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Journal of Applied Mathematics
Volume 2012, Article ID 237430, 13 pages
Research Article

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

1Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand

Received 5 April 2012; Accepted 11 July 2012

Academic Editor: B. V. Rathish Kumar

Copyright © 2012 Kanit Mukdasai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 [115] 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. + denotes the set of non negative integer numbers; 𝑛 denotes the 𝑛-dimensional space with the vector norm ; 𝑥 denotes the Euclidean vector norm of 𝑥𝑛; that is, 𝑥2=𝑥𝑇𝑥; 𝑀𝑛×𝑟 denotes the space of all matrices of (𝑛×𝑟)-dimensions; 𝐴𝑇 denotes transpose of the Matrix 𝐴; 𝐴 is symmetric if 𝐴=𝐴𝑇; 𝐼 denotes the identity matrix; 𝜆(𝐴) denotes the set of all eigenvalues of 𝐴; 𝜆max(𝐴)=max{Re𝜆𝜆𝜆(𝐴)}; Matrix 𝐴 is called semi positive definite (𝐴0) if 𝑥𝑇𝐴𝑥0, for all 𝑥𝑛;𝐴 is positive definite (𝐴>0) if 𝑥𝑇𝐴𝑥>0 for all 𝑥0; Matrix 𝐵 is called semi-negative definite (𝐵0) if 𝑥𝑇𝐵𝑥0, for all 𝑥𝑛;𝐵 is negative definite (𝐵<0) if 𝑥𝑇𝐵𝑥<0 for all 𝑥0; 𝐴>𝐵 means 𝐴𝐵>0; 𝐴𝐵 means 𝐴𝐵0; 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 [][]𝑥𝑥(𝑘+1)=𝐴(𝛼)+Δ𝐴(𝑘)𝑥(𝑘)+𝐵(𝛼)+Δ𝐵(𝑘)𝑥(𝑘(𝑘)),(𝑠)=𝜙(𝑠),𝑠=2,,1,0,(2.1) where 𝑘+, 𝑥(𝑘)𝑛 is the system state and 𝜙(𝑠) is a initial value at 𝑠. 𝐴(𝛼), 𝐵(𝛼)𝑀𝑛×𝑛 are uncertain matrices belonging to the polytope of the form 𝐴(𝛼)=𝑁𝑖=1𝛼𝑖𝐴𝑖,𝐵(𝛼)=𝑁𝑖=1𝛼𝑖𝐵𝑖,𝑁𝑖=1𝛼𝑖=1,𝛼𝑖0,𝐴𝑖,𝐵𝑖𝑀𝑛×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 Δ𝐴(𝑘)=𝐾(𝛼)Δ(𝑘)𝐴1(𝛼),Δ𝐵(𝑘)=𝐾(𝛼)Δ(𝑘)𝐵1𝐴(𝛼),1(𝛼)=𝑁𝑖=1𝛼𝑖𝐴1𝑖,𝐵1(𝛼)=𝑁𝑖=1𝛼𝑖𝐵1𝑖,𝐾(𝛼)=𝑁𝑖=1𝛼𝑖𝐾𝑖,𝑁𝑖=1𝛼𝑖=1,𝛼𝑖0,𝐴1𝑖,𝐵1𝑖𝑀𝑛×𝑛,𝑖=1,,𝑁.(2.3) The class of parametric uncertainties Δ(𝑘), which satisfies []Δ(𝑘)=𝐹(𝑘)𝐼𝐽𝐹(𝑘)1,(2.4) is said to be admissible where 𝐽 is a known matrix satisfying 𝐼𝐽𝐽𝑇>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 (𝑘) is upper and lower bounded. It satisfies the following assumption of the form 1(𝑘)2,(2.7) where 1 and 2 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)2𝑏𝑎𝑘sup2𝑙0𝜙(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+𝑍𝑇𝑌1𝑍<0 if and only if 𝑋𝑍𝑇𝑍𝑍𝑌<0or𝑌𝑍𝑇𝑋<0.(2.9)

Lemma 2.3 (see [2]). Given constant matrices 𝑀1,𝑀2, and 𝑀3 of appropriate dimensions with 𝑀1=𝑀𝑇1. Then, 𝑀1+𝑀2Δ(𝑘)𝑀3+𝑀𝑇3Δ(𝑘)𝑇𝑀𝑇2<0,(2.10) where Δ(𝑘)=𝐹(𝑘)[𝐼𝐽𝐹(𝑘)]1, 𝐹(𝑘)𝑇𝐹(𝑘)𝐼,forall𝑘+ if and only if 𝑀1+𝜖1𝑀𝑇3𝜖𝑀2𝐼𝐽𝐽𝑇𝐼1𝜖1𝑀𝑇3𝜖𝑀2𝑇<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),=21+1.(3.1)

Lemma 3.1. For any 𝐴(𝛼),𝐵(𝛼), in (3.1), 𝑃(𝛼) and 𝑄(𝛼) given by 𝑃(𝛼)=𝑁𝑖=1𝛼𝑖𝑃𝑖,𝑄(𝛼)=𝑁𝑖=1𝛼𝑖𝑄𝑖,𝑁𝑖=1𝛼𝑖=1,𝛼𝑖0,𝑖=1,,𝑁,(3.2) are parameter-dependent positive definite Lyapunov matrices such that 𝐴𝑇𝐴(𝛼)𝑃(𝛼)𝐴(𝛼)𝑃(𝛼)+𝑄(𝛼)𝑇𝐵(𝛼)𝑃(𝛼)𝐵(𝛼)𝑇(𝐵𝛼)𝑃(𝛼)𝐴(𝛼)𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)𝑄(𝛼)<0,(3.3) if and only if 𝑃(𝛼)+𝑄(𝛼)0𝐴(𝛼)𝑇𝑃(𝛼)𝜖1𝐴1(𝛼)𝑇0𝑄(𝛼)𝐵(𝛼)𝑇𝑃(𝛼)𝜖1𝐵1(𝛼)𝑇0𝑃(𝛼)0𝜖𝑃(𝛼)𝐾(𝛼)𝐼𝐽𝐼<0.(3.4)

Proof. Consider 𝐴𝑇𝐴(𝛼)𝑃(𝛼)𝐴(𝛼)𝑃(𝛼)+𝑄(𝛼)𝑇𝐵(𝛼)𝑃(𝛼)𝐵(𝛼)𝑇(𝐵𝛼)𝑃(𝛼)𝐴(𝛼)𝑇(=+𝐴𝛼)𝑃(𝛼)𝐵(𝛼)𝑄(𝛼)𝑃(𝛼)+𝑄(𝛼)00𝑄(𝛼)𝑇𝐴(𝛼)𝑃(𝛼)𝐴(𝛼)𝑇𝐵(𝛼)𝑃(𝛼)𝐵(𝛼)𝑇𝐵(𝛼)𝑃(𝛼)𝐴(𝛼)𝑇=+𝐴(𝛼)𝑃(𝛼)𝐵(𝛼)𝑃(𝛼)+𝑄(𝛼)00𝑄(𝛼)𝑇𝐵(𝛼)𝑇.(𝛼)𝑃(𝛼)𝐴(𝛼)𝐵(𝛼)(3.5) We assume that +𝑃(𝛼)+𝑄(𝛼)00𝑄(𝛼)𝐴𝑇(𝛼)𝐵𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝐵(𝛼)<0.(3.6) Using Lemma 2.2, we obtain 𝑃(𝛼)+𝑄(𝛼)0𝐴(𝛼)𝑇+𝐾(𝛼)Δ(𝑘)𝐴1(𝛼)𝑇𝑄(𝛼)𝐵(𝛼)𝑇+𝐾(𝛼)Δ(𝑘)𝐵1(𝛼)𝑇𝑃(𝛼)1<0.(3.7) We rewrite the latter inequality as 𝑃(𝛼)+𝑄(𝛼)0𝐴(𝛼)𝑇𝑄(𝛼)𝐵(𝛼)𝑇𝑃(𝛼)1+00Δ𝐴𝐾(𝛼)(𝑘)1(𝛼)𝐵1(+𝐴𝛼)01(𝛼)𝐵1(𝛼)0𝑇Δ(𝑘)𝑇00𝐾(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)+𝑄(𝛼)0𝐴(𝛼)𝑇𝑄(𝛼)𝐵(𝛼)𝑇𝑃(𝛼)1+𝜖1𝐴1(𝛼)𝑇0𝜖1𝐵1(𝛼)𝑇00𝜖𝐾(𝛼)𝐼𝐽𝐽𝐼1𝜖1𝐴1(𝛼)𝑇0𝜖1𝐵1(𝛼)𝑇00𝜖𝐾(𝛼)𝑇<0.(3.9) If we apply to (3.9), then we obtain 𝑃(𝛼)+𝑄(𝛼)0𝐴(𝛼)𝑇𝜖1𝐴1(𝛼)𝑇0𝑄(𝛼)𝐵(𝛼)𝑇𝜖1𝐵1(𝛼)𝑇0𝑃(𝛼)10𝜖𝐾(𝛼)𝐼𝐽𝐼<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 𝑃𝑖, 𝑄𝑖, 𝑖=1,2,,𝑁, and positive real numbers 𝜖,𝜁 such that 𝑃𝑖+𝑄𝑖0𝐴𝑇𝑖𝑃𝑖𝜖1𝐴1𝑖𝑇0𝑄𝑖𝐵𝑇𝑖𝑃𝑖𝜖1𝐵1𝑖𝑇0𝑃𝑖0𝜖𝑃𝑖𝐾𝑖𝐼𝐽𝐼<𝜁𝐼,𝑖=1,2,,𝑁,𝑃𝑖+𝑄𝑖0𝐴𝑇𝑖𝑃𝑗𝜖1𝐴1𝑖𝑇0𝑄𝑖𝐵𝑇𝑖𝑃𝑗𝜖1𝐵1𝑖𝑇0𝑃𝑖0𝜖𝑃𝑖𝐾𝑗+𝐼𝐽𝐼𝑃𝑗+𝑄𝑗0𝐴𝑇𝑗𝑃𝑖𝜖1𝐴1𝑗𝑇0𝑄𝑗𝐵𝑇𝑗𝑃𝑖𝜖1𝐵1𝑗𝑇0𝑃𝑗0𝜖𝑃𝑗𝐾𝑖<𝐼𝐽𝐼2𝜁𝐼,𝑁1𝑖=1,,𝑁1,𝑗=𝑖+1,,𝑁,(3.11) then, for any 𝐴(𝛼),𝐴1(𝛼),𝐵(𝛼),𝐵1(𝛼),𝐾(𝛼), 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)+𝑄(𝛼)0𝐴(𝛼)𝑇𝑃(𝛼)𝜖1𝐴1(𝛼)𝑇0𝑄(𝛼)𝐵(𝛼)𝑇𝑃(𝛼)𝜖1𝐵1(𝛼)𝑇0=𝑃(𝛼)0𝜖𝑃(𝛼)𝐾(𝛼)𝐼𝐽𝐼𝑁𝑁𝑖=1𝑗=1𝛼𝑖𝛼𝑗𝑃𝑖+𝑄𝑖0𝐴𝑇𝑖𝑃𝑗𝜖1𝐴1𝑖𝑇0𝑄𝑖𝐵𝑇𝑖𝑃𝑗𝜖1𝐵1𝑖𝑇0𝑃𝑖0𝜖𝑃𝑖𝐾𝑗.𝐼𝐽𝐼(3.12) Using the fact that 𝑁𝑖=1𝛼𝑖=1, we obtain the following identities: 𝑁𝑁𝑖=1𝑗=1𝛼𝑖𝛼𝑗𝐴𝑖𝐵𝑗=𝑁𝑖=1𝛼2𝐴𝑖𝐵𝑖+𝑁1𝑁𝑖=1𝑗=𝑖+1𝛼𝑖𝛼𝑗𝐴𝑖𝐵𝑗+𝐴𝑗𝐵𝑖,(𝑁1)𝑁𝑖=1𝛼2𝑖𝜁2𝑁1𝑁𝑖=1𝑗=𝑖+1𝛼𝑖𝛼𝑗𝜁=𝑁1𝑁𝑖=1𝑗=𝑖+1𝛼𝑖𝛼𝑗2𝜁0.(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))=𝑥𝑇(𝑘)𝑃(𝛼)𝑥(𝑘),𝑉2(𝑥(𝑘))=𝑘1𝑖=𝑘(𝑘)𝑥𝑇𝑉(𝑖)𝑄(𝛼)𝑥(𝑖),3(𝑥(𝑘))=1+1𝑗=2+2𝑘1𝑙=𝑘+𝑗1𝑥𝑇(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))=𝑉1(𝑥(𝑘+1))𝑉1(𝑥(𝑘))=𝑥𝑇(𝑘+1)𝑃(𝛼)𝑥(𝑘+1)𝑥𝑇(𝑘)𝑃(𝛼)𝑥(𝑘)=𝑥𝑇𝐴(𝑘)𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝑥(𝑘)+𝑥𝑇𝐵(𝑘(𝑘))𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝑥(𝑘)+𝑥𝑇𝐵(𝑘(𝑘))𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)𝑥(𝑘(𝑘))+𝑥𝑇(𝐴𝑘)𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)𝑥(𝑘(𝑘))𝑥𝑇(𝑘)𝑃(𝛼)𝑥(𝑘),(3.17)Δ𝑉2(𝑥(𝑘))=𝑉2(𝑥(𝑘+1))𝑉2(=𝑥(𝑘))𝑘𝑖=𝑘+1(𝑘+1)𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)𝑘1𝑖=𝑘(𝑘)𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)=𝑥𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)𝑥𝑇+(𝑘(𝑘))𝑄(𝛼)𝑥(𝑘(𝑘))𝑘1𝑖=𝑘+1(𝑘+1)𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)𝑘1𝑖=𝑘+1(𝑘+1)𝑥𝑇(+𝑖)𝑄(𝛼)𝑥(𝑖)𝑘1𝑖=𝑘+11𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖).(3.18) Form (𝑘)1, the two last terms of the right-hand side of the latter equality yield 𝑘1𝑖=𝑘+11𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)𝑘1𝑖=𝑘+1(𝑘+1)𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)0.(3.19) Thus, we obtain Δ𝑉2(𝑥(𝑘))𝑥𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)𝑥𝑇+(𝑘(𝑘))𝑄(𝛼)𝑥(𝑘(𝑘))𝑘1𝑖=𝑘+1(𝑘+1)𝑥𝑇(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))1+1𝑗=2+2𝑥𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)+𝑘1𝑙=𝑘+𝑗1𝑥𝑇(𝑙)𝑄(𝛼)𝑥(𝑙)𝑘1𝑙=𝑘+𝑗1𝑥𝑇=(𝑙)𝑄(𝛼)𝑥(𝑙)21𝑥𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)𝑘1𝑖=𝑘+12𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖).(3.21) It is easy to see that Δ𝑉2(𝑥(𝑘))+Δ𝑉3(𝑥(𝑘))21𝑥+1𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)𝑥𝑇𝑘𝑄(𝛼)𝑥𝑘+𝑘1𝑖=𝑘+1(𝑘+1)𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)𝑘1𝑖=𝑘+12𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖),(3.22) for simplicity, we let 𝑥(𝑘(𝑘))=𝑥𝑘. Since, (𝑘)2, we obtain that 𝑘1𝑖=𝑘+1(𝑘+1)𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)𝑘1𝑖=𝑘+12𝑥𝑇(𝑖)𝑄(𝛼)𝑥(𝑖)0.(3.23) Therefore, we conclude that Δ𝑉(𝑥(𝑘))𝑥𝑇𝐴(𝑘)𝑇𝐴(𝛼)𝑃(𝛼)(𝛼)𝑥(𝑘)+𝑥𝑇𝑘𝐵𝑇𝐴(𝛼)𝑃(𝛼)(𝛼)𝑥(𝑘)+𝑥𝑇𝐴(𝑘)𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)𝑥𝑘+𝑥𝑇𝑘𝐵𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)𝑥𝑘𝑥𝑇(𝑘)𝑃(𝛼)𝑥(𝑘)+21𝑥+1𝑇(𝑘)𝑄(𝛼)𝑥(𝑘)𝑥𝑇𝑘𝑄(𝛼)𝑥𝑘.(3.24) It follows form (3.24) that Δ𝑉(𝑥(𝑘))𝑌𝑇Δ11𝐴(𝛼)𝑇𝐵(𝛼)𝑃(𝛼)𝐵(𝛼)𝑇(𝐵𝛼)𝑃(𝛼)𝐴(𝛼)𝑇(𝛼)𝑃(𝛼)𝐵(𝛼)𝑄(𝛼)𝑌,(3.25) where Δ11𝐴(𝛼)=𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)𝑃(𝛼)+𝑄(𝛼) and 𝑌𝑇=[𝑥(𝑘)𝑇𝑥(𝑘(𝑘))𝑇]. By (3.11), (3.25), and Lemma 3.1, and 3.2, we obtain Δ𝑉(𝑥(𝑘))<𝜔𝑥2,(3.26) where 𝜔>0. By (3.14), it is easy to see that 𝑉(𝑥(𝑘))𝛽1𝑥2+𝛽1𝑘1𝑖=𝑘2𝑥(𝑖)2,(3.27) where 𝛽1𝜆=maxmax𝑃𝑖,𝜆max𝑄𝑖;𝑖=1,2,,𝑁.(3.28) It can be shown that there always exists a scalar 𝜃>1 satisfying (𝜃1)𝛽1𝜆𝜃+2𝜃2(𝜃1)𝛽1=0.(3.29) For any scalar 𝜃>1, it follows from (3.26) and (3.27) that 𝜃𝑘+1𝑉(𝑥(𝑘+1))𝜃𝑘+1𝑉(𝑥(𝑘))=𝜃𝑘+1(𝑉(𝑥(𝑘+1))𝑉(𝑥(𝑘)))+𝜃𝑘(𝜃1)𝑉(𝑥(𝑘))<𝛼1(𝜃)𝜃𝑘(𝑥𝑘)2+𝛼2(𝜃)𝜃𝑘𝑘1𝑖=𝑘2(𝑥𝑖)2,(3.30) where 𝛼1(𝜃)=(𝜃1)𝛽1𝜆𝜃,𝛼2(𝜃)=(𝜃1)𝛽1.(3.31) Therefore, for any integer 𝑇2+1, summing up both sides of (3.30) from 0 to 𝑇1 gives 𝜃𝑇𝑉(𝑥(𝑇))𝑉(𝑥(0))𝛼1(𝜃)𝑇1𝑖=0𝜃𝑖𝑥(𝑖)2+𝛼2(𝜃)𝑇1𝑖=0𝑖1𝑙=𝑖2𝜃𝑖𝑥(𝑙)2.(3.32) For 21, 𝑇1𝑖=0𝑖1𝑙=𝑖2𝜃𝑖𝑥(𝑙)21𝑙=2𝑙+2𝑖=0𝜃𝑖𝑥(𝑙)2+𝑇12𝑙=0𝑙+2𝑖=𝑙+1𝜃𝑖𝑥(𝑙)2+𝑇1𝑙=𝑇2𝑇1𝑖=𝑙+1𝜃𝑖𝑥(𝑙)221𝑙=2𝜃𝑙+2𝑥(𝑙)2+2𝑇12𝑙=0𝜃𝑙+2𝑥(𝑙)2+2𝑇1𝑙=𝑇2𝜃𝑙+2𝑥(𝑙)222𝜃+12sup2𝑙0𝜙(𝑙)2+2𝜃2𝑇1𝑙=1𝜃𝑙𝑥(𝑙)2.(3.33) From (3.32) and (3.33), we obtain 𝜃𝑇𝑉(𝑥(𝑇))𝑉(𝑥(0))+22𝜃+12𝛼2(𝜃)sup2𝑙0𝜙(𝑙)2+𝛼1(𝜃)+𝛼2(𝜃)2𝜃2𝑇1𝑙=0𝜃𝑙𝑥(𝑙)2.(3.34) Observe 𝑉(𝑥(𝑇))𝛾𝑥(𝑇)2𝛽,𝑉(𝑥(0))1+𝛽12sup2𝑙0𝜙(𝑙)2,𝜆𝛾=minmin𝑃𝑖.;𝑖=1,2,,𝑁(3.35) Then, it follows from (3.29), (3.33), and (3.35) that 𝑥(𝑇)222𝜃+12𝛼2(𝜃)+𝛽1+𝛽12𝛾1𝜃𝑇sup2𝑙0𝜙(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 (𝑘)=2+cos(𝑘𝜋/2), that is, 1=1,2=3 and 𝐴1=,𝐴2=,𝐵1=,𝐵0.,𝐴11=0.0050.00010.00010.005,𝐴12=,𝐵0.0060.00020.00020.00611=0.0070.00050.00050.007,𝐵12=,𝐾0.0040.00020.00020.0041=0.010.0030.0030.01,𝐾2=,0.020.0010.0010.02(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(=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 𝐴1=0.6000.010.60,𝐴2=,𝐵0.8000.050.701=0.1000.200.10,𝐵2=.0.1000.200.10(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 2.


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.


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