New Results on Robust Stability and Stabilization of Linear Discrete-Time Stochastic Systems with Convex Polytopic Uncertainties

Abstract

This paper addresses the robust stability for a class of linear discrete-time stochastic systems with convex polytopic uncertainties. The system to be considered is subject to both interval time-varying delays and convex polytopic type uncertainties. Based on the augmented parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of linear discrete-time stochastic control systems is given. Numerical examples are included to illustrate the effectiveness of our results.

1. Introduction

In the past decades, the problem of stability for neutral differential systems, which have delays in both its state and the derivatives of its states, has been widely investigated by many researchers. Such systems are often encountered in engineering, biology, and economics. The existence of time delay is frequently a source of instability or poor performances in the systems. Recently, some stability criteria for neutral system with time delay have been given in [1–8] and the references therein. Some delay-dependent stability criteria for discrete-time systems with time-varying delay are investigated in [2, 6, 9–11], where the discrete Lyapunov functional method is employed to prove stability conditions in terms of linear matrix inequalities (LMIs). A number of research works for dealing with asymptotic stability problem for discrete systems with interval time-varying delays have been presented in [12–24]. Theoretically, stability analysis of the systems with time-varying delays is more complicated, especially for the case where the system matrices belong to some convex polytope. In this case, the parameter-dependent Lyapunov-Krasovskii functionals are constructed as the convex combination of a set of functions assures the robust stability of the nominal systems, and the stability conditions must be solved upon a grid on the parameter space, which results in testing a finite number of linear matrix inequalities (LMIs) [11, 25, 26]. To the best of the authors' knowledge, the stability for linear discrete-time systems with both time-varying delays and polytopic uncertainties has not been fully investigated. The papers [27, 28] propose sufficient conditions for robust stability of discrete and continuous polytopic systems without time delays. More recently, combining the ideas in [25, 26], improved conditions for 𝒟-stability and 𝒟-stabilization of linear polytopic delay-difference equations with constant delays have been proposed in [29]. To the best of our knowledge, the stability and stabilization of linear discrete-time stochastic systems with convex polytopic uncertainties, nondifferentiable time-varying delays has not been fully studied yet (see, e.g., [1, 3–11, 13–36] and the references therein), which are important in both theories and applications. This motivates our research.

In this paper, we consider polytopic discrete-time stochastic equations with interval time-varying delays. By using the parameter-dependent Lyapunov-Krasovskii functional combined with LMI techniques, we propose new criteria for the robust stability of the stochastic system. The delay-dependent stability conditions are formulated in terms of LMIs, being thus solvable by the numeric technology available in the literature to date. The result is applied to robust stabilization of linear discrete-time stochastic control systems. Compared to other results, our result has its own advantages. First, it deals with the delay-difference stochastic system, where the state-space data belong to the convex polytope of uncertainties and the rate of change of the state depends not only on the current state of the systems, but also its state at some times in the past. Second, the time delay is assumed to be a time-varying function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available. Third, our approach allows us to apply in robust stabilization of the linear discrete-time stochastic system subjected to polytopic uncertainties and external controls. Therefore, our results are more general than the related previous results.

The paper is organized as follows. Section 2 introduces the main notations, definitions, and some lemmas needed for the development of the main results. In Section 3, sufficient conditions are derived for robust stability, stabilization of discrete-time stochastic systems with interval time-varying delays, and polytopic uncertainties. They are followed by some remarks. Illustrative examples are given in Section 4.

2. Preliminaries

The following notations will be used throughout this paper. 𝑅⁺ denotes the set of all real nonnegative numbers; 𝑅^𝑛 denotes the 𝑛-dimensional space with the scalar product ⟨⋅,⋅⟩ and the vector norm ‖⋅‖; 𝑅^𝑛×𝑟 denotes the space of all real matrices of (𝑛×𝑟)-dimension. 𝐴^𝑇 denotes the transpose of 𝐴; a matrix 𝐴 is symmetric if 𝐴=𝐴^𝑇, and a matrix 𝐼 is the identity matrix of appropriate dimension.

Matrix 𝐴 is semipositive definite (𝐴≥0) if ⟨𝐴𝑥,𝑥⟩≥0, for all 𝑥∈𝑅^𝑛;𝐴 is positive definite (𝐴>0) if ⟨𝐴𝑥,𝑥⟩>0 for all 𝑥≠0; 𝐴≥𝐵 means that 𝐴−𝐵≥0.

Consider delay-difference stochastic systems with polytopic uncertainties of the form 𝑥()(𝜉)𝑥(𝑘)(𝜉)𝑥((𝑘(𝑥(𝑘)((𝑘)𝜔(𝑘),𝑥(𝑘)𝑘+1=𝐴+𝐷𝑘−ℎ))+𝜎,𝑥𝑘−ℎ)),𝑘𝑘=0,1,2,…,=𝑣_𝑘,𝑘=−ℎ₂,−ℎ₂+1,…,0, (1) where 𝑥(𝑘)∈𝑅^𝑛 is the state (Figures 1 and 2), and the system matrices are subjected to uncertainties and belong to the polytope Ω given by Ω=[𝐴,𝐷](𝜉)∶=^𝑝_𝑖=1𝜉_𝑖𝐴_𝑖,𝐷_𝑖,^𝑝_𝑖=1𝜉_𝑖=1,𝜉_𝑖≥0, (2) where 𝐴_𝑖, 𝐷_𝑖, 𝑖=1,2,…,𝑝, are given constant matrices with appropriate dimensions, 𝜔(𝑘) is a scalar Wiener process (Brownian Motion) on (Ω,ℱ,𝒫) with 𝐸[𝜔(𝑘)]=0,𝐸𝜔²(𝑘𝐸𝜔(𝑖)𝜔𝑗,)=1,=0𝑖≠𝑗 (3) and 𝜎: 𝑅^𝑛×𝑅^𝑛×𝑅→𝑅^𝑛 is the continuous function and is assumed to satisfy that 𝜎^𝑇(𝑥(𝑘),𝑥(𝑘−ℎ(𝑘)),𝑘)𝜎(𝑥(𝑘),𝑥(𝑘−ℎ(𝑘)),𝑘)≤𝜌₁𝑥^𝑇(𝑘)𝑥(𝑘)+𝜌₂𝑥^𝑇((𝑘𝑥((𝑘,𝑥(𝑘)((𝑘𝑘−ℎ))𝑘−ℎ)),𝑥𝑘−ℎ))∈𝑅^𝑛, (4)

Figure 1 

The simulation of the solutions 𝑥1(𝑘) and 𝑥2(𝑘) with the initial condition 𝜙(𝑘)=[105]𝑇, 𝑘∈[0,10].

Figure 2 

The simulation of the solutions 𝑥1(𝑘) and 𝑥2(𝑘) with the initial condition 𝜙(𝑘)=[105]𝑇, 𝑘∈[0,10].

where 𝜌₁>0 and 𝜌₂>0 are known constant scalars.

For simplicity, we denote 𝜎(𝑥(𝑘),𝑥(𝑘−ℎ(𝑘)),𝑘) by 𝜎, respectively.

The time-varying function ℎ(𝑘) satisfies the condition: 0<ℎ₁≤ℎ(𝑘)≤ℎ₂,∀𝑘=0,1,2,…. (5)

Remark 1. It is worth noting that the time delay is a time-varying function belonging to a given interval, which allows the time delay to be a fast time-varying function, and the lower bound is not restricted to being zero as considered in [2, 6, 9–11, 18–24, 30–33].

Definition 2. The system (1) is robustly stable in the mean square if there exists a positive definite scalar function 𝑉(𝑘,𝑥(𝑘)): 𝑅^𝑛×𝑅^𝑛→𝑅 such that 𝐸[((𝑘[𝑉((((𝑘Δ𝑉𝑘,𝑥))]=𝐸𝑘+1,𝑥𝑘+1))−𝑉𝑘,𝑥))]<0, (6) along any trajectory of zero solution of the system (1) for all uncertainties in Ω.

Proposition 3. For real numbers 𝜉_𝑖≥0, 𝑖=1,2,…,𝑝, ∑^𝑝_𝑖=1𝜉_𝑖=1, the following inequality holds: 𝑝−1^𝑝_𝑖=1𝜉²_𝑖−2^𝑝−1𝑝_{𝑖=1𝑗=𝑖+1}𝜉_𝑖𝜉_𝑗≥0. (7)

Proof. The proof is followed from completing the square 𝑝−1^𝑝_𝑖=1𝜉²_𝑖−2^𝑝−1𝑝_{𝑖=1𝑗=𝑖+1}𝜉_𝑖𝜉_𝑗=^𝑝−1𝑝_{𝑖=1𝑗=𝑖+1}𝜉_𝑖−𝜉_𝑗²≥0. (8)

3. Main Results

3.1. Robust Stability

In this section, we present sufficient delay-dependent conditions for the robust stability of system (1). Let us set𝑥_𝑘=sup_{𝑠∈−ℎ2,0}‖𝑥(,ℳ𝑘+𝑠)‖_𝑖𝑗𝑃,𝑄,𝑆₁,𝑆₂=ℎ₂−ℎ₁+1𝑄_𝑖−𝑃_𝑖−𝑆_1𝑖𝐴_𝑗−𝐴^𝑇_𝑗𝑆^𝑇_1𝑖+2𝜌₁𝐼𝑆_1𝑖−𝑆_1𝑖𝐴_𝑗−𝑆_1𝑖𝐷_𝑗−𝑆_2𝑖𝐴_𝑗𝑆^𝑇_1𝑖−𝐴^𝑇_𝑗𝑆^𝑇_1𝑖𝑃_𝑖+𝑆_1𝑖+𝑆^𝑇_1𝑖𝑆_2𝑖−𝑆_1𝑖𝐷_𝑗−𝐷^𝑇_𝑗𝑆^𝑇_1𝑖−𝐴^𝑇_𝑗𝑆^𝑇_2𝑖𝑆^𝑇_2𝑖−𝐷^𝑇_𝑗𝑆^𝑇_1𝑖−𝑄_𝑖−𝑆_2𝑖𝐷_𝑗−𝐷^𝑇_𝑗𝑆^𝑇_2𝑖+2𝜌₂𝐼,𝒮=(𝜉)=𝑆00000000,𝑃∑^𝑝_𝑖=1𝜉_𝑖𝑃_𝑖,𝑄(𝜉)=∑^𝑝_𝑖=1𝜉_𝑖𝑄_𝑖,𝑆₁(𝜉)=∑^𝑝_𝑖=1𝜉_𝑖𝑆_1𝑖,𝑆₂(𝜉)=∑^𝑝_𝑖=1𝜉_𝑖𝑆_2𝑖. (9)

Theorem 4. The system (1) is robustly stable in the mean square if there exist symmetric matrices 𝑃_𝑖>0, 𝑄_𝑖>0, 𝑖=1,2…,𝑝, and constant matrices 𝑆≥0, 𝑆_1𝑖, 𝑆_2𝑖, 𝑖=1,2…,𝑝, satisfying the following LMIs:(i)ℳ_𝑖𝑖(𝑃,𝑄,𝑆₁,𝑆₂)+𝒮<0, 𝑖=1,2,…,𝑝;(ii)ℳ_𝑖j(𝑃,𝑄,𝑆₁,𝑆₂)+ℳ_𝑗𝑖(𝑃,𝑄,𝑆₁,𝑆₂)−(2/(𝑝−1))𝒮<0, 𝑖=1,2,…,𝑝−1; 𝑗=𝑖+1,…,𝑝.

Proof. Consider the following parameter-dependent Lyapunov-Krasovskii functional for system (1): 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘), (10) where 𝑉₁(𝑘)(𝑘)𝑃(𝜉)𝑥(𝑘),𝑉=𝑥₂(𝑘)=^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖),𝑉₃(𝑘)=^−ℎ1+1_{𝑗=−ℎ2+2}^𝑘−1_{𝑙=𝑘+𝑗+1}𝑥^𝑇(𝑙)𝑄(𝜉)𝑥(𝑙). (11) We can verify that 𝜆₁‖𝑥(𝑘)‖²≤𝑉(𝑘)≤𝜆₂𝑥_𝑘². (12) Let us set 𝑧(𝑘)=[𝑥^𝑇(𝑘)𝑥^𝑇(𝑘+1)𝑥^𝑇(𝑘−ℎ(𝑘))𝜔^𝑇(𝑘)], and 𝐺(𝜉)=(𝜉)(𝜉)=0000𝑃(𝜉)0𝑃0000000000,𝐹000𝐼𝐼0000𝐼0000𝐼. (13) Then, with the difference of 𝑉₁(𝑘) along the solution of the system (1) and taking the mathematical expectation, we obtained 𝐸Δ𝑉₁(𝑘𝑥)=𝐸^𝑇(𝑘+1)𝑃(𝜉)𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃(𝜉)𝑥(𝑘𝑧(𝑘))=𝐸^𝑇𝐺(𝜉)𝑧(𝑘)−2𝑧^𝑇(𝑘)𝐹^𝑇(𝜉)(𝑘)000,0.5𝑥 (14) because of 𝑧^𝑇(𝑘)𝐺(𝜉)𝑧(𝑘)=𝑥(𝑘+1)^𝑇𝑃(𝜉)𝑥(𝑘+1),2𝑧^𝑇(𝑘)𝐹^𝑇(𝜉)(𝑘)0000.5𝑥=𝑥^𝑇(𝑘)𝑃(𝜉)𝑥(𝑘). (15) Using the expression of system (1) 0=−𝑆₁(𝜉)𝑥(𝑘+1)+𝑆₁(𝜉)𝐴(𝜉)𝑥(𝑘)+𝑆₁(𝜉)𝐷(𝜉)𝑥(𝑘−ℎ(𝑘))+𝑆₁(𝜉)𝜎𝜔(𝑘),0=−𝑆₂(𝜉)𝑥(𝑘+1)+𝑆₂(𝜉)𝐴(𝜉)𝑥(𝑘)+𝑆₂(𝜉)𝐷(𝜉)𝑥(𝑘−ℎ(𝑘))+𝑆₂(𝜉)𝜎𝜔(𝑘),0=−𝜎^𝑇𝑥(𝑘+1)+𝜎^𝑇𝐴(𝜉)𝑥(𝑘)+𝜎^𝑇𝐷(𝜉)𝑥(𝑘−ℎ(𝑘))+𝜎^𝑇𝜎𝜔(𝑘), (16) we have−2𝑧^𝑇(𝑘)𝐹^𝑇(𝜉)0.5𝑥(𝑘)−𝑆₁(𝜉)𝑥(𝑘+1)+𝑆₁(𝜉)𝐴(𝜉)𝑥(𝑘)+𝑆₁(𝜉)𝐷(𝜉)𝑥(𝑘−ℎ(𝑘))+𝑆₁(𝜉)𝜎𝜔(𝑘)−𝑆₂(𝜉)𝑥(𝑘+1)+𝑆₂(𝜉)𝐴(𝜉)𝑥(𝑘)+𝑆₂(𝜉)𝐷(𝜉)𝑥(𝑘−ℎ(𝑘))+𝑆₂(𝜉)𝜎𝜔(𝑘)−𝜎^𝑇𝑥(𝑘+1)+𝜎^𝑇𝐴(𝜉)𝑥(𝑘)+𝜎^𝑇𝐷(𝜉)𝑥(𝑘−ℎ(𝑘))+𝜎^𝑇(𝑘)𝑧(𝑘)𝜎𝜔=−𝑧^𝑇(𝑘)𝐹^𝑇(𝜉)𝑆0.5𝐼00₁(𝜉)𝐴(𝜉)−𝑆₁(𝜉)𝑆₁(𝜉)𝐷(𝜉)𝑆₁(𝜉)𝜎𝑆₂(𝜉)𝐴(𝜉)−𝑆₂(𝜉)𝑆₂(𝜉)𝐷(𝜉)𝑆₂(𝜉)𝜎𝜎^𝑇𝐴(𝜉)−𝜎^𝑇𝜎^𝑇𝐷(𝜉)𝜎^𝑇𝜎𝑧(𝑘)−𝑧^𝑇(𝑘)𝑆0.5𝐼00₁(𝜉)𝐴(𝜉)−𝑆₁(𝜉)𝑆₁(𝜉)𝐷(𝜉)𝑆₁(𝜉)𝜎𝑆₂(𝜉)𝐴(𝜉)−𝑆₂(𝜉)𝑆₂(𝜉)𝐷(𝜉)𝑆₂(𝜉)𝜎𝜎^𝑇𝐴(𝜉)−𝜎^𝑇𝜎^𝑇𝐷(𝜉)𝜎^𝑇𝜎^𝑇𝐹(𝜉)𝑧(𝑘). (17)
Therefore, from (14), it follows that 𝐸Δ𝑉₁(𝑘𝑥)=𝐸^𝑇(𝑘)−𝑃(𝜉)−𝑆₁(𝜉)𝐴(𝜉)−𝐴(𝜉)^𝑇𝑆^𝑇₁(𝜉)𝑥(𝑘)+2𝑥^𝑇(𝑘)𝑆₁(𝜉)−𝑆₁(𝜉)𝐴(𝜉)𝑥(𝑘+1)+2𝑥^𝑇(𝑘)−𝑆₁(𝜉)𝐷(𝜉)−𝑆₂(𝜉)𝐴(𝜉)𝑥(𝑘−ℎ(𝑘))+2𝑥^𝑇(𝑘)−𝑆₁(𝜉)𝜎−𝜎^𝑇𝐴(𝜉𝜔(𝑘)(𝑃(𝜉))+𝑥𝑘+1)+𝑆₁(𝜉)+𝑆^𝑇₁(𝜉()(𝑆)×𝑥𝑘+1+2𝑥𝑘+1)₂(𝜉)−𝑆₁(𝜉)𝐷(𝜉((𝑘(𝜎)×𝑥𝑘−ℎ))+2𝑥𝑘+1)^𝑇−𝑆₁(𝜉)𝜎𝜔(𝑘)+𝑥^𝑇(𝑘−ℎ(𝑘))−𝑆₂(𝜉)𝐷(𝜉)−𝐷^𝑇(𝜉)𝑆^𝑇₂(𝜉((𝑘)×𝑥𝑘−ℎ))+𝑥^𝑇(𝑘−ℎ(𝑘))−𝑆₂(𝜉)𝜎−𝜎^𝑇𝐷(𝜉)𝜔(𝑘)+𝜔^𝑇(𝑘)−2𝜎^𝑇𝜎𝜔(𝑘). (18) By assumption (3), we have 𝐸Δ𝑉₁(𝑘𝑥)=𝐸^𝑇(𝑘)−𝑃(𝜉)−𝑆₁(𝜉)𝐴(𝜉)−𝐴(𝜉)^𝑇𝑆^𝑇₁(𝜉)𝑥(𝑘)+2𝑥^𝑇(𝑘)𝑆₁(𝜉)−𝑆₁(𝜉)𝐴(𝜉)𝑥(𝑘+1)+2𝑥^𝑇(𝑘)−𝑆₁(𝜉)𝐷(𝜉)−𝑆₂(𝜉)𝐴(𝜉((𝑘(𝑃(𝜉))×𝑥𝑘−ℎ))+𝑥𝑘+1)+𝑆₁(𝜉)+𝑆^𝑇₁(𝜉𝑥()(𝑆)𝑘+1+2𝑥𝑘+1)₂(𝜉)−𝑆₁(𝜉)𝐷(𝜉)𝑥(𝑘−ℎ(𝑘))+𝑥^𝑇(𝑘−ℎ(𝑘))−𝑆₂(𝜉)𝐷(𝜉)−𝐷^𝑇(𝜉)𝑆^𝑇₂(𝜉((𝑘)×𝑥𝑘−ℎ))+𝜔^𝑇(𝑘)−2𝜎^𝑇𝜎𝜔(𝑘). (19) Applying assumption (4), the following estimations holds: −𝜎^𝑇(𝑥(𝑘),𝑥(𝑘−ℎ(𝑘)),𝑘)𝜎_𝑖(𝑥(𝑘),𝑥(𝑘−ℎ(𝑘)),𝑘)≤𝜌₁𝑥^𝑇(𝑘)𝑥(𝑘)+𝜌₂𝑥^𝑇(𝑘−ℎ(𝑘))𝑥(𝑘−ℎ(𝑘)). (20) Therefore, we have 𝐸Δ𝑉₁(𝑘𝑥)=𝐸^𝑇(𝑘)−𝑃(𝜉)−𝑆₁(𝜉)𝐴(𝜉)−𝐴(𝜉)^𝑇𝑆^𝑇₁(𝜉)+2𝜌₁𝐼𝑥(𝑘)+2𝑥^𝑇(𝑘)𝑆₁(𝜉)−𝑆₁(𝜉)𝐴(𝜉)𝑥(𝑘+1)+2𝑥^𝑇(𝑘)−𝑆₁(𝜉)𝐷(𝜉)−𝑆₂(𝜉)𝐴(𝜉((𝑘(𝑃(𝜉))×𝑥𝑘−ℎ))+𝑥𝑘+1)+𝑆₁(𝜉)+𝑆^𝑇₁(𝜉()(𝑆)×𝑥𝑘+1+2𝑥𝑘+1)₂(𝜉)−𝑆₁(𝜉)𝐷(𝜉((𝑘)×𝑥𝑘−ℎ))+𝑥^𝑇(𝑘−ℎ(𝑘))−𝑆₂(𝜉)𝐷(𝜉)−𝐷^𝑇(𝜉)𝑆^𝑇₂(𝜉)+2𝜌₂𝐼𝑥(𝑘−ℎ(𝑘)). (21) The expectation of the difference of 𝑉₂(𝑘) is given by 𝐸Δ𝑉₂(𝑘)=𝐸^𝑘_{()𝑖=𝑘+1−ℎ𝑘+1}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖)−^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖)=𝐸^𝑘−ℎ1_{()𝑖=𝑘+1−ℎ𝑘+1}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖)+𝑥^𝑇(𝑘)𝑄(𝜉)𝑥(𝑘)−𝑥^𝑥((𝑘𝑄(𝜉)𝑥((𝑘+𝑘−ℎ))𝑘−ℎ))_{𝑘−1𝑖=𝑘+1−ℎ1}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖)−^𝑘−1_{(𝑘)𝑖=𝑘+1−ℎ}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖). (22) Since ℎ(𝑘)≥ℎ₁, we have ^𝑘−1_{𝑖=𝑘+1−ℎ1}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖)−^𝑘−1_{(𝑘)𝑖=𝑘+1−ℎ}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖)≤0, (23) and, hence, from (22), we have 𝐸Δ𝑉₂(𝑘)≤𝐸^𝑘−ℎ1_{()𝑖=𝑘+1−ℎ𝑘+1}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖)+𝑥^𝑇(𝑘)𝑄(𝜉)𝑥(𝑘)−𝑥^𝑇(𝑘−ℎ(𝑘))𝑄(𝜉)𝑥(𝑘−ℎ(𝑘)). (24) The difference of 𝑉₃(𝑘) is given by 𝐸Δ𝑉₃(𝑘)=𝐸^−ℎ1+1_{𝑗=−ℎ2+2}𝑥^𝑇(𝑘)𝑄(𝜉)𝑥(𝑘)−𝑥^𝑇𝑄(𝜉)ℎ𝑘+𝑗−1×𝑥𝑘+𝑗−1=𝐸₂−ℎ₁𝑥^𝑇(𝑘)𝑄(𝜉)𝑥(𝑘)−^𝑘−ℎ1_{𝑙=𝑘+1−ℎ2}𝑥^𝑇(𝑙)𝑄(𝜉)𝑥(𝑙). (25) Since ^𝑘−ℎ1_{()𝑖=𝑘=1−ℎ𝑘+1}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖)−^𝑘−ℎ1_{𝑖=𝑘+1−ℎ2}𝑥^𝑇(𝑖)𝑄(𝜉)𝑥(𝑖)≤0, (26) we obtain from (24) and (25) that 𝐸Δ𝑉₂(𝑘)+Δ𝑉₃(𝑘ℎ)≤𝐸₂−ℎ₁+1𝑥^𝑇(𝑘)𝑄(𝜉)𝑥(𝑘)−𝑥^𝑇(𝑘−ℎ(𝑘))𝑄(𝜉)𝑥(𝑘−ℎ(𝑘)). (27) Therefore, combining the inequalities (21), (27) gives 𝐸[Δ𝑉(𝑘)]≤𝐸𝜓^𝑇(𝑘)𝑇(𝜉)𝜓(𝑘), (28) where𝜓(𝑘)=[𝑥(𝑘)𝑥(𝑘+1)𝑥(𝑘−ℎ(𝑘))]^𝑇,𝑇(𝜉)=𝑀(𝜉)𝑆₁(𝜉)−𝑆₁(𝜉)𝐴(𝜉)−𝑆₁(𝜉)𝐷(𝜉)−𝑆₂(𝜉)𝐴(𝜉)𝑆^𝑇₁(𝜉)−𝐴^𝑇(𝜉)𝑆^𝑇₁(𝜉)𝑃(𝜉)+𝑆₁(𝜉)+𝑆^𝑇₁(𝜉)𝑆₂(𝜉)−𝑆₁(𝜉)𝐷(𝜉)−𝐷^𝑇(𝜉)𝑆^𝑇₁(𝜉)−𝐴^𝑇(𝜉)𝑆^𝑇₂(𝜉)𝑆^𝑇₂(𝜉)−𝐷^𝑇(𝜉)𝑆^𝑇₁(𝜉)−𝑄(𝜉)−𝑆₂(𝜉)𝐷(𝜉)−𝐷^𝑇(𝜉)𝑆^𝑇₂(𝜉)+2𝜌₂𝐼,𝑀(𝜉)=ℎ₂−ℎ₁+1𝑄(𝜉)−𝑃(𝜉)−𝑆₁(𝜉)𝐴(𝜉)−𝐴(𝜉)^𝑇𝑆₁(𝜉)^𝑇+2𝜌₁𝐼. (29)
Let us denote that 𝑀_𝑖𝑗∶=ℎ₂−ℎ₁+1𝑄_𝑖−𝑃_𝑖−𝑆_1𝑖𝐴_𝑗−𝐴^𝑇_𝑗𝑆^𝑇_1𝑖+2𝜌₁𝑆𝐼,₁𝐴_𝑖𝑗∶=𝑆_1𝑗𝐴_𝑖+𝑆_1𝑖𝐴_𝑗,𝑆₂𝐴_𝑖𝑗∶=𝑆_2𝑗𝐴_𝑖+𝑆_2𝑖𝐴_𝑗,𝑆₁𝐷_𝑖𝑗∶=𝑆_1𝑗𝐷_𝑖+𝑆_1𝑖𝐷_𝑗,𝑆₂𝐷_𝑖𝑗∶=𝑆_2𝑗𝐷_𝑖+𝑆_2𝑖𝐷_𝑗,𝑃_𝑖𝑗=𝑃_𝑖+𝑃_𝑗,𝑄_𝑖𝑗=𝑄_𝑖+𝑄_𝑗,𝑆_1𝑖𝑗=𝑆_1𝑖+𝑆_1𝑗,𝑆_2𝑖𝑗=𝑆_2𝑖+𝑆_2𝑗. (30) From the convex combination of the expression of 𝑃(𝜉), 𝑄(𝜉), 𝑆₁(𝜉), 𝑆₂(𝜉), 𝐴(𝜉), 𝐷(𝜉), we have𝑇(𝜉)=^𝑝_𝑖=1𝜉²_𝑖𝑀_𝑖𝑖𝑆_1𝑖−𝑆_1𝑖𝐴_𝑖−𝑆_1𝑖𝐷_𝑖−𝑆_2𝑖𝐴_𝑖𝑆^𝑇_1𝑖−𝐴^𝑇_𝑖𝑆^𝑇_1𝑖𝑃_𝑖+𝑆_1𝑖+𝑆^𝑇_1𝑖𝑆_2𝑖−𝑆_1𝑖𝐷_𝑖−𝐷^𝑇_𝑖𝑆^𝑇_1𝑖−𝐴^𝑇_𝑖𝑆^𝑇_2𝑖𝑆^𝑇_2𝑖−𝐷^𝑇_𝑖𝑆^𝑇_1𝑖−𝑄_𝑖−𝑆_2𝑖𝐷_𝑖−𝐷^𝑇_𝑖𝑆^𝑇_2𝑖+2𝜌₂𝐼+^𝑝−1𝑝_{𝑖=1𝑗=𝑖+1}𝜉_𝑖𝜉_𝑗𝑀_𝑖𝑗+𝑀_𝑗𝑖𝑆_1𝑖𝑗−𝑆₁𝐴_𝑖𝑗−𝑆₁𝐷_𝑖𝑗−𝑆₂𝐴_𝑖𝑗𝑆^𝑇_1𝑖𝑗−𝐴^𝑇𝑆^𝑇_1𝑖𝑗𝑃_𝑖𝑗+𝑆_1𝑖𝑗+𝑆^𝑇_1𝑖𝑗𝑆_2𝑖𝑗−𝑆₁𝐷_𝑖𝑗−𝐷^𝑇𝑆^𝑇_1𝑖𝑗−𝐴^𝑇𝑆^𝑇_2𝑖𝑗𝑆^𝑇_2𝑖𝑗−𝐷^𝑇𝑆^𝑇_1𝑖𝑗−𝑄_𝑖𝑗−𝑆₂𝐷_𝑖𝑗−𝐷^𝑇𝑆^𝑇_2𝑖𝑗+2𝜌₂𝐼=^𝑝_𝑖=1𝜉²_𝑖ℳ_𝑖𝑖𝑃,𝑄,𝑆₁,𝑆₂+^𝑝−1𝑝_{𝑖=1𝑗=𝑖+1}𝜉_𝑖𝜉_𝑗ℳ_𝑖𝑗𝑃,𝑄,𝑆₁,𝑆₂+ℳ_𝑗𝑖𝑃,𝑄,𝑆₁,𝑆₂. (31)
Then, the conditions (i) and (ii) give 𝑇(𝜉)<−^𝑝_𝑖=1𝜉²_𝑖2𝒮+𝑝−1^𝑝−1𝑝_{𝑖=1𝑗=𝑖+1}𝜉_𝑖𝜉_𝑗𝒮≤0, (32) because of Proposition 3 as 𝑝−1^𝑝_𝑖=1𝜉²_𝑖−2^𝑝−1𝑝_{𝑖=1𝑗=𝑖+1}𝜉_𝑖𝜉_𝑗=^𝑝−1𝑝_{𝑖=1𝑗=𝑖+1}𝜉_𝑖−𝜉_𝑗²≥0, (33) and, hence, we finally obtain from (28) that 𝐸[Δ𝑉(𝑘)]≤𝐸𝜓^𝑇(𝑘)𝑇(𝜉)𝜓(𝑘)<0,∀𝑘=0,1,2,…, (34) which together with (12) and Definition 2 implies that the system (1) is robustly stable in the mean square. This completes the proof of the theorem.

Remark 5. The stability conditions of Theorem 4 are more appropriate for most of real systems since it is usually impossible in practice to know exactly the delay but lower and upper bounds are always possible.

3.2. Robust Stabilization

This section deals with a stabilization problem considered in [15] for constructing a delayed feedback controller, which stabilizes the resulting closed-loop system. The robust stability condition obtained in previous section will be applied to design a time-delayed state feedback controller for the discrete-time control system described by 𝑥()(𝜉)𝑥(𝑘)(𝜉)𝑢(𝑘)(𝑥(𝑘)((𝑘)𝜔(𝑘),𝑘+1=𝐴+𝐵+𝜎,𝑥𝑘−ℎ)),𝑘𝑘=0,1,2,…, (35) where 𝑢(𝑘)∈𝑅^𝑛 is the control input, and the system matrices are subjected to uncertainties and belong to the polytope Ω given by Ω=[𝐴,𝐵](𝜉)∶=^𝑝_𝑖=1𝜉_𝑖𝐴_𝑖,𝐵_𝑖,^𝑝_𝑖=1𝜉_𝑖=1,𝜉_𝑖≥0, (36) where 𝐴_𝑖,𝐵_𝑖,𝑖=1,2,…,𝑝, are given constant matrices with appropriate dimensions. As in [8], we consider a parameter-dependent delayed feedback control law 𝑢(𝑘)=𝐹(𝜉)𝑥(𝑘−ℎ(𝑘)),𝑘=−ℎ₂,…,0, (37) where ℎ(𝑘) is the time-varying delay function satisfying 0<ℎ₁≤ℎ(𝑘)≤ℎ₂, and 𝐹(𝜉) is the controller gain to be determined. Applying the feedback controller (37) to the system (35), the closed-loop time-delay system is 𝑥()(𝜉)𝑥(𝑘)(𝜉)𝐹(𝜉)𝑥((𝑘(𝑥(𝑘)((𝑘)𝜔(𝑘),𝑘+1=𝐴+𝐵𝑘−ℎ))+𝜎,𝑥𝑘−ℎ)),𝑘𝑘=0,1,2,… (38)

Definition 6. The system (35) is robustly stabilizable in the mean square if there is a delayed feedback control (37) such that the closed-loop delay system (38) is robustly stable in the mean square.
Let ℳ_𝑖𝑗𝑃,𝑄,𝑆₁=ℎ₂−ℎ₁+1𝑄_𝑖−𝑃_𝑖−𝑆_1𝑖𝐴_𝑗−𝐴^𝑇_𝑗𝑆^𝑇_1𝑖+2𝜌₁𝐼𝑆_1𝑖−𝑆_1𝑖𝐴_𝑗−𝑃_𝑖−𝑆_1𝑖𝐴_𝑗𝑆^𝑇_1𝑖−𝐴^𝑇_𝑗𝑆^𝑇_1𝑖𝑃_𝑖+𝑆_1𝑖+𝑆^𝑇_1𝑖𝑆_1𝑖−𝑃_𝑖−𝑃_𝑖−𝐴^𝑇_𝑗𝑆^𝑇_1𝑖𝑆^𝑇_1𝑖−𝑃_𝑖−𝑄_𝑖−𝑃_𝑖−𝑃_𝑖+2𝜌₂𝐼,𝒮=.𝑆00000000 (39)

The following theorem can be derived from Theorem 4.

Theorem 7. The system (35) is robustly stabilizable in the mean square by the delayed feedback control (37), where 𝐹(𝜉)=𝐵^𝑇(𝜉)𝐵(𝜉)𝐵^𝑇(𝜉)⁻¹𝑆^𝑇₁(𝜉)𝑆₁(𝜉)𝑆^𝑇₁(𝜉)⁻¹𝑃(𝜉), (40)
if there exist symmetric matrices 𝑃_𝑖>0, 𝑄_𝑖>0, 𝑖=1,2…,𝑝, and constant matrices 𝑆_1𝑖, 𝑖=1,2,…,𝑝, 𝑆≥0, satisfying the following LMIs:(i)ℳ_𝑖𝑖(𝑃,𝑄,𝑆₁)+𝒮<0, 𝑖=1,2,…,𝑝;(ii)ℳ_𝑖𝑗(𝑃,𝑄,𝑆₁)+ℳ_𝑗𝑖(𝑃,𝑄,𝑆₁)−(2/(𝑝−1))𝒮<0, 𝑖=1,2,…,𝑝−1; 𝑗=𝑖+1,…,𝑝.

Proof. Taking 𝑆_1𝑖=𝑆_2𝑖 and using the feedback control (37), the closed-loop system becomes system (Σ_𝜉), where 𝐷(𝜉)=𝐵(𝜉)𝐹(𝜉)=𝑆^𝑇₁(𝜉)[𝑆₁(𝜉)𝑆^𝑇₁(𝜉)]⁻¹𝑃(𝜉). Since 𝑆₁(𝜉)𝐷(𝜉)=𝑃(𝜉), the robust stability condition of the closed-loop system (38), by Theorem 4, is immediately derived.

Remark 8. The stabilization conditions of Theorem 7 are more appropriate for most of real systems since it is usually impossible in practice to know exactly the delay but lower and upper bounds are always possible.

4. Numerical Examples

To illustrate the effectiveness of the previous theoretical results, we consider the following numerical examples.

Example 9 (robust stability). Consider system Σ_𝜉 for 𝑝=2, where the delay function ℎ(𝑘) is given by ℎ(𝑘)=1+28sin²𝑘𝜋2𝐴,𝑘=0,1,2,…,₁=−30.512−3.5,𝐴₂=,𝐷−35.513−4.5₁=−1.50.10.4−2.15,𝐷₂=.−2.50.20.3−1.85 (41)
By using the LMI Toolbox in MATLAB, the LMIs (i) and (ii) of Theorem 4 are feasible with ℎ₁=1, ℎ₂=29, 𝜌₁=0.011, 𝜌₂=0.015, and we use the condition in the Theorem 4 for this example. The solutions of LMI verify as follow of the form (42) Therefore, the system is robustly stable.

Example 10 (robust stabilization). Consider system (35) for 𝑝=2, where the delay function ℎ(𝑘) is given by ℎ(𝑘)=1+34sin²𝑘𝜋2𝐴,𝑘=0,1,2,…,₁=−30.512−3.5,𝐴₂=,𝐵−35.513−4.5₁=−1.50.10.4−2.15,𝐵₂=.−2.50.20.3−1.85 (43) By using the LMI Toolbox in MATLAB, the LMIs (i) and (ii) of Theorem 7 are feasible with ℎ₁=1, ℎ₂=35, 𝜌₁=0.011, 𝜌₂=0.015, and we use the condition in the Theorem 7 for this example. The solutions of LMI verify as follow of the form 𝑃₁=1.3886−0.0760−0.07601.3559,𝑃₂=,𝑄1.62860.06490.06491.5243₁=,𝑄0.0097−0.0048−0.00480.0057₂=,𝑆0.0728−0.0159−0.01590.0621₁₁=,𝑆−0.02740.0827−0.0133−0.2222₁₂=,.−0.02090.0619−0.0226−0.1942𝑆=0.5954−0.0672−0.06720.5469 (44) Therefore, the system is robustly stabilizable with the feedback control 𝑢(𝑘)=𝐵^𝑇(𝜉)𝐵(𝜉)𝐵^𝑇(𝜉)⁻¹𝑆^𝑇₁(𝜉)𝑆₁(𝜉)𝑆^𝑇₁(𝜉)⁻¹𝑃(𝜉)𝑥((𝑘=𝜉𝑘−ℎ))₁𝐵₁+𝜉₂𝐵₂^𝑇𝜉₁𝐵₁+𝜉₂𝐵₂𝜉₁𝐵₁+𝜉₂𝐵₂^𝑇−1×𝜉₁𝑆₁₁+𝜉₂𝑆₁₂^𝑇𝜉₁𝑆₁₁+𝜉₂𝑆₁₂𝜉₁𝑆₁₁+𝜉₂𝑆₁₂^𝑇−1𝜉₁𝑃₁+𝜉₂𝑃₂𝜉)𝑥((𝑘=(𝑘−ℎ))−1.5𝜉₁−2.5𝜉₂0.4𝜉₁+0.3𝜉₂0.1𝜉₁+0.2𝜉₂−2.15𝜉₁−1.85𝜉₂×−1.5𝜉₁−2.5𝜉₂0.4𝜉₁+0.3𝜉₂0.1𝜉₁+0.2𝜉₂−2.15𝜉₁−1.85𝜉₂⁻¹×−1.5𝜉₁−2.5𝜉₂0.1𝜉₁+0.2𝜉₂0.4𝜉₁+0.3𝜉₂−2.15𝜉₁−1.85𝜉₂⁻¹×−0.0274𝜉₁−0.0209𝜉₂−0.0133𝜉₁−0.0226𝜉₂0.0827𝜉₁+0.0619𝜉₂−0.2222𝜉₁−0.1942𝜉₂×−0.0274𝜉₁−0.0209𝜉₂−0.0133𝜉₁−0.0226𝜉₂0.0827𝜉₁+0.0619𝜉₂−0.2222𝜉₁−0.1942𝜉₂⁻¹×−0.0274𝜉₁−0.0209𝜉₂0.0827𝜉₁+0.0619𝜉₂−0.0133𝜉₁−0.0226𝜉₂−0.2222𝜉₁−0.1942𝜉₂⁻¹×1.3886𝜉₁+1.6286𝜉₂−0.0760𝜉₁+0.0649𝜉₂−0.0760𝜉₁+0.0649𝜉₂1.3559𝜉₁+1.5243𝜉₂=−2.0829𝜉²₁−5.9144𝜉₁𝜉₂−4.0715𝜉²₂−0.0304𝜉²₁+0.0260𝜉₁𝜉₂+0.0195𝜉²₂−0.0076𝜉²₁−0.0087𝜉₁𝜉₂+0.0128𝜉²₂−2.9152𝜉²₁−5.7856𝜉₁𝜉₂−2.8200𝜉²₂𝑥(𝑘−ℎ(𝑘)). (45)

Therefore, the feedback delayed controller is 𝑢₁(𝑘)=−2.0829𝜉²₁−5.9144𝜉₁𝜉₂−4.0715𝜉²₂×𝑥₁((𝑘+𝑘−ℎ))−0.0304𝜉²₁+0.0260𝜉₁𝜉₂+0.0195𝜉²₂×𝑥₂((𝑘,𝑢𝑘−ℎ))₂(𝑘)=−0.0076𝜉²₁−0.0087𝜉₁𝜉₂+0.0128𝜉²₂×𝑥₁((𝑘+𝑘−ℎ))−2.9152𝜉²₁−5.7856𝜉₁𝜉₂−2.8200𝜉²₂×𝑥₂(𝑘−ℎ(𝑘)). (46)

5. Conclusion

In this paper, new delay-dependent mean square robust stability conditions for linear polytopic delay-difference stochastic equations with interval time-varying delays have been presented in terms of LMIs. An application to mean square robust stabilization of discrete stochastic control systems with time-delayed feedback controllers has been studied. Numerical examples have been given to demonstrate the effectiveness of the proposed conditions.

Acknowledgments

This work was supported by the Office of Agricultural Research and Extension Maejo University, the Thailand Research Fund Grant, the Higher Education Commission, and Faculty of Science, Maejo University, Thailand. The first author is supported by the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education, Thailand. The authors thank anonymous reviewers for valuable comments and suggestions, which allowed them to improve the paper.