𝐻_∞ Control for Stochastic Systems with Markovian Switching and Time-Varying Delay via Sliding Mode Design

Abstract

This paper addresses the problem of 𝐻_∞ 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.

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 robust 𝐻_∞ 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∈𝑅^𝑛, 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 {𝑟_𝑡,𝑡≥0} represents the switching between the different modes taking values in a finite state space 𝑆={1,2,…,𝑁} with generator 𝜋=(𝜋_𝑖𝑗)_𝑁×𝑁 given by Pr𝑟_𝑡+Δ=𝑗∣𝑟_𝑡=𝜋=𝑖^𝑖𝑗Δ+o(Δ),if𝑖≠𝑗1+𝜋_𝑖𝑖Δ+o(Δ),if𝑖=𝑗, (1)

where 𝜋_𝑖𝑗 is the transition rate from mode 𝑖 to 𝑗 and satisfies the following relations: 𝜋_𝑖𝑗≥0,𝜋_𝑖𝑖=−_𝑗≠𝑖𝜋_𝑖𝑗, (2)

and o(Δ) is such that lim_Δ→0o(Δ)/Δ=0.

Consider the following stochastic system with Markovian switching and time-varying delay: 𝑑𝑥(𝑡)=𝐴𝑟_𝑡+𝐴𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑟_𝑡+𝐴_𝑑𝑟_𝑡𝑥((𝑡𝑟𝑡−𝜏))+𝐵_𝑡𝑢(𝑡)+𝑓𝑡,𝑥(𝑡),𝑟_𝑡+𝐵_𝑣𝑟_𝑡𝑣(𝑡𝑟)𝑑𝑡+𝐷_𝑡𝐶𝑟_𝑡+𝐶𝑟_𝑡𝑥(𝑡)+𝐶_𝑑𝑟_𝑡+𝐶_𝑑𝑟_𝑡𝑥((𝑡(𝑡),𝑧(𝑡)=𝐸𝑟𝑡−𝜏))𝑑𝜔_𝑡+𝐸𝑟_𝑡𝑥(𝑡)+𝐹𝑟_𝑡𝑣(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈[−𝑑,0], (3)

where 𝑥(𝑡)∈𝑅^𝑛 is the system state, 𝑢(𝑡)∈𝑅^𝑚 is the control input, 𝑧(𝑡)∈𝑅^𝑝 is the controlled output, 𝑣(𝑡)∈𝑅^𝑞 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(𝑟_𝑡), 𝐴_𝑑(𝑟_𝑡), 𝐵(𝑟_𝑡), 𝐵_𝑣(𝑟_𝑡), 𝐶(𝑟_𝑡), 𝐶_𝑑(𝑟_𝑡), 𝐷(𝑟_𝑡), 𝐸(𝑟_𝑡), and 𝐹(𝑟_𝑡) are known real constant matrices with appropriate dimensions. In general, it is assumed that the pair (𝐴(𝑟_𝑡),𝐵(𝑟_𝑡)) is controllable and the input matrix 𝐵(𝑟_𝑡) has full column rank. 𝜏(𝑡) is the time-varying delay satisfying 0<𝜏(𝑡)≤𝑑<∞,𝜏̇(𝑡)≤ℎ<1, (4)

where 𝑑 and ℎ are known real constant scalars, and 𝜑(𝑡) is a continuous vector-valued initial function. 𝐴(𝑟_𝑡), 𝐴_𝑑(𝑟_𝑡), 𝐶(𝑟_𝑡), 𝐶_𝑑(𝑟_𝑡), and 𝐸(𝑟_𝑡) are parameter uncertainties, and unknown function 𝑓(𝑡,𝑥(𝑡),𝑟_𝑡) is an unknown nonlinear uncertainties.

For each 𝑟_𝑡=𝑖∈𝑆, for the sake of convenience, denote 𝐴(𝑟_𝑡)=𝐴_𝑖 and 𝐴(𝑟_𝑡)=𝐴_𝑖(𝑡). Other matrices are defined as above. Then system (3) becomes 𝑑𝑥(𝑡)=𝐴_𝑖+𝐴_𝑖(𝑡)𝑥(𝑡)+𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡))+𝐵_𝑖𝑢(𝑡)+𝑓(𝑡,𝑥(𝑡),𝑖),+𝐵_𝑣𝑖𝑣(𝑡)𝑑𝑡+𝐷_𝑖𝐶_𝑖+𝐶_𝑖(𝑡)𝑥(𝑡),+𝐶_𝑑𝑖+𝐶_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡))𝑑𝜔(𝑡),𝑧(𝑡)=𝐸_𝑖+𝐸_𝑖(𝑡)𝑥(𝑡)+𝐹_𝑖𝑣(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈[−𝑑,0]. (5)(6)(7)

For 𝑟_𝑡=𝑖, the admissible uncertainties are assumed to be norm bounded and can be described as 𝐴_𝑖(𝑡)𝐴_𝑑𝑖(𝑡)𝐸_𝑖(𝑡)=𝑁_1𝑖𝐹_1𝑖(𝑡)𝐻_𝑎𝑖𝐻_𝑎𝑑𝑖𝐻_𝑒𝑖,𝐶_𝑖(𝑡)𝐶_𝑑𝑖(𝑡)=𝑁_2𝑖𝐹_2𝑖(𝑡)𝐻_𝑐𝑖𝐻_𝑐𝑑𝑖,𝑓(𝑡,𝑥(𝑡),𝑖)≤𝜂_𝑖‖𝑥(𝑡)‖, (8)(9)

where 𝜂_𝑖>0 is a constant scalar, 𝐻_𝑎𝑖, 𝐻_𝑎𝑑𝑖, 𝐻_𝑐𝑖, 𝐻_𝑐𝑑𝑖, 𝐻_𝑒𝑖, 𝑁_1𝑖, and 𝑁_2𝑖 are known real constant matrices, and 𝐹_1𝑖(𝑡) and 𝐹_2𝑖(𝑡) are unknown matrix functions satisfying 𝐹^𝑇_1𝑖(𝑡)𝐹_1𝑖(𝑡)≤𝐼,𝐹^𝑇_2𝑖(𝑡)𝐹_2𝑖(𝑡)≤𝐼,∀𝑡. (10)

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)𝐸^∞₀𝑥^𝑇(𝑡)𝑥(𝑡)𝑑𝑡∣𝑥₀,𝜎(0)=𝑟₀≤𝑥^𝑇₀𝐺𝑥₀. (11)

Definition 2. For given scalar 𝛾>0, the system (5)–(7) is said to stochastically stabilizable and satisfy ‖𝑇_𝑧𝑣(𝑠)‖_∞≤𝛾 if the closed-loop system is stochastically stable for 𝑣(𝑡)=0 and for all mismatched uncertainties, ‖𝑧(𝑡)‖_𝐸2<𝛾‖𝑣(𝑡)‖_𝐸2, (12)
holds, where ‖𝑧(𝑡)‖_𝐸2=(𝐸{∫^𝑡₀|𝑧(𝑡)|²𝑑𝑡})^1/2.

Definition 3. Let 𝐶^2,1(𝑅^𝑛×[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)∈𝐶^2,1(𝑅^𝑛×[𝑡₀−𝑟,∞]×𝑆;𝑅⁺), define an infinitesimal operator 𝐿𝑉 from 𝑅^𝑛×[𝑡₀−𝑟,∞]×𝑆 to 𝑅⁺ as follows: 𝐿𝑉(𝑥,𝑡,𝑖)=𝐿𝑉(𝑥,𝑡,𝑖)=lim_Δ→01Δ𝐸𝑉𝑥(𝑡+Δ),𝑟_𝑡+Δ∣𝑥(𝑡),𝑡+Δ=𝑥,𝑟_𝑡=𝑖,𝑡−𝑉(𝑥,𝑖,𝑡). (13)
The following matrix inequalities will be essential for the proofs in Section 3.

Lemma 4 (see [18]). Let 𝑄, 𝐻, 𝐹, and 𝐺 be real matrices of appropriate dimensions with 𝐹^𝑇𝐹≤𝐼, then, for any real matrix 𝑄=𝑄^𝑇, there exists scalar 𝜀>0, and one has the following: 𝑄+𝐻𝐹(𝑡)𝐺+𝐺^𝑇𝐹^𝑇(𝑡)𝐻^𝑇(𝑡)<0,∀𝐹𝑠.𝑡.𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼, (14)
if and only if there exists some scalar 𝜀>0 such that 𝑄+𝜀𝐻𝐻^𝑇+𝜀⁻¹𝐺^𝑇𝐺<0. (15)

Lemma 5 (see [18]). For any real vectors 𝑎,𝑏 and matrix 𝑋>0 of compatible dimensions 𝑎^𝑇𝑏+𝑏^𝑇𝑎≤𝑎^𝑇𝑋𝑎+𝑏^𝑇𝑋⁻¹𝑏. (16)

3. Main Results

This section constructs an SMC law 𝑢(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: 𝑠(𝑥(𝑡),𝑖)=𝐵^𝑇_𝑖𝑋_𝑖𝑥(𝑡)−^𝑡₀𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖𝑥(𝜏)𝑑𝜏, (17)

for each 𝑟_𝑡=𝑖∈𝑆. In (17), the matrix 𝐾_𝑖∈𝑅^𝑚×𝑛 is chosen such that the matrix 𝐴_𝑖+𝐵_𝑖𝐾_𝑖 is Hurwitz. In particular, the matrix 𝑋_𝑖∈𝑅^𝑛×𝑛 is to be designed so that 𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖 is nonsingular and 𝐵^𝑇_𝑖𝑋_𝑖𝐷_𝑖=0. Under the assumption that 𝐵_𝑖 is full rank, it can be easily shown that the nonsingularity of 𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖 can be guaranteed if 𝑋_𝑖 is symmetric positive definite, that is, 𝑋_𝑖>0. In addition, the condition 𝐵^𝑇_𝑖𝑋_𝑖𝐷_𝑖=0 will be incorporated in Theorem 6.

The solution 𝑥(𝑡) of the stochastic system (5)–(7) is expressed as follows: 𝑥(𝑡)(0)+=𝜑_𝑡0𝐴_𝑖+𝐴_𝑖(𝑠)𝑥(𝑠)+𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑠)𝑥(𝑠−𝜏(𝑠))+𝐵_𝑖𝑢(𝑠)+𝑓(𝑠,𝑥(𝑠),𝑖)+𝐵_𝑣𝑖𝑣(𝑠+)𝑑𝑠_𝑡0𝐷_𝑖𝐶_𝑖+𝐶_𝑖(𝑠)𝑥(𝑠)+𝐶_𝑑𝑖+𝐶_𝑑𝑖(𝑠)𝑥(𝑠−𝜏(𝑠))𝑑𝜔(𝑠). (18)

Under the condition that 𝐵^𝑇_𝑖𝑋_𝑖𝐷_𝑖=0, it can be obtained from (17) and (18) that 𝑠(𝑥(𝑡),𝑖)=𝐵^𝑇_𝑖𝑋_𝑖𝜑(0)+^𝑡₀𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑖+𝐴_𝑖(𝑠)𝑥(𝑠)+𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑠)𝑥(𝑠−𝜏(𝑠))+𝐵_𝑖𝑢(𝑠)+𝑓(𝑠,𝑥(𝑠),𝑖)+𝐵_𝑣𝑖𝑣(𝑠)𝑑𝑠. (19)

According to the sliding mode theory, we have 𝑠(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 𝑢_eq(𝑡)=𝐾_𝑖𝑥(𝑡)−𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖×𝐴_𝑖(𝑡)𝑥(𝑡)+𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡))+𝐵_𝑣𝑖𝑣(𝑡)−𝑓(𝑡,𝑥(𝑡),𝑖). (20)

Substituting (20) into (5), the sliding mode dynamics in 𝑠(𝑥(𝑡),𝑖)=0 can be written as 𝑑𝑥(𝑡)=𝐴_𝑖+𝐵_𝑖𝐾_𝑖+𝐴_𝑖(𝑡)−𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑖(𝑡𝑥(𝑡)+𝐴_𝑑𝑖)+𝐴_𝑑𝑖(𝑡)−𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹×𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)𝑥((𝑡+𝐵𝑡−𝜏))_𝑣𝑖𝑣(𝑡)−𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑣𝑖𝑣(𝑡)𝑑𝑡+𝐷_𝑖𝐶_𝑖+𝐶_𝑖(𝑡)𝑥(𝑡)+𝐶_𝑑𝑖+𝐶_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡))𝑑𝜔(𝑡). (21)

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 𝑋_𝑖>0 and 𝑄>0, scalars 𝜀_1𝑖>0, 𝜀_2𝑖>0, 𝜀_3𝑖>0, and 𝜀_4𝑖>0 (𝑖=1,2,…,𝑁) satisfy the following LMIs: ^{Π1𝑖Π∗∗∗∗∗∗𝑇5𝑖Π2𝑖𝐵∗∗∗∗∗𝑇𝑣𝑖𝑋𝑖0Π3𝑖𝑋∗∗∗∗𝑖𝐷𝑖𝐶𝑖𝑋𝑖𝐷𝑖𝐶𝑑𝑖0−𝑋𝑖𝑁∗∗∗𝑇2𝑖𝐷𝑇𝑖𝑋𝑖𝐷𝑖𝐶𝑖𝑁𝑇2𝑖𝐷𝑇𝑖𝑋𝑖𝐷𝑖𝐶𝑑𝑖00Π4𝑖𝐸∗∗𝑖0𝐹𝑖00−𝐼∗000000−𝑋𝑖0𝑋𝑖𝐴𝑇𝑑𝑖√000002𝐵𝑇𝑖𝑋𝑖𝐵000000𝑇𝑖𝑋𝑖𝑁000000𝑇1𝑖𝑋𝑖𝑁000000𝑇1𝑖𝑋𝑖00000000000𝑁𝑇1𝑖}𝐵^{𝑋𝑖0000000𝑁𝑇1𝑖𝑋𝑖0000000∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−𝑋𝑖∗∗∗∗∗∗∗0−𝐵𝑇𝑖𝑋𝑖𝐵𝑖∗∗∗∗∗∗00−𝐵𝑇𝑖𝑋𝑖𝐵𝑖∗∗∗∗∗000−𝜀1𝑖𝐼∗∗∗∗0000−𝜀2𝑖𝐼∗∗∗00000−𝜀4𝑖𝐼∗∗000000−𝜀4𝑖𝑁𝐼∗𝑇1𝑖𝑋𝑖000000−𝜀4𝑖𝐼,𝑇}_𝑖𝑋_𝑖𝐷_𝑖=0, (22)(23)
with Π_1𝑖=𝑋_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖+𝐴_𝑖+𝐵_𝑖𝐾_𝑖^𝑇𝑋_𝑖+𝑄+𝜀_1𝑖𝐻^𝑇_𝑎𝑖𝐻_𝑎𝑖+𝜀_3𝑖𝐻^𝑇_𝑐𝑖𝐻_𝑐𝑖+𝜀_4𝑖𝐻^𝑇_𝑎𝑖𝐻_𝑎𝑖+𝜀_4𝑖𝐻^𝑇_𝑒𝑖𝐻_𝑒𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝑋_𝑗,Π_2𝑖=−(1−ℎ)𝑄+𝜀_2𝑖𝐻^𝑇_𝑎𝑑𝑖𝐻_𝑎𝑑𝑖+𝜀_3𝑖𝐻^𝑇_𝑐𝑖𝐻_𝑐𝑖+𝜀_4𝑖𝐻^𝑇_𝑎𝑑𝑖𝐻_𝑎𝑑𝑖,Π_3𝑖=𝐵^𝑇_𝑣𝑖𝑋_𝑖𝐵_𝑣𝑖−𝛾²Π𝐼,_4𝑖=−𝜀_3𝑖𝐼+𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖,Π_5𝑖=𝑋_𝑖𝐴_𝑑𝑖+𝜀_3𝑖𝐻^𝑇_𝑐𝑖𝐻_𝑐𝑑𝑖, (24)
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: 𝑉(𝑥(𝑡),𝑖)=𝑥^𝑇(𝑡)𝑋_𝑖𝑥(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠. (25)
By Definition 2, 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∈𝑆𝐿𝑉(𝑥(𝑡),𝑖)=2𝑥^𝑇(𝑡)𝑋_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖+𝐴_𝑖(𝑡)−𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑖(𝑡)𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑋_𝑖×𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)−𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)𝑥((𝑡+𝐶𝑡−𝜏))_𝑖+𝐶_𝑖(𝑡)𝑥(𝑡)+𝐶_𝑑𝑖+𝐶_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡))^𝑇×𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖×𝐶_𝑖+𝐶_𝑖(𝑡)𝑥(𝑡)+𝐶_𝑑𝑖+𝐶_𝑑𝑖(𝑡)𝑥((𝑡+𝑡−𝜏))_𝑁𝑗=1𝜋_𝑖𝑗𝑥^𝑇(𝑡)𝑋_𝑗𝑥(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−(1−𝜏̇(𝑡))𝑥^𝑇(𝑡−𝜏(𝑡))𝑄𝑥(𝑡−𝜏(𝑡)). (26)
Using Lemma 5, one has 2𝑥^𝑇(𝑡)𝑋_𝑖𝐴_𝑖(𝑡)𝑥(𝑡)≤𝜀⁻¹_1𝑖𝑥^𝑇(𝑡)𝑋_𝑖𝑁_1𝑖𝑁^𝑇_1𝑖𝑋_𝑖𝑥(𝑡)+𝜀_1𝑖𝑥^𝑇(𝑡)𝐻^𝑇_𝑎𝑖𝐻_𝑎𝑖𝑥(𝑡),2𝑥^𝑇(𝑡)𝑋_𝑖𝐴_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡))≤𝜀⁻¹_2𝑖𝑥^𝑇(𝑡)𝑋_𝑖𝑁_1𝑖𝑁^𝑇_1𝑖𝑋_𝑖𝑥(𝑡)+𝜀_2𝑖𝑥^𝑇(𝑡−𝜏(𝑡))𝐻^𝑇_𝑎𝑑𝑖𝐻_𝑎𝑑𝑖𝑥(𝑡−𝜏(𝑡)),−2𝑥^𝑇(𝑡)𝑋_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑖(𝑡)𝑥(𝑡)≤𝑥^𝑇(𝑡)𝑋_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝐴^𝑇_𝑖(𝑡)𝑋_𝑖𝐴_𝑖(𝑡)𝑥(𝑡),−2𝑥^𝑇(𝑡)𝑋_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡))≤𝑥^𝑇(𝑡)𝑋_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡−𝜏(𝑡)𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)^𝑇×𝑋_𝑖𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡)). (27)
From (22), we can obtain 𝜀_3𝑖𝐼−𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖>0. According to assumption (8), it has 𝐷_𝑖𝐶_𝑖+𝐷_𝑖𝑁_2𝑖𝐹_2𝑖(𝑡)𝐻_𝑐𝑖^𝑇𝑋_𝑖𝐷_𝑖𝐶_𝑖+𝐷_𝑖𝑁_2𝑖𝐹_2𝑖(𝑡)𝐻_𝑐𝑖≤𝐶^𝑇_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑖+𝐶^𝑇_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖×𝜀_3𝑖𝐼−𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖⁻¹𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑖+𝜀_3𝑖𝐻^𝑇_𝑐𝑖𝐻_𝑐𝑖, (28)
where 𝐶_𝑖=𝐶_𝑖𝐶_𝑑𝑖, 𝐻_𝑐𝑖=𝐻_𝑐𝑖𝐻_𝑐𝑑𝑖. Define Φ=[𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−𝜏(𝑡))]^𝑇. Then it follows from (26)–(28) that 𝐿𝑉(𝑥(𝑡),𝑖)≤Φ^𝑇Γ_𝑖Φ, (29)
in which Γ_𝑖=Ξ_1𝑖𝑋_𝑖𝐴_𝑑𝑖𝐴^𝑇_𝑑𝑖𝑋_𝑖Ξ_2𝑖+𝐶^𝑇_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑖+𝐶^𝑇_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖𝜀_3𝑖𝐼−𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖⁻¹×𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑖+𝜀_3𝑖𝐻^𝑇_𝑐𝑖𝐻_𝑐𝑖, (30)
with Ξ_1𝑖=𝑋_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖+𝐴_𝑖+𝐵_𝑖𝐾_𝑖^𝑇𝑋_𝑖+𝑄+𝜀_1𝑖𝐻^𝑇_𝑎𝑖𝐻_𝑎𝑖+𝜀⁻¹_1𝑖𝑋_𝑖𝑁_1𝑖𝑁^𝑇_1𝑖𝑋_𝑖+𝜀⁻¹_2𝑖𝑋_𝑖𝑁_1𝑖𝑁^𝑇_1𝑖𝑋_𝑖+𝐴^𝑇_𝑖(𝑡)𝑋_𝑖𝐴_𝑖(𝑡)+2𝑋_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖,Ξ_2𝑖=−(1−ℎ)𝑄+𝜀_2𝑖𝐻^𝑇_𝑎𝑑𝑖𝐻_𝑎𝑑𝑖+𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)^𝑇𝑋_𝑖𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡). (31)
So, if Γ_𝑖<0, for any Φ≠0, we have 𝐿𝑉(𝑥(𝑡),𝑖)<0. (32)
According to Definition 1, the sliding mode dynamics (21) is stochastically stable.
By Schur’s complement, Γ_𝑖<0 is equivalent to the following LMI: +𝐺𝐹(^{Ξ∗1𝑖Π∗∗∗∗∗∗∗∗𝑇5𝑖Ξ∗2𝑖𝑋∗∗∗∗∗∗∗𝑖𝐷𝑖𝐶𝑖𝑋𝑖𝐷𝑖𝐶𝑑𝑖−𝑋𝑖𝐸∗∗∗∗∗∗𝑇2𝑖𝐷𝑇𝑖𝑋𝑖𝐷𝑖𝐶𝑖𝐸𝑇2𝑖𝐷𝑇𝑖𝑋𝑖𝐷𝑖𝐶𝑑𝑖0Π3𝑖∗∗∗∗∗}_{0000−𝑋𝑖∗∗∗∗0𝑋𝑖𝐴𝑑𝑖000−𝑋𝑖√∗∗∗2𝐵𝑇𝑖𝑋𝑖00000−𝐵𝑇𝑖𝑋𝑖𝐵𝑖𝑁∗∗𝑇1𝑖𝑋𝑖000000−𝜀1𝑖𝑁𝐼∗𝑇1𝑖𝑋𝑖0000000−𝜀2𝑖𝐼}𝑡)𝐻+𝐻^𝑇𝐹^𝑇(𝑡)𝐺^𝑇<0, (33)
where Ξ^∗_1𝑖=𝑋_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖+𝐴_𝑖+𝐵_𝑖𝐾_𝑖^𝑇𝑋_𝑖+𝑄+𝜀_1𝑖𝐻^𝑇_𝑎𝑖𝐻_𝑎𝑖+𝜀_3𝑖𝐻^𝑇_𝑐𝑖𝐻_𝑐𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝑋_𝑗,Ξ^∗_2𝑖=−(1−ℎ)𝑄+𝜀_2𝑖𝐻^𝑇_𝑎𝑑𝑖𝐻_𝑎𝑑𝑖+𝜀_3𝑖𝐻^𝑇_𝑐𝑑𝑖𝐻_𝑐𝑑𝑖,𝐻𝐺=^𝑎𝑖000000000𝐻_𝑎𝑑𝑖0000000^𝑇,𝐹(𝑡)=𝐹^𝑇_1𝑖(𝑡)00𝐹^𝑇_2𝑖(𝑡),𝐻=0000𝑁^𝑇_1𝑖𝑋_𝑖000000000𝑁^𝑇_1𝑖𝑋_𝑖.000 (34)
According to Lemma 4, there exists positive scalar 𝜀_4𝑖 such that the following LMI holds:Ξ_3𝑖Π∗∗∗∗∗∗∗∗∗∗_𝑇5𝑖Ξ_4𝑖𝑋∗∗∗∗∗∗∗∗∗_𝑖𝐷_𝑖𝐶_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑑𝑖−𝑋_𝑖𝐸∗∗∗∗∗∗∗∗_𝑇2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑖𝐸^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑑𝑖0Π_3𝑖∗∗∗∗∗∗∗0000−𝑋_𝑖∗∗∗∗∗∗0𝑋_𝑖𝐴_𝑑𝑖000−𝑋_𝑖√∗∗∗∗∗2𝐵^𝑇_𝑖𝑋_𝑖00000−𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖𝑁∗∗∗∗_𝑇1𝑖𝑋_𝑖000000−𝜀_1𝑖𝑁𝐼∗∗∗_𝑇1𝑖𝑋_𝑖0000000−𝜀_2𝑖𝐼∗∗0000𝑁^𝑇_1𝑖𝑋_𝑖0000−𝜀_4𝑖𝐼∗00000𝑁^𝑇_1𝑖𝑋_𝑖0000−𝜀_4𝑖𝐼<0, (35)
with Ξ_3𝑖=𝑋_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖+𝐴_𝑖+𝐵_𝑖𝐾_𝑖^𝑇𝑋_𝑖+𝑄+𝜀_1𝑖𝐻^𝑇_𝑎𝑖𝐻_𝑎𝑖+𝜀_3𝑖𝐻^𝑇_𝑐𝑖𝐻_𝑐𝑖+𝜀_4𝑖𝐻^𝑇_𝑎𝑖𝐻_𝑎𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝑋_𝑗,Ξ_4𝑖=−(1−ℎ)𝑄+𝜀_2𝑖𝐻^𝑇_𝑎𝑑𝑖𝐻_𝑎𝑑𝑖+𝜀_3𝑖𝐻^𝑇_𝑐𝑑𝑖𝐻_𝑐𝑑𝑖+𝜀_4𝑖𝐻^𝑇_𝑎𝑑𝑖𝐻_𝑎𝑑𝑖. (36)
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 ‖𝑧(𝑡)‖_𝐸2<𝛾‖𝑣(𝑡)‖₂, (37)
for all nonzero 𝑣(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: 𝐿𝑉(𝑥(𝑡),𝑖)=2𝑥^𝑇(𝑡)𝑋_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖+𝐴_𝑖(𝑡)−𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑖(𝑡)𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑋_𝑖𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)−𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑑𝑖+𝐴_𝑑𝑖(𝑡)×𝑥(𝑡−𝜏(𝑡))+2𝑥^𝑇(𝑡)𝑋_𝑖×𝐼−𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑣𝑖𝑣(𝑡)+𝐶_𝑖+𝐶_𝑖(𝑡)𝑥(𝑡)+𝐶_𝑑𝑖+𝐶_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡))^𝑇×𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖×𝐶_𝑖+𝐶_𝑖(𝑡)𝑥(𝑡)+𝐶_𝑑𝑖+𝐶_𝑑𝑖(𝑡)𝑥((𝑡+𝑡−𝜏))_𝑁𝑗=1𝜋_𝑖𝑗𝑥^𝑇(𝑡)𝑋_𝑗𝑥(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−(1−𝜏̇(𝑡))𝑥^𝑇(𝑡−𝜏(𝑡))𝑄𝑥(𝑡−𝜏(𝑡)). (38)
Let 𝐽(𝑡)≜𝐸^𝑡_𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠. (39)
So, under zero initial condition, we have 𝑉(𝑥(𝑡₀),𝑖)=0 for 𝑥(𝑡₀)=0. From Dynkin’s formula, one has 𝐸{𝑉(𝑥(𝑡),𝑡)}=𝐸^𝑡_𝑡0𝐿𝑉(𝑥(𝑠),𝑠)𝑑𝑠. (40)
Then, similar to the (26)–(29), we can obtain 𝐽(𝑡)=𝐸^𝑡_𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)(𝑥(𝑠){𝑉(𝑥(𝑡)+𝐿𝑉,𝑠)𝑑𝑠−𝐸,𝑡)}≤𝐸^𝑡₀𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)+𝐿𝑉(𝑥(𝑠),𝑠)𝑑𝑠=𝐸^𝑡₀𝑥^𝑇(𝑠)𝑥^𝑇(𝑠−𝜏(𝑠))𝑣^𝑇(𝑠)×Ω_𝑖𝑥^𝑇(𝑠)𝑥^𝑇(𝑠−𝜏(𝑠))𝑣^𝑇(𝑠)^𝑇𝑑𝑠, (41)
where Ω_𝑖=Θ_1𝑖Θ_3𝑖𝑋_𝑖𝐵_𝑣𝑖+𝐸_𝑖+𝐸_𝑖^𝑇𝐹_𝑖Θ^𝑇_3𝑖Θ_2𝑖0𝐵^𝑇_𝑣𝑖𝑋_𝑖+𝐹^𝑇_𝑖𝐸_𝑖+𝐸_𝑖0𝐵^𝑇_𝑣𝑖𝑋_𝑖𝐵_𝑣𝑖+𝐹^𝑇_𝑖𝐹_𝑖−𝛾²𝐼, (42)
with Θ_1𝑖=Ξ_1𝑖+𝑋_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖+𝐸_𝑖+𝐸_𝑖^𝑇𝐸_𝑖+𝐸_𝑖+𝐶^𝑇_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑖+𝐶^𝑇_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖×𝜀_3𝑖𝐼−𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖⁻¹𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑖,Θ_2𝑖=Ξ_2𝑖+𝐶^𝑇_𝑑𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑑𝑖+𝐶^𝑇_𝑑𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖×𝜀_3𝑖𝐼−𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖⁻¹𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑑𝑖,Θ_3𝑖=𝑋_𝑖𝐴_𝑑𝑖+𝐶^𝑇_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖×𝜀_3𝑖𝐼−𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝑁_2𝑖⁻¹×𝑁^𝑇_2𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐶_𝑑𝑖, (43)
and Ξ_1𝑖 and Ξ_2𝑖 are defined as in (30).
It is observed that Ω_𝑖<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 Theorem 6 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 𝐵^𝑇_𝑖𝑋_𝑖𝐷_𝑖=0 with 𝑋_𝑖 satisfying inequality (22) can be equivalently converted to tr[(𝐵^𝑇_𝑖𝑋_𝑖𝐷_𝑖)^𝑇𝐵^𝑇_𝑖𝑋_𝑖𝐷_𝑖]=0.
Consider the following matrix inequality for 𝛼>0: 𝐵^𝑇_𝑖𝑋_𝑖𝐷_𝑖^𝑇𝐵^𝑇_𝑖𝑋_𝑖𝐷_𝑖≤𝛼𝐼,for𝑖∈𝑆. (44)
By Schur’s complement, one has −𝛼𝐼𝐵^𝑇_𝑖𝑋_𝑖𝐷_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝐵_𝑖−𝐼≤0,for𝑖∈𝑆. (45)
Now, the problem is changed to a problem of finding a global solution of the following minimization problem: min𝛼,subjectto(22)and(45). (46)
It can be seen that if the global infimum 𝛼 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 𝑋_𝑖, (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: 𝑢_𝑖(𝑡)=𝐾_𝑖𝑥(𝑡)−𝜇_𝑖𝑠(𝑥(𝑡))−𝐵,𝑖_𝑇𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑑𝑖𝑥((𝑡−1𝑡−𝜏))2^𝑁_𝑗=1𝜋_𝑖𝑗𝐵^𝑇_𝑗𝑋_𝑗𝐵_𝑗⁻¹𝑠(𝑥(𝑡),𝑖)−𝜌_𝑖(𝑡)sgn(𝑠(𝑥(𝑡),𝑖)), (47)
where 𝜌_𝑖(𝑡)=𝜂_𝑖‖𝑥(𝑡)‖+𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝑁_1𝑖×𝐻_𝑎𝑖𝑥(𝑡)+𝐻_𝑎𝑑𝑖𝑥((𝑡+𝐵𝑡−𝜏))_𝑇𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑣𝑖‖𝑣(𝑡)‖, (48)
with 𝜇_𝑖 are a small positive constant for each 𝑖∈𝑆.

Proof. By (19) and (47), and by observing that 𝐵^𝑇_𝑖𝑋_𝑖𝐷_𝑖=0, it follows that 𝑠̇(𝑥(𝑡),𝑖)=𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑖(𝑡)𝑥(𝑡)+𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡))−𝜇_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖𝑠(𝑥(𝑡),𝑖)+𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑣𝑖𝑣(𝑡)−12^𝑁_𝑗=1𝜋_𝑖𝑗𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖𝐵^𝑇_𝑗𝑋_𝑗𝐵_𝑗⁻¹𝑠(𝑥(𝑡),𝑖)−𝜌_𝑖(𝑡)𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖sgn(𝑠(𝑥(𝑡),𝑖))+𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖𝑓(𝑡,𝑥(𝑡),𝑖). (49)
Choose the following Lyapunov function candidate: 𝑉(𝑡)=12𝑠^𝑇(𝑡)𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝑠(𝑡). (50)
Utilizing expression (49) yields 𝑉̇(𝑡)=𝑠^𝑇(𝑥(𝑡),𝑖)𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝑠̇(𝑥(𝑡))+1,𝑖2^𝑁_𝑗=1𝜋_𝑖𝑗𝑠^𝑇(𝑥(𝑡),𝑖)𝐵^𝑇_𝑗𝑋_𝑗𝐵_𝑗⁻¹𝑠(𝑥(𝑡),𝑖)=𝑠^𝑇(𝑥(𝑡),𝑖)𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑖(𝑡)𝑥(𝑡)+𝑠^𝑇(𝑥(𝑡),𝑖)𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑑𝑖(𝑡)𝑥(𝑡−𝜏(𝑡))−𝜇_𝑖𝑠^𝑇(𝑥(𝑡),𝑖)𝑠(𝑥(𝑡),𝑖)−𝜌_𝑖(𝑡)𝑠^𝑇(𝑥(𝑡),𝑖)sgn(𝑠(𝑥(𝑡),𝑖))+𝑠^𝑇(𝑥(𝑡),𝑖)𝑓(𝑡,𝑥(𝑡),𝑖)+𝑠^𝑇(𝑥(𝑡),𝑖)𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑣𝑖𝑣(𝑡)≤‖𝑠(𝑥(𝑡),𝑖)‖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝑁_1𝑖×𝐻_𝑎𝑖𝑥(𝑡)+𝐻_𝑎𝑑𝑖𝑥(𝑡−𝜏(𝑡))−𝜌_𝑖(𝑡)‖𝑠(𝑥(𝑡),𝑖)‖₁+‖𝑠(𝑥(𝑡)×𝐵,𝑖)‖_𝑇𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑣𝑖‖𝑣(𝑡)‖−𝜇_𝑖‖𝑠(𝑥(𝑡),𝑖)‖²+𝜂_𝑖‖𝑠(𝑥(𝑡),𝑖)‖‖𝑥(𝑡)‖. (51)
Noting that ‖𝑠(𝑥(𝑡),𝑖)‖₁≥‖𝑠(𝑥(𝑡),𝑖)‖ and substituting (48) into (51), one has 𝑉̇(𝑡)≤−𝜇_𝑖‖𝑠(𝑥(𝑡),𝑖)‖²<0,for‖𝑠(𝑥(𝑡),𝑖)‖≠0. (52)
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 function sgn(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 𝜂_𝑖 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 𝜂_𝑖 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 𝑋_𝑖, (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: 𝑢_𝑖(𝑡)=𝐾_𝑖𝑥(𝑡)−𝜇_𝑖𝑠(𝑥(𝑡))−𝐵,𝑖_𝑇𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐴_𝑑𝑖𝑥((𝑡−1𝑡−𝜏))2^𝑁_𝑗=1𝜋_𝑖𝑗𝐵^𝑇_𝑗𝑋_𝑗𝐵_𝑗⁻¹𝑠(𝑥(𝑡))−𝜌,𝑖_𝑖(𝑡)sgn(𝑠(𝑥(𝑡),𝑖)), (53)
where 𝜌_𝑖(𝑡)=𝜂_𝑖‖𝑥(𝑡)‖+𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝑁_1𝑖×𝐻_𝑎𝑖𝑥(𝑡)+𝐻_𝑎𝑑𝑖𝑥((𝑡+𝐵𝑡−𝜏))_𝑇𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑣𝑖‖𝑣(𝑡)‖, (54)
and adaptive law as 𝜂̇_𝑖=𝛽_𝑖‖𝑠(𝑡)‖‖𝑥(𝑡)‖, (55)
with 𝜇_𝑖, 𝛽_𝑖 as small positive constants for each 𝑖∈𝑆.

Proof. Choose the following Lyapunov function: 𝑉(𝑡)=12𝑠^𝑇(𝑡)𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝑠(𝑡)+12𝛽_𝑖⁻¹𝜂²_𝑖(𝑡), (56)
where 𝜂_𝑖(𝑡)=𝜂_𝑖(𝑡)−𝜂_𝑖. In line with the proof of Theorem 8, it can easily obtained that 𝑉̇(𝑡)≤‖𝑠(𝑥(𝑡),𝑖)‖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝑁_1𝑖×𝐻_𝑎𝑖𝑥(𝑡)+𝐻_𝑎𝑑𝑖𝑥((𝑡−𝜌𝑡−𝜏))_𝑖(𝑡)‖𝑠(𝑥(𝑡),𝑖)‖₁+‖𝑠(𝑥(𝑡),𝑖)‖𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑖⁻¹𝐵^𝑇_𝑖𝑋_𝑖𝐵_𝑣𝑖‖𝑣(𝑡)‖−𝜇_𝑖‖𝑠(𝑥(𝑡),𝑖)‖²+𝜂_𝑖‖𝑠(𝑥(𝑡)𝑥(𝑡+𝜂,𝑖)‖‖)‖_𝑖(𝑡)‖𝑠(𝑥(𝑡),𝑖)‖‖𝑥(𝑡)‖. (57)
Noting that ‖𝑠(𝑥(𝑡),𝑖)‖₁≥‖𝑠(𝑥(𝑡),𝑖)‖ and substituting (54) into (57), one has 𝑉̇(𝑡)≤−𝜇_𝑖‖𝑠(𝑥(𝑡),𝑖)‖²<0,for‖𝑠(𝑥(𝑡),𝑖)‖≠0. (58)
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: 𝐴₁=−10.6−2.420.1−0.50.12.20.5,𝐴_𝑑1=,𝐵0.20.10.1−0.10.1000.1−0.5₁=1−243.52.45.5,𝐵_𝑣1=0.20.10.3,𝐷₁=,𝐶0.20.3₁=−0.20.1010.10.04^𝑇,𝐸₁=,𝐹0.40.400.50.10.20.10.20.3₁=0.20.30.1,𝑁₁₁=0,𝑁0.10.1₂₁=0.10.1,𝐻_𝑎1=0,𝐻0.10.2_𝑎𝑑1=00.10.2,𝐻_𝑒1=,𝐻0.010.10.1_𝑐1=0.10.20.11−0.10.3−0.20.30,𝐻_𝑐𝑑1=,𝐶−0.200.20.10.10.10.2−0.10_𝑑1=00.30.20.20.010.1^𝑇,𝑓(𝑥(𝑡),𝑡,1)=0.6sin𝑥²₁(𝑡)+𝑥²₂(𝑡)0.6𝑥₂(𝑡)^𝑇,𝑣(𝑡)=31+𝑡²,𝐹₁₁(𝑡)𝐹=0.5sin𝑡,₂₁(𝑡)=0.2sin𝑡0.2cos𝑡0. (59)

Mode 2. We have the following: 𝐴₂=10.8100.5−0.60.30.4−0.5,𝐴_𝑑2=,𝐵−0.20.20.300.50.10.10.40.5₂=1.23400.42,𝐵_𝑣2=,𝐷0.10.20.2₂=0.10.4,𝐶₂=0.2−0.101−0.10.02^𝑇,𝐸₂=0.30.50.200.80.30.200.3,𝐹₂=,𝑁0.30.40.2₂₁=00.2−0.1,𝑁₂₂=,𝐻0.20.1_𝑎2=0,𝐻0.20.4_𝑎𝑑2=00.10.1,𝐻_𝑒2=0,𝐻0.10.2_𝑐2=0.20.10.20.1−0.10.2−0.10.20,𝐻_𝑐𝑑2=,𝐶−0.100.10.20.10.20.1−0.10_𝑑2=0.10.2−0.20.10.01−0.1^𝑇,𝑓(𝑥(𝑡),𝑡,2)=sin𝑥₁(𝑡)2𝑥₂(𝑡)𝑥₃(𝑡)^𝑇,𝐹₁₂(𝑡)=0.5cos𝑡,𝐹₂₂(𝑡)=0.2cos𝑡0.2sin𝑡0. (60)

Moreover, the time-varying delay 𝜏(𝑡)=0.5sin𝑡+0.5 with 𝑑=1, ℎ=0.5, and the transition rate matrix 𝜋 of Markov chain {𝑟_𝑡} is defined as 𝜋=⁻¹¹_1.5−1.5. Choose 𝛾=1. Now, for the bounds 𝜂₁=0.6 and 𝜂₂=1, solving LMIs (22)-(23) yields 𝑋₁=,𝑋1.7215−1.00691.1293−1.00690.9575−0.80571.1293−0.80571.1127₂=,𝜀1.9635−0.6287−1.30140.62870.7440−0.0362−1.3014−0.03624.0151₁₁=1.0867,𝜀₁₂𝜀=1.2007,₂₁=4.2260,𝜀₂₂=2.2260,𝜀₃₁𝜀=1.2260,₃₂=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 (61)

Furthermore, controller gain 𝐾₁ and 𝐾₂ are chosen as 𝐾₁=−1.5−4.5−433.5−1.2,𝐾₂=.1−32−3.50.2−0.8 (62)

Then, the desired SMC law can be obtained as follows.

Mode 1. We have the following: 𝑢(𝑡)=𝑥(𝑡)−(𝑥(𝑡))−𝑥((𝑡−‖𝑥(𝑡𝑥(𝑡+𝑥((𝑡(𝑠(𝑥(𝑡).−1.5−4.5−433.5−1.20.36420.11270.11270.4264×𝑠,10.04790.04960.0252−0.0483−0.0143−0.0699𝑡−𝜏))0.6)‖+0.020.10.20)00.10.2𝑡−𝜏))+0.0867×sgn,1)) (63)

Mode 2. We have the following: 𝑢(𝑡)=𝑥(𝑡)−𝑠(𝑥(𝑡))−𝑥((𝑡−𝑥(𝑡𝑥(𝑡+𝑥((𝑡](𝑠(𝑥(𝑡),1−32−3.50.2−0.80.7037−0.1691−0.16910.6103,20.01240.14190.0416−0.01200.09090.1627𝑡−𝜏))‖)‖+0.05990.20.40)0.100.1𝑡−𝜏))+0.0763sgn,2)) (64)

where 𝜇₁=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 vector .

Figure 2 

Mode 1: sliding mode .

Figure 3 

Mode 1: control signal .

Figure 4 

Mode 2: state vector .

Figure 5 

Mode 2: sliding mode .

Figure 6 

Mode 2: control signal .

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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China 61273091 and 61104007, the Young and Middle-Aged Scientists Research Foundation of Shandong Province under Grant BS2011DX013 and BS2012SF008, Taishan Scholar Project of Shandong Province, and the Natural Science Foundation of Shandong province ZR2011FM033.