On delay-dependent robust stability of a class of uncertain mixed neutral and Lur’e dynamical systems with interval time-varying delays

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Abstract

This paper deals with the problem of a new delay-dependent robust stability criteria for a class of mixed neutral and Lur’e systems. The system has time-varying uncertainties, interval time-varying delays and sector-bounded nonlinearity. The proposed method is based on Lyapunov method, a delay-dependent criterion for asymptotic stability is established in terms of linear matrix inequality (LMI). Numerical examples show the effectiveness of the proposed method.

Introduction

Many nonlinear physical systems can be expressed as a feedback connection of a linear dynamical system and a nonlinear element. One of the important classes of nonlinear systems is the Lur’e system whose nonlinear element satisfies certain sector constraints. Since the notion of absolute stability was introduced by Lur’e [1], stability analysis for the Lur’e systems has been extensively studied in [2], [3], [4], [5]. For neutral systems, the stability analysis has been also presented in [6], [7], [8], [9]. Since time-delay frequently occurrs in practical systems and is often the source of instability, many researchers have focused on the stability analysis of time-delay systems. For more details, see the works [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] and references cited therein.

During the last several decades, a great deal of effort has been made for the stability analysis of time-delay systems with sector-bounded nonlinearity, since occurrence of time-delay, nonlinear perturbations and parameter uncertainties often cause instability and poor performance of systems. To enlarge the feasibility region of the stability criteria, a new bounding technique was studied in [10] by introducing variables in cross-terms. Concerning the descriptor approach for time-delayed systems, Fridman et al. [13] developed an extensive work. By employing the approach of the linear matrix inequalities (LMIs), many novel conditions were obtained for absolute stability of Lur’e systems with time-delay [2], [3], [4]. Based on the method of decomposing the matrices and using the descriptor system approach, the absolute stability for Lur’e systems was studied in [5]. On the other hand, the stability problem for neutral systems with nonlinearity perturbation and time-delay was thus of interest to a great number of researchers. Souza et al. [6] dealt with the robust stability analysis by extending the discretisation technique. Yu and Lien [8] solved the stability and stabilization problem for uncertain neutral systems by using genetic algorithm. Especially, in [7], the paper dealt with the absolute stability criterion for mixed neutral and Lur’e system with time-delay. Recently, the free-weighting matrix approach [20], [21], [22], [23], [24] was also employed to further reduce the detailed conservatism, which brought some novel results.

In this paper, a new delay-dependent robust stability criterion for uncertain mixed neutral and Lur’e systems with interval time-varying delays is proposed. By constructing a suitable augmented Lyapunov functional, a delay-dependent criterion is derived in terms of LMIs which can be solved efficiently by using the interior-point algorithms [25]. In order to obtain less conservative results, a new integral inequality lemma which utilizes free weighting matrices is proposed. Finally, numerical examples are included to show the effectiveness of the proposed method.

Notations: Rn denotes n-dimensional Euclidean space, and Rn×m is the set of all real matrices. A<0 means that it is a real symmetric negative definitive matrix. I is the identity matrix with appropriate dimensions. · denotes Euclidean vector norm or includes the matrix 2-norm. diag{} denotes the block diagonal matrix. ρ(·) denotes the spectral radius of given matrix. L2 [a,b] is the space of the square integral function on the interval [a,b]. C([0,)Rn) denotes the Banach space of continuous vector functions from [0,) to Rn.

Section snippets

Problem statements and preliminaries

Consider the following uncertain mixed neutral and Lur’e system with interval time-varying delay:x˙(t)Cx˙(tτ(t))=(A+δA(t))x(t)+(A1+δA1(t))x(th(t))+(B+δB(t))f(σ(t)),σ(t)=HTx(t)=[h1h2hm]Tx(t),t0,x(s)=ϕ(s),s[max{hu,τu},0],where x(t)Rn is the state vector, σ(t)Rm is the output vector, ARn×n, BRn×m, CRn×n, A1Rn×n, HRn×m are constant known matrices. f(σ(t))Rm is the nonlinear function in the feedback path, which is denoted as f for simplicity in the sequel. Its form is formulated as f(

Main result

In this section, we propose a new stability criterion for uncertain mixed neutral and Lur’e systems (1) with interval time-varying delays.

For simplicity, define the matrices: E¯=[E1E20E30000000],F¯1=[F11F12F13F14F15F16F17F18F19F1AF1B],F¯2=[F21F22F23F24F25F26F27F28F29F2AF2B],F¯3=[F31F32F33F34F35F36F37F38F39F3AF3B],N¯=[N1TN2TN3TN4TN5TN6TN7TN8TN9TNATNBT]T,M¯1=[M11TM12TM13TM14TM15TM16TM17TM18TM19TM1ATM1BT]T,M¯2=[M21TM22TM23TM24TM25TM26TM27TM28TM29TM2ATM2BT]T,M¯3=[M31TM32TM33TM34TM35TM36TM37TM38TM39T

Numerical examples

The following numerical examples are presented to illustrate the less conservative of the proposed theoretical results in the paper.

Example 1

Consider the uncertain neutral system with parameters [8]:x˙(t)Cx˙(tτ(t))=(A+δA(t))x(t)+(A1+δA1(t))x(th(t)),where A=2001,A1=1011,C=ω00ω,δA(t)=α100α2,δA1(t)=1002,and τ(t)=τ is constant, ω,αi and i (i=1,2) denote the uncertainties which satisfy: 1<ω<1,1.6<α1<1.6,0.05<α2<0.05,0.1<1<0.1,0.3<2<0.3.For hL=0.5,ω=±0.1, our results obtained by applying

Conclusions

In this paper, delay-dependent robust stability of a class of uncertain mixed neutral and Lur’e systems with interval time-varying delays and sector-bounded nonlinearity has been proposed. Differing with the previous work on this topic, the suitable augmented Lyapunov–Krasovskii functional and free-weighting matrices were utilized to derive less conservative results for the considered systems. A new delay-dependent robust stability criterion of the class of uncertain mixed neutral and Lur’e

References (30)

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This work was supported by National Basic Research Program of China (2010CB732501) and the National Natural Science Foundation of China (NSFC-60873102).

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