Robust Exponential Stability for LPD Discrete-Time System with Interval Time-Varying Delay

Abstract

This paper investigates the problem of robust exponential stability for uncertain linear-parameter dependent (LPD) discrete-time system with delay. The delay is of an interval type, which means that both lower and upper bounds for the time-varying delay are available. The uncertainty under consideration is norm-bounded uncertainty. Based on combination of the linear matrix inequality (LMI) technique and the use of suitable Lyapunov-Krasovskii functional, new sufficient conditions for the robust exponential stability are obtained in terms of LMI. Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods.

1. Introduction

Over the past decades, the problem of stability analysis of delay discrete-time systems has been widely investigated by many researchers. Because the existence of time delay is frequent, a source of oscillation instability performances degradation of systems. Stability criteria for discrete-time systems with time delay is generally divided into two classes: delay-independent ones and delay-dependent ones. Delay-independent stability criteria tend to be more conservative, especially for small-size delay; such criteria do not give any information on the size of the delay. On the other hand, delay-dependent stability criteria is concerned with the size of the delay and usually provide a maximal delay size. Moreover, robust stability of linear 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 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 [1–15] and references cited therein. Most of these conditions have been obtained via the Lyapunov theory approaches in which parameter dependent Lyapunov functions have been employed. These conditions are always expressed in terms of linear matrix inequalities (LMIs) which can be solved numerically by using available tools such as LMI toolbox in MATLAB. The results have been obtained for robust stability for LPD systems in which time delay occur in state variable such as [6, 11, 14] present sufficient conditions for robust stability of LPD continuous-time system with delays. However, a few results have been obtained for robust stability for LPD discrete-time systems with delay.

In this paper, we deal with the problem of robust exponential stability for uncertain LPD discrete-time system with interval time-varying delay. Combined with the linear matrix inequality technique and the use of suitable Lyapunov-Krasovskii functional, new sufficient conditions for the robust exponential stability are obtained in terms of LMI. Finally, numerical examples have demonstrated the effectiveness of the criteria.

2. Problem Formulation and Preliminaries

We introduce some notations and definitions that will be used throughout the paper. ℤ⁺ denotes the set of non negative 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 matrices of (𝑛×𝑟)-dimensions; 𝐴^𝑇 denotes 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)}; Matrix 𝐴 is called semi positive definite (𝐴≥0) if 𝑥^𝑇𝐴𝑥≥0, for all 𝑥∈ℝ^𝑛; 𝐴 is positive definite (𝐴>0) if 𝑥^𝑇𝐴𝑥>0 for all 𝑥≠0; Matrix 𝐵 is called semi-negative definite (𝐵≤0) if 𝑥^𝑇𝐵𝑥≤0, for all 𝑥∈ℝ^𝑛;𝐵 is 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 delay in the state 𝑥(𝑘+1)=[𝐴(𝛼𝑥)(+𝑠)Δ=𝐴𝜙(𝑘(𝑠))],𝑥(𝑘𝑠)=+−[𝐵ℎ₂(𝛼)+Δ𝐵(𝑘)]𝑥(𝑘−ℎ(𝑘)),,…,−1,0,(2.1) where 𝑘∈ℤ⁺, 𝑥(𝑘)∈ℝ^𝑛 is the system state and 𝜙(𝑠) is a 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,_𝑁_𝑖=1𝛼_𝑖=1,𝛼_𝑖≥0,𝐴_𝑖,𝐵_𝑖∈𝑀^𝑛×𝑛,𝑖=1,…,𝑁.(2.2)Δ𝐴(𝑘) and Δ𝐵(𝑘) are unknown matrices representing time-varying parameter uncertainties, we assumed to be of the form Δ𝐴𝐴(𝑘)=𝐾(𝛼)Δ(𝑘)𝐴₁(𝛼),Δ𝐵(𝑘)=𝐾(𝛼)Δ(𝑘)𝐵₁(𝛼),₁(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐴^1𝑖,𝐵₁(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐵^1𝑖,𝐾(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐾_𝑖,_𝑁_𝑖=1𝛼_𝑖=1,𝛼_𝑖≥0,𝐴^1𝑖,𝐵^1𝑖∈𝑀^𝑛×𝑛,𝑖=1,…,𝑁.(2.3) The class of parametric uncertainties Δ(𝑘), which satisfies Δ(𝑘)=𝐹(𝑘)[𝐼−𝐽𝐹(𝑘)]⁻¹,(2.4) is said to be admissible where 𝐽 is a known matrix satisfying 𝐼−𝐽𝐽^𝑇>0,(2.5) and 𝐹(𝑘) is uncertain matrix satisfying 𝐹(𝑘)^𝑇𝐹(𝑘)≤𝐼.(2.6) In addition, we assume that the time-varying delay ℎ(𝑘) is upper and lower bounded. It satisfies the following assumption of the form ℎ₁≤ℎ(𝑘)≤ℎ₂,(2.7) where ℎ₁ and ℎ₂ are known positive integers.

Definition 2.1. The uncertain LPD discrete-time-delayed system in (2.1) is said to be robustly exponentially stable if there exist constant scalars 0<𝑎<1 and 𝑏>0 such that ‖𝑥(𝑘)‖²≤𝑏𝑎^𝑘sup_{−ℎ2≤𝑙≤0}‖𝜙(𝑙)‖²,(2.8) for all admissible uncertainties.

Lemma 2.2 (see [5] (Schur complement lemma)). Given constant matrices 𝑋,𝑌,𝑍 of appropriate dimensions with 𝑌>0. Then 𝑋+𝑍^𝑇𝑌⁻¹𝑍<0 if and only if ⎛⎜⎜⎝𝑋𝑍^𝑇𝑍−𝑌⎞⎟⎟⎠<0or⎛⎜⎜⎝−𝑍𝑌𝑋𝑍^𝑇⎞⎟⎟⎠<0.(2.9)

Lemma 2.3 (see [2]). Given constant matrices 𝑀₁,𝑀₂, and 𝑀₃ of appropriate dimensions with 𝑀₁=𝑀^𝑇1. Then, 𝑀₁+𝑀₂Δ(𝑘)𝑀₃+𝑀^𝑇3Δ(𝑘)^𝑇𝑀^𝑇2<0,(2.10) where Δ(𝑘)=𝐹(𝑘)[𝐼−𝐽𝐹(𝑘)]⁻¹, 𝐹(𝑘)^𝑇𝐹(𝑘)≤𝐼,forall𝑘∈ℤ⁺ if and only if 𝑀₁+𝜖⁻¹𝑀^𝑇3𝜖𝑀₂⎡⎢⎢⎣−𝐼𝐽_𝑇−𝐽𝐼⎤⎥⎥⎦⁻¹𝜖⁻¹𝑀^𝑇3𝜖𝑀₂^𝑇<0,(2.11) for some scalar 𝜖>0.

3. Main Results

In this section, we present our main results on the robust exponential stability criteria for uncertain LPD discrete-time system with interval time-varying delays. We introduce the following notation for later use: 𝐴(𝛼)=𝐴(𝛼)+Δ𝐴(𝑘),𝐵(𝛼)=𝐵(𝛼)+Δ𝐵(𝑘),ℎ=ℎ₂−ℎ₁+1.(3.1)

Lemma 3.1. For any 𝐴(𝛼),𝐵(𝛼),ℎ in (3.1), 𝑃(𝛼) and 𝑄(𝛼) given by 𝑃(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑃_𝑖,𝑄(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝑄_𝑖,^𝑁_𝑖=1𝛼_𝑖=1,𝛼_𝑖≥0,𝑖=1,…,𝑁,(3.2) are parameter-dependent positive definite Lyapunov matrices such that ⎡⎢⎢⎣𝐴^𝑇(𝛼)𝑃𝐵(𝛼)𝐴(𝛼)−𝑃(𝛼)+ℎ𝑄(𝛼)𝐴^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)_𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝐵^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)−𝑄(𝛼)⎤⎥⎥⎦<0,(3.3) if and only if ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃(𝛼)+∗ℎ𝑄(𝛼)−𝑄0(𝛼)𝐵𝐴((𝛼𝛼))^𝑇_𝑇𝑃(𝛼)𝜖⁻¹𝐴₁(𝛼)^𝑇0∗∗∗∗∗∗−𝑃∗∗𝑃(𝛼)(𝛼)𝜖⁻¹𝐵−∗0₁𝐼(𝛼)^𝑇𝜖𝑃(𝛼−𝐽0)𝐼𝐾(𝛼)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(3.4)

Proof. Consider ⎡⎢⎢⎣𝐴^𝑇(𝛼)𝑃𝐵(𝛼)𝐴(𝛼)−𝑃(𝛼)+ℎ𝑄(𝛼)𝐴^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)_𝑇=⎡⎢⎢⎣−𝑃(𝛼)+0(𝛼ℎ)𝑄𝑃((𝛼𝛼))𝐴−(𝑄𝛼0)(𝛼)⎤⎥⎥⎦+⎡⎢⎢⎣𝐴_𝑇𝐵^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)−𝑄(𝛼)⎤⎥⎥⎦𝐵(𝛼)𝑃(𝛼)𝐴(𝛼)𝐴^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)_𝑇=⎡⎢⎢⎣−𝑃(𝛼)+0ℎ𝑄(𝛼)−𝑄0(𝛼)⎤⎥⎥⎦+⎡⎢⎢⎣𝐴_𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝐵^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)⎤⎥⎥⎦𝐵(𝛼)_𝑇(𝛼)⎤⎥⎥⎦𝑃(𝛼)𝐴(𝛼)𝐵(𝛼).(3.5) We assume that ⎡⎢⎢⎣−𝑃(𝛼)+0ℎ𝑄(𝛼)−𝑄0(𝛼)⎤⎥⎥⎦+⎡⎢⎢⎣𝐴^𝑇(𝛼)𝐵^𝑇(𝛼)⎤⎥⎥⎦𝑃(𝛼)𝐴(𝛼)𝐵(𝛼)<0.(3.6) Using Lemma 2.2, we obtain ⎡⎢⎢⎢⎢⎣−𝑃(𝛼)+ℎ𝑄(𝛼)0𝐴(𝛼)^𝑇+𝐾(𝛼)Δ(𝑘)𝐴₁(𝛼)^𝑇∗−𝑄(𝛼)𝐵(𝛼)^𝑇+𝐾(𝛼)Δ(𝑘)𝐵₁(𝛼)^𝑇∗∗−𝑃(𝛼)⁻¹⎤⎥⎥⎥⎥⎦<0.(3.7) We rewrite the latter inequality as ⎡⎢⎢⎢⎢⎣−𝑃(𝛼)+ℎ𝑄(𝛼)0𝐴(𝛼)^𝑇∗−𝑄(𝛼)𝐵(𝛼)^𝑇∗∗−𝑃(𝛼)⁻¹⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣𝐾00(𝛼)⎤⎥⎥⎥⎥⎦Δ(𝑘)𝐴₁+𝐴(𝛼)𝐵₁(𝛼)0₁(𝛼)𝐵₁(𝛼)0^𝑇Δ(𝑘)^𝑇⎡⎢⎢⎢⎢⎣𝐾00(𝛼)⎤⎥⎥⎥⎥⎦^𝑇<0.(3.8) Using Lemma 2.3, inequality (3.8) holds if and only if there exists 𝜖>0 such that ⎡⎢⎢⎢⎢⎣−𝑃(𝛼)+ℎ𝑄(𝛼)0𝐴(𝛼)^𝑇∗−𝑄(𝛼)𝐵(𝛼)^𝑇∗∗−𝑃(𝛼)⁻¹+⎡⎢⎢⎢⎢⎣𝜖₋₁⎤⎥⎥⎥⎥⎦𝜖₋₁𝐴₁(𝛼)^𝑇0𝐵0₁(𝛼)^𝑇𝜖𝐾0(𝛼)⎤⎥⎥⎥⎥⎦⎡⎢⎢⎣−𝐼𝐽−𝐼𝐽⎤⎥⎥⎦⁻¹⎡⎢⎢⎢⎢⎣𝜖𝜖⁻¹₋₁𝐴₁(𝛼)^𝑇0𝐵0₁(𝛼)^𝑇𝜖𝐾0(𝛼)⎤⎥⎥⎥⎥⎦^𝑇<0.(3.9) If we apply to (3.9), then we obtain ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃(𝛼)+∗ℎ𝑄(𝛼)−𝑄0(𝛼)𝐵𝐴((𝛼𝛼))^𝑇_𝑇𝜖⁻¹𝐴₁(𝛼)^𝑇0∗∗−𝑃(𝛼)₋₁𝜖⁻¹𝐵₁(𝛼)^𝑇0∗∗∗∗∗∗−∗0𝐼𝜖𝐾−𝐽𝐼(𝛼)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(3.10) Premultiplying (3.10) by diag{𝐼,𝐼,𝑃(𝛼),𝐼,𝐼} and postmultiplying by diag{𝐼,𝐼,𝑃(𝛼),𝐼,𝐼}, we get that (3.4) and the lemma is proved.

Lemma 3.2. If there exist positive definite symmetric matrices 𝑃_𝑖, 𝑄_𝑖, 𝑖=1,2,…,𝑁, and positive real numbers 𝜖,𝜁 such that ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃_𝑖+∗ℎ𝑄_𝑖−0𝑄_𝑖𝐴^𝑇𝑖𝑃_𝑖𝜖⁻¹𝐴^1𝑖𝑇0∗∗𝐵−^𝑇𝑖𝑃𝑃_𝑖𝑖𝜖⁻¹𝐵^1𝑖𝑇00𝜖𝑃_𝑖𝐾_𝑖⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃_𝑖∗∗∗∗∗∗−∗𝐼−𝐽𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦<−𝜁𝐼,𝑖=1,2,…,𝑁,+∗ℎ𝑄_𝑖−0𝑄_𝑖𝐴^𝑇𝑖𝑃_𝑗𝜖⁻¹𝐴^1𝑖𝑇0∗∗𝐵−^𝑇𝑖𝑃𝑃_𝑗𝑖𝜖⁻¹𝐵^1𝑖𝑇00𝜖𝑃_𝑖𝐾_𝑗∗∗∗∗∗∗−∗𝐼−𝐽𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃^𝑗+∗ℎ𝑄_𝑗−0𝑄_𝑗𝐴^𝑇𝑗𝑃_𝑖𝜖⁻¹𝐴^1𝑗𝑇0∗∗𝐵−^𝑇𝑗𝑃𝑃_𝑖𝑗𝜖⁻¹𝐵^1𝑗𝑇00𝜖𝑃_𝑗𝐾_𝑖∗∗∗∗∗∗−∗𝐼−𝐽𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦<2𝜁𝐼𝑖=1,…,𝑁−1,𝑗=𝑖+1,…𝑁,𝑁−,1,(3.11) then, for any 𝐴(𝛼),𝐴₁(𝛼),𝐵(𝛼),𝐵₁(𝛼),𝐾(𝛼),ℎ in (3.1), 𝑃(𝛼) and 𝑄(𝛼) are parameter-dependent positive definite Lyapunov matrices in Lemma 3.1 such that (3.4) holds.

Proof. Consider ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃(𝛼)+∗ℎ𝑄(𝛼)−𝑄0(𝛼)𝐵𝐴((𝛼𝛼))^𝑇_𝑇𝑃(𝛼)𝜖⁻¹𝐴₁(𝛼)^𝑇0=∗∗∗∗∗∗−𝑃∗∗𝑃(𝛼)(𝛼)𝜖⁻¹𝐵−∗0₁𝐼(𝛼)^𝑇𝜖𝑃(𝛼−𝐽0)𝐼_𝑁𝐾(𝛼)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦_𝑁𝑖=1_𝑗=1𝛼_𝑖𝛼_𝑗⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃_𝑖+∗ℎ𝑄_𝑖−0𝑄_𝑖𝐴^𝑇𝑖𝑃_𝑗𝜖⁻¹𝐴^1𝑖𝑇0∗∗𝐵−^𝑇𝑖𝑃𝑃_𝑗𝑖𝜖⁻¹𝐵^1𝑖𝑇00𝜖𝑃_𝑖𝐾_𝑗∗∗∗∗∗∗−∗𝐼−𝐽𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(3.12) Using the fact that ∑^𝑁𝑖=1𝛼_𝑖=1, we obtain the following identities: ^𝑁_𝑁𝑖=1_𝑗=1𝛼_𝑖𝛼_𝑗𝐴_𝑖𝐵_𝑗=^𝑁_𝑖=1𝛼²𝐴_𝑖𝐵_𝑖+^𝑁−1_𝑁𝑖=1_𝑗=𝑖+1(𝑁−1)_𝑁𝛼_𝑖𝛼_𝑗𝐴_𝑖𝐵_𝑗+𝐴_𝑗𝐵_𝑖,_𝑖=1𝛼^2𝑖𝜁−2^𝑁−1_𝑁𝑖=1_𝑗=𝑖+1𝛼_𝑖𝛼_𝑗𝜁=^𝑁−1_𝑁𝑖=1_𝑗=𝑖+1𝛼_𝑖−𝛼_𝑗²𝜁≥0.(3.13) Then, it follows from (3.11), (3.12), and (3.13) that (3.4) holds. The proof of the lemma is complete.

Theorem 3.3. The system (2.1) is robustly exponentially stable if the LMI conditions (3.11) are feasible.

Proof. Consider the following Lyapunov-Krasovskii function for system (2.1) of the form 𝑉(𝑥(𝑘))=𝑉₁(𝑥(𝑘))+𝑉₂(𝑥(𝑘))+𝑉₃(𝑥(𝑘)),(3.14) where 𝑉₁(𝑥(𝑘))=𝑥^𝑇(𝑘)𝑃(𝛼)𝑥(𝑘),𝑉₂(𝑥(𝑘))=^𝑘−1_𝑖=𝑘_−ℎ(𝑘)𝑉𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖),₃(𝑥(𝑘))=^−ℎ1+1_{𝑗=−ℎ2+2𝑘−1𝑙=𝑘+𝑗−1}𝑥^𝑇(𝑙)𝑄(𝛼)𝑥(𝑙).(3.15) A Lyapunov-Krasovskii difference for the system (2.1) is defined as Δ𝑉(𝑥(𝑘))=Δ𝑉₁(𝑥(𝑘))+Δ𝑉₂(𝑥(𝑘))+Δ𝑉₃(𝑥(𝑘)).(3.16) Taking the difference of 𝑉₁(𝑥(𝑘)) and 𝑉₂(𝑥(𝑘)), the increments of 𝑉₁(𝑥(𝑘)) and 𝑉₂(𝑥(𝑘)) are Δ𝑉₁(𝑥(𝑘))==𝑉𝑥_1𝑇(𝑥(𝑘+1))−𝑉₁(𝑥(𝑘))=𝑥_𝑇(𝑘+1)𝑃(𝛼)𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃(𝛼)𝑥(𝑘)+𝑥(𝑘_𝑇)𝐴^𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝑥(𝑘)+𝑥^𝑇(𝑘−ℎ(𝑘))𝐵^𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝑥(𝑘)+𝑥_𝑇(𝑘−ℎ(𝑘))𝐵^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)𝑥(𝑘−ℎ(𝑘))Δ𝑉₂(𝑘)𝐴^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)𝑥(𝑘−ℎ(𝑘))−𝑥^𝑇(𝑘)𝑃(𝛼)𝑥(𝑘),(3.17)(𝑥(𝑘))==𝑉₂(𝑥(𝑘+1))−𝑉₂(𝑥(𝑘))_{𝑘𝑖=𝑘+1}_{−ℎ(𝑘+1)}𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)−^𝑘−1_𝑖=𝑘_−ℎ(𝑘)=𝑥_𝑇𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)+(𝑘)𝑄(𝛼)𝑥(𝑘)−𝑥^𝑇(𝑘−ℎ(𝑘))𝑄(𝛼)𝑥(𝑘−ℎ(𝑘))_{𝑘−ℎ1𝑖=𝑘+1}_{−ℎ(𝑘+1)}𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)−^𝑘−1_𝑖=𝑘+1_{−ℎ(𝑘+1)}+𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)_{𝑘−1𝑖=𝑘}_+1−ℎ1𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖).(3.18) Form ℎ(𝑘)≥ℎ₁, the two last terms of the right-hand side of the latter equality yield ^𝑘−1_{𝑖=𝑘+1−ℎ1}𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)−^𝑘−1_𝑖=𝑘+1_{−ℎ(𝑘+1)}𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)≤0.(3.19) Thus, we obtain Δ𝑉₂(𝑥(𝑘))≤𝑥+^𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)−𝑥^𝑇(𝑘−ℎ(𝑘))𝑄(𝛼)𝑥(𝑘−ℎ(𝑘))_{𝑘−ℎ1𝑖=𝑘+1}_{−ℎ(𝑘+1)}𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖).(3.20) The increment of 𝑉₃(𝑥(𝑘)) is easily computed as Δ𝑉₃(𝑥(𝑘))==𝑉₃(𝑥(𝑘+1))−𝑉₃(𝑥(𝑘))_{−ℎ1+1𝑗=}_−ℎ2+2𝑥^𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)+^𝑘−1_𝑙=_{𝑘+𝑗−1}𝑥^𝑇(𝑙)𝑄(𝛼)𝑥(𝑙)−^𝑘−1_𝑙=_{𝑘+𝑗−1}=ℎ𝑥^𝑇(𝑙)𝑄(𝛼)𝑥(𝑙)₂−ℎ₁𝑥^𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)−^𝑘−ℎ1_𝑖=𝑘_+1−ℎ2𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖).(3.21) It is easy to see that Δ𝑉₂(𝑥(𝑘))+Δ𝑉₃(𝑥(𝑘))≤ℎ₂−ℎ₁+1𝑥^𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)−𝑥^{𝑇𝑘−ℎ}𝑄(𝛼)𝑥_𝑘−ℎ+^𝑘−ℎ1_𝑖=𝑘+1_{−ℎ(𝑘+1)}𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)−^𝑘−ℎ1_𝑖=𝑘_+1−ℎ2𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖),(3.22) for simplicity, we let 𝑥(𝑘−ℎ(𝑘))=𝑥_𝑘−ℎ. Since, ℎ(𝑘)≤ℎ₂, we obtain that ^𝑘−ℎ1_𝑖=_{𝑘+1−ℎ(𝑘+1)}𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)−^𝑘−ℎ1_𝑖=𝑘_+1−ℎ2𝑥^𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)≤0.(3.23) Therefore, we conclude that Δ𝑉(𝑥(𝑘))≤𝑥+^𝑇𝑥(𝑘_𝑇)𝐴^𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝑥(𝑘)+𝑥^{𝑇𝑘−ℎ}𝐵^𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝑥(𝑘)(𝑘)𝐴^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)𝑥_𝑘−ℎ+𝑥^{𝑇𝑘−ℎ}𝐵^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)𝑥_𝑘−ℎ−−𝑥𝑥^𝑇(_{𝑇𝑘−ℎ}𝑘)𝑃(𝛼)𝑥(𝑘)+ℎ₂−ℎ₁+1𝑥^𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)𝑄(𝛼)𝑥_𝑘−ℎ.(3.24) It follows form (3.24) that Δ𝑉(𝑥(𝑘))≤𝑌^𝑇⎡⎢⎢⎣𝐵Δ₁₁(𝛼)𝐴^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)_𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝐵^𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)−𝑄(𝛼)⎤⎥⎥⎦𝑌,(3.25) where Δ₁₁(𝛼)=𝐴^𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)−𝑃(𝛼)+ℎ𝑄(𝛼) and 𝑌^𝑇=[𝑥(𝑘)^𝑇𝑥(𝑘−ℎ(𝑘))^𝑇]. By (3.11), (3.25), and Lemma 3.1, and 3.2, we obtain Δ𝑉(𝑥(𝑘))<−𝜔‖𝑥‖²,(3.26) where 𝜔>0. By (3.14), it is easy to see that 𝑉(𝑥(𝑘))≤𝛽₁‖𝑥‖²+𝛽₁ℎ^𝑘−1_𝑖=_𝑘−ℎ2‖𝑥(𝑖)‖²,(3.27) where 𝛽₁=max𝜆_max𝑃_𝑖,𝜆_max𝑄_𝑖;𝑖=1,2,…,𝑁.(3.28) It can be shown that there always exists a scalar 𝜃>1 satisfying (𝜃−1)𝛽₁−𝜆𝜃+ℎ₂𝜃^ℎ2(𝜃−1)𝛽₁ℎ=0.(3.29) For any scalar 𝜃>1, it follows from (3.26) and (3.27) that 𝜃^𝑘+1𝑉(𝑥(𝑘+1))−𝜃=𝜃^𝑘+1𝑉_𝑘+1(𝑥(𝑘))<𝛼₁(𝑉(𝑥(𝑘+1))−𝑉(𝑥(𝑘)))+𝜃^𝑘(𝜃−1)𝑉(𝑥(𝑘))(𝜃)𝜃^𝑘‖𝑥(𝑘)‖²+𝛼₂(𝜃)𝜃^𝑘𝑘−1_𝑖=_𝑘−ℎ2‖𝑥(𝑖)‖²,(3.30) where 𝛼₁(𝜃)=(𝜃−1)𝛽₁−𝜆𝜃,𝛼₂(𝜃)=(𝜃−1)𝛽₁ℎ.(3.31) Therefore, for any integer 𝑇≥ℎ₂+1, summing up both sides of (3.30) from 0 to 𝑇−1 gives 𝜃^𝑇𝑉(𝑥(𝑇))−𝑉(𝑥(0))≤𝛼₁(𝜃)^𝑇−1_𝑖=0𝜃^𝑖‖𝑥(𝑖)‖²+𝛼₂(𝜃)^𝑇−1_{𝑖=0𝑖−1𝑙=𝑖−ℎ2}𝜃^𝑖‖𝑥(𝑙)‖².(3.32) For ℎ₂≥1, ^𝑇−1_{𝑖=0𝑖−1𝑙=𝑖−ℎ2}𝜃^𝑖‖𝑥(𝑙)‖²≤⁻¹_𝑙=−ℎ2^𝑙+ℎ2_𝑖=0𝜃^𝑖‖𝑥(𝑙)‖²+^{𝑇−1−ℎ2}_{𝑙=0𝑙+ℎ}²_𝑖=𝑙+1𝜃^𝑖‖𝑥(𝑙)‖²+^𝑇−1_{𝑙=𝑇−ℎ2}^𝑇−1_𝑖=𝑙+1𝜃^𝑖‖𝑥(𝑙)‖²≤ℎ_2−1𝑙=_−ℎ2𝜃^𝑙+ℎ2‖𝑥(𝑙)‖²+ℎ_{2𝑇−1−ℎ}²_𝑙=0𝜃^𝑙+ℎ2‖𝑥(𝑙)‖²+ℎ_{2𝑇−1𝑙=}_𝑇−ℎ2𝜃^𝑙+ℎ2‖𝑥(𝑙)‖²≤ℎ₂ℎ₂+1𝜃^ℎ2sup_{−ℎ2≤𝑙≤0}‖𝜙(𝑙)‖²+ℎ₂𝜃^ℎ2𝑇−1_𝑙=1𝜃^𝑙‖𝑥(𝑙)‖².(3.33) From (3.32) and (3.33), we obtain 𝜃^𝑇𝑉(𝑥(𝑇))≤𝑉(𝑥(0))+ℎ₂ℎ₂+1𝜃^ℎ2𝛼₂(𝜃)sup_{−ℎ2≤𝑙≤0}‖𝜙(𝑙)‖²+𝛼₁(𝜃)+𝛼₂(𝜃)ℎ₂𝜃^ℎ2^𝑇−1_𝑙=0𝜃^𝑙‖𝑥(𝑙)‖².(3.34) Observe 𝑉(𝑥(𝑇))≥𝛾‖𝑥𝛾(=𝑇)m‖i²n,𝜆𝑉_min(𝑥(0))≤𝛽₁+𝛽₁ℎℎ₂sup_{−ℎ2≤𝑙≤0}‖𝜙(𝑙)‖²,𝑃_𝑖;𝑖=1,2,…,𝑁.(3.35) Then, it follows from (3.29), (3.33), and (3.35) that ‖𝑥(𝑇)‖²≤ℎ²ℎ²+1𝜃^ℎ2𝛼₂(𝜃)+𝛽₁+𝛽₁ℎℎ₂𝛾𝜃1^𝑇sup_{−ℎ2≤𝑙≤0}‖𝜙(𝑙)‖².(3.36) By Definition 2.1, this means that the system (2.1) is robustly exponentially stable. The proof of the theorem is complete.

4. Numerical Example

Example 4.1. Consider the following uncertain LPD discrete-time system with time-varying delays (2.1) where ℎ(𝑘)=2+cos(𝑘𝜋/2), that is, ℎ₁=1,ℎ₂=3 and 𝐵𝐴₁=⎡⎢⎢⎣−0.00.26−0.00.26⎤⎥⎥⎦,𝐴₂=⎡⎢⎢⎣−0.00.37−0.00.37⎤⎥⎥⎦,𝐵₁=⎡⎢⎢⎣−0.00.26−00.0.028₂⎤⎥⎥⎦,=⎡⎢⎢⎣−0.00.38𝐵−00.0.039⎤⎥⎥⎦,𝐴¹¹=⎡⎢⎢⎣00.0.00005100.0.000051⎤⎥⎥⎦,𝐴¹²=⎡⎢⎢⎣00..000006200..000006₁₁2⎤⎥⎥⎦,𝐾=⎡⎢⎢⎣−0.00.000057−0.00.000057⎤⎥⎥⎦,𝐵¹²=⎡⎢⎢⎣−0.00.000024−0.00.000024₁⎤⎥⎥⎦,=⎡⎢⎢⎣00..0001300..00013⎤⎥⎥⎦,𝐾₂=⎡⎢⎢⎣00.0.002100.0.0021⎤⎥⎥⎦,(4.1) and 𝐽=^0.001000.001. By using LMI Toolbox in MATLAB, we use condition (3.11) in Theorem 3.3 for this example. The solutions of LMI verify as follows of the form 𝜖=1, 𝑃₁=^{31.36351.23651.236529.4763},𝑃₂=^{37.63540.25430.254341.3745},𝑄₁=^{9.43250.55870.558711.4534}, and 𝑄₂=^{10.85641.38561.385611.9781} (see Figure 1).

Figure 1 

The simulation solution of the states 𝑥1(𝑘) and 𝑥2(𝑘) in Example 4.1 for uncertain LPD discrete-time delayed system with initial conditions 𝑥1(𝑘)=2 and 𝑥2(𝑘)=4, 𝑘=−3,−2,−1,0, and 𝛼1=𝛼2=1/2 by using the method of Runge-Kutta order 4(ℎ=0.01) with Matlab.

Example 4.2. Consider the following the LPD discrete-time system with time-varying delays (2.1) where, Δ𝐴(𝑘)=Δ𝐵(𝑘)=0 with 𝐵𝐴₁=⎡⎢⎢⎣00..60010.060⎤⎥⎥⎦,𝐴₂=⎡⎢⎢⎣00..80050.070₁⎤⎥⎥⎦,=⎡⎢⎢⎣00..12000.010⎤⎥⎥⎦,𝐵₂=⎡⎢⎢⎣−−00..1200−00.10⎤⎥⎥⎦.(4.2) Table 1 lists the comparison of the upper-bound delay for asymptotic stability of system (2.1) where Δ𝐴(𝑘)=Δ𝐵(𝑘)=0 by different method. We apply Theorem 3.3 and see from Table 1 that our result is superior to those in [7, Theorem 3.2].

Table 1 

Comparison of the maximum allowed time delay ℎ₂.

Acknowledgments

This work was supported by the Thailand Research Fund (TRF), the Office of the Higher Education Commission (OHEC), and the Khon Kaen University (Grant number MRG5580006), and the author would like to thank his advisor Associate Professor Dr. Piyapong Niamsup from the Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand.