- Research Article
- Open Access
- Published:

# Stability Criterion for Discrete-Time Systems

*Journal of Inequalities and Applications*
**volumeÂ 2010**, ArticleÂ number:Â 201459 (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.

For any positive scalar and vectors and the following inequality holds:

Let us denote

Lemma 2.1 (see [13]).

The zero solution of difference system is asymptotic stability if there exists a positive definite function such that

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]).

For any constant symmetric matrix , scalar , vector function , one has

## 3. Improved Stability Criterion

In this section, we give a novel delay-dependent stability condition for discrete-time systems with interval-like time-varying delays. Now, consider the following system:

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.

Give integers and . Then, the discrete time-delay system (3.1) is asymptotically stable for any time delay satisfying , if there exist symmetric positive definite matrices satisfying the following matrix inequalities:

where , .

Proof.

Consider the Lyapunov function , where

with being symmetric positive definite solutions of (3.2) and

Then difference of along trajectory of solution of (3.1) is given by

where

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

Note that

and hence

Then we have

Using Lemma 2.2, we obtain

From the above inequality it follows that

where , , and

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.

## 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):412â€“416. 10.1049/ip-cta:20030572He Y, Wang Q-G, Lin C, Wu M: Delay-range-dependent stability for systems with time-varying delay.

*Automatica*2007, 43(2):371â€“376. 10.1016/j.automatica.2006.08.015Lin 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):87â€“91. 10.1109/TAC.2005.861720Park P: A delay-dependent stability criterion for systems with uncertain time-invariant delays.

*IEEE Transactions on Automatic Control*1999, 44(4):876â€“877. 10.1109/9.754838Xu S, Lam J: Improved delay-dependent stability criteria for time-delay systems.

*IEEE Transactions on Automatic Control*2005, 50(3):384â€“387.Fridman E, Shaked U: Stability and guaranteed cost control of uncertain discrete delay systems.

*International Journal of Control*2005, 78(4):235â€“246. 10.1080/00207170500041472Gao H, Chen T: New results on stability of discrete-time systems with time-varying state delay.

*IEEE Transactions on Automatic Control*2007, 52(2):328â€“334.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):691â€“698. 10.1049/ip-cta:20040822Jiang 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*2817â€“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):327â€“339. 10.1016/0167-6911(94)90030-2Xiong J, Lam J: Stabilization of discrete-time Markovian jump linear systems via time-delayed controllers.

*Automatica*2006, 42(5):747â€“753. 10.1016/j.automatica.2005.12.015Zhang B, Xu S, Zou Y: Improved stability criterion and its applications in delayed controller design for discrete-time systems.

*Automatica*2008, 44(11):2963â€“2967. 10.1016/j.automatica.2008.04.017Agarwal 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.Callier FM, Desoer CA:

*Linear System Theory, Springer Texts in Electrical Engineering*. Springer, New York, NY, USA; 1991:xiv+509.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Ratchagit, K., Phat, V. Stability Criterion for Discrete-Time Systems.
*J Inequal Appl* **2010**, 201459 (2010). https://doi.org/10.1155/2010/201459

Received:

Accepted:

Published:

DOI: https://doi.org/10.1155/2010/201459

### Keywords

- Lyapunov Functional
- Symmetric Positive Definite Matrice
- Constant Symmetric Matrix
- Follow Matrix Inequality
- Improve Stability Criterion