Delay-dependent exponential stability criteria for neutral systems with interval time-varying delays and nonlinear perturbations

Abstract

This paper investigates the problem of global exponential stability for neutral systems with interval time varying delays and nonlinear perturbations. It is assumed that the state delay belongs to a given interval, which means that both the lower and upper bounds of the time-varying delay are available. The uncertainties under consideration are norm-bounded. Based on the Lyapunov–Krasovskii stability theory, delay-partitioning technique and lower bounds lemma, less conservative delay-dependent exponential stability criteria are derived in terms of linear matrix inequalities (LMIs) with fewer decision variables than the existing ones. Numerical examples are given to show the effectiveness of the proposed method.

Introduction

It is well known that time delays are frequently encountered in various practical engineering systems, such as biological systems, power systems and networked control systems. Since the existence of time delays is an important source of oscillation divergence, and instability in systems, it is very important to investigate stability analysis for systems with time-delays. In particular, the interest in the stability of neutral time-delay systems, which contain delays in both its state and its derivatives of the states, has been increasing rapidly due to their successful applications in various fields such as population ecology [1], distributed networks containing lossless transmission lines [2], and partial element equivalent circuit (PEEC) [3]. It should be pointed out that neutral delay systems constitute a more general class than those of the retarded type. The stability of these systems proves to be a more complex issue because the systems involve the derivative of the delayed state.

On the other hand, because of the inaccuracies and changes in the environment of the model, nonlinear perturbations which can be commonly encountered will break the stability of the systems. Therefore, the problem of stability analysis for neutral systems with time delays and nonlinear perturbations has been studied by many researchers in recent years [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. The stability criteria for time-delay systems can be divided into two categories, delay-independent criteria and delay-dependent criteria. It is well known that the delay-dependent stability criteria are less conservative than delay-independent ones, particularly when the size of time delay is small. Thus, more attentions have been focused on the derivation of delay-dependent stability results. In delay-dependent stability analysis, an important issue is to enlarge the feasibility region of the criteria or to obtain a maximum allowable upper bound of time delays as large as possible. In order to increase the delay bounds, various methods are employed, such as model transformation technique [9], integral-inequality method [14], free-weighting matrices technique [17], input–output approach [25] and delay-partitioning method [24], [28], [29]. Furthermore, some researchers put their effort on reducing the complexity of its calculation. For example, in [30], based on the scaled small gain theorem, less conservative stability criteria were derived, and much fewer variables were needed than those in the cited literatures.

As pointed out in [26], [27], interval time-varying delay, which can be found in real dynamic systems in networked control system, means that the lower bound of delay is not restricted to zero. Recently, the stability analysis of neutral systems with interval time-varying delay has been studied by some researchers. Yu et al. [6] investigated the asymptotic stability of neutral system with interval time varying delays and two classes of uncertainties based on Lyapunov–Krasovskii functional combined with Leibniz–Newton formula. Rakkiyappan et al. [19] proposed delay-range-dependent stability criteria for systems with interval time-varying delay by constructing new Lyapunov functional and using integral inequalities. Lakshmanan et al. [20] obtained less conservative results than that of [6], [19] by using a convex combination technique.

The results mentioned above are only concerned with the asymptotic stability. However, the exponential stability problem is also important since it can determine the convergence rate of system states to equilibrium points. Chen et al. [16] derived the robust exponential stability criteria for uncertain neutral systems with time-varying delays and nonlinear perturbations by using an integral inequality. Zhang et al. [18] studied the problem for neutral switched systems with interval time-varying mixed delays and nonlinear perturbations by using the piecewise Lyapunov–Krasovskii functional technique and the free-weighting matrix method. Ail [21] obtained the exponential stability criteria by using the free-weighting matrix method and the generalized eigenvalue problem approach. However, in [16], [21], the lower bound of time delay was fixed to zero. Furthermore, the nonlinear perturbation such as $f_3(t,\dot{y}(t-\tau \left(t\right))$ is not considered in [16], [18]. To the best of authors' knowledge, there are no results in the existing literatures concerning the problem of the exponential stability of the neutral systems with interval time varying delays and nonlinear perturbations which include $f_3(t,\dot{y}(t-\tau \left(t\right))$. It is worth pointing out that the Jensen integral inequality has been adopted to handle the integral term in [16], [18], [21]. However, the lower bounds lemma, which is investigated for a linear combination of positive functions weighted by the inverses of convex parameters [27], can obtain better results than those based on the Jensen integral inequality. Furthermore, it can decrease the number of decision variables.

Motivated by the above statement, in this paper, the problem of robust exponential stability for neutral systems with interval time-varying delays and nonlinear perturbations is investigated. Based on the Lyapunov stability theory, some new exponential delay-dependent stability conditions are derived. In order to get less conservative stability criteria and reduce the complexity of its calculation, delay-partitioning technique [24], [27], [28] and reciprocally convex method [27] are used. Finally, three numerical examples are given to illustrate the effectiveness of the proposed method.

Notations: Throughout this paper, I denotes the identity matrix with appropriate dimensions, ${\mathcal{R}}^n$ denotes the n-dimensional Euclidean space, and ${\mathcal{R}}^{m\times n}$ is the set of all $m\times n$ real matrices, $\parallel .\parallel$ refers to the Euclidean vector norm and the induced matrix norm. For symmetric matrices A and B, the notation $A>B$ (respectively, $A\ge B$) means that the matrix $A-B$ is positive definite (respectively, nonnegative). ${\lambda }_{\mathit{max}}(·)$ and ${\lambda }_{\mathit{min}}(·)$ stand for the largest and smallest eigenvalue of given square matrix, respectively. $\mathit{diag}\{\dots \}$ denotes the block diagonal matrix. $\parallel \phi \parallel ={\sup }_{-\mathit{max}\{h_M,{\tau }_M\}\le s\le 0}\left\{\parallel y\right(t+s)\parallel ,\parallel \dot{y}(t+s\left)\parallel \right\}$, for positive constants h_M and ${\tau }_M$.