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Delay-Dependent Stability Criteria for Systems with Interval Time-Varying Delay
Boren Li
Journal of Control Science and Engineering , 2011, DOI: 10.1155/2011/938749
Abstract: This paper is concerned with robust stability of uncertain linear systems with interval time-varying delay. The time-varying delay is assumed to belong to an interval, which means that the derivative of the time-varying delay has an upper bound or a restriction. On other occasions, if we do not take restriction on the derivative of the time-varying delay into consideration, it allows the delay to be a fast time-varying function. The uncertainty under consideration includes a polytopic-type uncertainty and a linear fractional norm-bounded uncertainty. In order to obtain much less conservative results, a new Lyapunov-Krasovskii functional, which makes use of the information of both the lower and upper bounds of the interval time-varying delay, is proposed to derive some new stability criteria. Numerical examples are given to demonstrate the effectiveness of the proposed stability criteria.
Novel delay-dependent stability criteria for neural networks with interval time-varying delay

Wang Jian-An,

中国物理 B , 2011,
Abstract: The problem of delay-dependent asymptotic stability for neural networks with interval time-varying delay is investigated. Based on the idea of delay decomposition method, a new type of Lyapunov-Krasovskii functional is constructed. Several novel delay-dependent stability criteria are presented in terms of linear matrix inequality by using the Jensen integral inequality and a new convex combination technique. Numerical examples are given to demonstrate that the proposed method is effective and less conservative.
Improved Criteria on Delay-Dependent Stability for Discrete-Time Neural Networks with Interval Time-Varying Delays
O. M. Kwon,M. J. Park,Ju H. Park,S. M. Lee,E. J. Cha
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/285931
Abstract: The purpose of this paper is to investigate the delay-dependent stability analysis for discrete-time neural networks with interval time-varying delays. Based on Lyapunov method, improved delay-dependent criteria for the stability of the networks are derived in terms of linear matrix inequalities (LMIs) by constructing a suitable Lyapunov-Krasovskii functional and utilizing reciprocally convex approach. Also, a new activation condition which has not been considered in the literature is proposed and utilized for derivation of stability criteria. Two numerical examples are given to illustrate the effectiveness of the proposed method.
New stability criteria for discrete-time systems with interval time-varying delay and polytopic uncertainty
Zhang,W.; Xie,Q. Y.; Cai,X. S.; Han,Z. Z.;
Latin American applied research , 2010,
Abstract: this paper is considered with the robust stability problem for linear discrete-time systems with polytopic uncertainty and an interval time-varying delay in the state. on the basis of a novel lyapunov-krasovskii functional, new delay-range-dependent stability criteria are established by employing the free-weighting matrix approach and a jensen-type sum inequality. it is shown that the newly proposed criteria can provide less conservative results than some existing ones. numerical examples are given to illustrate the effectiveness of the proposed approach.
New stability criteria for discrete-time systems with interval time-varying delay and polytopic uncertainty  [cached]
W. Zhang,Q. Y. Xie,X. S. Cai,Z. Z. Han
Latin American applied research , 2010,
Abstract: This paper is considered with the robust stability problem for linear discrete-time systems with polytopic uncertainty and an interval time-varying delay in the state. On the basis of a novel Lyapunov-Krasovskii functional, new delay-range-dependent stability criteria are established by employing the free-weighting matrix approach and a Jensen-type sum inequality. It is shown that the newly proposed criteria can provide less conservative results than some existing ones. Numerical examples are given to illustrate the effectiveness of the proposed approach.
Robust Stability Criteria for Uncertain Neutral Systems with Interval Nondifferentiable Time-Varying Delay and Nonlinear Perturbations  [PDF]
W. Weera,P. Niamsup
Journal of Applied Mathematics , 2011, DOI: 10.1155/2011/138912
Abstract: We study the robust stability criteria for uncertain neutral systems with interval time-varying delays and time-varying nonlinear perturbations simultaneously. The constraint on the derivative of the time-varying delay is not required, which allows the time-delay to be a fast time-varying function. Based on the Lyapunov-Krasovskii theory, we derive new delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) which can be solved by various available algorithms. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature. 1. Introduction It is well known that the existence of time delay in a system may cause instability and oscillations. Example, of time-delay systems are chemical engineering systems, biological modeling, electrical networks, physical networks, and many others, [7–16]. The stability criteria for system with time delays can be classified into two categories: delay-independent and delay-dependent. Delay-independent criteria do not employ any information on the size of the delay; while delay-dependent criteria make use of such information at different levels. Delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the delay is small. In many practical systems, models of system are described by neutral differential equations, in which the models depend on the delays of state and state derivatives. Heat exchanges, distributed networks containing lossless transmission lines and population ecology are examples of neutral systems because of its wider application. Therefore, several researchers have studied neutral systems and provided sufficient conditions to guarantee the asymptotic stability of neutral time delay systems, see [5, 9, 11–14, 16, 17] and references cited therein. Well-known nonlinearities, as the delays, may cause instability and poor performance of practical systems, which have driven many researchers to study the problem of nonlinear perturbed systems with state delays during the recent years [5, 7, 9, 18]. In [18], the delay-dependent robust stability for linear time-varying systems with nonlinear perturbations is given, by using the Newton-Leibniz formula which has been taken into account instead of applying an integral inequality. In [7], a model transformation technique is used to deal with the stability of system with time varying for delays and nonlinear perturbations. In [9], based on a descriptor model transformation combined with a matrix decomposition approach, the
Exponential Admissibility and Dynamic Output Feedback Control of Switched Singular Systems with Interval Time-Varying Delay  [PDF]
Jinxing Lin,Chunxia Fan
Mathematical Problems in Engineering , 2010, DOI: 10.1155/2010/680382
Abstract: This paper is concerned with the problems of exponential admissibility and dynamic output feedback (DOF) control for a class of continuous-time switched singular systems with interval time-varying delay. A full-order, dynamic, synchronously switched DOF controller is considered. First, by using the average dwell time approach, a delay-range-dependent exponential admissibility criterion for the unforced switched singular time-delay system is established in terms of linear matrix inequalities (LMIs). Then, based on this criterion, a sufficient condition on the existence of a desired DOF controller, which guarantees that the closed-loop system is regular, impulse free and exponentially stable, is proposed by employing the LMI technique. Finally, some illustrative examples are given to show the effectiveness of the proposed approach. 1. Introduction The past decades have witnessed an enormous interest in switched systems, due to their powerful ability in modeling of event-driven systems, logic-based systems, parameter- or structure-varying systems, and so forth; for details, see [1–4] and the references therein. Switched systems are a class of hybrid systems, which consist of a collection of continuous- or discrete-time subsystems and a switching rule specifying the switching between them. When focusing on the classification problems in switched systems, it is commonly recognized that there exist three basic problems [1]: (i) finding conditions for stability under arbitrary switching; (ii) identifing the limited but useful class of stabilizing switching signals, and (iii) construct a stabilizing switching signal. Many effective methods have been presented to tackle these three basic problems such as the multiple Lyapunov function approach [5], the piecewise Lyapunov function approach [6], the switched Lyapunov function approach [7], the convex combination technique [8], and the dwell time or average dwell time scheme [9–12]. On the other hand, time-delay is very common in engineering systems and is frequently a source of instability and poor performance [13]. Therefore, control of switched time-delay systems has received more and more attention in the past few years; see [14–23] and the references therein. As far as we know, singular systems (known also as descriptor, implicit or differential algebraic systems) also provide a natural framework for modeling of dynamic systems and describe a larger class of systems than the regular system models [24]. Switched singular systems have strong engineering background such as electrical networks [25], economic
Synchronization and Pinning Control in Complex Networks with Interval Time-Varying Delay
Hai-Feng Jiang,Tao Li
Mathematical Problems in Engineering , 2012, DOI: 10.1155/2012/948495
Abstract: The problems on synchronization and pinning control for complex dynamical networks with interval time-varying delay are investigated and two less conservative criteria are established based on reciprocal convex technique. Pinning control strategies are designed to make the complex networks synchronized. Moreover, the problem of designing controllers can be converted into solving a series of NMIs (nonlinear matrix inequalities) and LMIs (linear matrix inequalities), which reduces the computation complexity when comparing with those present results. Finally, numerical simulations can verify the effectiveness of the derived methods.
Delay-Dependent Robust Exponential Stability for Uncertain Neutral Stochastic Systems with Interval Time-Varying Delay
Weihua Mao,Feiqi Deng,Anhua Wan
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/593780
Abstract: This paper discusses the mean-square exponential stability of uncertain neutral linear stochastic systems with interval time-varying delays. A new augmented Lyapunov-Krasovskii functional (LKF) has been constructed to derive improved delay-dependent robust mean-square exponential stability criteria, which are forms of linear matrix inequalities (LMIs). By free-weight matrices method, the usual restriction that the stability conditions only bear slow-varying derivative of the delay is removed. Finally, numerical examples are provided to illustrate the effectiveness of the proposed method.
Delay-Range-Dependent Stability Criteria for Takagi-Sugeno Fuzzy Systems with Fast Time-Varying Delays
Pin-Lin Liu
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/475728
Abstract: The problem of delay-range-dependent stability for T-S fuzzy system with interval time-varying delay is investigated. The constraint on the derivative of the time-varying delay is not required, which allows the time delay to be a fast time-varying function. By developing delay decomposition approach, integral inequalities approach (IIA), and Leibniz-Newton formula, the information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMIs) which can be easily solved by various optimization algorithms. Simulation examples show resulting criteria outperform all existing ones in the literature. It is worth pointing out that our criteria are carried out more efficiently for computation and less conservatism of the proposed criteria.
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