Sliding Mode Control of Uncertain Neutral Stochastic Systems with Multiple Delays

Abstract

This paper is concerned with the sliding mode control for uncertain stochastic neutral systems with multiple delays. A switching surface is adopted first. Then, by means of linear matrix inequalities (LMIs), a sufficient condition is derived to ensure the global stochastic stability of the stochastic system in the sliding mode for all admissible uncertainties. The synthesized sliding mode controller guarantees the existence of the sliding mode.

1. Introduction

Time delay occurs due to the finite capabilities of information processing and data transmission among various parts of the system. The phenomena of time delay are often encountered in various relevant systems, such as HIV infection with drug therapy, aircraft stabilization, chemical engineering systems, inferred grinding model, manual control, neural network, nuclear reactor, population dynamic model, rolling mill, ship stabilization, and systems with lossless transmission lines. It is well known that time delay factors always lead to poor performance. Hence, problems of stability analysis and stabilization of dynamical systems with time delays in the state variables and/or control inputs have received considerable interest for more than three decades [1–6].

In practice, systems are almost always innately “noisy”. Therefore, in order to model a system realistically, a degree of randomness must be incorporated into the model. Thus, a class of stochastic systems has received great attention in the past decade [7]. On the other hand, it has been shown that a lot of practical systems can be modeled by using functional differential equations of the neutral type [8, 9]. However, the mathematical model always contains some uncertain elements. Therefore, uncertain systems have been extensively studied in the past years [10–12].

To cope with the problem of stability of uncertain stochastic neutral delay systems, most of the research focused on the retarded functional differential equations and it also seems that few results are available on the variable structure control.

Sliding mode control (SMC) is a particular type of variable structure control. It provides an effective alternative to deal with the nonlinear dynamic systems. The main feature of SMC is its easy realization, control of independent motion, insensitivity to variation in plant parameters or external perturbations, and wide variety of operational models [13–15].

The purpose of this paper lies in the design of SMC for a class of uncertain stochastic neutral delay systems. A switching surface, which makes it easy to guarantee the stability of the uncertain stochastic neutral delay systems in the sliding mode, is first proposed. By means of linear matrix inequalities (LMIs), a sufficient condition is given such that the stochastic dynamics in the specified switching surface is globally stochastically stable. And then, based on this switching surface, a synthesized SMC law is derived to guarantee the existence of the composite sliding motion. Finally, a numerical example is illustrated to demonstrate the validity of the proposed SMC.

2. Problem Formulation

Consider the following neutral stochastic system with uncertainties and multiple delays:𝑑[𝐸𝑥(𝑡)−𝐶𝑥(𝑡−𝜏)]=[(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+(𝐴_𝑑+Δ𝐴_𝑑(𝑡))𝑥(𝑡−ℎ(𝑡))+𝐵𝑢(𝑡)]𝑑𝑡+[Δ𝐸(𝑡)𝑥(𝑡)+Δ𝐸_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡))]𝑑𝑤(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡∈[−𝐻,0], (2.1)where 𝑥(𝑡)∈ℝ^𝑛 is the state, 𝑢(𝑡)∈ℝ^𝑚 is the control input, 𝜏 is the constant delay, ℎ(𝑡) is the time-varying differentiable bounded delay satisfying 0≤ℎ(𝑡)≤ℎ_𝑀, ℎ(^Δ𝑡)≤ℎ_𝐷<1, 𝑤(𝑡) is an 𝑚-dimensional Brownian motion, 𝐻=max{𝜏,ℎ_𝑀}. It is assumed that 𝜙(𝑡) is the initial condition which is continuous, 𝑡∈[−𝐻,0]. In system (2.1), 𝐸∈ℝ^𝑛×𝑛, 𝐶∈ℝ^𝑛×𝑛, 𝐴∈ℝ^𝑛×𝑛, 𝐴_𝑑∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑚 are known real constant matrices. Δ𝐴(𝑡), Δ𝐴_𝑑(𝑡), Δ𝐸(𝑡), and Δ𝐸_𝑑(𝑡) represent the structured uncertainties in (2.1), which are assumed to be of the formsΔ𝐴(𝑡)=𝑀𝐹₁(𝑡)𝑁_𝐴,Δ𝐴_𝑑(𝑡)=𝑀𝐹₂(𝑡)𝑁_𝐴𝑑,[Δ𝐸(𝑡),Δ𝐸_𝑑(𝑡)]=𝑀𝐹₃(𝑡)[𝑁_𝐸,𝑁_𝐸𝑑], (2.2)𝑀, 𝑁_𝐴, 𝑁_𝐴𝑑, 𝑁_𝐸, and 𝑁_𝐸𝑑 are some given constant matrices, 𝐹_𝑙(𝑡)(𝑙=1,2,3) are unknown real time-varying matrices which have the following structure:

𝐹_𝑙(𝑡)=blockdiag{𝛿_𝑙1(𝑡)𝐼_𝑟𝑙1,…,𝛿_𝑙𝑘(𝑡)𝐼_𝑟𝑙𝑘,𝐹_𝑙1(𝑡),…,𝐹_𝑙𝑠(𝑡)}, 𝛿_𝑙𝑖∈ℝ, |𝛿_𝑙𝑖|≤1, 1≤𝑖≤𝑘, and 𝐹^⊤_𝑙𝑗𝐹_𝑙𝑗≤𝐼, 1≤𝑗≤𝑠.

We define the sets Δ_𝑙 asΔ_𝑙={𝐹^⊤_𝑙(𝑡)𝐹_𝑙(𝑡)≤𝐼,𝐹_𝑙𝑁_𝑙=𝑁_𝑙𝐹_𝑙,∀𝑁_𝑙∈Σ_𝑙}, (2.3)where Σ_𝑙={𝑁_𝑙=blockdiag[𝑁_𝑙1,…,𝑁_𝑙𝑘,𝑛_𝑙1𝐼_𝑓𝑙1,…,𝑛_𝑙𝑠𝐼_𝑓𝑙𝑠]}, 𝑁_𝑙𝑖 are invertible for 1≤𝑖≤𝑘, and 𝑛_𝑙𝑗∈ℝ, 𝑛_𝑙𝑗≠0 for 1≤𝑗≤𝑠.

The following useful lemmas will be used to derive the desired LMI-based stability criteria.

Lemma 2.1 (See [1]). The LMI [^𝑆11𝑆12_∗𝑆22]<0, with 𝑆₁₁=𝑆^⊤₁₁, 𝑆₂₂=𝑆^⊤₂₂, is equivalent to𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^⊤₁₂<0. (2.4)

Lemma 2.2 (See [11]). Let 𝐷, 𝐸, 𝐹_𝑙 be real matrices of appropriate dimensions and 𝐹_𝑙∈Δ_𝑙. Then for any block-structured matrix 𝑁_𝑙∈Σ_𝑙,𝐷𝐹_𝑙𝐸+(𝐷𝐹_𝑙𝐸)^⊤≤𝐷(𝑁_𝑙𝑁^⊤_𝑙)𝐷^⊤+𝐸^⊤(𝑁_𝑙𝑁^⊤_𝑙)⁻¹𝐸. (2.5)

Lemma 2.3 (See [11]). Let 𝐴, 𝐷, 𝐸, 𝐹_𝑙, and 𝑃 be real matrices of appropriate dimensions with 𝑃=𝑃^⊤>0 and 𝐹_𝑙∈Δ_𝑙. Then for any block-structured matrix 𝑁_𝑙∈Σ_𝑙 satisfying 𝑃⁻¹−𝐷(𝑁_𝑙𝑁^⊤_𝑙)𝐷^⊤>0, one has(𝐴+𝐷𝐹_𝑙𝐸)^⊤𝑃(𝐴+𝐷𝐹_𝑙𝐸)≤𝐴^⊤(𝑃⁻¹−𝐷𝑁_𝑙𝑁^⊤_𝑙𝐷^⊤)⁻¹𝐴+𝐸^⊤(𝑁_𝑙𝑁^⊤_𝑙)⁻¹𝐸. (2.6)

Lemma 2.4 (See [11]). For any 𝑧,𝑦∈ℝ^𝑛 and for any symmetric positive-definite matrix 𝑋∈ℝ^𝑛×𝑛,  −2𝑧^⊤𝑦≤𝑧^⊤𝑋⁻¹𝑧+𝑦^⊤𝑋𝑦. (2.7)

Definition 2.5 (See [14]). The nominal stochastic time-delay system of form (2.1) with 𝑢(𝑡)=0 is said to be mean-square asymptotically stable iflim_𝑡→∞𝔼|𝑥(𝑡)|²=0. (2.8)

Definition 2.6 (See [14]). The uncertain time delay system of the form (2.1) is robustly mean square stabilized if the nominal system is mean-square asymptotically stable for all admissible uncertainties.

In order to simplify the treatment of the problem, the operator ℑ∶𝐶([−𝜏,0],ℝ^𝑛)→ℝ^𝑛 is defined to beℑ(𝑥_𝑡)=𝐸𝑥(𝑡)−𝐶𝑥(𝑡−𝜏). (2.9)The stability of the operator ℑ is defined as follows.

Definition 2.7 (See [9]). The operator ℑ is said to be stable if the zero solution of the homogeneous difference equationℑ(𝑥_𝑡𝑋)=0,𝑡≥0,₀=𝜑∈{𝜓∈𝐶[−𝜏,0]∶ℑ𝜓=0} (2.10)is uniformly asymptotically stable.

If rank(𝐸)=𝑚<𝑛, then it is easy to find that there exist nonsingular constant matrices 𝐾 and 𝑆, such that𝐾𝐸𝑆⁻¹=𝐸₁000,𝐾𝐶𝑆⁻¹=𝐶₁00𝐶₂, (2.11)where 𝐸₁ is an 𝑚×𝑚 nonsingular matrix, 𝐶₁ and 𝐶₂ are 𝑚×𝑚 and (𝑛−𝑚)×(𝑛−𝑚) constant matrices, respectively.

Lemma 2.8 (See [9]). The operator ℑ is stable if ‖𝐸₁⁻¹𝐶₁‖<1 and |𝐶₂|≠0, where 𝐸₁, 𝐶₁, and 𝐶₂ are defined as in (2.11) and ‖⋅‖ is any matrix norm.

3. Switching Surface and Controller Design

In this work, we choose the switching function as follows:𝑆(𝑡)=𝐷[𝐸𝑥(𝑡)−𝐶𝑥(𝑡−𝜏)]+𝜎(𝑡), (3.1)where the auxiliary variable 𝜎(𝑡) satisfies the following:𝑑𝜎=−𝐷[(𝐴+𝐵𝐾)𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−ℎ(𝑡))]𝑑𝑡−𝐷[Δ𝐸𝑥(𝑡)+Δ𝐸_𝑑𝑥(𝑡−ℎ(𝑡))]𝑑𝑤(𝑡), (3.2)where 𝐷∈ℝ^𝑚×𝑛 and 𝐾∈ℝ^𝑚×𝑛 are constant matrices. The matrix 𝐾 is chosen such that the matrix 𝐴+𝐵𝐾 is Hurwitz, and the matrix 𝐷 is to be designed later so that 𝐷𝐵 is nonsingular. As long as the system operates in the sliding mode, it satisfies the equations 𝑆(𝑡)=0 and 𝑆^Δ(𝑡)=0 [13].

Therefore, the equivalent control 𝑢_eq(𝑡) in the sliding manifold is given by𝑢_eq=−(𝐷𝐵)⁻¹𝐷[(Δ𝐴(𝑡)−𝐵𝐾)𝑥(𝑡)+Δ𝐴_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡))]. (3.3)Substituting (3.3) into system (2.1), the following equivalent sliding mode dynamics can be obtained:𝑑[𝐸𝑥(𝑡)−𝐶𝑥(𝑡−𝜏)]=[(𝐴+𝐵𝐾+Δ𝐴(𝑡)−𝐵(𝐷𝐵)⁻¹𝐷Δ𝐴(𝑡))𝑥(𝑡)+(𝐴_𝑑+Δ𝐴_𝑑(𝑡)−𝐵(𝐷𝐵)⁻¹𝐷Δ𝐴_𝑑)𝑥(𝑡−ℎ(𝑡))]𝑑𝑡+[Δ𝐸𝑥(𝑡)+Δ𝐸_𝑑𝑥(𝑡−ℎ(𝑡))]𝑑𝑤(𝑡). (3.4)Now, we proceed to the first task which is to analyze the robustly stochastic stability of the sliding motion described by (3.4), and derive a sufficient condition by means of the linear matrix inequality method.

4. Robust Stabilization in the Mean Square Sense

Theorem 4.1. Consider the equivalent sliding mode dynamics (3.4). If the operator ℑ is stable and there exist symmetric positive-definite matrices 𝑋, 𝑄₁, 𝑄₂, 𝑇₁, 𝑇₂, 𝑇₃, 𝑇₄, 𝑇₅, and 𝑇₆ satisfying the following LMIs: ΠΠ=¹¹Π₁₂Π₁₃𝐸^⊤𝑋𝐵𝐸^⊤𝑋𝑀𝐸^⊤𝑋𝑀000∗Π₂₂−𝐶^⊤𝑋𝐴_𝑑000𝐶^⊤𝑋𝐵𝐶^⊤𝑋𝑀𝐶^⊤𝑋𝑀∗∗Π₃₃𝐵000000∗∗∗−^⊤𝑋𝐵200000∗∗∗∗−𝑇₁0000∗∗∗∗∗−𝑇₂𝐵000∗∗∗∗∗∗−^⊤𝑋𝐵200∗∗∗∗∗∗∗−𝑇₃0∗∗∗∗∗∗∗0−𝑇₄, (4.1)where Π₁₁=𝐸^⊤𝑋(𝐴+𝐵𝐾)+(𝐴+𝐵𝐾)^⊤𝑋𝐸+𝑄₁+𝑄₂+𝑁^⊤_𝐴(𝑇₁+𝑇₃+𝑇₅)𝑁_𝐴+𝑁^⊤_𝐸𝑇₆𝑁_𝐸, Π₁₂=−(𝐴+𝐵𝐾)^⊤𝑋𝐶, Π₁₃=𝐸^⊤𝑋𝐴_𝑑+𝑁^⊤_𝐸𝑑𝑇₄𝑁_𝐸𝑑, Π₂₂=−𝑄₁+𝑁^⊤_𝐴𝑇₂𝑁_𝐴, Π₃₃=−(1−ℎ_𝐷)𝑄₁+𝑁^⊤_𝐴𝑑(𝑇₁+𝑇₃+𝑇₅)𝑁_𝐴𝑑+𝑁^⊤_𝐸𝑑𝑇₄𝑁_𝐸𝑑, −𝑋𝑋𝑀∗−𝑇₆−𝑋<0,2𝑋𝑀∗−𝑇₅<0, (4.2)(4.3)then the uncertain time delay system of the form (2.1) with the switching surface (3.1) is robustly stochastically stable and sliding mode matrix 𝐷=𝐵^⊤𝑋. In the above LMIs, 𝑇 takes the form of 𝑁_𝑙𝑁^𝑇_𝑙 for 𝑁_𝑙∈Σ_𝑙.

Proof. Choose a Lyapunov functional candidate 𝑉(𝑥(𝑡),𝑡) as𝑉(𝑥_𝑡)=(ℑ𝑥_𝑡)^⊤(𝑡)𝑃(ℑ𝑥_𝑡)+^𝑡_𝑡−𝜏𝑥^⊤(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−ℎ(𝑡)}𝑥^⊤(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠. (4.4)Then, the averaged derivative is given by the following expression:𝐿𝑉(𝑥_𝑡)=2[𝐸𝑥(𝑡)−𝐶𝑥(𝑡−𝜏)]^⊤𝑋[(𝐴+𝐵𝐾+Δ𝐴(𝑡)−𝐵(𝐷𝐵)⁻¹𝐷Δ𝐴(𝑡))𝑥(𝑡)+(𝐴_𝑑+Δ𝐴_𝑑(𝑡)−𝐵(𝐷𝐵)⁻¹𝐷Δ𝐴_𝑑(𝑡))𝑥(𝑡−ℎ(𝑡))]+[Δ𝐸(𝑡)𝑥(𝑡)+Δ𝐸_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡))]^⊤𝑃[Δ𝐸(𝑡)𝑥(𝑡)+Δ𝐸_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡))]+𝑥^⊤(𝑡)𝑄₁𝑥(𝑡)−𝑥^⊤(𝑡−𝜏)𝑄₁𝑥(𝑡−𝜏)+𝑥^⊤(𝑡)𝑄₂𝑥(𝑡)−(1−ℎ^Δ(𝑡))𝑥^⊤(𝑡−ℎ(𝑡))𝑄₂𝑥(𝑡−ℎ(𝑡)). (4.5)Using Lemma 2.1, inequality (4.2) is equivalent to𝑋⁻¹−𝑀𝑇₆⁻¹𝑀^⊤>0. (4.6)Hence, it follows from Lemma 2.3 that[Δ𝐸(𝑡)𝑥(𝑡)+Δ𝐸_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡))]^⊤𝑋[Δ𝐸(𝑡)𝑥(𝑡)+Δ𝐸_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡))]={𝑀𝐹₃(𝑡)[𝑁_𝐸𝑥(𝑡)+𝑁_𝐸𝑑𝑥(𝑡−ℎ(𝑡))]}^⊤𝑋{𝑀𝐹₃(𝑡)[𝑁_𝐸𝑥(𝑡)+𝑁_𝐸𝑑𝑥(𝑡−ℎ(𝑡))]}≤[𝑁_𝐸𝑥(𝑡)+𝑁_𝐸𝑑𝑥(𝑡−ℎ(𝑡))]^⊤𝑇₆[𝑁_𝐸𝑥(𝑡)+𝑁_𝐸𝑑𝑥(𝑡−ℎ(𝑡))]=𝑥^⊤(𝑡)𝑁^⊤_𝐸𝑇₆𝑁_𝐸𝑥(𝑡)+𝑥^⊤(𝑡)𝑁^⊤_𝐸𝑇₆𝑁_𝐸𝑑𝑥(𝑡−ℎ(𝑡))+𝑥^⊤(𝑡−ℎ(𝑡))𝑁^⊤_𝐸𝑑𝑇₆𝑁_𝐸𝑥(𝑡)+𝑥^⊤(𝑡−ℎ(𝑡))𝑁^⊤_𝐸𝑑𝑇₆𝑁_𝐸𝑑𝑥(𝑡−ℎ(𝑡)). (4.7)Note that 𝐷=𝐵^⊤𝑋, and it follows form Lemma 2.4 that−2𝑥^⊤(𝑡)𝐸^⊤𝑋𝐵(𝐷𝐵)⁻¹𝐷Δ𝐴(𝑡)𝑥(𝑡)≤𝑥^⊤[𝐸^⊤𝑋𝐵(𝐵^⊤𝑋𝐵)⁻¹𝐵^⊤𝑋𝐸+Δ𝐴^⊤(𝑡)𝑋Δ𝐴(𝑡)]𝑥(𝑡),−2𝑥^⊤(𝑡)𝐸^⊤𝑋𝐵(𝐷𝐵)⁻¹𝐷Δ𝐴_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡))≤𝑥^⊤𝐸^⊤𝑋𝐵(𝐵^⊤𝑋𝐵)⁻¹𝐵^⊤𝑋𝐸𝑥(𝑡)+𝑥^⊤(𝑡−ℎ(𝑡))Δ𝐴^⊤_𝑑(𝑡)𝑋Δ𝐴_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡)),2𝑥^⊤(𝑡−𝜏)𝐶^⊤𝑋𝐵(𝐷𝐵)⁻¹𝐷Δ𝐴(𝑡)𝑥(𝑡)≤𝑥^⊤(𝑡−𝜏)𝐶^⊤𝑋𝐵(𝐵^⊤𝑋𝐵)⁻¹𝐵^⊤𝑋𝐶𝑥(𝑡−𝜏)+𝑥^⊤(𝑡)Δ𝐴^⊤(𝑡)𝑋Δ𝐴(𝑡)𝑥(𝑡),2𝑥^⊤(𝑡−𝜏)𝐶^⊤𝑋𝐵(𝐷𝐵)⁻¹𝐷Δ𝐴_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡))≤𝑥^⊤(𝑡−𝜏)𝐶^⊤𝑋𝐵(𝐵^⊤𝑋𝐵)⁻¹𝐵^⊤𝑋𝐶𝑥(𝑡−𝜏)+𝑥^⊤(𝑡−ℎ(𝑡))Δ𝐴^⊤_𝑑(𝑡)𝑋Δ𝐴_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡)). (4.8)Substituting (4.7) and (4.8) into (4.5), we obtain𝐿𝑉(𝑥(𝑡))≤𝑋^⊤(𝑡)Ξ𝑋(𝑡), (4.9)where Ξ=Ξ₁+𝑀₁𝐹₁(𝑡)𝑁₁+𝑁^⊤₁𝐹₁(𝑡)𝑀^⊤₁+𝑀₁𝐹₂(𝑡)𝑁₂+𝑁^⊤₂𝐹₂(𝑡)𝑀^⊤₁+𝑀₂𝐹₁(𝑡)𝑁₁+𝑁^⊤₁𝐹₁(𝑡)𝑀^⊤₂+𝑀₂𝐹₂(𝑡)𝑁₂+𝑁^⊤₂𝐹₂(𝑡)𝑀^⊤₂, 𝑋=[𝑥^⊤(𝑡),𝑥^⊤(𝑡−𝜏),𝑥^⊤(𝑡−ℎ(𝑡))]^⊤, Ξ₁=Ξ₁₁−(𝐴+𝐵𝐾)^⊤𝑋𝐶𝐸^⊤𝑋𝐴_𝑑+𝑁^⊤_𝐸𝑑𝑇₆𝑁_𝐸𝑑∗Ξ₂₂−𝐶^⊤𝑋𝐴_𝑑∗∗Ξ₃₃,Ξ₁₁=𝐸^⊤𝑋(𝐴+𝐵𝐾)+(𝐴+𝐵𝐾)^⊤𝑋𝐸+𝑁^⊤_𝐸𝑇₆𝑁_𝐸+2Δ𝐴^⊤(𝑡)𝑋Δ𝐴(𝑡)+𝑄₁+𝑄₂+2𝐸^⊤𝑋𝐵(𝐵^⊤𝑋𝐵)⁻¹𝐵^⊤Ξ𝑋𝐸,₂₂=−𝑄₁+2𝐶^⊤𝑋𝐵(𝐵^⊤𝑋𝐵)⁻¹𝐵^⊤Ξ𝑋𝐶,₃₃=−(1−ℎ_𝐷)𝑄₂+2Δ𝐴^⊤_𝑑(𝑡)𝑋Δ𝐴_𝑑(𝑡)+𝑁^⊤_𝐸𝑑𝑇₆𝑁_𝐸𝑑,𝑀₁=𝐸^⊤00𝑋𝑀,𝑀₂=0−𝐶^⊤0𝑋𝑀,𝑁₁=𝑁_𝐴00,𝑁₂=00𝑁_𝐴𝑑. (4.10)Using Lemmas 2.1 and 2.2, we haveΞ≤Ξ^′=Ξ^′₁₁Π₁₂Π₁₃𝐸^⊤𝑋𝐵𝐸^⊤𝑋𝑀𝐸^⊤𝑋𝑀00000∗Π₂₂−𝐶^⊤𝑋𝐴_𝑑000𝐶^⊤𝑋𝐵𝐶^⊤𝑋𝑀𝐶^⊤𝑋𝑀00∗∗Ξ^′₃₃𝐵00000000∗∗∗−^⊤𝑋𝐵20000000∗∗∗∗−𝑇₁000000∗∗∗∗∗−𝑇₂𝐵00000∗∗∗∗∗∗−^⊤𝑋𝐵20000∗∗∗∗∗∗∗−𝑇₃000∗∗∗∗∗∗∗∗−𝑇₄𝑋00∗∗∗∗∗∗∗∗∗−20𝑋∗∗∗∗∗∗∗∗∗0−2+𝑀₃𝐹(𝑡)𝑁₃+𝑁^⊤₃𝐹^⊤(𝑡)𝑀^𝑇₃,Ξ^′₁₁=𝐸^⊤𝑋(𝐴+𝐵𝐾)+(𝐴+𝐵𝐾)^⊤𝑋𝐸+𝑄₁+𝑄₂+𝑁^⊤_𝐴(𝑇₁+𝑇₃)𝑁_𝐴+𝑁^⊤_𝐸𝑇₆𝑁_𝐸,Ξ^′₃₃=−(1−ℎ_𝐷)𝑄₁+𝑁_𝐴𝑑(𝑇₂+𝑇₄)𝑁_𝐴𝑑+𝑁^⊤_𝐸𝑑𝑇₆𝑁_𝐸𝑑,𝑀₃=𝑁_𝐴000000000000𝑁_𝐴𝑑00000000^⊤,𝐹𝐹(𝑡)=^⊤1(𝑡)00𝐹^⊤₂,𝑁(𝑡)₃=000000000𝑀^⊤𝑋00000000000𝑀^⊤𝑋. (4.11)With Lemma 2.1, we can see that Ξ^′<0 is equivalent to LMIs (4.1)–(4.3).
According to Ito's formula, system (2.1) is robustly stochastically stable. This completes the proof.

5. Sliding Mode Control

We now design an SMC law such that the reachability of the specified switching surface is ensured.

Theorem 5.1. Consider the uncertain stochastic time delay system (2.1). Suppose that the switching function is given as (3.1) with 𝐷=𝐵^⊤𝑋, where 𝑋 is the solution of LMIs (4.1)–(4.3). Then the reachability of the sliding surface 𝑠(𝑡)=0 can be guaranteed by the following SMC law:𝑢(𝑡)=𝐾𝑥(𝑡)−𝜌(𝑡)𝑠𝑔𝑛(𝑠(𝑡)), (5.1)where the switching gain 𝜌(𝑡) is given as𝜌(𝑡)=𝜆+‖(𝐷𝐵)⁻¹𝐷𝑀‖×(‖𝑁_𝐴𝑥(𝑡)‖+‖𝑁_𝐴𝑑𝑥(𝑡−ℎ(𝑡))‖) (5.2)with 𝜆>0.

Proof. A Lyapunov functional candidate 𝑉(𝑡) is defined as1𝑉(𝑡)=2𝑆^⊤(𝑡)(𝐷𝐵)⁻¹𝑆(𝑡). (5.3)Hence we have𝑉^Δ(𝑡)=𝑆^⊤(𝑡)(𝐷𝐵)^Δ−1𝑆(𝑡)=𝑆^⊤(𝑡)(𝐷𝐵)⁻¹𝐷[Δ𝐴(𝑡)𝑥(𝑡)+Δ𝐴_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡))−𝐵𝜌(𝑡)sgn(𝑠(𝑡))]≤‖𝑆(𝑡)‖‖(𝐷𝐵)⁻¹𝐷𝑀‖×(‖𝑁_𝐴𝑥(𝑡)‖+‖𝑁_𝐴𝑑𝑥(𝑡−ℎ(𝑡))‖)−𝜌(𝑡)‖𝑆(𝑡)‖≤−𝜆‖𝑆(𝑡)‖<0for‖𝑆(𝑡)‖≠0. (5.4)This completes the proof.

6. An Illustrative Example

Consider neutral stochastic systems (2.1) with𝐴=,2.31.223.4,𝐵=1001,𝐶=0.10.300.1𝑀=0.200.3−0.01,𝐸=𝐼,𝐴_𝑑=,𝑁0.500.20.3_𝐴=0.100.10.2,𝑁_𝐴𝑑=−0.100.30.3,𝑁_𝐸=,𝑁0.40−0.010.4_𝐸𝑑=0.2100.1−0.1,ℎ(𝑡)=0.1sin𝑡. (6.1)We select matrix 𝐾=[^{−11.3000−1.2000}_{−2.0000−12.4000}]. Using Matlab LMI control toolbox to solve the LMIs (4.1)–(4.3), we obtain the following:𝑋=0.1752−0.0038−0.00380.1985,𝑄₁=1.1255−0.0186−0.01861.2138,𝑄₂=,𝑇1.0613−0.0605−0.06051.2145₁=1.16700.00010.00011.1623,𝑇₂=1.1661−0.0007−0.00071.1652,𝑇₃=,𝑇1.16700.00010.00011.1623₄=1.16470.00250.00251.1700,𝑇₅=1.16700.00010.00011.1623,𝑇₆=.1.1580−0.0046−0.00461.1705 (6.2)

7. Conclusions

In this paper, we have investigated the sliding mode control problem for uncertain stochastic neutral systems with multiple delays. The stability criteria are expressed by means of LMIs, which can be readily tested by some standard numerical packages. Therefore, the developed result is practical.

Acknowledgments

This work is supported in part by National Science Foundation of China (60474031), NCET (04-0383), National 973 Key Fundamental Research Program (2002cb312200-3), and Australia-China Special Fund for Scientific and Technological Cooperation.