Abstract

In this study, we focus on stability analysis for systems with time-varying delay and nonlinear perturbations. In order to cut down the conservatism of the existing stability criteria, we utilize the triple integral forms of Lyapunov-Krasovskii functional (LKF). In addition, by using single and double integral forms of Wirtinger-based inequality, we overcome some conservatism which come from Jensen’s inequality. Three well-known numerical examples are given at the end. Compared with some existing results, our results have less conservatism.

1. Introduction

Time-delay has attracted a lot of interests because it is widely encountered in communication systems, neural networks, economic systems, biological systems, and networked control systems [1–6]. Since time-delay will cause serious degradation of system performance, a lot of researchers are making considerable effort for stability analysis of time-delay systems in the last few decades. The LKF approach and LMI method are the efficient instrument to get the delay-dependent stability criterion. However, only a sufficient condition we can get by utilizing these methods. Therefore, we focus on cutting down the conservatism of stability criterion. As we know, the maximum allowable delay bound (MADB) of the time-delay can measure the conservatism of stability criterion. We can get the larger MADB of time-varying delay according to the stability criterion with less conservatism.

In a lot of existing results, Jensen’s inequality has played an important role. However, it will induce some conservatism hard to overcome. To cut down the conservatism, Wirtinger-based integral inequality [7], which can be used to obtain much tighter lower bound of single integral terms, was proposed. Very recently, based on Wirtinger-based integral inequality, a Wirtinger-based double integral inequality was proposed to get much tighter lower bound of double integral terms [8].

In a lot of recent literatures [9–11], researchers only utilize single and double integral forms of LKF to derive delay-dependent stability criterion. We believe that triple integral form of LKF is helpful for improving the performance of former criteria.

Motivated by the previous discussions, we are concerned about the stability analysis for the time-varying delay systems with nonlinear perturbations. We introduce the triple integral forms of LKF to cut down the conservatism. Taking the time derivative of , we obtain . Instead of Jensen’s inequality, Wirtinger-based double integral inequality was utilized to find much tighter upper bound of . Compared with some existing results, our results have less conservatism.

The organization of this paper is as follows. The system is presented in Section 2, and some useful lemmas are given. Then the new stability criterion and proof are given in Section 3. In Section 4, the effectiveness of our results can be illustrated by three examples. Section 5 is the conclusion of our investigation.

Notations. In this paper, the superscript means the transpose of a matrix; denotes the column vector.

2. Problem Formulation and Preliminaries

The time-varying delay systems with nonlinear perturbations are considered as follows:with as the state variate; are predefined constant matrices; represent a time-varying delay satisfyingThe nonlinear perturbations and can be abbreviated as and , assumed as follows:where and are positive scalars and are constant matrices.

In order to derive improved stability criterion, the following lemmas will be used.

Lemma 1 (see [7]). is a symmetric positive definite matrix, for differentiable function , and we can obtain where

Lemma 2 (see [28]). For positive definite , , reciprocally convex combination of can be written as

Lemma 3 (see [8]). is a symmetric positive definite matrix, for differentiable function , and we havewhere

3. Stability Criterion

Theorem 4. System (1) satisfying (2)-(3) will be asymptotically stable if there exist given scalars , and and and and positive symmetric matrices , , and and appropriate dimensions matrices , satisfying the following LMIs:where

Proof. Construct the following LKF: whereThe time derivative of is as follows:wherewhereAccording to Lemma 1, we obtainwhereand inequality (16) can be denoted asIt is clear that the real numbers and satisfy , , and . Then introduce appropriate dimensions matrices and , such thatApplying Lemma 2 to (17) and (18)Using Lemma 3 can lead to whereForm (3), we haveCombining (13), (14), (20), (22), (23), and (25), we obtainUsing Schur complement, (26) can be transform to the first LMI of (9), which completed the proof of Theorem 4.