On the Lyapunov matrices for integral delay systems

We study Lyapunov matrices for the class of integral delay systems with constant kernel and one delay. The uniqueness and computational issues of these Lyapunov matrices for exponentially stable systems are investigated.     

Stability conditions for integral delay systems

In this paper we consider a special class of integral delay systems arising in several stability problems of time‐delay systems. For these integral systems we derive stability and robust stability conditions in terms of Lyapunov–Krasovskii functionals. More explicitly, after providing the stability conditions we compute quadratic functionals and apply them to derive exponential estimates for solutions, and robust stability conditions for perturbed integral delay systems. Copyright © 2008 John Wiley & Sons, Ltd.     

An augmented model for robust stability analysis of time-varying delay systems

Stability analysis of linear systems with time-varying delay is investigated. In order to highlight the relations between the variation of the delay and the states, redundant equations are introduced to construct a new modelling of the delay system. New types of Lyapunov–Krasovskii functionals are then proposed allowing to reduce the conservatism of the stability criterion. Delay-dependent stability conditions are then formulated in terms of linear matrix inequalities. Finally, several examples show the effectiveness of the proposed methodology.     

Necessary stability conditions for linear delay systems

We present necessary conditions for the exponential stability of linear systems with multiple delays. They are expressed in terms of the delay Lyapunov matrix of the Lyapunov–Krasovskii functionals of complete type approach. New properties of independent interest, that establish connections of the system fundamental matrix with its Lyapunov matrix, are crucial elements of our proof. We illustrate our work with a number of examples.     

On exponential stability of integral delay systems

This paper is concerned with exponential stability of a class of integral delay systems with a prescribed decay rate. First, by carefully exploring the literature on this topic, a delay decomposition approach is established to reduce the conservatism in the existing sufficient conditions by constructing new Lyapunov–Krasovskii (LK) functionals. It is proven that the proposed sufficient conditions are less conservative than a recently established set of sufficient conditions. Second, by analyzing the characteristic equation of the considered integral delay system, necessary and sufficient conditions for the stability are obtained by computing the right-most zeros of a certain auxiliary point-delay linear system, for which stability criteria that are easy to test are also established based on this method. Numerical examples illustrate the effectiveness of the obtained results.     

New stability criteria of singular systems with time-varying delay via free-matrix-based integral inequality

This paper concerns the stability problem of singular systems with time-varying delay. First, the singular system with time-varying delay is transformed into the neutral system with time-varying delay. Second, a more proper Lyapunov–Krasovskii functional (LKF) is constructed by adding some integral terms to quadratic forms. Then, to obtain less conservative conditions, the free-matrix-based integral inequality is adopted to estimate the derivative of LKF. As a result, some delay-dependent stability criteria are given in terms of linear matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness and superiority of the proposed method.     

Stability analysis of systems via a new double free-matrix-based integral inequality with interval time-varying delay

ABSTRACT

This paper investigates the problem of delay-dependent stability analysis for systems with interval time-varying delay. By means of a new double free-matrix-based integral inequality and augmented Lyapunov–Krasovskii functionals containing as much information of time-varying delay as possible, a new stability criterion for systems is established. Firstly, by a double integral term two-step estimation approach and combined with single free-matrix-based integral inequalities, a stability criteria is presented. Then, compared with the double integral term two-step estimation approach, the proposed new double free-matrix-based integral inequality with more related time delays has potential to lead to a criterion with less conservatism. Finally, the validity of the presented method is demonstrated by two numerical examples.     

Necessary exponential stability conditions for linear periodic time‐delay systems

Exponential stability necessary conditions for linear periodic time‐delay systems are presented. They are obtained with the help of new properties of the Lyapunov matrix in the framework of Lyapunov–Krasvoskii functionals of complete type. An academic example illustrates our results. Copyright © 2016 John Wiley & Sons, Ltd.     

Wirtinger-based multiple integral inequality for stability of time-delay systems

Note that the conservatism of the delay-dependent stability criteria can be reduced by increasing the integral terms in Lyapunov–Krasovskii functional (LKF). This brief revisits the stability problem for a class of linear time-delay systems via multiple integral approach. The novelty of this brief lies in that a Wirtinger-based multiple integral inequality is employed to estimate the derivative of a class of LKF with multiple integral terms. Based on these innovations, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. Two numerical examples are exploited to demonstrate the effectiveness and superiority of the proposed method.     

Necessary stability conditions for neutral-type systems with multiple commensurate delays

Necessary stability conditions depending on the Lyapunov matrix for neutral-type systems with commensurate delays are presented. In order to obtain them, further results are introduced: the computation of the Lyapunov–Krasovskii functional of complete-type by a new Cauchy formula and the introduction of identities that relate the fundamental and the Lyapunov matrix of the system. The main result is illustrated by some examples.     

New robust exponential stability analysis for uncertain neural networks with time-varying delay

In this paper, the global robust exponential stability is considered for a class of neural networks with parametric uncer-tainties and time-varying delay. By using Lyapunov functional method, and by resorting to the new technique for estimating the upper bound of the derivative of the Lyapunov functional, some less conservative exponential stability criteria are derived in terms of linear matrix inequalities (LMIs). Numerical examples are presented to show the effectiveness of the proposed method.     

具有时滞脉冲的线性随机时滞系统的均方指数稳定

本文研究了具有时滞脉冲的线性随机时滞系统的稳定性问题,基于Lyapunov函数和Razumikhin技巧,针对具有镇定型脉冲和反镇定型脉冲的线性随机时滞系统分别建立了系统均方指数稳定的充分条件,最后给出两个数值例子论证结果的有效性.     

Necessary conditions of exponential stability for a class of linear neutral-type time-delay systems

For a class of linear neutral-type time-delay systems (NTTDSs), this paper will present necessary exponential stability conditions by employing the Lyapunov--Krasovskii functional approach. Since these conditions are represented by the Lyapunov matrix and the neutral coefficient matrix, they not only offer a novel tool for analysing stability of linear NTTDSs by characterising instability domains, but also extend the existing results of the neutral-delay-free systems. As a medium step, the relations between the Lyapunov matrix and the fundamental matrix are characterised. The validation of the obtained results is explained by numerical examples and comparison with some existing results.     

Critical parameters of integral delay systems

The critical frequencies of integral delay systems with a class of analytic kernels are determined via an auxiliary delay‐free system, and an upper bound on the number of critical frequencies is obtained. The critical delays are obtained by substituting these frequencies into the characteristic equation of the system. The procedure is validated with a number of nontrivial examples. Copyright © 2013 John Wiley & Sons, Ltd.     

An analysis of the exponential stability of linear stochastic neutral delay systems

This paper is concerned with the analysis of the mean square exponential stability and the almost sure exponential stability of linear stochastic neutral delay systems. A general stability result on the mean square and almost sure exponential stability of such systems is established. Based on this stability result, the delay partitioning technique is adopted to obtain a delay‐dependent stability condition in terms of linear matrix inequalities (LMIs). In obtaining these LMIs, some basic rules of the Ito calculus are also utilized to introduce slack matrices so as to further reduce conservatism. Some numerical examples borrowed from the literature are used to show that, as the number of the partitioning intervals increases, the allowable delay determined by the proposed LMI condition approaches h_max, the maximal allowable delay for the stability of the considered system, indicating the effectiveness of the proposed stability analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Delay-dependent exponential stability for uncertain neutral stochastic neural networks with interval time-varying delay

This paper is mainly concerned with the problem for the robustly exponential stability in mean square moment of uncertain neutral stochastic neural networks with interval time-varying delay. With an appropriate augmented Lyapunov–Krasovskii functional (LKF) formulated, the convex combination method is utilised to estimate the derivative of the LKF. Some new delay-dependent exponential stability criteria for such systems are obtained in terms of linear matrix inequalities, which involve fewer matrix variables and have less conservatism. Finally, two illustrative numerical examples are given to show the effectiveness of our obtained results.     

New criteria for exponential stability of nonlinear difference systems with time-varying delay

General nonlinear time-varying difference systems with time-varying delay are considered. Some new explicit criteria for global exponential stability are given. Two examples are given to illustrate the obtained results.     

A dissipative dynamical systems approach to stability analysis of time delay systems

In this paper the concepts of dissipativity and the exponential dissipativity are used to provide sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite‐dimensional linear dynamical system and an infinite‐dimensional time delay operator, we show that the time delay operator is dissipative with respect to a quadratic supply rate and with a storage functional involving an integral term identical to the integral term appearing in standard Lyapunov–Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides a dissipativity theoretic interpretation of Lyapunov–Krasovskii functionals for asymptotically stable dynamical systems with arbitrary time delay. Copyright © 2004 John Wiley & Sons, Ltd.     

New stability criteria of linear singular systems with time-varying delay

Most of the existing results on the stability problem of delayed singular systems only pertain to the case of constant delay. This is due to the fact that time-varying delay makes it hardly possible to explicitly express the fast variables. In this paper, aiming at dealing with the case of time-varying delay, we create a way to prove the stability by using a perturbation approach. Rather, we first get the decay rate for slow variables by using Lyapunov functional approach and, furthermore, guarantee that the fast variables fall into decay by characterising their effect on the derived decay rate. Also, we present a convexity technique in computing the constructed Lyapunov functional which contributes to the elimination of the possible conservatism caused by the varying rate of delay. Finally, we provide two numerical examples to demonstrate the effectiveness of the method.     

Necessary conditions for the exponential stability of time‐delay systems via the Lyapunov delay matrix

Exponential necessary stability conditions for linear systems with multiple delays are presented. The originality of these conditions is that, in analogy with the case of delay free systems, they depend on the Lyapunov matrix function of the delay system. They are validated by examples for which the analytic characterization of the stability region is known. Copyright © 2013 John Wiley & Sons, Ltd.