New Delay-Dependent Robust Stability Criterion for LPD Discrete-Time Systems with Interval Time-Varying Delays

Abstract

This paper investigates the problem of robust stability for linear parameter-dependent (LPD) discrete-time systems with interval time-varying delays. Based on the combination of model transformation, utilization of zero equation, and parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent robust stability conditions are obtained and formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods.

1. Introduction

Systems with time delay exist in many fields such as electric systems, chemical processes systems, networked control systems, telecommunication systems, and economical systems. Over the past decades, the problem of robust stability analysis for uncertain systems with time delay has been widely investigated by many researchers. Commonly, stability criteria for uncertain systems with time delay are generally divided into two classes: a delay-independent one and a delay-dependent one. The delay-independent stability criteria tends to be more conservative, especially for a small size delay; such criteria do not give any information on the size of delay. On the other hand, delay-dependent stability criteria are concerned with the size of delay and usually provide a maximal delay size.

Discrete-time systems with state delay have strong background in engineering applications, among which network-based control has been well recognized to be a typical example. If the delay is constant in discrete systems, one can transform a delayed system into a delay-free one by using state augmentation techniques. However, when the delay is large, the augmented system will become much complex and thus difficult to analyze and synthesize [1]. In recent years, robust stability analysis of continuous-time and discrete-time systems subject to time-invariant parametric uncertainty has received considerable attention. An important class of linear time-invariant parametric uncertain system is a linear parameter-dependent (LPD) system in which the uncertain state matrices are in the polytope consisting of all convex combination of known matrices. To address this problem, several results have been obtained in terms of sufficient (or necessary and sufficient) conditions, see [1–24] and references cited therein. Most of these conditions have been obtained via the Lyapunov theory approaches in which the parameter-dependent Lyapunov functions have been employed. These conditions are always expressed in terms of LMIs which can be solved numerically by using available tools such as the LMI Toolbox in MATLAB. Recently, delay-dependent robust stability criteria for LPD continuous-time systems with time delay have been taken into consideration. Sufficient conditions for robust stability of time-delay systems have been presented via Lyapunov approaches [8, 16, 21]. However, much attention has been focused on the problem of robust stability analysis for LPD discrete-time systems with time delay [10, 13, 22].

In this paper, we focus on the delay-dependent robust stability criterion for LPD discrete-time systems with interval time-varying delays. Based on the combination of model transformation, utilization of zero equation, and parameter-dependent Lyapunov functional, new delay-dependent robust stability conditions are obtained and formulated in terms of linear matrix inequalities (LMIs). Finally, numerical examples are given to illustrate that the resulting criterion outperforms the existing stability condition.

2. Problem Formulation and Preliminaries

We introduce some notations, definitions, and propositions that will be used throughout the paper. ℤ⁺ denotes the set of nonnegative integer numbers; ℝ^𝑛 denotes the 𝑛-dimensional space with the vector norm ‖⋅‖; ‖𝑥‖ denotes the Euclidean vector norm of 𝑥∈ℝ^𝑛, that is, ‖𝑥‖²=𝑥^𝑇𝑥; 𝑀^𝑛×𝑟 denotes the space of all real matrices of (𝑛×𝑟)-dimensions; 𝐴^𝑇 denotes the transpose of the matrix 𝐴; 𝐴 is symmetric if 𝐴=𝐴^𝑇; 𝐼 denotes the identity matrix; 𝜆(𝐴) denotes the set of all eigenvalues of 𝐴; 𝜆_max(𝐴)=max{Re𝜆∶𝜆∈𝜆(𝐴)}; 𝜆_min(𝐴)=min{Re𝜆∶𝜆∈𝜆(𝐴)}; 𝜆_max(𝐴(𝛼))=max{𝜆_max(𝐴_𝑖)∶𝑖=1,2,…,𝑁}; 𝜆_min(𝐴(𝛼))=min{𝜆_min(𝐴_𝑖)∶𝑖=1,2,…,𝑁}; matrix 𝐴 is called a semipositive definite (𝐴≥0) if 𝑥^𝑇𝐴𝑥≥0, for all 𝑥∈ℝ^𝑛; 𝐴 is a positive definite (𝐴>0) if 𝑥^𝑇𝐴𝑥>0 for all 𝑥≠0; matrix 𝐵 is called a seminegative definite (𝐵≤0) if 𝑥^𝑇𝐵𝑥≤0, for all 𝑥∈ℝ^𝑛;  𝐵 is a negative definite (𝐵<0) if 𝑥^𝑇𝐵𝑥<0 for all 𝑥≠0; 𝐴>𝐵 means 𝐴−𝐵>0; 𝐴≥𝐵 means 𝐴−𝐵≥0; ∗ represents the elements below the main diagonal of a symmetric matrix.

Consider the following uncertain LPD discrete-time system with interval time-varying delays of the form 𝑥()(𝛼)𝑥(𝑘)(𝛼)𝑥((𝑘,𝑥(𝑠)(𝑠)𝑘+1=𝐴+𝐵𝑘−ℎ))=𝜙,𝑠∈−ℎ₂,−ℎ₂+1,…,−1,0,, (1)(2) where 𝑘∈ℤ⁺, 𝑥(𝑘)∈ℝ^𝑛 is the system state and 𝜙(𝑠) is an initial value at 𝑠. 𝐴(𝛼), 𝐵(𝛼)∈𝑀^𝑛×𝑛 are uncertain matrices belonging to the polytope of the form 𝐴(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐴_𝑖,𝐵(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐵_𝑖,_𝑁𝑖=1𝛼_𝑖=1,𝛼_𝑖≥0,𝐴_𝑖,𝐵_𝑖∈𝑀^𝑛×𝑛,𝑖=1,…,𝑁. (3) In addition, we assume that the time-varying delay ℎ(𝑘) is upper and lower bounded. It satisfies the following assumption of the form 0<ℎ₁≤ℎ(𝑘)≤ℎ₂, (4) where ℎ₁ and ℎ₂ are known positive integers.

Definition 1 (see [19]). The system (1) is said to be robustly stable if there exists a positive definite function 𝑉(𝑘,𝑥(𝑘))∶ℤ⁺×ℝ^𝑛→ℝ such that Δ𝑉(𝑘,𝑥(𝑘))=𝑉(𝑘+1,𝑥(𝑘+1))−𝑉(𝑘,𝑥(𝑘))<0, (5) along any trajectory of the solution of the system (1).

Definition 2 (see [19]). The system (1) when 𝐴(𝛼)=𝐴, 𝐵(𝛼)=𝐵, and 𝐴,𝐵∈𝑀^𝑛×𝑛 is said to be asymptotically stable if there exists a positive definite function 𝑉(𝑘,𝑥(𝑘))∶ℤ⁺×ℝ^𝑛→ℝ such that Δ𝑉(𝑘,𝑥(𝑘))=𝑉(𝑘+1,𝑥(𝑘+1))−𝑉(𝑘,𝑥(𝑘))<0, (6) along any trajectory of the solution of the system (1) when 𝐴(𝛼)=𝐴, 𝐵(𝛼)=𝐵.

Proposition 3 ([7, the Schur complement lemma]). Given constant symmetric matrices 𝑋,𝑌, and 𝑍 of appropriate dimensions with 𝑌>0, then 𝑋+𝑍^𝑇𝑌⁻¹𝑍<0 if and only if 𝑋𝑍^𝑇𝑍−𝑌𝑍<0or−𝑌𝑍_𝑇𝑋<0. (7)

Proposition 4 (see [9]). For any constant matrix 𝑊∈𝑀^𝑛×𝑛,𝑊=𝑊^𝑇>0, two integers 𝑟_𝑀 and 𝑟_𝑚 satisfying 𝑟_𝑀≥𝑟_𝑚, and vector function 𝑥∶[𝑟_𝑚,𝑟_𝑀]→ℝ^𝑛, the following inequality holds: ^𝑟𝑀_{𝑖=𝑟𝑚}𝑥(𝑖)^𝑇𝑊^𝑟𝑀_{𝑖=𝑟𝑚}𝑥(𝑖)≤𝑟_𝑀−𝑟_𝑚+1^𝑟𝑀_{𝑖=𝑟𝑚}𝑥^𝑇(𝑖)𝑊𝑥(𝑖). (8)

Rewrite the system (1) in the following system: 𝑥()(𝑘)(𝑘),𝑦(𝑘)=[𝐴(𝛼)(𝛼)]𝑥(𝑘)(𝛼)𝑘+1=𝑥+𝑦+𝐵−𝐼−𝐵^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑦(𝑖). (9)

3. Robust Stability Conditions

In this section, we study the robust stability criteria for the system (1) by using the combination of model transformation, the linear matrix inequality (LMI) technique, and the Lyapunov method. We introduce the following notations for later use: 𝐶_𝑗(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐶^𝑗_𝑖,𝐷_𝑗(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐷^𝑗_𝑖,𝐸_𝑗(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐸^𝑗_𝑖,𝐺_𝑗(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐺^𝑗_𝑖,𝐿_𝑗(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐿^𝑗_𝑖,𝑃(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑃_𝑖,𝑄(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑄_𝑖,𝑅(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑅_𝑖,𝑆(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑆_𝑖,𝑇(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑇_𝑖,𝑈(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑈_𝑖,𝑉(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑉_𝑖,𝑊(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑊_𝑖,𝑋(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑋_𝑖,𝑌(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑌_𝑖,𝑍(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑍_𝑖,_𝑁𝑖=1𝛼_𝑖=1,𝛼_𝑖𝐶≥0,_𝑗𝑖,𝐷^𝑗_𝑖,𝐸^𝑗_𝑖,𝐺^𝑗_𝑖,𝐿^𝑗_𝑖,𝑃_𝑖,𝑄_𝑖,𝑅_𝑖,𝑆_𝑖,𝑇_𝑖,𝑈_𝑖,𝑉_𝑖,𝑊_𝑖,𝑋_𝑖,𝑌_𝑖,𝑍_𝑖∈𝑀^𝑛×𝑛,𝑗=1,2,3,𝑖=1,2,…,𝑁,𝜌=ℎ₂−ℎ₁.Moreover,_𝑖,𝑗=Σ¹¹_𝑖,𝑗Σ¹²_𝑖,𝑗Σ¹³_𝑖,𝑗Σ¹⁴_𝑖,𝑗Σ¹⁵_𝑖,𝑗Σ_𝑖¹⁶Σ¹⁷_𝑖,𝑗Σ¹⁸_𝑖,𝑗0∗Σ²²_𝑖,𝑗00000Σ²⁸_𝑖,𝑗0∗∗Σ_𝑖³³Σ_𝑖³⁴0Σ³⁶_𝑖,𝑗00Σ¹⁸_𝑖,𝑗∗∗∗Σ_𝑖⁴⁴00Σ⁴⁷_𝑖,𝑗0Σ¹⁸_𝑖,𝑗∗∗∗∗Σ_𝑖⁵⁵00Σ¹⁸_𝑖,𝑗0∗∗∗∗∗Σ_𝑖⁶⁶000∗∗∗∗∗∗Σ⁷⁷_𝑖,𝑗00∗∗∗∗∗∗∗Σ⁸⁸_𝑖,𝑗0∗∗∗∗∗∗∗∗Σ⁹⁹_𝑖,𝑗, (10) where Σ¹¹_𝑖,𝑗=𝑄_𝑖+𝑅_𝑖+𝑇_𝑖+ℎ₁𝑀_𝑖+𝑁_𝑖+𝑁^𝑇_𝑖+ℎ₂𝐾_𝑖+𝐿_𝑖+𝐿^𝑇_𝑖+𝐶¹_𝑖+𝐶¹_𝑖^𝑇+𝐷¹_𝑖+𝐷¹_𝑖^𝑇+𝐸¹_𝑖+𝐸¹_𝑖^𝑇+𝐿²_𝑖^𝑇𝐴_𝑗+𝐴^𝑇_𝑖𝐿²_𝑗+𝐿²_𝑖^𝑇𝐵_𝑗+𝐵^𝑇_𝑖𝐿²_𝑗−𝐿²_𝑖−𝐿²_𝑖^𝑇,Σ¹²_𝑖,𝑗=𝐴^𝑇_𝑖𝐿¹_𝑗+𝐵^𝑇_𝑖𝐿¹_𝑗−𝐿¹_𝑖−𝐿²_𝑖^𝑇+𝑃_𝑖,Σ_𝑖¹³=𝐶²_𝑖−𝑁_𝑖−𝐶¹_𝑖^𝑇,Σ_𝑖¹⁴=𝐷²_𝑖−𝐿_𝑖−𝐷¹_𝑖^𝑇,Σ_𝑖¹⁵=−𝐸¹_𝑖^𝑇+𝐸²_𝑖,Σ_𝑖¹⁶=−𝐶¹_𝑖^𝑇+𝐶³_𝑖,Σ_𝑖¹⁷=−𝐷¹_𝑖^𝑇+𝐷³_𝑖,Σ¹⁸_𝑖,𝑗=−𝐸¹_𝑖^𝑇+𝐸³_𝑖−𝐿²_𝑖^𝑇𝐵_𝑗+𝐴^𝑇_𝑖𝐿³_𝑗+𝐵^𝑇_𝑖𝐿³_𝑗−𝐿³_𝑖,Σ_𝑖²²=𝑃_𝑖+ℎ²₁𝑈_𝑖+ℎ²₂𝑉_𝑖+ℎ²₂𝑊_𝑖+𝜌²𝑋_𝑖+ℎ₁𝑌_𝑖+ℎ₂𝑍_𝑖−𝐿¹_𝑖−𝐿¹_𝑖^𝑇,Σ²⁸_𝑖,𝑗=−𝐿¹_𝑖^𝑇𝐵_𝑗−𝐿³_𝑖,Σ_𝑖³³=−𝑄_𝑖+𝑆_𝑖−𝐶²_𝑖−𝐶²_𝑖^𝑇+𝐺¹_𝑖+𝐺¹_𝑖^𝑇,Σ_𝑖³⁴=𝐺²_𝑖−𝐺¹_𝑖^𝑇,Σ_𝑖³⁶=−𝐶³_𝑖−𝐶²_𝑖^𝑇,Σ_𝑖³⁹=𝐺³_𝑖−𝐺¹_𝑖^𝑇,Σ_𝑖⁴⁴=−𝑅_𝑖−𝑆_𝑖−𝐷²_𝑖−𝐷²_𝑖^𝑇−𝐺²_𝑖−𝐺²_𝑖^𝑇,Σ_𝑖⁴⁷=−𝐷³_𝑖−𝐷²_𝑖^𝑇,Σ_𝑖⁴⁹=−𝐺³_𝑖−𝐺²_𝑖^𝑇,Σ_𝑖⁵⁵=−𝑇_𝑖−𝐸²_𝑖−𝐸²_𝑖^𝑇,Σ_𝑖⁵⁸=−𝐸³_𝑖−𝐸²_𝑖^𝑇,Σ_𝑖⁶⁶=−𝑈_𝑖−𝐶³_𝑖−𝐶³_𝑖^𝑇,Σ_𝑖⁷⁷=−𝑉_𝑖−𝐷³_𝑖−𝐷³_𝑖^𝑇,Σ⁸⁸_𝑖,𝑗=−𝑊_𝑖−𝐸³_𝑖−𝐸³_𝑖^𝑇−𝐿³_𝑖^𝑇𝐵_𝑗−𝐵^𝑇_𝑖𝐿³_𝑗,Σ⁹⁹_𝑖,𝑗=−𝑋_𝑖−𝐺³_𝑖−𝐺³_𝑖^𝑇. (11)

Theorem 5. The system (1)-(2) is robustly stable if there exist positive definite symmetric matrices 𝑃_𝑖,𝑄_𝑖,𝑅_𝑖,𝑆_𝑖,𝑇_𝑖,𝑈_𝑖,𝑉_𝑖,𝑊_𝑖,𝑋_𝑖,𝑌_𝑖, and 𝑍_𝑖, any appropriate dimensional matrices 𝑀_𝑖,𝑁_𝑖,𝐾_𝑖,𝐿_𝑖,𝐶^𝑗_𝑖,𝐷^𝑗_𝑖,𝐸^𝑗_𝑖,𝐺^𝑗_𝑖, and 𝐿^𝑗_𝑖, 𝑗=1,2,3, 𝑖=1,2,…,𝑁 satisfying the following LMIs: _𝑖,𝑖<−𝐼,𝑖=1,2,…,𝑁,_𝑖,𝑗+_𝑗,𝑖<2𝑀𝑁−1𝐼,𝑖=1,…,𝑁−1,𝑗=𝑖+1,…,𝑁,_𝑖𝑁_𝑖𝑁^𝑇_𝑖𝑌_𝑖𝐾≥0,𝑖=1,2,…,𝑁,_𝑖𝐿_𝑖𝐿^𝑇_𝑖𝑍_𝑖≥0,𝑖=1,2,…,𝑁. (12)(13)(14)(15)

Proof. Consider the following parameter-dependent Lyapunov-Krasovskii function for the system (9) of the form 𝑉(𝑘)=⁵_𝑖=1𝑉_𝑖(𝑘), (16) where 𝑉₁(𝑘)=𝑥^𝑇(𝑘)𝑃(𝛼)𝑥(𝑘),𝑉₂(𝑘)=^𝑘−1_{𝑖=𝑘−ℎ1}𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)+^𝑘−1_{𝑖=𝑘−ℎ2}𝑥^𝑇(𝑖)𝑅(𝛼)𝑥(𝑖)+^{𝑘−ℎ1−1}_{𝑖=𝑘−ℎ2}𝑥^𝑇(𝑖)𝑆(𝛼)𝑥(𝑖)+^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑥^𝑇(𝑖)𝑇(𝛼)𝑥(𝑖),𝑉₃(𝑘)=ℎ₁⁻¹_𝑗=−ℎ1^𝑘−1_{𝑖=𝑘+𝑗}𝑦^𝑇(𝑖)𝑈(𝛼)𝑦(𝑖)+ℎ₂⁻¹_𝑗=−ℎ2^𝑘−1_{𝑖=𝑘+𝑗}𝑦^𝑇(𝑖)𝑉(𝛼)𝑦(𝑖)+ℎ₂⁻¹_𝑗=−ℎ2^𝑘−1_{𝑖=𝑘+𝑗}𝑦^𝑇(𝑖)𝑊(𝛼)𝑦(𝑖),𝑉₄(𝑘)=𝜌^−ℎ1−1_𝑗=−ℎ2^𝑘−1_{𝑖=𝑘+𝑗}𝑦^𝑇(𝑖)𝑋(𝛼)𝑦(𝑖),𝑉₅(𝑘)=⁰_{𝑗=−ℎ1+1}^𝑘−1_{𝑖=𝑘−1+𝑗}𝑦^𝑇(𝑖)𝑌(𝛼)𝑦(𝑖)+⁰_{𝑗=−ℎ2+1}^𝑘−1_{𝑖=𝑘−1+𝑗}𝑦^𝑇(𝑖)𝑍(𝛼)𝑦(𝑖). (17) Evaluating the forward deference of 𝑉(𝑘), it is defined as Δ𝑉(𝑘)=𝑉(𝑘+1)−𝑉(𝑘). (18) Let us define, for 𝑖=1,2,…,5, Δ𝑉_𝑖(𝑘)=𝑉_𝑖(𝑘+1)−𝑉_𝑖(𝑘). (19) Then along the solution of the system (9), we obtain Δ𝑉₁(𝑘)=𝑉₁(𝑘+1)−𝑉₁(𝑘)=𝑥^𝑇(𝑘+1)𝑃(𝛼)𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃(𝛼)𝑥(𝑘)=2𝑥^𝑇(𝑘)𝑃(𝛼)𝑦(𝑘)+𝑦^𝑇(𝑘)𝑃(𝛼)𝑦(𝑘)+2𝑥^𝑇(𝑘)𝐿^𝑇₁(𝛼)(𝑘)+[𝐴(𝛼)(𝛼)](𝑘)(𝛼)−𝑦+𝐵−𝐼×𝑥−𝐵^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑦(𝑖)+2𝑦^𝑇(𝑘)𝐿^𝑇₂(𝛼)(𝑘)+[𝐴(𝛼)(𝛼)](𝑘)(𝛼)−𝑦+𝐵−𝐼×𝑥−𝐵^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑦(𝑖)+2^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑦(𝑖)^𝑇𝐿^𝑇₃(𝛼)(𝑘)+[𝐴(𝛼)(𝛼)](𝑘)(𝛼)−𝑦+𝐵−𝐼×𝑥−𝐵^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑦(𝑖),Δ𝑉₂(𝑘)=𝑉₂(𝑘+1)−𝑉₂(𝑘)=𝑥^𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)−𝑥^𝑇𝑘−ℎ₁𝑄(𝛼)𝑥𝑘−ℎ₁+𝑥^𝑇(𝑘)𝑅(𝛼)𝑥(𝑘)−𝑥^𝑇𝑘−ℎ₂𝑅(𝛼)𝑥𝑘−ℎ₂+𝑥^𝑇𝑘−ℎ₁𝑆(𝛼)𝑥𝑘−ℎ₁−𝑥^𝑇𝑘−ℎ₂×𝑆(𝛼)𝑥𝑘−ℎ₂+𝑥^𝑇(𝑘)𝑇(𝛼)𝑥(𝑘)−𝑥^𝑇(𝑘−ℎ(𝑘))𝑇(𝛼)𝑥(𝑘−ℎ(𝑘)). (20) From Proposition 4, we have Δ𝑉₃(𝑘)=𝑉₃(𝑘+1)−𝑉₃(𝑘)=ℎ²₁𝑦^𝑇(𝑘)𝑈(𝛼)𝑦(𝑘)−ℎ₁^𝑘−1_{𝑖=𝑘−ℎ1}𝑦^𝑇(𝑖)𝑈(𝛼)𝑦(𝑖)+ℎ²₂𝑦^𝑇(𝑘)𝑉(𝛼)𝑦(𝑘)−ℎ₂^𝑘−1_{𝑖=𝑘−ℎ2}𝑦^𝑇(𝑖)𝑉(𝛼)𝑦(𝑖)+ℎ²₂𝑦^𝑇(𝑘)𝑊(𝛼)𝑦(𝑘)−ℎ₂^𝑘−1_{𝑖=𝑘−ℎ2}𝑦^𝑇(𝑖)𝑊(𝛼)𝑦(𝑖)≤ℎ²₁𝑦^𝑇(𝑘)𝑈(𝛼)𝑦(𝑘)−^𝑘−1_{𝑖=𝑘−ℎ1}𝑦(𝑖)^𝑇×𝑈(𝛼)^𝑘−1_{𝑖=𝑘−ℎ1}𝑦(𝑖)+ℎ²₂𝑦^𝑇(𝑘)𝑉(𝛼)𝑦(𝑘)−^𝑘−1_{𝑖=𝑘−ℎ2}𝑦(𝑖)^𝑇𝑉(𝛼)^𝑘−1_{𝑖=𝑘−ℎ2}𝑦(𝑖)+ℎ²₂𝑦^𝑇(𝑘)𝑊(𝛼)𝑦(𝑘)−^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑦(𝑖)^𝑇×𝑊(𝛼)^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑦(𝑖),Δ𝑉₄(𝑘)=𝑉₄(𝑘+1)−𝑉₄(𝑘)=𝜌²𝑦^𝑇(𝑘)𝑋(𝛼)𝑦(𝑘)−𝜌^{𝑘−ℎ1−1}_{𝑖=𝑘−ℎ2}𝑦^𝑇(𝑖)𝑋(𝛼)𝑦(𝑖)≤𝜌²𝑦^𝑇(𝑘)𝑋(𝛼)𝑦(𝑘)−^{𝑘−ℎ1−1}_{𝑖=𝑘−ℎ2}𝑦(𝑖)^𝑇𝑋(𝛼)×^{𝑘−ℎ1−1}_{𝑖=𝑘−ℎ2}𝑦(𝑖),Δ𝑉₅(𝑘)=𝑉₅(𝑘+1)−𝑉₅(𝑘)=ℎ₁𝑦^𝑇(𝑘)𝑌(𝛼)𝑦(𝑘)−⁰_{𝑖=−ℎ1+1}𝑦^𝑇()𝑌(𝛼)()𝑘−1+𝑖×𝑦𝑘−1+𝑖+ℎ₂𝑦^𝑇(𝑘)𝑍(𝛼)𝑦(𝑘)−⁰_{𝑖=−ℎ2+1}𝑦^𝑇()𝑍(𝛼)().𝑘−1+𝑖×𝑦𝑘−1+𝑖 (21) We can show that 2𝑥^𝑇(𝑘)𝑁(𝛼)⁰_{𝑖=−ℎ1+1}𝑦()+𝑘−1+𝑖_{0𝑖=−ℎ1+1}𝑦^𝑇(𝑘−1+𝑖)𝑌(𝛼)𝑦(𝑘−1+𝑖)+ℎ₁𝑥^𝑇(𝑘)𝑀(𝛼)𝑥(𝑘)=⁰_{𝑖=−ℎ1+1}𝑥(𝑘)𝑦(𝑘−1+𝑖)^𝑇𝑀(𝛼)𝑁(𝛼)𝑁^𝑇(𝛼)𝑌(𝛼)×𝑥(𝑘)𝑦(𝑘−1+𝑖)≥0. (22) It is easy to see that 2𝑥^𝑇(𝑘)𝐿(𝛼)⁰_{𝑖=−ℎ1+1}𝑦()+𝑘−1+𝑖_{0𝑖=−ℎ2+1}𝑦^𝑇(𝑘−1+𝑖)𝑍(𝛼)𝑦(𝑘−1+𝑖)+ℎ₂𝑥^𝑇(𝑘)𝐾(𝛼)𝑥(𝑘)=⁰_{𝑖=−ℎ2+1}𝑥(𝑘)𝑦(𝑘−1+𝑖)^𝑇𝐾(𝛼)𝐿(𝛼)𝐿^𝑇(𝛼)𝑍(𝛼)×𝑥(𝑘)𝑦(𝑘−1+𝑖)≥0. (23) By (22) and (23), we can obtain −⁰_{𝑖=−ℎ1+1}𝑦^𝑇(𝑘−1+𝑖)𝑌(𝛼)𝑦(𝑘−1+𝑖)≤ℎ₁𝑥^𝑇(𝑘)𝑀(𝛼)𝑥(𝑘)+2𝑥^𝑇(𝑘)𝑁(𝛼)×⁰_{𝑖=−ℎ1+1}𝑦(𝑘−1+𝑖)=ℎ₁𝑥^𝑇(𝑘)𝑀(𝛼)𝑥(𝑘)+2𝑥^𝑇(𝑘)𝑁(𝛼)𝑥(𝑘)−2𝑥^𝑇(𝑘)𝑁(𝛼)𝑥𝑘−ℎ₁, (24) and we conclude that −⁰_{𝑖=−ℎ2+1}𝑦^𝑇(𝑘−1+𝑖)𝑍(𝛼)𝑦(𝑘−1+𝑖)≤ℎ₂𝑥^𝑇(𝑘)𝐾(𝛼)𝑥(𝑘)+2𝑥^𝑇(𝑘)𝐿(𝛼)×⁰_{𝑖=−ℎ2+1}𝑦(𝑘−1+𝑖)=ℎ₂𝑥^𝑇(𝑘)𝐾(𝛼)𝑥(𝑘)+2𝑥^𝑇(𝑘)𝐿(𝛼)𝑥(𝑘)−2𝑥^𝑇(𝑘)𝐿(𝛼)𝑥𝑘−ℎ₂. (25) It is obvious that Υ≡𝑥(𝑘)−𝑥𝑘−ℎ₁−^𝑘−1_{𝑖=𝑘−ℎ1}𝑦(𝑖)(𝑘)=0,Φ≡𝑥−𝑥𝑘−ℎ₂−^𝑘−1_{𝑖=𝑘−ℎ2}𝑦(𝑖)(𝑘)((𝑘−=0,Ψ≡𝑥−𝑥𝑘−ℎ))^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑦(𝑖)=0,Ω≡𝑥𝑘−ℎ₁−𝑥𝑘−ℎ₂−^{𝑘−ℎ1−1}_{𝑖=𝑘−ℎ2}𝑦(𝑖)=0. (26) The following equations are true for any polytopic matrices with appropriate dimensions: 2𝑥^𝑇(𝑘)𝐶^𝑇₁(𝛼)+2𝑥^𝑇𝑘−ℎ₁𝐶^𝑇₂(𝛼)+2^𝑘−1_{𝑖=𝑘−ℎ1}𝑦(𝑖)^𝑇𝐶^𝑇₃(𝛼)×Υ=0,2𝑥^𝑇(𝑘)𝐷^𝑇₁(𝛼)+2𝑥^𝑇𝑘−ℎ₂𝐷^𝑇₂(𝛼)+2^𝑘−1_{𝑖=𝑘−ℎ2}𝑦(𝑖)^𝑇𝐷^𝑇₃(𝛼)×Φ=0,2𝑥^𝑇(𝑘)𝐸^𝑇₁(𝛼)+2𝑥^𝑇(𝑘−ℎ(𝑘))𝐸^𝑇₂(𝛼)+2^𝑘−1_{(𝑘)𝑖=𝑘−ℎ}𝑦(𝑖)^𝑇𝐸^𝑇₃(𝛼)×Ψ=0,2𝑥^𝑇𝑘−ℎ₁𝐺^𝑇₁(𝛼)+2𝑥^𝑇𝑘−ℎ₂𝐺^𝑇₂(𝛼)+2^{𝑘−ℎ1−1}_{𝑖=𝑘−ℎ2}𝑦(𝑖)^𝑇𝐺^𝑇₃(𝛼)×Ω=0. (27)(28)(29)(30) It follows from (18)–(30) that Δ𝑉(𝑘)≤𝜉^𝑇(𝑘)^𝑁𝑁_𝑖=1𝑗=1𝛼_𝑖𝛼_{𝑗𝑖,𝑗}𝜉(𝑘), (31) where 𝜉^𝑇(𝑘)=[𝑥(𝑘)^𝑇𝑦(𝑘)^𝑇𝑥(𝑘−ℎ₁)^𝑇𝑥(𝑘−ℎ₂)^𝑇𝑥(𝑘−ℎ(𝑘))^𝑇∑^𝑘−1_{𝑖=𝑘−ℎ1}𝑦^𝑇(𝑖)∑^𝑘−1_{𝑖=𝑘−ℎ2}𝑦^𝑇(𝑖)∑^𝑘−1_{𝑖=𝑘−ℎ(𝑘)}𝑦(𝑖)^𝑇∑^{𝑘−ℎ1−1}_{𝑖=𝑘−ℎ2}𝑦^𝑇(𝑖)] and ∏_𝑖,𝑗 is defined in (10). Due to the fact that ∑^𝑁_𝑖=1𝛼_𝑖=1, we obtain the following identities: ^𝑁𝑁_𝑖=1𝑗=1𝛼_𝑖𝛼_{𝑗𝑖,𝑗}=^𝑁_𝑖=1𝛼²_𝑖,𝑖+^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝛼_𝑖𝛼_{𝑖𝑖,𝑗}+_𝑗,𝑖(),𝑁−1^𝑁_𝑖=1𝛼²_𝑖−2^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝛼_𝑖𝛼_𝑗=^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝛼_𝑖−𝛼_𝑗²≥0. (32)(33) By (31)–(33), if the conditions (12)–(15) are true, then Δ𝑉(𝑘)<−𝜔‖𝑥‖², (34) where 𝜔>0. This means that the system (1)-(2) is robustly stable. The proof of the theorem is complete.

If 𝐴(𝛼)=𝐴 and 𝐵(𝛼)=𝐵 when 𝐴,𝐵∈𝑀^𝑛×𝑛 then the system (1)-(2) reduces to the following system: 𝑥()(𝑘)((𝑘,𝑥(𝑠)(𝑠)𝑘+1=𝐴𝑥+𝐵𝑥𝑘−ℎ))=𝜙,𝑠∈−ℎ₂,…,−1,0. (35)(36) Take the Lyapunov-Krasovskii functional as (16), where 𝑃(𝛼)=𝑃, 𝑄(𝛼)=𝑄, 𝑅(𝛼)=𝑅, 𝑆(𝛼)=𝑆, 𝑇(𝛼)=𝑇, 𝑈(𝛼)=𝑈, 𝑉(𝛼)=𝑉, 𝑊(𝛼)=𝑊, 𝑋(𝛼)=𝑋, 𝑌(𝛼)=𝑌, and 𝑍(𝛼)=𝑍 when 𝑃,𝑄,𝑅,𝑆,𝑇,𝑈,𝑉,𝑊,𝑋,𝑌,𝑍∈𝑀^𝑛×𝑛. Moreover, let us set polytopic matrices with appropriate dimensions of the forms 𝐶_𝑗(𝛼)=𝐶^𝑗,𝐷_𝑗(𝛼)=𝐷^𝑗,𝐸_𝑗(𝛼)=𝐸^𝑗,𝐺_𝑗(𝛼)=𝐺^𝑗, and 𝐿_𝑗(𝛼)=𝐿^𝑗 when 𝐶^𝑗,𝐷^𝑗,𝐸^𝑗,𝐺^𝑗,𝐿^𝑗∈𝑀^𝑛×𝑛,𝑗=1,2,3. According to Theorem 5, we have Corollary 6 for the delay-dependent asymptotically stability criterion of the system (35)-(36). We introduce the following notations for later use: Σ=¹¹Σ¹²Σ¹³Σ¹⁴Σ¹⁵Σ¹⁶Σ¹⁷Σ¹⁸0∗Σ²²00000Σ²⁸0∗∗Σ³³Σ³⁴0Σ³⁶00Σ¹⁸∗∗∗Σ⁴⁴00Σ⁴⁷0Σ¹⁸∗∗∗∗Σ⁵⁵00Σ¹⁸0∗∗∗∗∗Σ⁶⁶000∗∗∗∗∗∗Σ⁷⁷00∗∗∗∗∗∗∗Σ⁸⁸0∗∗∗∗∗∗∗∗Σ⁹⁹, (37) where Σ¹¹=𝑄+𝑅+𝑇+ℎ₁𝑀+𝑁+𝑁^𝑇+ℎ₂𝐾+𝐿+𝐿^𝑇+𝐶¹+𝐶^1𝑇+𝐷¹+𝐷^1𝑇+𝐸¹+𝐸^1𝑇+𝐿^2𝑇𝐴+𝐴^𝑇𝐿²+𝐿^2𝑇𝐵+𝐵^𝑇𝐿²−𝐿²−𝐿^2𝑇,Σ¹²=𝐴^𝑇𝐿¹+𝐵^𝑇𝐿¹−𝐿¹−𝐿^2𝑇Σ+𝑃,₁₃=𝐶²−𝑁−𝐶^1𝑇,Σ¹⁴=𝐷²−𝐿−𝐷^1𝑇,Σ¹⁵=−𝐸^1𝑇+𝐸²,Σ¹⁶=−𝐶^1𝑇+𝐶³,Σ¹⁷=−𝐷^1𝑇+𝐷³,Σ¹⁸=−𝐸^1𝑇+𝐸³−𝐿^2𝑇𝐵+𝐴^𝑇𝐿³+𝐵^𝑇𝐿³−𝐿³,Σ²²=𝑃+ℎ²₁𝑈+ℎ²₂𝑉+ℎ²₂𝑊+𝜌²𝑋+ℎ₁𝑌+ℎ₂𝑍−𝐿¹−𝐿^1𝑇,Σ²⁸=−𝐿^1𝑇𝐵−𝐿³,Σ³³=−𝑄+𝑆−𝐶²−𝐶^2𝑇+𝐺¹+𝐺^1𝑇,Σ³⁴=𝐺²−𝐺^1𝑇,Σ³⁶=−𝐶³−𝐶^2𝑇,Σ³⁹=𝐺³−𝐺^1𝑇,Σ⁴⁴=−𝑅−𝑆−𝐷²−𝐷^2𝑇−𝐺²−𝐺^2𝑇,Σ⁴⁷=−𝐷³−𝐷^2𝑇,Σ⁴⁹=−𝐺³−𝐺^2𝑇,Σ⁵⁵=−𝑇−𝐸²−𝐸^2𝑇,Σ⁵⁸=−𝐸³−𝐸^2𝑇,Σ⁶⁶=−𝑈−𝐶³−𝐶^3𝑇,Σ⁷⁷=−𝑉−𝐷³−𝐷^3𝑇,Σ⁸⁸=−𝑊−𝐸³−𝐸^3𝑇−𝐿^3𝑇𝐵−𝐵^𝑇𝐿³,Σ⁹⁹=−𝑋−𝐺³−𝐺^3𝑇. (38)

Corollary 6. The system (35)-(36) is asymptotically stable if there exist positive definite symmetric matrices 𝑃,𝑄,𝑅,𝑆,𝑇,𝑈,𝑉,𝑊,𝑋,𝑌, and 𝑍, any appropriate dimensional matrices 𝑀,𝑁,𝐾,𝐿,𝐶^𝑗,𝐷^𝑗,𝐸^𝑗,𝐺^𝑗, and 𝐿^𝑗, 𝑗=1,2,3, satisfying the following LMIs: 𝑁<0,𝑀𝑁_𝑇𝑌𝐿≥0,𝐾𝐿_𝑇𝑍≥0. (39)

4. Numerical Examples

Example 7. Consider the following LPD discrete-time system with interval time-varying delays (1)-(2) with 𝐴₁=0.8000.010.60,𝐴₂=,𝐵0.9000.050.90₁=0.1000.200.10,𝐵₂=,−0.100−0.20−0.10 (40)ℎ(𝑘)=2+14cos²(𝑘𝜋/2) with initial condition 𝜙(𝑘)=^−5.5_8.5, 𝑘∈[−16,0]. The numerical solutions 𝑥₁(𝑘) and 𝑥₂(𝑘) of (1)-(2) with (40) are plotted in Figure 1.

Figure 1 

The Simulation solution of the states 𝑥1(𝑘) and 𝑥2(𝑘) in Example 7 for LPD discrete-time delayed system with initial conditions 𝑥1(𝑘)=−5.5 and 𝑥2(𝑘)=8.5, 𝑘=−16,…,−2,−1,0 and 𝛼1=1/3, 𝛼2=2/3 by using method of Runge-Kutta order 4 (ℎ=0.01) with Matlab.

Solution
By using the LMI Toolbox in MATLAB (with accuracy 0.01) and conditions (12)–(15) of Theorem 5, this system is robustly stable for discrete delay time satisfying ℎ₁=2, ℎ₂=16, and 𝑃₁=10⁶×,𝑃4.3970−0.4104−0.41040.3531₂=10⁵×,𝑄5.2274−0.0613−0.06130.3488₁=10⁴×,𝑄4.9438−1.3988−1.39881.3372₂=10⁴×,𝑅1.6042−0.3173−0.31730.1431₁=10⁴×,𝑅1.8144−0.5737−0.57370.4286₂=10³×,𝑆4.2669−0.8605−0.86050.3167₁=10⁴×,𝑆2.3161−0.7409−0.74090.5638₂=10³×,𝑇5.5557−1.1238−1.12380.4007₁=10⁵×,𝑇2.2862−0.3474−0.34740.2764₂=10⁴×,𝑈5.08670.31980.31980.2709₁=10⁴×,𝑈9.6266−0.3736−0.37362.6793₂=10⁴×,𝑉6.7104−1.2896−1.28960.6493₁=10³×,𝑉2.6354−0.1810−0.18100.2286₂=10³×,𝑊1.1098−0.2116−0.21160.0795₁=10⁴×,𝑊1.25810.00310.00310.0279₂=10³×,𝑋9.88600.99470.99470.4654₁=10³×,𝑋3.6535−0.2316−0.23160.3050₂=10³×,𝑌1.5049−0.2858−0.28580.1068₁=10⁵×,𝑌1.0677−0.0668−0.06680.3783₂=10⁴×,𝑍8.4989−1.5451−1.54511.0863₁=10⁴×,𝑍4.8154−0.3746−0.37460.6475₂=10⁴×,𝐶2.5697−0.4901−0.49010.2043₁₁=10⁴×,𝐶−1.5983−0.1755−0.1755−1.6231₁₂=10⁴×,𝐶−2.14480.11270.1127−1.6904₂₁=10⁴×,𝐶1.28370.17100.17101.5002₂₂=10⁴×,𝐶1.6947−0.0260−0.02601.6512₃₁=10³×,𝐶−7.30490.24910.24919.7027₃₂=10⁴×,𝐷−0.13740.33890.33891.4742₁₁=10⁴×,𝐷−0.9350−0.1962−0.1962−1.4853₁₂=10⁴×,𝐷−1.70990.02290.0229−1.6489₂₁=10⁴×,𝐷0.57140.29500.29501.3754₂₂=10⁴×,𝐷1.39280.03930.03931.6269₃₁=10⁴×,𝐷1.9066−0.0891−0.08911.7055₃₂=10⁴×,𝐸1.7454−0.0290−0.02901.6519₁₁=10⁴×,𝐸−4.2910−1.6701−1.6701−1.5104₁₂=10⁴×,𝐸−3.7600−0.2382−0.2382−1.7532₂₁=10⁴×,𝐸−7.02370.57630.57630.8408₂₂=10⁴×,𝐸−0.7534−0.1269−0.12691.4957₃₁=10⁵×,𝐸1.0553−0.0488−0.04880.2403₃₂=10⁴×,𝐺3.84460.27210.27211.7426₁₁=10⁴×,𝐺−1.1174−0.1492−0.1492−1.4929₁₂=10⁴×,𝐺−1.6145−0.0032−0.0032−1.6397₂₁=10⁴×,𝐺0.68490.27380.27381.3873₂₂=10⁴×,𝐺1.35460.05450.05451.6203₃₁=10⁴×,𝐺1.8742−0.0945−0.09451.7071₃₂=10⁴×,𝐿1.7014−0.0146−0.01461.6454₁₁=10⁶×,𝐿5.3779−0.2824−0.28240.4796₁₂=10⁶×,𝐿2.31170.02940.02940.3636₂₁=10⁶×,𝐿4.0552−0.3696−0.36960.2311₂₂=10⁴×,𝐿9.42670.91100.91107.4877₃₁=10⁵×,𝐿−2.9313−0.2209−0.2209−0.4362₃₂=10⁵×,𝐾2.1784−0.0837−0.0837−0.1597₁=,𝐾897.9190−266.0787−266.0787187.1557₂=,𝑀236.7231−47.9750−47.975019.2914₁=10³×,𝑀6.1473−1.9404−1.94041.4158₂=10³×,𝑁1.6184−0.3310−0.33100.1305₁=10³×,𝑁−5.45980.96280.9628−2.2399₂=10³×,𝐿−3.99630.77620.7762−0.4562₁=10³×,𝐿−2.49820.31020.3102−0.3738₂=10³×.−1.19120.23240.2324−0.0945 (41) For the given ℎ₁, Table 1 lists the comparison of the upper bounds delay ℎ₂ for robust stability of the system (1)-(2) with (40) by the different method. By a conditions in Theorem 5, we can see from Table 1 that our result is superior to those in [22, Theorem 1].

Table 1 

Comparison of the maximum allowed time delay ℎ₂ for different conditions.

Example 8. Consider the system (35) with 𝐴=0.8000.050.90,𝐵=−0.100−0.20−0.10. (42) For the given ℎ₁, we calculate the allowable maximum value of ℎ₂ that guarantees the asymptotic stability of the system (35) with (42). By using different methods, the calculated results are presented in Table 2. From the table, we can see that Corollary 6 in this paper provides the less conservative results.

Table 2 

Comparison of the maximum allowed time delay ℎ₂ for different conditions.

Acknowledgments

This work is supported by National Research Council of Thailand and Khon Kaen University, Thailand (Grant no. kku fmis 121740).