Mathematical Problems in Engineering

Volume 2017, Article ID 5731325, 13 pages

https://doi.org/10.1155/2017/5731325

## Novel Stability Analysis for Uncertain Neutral-Type Lur’e Systems with Time-Varying Delays Using New Inequality

^{1}College of Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing, China^{2}Mathematics and Computer Science, Yunnan Minzu University, Kunming, China^{3}School of Mathematics and Statistics, Yunnan University, Kunming, China^{4}School of Science, Nanjing University of Science and Technology, Nanjing, China

Correspondence should be addressed to Lianglin Xiong; moc.621@8135_nilgnail

Received 8 October 2016; Revised 10 January 2017; Accepted 18 January 2017; Published 21 February 2017

Academic Editor: Sabri Arik

Copyright © 2017 Yanmeng Wang et al. 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 considers the delay-dependent stability analysis of neutral-type Lur’e systems with time-varying delays and sector bounded nonlinearities. First of all, using constructed function methods, a new Jensen-like inequality is introduced to obtain less conservative results. Second, a new class of Lyapunov-Krasovskii functional (LKF) is constructed according to the characteristic of the considered systems. Third, combining with the new inequality and reciprocal convex approach and some other inequality techniques, the new less conservative robust stability criteria are shown in the form of linear matrix inequalities (LMIs). Finally, three examples demonstrate the feasibility and the superiority of our methods.

#### 1. Introduction

Delay phenomenon is often encountered in many practical systems, such as biological systems, chemical systems, electronic systems, and network control systems. However, time-delay is usually the main cause of instability and bad performance. Hence, many authors devote themselves to studying the stability and many effective methods of the time-varying system to gain less conservative delay-dependent stability criteria [1–11], which include linear systems with the delay-fraction theory [2, 3] and nonlinear systems Lur’e systems [6–8]. As is known to all, delay-dependent stability results are less conservative than the delay-independent ones if delay size is very small. Therefore, a lot of articles were published recently which studied the delay-dependent stability for a class of neutral-type Lur’e systems with time-varying delays and sector bounded nonlinearities, and lots of significant results have been developed [12–30].

Delay-dependent stability criteria were presented for nominal and uncertain neutral-type Lur’e systems with constant time delays and sector bounded nonlinearities in [27]. The robust stability problems for neutral-type Lur’e systems with time-varying delays were considered because time delays vary always depending on time in [12–21, 27–30]. The free-weighting matrix method was applied to get less conservative stability criteria and to deal with the robust stability problems for neutral-type Lur’e systems with time-varying delays in [15, 16]. However, the free-weighting matrix method brings more variables which make the computation quite complex. So, it is the right time to improve the disadvantage of the free-weighting matrix method and to get less conservative stability criteria for neutral-type Lur’e systems with time-varying delays. Some new robust stability criteria were proposed without using the general free-weighting matrix method which are less conservative and easier to calculate than some previous ones in [15, 19]. Reference [31] can reduce conservatism by reducing the conservatism of the Jensen-like inequality. Motivated by [31], developing the Jensen-like inequality with double-integral term may reduce the conservativeness. As a result, less conservative criteria may be also got by constructing new integral inequalities used in LKF.

This paper studies the stability for a class of neutral-type Lur’e systems with time-varying delays and sector bounded nonlinearities. To investigate the neutral-type Lur’e system, this paper introduces a new triple-integral inequality used in the following LKFs and gets less conservative criteria. The LKF contains not only double-integral terms but also triple-integral terms. Using some effective techniques, such as a novel integral inequality, a piecewise analysis method, and the reciprocally convex combination inequality, instead of the general free-weighting matrix method, the delay-dependent stability criteria derived in the form of LMIs are less conservative than some existing results in other papers. The effectiveness and the less conservatism of stability criteria proposed in this paper are demonstrated by using numerical examples in Section 4.

*Notation*. denotes the -dimensional Euclidean vector space, and denotes the set of all real matrices. For a symmetric matrix , (resp., ) shows that is a positive (resp., negative) definite matrix. represents a diagonal matrix with diagonal elements . denotes a symmetric term in a symmetric matrix.

#### 2. Problem Statement and Preliminaries

Consider a class of Lur’e systems of uncertain neutral type with time-varying delays and sector-bound nonlinearities described by the following equation:where and stand for the output and state vectors, respectively. is a real-valued continuous initial function on . , , , , and are known real constant matrices with appropriate dimensions. , , and represent the time-varying uncertainty parameters. is the nonlinear function such aswhere is the th component of the output vector , and each term satisfies the finite sector condition shown in Figure 1(a) [32]with known positive scalars or the infinite sector condition shown in Figure 1(b) [32]