#### 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. denotes the set of nonnegative integer numbers; denotes the -dimensional space with the vector norm ; denotes the Euclidean vector norm of , that is, ; denotes the space of all real matrices of -dimensions; denotes the transpose of the matrix ; is symmetric if ; denotes the identity matrix; denotes the set of all eigenvalues of ; ; ; ; ; matrix is called a semipositive definite () if , for all ; is a positive definite () if for all ; matrix is called a seminegative definite () if , for all ; is a negative definite () if for all ; means ; means ; 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 where , is the system state and is an initial value at . , are uncertain matrices belonging to the polytope of the form In addition, we assume that the time-varying delay is upper and lower bounded. It satisfies the following assumption of the form where and are known positive integers.

*Definition 1 (see [19]). * The system (1) is said to be robustly stable if there exists a positive definite function such that
along any trajectory of the solution of the system (1).

*Definition 2 (see [19]). *The system (1) when , , and is said to be asymptotically stable if there exists a positive definite function such that
along any trajectory of the solution of the system (1) when , .

Proposition 3 ([7, the Schur complement lemma]). *Given constant symmetric matrices , and of appropriate dimensions with , then if and only if
*

Proposition 4 (see [9]). * For any constant matrix , two integers and satisfying , and vector function , the following inequality holds:
*

Rewrite the system (1) in the following system:

#### 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: where

Theorem 5. *The system (1)-(2) is robustly stable if there exist positive definite symmetric matrices , and , any appropriate dimensional matrices ,, and , , satisfying the following LMIs:
*

*Proof. *Consider the following parameter-dependent Lyapunov-Krasovskii function for the system (9) of the form
where
Evaluating the forward deference of , it is defined as
Let us define, for ,
Then along the solution of the system (9), we obtain
From Proposition 4, we have
We can show that
It is easy to see that
By (22) and (23), we can obtain
and we conclude that
It is obvious that
The following equations are true for any polytopic matrices with appropriate dimensions:
It follows from (18)–(30) that
where and is defined in (10). Due to the fact that , we obtain the following identities:
By (31)–(33), if the conditions (12)–(15) are true, then
where . This means that the system (1)-(2) is robustly stable. The proof of the theorem is complete.

If and when then the system (1)-(2) reduces to the following system: Take the Lyapunov-Krasovskii functional as (16), where , , , , , , , , , , and when . Moreover, let us set polytopic matrices with appropriate dimensions of the forms , and when . 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: where

Corollary 6. *The system (35)-(36) is asymptotically stable if there exist positive definite symmetric matrices , and , any appropriate dimensional matrices , and , , satisfying the following LMIs:
*

#### 4. Numerical Examples

*Example 7. *Consider the following LPD discrete-time system with interval time-varying delays (1)-(2) with
with initial condition , . The numerical solutions and of (1)-(2) with (40) are plotted in Figure 1.

*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 , , and
For the given , Table 1 lists the comparison of the upper bounds delay 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].

*Example 8. *Consider the system (35) with
For the given , we calculate the allowable maximum value of 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.

#### Acknowledgments

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