# Sliding Mode Control for Markovian Switching Singular Systems with Time-Varying Delays and Nonlinear Perturbations

Authors: Guowei Yang; Yonggui Kao; Wei Li
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This paper is devoted to investigating sliding mode control (SMC) for Markovian switching singular
systems with time-varying delays and nonlinear perturbations. The sliding mode controller
is designed to guarantee that the nonlinear singular system is stochastically admissible and its trajectory
can reach the sliding surface in finite time. By using Lyapunov functional method, some criteria on
stochastically admissible are established in the form of linear matrix inequalities (LMIs). A numerical
example is presented to illustrate the effectiveness and efficiency of the obtained results.


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## Body

## 1. Introduction

The sliding mode control (SMC) theory has made rapid progress since it was proposed by Utkin [1]. As an effective robust control strategy, SMC has been successfully applied to a wide variety of practical engineering systems such as robot manipulators, aircrafts, underwater vehicles, spacecrafts, flexible space structures, electrical motors, power systems, and automotive engines [2]. The SMC system has various attractive features such as fast response, good transient performance, and insensitiveness to the uncertainties on the sliding surface [2]. These advantages provide more freedom in designing the controllers for the system models which can be easily modified by introducing virtual disturbances to satisfy some requirements. The SMC strategy has been successfully applied to many kinds of systems, such as uncertain time-delay systems and Markovian jump system [3–14].Singular systems, also referred to as descriptor systems, generalized state-space systems, differential-algebraic systems, or semistate systems, are more appropriate to describe the behavior of some practical systems, such as economic systems, power systems, and circuits systems, because singular systems mix up dynamic equations and static equations. Basic control theory for singular systems has been widely studied, such as stability and stabilization [14–18], H∞ control problem [19–22], and optimal control [23] and filtering problem [24–26]. Xu et al. have designed an integral sliding mode controller for singular stochastic hybrid systems [27]. They put up with new sufficient conditions in terms of strict LMIs, which guarantees stochastic stability of the sliding mode dynamics.In practice, many physical systems may happen to have abrupt variations in their structure, due to random failures or repair of components, sudden environmental disturbances, changing subsystem interconnections, and abrupt variations in the operating points of a nonlinear plant. Therefore, Markovian jump systems have received increasing attention, see [28–33] and the references therein. Wu et al. [28] have probed sliding mode control with bounded H2 gain performance of Markovian jump singular time-delay systems. Kao et al. [29] have investigated delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen-Grossberg neural networks with mixed delays. Zhang and Boukas have discussed mode-dependent H∞ filtering for discrete-time Markovian jump linear systems with partly unknown transition probability. To the authors' best knowledge, sliding mode control for a class of Markovian switching singular systems with time-varying delays and nonlinear perturbations has not been properly investigated. Especially few consider exponential stabilization for this kind of nonlinear singular systems by sliding mode control.Motivated by the above discussion, we consider exponential stabilization for Markovian switching nonlinear singular systems via sliding mode control. First, we develop two lemmas. Based on these lemmas, delay-dependent sufficient condition on exponential stabilization for singular time-varying delay systems is given in terms of nonstrict LMIs. Some specified matrices are introduced and the non-strict LMIs are translated into strict LMIs which are easy to check by MATLAB LMI toolbox. Second, a sliding surface is derived using an equivalent control approach. A sliding mode controller is developed to drive the systems to the sliding surface in finite time and maintain a sliding motion thereafter. Finally, a numerical example is provided to show the effectiveness of the proposed result.Notations. (Ξ,ℱ,{ℱt}t≥0,𝒫) is a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions. Lℱ0P is the family of all ℱ0-measurable C([-τ,0];Rn) valued random variables ξ=ξ(θ):-τ≤θ≤0 such that sup-τ≤θ≤0ℰ∥ξ(θ)∥22<∞, where ℰ{·} stands for the mathematical expectation operator with respect to the given probability measure 𝒫. Rn and Rn×m denote, respectively, the n-dimensional Euclidean space and the set of n×m real matrices. The superscript T denotes the transpose, and the notation X,Y (resp., X>Y) where X and Y are symmetric matrices means that X-Y is positive semi-definite (resp., positive definite). L2 stands for the space of square integral vector functions. ∥·∥ will refer to the Euclidean vector norm, and * represents the symmetric form of matrix.


## 2. System Description and Definitions

Consider the following singular system with time-varying delays, nonlinear perturbations, and Markovian switching:(1)E(r(t))x˙(t)=A(r(t))x(t)+Ad(r(t))x(t-τ(t))22+B(r(t))u(t)+f(x(t),t,r(t)),x(t)=ϕ(t),t∈[-τ,0],
where x(t)∈Rn is the state vector; u(t)∈Rm is the control input; f(x(t),t,r(t))∈Rn represents the system nonlinearity and any model uncertainties in the systems including external disturbances with Markovian switching;ϕ(t)∈Lℱ0P([-τ,0];Rn) is a compatible vector valued continuous function. A(r(t)),Ad(r(t)), and B(r(t)) are real constant matrices with appropriate dimensions. The matrix E(r(t))∈Rn×n may be singular, and we assume that 0<rank(E(r(t)))=r≤n. τ(t) denotes the time-varying delay and satisfies
(2)0≤τ(t)≤τ,τ˙(t)≤d<1.Let{r(t),t≥0} be a continuous-time discrete-state Markovian process with right continuous trajectories taking value in a finite set 𝕊={1,2,…,N} with transition probability matrix ∏‍=(rij),(i,j∈𝕊),
(3)P{r(t+Δ)=j∣r(t)=i}={rij×Δ+o(Δ),i≠j1+rii×Δ+o(Δ),i=j,
where Δ>0 and limΔ→0(o(Δ))/Δ=0, rij>0 is the transition rate from i to j if i≠j and rii=-∑j=1,i≠jN‍rij.The nominal Markovian jump singular and time-delay system of system (1) are as follows:
(4)E(r(t))x˙(t)=A(r(t))x(t),(5)E(r(t))x˙(t)=A(r(t))x(t)+Ad(r(t))x(t-τ(t)).The initial Markovian jump singular system in (1) is assumed to be
(6)x(0,γ0)=ϕ(0).Recall that the Markovian process{r(t),t≥0} takes value in a finite set 𝕊={1,2,…,N}. For simplicity, we write r(t)=i∈𝕊, E(i)=Ei, A(i)=Ai, Ad(i)=AdiB(r(t))=Bi, f(x(t),t,r(t))=fi(x(t),t).Definition 1.
(i) The system (4) is said to be regular if det(sEi-Ai)≠0 for every i∈𝕊.
(ii) The system (4) is said to impulse free if deg(det(sEi-Ai))=rank(Ei) for every i∈𝕊.Definition 2.
For a given scalarτ>0, the Markovian jump singular delay system (5) is said to be regular and impulse free for any time delay τ(t) satisfying 0≤τ(t)≤τ, if the system (4) and the system E(r(t))X˙(t)=(A(r(t))+Ad(r(t)))X(t) are all regular and impulse free.
The system (5) is said to be stochastically stable if for any x0∈Rn and r0∈𝕊 there exists a scalar M~(x0,r0)>0 such that limt→∞ℰ(∫0t‍xT(s,x0,r0)x(s,x0,r0)ds∣x0,r0)≤M~(x0,r0), where x(s,x0,r0) denotes the solution of system (5) at time t under the initial condition x0 and r0.
The system (5) is said to be stochastically admissible if it is regular, impulse free, and stochastically stable.We will assume the followings to be valid.Assumption 1.
B
(
r
(
t
)
) is full-rank:rank (B(r(t)))=m.Assumption 2.
The perturbation termf(x(t),t,r(t)) is Lipshitz, continuous and satisfies the following matching conditions:
(7)f(x(t),t,r(t))=B(r(t))f-(x(t),t,r(t)),
withf-(x(t),t,r(t))∈Rm bounded by
(8)∥f-(x(t),t,r(t))∥≤ϵi∥x(t)∥,
where ϵ>0 is a constant.Lemma3 will support the non-strict LMI to be translated into strict LMI.Lemma 3 (see [34]).
LetX∈Rn×n be symmetric such that ELTXEL>0 and T∈R(n-r)×(n-r) nonsingular. Then, XE+MTTST is nonsingular and its inverse is expressed as
(9)(XE+MTTST)-1=𝕏ET+S𝕋M,
where 𝕏 is symmetric and 𝕋 is a singular matrix with
(10)ERT𝕏ER=(ELTXEL)-1,𝕋=(STS)-1T-1(MMT)-1,
where M and S are any matrices with full row rank and satisfy ME=0 and ES=0, respectively; E is decomposed as E=ELERT with EL∈Rn×r and ER∈Rn×r are of full column rank.Lemma 4 (see [35]).
There exists symmetric matrix X such that(11)[P1+XQ1Q1TR1]<0,[P2-XQ2Q2TR2]<0
if and only if
(12)[P1+P2Q1Q2*R10**R2]<0.Lemma 5.
LetQ=QT, S, R=RT be matrices of appropriate dimensions, then R<0,Q-SR-1ST<0 is equivalent to
(13)[QSSTR]<0.Lemma 6.
LetX∈Rn, Y∈Rn, and Q>0. Then we have XTY+YTX≤YTQY+XTQ-1X.We give the following result for the stochastic admissibility of the system (4) without proof, and readers are referred to [36] for detailed proof.Lemma 7.
The Markovian jump singular system (4) is stochastically admissible if and only if there exists matrices Pi,i=1,2,…,N such that
(14)EiTPi=PiTEi≥0∑j=1N‍rijEiTPj+PiTAi+AiTPi<0.Lemma 8.
For a prescribed scalarsτ>0, d, and any time delay τ(t) satisfying 0≤τ(t)≤τ, the Markovian jump singular time-delay system (5) is stochastically admissible, if there exist symmetric positive-definite matrices Q,R and nonsingular matrix Pi for every i=1,2,…,N, such that
(15)EiTPi=PiTEi≥0,(16)Φi=[Φi11PiTAdi0*-(1-d)Q0**-1τR]<0,
where
(17)Φi11=PiTAi+AiTPi+Q+τR+∑j=1N‍rijEjTPj.Proof.
First to prove the systemE(r(t))x˙(t)=[A(r(t))+Ad(r(t)]x(t) is regular and impulse free.
From (16), it is easy to know Φi11<0 and
(18)[Φi11PiTAdi*-(1-d)Q]<0.
By pre- and postmultiplying (18) by [I,I] and [I,I]T, we get
(19)PiT(Ai+Adi)+(Ai+Adi)TPi+dQ+τR+∑j=1N‍rijEjTPj<0,
where Q and R are symmetric positive-definite matrices, we have
(20)PiT(Ai+Adi)+(Ai+Adi)TPi+∑j=1N‍rijEjTPj<0,(21)∑j=1N‍rijEiTPj+PiTAi+AiTPi<0.
Based on Lemma7, (15) and (20) show that the system E(r(t))x˙(t)=[A(r(t))+Ad(r(t))]x(t) is regular and impulse free, and (15) and (21) ensure that the system E(r(t))X˙(t)=A(r(t))x(t) is regular and impulse free. Hence, according to Definition 2, the system (5) is regular and impulse free for any delay τ(t) satisfying 0≤τ(t)≤τ.
Second, to prove the system (5) is stochastically stable. Take a functional candidate for the system as follows:
(22)V(x(t),r(t))=XT(t)ET(r(t))P(r(t))X(t)22+∫t-τ(t)t‍XT(s)QX(s)ds22+∫-τ0‍∫t+θt‍XT(s)RX(s)dsdθ.
Then, letℒ be the weak infinitesimal generator of the random process x(t),r(t), and for each i∈𝕊, we have
(23)ℒV(x(t),r(t)=i)=limΔ→01Δ{ℰ[V(x(t+Δ),r(t+Δ))∣x(t),222222222222222r(t)=i]-V(x(t),r(t)=i)}≤2xT(t)PiTEix˙(t)+xT(t)(∑j=1N‍rijEjTPj)x(t)+xT(t)Qx(t)-(1-τ˙(t))xT(t-τ(t))Qx(t-τ(t))+τ(xT(t)Rx(t))-∫t-τt‍xT(s)Rx(s)ds≤2xT(t)PiTAix(t)+2xT(t)PiTAdix(t-τ(t))+xT(t)(∑j=1N‍rijEjTPj)x(t)+xT(t)Qx(t)-(1-d)xT(t-τ(t))Qx(t-τ(t))+τxT(t)Rx(t)-[∫t-τt‍xT(s)ds]×Rτ[∫t-τt‍x(s)ds]=ξ(t)ΦiξT(t),
where ξ(t)=[xT(t),xT(t-τ(t)),∫t-τt‍xT(s)ds].
With (16), it is easy to know that there exists a scalar μ>0, such that
(24)ℒV(x(t),r(t)=i)≤-μxT(t)x(t).
By Dynkin's formula, we get(25)ℰ{V((x(t),r(t))x(0),r(0))-V(x(0),r(0))}=ℰ{∫0t‍LV(x(s),r(s))ds∣x(0),r(0)}≤-μℰ{∫0t‍xT(s)x(s)ds∣x(0),r(0)},
and this means
(26)ℰ(∫0t‍xT(s)x(s)ds∣x(0),r(0))≤1μV(x(0),r(0)).
Therefore by Definition2, the Markovian jump singular system (5) is stochastically stable. This completes the proof.


## 3. Sliding Motion Stability Analysis

### 3.1. Sliding Surface Design

SMC design involves two basic steps: sliding surface design and controller design. For everyi∈𝕊, integral sliding surface with delay and Markovian switching is considered as follows:
(27)S(t)=GiEix(t)-∫0t‍Gi(Ai+BiKi)x(θ)dθ22-∫0t‍GiAdix(θ-τ(θ))dθ,
where Ki∈Rm×n is real matrix to be designed and Gi∈Rm×n is designed to satisfy that GiBi is nonsingular. According to SMC theory, when the system trajectories reach onto the sliding surface, it follows that S(t)=0 and S˙(t)=0.Therefore, byS˙(t)=0, we get the equivalent control as
(28)ueqi(t)=Kix(t)-f-(x(t),t,i).Substitutingueq(r(t)=K(r(t))x(t)-f-(x(t),t,r(t)) into (1), we obtain the following sliding mode dynamics:
(29)E(r(t))x˙(t)=[A(r(t))+B(r(t))K(r(t))]x(t)22+Ad(r(t))x(t-τ(t)).


### 3.2. Sliding Mode Dynamics Analysis

In this section, we pay attention to establishingLMIs conditions to check the stochastical admissibility of the system (29). Based on Lemmas 4, it is easy to get the sufficient condition provided in the following theorem.Theorem 9.
Given scalarsτ and d, for any delays τ(t) satisfying (2), the system (29) is stochastically admissible if there exist nonsingular matrices Pi and symmetric positive-definite matrix Q and R such that
(30)EiTPi=PiTEi≥0,(31)[Φi11PiTAdi0*-(1-d)Q0**-1τR]<0,
where
(32)Φi11=PiT(Ai+BiKi)+(Ai+BiKi)TPi+Q+τR+∑j=1N‍rijEjTPj.Remark 10.
Note that the conditions in Theorem9 are not strict LMI conditions due to matrix equality constraint of (30). According to Lemma 3 and Theorem 9, the strict LMI conditions are given as follows.Theorem 11.
Given scalarsτ and d, for any delays τ(t) satisfying (2), the system (29) is stochastically admissible if there exist symmetric positive-definite matrices 𝕏,Q,Yi,H, R, and symmetric matrix 𝕋∈R(n-r)×(n-r), matrix 𝕃i∈Rm×n, and any matrices with full row rank Mi,Si satisfying MiEi=0,EiSi=0, respectively, such that
(33)[Γi11AdiN1*-(1-d)Q0**N2]<0,(34)[Q+τR-H0N3*-1τR0**N4]<0,(35)[-Yi0Ei𝕏+MiT𝕋SiT0*-H0H**-I0***-I]<0,
where
(36)Γi11=Ai(𝕏EiT+Si𝕋Mi)+(𝕏EiT+Si𝕋Mi)TAiT+Bi𝕃iEiT+(Bi𝕃iEiT)T+rii(𝕏EiT+Si𝕋Mi)TEiT+Yi,N1=[Bi𝕃i,MiT𝕋SiT],N2=diag[-𝕏,-𝕏],N3=[E1R,…,Ei-1,R,Ei+1,R,…,EnR],N4=diag[-1ri1E1RT𝕏E1R,…,-1ri,i-1Ei-1,RT𝕏Ei-1,R,22222-1ri,i+1Ei+1RT𝕏Ei+1R,…,-1rinEnRT𝕏EnR],𝕃i=Ki𝕏.Proof.
LetPi≜XEi+MiTTSiT in Theorem 9. We can get
(37)[Φi11(XEi+MiTTSiT)TAdi0*-(1-d)Q0**-1τR]<0,
where
(38)Φi11=(XEi+MiTTSiT)T(Ai+BiKi)+(Ai+BiKi)T(XEi+MiTTSiT)+Q+τR+∑j=1N‍rijEjTXEj.
Using Lemma3, we get
(39)(XEi+MiTTSiT)-1=𝕏EiT+Si𝕋Mi≜Li.
By pre- and postmultiplying (37) by diag[LiT,I,I] and diag[Li,I,I], we get
(40)[Ψi′11Adi0*-(1-d)Q0**-1τR]<0,
where Ψi′11=(Ai+BiKi)Li+LiT(Ai+BiKi)T+LiTQLi+τLiTRLi+∑j=1N‍rijLiTEjTXEj.
In light of Lemma4, there exists symmetric matrix Yi such that
(41)[(Ai+BiKi)Li+LiT(Ai+BiKi)T+riiLiTEiT+YiAdi*-(1-d)Q]<0,(42)[LiTQLi+τLiTRLi+∑i≠j=1N‍rijLiTEjTXEjLi-Yi0*-1τR]<0.
It is easy to know that(43)BiKiSi𝕋Mi+(BiKiSi𝕋Mi)T=BiKi𝕏𝕏-1Si𝕋Mi22+(BiKi𝕏𝕏-1Si𝕋Mi)T≤(BiKi𝕏)𝕏-1(𝕏KiTBiT)22+(MiT𝕋SiT)𝕏-1(Si𝕋Mi).
Using Shur's complement, the following inequality can ensure (41) as
(44)[AiLi+LiTAiT+BiKi𝕏EiT+(BiKi𝕏EiT)T+riiLiTEiT+YiAdiN1*-(1-d)Q0**N2]>0,
where N1=[BiKi𝕏,MiT𝕋SiT] and N2=diag[-𝕏,-𝕏]. Equation (42) is equivalent to
(45)[Q+τR+∑i≠j=1N‍rijEjTXEj-PiTYiPi0*-1τR]<0.
There exists a matrixHT=H>0 such that
(46)[Q+τR+∑i≠j=1N‍rijEjTXEj-PiTYiPi0*-1τR]<[Q+τR+∑i≠j=1N‍rijEjTXEj-H0*-1τR]<0
if and only if
(47)-PiTYiPi<-H<0.
The above inequality is equivalent to the following inequality:(48)[-YiLiTH*-H]<0.
And the following inequality can guarantee that (48) is true:
(49)[-Yi0LiT0*-H0H**-I0***-I]<0.
By Shur's complement, the right of (46) is equivalent to
(50)[Q+τR-H0N3*-1τR0**N4]<0,
where N3=[E1R,…,Ei-1,R,Ei+1,R,…,EnR] and N4=diag[-(1/ri1)E1RT𝕏E1R,…,-(1/ri,i-1)Ei-1,RT𝕏Ei-1,R, -(1/ri,i+1)Ei+1RT𝕏Ei+1R,…,-(1/rin)EnRT𝕏EnR].Ei is decomposed as Ei=EiLEiRT with EiL∈Rn×r and EiR∈Rn×r are of full column rank. This completes the proof.Remark 12.
From the proof of the Theorem11, it is not difficult to know that Theorem 11 is more easy to compute than Theorem 9.


### 3.3. Sliding Mode Control Design

After switching surface design, the next important part of sliding mode control is to design a slide mode controller to guarantee the existence of a sliding mode. Now, we design an SMC law, by which the trajectories of singular system (1) can be driven onto the designed sliding surface S(t)=0 in a finite time.Theorem 13.
With the constant matrixKi mentioned in Theorem 11 and the integral sliding surface given by (29), the trajectory of the closed-loop system (1) can be driven onto the sliding surface in finite time with the control (51) as
(51)ui(t)=Kix(t)-(ρi+ϵi∥x(t)∥)sign(BiTGiTS(t)),
where ρi is a positive constant.Proof.
ChooseGi under the condition of GiBi is nonsingular. Consider the following Lyapunov function:
(52)V(S(t),t)=12ST(t)S(t).
Due to (29), we have
(53)dS=GiBi[-Kix(t)+ui(t)+f-i]dt.
DifferentiatingVi(t) along the closed-loop trajectories and using (53), we have
(54)ℒV(S(t),(t))=SiT(t)dSi(t)=SiT(t)GiBi[-Kix(t)+ui(t)+f-i]=SiT(t)GiBi[(BiTGiTS(t))-(ρ+ϵ∥x(t)∥)signSiT22GiBi222×(BiTGiTS(t))+f-i]≤-(ρ+ϵ∥x(t)∥)∥BiTGiTSi(t)∥22+∥f-i∥∥BiTGiTSi(t)∥≤-ρ∥BiTGiTSi(t)∥≤-ρ-∥Si(t)∥<0,ifSi(t)≠0,
where ρ-i>ρiλmin(GiBiBiTGiT). Then the state trajectory converges to the surface and is restricted to the surface for all subsequent time. This completes the proof.


## 3.1. Sliding Surface Design

SMC design involves two basic steps: sliding surface design and controller design. For everyi∈𝕊, integral sliding surface with delay and Markovian switching is considered as follows:
(27)S(t)=GiEix(t)-∫0t‍Gi(Ai+BiKi)x(θ)dθ22-∫0t‍GiAdix(θ-τ(θ))dθ,
where Ki∈Rm×n is real matrix to be designed and Gi∈Rm×n is designed to satisfy that GiBi is nonsingular. According to SMC theory, when the system trajectories reach onto the sliding surface, it follows that S(t)=0 and S˙(t)=0.Therefore, byS˙(t)=0, we get the equivalent control as
(28)ueqi(t)=Kix(t)-f-(x(t),t,i).Substitutingueq(r(t)=K(r(t))x(t)-f-(x(t),t,r(t)) into (1), we obtain the following sliding mode dynamics:
(29)E(r(t))x˙(t)=[A(r(t))+B(r(t))K(r(t))]x(t)22+Ad(r(t))x(t-τ(t)).


## 3.2. Sliding Mode Dynamics Analysis

In this section, we pay attention to establishingLMIs conditions to check the stochastical admissibility of the system (29). Based on Lemmas 4, it is easy to get the sufficient condition provided in the following theorem.Theorem 9.
Given scalarsτ and d, for any delays τ(t) satisfying (2), the system (29) is stochastically admissible if there exist nonsingular matrices Pi and symmetric positive-definite matrix Q and R such that
(30)EiTPi=PiTEi≥0,(31)[Φi11PiTAdi0*-(1-d)Q0**-1τR]<0,
where
(32)Φi11=PiT(Ai+BiKi)+(Ai+BiKi)TPi+Q+τR+∑j=1N‍rijEjTPj.Remark 10.
Note that the conditions in Theorem9 are not strict LMI conditions due to matrix equality constraint of (30). According to Lemma 3 and Theorem 9, the strict LMI conditions are given as follows.Theorem 11.
Given scalarsτ and d, for any delays τ(t) satisfying (2), the system (29) is stochastically admissible if there exist symmetric positive-definite matrices 𝕏,Q,Yi,H, R, and symmetric matrix 𝕋∈R(n-r)×(n-r), matrix 𝕃i∈Rm×n, and any matrices with full row rank Mi,Si satisfying MiEi=0,EiSi=0, respectively, such that
(33)[Γi11AdiN1*-(1-d)Q0**N2]<0,(34)[Q+τR-H0N3*-1τR0**N4]<0,(35)[-Yi0Ei𝕏+MiT𝕋SiT0*-H0H**-I0***-I]<0,
where
(36)Γi11=Ai(𝕏EiT+Si𝕋Mi)+(𝕏EiT+Si𝕋Mi)TAiT+Bi𝕃iEiT+(Bi𝕃iEiT)T+rii(𝕏EiT+Si𝕋Mi)TEiT+Yi,N1=[Bi𝕃i,MiT𝕋SiT],N2=diag[-𝕏,-𝕏],N3=[E1R,…,Ei-1,R,Ei+1,R,…,EnR],N4=diag[-1ri1E1RT𝕏E1R,…,-1ri,i-1Ei-1,RT𝕏Ei-1,R,22222-1ri,i+1Ei+1RT𝕏Ei+1R,…,-1rinEnRT𝕏EnR],𝕃i=Ki𝕏.Proof.
LetPi≜XEi+MiTTSiT in Theorem 9. We can get
(37)[Φi11(XEi+MiTTSiT)TAdi0*-(1-d)Q0**-1τR]<0,
where
(38)Φi11=(XEi+MiTTSiT)T(Ai+BiKi)+(Ai+BiKi)T(XEi+MiTTSiT)+Q+τR+∑j=1N‍rijEjTXEj.
Using Lemma3, we get
(39)(XEi+MiTTSiT)-1=𝕏EiT+Si𝕋Mi≜Li.
By pre- and postmultiplying (37) by diag[LiT,I,I] and diag[Li,I,I], we get
(40)[Ψi′11Adi0*-(1-d)Q0**-1τR]<0,
where Ψi′11=(Ai+BiKi)Li+LiT(Ai+BiKi)T+LiTQLi+τLiTRLi+∑j=1N‍rijLiTEjTXEj.
In light of Lemma4, there exists symmetric matrix Yi such that
(41)[(Ai+BiKi)Li+LiT(Ai+BiKi)T+riiLiTEiT+YiAdi*-(1-d)Q]<0,(42)[LiTQLi+τLiTRLi+∑i≠j=1N‍rijLiTEjTXEjLi-Yi0*-1τR]<0.
It is easy to know that(43)BiKiSi𝕋Mi+(BiKiSi𝕋Mi)T=BiKi𝕏𝕏-1Si𝕋Mi22+(BiKi𝕏𝕏-1Si𝕋Mi)T≤(BiKi𝕏)𝕏-1(𝕏KiTBiT)22+(MiT𝕋SiT)𝕏-1(Si𝕋Mi).
Using Shur's complement, the following inequality can ensure (41) as
(44)[AiLi+LiTAiT+BiKi𝕏EiT+(BiKi𝕏EiT)T+riiLiTEiT+YiAdiN1*-(1-d)Q0**N2]>0,
where N1=[BiKi𝕏,MiT𝕋SiT] and N2=diag[-𝕏,-𝕏]. Equation (42) is equivalent to
(45)[Q+τR+∑i≠j=1N‍rijEjTXEj-PiTYiPi0*-1τR]<0.
There exists a matrixHT=H>0 such that
(46)[Q+τR+∑i≠j=1N‍rijEjTXEj-PiTYiPi0*-1τR]<[Q+τR+∑i≠j=1N‍rijEjTXEj-H0*-1τR]<0
if and only if
(47)-PiTYiPi<-H<0.
The above inequality is equivalent to the following inequality:(48)[-YiLiTH*-H]<0.
And the following inequality can guarantee that (48) is true:
(49)[-Yi0LiT0*-H0H**-I0***-I]<0.
By Shur's complement, the right of (46) is equivalent to
(50)[Q+τR-H0N3*-1τR0**N4]<0,
where N3=[E1R,…,Ei-1,R,Ei+1,R,…,EnR] and N4=diag[-(1/ri1)E1RT𝕏E1R,…,-(1/ri,i-1)Ei-1,RT𝕏Ei-1,R, -(1/ri,i+1)Ei+1RT𝕏Ei+1R,…,-(1/rin)EnRT𝕏EnR].Ei is decomposed as Ei=EiLEiRT with EiL∈Rn×r and EiR∈Rn×r are of full column rank. This completes the proof.Remark 12.
From the proof of the Theorem11, it is not difficult to know that Theorem 11 is more easy to compute than Theorem 9.


## 3.3. Sliding Mode Control Design

After switching surface design, the next important part of sliding mode control is to design a slide mode controller to guarantee the existence of a sliding mode. Now, we design an SMC law, by which the trajectories of singular system (1) can be driven onto the designed sliding surface S(t)=0 in a finite time.Theorem 13.
With the constant matrixKi mentioned in Theorem 11 and the integral sliding surface given by (29), the trajectory of the closed-loop system (1) can be driven onto the sliding surface in finite time with the control (51) as
(51)ui(t)=Kix(t)-(ρi+ϵi∥x(t)∥)sign(BiTGiTS(t)),
where ρi is a positive constant.Proof.
ChooseGi under the condition of GiBi is nonsingular. Consider the following Lyapunov function:
(52)V(S(t),t)=12ST(t)S(t).
Due to (29), we have
(53)dS=GiBi[-Kix(t)+ui(t)+f-i]dt.
DifferentiatingVi(t) along the closed-loop trajectories and using (53), we have
(54)ℒV(S(t),(t))=SiT(t)dSi(t)=SiT(t)GiBi[-Kix(t)+ui(t)+f-i]=SiT(t)GiBi[(BiTGiTS(t))-(ρ+ϵ∥x(t)∥)signSiT22GiBi222×(BiTGiTS(t))+f-i]≤-(ρ+ϵ∥x(t)∥)∥BiTGiTSi(t)∥22+∥f-i∥∥BiTGiTSi(t)∥≤-ρ∥BiTGiTSi(t)∥≤-ρ-∥Si(t)∥<0,ifSi(t)≠0,
where ρ-i>ρiλmin(GiBiBiTGiT). Then the state trajectory converges to the surface and is restricted to the surface for all subsequent time. This completes the proof.


## 4. Numerical Example

In this section, a numerical example is presented to illustrate the effectiveness of the main results in this paper.Example 14.
Let us consider the system (1) with Markovian process that governs that the mode switching has generator ∏‍=(rij),(i,j=1,2), r12=0.25,r21=0.2. The system data are as follows:
(55)E1=[100010000],A1=[-6.71.600-500.52.23.2],Ad1=[-0.2000-0.1000-0.1],B1=[110],M1=[001],S1=[001],E1R=[100100],E2=[100000001],A2=[-5.7-4.20-3.74.81.302.4-6],Ad2=[-0.1000-0.10000],B2=[111],M2=[010],S2=[010]E2R=[100001].
In addition,τ=0.3,d=0.3,τ(t)=0.3e-t and f1(x)=0.65B1x(t)sinx(t),f2(x)=0.65B2x(t)sinx(t). Solve the LMI (33), (34), and (35) as follows:
(56)K1=[0.0963-0.0506-0.003]K2=[0.0953-0.00390.1793].
Fordet(GiBi)≠0, we can choose G1 and G2 as
(57)G1=[1.21.61.1];G2=[1.31.21.5].
Thus the sliding surface function is(58)s1(t)={s1(t)=[1.2,1.6,0]x(t)22-∫0t‍[-7.2204,-3.8017,-3.5116]x(θ)dθ22-∫0t‍[-0.24,-0.016,-0.11]x(θ-τ(θ))dθ,2222222222222222222222222222i=1s2(t)=[1.3,0,1.5]x(t)22-∫0t‍[-11.4688,3.8852,-6.7228]x(θ)dθ22-∫0t‍[-0.13,-0.12,0]x(θ-τ(θ))dθ,2222222222222222222222222222i=2.
Takeρ1=ρ2=0.64, then the SMC law designed in (51) can be described as
(59)u(t)={u1(t)=[0.0963,-0.0506,-0.003]x(t)-ρ(t)sign(2.8s1(t)),i=1u2(t)=[0.0953,-0.0037,0.1793]x(t)-ρ(t)sign(4s2(t)),i=2,
where ρ(t)=0.64+0.65∥x(t)∥. The simulation results are given in Figures 1, 2, 3, 4, and 5, which show the validity of the proposed method.Figure 1
State trajectories of the open-loop system.Figure 2
Switching surfaces(t).Figure 3
State trajectories of the closed-loop system.Figure 4
Controller inputu(t).Figure 5
State trajectories of sliding mode.Remark 15.
Obviously, our results include Markovian switching and nonlinear perturbation effects, and this model can not be dealt with by the results of [6, 8, 10–14, 16, 18, 20, 22, 26, 27, 34, 37], which show that our results are new.


## 5. Conclusion

In this paper, the stochastically admissible using sliding mode control for singular system with time-varying delay and nonlinear perturbations is studied by LMI method. The sliding mode control is designed to ensure that the closed-loop system is stochastically admissible. A numerical example demonstrates the effectiveness of the method mentioned above.



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*Source: 507828-2013-02-27.xml*