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

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.

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,{ℱ_𝑡}_𝑡≥0,𝒫) is a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions. 𝐿^𝑃_ℱ0 is the family of all ℱ₀-measurable 𝐶([−𝜏,0];𝑅^𝑛) valued random variables 𝜉=𝜉(𝜃)∶−𝜏≤𝜃≤0 such that sup_{−𝜏≤𝜃≤0}ℰ‖𝜉(𝜃)‖²₂<∞, where ℰ{⋅} stands for the mathematical expectation operator with respect to the given probability measure 𝒫. 𝑅^𝑛 and 𝑅^𝑛×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:𝐸(𝑟(𝑡))𝑥̇(𝑡)=𝐴(𝑟(𝑡))𝑥(𝑡)+𝐴_𝑑(𝑟(𝑡𝑥((𝑡(𝑟(𝑡𝑢(𝑡)(𝑥(𝑡)(𝑡,𝑥(𝑡)(𝑡)[],))𝑡−𝜏))+𝐵))+𝑓,𝑡,𝑟))=𝜙,𝑡∈−𝜏,0 (1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector; 𝑢(𝑡)∈𝑅^𝑚 is the control input; 𝑓(𝑥(𝑡),𝑡,𝑟(𝑡))∈𝑅^𝑛 represents the system nonlinearity and any model uncertainties in the systems including external disturbances with Markovian switching;𝜙(𝑡)∈𝐿^𝑃_ℱ0([−𝜏,0];𝑅^𝑛) is a compatible vector valued continuous function. 𝐴(𝑟(𝑡)),𝐴_𝑑(𝑟(𝑡)), and 𝐵(𝑟(𝑡)) are real constant matrices with appropriate dimensions. The matrix 𝐸(𝑟(𝑡))∈𝑅^𝑛×𝑛 may be singular, and we assume that 0<rank(𝐸(𝑟(𝑡)))=𝑟≤𝑛. 𝜏(𝑡) denotes the time-varying delay and satisfies 0≤𝜏(𝑡)≤𝜏,𝜏̇(𝑡)≤𝑑<1. (2)

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 ∏=(𝑟_𝑖𝑗),(𝑖,𝑗∈𝕊), 𝑃𝑟()(𝑡)=𝑟𝑡+Δ=𝑗∣𝑟=𝑖^𝑖𝑗×Δ+𝑜(Δ),𝑖≠𝑗1+𝑟_𝑖𝑖×Δ+𝑜(Δ),𝑖=𝑗, (3) where Δ>0 and lim_Δ→0(𝑜(Δ))/Δ=0, 𝑟_𝑖𝑗>0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 and 𝑟_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑖≠𝑗}𝑟_𝑖𝑗.

The nominal Markovian jump singular and time-delay system of system (1) are as follows: 𝐸(𝑟(𝑡𝑥̇(𝑡)(𝑟(𝑡𝑥(𝑡),𝐸(𝑟(𝑡𝑥̇(𝑡)(𝑟(𝑡𝑥(𝑡)))=𝐴))))=𝐴))+𝐴_𝑑(𝑟(𝑡))𝑥(𝑡−𝜏(𝑡)). (4)(5)

The initial Markovian jump singular system in (1) is assumed to be 𝑥0,𝛾₀=𝜙(0). (6)

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)=𝐸_𝑖, 𝐴(𝑖)=𝐴_𝑖, 𝐴_𝑑(𝑖)=𝐴_𝑑𝑖𝐵(𝑟(𝑡))=𝐵_𝑖, 𝑓(𝑥(𝑡),𝑡,𝑟(𝑡))=𝑓_𝑖(𝑥(𝑡),𝑡).

Definition 1. (i) The system (4) is said to be regular if det(𝑠𝐸_𝑖−𝐴_𝑖)≠0 for every 𝑖∈𝕊.
(ii) The system (4) is said to impulse free if deg(det(𝑠𝐸_𝑖−𝐴_𝑖))=rank(𝐸_𝑖) for every 𝑖∈𝕊.

Definition 2. For a given scalar 𝜏>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))𝑋̇(𝑡)=(𝐴(𝑟(𝑡))+𝐴_𝑑(𝑟(𝑡)))𝑋(𝑡) are all regular and impulse free.
The system (5) is said to be stochastically stable if for any 𝑥₀∈𝑅^𝑛 and 𝑟₀∈𝕊 there exists a scalar 𝑀(𝑥₀,𝑟₀)>0 such that lim_𝑡→∞ℰ(∫^𝑡₀𝑥^𝑇(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 term 𝑓(𝑥(𝑡),𝑡,𝑟(𝑡)) is Lipshitz, continuous and satisfies the following matching conditions: 𝑓(𝑥(𝑡),𝑡,𝑟(𝑡))=𝐵(𝑟(𝑡))𝑓(𝑥(𝑡),𝑡,𝑟(𝑡)), (7)
with 𝑓(𝑥(𝑡),𝑡,𝑟(𝑡))∈𝑅^𝑚 bounded by 𝑓(𝑥(𝑡),𝑡,𝑟(𝑡))≤𝜖_𝑖‖𝑥(𝑡)‖, (8) where 𝜖>0 is a constant.

Lemma 3 will support the non-strict LMI to be translated into strict LMI.

Lemma 3 (see [34]). Let 𝑋∈𝑅^𝑛×𝑛 be symmetric such that 𝐸^𝑇_𝐿𝑋𝐸_𝐿>0 and 𝑇∈𝑅^{(𝑛−𝑟)×(𝑛−𝑟)} nonsingular. Then, 𝑋𝐸+𝑀^𝑇𝑇𝑆^𝑇 is nonsingular and its inverse is expressed as 𝑋𝐸+𝑀^𝑇𝑇𝑆^𝑇−1=𝕏𝐸^𝑇+𝑆𝕋𝑀, (9) where 𝕏 is symmetric and 𝕋 is a singular matrix with 𝐸^𝑇_𝑅𝕏𝐸_𝑅=𝐸^𝑇_𝐿𝑋𝐸_𝐿⁻¹,𝕋=𝑆^𝑇𝑆⁻¹𝑇⁻¹𝑀𝑀^𝑇−1, (10) where 𝑀 and 𝑆 are any matrices with full row rank and satisfy 𝑀𝐸=0 and 𝐸𝑆=0, respectively; 𝐸 is decomposed as 𝐸=𝐸_𝐿𝐸^𝑇_𝑅 with 𝐸_𝐿∈𝑅^𝑛×𝑟 and 𝐸_𝑅∈𝑅^𝑛×𝑟 are of full column rank.

Lemma 4 (see [35]). There exists symmetric matrix X such that 𝑃₁+𝑋𝑄₁𝑄^𝑇₁𝑅₁𝑃<0,²−𝑋𝑄₂𝑄^𝑇₂𝑅₂<0 (11) if and only if 𝑃₁+𝑃₂𝑄₁𝑄₂∗𝑅₁0∗∗𝑅₂<0. (12)

Lemma 5. Let 𝑄=𝑄^𝑇, 𝑆, 𝑅=𝑅^𝑇 be matrices of appropriate dimensions, then 𝑅<0,𝑄−𝑆𝑅⁻¹𝑆^𝑇<0 is equivalent to 𝑆𝑄𝑆_𝑇𝑅<0. (13)

Lemma 6. Let 𝑋∈𝑅^𝑛, 𝑌∈𝑅^𝑛, and 𝑄>0. Then we have 𝑋^𝑇𝑌+𝑌^𝑇𝑋≤𝑌^𝑇𝑄𝑌+𝑋^𝑇𝑄⁻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 𝑃_𝑖,𝑖=1,2,…,𝑁 such that 𝐸^𝑇_𝑖𝑃_𝑖=𝑃^𝑇_𝑖𝐸_𝑖≥0_𝑁𝑗=1𝑟_𝑖𝑗𝐸^𝑇_𝑖𝑃_𝑗+𝑃^𝑇_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖<0. (14)

Lemma 8. For a prescribed scalars 𝜏>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 𝑃_𝑖 for every 𝑖=1,2,…,𝑁, such that 𝐸^𝑇_𝑖𝑃_𝑖=𝑃^𝑇_𝑖𝐸_𝑖Φ≥0,_𝐢=Φ_𝐢^𝟏𝟏𝑃^𝑇_𝑖𝐴_𝑑𝑖0()1∗−1−𝑑𝑄0∗∗−𝜏𝑅<0, (15)(16) where Φ_𝐢^𝟏𝟏=𝑃^𝑇_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝑄+𝜏𝑅+^𝑁_𝑗=1𝑟_𝑖𝑗𝐸^𝑇_𝑗𝑃_𝑗. (17)

Proof. First to prove the system 𝐸(𝑟(𝑡))𝑥̇(𝑡)=[𝐴(𝑟(𝑡))+𝐴_𝑑(𝑟(𝑡)]𝑥(𝑡) is regular and impulse free.
From (16), it is easy to know Φ_𝐢^𝟏𝟏<0 and Φ_𝐢^𝟏𝟏𝑃^𝑇_𝑖𝐴_𝑑𝑖∗−(1−𝑑)𝑄<0. (18)
By pre- and postmultiplying (18) by [𝐼,𝐼] and [𝐼,𝐼]^𝑇, we get 𝑃^𝑇_𝑖𝐴_𝑖+𝐴_𝑑𝑖+𝐴_𝑖+𝐴_𝑑𝑖^𝑇𝑃_𝑖+𝑑𝑄+𝜏𝑅+^𝑁_𝑗=1𝑟_𝑖𝑗𝐸^𝑇_𝑗𝑃_𝑗<0, (19) where 𝑄 and 𝑅 are symmetric positive-definite matrices, we have 𝑃^𝑇_𝑖𝐴_𝑖+𝐴_𝑑𝑖+𝐴_𝑖+𝐴_𝑑𝑖^𝑇𝑃_𝑖+^𝑁_𝑗=1𝑟_𝑖𝑗𝐸^𝑇_𝑗𝑃_𝑗<0,_𝑁𝑗=1𝑟_𝑖𝑗𝐸^𝑇_𝑖𝑃_𝑗+𝑃^𝑇_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖<0. (20)(21)
Based on Lemma 7, (15) and (20) show that the system 𝐸(𝑟(𝑡))𝑥̇(𝑡)=[𝐴(𝑟(𝑡))+𝐴_𝑑(𝑟(𝑡))]𝑥(𝑡) is regular and impulse free, and (15) and (21) ensure that the system 𝐸(𝑟(𝑡))𝑋̇(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: 𝑉(𝑥(𝑡),𝑟(𝑡))=𝑋^𝑇(𝑡)𝐸^𝑇(𝑟(𝑡))𝑃(𝑟(𝑡))𝑋(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}𝑋^𝑇(𝑠)(𝑠)+𝑄𝑋𝑑𝑠_{0−𝜏𝑡𝑡+𝜃}𝑋^𝑇(𝑠)𝑅𝑋(𝑠)𝑑𝑠𝑑𝜃. (22)
Then, let ℒ be the weak infinitesimal generator of the random process 𝑥(𝑡),𝑟(𝑡), and for each 𝑖∈𝕊, we have ℒ𝑉(𝑥(𝑡),𝑟(𝑡)=𝑖)=lim_Δ→01Δ{ℰ[𝑉(𝑥()((𝑡),𝑟(𝑡)](𝑥(𝑡)(𝑡)𝑡+Δ,𝑟𝑡+Δ))∣𝑥=𝑖−𝑉,𝑟=𝑖)}≤2𝑥^𝑇(𝑡)𝑃^𝑇_𝑖𝐸_𝑖𝑥̇(𝑡)+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝑟_𝑖𝑗𝐸^𝑇_𝑗𝑃_𝑗𝑥(𝑡)+𝑥^𝑇(𝑡)(𝑡)−(𝜏̇(𝑡𝑥𝑄𝑥1−))^𝑇((𝑡((𝑡𝑥𝑡−𝜏))𝑄𝑥𝑡−𝜏))+𝜏^𝑇(𝑡)𝑅𝑥(𝑡)−^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑅𝑥(𝑠)𝑑𝑠≤2𝑥^𝑇(𝑡)𝑃^𝑇_𝑖𝐴_𝑖𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃^𝑇_𝑖𝐴_𝑑𝑖𝑥(𝑡−𝜏(𝑡))+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝑟_𝑖𝑗𝐸^𝑇_𝑗𝑃_𝑗𝑥(𝑡)+𝑥^𝑇(𝑡)(𝑡)−()𝑥𝑄𝑥1−𝑑^𝑇(𝑡−𝜏(𝑡))𝑄𝑥(𝑡−𝜏(𝑡))+𝜏𝑥^𝑇(𝑡)𝑅𝑥(𝑡)−^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)×𝑅𝑑𝑠𝜏^𝑡_𝑡−𝜏𝑥(𝑠)(𝑡)Φ𝑑𝑠=𝜉_𝑖𝜉^𝑇(𝑡), (23) where 𝜉(𝑡)=[𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−𝜏(𝑡)),∫^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑑𝑠].
With (16), it is easy to know that there exists a scalar 𝜇>0, such that ℒ𝑉(𝑥(𝑡),𝑟(𝑡)=𝑖)≤−𝜇𝑥^𝑇(𝑡)𝑥(𝑡). (24)
By Dynkin's formula, we get ℰ{𝑉((𝑥(𝑡),𝑟(𝑡))𝑥(0),𝑟(0))−𝑉(𝑥(0),𝑟(0))}=ℰ^𝑡₀𝐿𝑉(𝑥(𝑠),𝑟(𝑠))𝑑𝑠∣𝑥(0),𝑟(0)≤−𝜇ℰ^𝑡₀𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠∣𝑥(0),𝑟(0), (25) and this means ℰ^𝑡₀𝑥^𝑇(𝑠)𝑥(𝑠)(0)(0)≤1𝑑𝑠∣𝑥,𝑟𝜇𝑉(𝑥(0),𝑟(0)). (26)
Therefore by Definition 2, 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 every 𝑖∈𝕊, integral sliding surface with delay and Markovian switching is considered as follows: 𝑆(𝑡)=𝐺_𝑖𝐸_𝑖𝑥(𝑡)−^𝑡₀𝐺_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖𝑥(𝜃)−𝑑𝜃_𝑡0𝐺_𝑖𝐴_𝑑𝑖𝑥(𝜃−𝜏(𝜃))𝑑𝜃, (27) where 𝐾_𝑖∈𝐑^𝑚×𝑛 is real matrix to be designed and 𝐺_𝑖∈𝐑^𝑚×𝑛 is designed to satisfy that 𝐺_𝑖𝐵_𝑖 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, by 𝑆(̇𝑡)=0, we get the equivalent control as 𝑢_eqi(𝑡)=𝐾_𝑖𝑥(𝑡)−𝑓(𝑥(𝑡),𝑡,𝑖). (28)

Substituting 𝑢_eq(r(𝑡)=𝐾(𝑟(𝑡))𝑥(𝑡)−𝑓(𝑥(𝑡),𝑡,𝑟(𝑡)) into (1), we obtain the following sliding mode dynamics: 𝐸(𝑟(𝑡))𝑥̇(𝑡)=[𝐴(𝑟(𝑡))+𝐵(𝑟(𝑡))𝐾(𝑟(𝑡))]𝑥(𝑡)+𝐴_𝑑(𝑟(𝑡))𝑥(𝑡−𝜏(𝑡)). (29)

3.2. Sliding Mode Dynamics Analysis

In this section, we pay attention to establishing 𝐿𝑀𝐼𝑠 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 scalars 𝜏 and 𝑑, for any delays 𝜏(𝑡) satisfying (2), the system (29) is stochastically admissible if there exist nonsingular matrices 𝑃_𝑖 and symmetric positive-definite matrix 𝑄 and 𝑅 such that 𝐸^𝑇_𝑖𝑃_𝑖=𝑃^𝑇_𝑖𝐸_𝑖Φ≥0,_𝑖11𝑃^𝑇_𝑖𝐴_𝑑𝑖0()1∗−1−𝑑𝑄0∗∗−𝜏𝑅<0, (30)(31) where Φ_𝑖¹¹=𝑃^𝑇_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖+𝐴_𝑖+𝐵_𝑖𝐾_𝑖^𝑇𝑃_𝑖+𝑄+𝜏𝑅+^𝑁_𝑗=1𝑟_𝑖𝑗𝐸^𝑇_𝑗𝑃_𝑗. (32)

Remark 10. Note that the conditions in Theorem 9 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 scalars 𝜏 and 𝑑, for any delays 𝜏(𝑡) satisfying (2), the system (29) is stochastically admissible if there exist symmetric positive-definite matrices 𝕏,𝑄,𝑌_𝑖,𝐻, 𝑅, and symmetric matrix 𝕋∈𝐑^{(𝑛−𝑟)×(𝑛−𝑟)}, matrix 𝕃_𝑖∈𝑅^𝑚×𝑛, and any matrices with full row rank 𝑀_𝑖,𝑆_𝑖 satisfying 𝑀_𝑖𝐸_𝑖=0,𝐸_𝑖𝑆_𝑖=0, respectively, such that Γ_𝑖¹¹𝐴_𝑑𝑖𝑁₁∗−(1−𝑑)𝑄0∗∗𝑁₂<0,𝑄+𝜏𝑅−𝐻0𝑁₃1∗−𝜏𝑅0∗∗𝑁₄<0,−𝑌_𝑖0𝐸_𝑖𝕏+𝑀^𝑇_𝑖𝕋𝑆^𝑇_𝑖0∗−𝐻0𝐻∗∗−𝐼0∗∗∗−𝐼<0, (33)(34)(35) where Γ_𝑖¹¹=𝐴_𝑖𝕏𝐸^𝑇_𝑖+𝑆_𝑖𝕋𝑀_𝑖+𝕏𝐸^𝑇_𝑖+𝑆_𝑖𝕋𝑀_𝑖^𝑇𝐴^𝑇_𝑖+𝐵_𝑖𝕃_𝑖𝐸^𝑇_𝑖+𝐵_𝑖𝕃_𝑖𝐸^𝑇_𝑖^𝑇+𝑟_𝑖𝑖𝕏𝐸^𝑇_𝑖+𝑆_𝑖𝕋𝑀_𝑖^𝑇𝐸^𝑇_𝑖+𝑌_𝑖,𝑁₁=𝐵_𝑖𝕃_𝑖,𝑀^𝑇_𝑖𝕋𝑆^𝑇_𝑖,𝑁₂[],𝑁=diag−𝕏,−𝕏₃=𝐸_1𝑅,…,𝐸_{𝑖−1,𝑅},𝐸_𝑖+1,𝑅,…,𝐸_𝑛𝑅,𝑁₄−1=diag𝑟_𝑖1𝐸^𝑇_1𝑅𝕏𝐸_1𝑅1,…,−𝑟_{𝑖,𝑖−1}𝐸^𝑇_{𝑖−1,𝑅}𝕏𝐸_{𝑖−1,𝑅},−1𝑟_𝑖,𝑖+1𝐸^𝑇_𝑖+1𝑅𝕏𝐸_𝑖+1𝑅1,…,−𝑟_𝑖𝑛𝐸^𝑇_𝑛𝑅𝕏𝐸_𝑛𝑅,𝕃_𝑖=𝐾_𝑖𝕏. (36)

Proof. Let 𝑃_𝑖≜𝑋𝐸_𝑖+𝑀^𝑇_𝑖𝑇𝑆^𝑇_𝑖 in Theorem 9. We can get Φ_𝑖¹¹𝑋𝐸_𝑖+𝑀^𝑇_𝑖𝑇𝑆^𝑇_𝑖^𝑇𝐴_𝑑𝑖0()1∗−1−𝑑𝑄0∗∗−𝜏𝑅<0, (37) where Φ_𝑖¹¹=𝑋𝐸_𝑖+𝑀^𝑇_𝑖𝑇𝑆^𝑇_𝑖^𝑇𝐴_𝑖+𝐵_𝑖𝐾_𝑖+𝐴_𝑖+𝐵_𝑖𝐾_𝑖^𝑇𝑋𝐸_𝑖+𝑀^𝑇_𝑖𝑇𝑆^𝑇_𝑖+𝑄+𝜏𝑅+^𝑁_𝑗=1𝑟_𝑖𝑗𝐸^𝑇_𝑗𝑋𝐸_𝑗. (38)
Using Lemma 3, we get 𝑋𝐸_𝑖+𝑀^𝑇_𝑖𝑇𝑆^𝑇_𝑖⁻¹=𝕏𝐸^𝑇_𝑖+𝑆_𝑖𝕋𝑀_𝑖≜𝐿_𝑖. (39)
By pre- and postmultiplying (37) by diag[𝐿^𝑇_𝑖,𝐼,𝐼] and diag[𝐿_𝑖,𝐼,𝐼], we get Ψ_𝑖^′11𝐴_𝑑𝑖0()1∗−1−𝑑𝑄0∗∗−𝜏𝑅<0, (40) where Ψ_𝑖^′11=(𝐴_𝑖+𝐵_𝑖𝐾_𝑖)𝐿_𝑖+𝐿^𝑇_𝑖(𝐴_𝑖+𝐵_𝑖𝐾_𝑖)^𝑇+𝐿^𝑇_𝑖𝑄𝐿_𝑖+𝜏𝐿^𝑇_𝑖𝑅𝐿_𝑖+∑^𝑁_𝑗=1𝑟_𝑖𝑗𝐿^𝑇_𝑖𝐸^𝑇_𝑗𝑋𝐸_𝑗.
In light of Lemma 4, there exists symmetric matrix 𝑌_𝑖 such that 𝐴_𝑖+𝐵_𝑖𝐾_𝑖𝐿_𝑖+𝐿^𝑇_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖^𝑇+𝑟_𝑖𝑖𝐿^𝑇_𝑖𝐸^𝑇_𝑖+𝑌_𝑖𝐴_𝑑𝑖()𝑄𝐿∗−1−𝑑<0,_𝑇𝑖𝑄𝐿_𝑖+𝜏𝐿^𝑇_𝑖𝑅𝐿_𝑖+∑^𝑁_{𝑖≠𝑗=1}𝑟_𝑖𝑗𝐿^𝑇_𝑖𝐸^𝑇_𝑗𝑋𝐸_𝑗𝐿_𝑖−𝑌_𝑖01∗−𝜏𝑅<0. (41)(42)
It is easy to know that 𝐵_𝑖𝐾_𝑖𝑆_𝑖𝕋𝑀_𝑖+𝐵_𝑖𝐾_𝑖𝑆_𝑖𝕋𝑀_𝑖^𝑇=𝐵_𝑖𝐾_𝑖𝕏𝕏⁻¹𝑆_𝑖𝕋𝑀_𝑖+𝐵_𝑖𝐾_𝑖𝕏𝕏⁻¹𝑆_𝑖𝕋𝑀_𝑖^𝑇≤𝐵_𝑖𝐾_𝑖𝕏𝕏⁻¹𝕏𝐾^𝑇_𝑖𝐵^𝑇_𝑖+𝑀^𝑇_𝑖𝕋𝑆^𝑇_𝑖𝕏⁻¹𝑆_𝑖𝕋𝑀_𝑖. (43)
Using Shur's complement, the following inequality can ensure (41) as 𝐴_𝑖𝐿_𝑖+𝐿^𝑇_𝑖𝐴^𝑇_𝑖+𝐵_𝑖𝐾_𝑖𝕏𝐸^𝑇_𝑖+𝐵_𝑖𝐾_𝑖𝕏𝐸^𝑇_𝑖^𝑇+𝑟_𝑖𝑖𝐿^𝑇_𝑖𝐸^𝑇_𝑖+𝑌_𝑖𝐴_𝑑𝑖𝑁₁∗−(1−𝑑)𝑄0∗∗𝑁₂>0, (44) where 𝑁₁=[𝐵_𝑖𝐾_𝑖𝕏,𝑀^𝑇_𝑖𝕋𝑆^𝑇_𝑖] and 𝑁₂=diag[−𝕏,−𝕏]. Equation (42) is equivalent to 𝑄+𝜏𝑅+∑^𝑁_{𝑖≠𝑗=1}𝑟_𝑖𝑗𝐸^𝑇_𝑗𝑋𝐸_𝑗−𝑃^𝑇_𝑖𝑌_𝑖𝑃_𝑖01∗−𝜏𝑅<0. (45)
There exists a matrix 𝐻^𝑇=𝐻>0 such that 𝑄+𝜏𝑅+∑^𝑁_{𝑖≠𝑗=1}𝑟_𝑖𝑗𝐸^𝑇_𝑗𝑋𝐸_𝑗−𝑃^𝑇_𝑖𝑌_𝑖𝑃_𝑖01∗−𝜏𝑅<𝑄+𝜏𝑅+∑^𝑁_{𝑖≠𝑗=1}𝑟_𝑖𝑗𝐸^𝑇_𝑗𝑋𝐸_𝑗1−𝐻0∗−𝜏𝑅<0 (46) if and only if −𝑃^𝑇_𝑖𝑌_𝑖𝑃_𝑖<−𝐻<0. (47)
The above inequality is equivalent to the following inequality: −𝑌_𝑖𝐿^𝑇_𝑖𝐻∗−𝐻<0. (48)
And the following inequality can guarantee that (48) is true: −𝑌_𝑖0𝐿^𝑇_𝑖0∗−𝐻0𝐻∗∗−𝐼0∗∗∗−𝐼<0. (49)
By Shur's complement, the right of (46) is equivalent to 𝑄+𝜏𝑅−𝐻0𝑁₃1∗−𝜏𝑅0∗∗𝑁₄<0, (50) where 𝑁₃=[𝐸_1𝑅,…,𝐸_{𝑖−1,𝑅},𝐸_𝑖+1,𝑅,…,𝐸_𝑛𝑅] and 𝑁₄=diag[−(1/𝑟_𝑖1)𝐸^𝑇_1𝑅𝕏𝐸_1𝑅,…,−(1/𝑟_{𝑖,𝑖−1})𝐸^𝑇_{𝑖−1,𝑅}𝕏𝐸_{𝑖−1,𝑅}, −(1/𝑟_𝑖,𝑖+1)𝐸^𝑇_𝑖+1𝑅𝕏𝐸_𝑖+1𝑅,…,−(1/𝑟_𝑖𝑛)𝐸^𝑇_𝑛𝑅𝕏𝐸_𝑛𝑅].𝐸_𝑖 is decomposed as 𝐸_𝑖=𝐸_𝑖𝐿𝐸^𝑇_𝑖𝑅 with 𝐸_𝑖𝐿∈𝐑^𝑛×𝑟 and 𝐸_𝑖𝑅∈𝐑^𝑛×𝑟 are of full column rank. This completes the proof.

Remark 12. From the proof of the Theorem 11, 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 matrix 𝐾_𝑖 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 𝑢_𝑖(𝑡)=𝐾_𝑖𝑥(𝑡)−𝜌_𝑖+𝜖_𝑖‖𝑥(𝑡)‖sign𝐵^𝑇_𝑖𝐺^𝑇_𝑖𝑆(𝑡), (51) where 𝜌_𝑖 is a positive constant.

Proof. Choose 𝐺_𝑖 under the condition of 𝐺_𝑖𝐵_𝑖 is nonsingular. Consider the following Lyapunov function: 𝑉(𝑆(𝑡))=1,𝑡2𝑆^𝑇(𝑡)𝑆(𝑡). (52) Due to (29), we have 𝑑𝑆=𝐺_𝑖𝐵_𝑖−𝐾_𝑖𝑥(𝑡)+𝑢_𝑖(𝑡)+𝑓_𝑖𝑑𝑡. (53)
Differentiating 𝑉_𝑖(𝑡) along the closed-loop trajectories and using (53), we have ℒ𝑉(𝑆(𝑡),(𝑡))=𝑆^𝑇_𝑖(𝑡)𝑑𝑆_𝑖(𝑡)=𝑆^𝑇_𝑖(𝑡)𝐺_𝑖𝐵_𝑖−𝐾_𝑖𝑥(𝑡)+𝑢_𝑖(𝑡)+𝑓_𝑖=𝑆^𝑇_𝑖(𝑡)𝐺_𝑖𝐵_𝑖−‖𝑥(𝑡×𝐵𝜌+𝜖)‖sign_𝑇𝑖𝐺^𝑇_𝑖𝑆(𝑡)+𝑓_𝑖≤−𝜌+𝜖‖𝑥(𝑡)‖𝐵^𝑇_𝑖𝐺^𝑇_𝑖𝑆_𝑖(𝑡+)𝑓_𝑖𝐵^𝑇_𝑖𝐺^𝑇_𝑖𝑆_𝑖(𝑡𝐵)≤−𝜌^𝑇_𝑖𝐺^𝑇_𝑖𝑆_𝑖(𝑡)≤−𝜌𝑆_𝑖(𝑡)<0,if𝑆_𝑖(𝑡)≠0, (54) where 𝜌_𝑖>𝜌_𝑖𝜆_min(𝐺_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝐺^𝑇_𝑖). 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 ∏=(𝑟_𝑖𝑗),(𝑖,𝑗=1,2), 𝑟₁₂=0.25,𝑟₂₁=0.2. The system data are as follows: 𝐄₁=100010000,𝐀₁=,𝐀−6.71.600−500.52.23.2_𝑑1=−0.2000−0.1000−0.1,𝐁₁=110,𝐌₁=001,𝐒₁=001,𝐄_1𝑅=,𝐄100100₂=100000001,𝐀₂=,𝐀−5.7−4.20−3.74.81.302.4−6_𝑑2=−0.1000−0.10000,𝐁₂=111,𝐌₂=010,𝐒₂=010𝐄_2𝑅=.100001 (55)
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: 𝐊_𝟏=𝐊0.0963−0.0506−0.003_𝟐=0.0953−0.00390.1793. (56)
For det(𝐺_𝑖𝐵_𝑖)≠0, we can choose 𝐺₁ and 𝐺₂ as 𝐺₁=1.21.61.1;𝐺₂=1.31.21.5. (57)
Thus the sliding surface function is 𝑠₁(𝑡)=𝑠₁(𝑡)=[1.2,1.6,0]𝑥(𝑡)−∫^𝑡₀[−7.2204,−3.8017,−3.5116]𝑥(𝜃)𝑑𝜃−∫^𝑡₀[]𝑥((𝜃𝑠−0.24,−0.016,−0.11𝜃−𝜏))𝑑𝜃,𝑖=1₂(𝑡)=[1.3,0,1.5]𝑥(𝑡)−∫^𝑡₀[]𝑥(𝜃)−∫−11.4688,3.8852,−6.7228𝑑𝜃_𝑡0[−0.13,−0.12,0]𝑥(𝜃−𝜏(𝜃))𝑑𝜃,𝑖=2. (58)
Take 𝜌₁=𝜌₂=0.64, then the SMC law designed in (51) can be described as 𝑢(𝑡)=𝑢₁(𝑡)=[]𝑥(𝑡)(𝑡)0.0963,−0.0506,−0.003−𝜌sign2.8𝑠₁(𝑡)𝑢,𝑖=1₂(𝑡)=[]𝑥(𝑡)(𝑡)0.0953,−0.0037,0.1793−𝜌sign4𝑠₂(𝑡),𝑖=2, (59) 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 surface 𝑠(𝑡).

Figure 3 

State trajectories of the closed-loop system.

Figure 4 

Controller input 𝑢(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.

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and constructive suggestions. This research is supported by the Natural Science Foundation of Guangxi Autonomous Region (no. 2012GXNSFBA053003), the National Natural Science Foundations of China (60973048, 61272077, 60974025, 60673101, 60939003), National 863 Plan Project (2008 AA04Z401, 2009AA043404), the Natural Science Foundation of Shandong Province (no. Y2007G30), the Scientific and Technological Project of Shandong Province (no. 2007GG3WZ04016), the Science Foundation of Harbin Institute of Technology (Weihai) (HIT(WH) 200807), the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF. 2001120), the China Postdoctoral Science Foundation (2010048 1000), and the Shandong Provincial Key Laboratory of Industrial Control Technique (Qingdao University).