#### Abstract

This paper investigates the problem of robust exponential stability for linear parameter-dependent (LPD) systems with discrete and distributed time-varying delays and nonlinear perturbations. Parameter dependent Lyapunov-Krasovskii functional, Leibniz-Newton formula, and linear matrix inequality are proposed to analyze the stability. On the basis of the estimation and by utilizing free-weighting matrices, new delay-dependent exponential stability criteria are established in terms of linear matrix inequalities (LMIs). Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods.

#### 1. Introduction

Over the past decades, dynamical systems with state delays have attracted much interest in the literature over the half century, especially in the last decade. Since time delay is frequently a source of instability or poor performances in various systems such as electric, chemical processes, and long transmission line in pneumatic systems [1]. The problems of stability and stabilization for dynamical systems with or without state delays have been intensively studied in the past years by many researchers in mathematics and control communities [2, 3]. Stability criteria for dynamical systems with time delay is generally divided into two classes: delay-independent one and delay-dependent one. Delay-independent stability criteria tends 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 concerned with the size of the delay and usually provide a maximal delay size. Various stability of linear continuous-time and discrete-time systems subject to time-invariant parametric uncertainty have 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. 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 such as [4–6] which present sufficient conditions for robust stability of LPD continuous-time system with delays.

Recently, many researchers have studied the problem of stability for time-delay systems with nonlinear perturbations such as [7] which considers the robust stability for a class of linear systems with interval time-varying delay and nonlinear perturbations. In [8], exponential stability of time-delay systems with nonlinear uncertainties is studied. Based on the Lyapunov theory approach and the approaches of decomposing the matrix, a new exponential stability criterion is derived in terms of LMI. In [9], they propose a new delay-dependent stability criterion in terms of linear matrix inequality for dynamic systems with time-varying delays and nonlinear perturbations by using Lyapunov theory. However, many researchers have studied the problem of stability for systems with discrete and distributed delays such as [10] which presented some stability conditions for uncertain neutral systems with discrete and distributed delays. The robust stability of uncertain linear neutral systems with discrete and distributed delays has been studied in [11]. In [12, 13], they studied the problem of stability for linear switching system with discrete and distributed delays. Moreover, a descriptor model transformation and a corresponding Lyapunov-Krasovskii functionals have been introduced for stability analysis of systems with delays in [14, 15].

In this paper, we will investigate the problems of robust exponential stability for LPD system with mixed time-varying delays and nonlinear perturbations. Based on the combination of Leibniz-Newton formula and linear matrix inequality, the use of suitable Lyapunov-Krasovskii functional, new delay-dependent exponential stability criteria 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, 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 real matrices; 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 ; 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 equation of the form where is the state variable, , , are uncertain matrices belonging to the polytope and are discrete and distributed time-varying delays, respectively, satisfying where are given positive real constants. Consider the initial functions , with the norm and . The uncertainties represent the nonlinear parameter perturbations with respect to the current state and the delayed state , respectively, and are bounded in magnitude of the form where are given real constants.

*Definition 2.1. * The system (2.1) is robustly exponentially stable, if there exist positive real numbers and such that for each , the solution of the system (2.1) satisfies

Lemma 2.2 (Schur complement lemma, see [9]). * Given constant symmetric matrices where . Then if and only if
*

Lemma 2.3 (Jensen's inequality, see [1]). *For any constant matrix , , scalar , vector function such that the integrations concerned are well defined, then
*

*Rearranging the term with , one can yield the following inequality:*

Lemma 2.4 (see [16]). *Let be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices , and a scalar function :
**
where
*

#### 3. Main Results

In this section, we first study the robust exponential stability criteria for the system (2.1) by using the combination of linear matrix inequality (LMI) technique and Lyapunov theory method. We introduce the following notations for later use: where

Theorem 3.1. *For given positive real constants and , system (2.1) is robustly exponentially stable with a decay rate , if there exist positive definite symmetric matrices , any appropriate dimensional matrices , , , , , , , , 2, 3, , and positive real constants and satisfying the following LMIs:
*

*Moreover, the solution satisfies the inequality*

*where + .*

* Proof. *Choose a parameter-dependent Lyapunov-Krasovskii functional candidate for the system (2.1) of the form
where
Calculating the time derivatives of , , along the trajectory of (2.1) yields
where
Taking the time-derivative of leads to
Obviously, for any scalar , we get . Together with Lemma 2.3 (Jensen's inequality), we obtain
Following the estimation of , we have
From (3.16), it follows that
Taking the time derivative of and , we obtain
From the Leibinz-Newton formula, the following equation is true for any real matrices , , with appropriate dimensions
From the utilization of zero equation, the following equation is true for any real matrices , with appropriate dimensions
From (2.4), we obtain for any positive real constants and ,
By (3.5), Lemma 2.4 and the integral term of the right-hand side of and , we obtain
According to (3.12)–(3.22), it is straightforward to see that
where , , , , , , , and is defined in (3.2). The facts that , we obtain the following identities:
We define and as
From , we obtain
By (3.23)–(3.26), if the conditions (3.6)–(3.8) are true, then
which gives
From (3.28), it is easy to see that
where
Therefore, we get
where + . From (3.31), we get
This means that system (2.1) is robustly exponentially stable. The proof of the theorem is complete.

If , and when and are appropriate dimensional constant matrices, then system (2.1) reduces to the following system: Take the Lyapunov-Krasovskii functional as where According to Theorem 3.1, we have the following Corollary 3.2 for the delay-dependent exponential stability criteria of system (3.33).

Corollary 3.2. * For given positive real constants and , system (3.33) is exponentially stable with a decay rate , if there exist positive definite symmetric matrices , , any appropriate dimensional matrices , , and positive real constants and satisfying the following LMIs:*

*where*

*Moreover, the solution satisfies the inequality*

*where .*

#### 4. Numerical Examples

In order to show the effectiveness of the approaches presented in Section 3, four numerical examples are provided.

*Example 4.1. *Consider the LPD time-delay system (2.1) with the following parameters :
It is easy to see that , , , and . Find the discrete delay time to guarantee system (2.1) with the above parameters to be robustly exponentially stable with a decay rate .

*Solution 1. *By using the LMI Toolbox in Matlab (with accuracy 0.01) and conditions (3.4)–(3.8) of Theorem 3.1, this system is robustly exponentially stable for discrete delay satisfying and