Journal of Applied Mathematics

Volume 2013, Article ID 268905, 18 pages

http://dx.doi.org/10.1155/2013/268905

## New Delay-Dependent Robust Exponential Stability Criteria of LPD Neutral Systems with Mixed Time-Varying Delays and Nonlinear Perturbations

Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

Received 15 July 2013; Revised 30 September 2013; Accepted 2 October 2013

Academic Editor: Jitao Sun

Copyright © 2013 Sirada Pinjai and 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.

#### Abstract

This paper is concerned with the problem of robust exponential stability for linear parameter-dependent (LPD) neutral systems with mixed time-varying delays and nonlinear perturbations. Based on a new parameter-dependent Lyapunov-Krasovskii functional, Leibniz-Newton formula, decomposition technique of coefficient matrix, free-weighting matrices, Cauchy’s inequality, modified version of Jensen’s inequality, model transformation, and linear matrix inequality technique, new delay-dependent robust exponential stability criteria are established in terms of linear matrix inequalities (LMIs). Numerical examples are given to show the effectiveness and less conservativeness of the proposed methods.

#### 1. Introduction

Over the past decades, the problem of stability for neutral differential systems, which have delays in both their state and the derivatives of their states, has been widely investigated by many researchers, especially in the last decade. It is well known that nonlinearities, as time delays, may cause instability and poor performance of practical systems such as engineering, biology, and economics [1]. The problems of various stability and stabilization for dynamical systems with or without state delays and nonlinear perturbations have been intensively studied in the past years by many researchers of mathematics and control communities [1–35]. Stability criteria for dynamical systems with time delay are generally divided into two classes: delay-independent one and delay-dependent one. 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 are concerned with the size of the delay and usually provide a maximal delay size.

Recently, many researchers have studied the stability problem for neutral systems with time-varying delays and nonlinear perturbations have appeared [29, 31]. Furthermore, the convergence rates are essential for the practical system; then the exponential stability analysis of time delay systems has been favorably approved in the past decades; see, for example, [3, 9, 10, 14, 18–21, 25–28].

In addition, many researchers have paid attention to the problem of stability for linear systems with polytope uncertainties. The linear systems with polytopic-type uncertainties are called linear parameter-dependent (LPD) systems. That is, the uncertain state matrices are in the polytope consisting of all convex combination of known matrices. Most of sufficient (or necessary and sufficient) conditions have been obtained via Lyapunov-Krasovskii theory approaches in which parameter-dependent Lyapunov-Krasovskii functional has been employed. These conditions are always expressed in terms of linear matrix inequalities (LMIs). The results have been obtained for robust stability for LPD systems in which time-delay occurs in state variable; for example, [17, 18] presented sufficient conditions for robust stability of LPD discrete-time systems with delays. Moreover, robust stability of LPD continuous-time systems with delays was studied in [6, 19, 22, 30].

In consequence, it is important and interesting to study the problem of robust exponential stability for neutral systems with parametric uncertainties. This paper investigates the robust exponential stability analysis for LPD neutral systems with mixed time-varying delays and nonlinear perturbations. Based on combination of Leibniz-Newton formula, free-weighting matrices, Cauchy’s inequality, modified version of Jensen's inequality, decomposition technique of coefficient matrix, the use of suitable parameter-dependent Lyapunov-Krasovskii functional, model transformation, and linear matrix inequality technique, new delay-dependent robust exponential stability criteria for these systems will be obtained in terms of LMIs. Finally, numerical examples will be given to show the effectiveness of the obtained results.

#### 2. Problem Formulation and Preliminaries

We introduce some notations, a definition, and lemmas that will be used throughout the paper. denotes the set of all real nonnegative numbers; denotes the -dimensional space with the vector norm ; denotes the Euclidean vector norm of ; denotes the set of real matrices; denotes the transpose of the matrix ; is symmetric if ; denotes the identity matrix; denotes the set of all eigenvalues of ; ; ; ; ; denotes the space of all continuous vector functions mapping into , where , ; represents the elements below the main diagonal of a symmetric matrix.

Consider the system described by the following state equations of the form: where is the state variable and , , are uncertain matrices belonging to the polytope and are discrete and neutral time-varying delays, respectively, where , , , and are given positive real constants. Consider the initial functions , with the norm and . The uncertainties , , and are the nonlinear perturbations with respect to current state , discrete delayed state , and neutral delayed state , respectively, and are bounded in magnitude: where , and are given positive real constants.

In order to improve the bound of the discrete time-varying delayed in system (1), let us decompose the constant matrix as where , , and , with , being real constant matrices. By Leibniz-Newton formula, we have

By utilizing the following zero equation, we obtain where is a given positive real constant and will be chosen to guarantee the robust exponential stability of system (1). By (6), (7), and (8), system (1) can be represented by the form

*Definition 1. *The system (1) is robustly exponentially stable, if there exist positive real constants and such that, for each , the solution of the system (1) satisfies

Lemma 2 (Cauchy inequality). *For any constant symmetric positive definite matrix and , one has
*

Lemma 3 (see [15]). *The following inequality holds for any , , , , and :
**
where .*

Lemma 4. *For any constant symmetric positive definite matrix and which is discrete time-varying delays with (3), vector function such that the integrations concerned are well defined; then
*

*Proof. *From Lemma 2, it is easy to see that

Lemma 5. *Let be a vector-valued function with first order continuous derivative entries. Then, the following integral inequality holds for any matrices , , and is discrete time-varying delays with (3) and symmetric positive definite matrix :
**
where
*

*Proof. *From the Leibniz-Newton formula, one has
Therefore, for any , , the following equation is true:

Using Lemma 3 with , , , , and , we obtain

Substituting (19) into (18), we obtain

From (3), it is clear that

From (20) and (21), the integral inequality becomes
The proof of the theorem is complete.

*Remark 6. *In Lemma 4 and Lemma 5, we have modified the method from [8, 33], respectively.

#### 3. Main results

##### 3.1. Robust Exponential Stability Criteria

In this section, robust exponential stability criteria dependent on mixed time-varying delays of LPD neutral delayed system (1) with nonlinear perturbations via linear matrix inequality (LMI) approach will be presented. We introduce the following notations for later use: where

Theorem 7. *For , and given positive real constants , , , , , , , and , system (1) is robustly exponentially stable with a decay rate , if there exist symmetric positive definite matrices , any appropriate dimensional matrices , , , , , , , , , and positive real constants , , and such that the following symmetric linear matrix inequalities hold:
**Moreover, the solution satisfies the inequality
**
where + + + + + + .*

*Proof. *Choose a parameter-dependent Lyapunov-Krasovskii functional candidate for system (9) as
where

Calculating the time derivatives of , , along the trajectory of (9), yields

The time derivative of is

Obviously, for any scalar , we get and for any scalar , we obtain . Together with Lemma 4, we obtain

Taking the time derivative of , we obtain

By Lemma 5 and the integral term of the right hand side of and , we obtain

From the Leibniz-Newton formula, the following equations are true for any parameter-dependent real matrices , , with appropriate dimensions:

From the utilization of zero equation, the following equation is true for any parameter-dependent real matrices , with appropriate dimensions:

From (5), we obtain, for any positive real constants , , and ,

According to (29)–(36), it is straightforward to see that
where , , , , , , , and is defined in (23). From the fact that ,

It is true that if conditions (25) hold, then