisfy the following LMIs: (22)[Π1i******Π5iTΠ2i*****BviTXi0Π3i****XiDiCiXiDiCdi0-Xi***N2iTDiTXiDiCiN2iTDiTXiDiCdi00Π4i**Ei0Fi00-I*000000-Xi0XiAdiT000002BiTXi000000BiTXi000000N1iTXi000000N1iTXi00000000000N1iTXi0000000N1iTXi0000000********************************************************-Xi*******0-BiTXiBi******00-BiTXiBi*****000-ε1iI****0000-ε2iI***00000-ε4iI**000000-ε4iI*N1iTXi000000-ε4iI],(23)BiTXiDi=0,
with(24)Π1i=Xi(Ai+BiKi)+(Ai+BiKi)TXiΠ1i=+Q+ε1iHaiTHai+ε3iHciTHci+ε4iHaiTHaiΠ1i=+ε4iHeiTHei+∑j=1NπijXj,Π2i=-(1-h)Q+ε2iHadiTHadi+ε3iHciTHci+ε4iHadiTHadi,Π3i=BviTXiBvi-γ2I,Π4i=-ε3iI+N2iTDiTXiDiN2i,Π5i=XiAdi+ε3iHciTHcdi,
and then the sliding motion (21) is stochastically stable and satisfies H∞ performance for all v(t)∈L2[0,∞).Proof.
Consider the following Lyapunov function:(25)V(x(t),i)=xT(t)Xix(t)+∫t-τ(t)txT(s)Qx(s)ds.
By Definition2, the stochastic stability for systems (21) is firstly established with v(t)=0. From Definition 3, the infinitesimal operator LV(x(t),i) for (21) with v(t)=0 is obtained for each i∈S(26)LV(x(t),i)=2xT(t)Xi[(BiTXiBi)-1Ai+BiKi+A~i(t)-Bi(BiTXiBi)-1BiTXiA~i(t)]x(t)+2xT(t)Xi×[(BiTXiBi)-1Adi+A~di(t)-Bi(BiTXiBi)-1BiTXi(Adi+A~di(t))]x(t-τ(t))+[(Ci+C~i(t))x(t)+(Cdi+C~di(t))x(t-τ(t))]T×DiTXiDi×[(Ci+C~i(t))x(t)+(Cdi+C~di(t))x(t-τ(t))]+∑j=1NπijxT(t)Xjx(t)+xT(t)Qx(t)-(1-τ˙(t))xT(t-τ(t))Qx(t-τ(t)).
Using Lemma5, one has
(27)2xT(t)XiA~i(t)x(t)≤ε1i-1xT(t)XiN1iN1iTXix(t)+ε1ixT(t)HaiTHaix(t),2xT(t)XiA~di(t)x(t-τ(t))≤ε2i-1xT(t)XiN1iN1iTXix(t)+ε2ixT(t-τ(t))HadiTHadix(t-τ(t)),-2xT(t)XiBi(BiTXiBi)-1BiTXiA~i(t)x(t)≤xT(t)XiBi(BiTXiBi)-1BiTXix(t)+xT(t)A~iT(t)XiA~i(t)x(t),-2xT(t)XiBi(BiTXiBi)-1BiTXi(Adi+A~di(t))x(t-τ(t))≤xT(t)XiBi(BiTXiBi)-1BiTXix(t)+xT(t-τ(t)(Adi+A~di(t))T×Xi(Adi+A~di(t))x(t-τ(t)).
From (22), we can obtain ε3iI-N2iTDiTXiDiN2i>0. According to assumption (8), it has
(28)[DiC-i+DiN2iF2i(t)H-ci]TXi[DiC-i+DiN2iF2i(t)H-ci]≤C-iTDiTXiDiC-i+C-iTDiTXiDiN2i×(ε3iI-N2iTDiTXiDiN2i)-1N2iTDiTXiDiC-i+ε3iH-ciTH-ci,
whereC-i=[CiCdi], H-ci=[HciHcdi]. Define Φ=[xT(t),xT(t-τ(t))]T. Then it follows from (26)–(28) that
(29)LV(x(t),i)≤ΦTΓiΦ,
in which(30)Γi=[Ξ1iXiAdiAdiTXiΞ2i]+C-iTDiTXiDiC-i+C-iTDiTXiDiN2i(ε3iI-N2iTDiTXiDiN2i)-1×N2iTDiTXiDiC-i+ε3iH-ciTH-ci,
with(31)Ξ1i=Xi(Ai+BiKi)+(Ai+BiKi)TXi+Q+ε1iHaiTHaiΞ1i=+ε1i-1XiN1iN1iTXi+ε2i-1XiN1iN1iTXi+A~iT(t)XiA~i(t)Ξ1i=+2XiBi(BiTXiBi)-1BiTXi,Ξ2i=-(1-h)Q+ε2iHadiTHadiΞ2i=+(Adi+A~di(t))TXi(Adi+A~di(t)).
So, ifΓi<0, for any Φ≠0, we have
(32)LV(x(t),i)<0.
According to Definition1, the sliding mode dynamics (21) is stochastically stable.
By Schur’s complement,Γi<0 is equivalent to the following LMI: (33)[Ξ1i*********Π5iTΞ2i********XiDiCiXiDiCdi-Xi******E2iTDiTXiDiCiE2iTDiTXiDiCdi0Π3i*****0000-Xi****0XiAdi000-Xi***2BiTXi00000-BiTXiBi**N1iTXi000000-ε1iI*N1iTXi0000000-ε2iI]+GF(t)H+HTFT(t)GT<0,
where(34)Ξ1i*=Xi(Ai+BiKi)+(Ai+BiKi)TXiΞ1i*=+Q+ε1iHaiTHai+ε3iHciTHci+∑j=1NπijXj,Ξ2i*=-(1-h)Q+ε2iHadiTHadi+ε3iHcdiTHcdi,G=[Hai000000000Hadi0000000]T,F(t)=[F1iT(t)00F2iT(t)],H=[0000N1iTXi000000000N1iTXi000].
According to Lemma4, there exists positive scalar ε4i  such that the following LMI holds:(35)[Ξ3i**********Π5iTΞ4i*********XiDiCiXiDiCdi-Xi********E2iTDiTXiDiCiE2iTDiTXiDiCdi0Π3i*******0000-Xi******0XiAdi000-Xi*****2BiTXi00000-BiTXiBi****N1iTXi000000-ε1iI***N1iTXi0000000-ε2iI**0000N1iTXi0000-ε4iI*00000N1iTXi0000-ε4iI]<0,
with(36)Ξ3i=Xi(Ai+BiKi)+(Ai+BiKi)TXi+Q+ε1iHaiTHaiΞ3i=+ε3iHciTHci+ε4iHaiTHai+∑j=1NπijXj,Ξ4i=-(1-h)Q+ε2iHadiTHadi+ε3iHcdiTHcdi+ε4iHadiTHadi.
Moreover, by Schur’s complement lemma, the above inequality is implied by (22).
Then, we will prove that the stochastic system (21) with (6) satisfies
(37)∥z(t)∥E2<γ∥v(t)∥2,
for all nonzerov(t)∈L2[0,∞).
Choose the same Lyapunov function as (25). And the generator LV(x(t),i) with v(t)≠0 can be calculated as follows:
(38)LV(x(t),i)=2xT(t)Xi[(BiTXiBi)-1Ai+BiKi+A~i(t)-Bi(BiTXiBi)-1BiTXiA~i(t)]x(t)+2xT(t)Xi[(BiTXiBi)-1Adi+A~di(t)-Bi(BiTXiBi)-1BiTXi(Adi+A~di(t))]×x(t-τ(t))+2xT(t)Xi×[I-Bi(BiTXiBi)-1BiTXi]Bviv(t)+[(Ci+C~i(t))x(t)+(Cdi+C~di(t))x(t-τ(t))]T×DiTXiDi×[(Ci+C~i(t))x(t)+(Cdi+C~di(t))x(t-τ(t))]+∑j=1NπijxT(t)Xjx(t)+xT(t)Qx(t)-(1-τ˙(t))xT(t-τ(t))Qx(t-τ(t)).
Let(39)J(t)≜E{∫t0t[zT(s)z(s)-γ2vT(s)v(s)]ds}.
So, under zero initial condition, we haveV(x(t0),i)=0 for x(t0)=0. From Dynkin’s formula, one has
(40)E{V(x(t),t)}=E{∫t0tLV(x(s),s)ds}.
Then, similar to the (26)–(29), we can obtain
(41)J(t)=E{∫t0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s)]ds}J(t)=-E{V(x(t),t)}J(t)≤E{∫0t[zT(s)z(s)-γ2vT(s)v(s)+LV(x(s),s)]ds}J(t)=E{∫0t[xT(s)xT(s-τ(s))vT(s)]×Ωi[xT(s)xT(s-τ(s))vT(s)]Tds∫0t[xT(s)xT(s-τ(s))vT(s)]},
where(42)Ωi=[Θ1iΘ3iXiBvi+(Ei+E~i)TFiΘ3iTΘ2i0BviTXi+FiT(Ei+E~i)0BviTXiBvi+FiTFi-γ2I],
with(43)Θ1i=Ξ1i+XiBi(BiTXiBi)-1BiTXiΘ1i=+(Ei+E~i)T(Ei+E~i)+CiTDiTXiDiCiΘ1i=+CiTDiTXiDiN2iΘ1i×(ε3iI-N2iTDiTXiDiN2i)-1N2iTDiTXiDiCi,Θ2i=Ξ2i+CdiTDiTXiDiCdi+CdiTDiTXiDiN2iΘ2i×(ε3iI-N2iTDiTXiDiN2i)-1N2iTDiTXiDiCdi,Θ3i=XiAdi+CiTDiTXiDiN2iΘ3i=×(ε3iI-N2iTDiTXiDiN2i)-1Θ3i×N2iTDiTXiDiCdi,
andΞ1i and Ξ2i are defined as in (30).
It is observed thatΩi<0 from (22) and Schur’s complement. Hence, by (39), inequality ∥z(t)∥E2<γ∥v(t)∥2 holds for all nonzero v(t)∈L2[0,∞). The proof is complete.Remark 7.
Notice that the condition in Theorem6 is not a convex set due to the matrix equality constraints in (23). A simple algorithm is given to solve the feasibility problem. The equality condition BiTXiDi=0 with Xi satisfying inequality (22) can be equivalently converted to tr[(BiTXiDi)TBiTXiDi]=0.
Consider the following matrix inequality forα>0:
(44)(BiTXiDi)TBiTXiDi≤αI,fori∈S.
By Schur’s complement, one has(45)[-αIBiTXiDiDiTXiBi-I]≤0,fori∈S.
Now, the problem is changed to a problem of finding a global solution of the following minimization problem:(46)minα,subjectto(22)and(45).
It can be seen that if the global infimumα of (46) equals to zero, the corresponding solutions will satisfy the LMIs (22) and (23). So, the sliding control problem is solvable.


## 3.3. SMC Law Synthesis

Now, we will synthesize a SMC law to ensure that the trajectories of system (5)–(7) can be driven to reach and keep the predefined surface s(x(t),i)=0 from the initial time.Theorem 8.
For the uncertain stochastic delay system (5)–(7) with Markovian jumping and assumptions (4) and (8)–(10), the sliding surface is designed as (17) where Xi, (i∈S) are the solutions of LMI (22)-(23). Then the trajectories of system can be driven to the sliding surface s(x(t),i)=0 in finite time (with probability one) by employing the following SMC law:
(47)ui(t)=Kix(t)-μis(x(t),i)ui(t)=-(BiTXiBi)-1BiTXiAdix(t-τ(t))ui(t)=-12∑j=1Nπij(BjTXjBj)-1s(x(t),i)ui(t)=-ρi(t)sgn(s(x(t),i)),
where(48)ρi(t)=ηi∥x(t)∥+∥(BiTXiBi)-1BiTXiN1i∥ρi(t)×(∥Haix(t)∥+∥Hadix(t-τ(t))∥)ρi(t)=+∥(BiTXiBi)-1BiTXiBvi∥∥v(t)∥,
withμi are a small positive constant for each i∈S.Proof.
By (19) and (47), and by observing that BiTXiDi=0, it follows that
(49)s˙(x(t),i)=BiTXiA~i(t)x(t)+BiTXiA~di(t)x(t-τ(t))s˙(x(t),i)=-μiBiTXiBis(x(t),i)+BiTXiBviv(t)s˙(x(t),i)=-12∑j=1NπijBiTXiBi(BjTXjBj)-1s(x(t),i)s˙(x(t),i)=-ρi(t)BiTXiBisgn(s(x(t),i))s˙(x(t),i)=+BiTXiBif(t,x(t),i).
Choose the following Lyapunov function candidate:(50)V(t)=12sT(t)(BiTXiBi)-1s(t).
Utilizing expression (49) yields
(51)V˙(t)=sT(x(t),i)(BiTXiBi)-1s˙(x(t),i)V˙(t)=+12∑j=1NπijsT(x(t),i)(BjTXjBj)-1s(x(t),i)V˙(t)=sT(x(t),i)(BiTXiBi)-1BiTXiA~i(t)x(t)V˙(t)=+sT(x(t),i)(BiTXiBi)-1BiTXiA~di(t)x(t-τ(t))V˙(t)=-μisT(x(t),i)s(x(t),i)V˙(t)=-ρi(t)sT(x(t),i)sgn(s(x(t),i))V˙(t)=+sT(x(t),i)f(t,x(t),i)V˙(t)=+sT(x(t),i)(BiTXiBi)-1BiTXiBviv(t)V˙(t)≤∥s(x(t),i)∥∥(BiTXiBi)-1BiTXiN1i∥V˙(t)×(∥Haix(t)∥+∥Hadix(t-τ(t))∥)V˙(t)=-ρi(t)∥s(x(t),i)∥1+∥s(x(t),i)∥×∥(BiTXiBi)-1BiTXiBvi∥∥v(t)∥-μi∥s(x(t),i)∥2V˙(t)=+ηi∥s(x(t),i)∥∥x(t)∥.
Noting that∥s(x(t),i)∥1≥∥s(x(t),i)∥ and substituting (48) into (51), one has
(52)V˙(t)≤-μi∥s(x(t),i)∥2<0,for∥s(x(t),i)∥≠0.
This means that the trajectories of system (5)–(7) can be driven to remain in the sliding surface from the initial time.Remark 9.
The use of sign functionsgn(s(x(t),i)) may cause chattering behavior, which is undesired in practical engineering background [19]. To avoid the shortcomings, the function tanh(s(x(t),i)) is introduced to approximate function sgn(s(x(t),i)) [20].From SMC (47)-(48), it is seen that the bound ηi of f(t,x(t),i) is needed to synthesize the SMC law. It is well known that a practical engineering system may lose the exact knowledge of the bound due to the noise pollution and measurement error. Then, an adaptive SMC law is further presented for the case when the bound ηi is unknown.Theorem 10.
For the uncertain stochastic delay systems (5)–(7) with Markovian jumping and assumptions (4) and (8)–(10), the sliding surface is designed as (17) where Xi, (i∈S) are the solutions of LMI (22)-(23). Then the trajectories of system can be driven to the sliding surface s(x(t),i)=0 in finite time (with probability one) by employing the following SMC law:
(53)ui(t)=Kix(t)-μis(x(t),i)ui(t)=-(BiTXiBi)-1BiTXiAdix(t-τ(t))ui(t)=-12∑j=1Nπij(BjTXjBj)-1s(x(t),i)ui(t)=-ρ^i(t)sgn(s(x(t),i)),
where(54)ρ^i(t)=η^i∥x(t)∥+∥(BiTXiBi)-1BiTXiN1i∥ρ^i(t)×(∥Haix(t)∥+∥Hadix(t-τ(t))∥)ρ^i(t)×+∥(BiTXiBi)-1BiTXiBvi∥∥v(t)∥,
and adaptive law as(55)η^˙i=βi∥s(t)∥∥x(t)∥,
withμi, βi as small positive constants for each i∈S.Proof.
Choose the following Lyapunov function:(56)V(t)=12sT(t)(BiTXiBi)-1s(t)+12βi-1η~i2(t),
whereη~i(t)=η^i(t)-ηi. In line with the proof of Theorem 8, it can easily obtained that
(57)V˙(t)≤∥s(x(t),i)∥∥(BiTXiBi)-1BiTXiN1i∥×(∥Haix(t)∥+∥Hadix(t-τ(t))∥)-ρ^i(t)∥s(x(t),i)∥1V˙(t)≤+∥s(x(t),i)∥∥(BiTXiBi)-1BiTXiBvi∥∥v(t)∥V˙(t)≤-μi∥s(x(t),i)∥2+ηi∥s(x(t),i)∥∥x(t)∥V˙(t)≤+η~i(t)∥s(x(t),i)∥∥x(t)∥.
Noting that∥s(x(t),i)∥1≥∥s(x(t),i)∥ and substituting (54) into (57), one has
(58)V˙(t)≤-μi∥s(x(t),i)∥2<0,for∥s(x(t),i)∥≠0.
This means that the trajectories of system (5)–(7) can be driven to remain in the sliding surface from the initial time. This completes the proof.


## 4. Simulation Example

Consider the uncertain stochastic delay system (3) with N=2 and the following parameters.Mode 1. We have the following:
(59)A1=[-10.6-2.420.1-0.50.12.20.5],Ad1=[0.20.10.1-0.10.1000.1-0.5],B1=[1-243.52.45.5],Bv1=[0.20.10.3],D1=[0.20.3],C1=[-0.20.1010.10.04]T,E1=[0.40.400.50.10.20.10.20.3],F1=[0.20.30.1],N11=[00.10.1],N21=[0.10.1],Ha1=[0.10.20],Had1=[00.10.2],He1=[0.010.10.1],Hc1=[0.10.20.11-0.10.3-0.20.30],Hcd1=[-0.200.20.10.10.10.2-0.10],Cd1=[00.30.20.20.010.1]T,f(x(t),t,1)=[0.6sinx12(t)+x22(t)0.6x2(t)]T,v(t)=31+t2,F11(t)=0.5sint,F21(t)=[0.2sint0.2cost0].Mode 2. We have the following:
(60)A2=[10.8100.5-0.60.30.4-0.5],Ad2=[-0.20.20.300.50.10.10.40.5],B2=[1.23400.42],Bv2=[0.10.20.2],D2=[0.10.4],C2=[0.2-0.101-0.10.02]T,E2=[0.30.50.200.80.30.200.3],F2=[0.30.40.2],N21=[00.2-0.1],N22=[0.20.1],Ha2=[0.20.40],Had2=[0.100.1],He2=[0.10.20],Hc2=[0.20.10.20.1-0.10.2-0.10.20],Hcd2=[-0.100.10.20.10.20.1-0.10],Cd2=[0.10.2-0.20.10.01-0.1]T,f(x(t),t,2)=[sinx1(t)2|x2(t)x3(t)|]T,F12(t)=0.5cost,F22(t)=[0.2cost0.2sint0].Moreover, the time-varying delayτ(t)=0.5sint+0.5 with d=1, h=0.5, and the transition rate matrix π of Markov chain {rt} is defined as π=[-111.5-1.5]. Choose γ=1. Now, for the bounds η1=0.6 and η2=1, solving LMIs (22)-(23) yields
(61)X1=[1.7215-1.00691.1293-1.00690.9575-0.80571.1293-0.80571.1127],X2=[1.9635-0.6287-1.30140.62870.7440-0.0362-1.3014-0.03624.0151],ε11=1.0867,ε12=1.2007,ε21=4.2260,ε22=2.2260,ε31=1.2260,ε32=1.5600,ε41=3.2260,ε42=2.1200,β≈1.4797×10-11,Q=[6.7459-1.58853.5946-1.58853.3264-0.10673.5946-0.10674.4201].Furthermore, controller gainK1 and K2 are chosen as
(62)K1=[-1.5-4.5-433.5-1.2],K2=[1-32-3.50.2-0.8].Then, the desired SMC law can be obtained as follows.Mode 1. We have the following:
(63)u(t)=[-1.5-4.5-433.5-1.2]x(t)-[0.36420.11270.11270.4264]u(t)=×s(x(t),1)u(t)=-[0.04790.04960.0252-0.0483-0.0143-0.0699]x(t-τ(t))u(t)=-[0.6∥x(t)∥+0.02(∥[0.10.20]x(t)∥u(t)=+∥[00.10.2]x(t-τ(t))∥)u(t)=+0.08670.6∥x(t)∥+0.02(∥[0.10.20]x(t)∥]u(t)=×sgn(s(x(t),1)).Mode 2. We have the following:
(64)u(t)=[1-32-3.50.2-0.8]x(t)u(t)=-[0.7037-0.1691-0.16910.6103]s(x(t),2)u(t)=-[0.01240.14190.0416-0.01200.09090.1627]x(t-τ(t))u(t)=-[∥x(t)∥+0.0599(∥[0.20.40]x(t)∥u(t)=-d∥x(t)∥+0.0599+∥[0.100.1]x(t-τ(t))∥)+0.0763]sgn(s(x(t),2)),whereμ1=μ2=0.5.Using the discretization approach similar to that in [21], the simulation results are given in Figures 1, 2, 3, 4, 5, and 6 which show the effective of the proposed methods. Figures 1–6 demonstrated the simulation results for every mode, respectively.Figure 1
Mode 1: state vectorx(t).Figure 2
Mode 1: sliding modes(t).Figure 3
Mode 1: control signalu(t).Figure 4
Mode 2: state vectorx(t).Figure 5
Mode 2: sliding modes(t).Figure 6
Mode 2: control signalu(t).


## 5. Conclusions

This work has investigated the problem of SMC problem for stochastic systems with Markovian switching and time-varying delays. A sufficient condition for the stochastic stability of sliding motion has been proposed in terms of LMIs. An SMC law has been synthesized such that the state trajectories of the closed-lop systems are globally driven onto the specified switching surface corresponding to every mode and reduces the effect of the disturbance input on the controlled output to a prescribed level irrespective of all the admissible uncertainties. It is observed that the effects of Markovian switching and time delay have been considered in the design of both sliding surface and SMC law. Future works will consider the problem of SMC problem for singular systems with Markovian switching and time-varying delays. In that case, due to the existence of singular matrix, the sliding mode control law design is much more complicated.



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*Source: 898172-2013-05-16.xml*