# Stability Criterion for Discrete-Time Systems

- K Ratchagit
^{1}Email author and - VuN Phat
^{2}

**2010**:201459

**DOI: **10.1155/2010/201459

© K. Ratchagit and V. N. Phat. 2010

**Received: **21 November 2009

**Accepted: **18 January 2010

**Published: **2 February 2010

## Abstract

This paper is concerned with the problem of delay-dependent stability analysis for discrete-time systems with interval-like time-varying delays. The problem is solved by applying a novel Lyapunov functional, and an improved delay-dependent stability criterion is obtained in terms of a linear matrix inequality.

## 1. Introduction

Recently, the problem of delay-dependent stability analysis for time-delay systems has received considerable attention, and lots of significant results have been reported; see, for example, Chen et al. [1], He et al. [2], Lin et al. [3], Park [4], and Xu and Lam [5], and the references therein. Among these references, we note that the delay-dependent stability problem for discrete-time systems with interval-like time-varying delays (i.e., the delay satisfies ) has been studied by Fridman and Shaked [6], Gao and Chen [7], Gao et al. [8], and Jiang et al. [9], where some LMI-based stability criteria have been presented by constructing appropriate Lyapunov functionals and introducing free-weighting matrices. It should be pointed out that the Lyapunov functionals considered in these references are more restrictive due to the ignorance of the term Moreover, the term is also ignored in Gao and Chen [7] and Gao et al. [8]. The ignorance of these terms may lead to considerable conservativeness.

On the other hand, in the study of stabilization for the discrete-time linear systems, traditional idea of the control schemes is to construct a control signal according to the current system state [10]. However, as pointed out by Xiong and Lam [11], in practice there is often a system that itself is not time-delayed but time-delayed may exist in a channel from system to controller. A typical example for the existence of such delays is the measurement and the network transmission of signals. In this case, a time-delayed controller is naturally taken into account. It is worth noting that the closed-loop system resulting from a delayed controller is actually a time-delay system. Therefore, stability results of time-delay systems could be applied to design time-delayed controller.

The present study, based on a new Lyapunov functional, an improved delay-dependent stability criterion for discrete-time systems with time-varying delays is presented in terms of LMIs. It is shown that the obtained result is less conservative than those by Fridman and Shaked [6], Gao and Chen [7], Gao et al. [8], Jiang et al. [9], and Zhang et al. [12].

## 2. Preliminaries

Fact 1.

Lemma 2.1 (see [13]).

along the solution of the system. In the case the above condition holds for all , say one that the zero solution is locally asymptotically stable.

Lemma 2.2 (see [13]).

## 3. Improved Stability Criterion

where is the state vector, and are known constant matrices, and is a time-varying delay satisfying , where and are positive integers representing the lower and upper bounds of the delay. For (3.1), we have the following result.

Theorem 3.1.

Proof.

with being symmetric positive definite solutions of (3.2) and

where Fact 1 is utilized in (3.6), respectively.

By condition (3.2),
is negative definite; namely, there is a number
such that
and hence, the asymptotic stability of the system immediately follows from Lemma 2.1**.** This completes the proof.

Remark 3.2.

Theorem 3.1 gives a sufficient condition for stability criterion for discrete-time systems (3.1). These conditions are described in terms of certain diagonal matrix inequalities, which can be realized by using the linear matrix inequality algorithm proposed in [14]. But Zhang et al. in [12] proved that these conditions are described in terms of certain symmetric matrix inequalities, which can be realized by using the Schur complement lemma and linear matrix inequality algorithm proposed in [14].

## 4. Conclusions

In this paper, an improved delay-dependent stability condition for discrete-time linear systems with interval-like time-varying delays has been presented in terms of an LMI.

## Authors’ Affiliations

## References

- Chen W-H, Guan Z-H, Lu X: Delay-dependent guaranteed cost control for uncertain discrete-time systems with delay.
*IEE Proceedings: Control Theory and Applications*2003, 150(4):412–416. 10.1049/ip-cta:20030572Google Scholar - He Y, Wang Q-G, Lin C, Wu M: Delay-range-dependent stability for systems with time-varying delay.
*Automatica*2007, 43(2):371–376. 10.1016/j.automatica.2006.08.015MATHMathSciNetView ArticleGoogle Scholar - Lin C, Wang Q-G, Lee TH: A less conservative robust stability test for linear uncertain time-delay systems.
*IEEE Transactions on Automatic Control*2006, 51(1):87–91. 10.1109/TAC.2005.861720MathSciNetView ArticleGoogle Scholar - Park P: A delay-dependent stability criterion for systems with uncertain time-invariant delays.
*IEEE Transactions on Automatic Control*1999, 44(4):876–877. 10.1109/9.754838MATHView ArticleGoogle Scholar - Xu S, Lam J: Improved delay-dependent stability criteria for time-delay systems.
*IEEE Transactions on Automatic Control*2005, 50(3):384–387.MathSciNetView ArticleGoogle Scholar - Fridman E, Shaked U: Stability and guaranteed cost control of uncertain discrete delay systems.
*International Journal of Control*2005, 78(4):235–246. 10.1080/00207170500041472MATHMathSciNetView ArticleGoogle Scholar - Gao H, Chen T: New results on stability of discrete-time systems with time-varying state delay.
*IEEE Transactions on Automatic Control*2007, 52(2):328–334.MathSciNetView ArticleGoogle Scholar - Gao H, Lam J, Wang C, Wang Y: Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay.
*IEE Proceedings: Control Theory and Applications*2004, 151(6):691–698. 10.1049/ip-cta:20040822MathSciNetGoogle Scholar - Jiang X, Han QL, Yu X: Stability criteria for linear discrete-time systems with interval-like time-varying delay.
*Proceedings of the American Control Conference, 2005*2817–2822. - Garcia G, Bernussou J, Arzelier D: Robust stabilization of discrete-time linear systems with norm-bounded time-varying uncertainty.
*Systems & Control Letters*1994, 22(5):327–339. 10.1016/0167-6911(94)90030-2MATHMathSciNetView ArticleGoogle Scholar - Xiong J, Lam J: Stabilization of discrete-time Markovian jump linear systems via time-delayed controllers.
*Automatica*2006, 42(5):747–753. 10.1016/j.automatica.2005.12.015MATHMathSciNetView ArticleGoogle Scholar - Zhang B, Xu S, Zou Y: Improved stability criterion and its applications in delayed controller design for discrete-time systems.
*Automatica*2008, 44(11):2963–2967. 10.1016/j.automatica.2008.04.017MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP:
*Difference Equations and Inequalities: Theory, Methods and Applications, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 155*. Marcel Dekker, New York, NY, USA; 1992:xiv+777.Google Scholar - Callier FM, Desoer CA:
*Linear System Theory, Springer Texts in Electrical Engineering*. Springer, New York, NY, USA; 1991:xiv+509.Google Scholar

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