Exponential stability of slowly varying discrete systems with multiple state delays

We give sufficient conditions for the exponential stability of a class of perturbed time‐varying difference equations with multiple delays and slowly varying coefficients. Under appropriate growth conditions on the perturbations, combined with the ‘freezing’ technique, we establish explicit conditions for global exponential stability. Copyright © 2012 John Wiley & Sons, Ltd.     

Exponential stability of impulsive positive switched systems with discrete and distributed time‐varying delays

This paper studies the exponential stability problems of discrete‐time and continuous‐time impulsive positive switched systems with mixed (discrete and distributed) time‐varying delays, respectively. By constructing novel copositive Lyapunov‐Krasovskii functionals and using the average dwell time technique, delay‐dependent sufficient conditions for the solvability of considered problems are given in terms of fairly simple linear matrix inequalities. Compared with the most existing results, by introducing an extra real vector, restrictive conditions on derivative of the time‐varying delays (less than 1) are relaxed, thus the obtained improved stability criteria can deal with a wider class of continuous‐time positive switched systems with time‐varying delays. Finally, two simple examples are provided to verify the validity of theoretical results.     

Improved robust stability criteria for uncertain systems with time‐varying delay

This paper is concerned with the delay‐dependent stability and robust stability for uncertain systems with time‐varying delay. Through constructing an appropriate type of Lyapunov‐Krasovskii functional and proving its positive definiteness, using slack matrices and a convex combination condition, the delay‐dependent stability criteria, which are less conservative, are derived in terms of linear matrix inequalities. Numerical examples are also given to illustrate the improvement on the conservatism of the delay bound over some existing results. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

New results of stability analysis for systems with time‐varying delay

This paper studies the problem of stability analysis for continuous‐time systems with time‐varying delay. By developing a delay decomposition approach, 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. The merits of the proposed results lie in their less conservatism, which are realized by choosing different Lyapunov matrices in the decomposed integral intervals and estimating the upper bound of some cross term more exactly. Numerical examples are given to illustrate the effectiveness and less conservatism of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

New criteria for exponential stability of nonlinear time‐varying differential systems

General nonlinear time‐varying differential systems are considered. An explicit criterion for exponential stability is presented. Furthermore, an explicit robust stability bound for systems subjected to nonlinear time‐varying perturbations is given. In particular, it is shown that the generalized Aizerman conjecture holds for positive linear systems. Some examples are given to illustrate obtained results.Copyright © 2012 John Wiley & Sons, Ltd.     

On the stability issues of switched singular time‐delay systems with slow switching based on average dwell‐time

The issue of exponential stability analysis of continuous‐time switched singular systems consisting of a family of stable and unstable subsystems with time‐varying delay is investigated in this paper. It is very difficult to analyze the stability of such systems because of the existence of time‐delay and unstable subsystems. In this regard, on the basis of the free‐weighting matrix approach, by constructing the new Lyapunov‐like Krasovskii functional, and using the average dwell‐time approach, delay‐dependent sufficient conditions are derived and formulated in terms of LMIs to check the exponential stability of such systems. This paper also highlights the relationship between the average dwell‐time of the switched singular time‐delay system, its stability, exponential convergence rate of differential states, and algebraic states. Finally, a numerical example is given to confirm the analytical results and illustrate the effectiveness of the proposed strategy. Copyright © 2012 John Wiley & Sons, Ltd.     

Compensation of time‐varying input delay for discrete‐time nonlinear systems

We consider general discrete‐time nonlinear systems (of arbitrary nonlinear growth) with time‐varying input delays and design an explicit predictor feedback controller to compensate the input delay. Such results have been achieved in continuous time, but only under the restriction that the delay rate is bounded by unity, which ensures that the input signal flow does not get reversed, namely, that old inputs are not felt multiple times by the plant (because on such subsequent occasions, the control input acts as a disturbance). For discrete‐time systems, an analogous restriction would be that the input delay is non‐increasing. In this work, we do not impose such a restriction. We provide a design and a global stability analysis that allow the input delay to be arbitrary (containing intervals of increase, decrease, or stagnation) over an arbitrarily long finite period of time. Unlike in the continuous‐time case, the predictor feedback law in the discrete‐time case is explicit. We specialize the result to linear time‐invariant systems and provide an explicit estimate of the exponential decay rate. Carefully constructed examples are provided to illustrate the design and analytical challenges. Copyright © 2015 John Wiley & Sons, Ltd.     

Exponential H∞ filtering for switched linear systems with interval time‐varying delay

This paper deals with the problem of exponential H_∞ filtering for a class of continuous‐time switched linear system with interval time‐varying delay. The time delay under consideration includes two cases: one is that the time delay is differentiable and bounded with a constant delay‐derivative bound, whereas the other is that the time delay is continuous and bounded. Switched linear filters are designed to ensure that the filtering error systems under switching signal with average dwell time are exponentially stable with a prescribed H_∞ noise attenuation level. Based on the free‐weighting matrix approach and the average dwell technology, delay‐dependent sufficient conditions for the existence of such a filter are derived and formulated in terms of linear matrix inequalities (LMIs). By solving that corresponding LMIs, the desired filter parameterized matrices and the minimal average dwell time are obtained. Finally, two numerical examples are presented to demonstrate the effectiveness of the developed results. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal partitioning method for stability analysis of continuous/discrete delay systems

This paper is concerned with the problem of stability analysis for continuous‐time/discrete‐time systems with interval time‐varying delay. Based on the idea of partitioning the delay interval into l nonuniform subintervals, new Lyapunov functionals are established. By utilizing the reciprocally convex approach to deal with the delay information in each subinterval, sufficient stability conditions are proposed in terms of linear matrix inequalities. Based on these criteria, the optimal partitioning method is given on the basis of the genetic algorithm. Finally, the reduced conservatism of the results in this paper is illustrated by numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust stability of nonlinear time‐delay systems with interval time‐varying delay

This paper deals with the problem of obtaining delay‐dependent stability conditions and L₂‐gain analysis for a class of nonlinear time‐delay systems with norm‐bounded and possibly time‐varying uncertainties. No restrictions on the derivative of the time‐varying delay are imposed, though lower and upper bounds of the delay interval are assumed to be known. A Lyapunov–Krasovskii functional approach is proposed to derive novel delay‐dependent stability conditions which are expressed in terms of linear matrix inequalities (LMIs). To reduce conservatism, the work exploits the idea of splitting the delay interval in multiple regions, so that specific conditions can be imposed to a unique functional in the different regions. This improves the computed bounds for certain delay‐dependent integral terms, providing less conservative LMI conditions. Examples are provided to demonstrate the reduced conservatism with respect to the available results in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

Linear quadratic regulation for systems with time‐varying delay

In this paper we study the linear quadratic regulation (LQR) problem for discrete‐time systems with time‐varying delay in the control input channel. We assume that the time‐varying delay is of a known upper bound, then the LQR problem is transformed into the optimal control problem for systems with multiple input channels, each of which has single constant delay. The optimal controller is derived by establishing a duality between the LQR and a smoothing estimation for an associated system with a multiple delayed measurement. Copyright © 2009 John Wiley & Sons, Ltd.     

On l1 stability of switched positive singular systems with time‐varying delay

This paper investigates the problem of exponential stability and l₁‐gain performance analysis for a class of discrete‐time switched positive singular systems with time‐varying delay. Firstly, a necessary and sufficient condition of positivity for the system is established by using the singular value decomposition method. Then by constructing an appropriate co‐positive Lyapunov functional and using the average dwell time scheme, we develop a sufficient delay‐dependent condition and identify a class of switching signals for the switched positive singular system to be exponentially stable and meet a prescribed l₁‐gain performance level under the switching signal. Based on this condition, the decay rate of the system can be tuned and the optimal system performance level can be determined by solving a convex optimization problem. All of the criteria obtained in this paper are presented in terms of linear programming, which suggests a good scalability and applicability to high dimensional systems. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite‐horizon robust Kalman filtering for uncertain discrete time‐varying systems with state‐delay

A finite‐horizon robust estimator design approach is developed for a class of discrete time‐varying uncertain systems with state‐delay. It extends the Kalman filter to the case in which the considered system is subject to norm‐bounded uncertainties in both state and output matrices. The state and gain matrices of the designed filter are optimized to give a minimal upper bound such that the estimation error variance is guaranteed to lie within a certain bound for all admissible uncertainties. A simulation example is presented to show the effectiveness of the proposed approach by comparing to the traditional Kalman filtering method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Finite‐time stability of switched nonlinear time‐delay systems

This article addresses the problem of finite‐time stability (FTS) and finite‐time contractive stability (FTCS) for switched nonlinear time‐delay systems (SNTDSs). By virtues of the Lyapunov‐Razumikhin method, Lyapunov functionals approach, and the comparison principle technique, we obtain some improved Razumikhin‐type theorems that verify FTS and FTCS property for SNTDSs. Moreover, our results allow the estimate of the upper bound of the derivatives for Lyapunov functions to be mode dependent functions which can be positive and negative. Meanwhile, the proposed results also improve the related existing results on the same topic by removing some restrictive conditions. Finally, two examples are presented to verify the effectiveness of our methods.     

New stability criterion for linear switched systems with time‐varying delay

This paper studies the stability problem of a class of linear switched systems with time‐varying delay in the sense of Hurwitz convex combination. By designing a parameter‐dependent switching law and using a new convex combination technique to deal with delay terms, a new stability criterion is established in terms of LMIs, which is dependent on the parameters of Hurwitz convex combination. The advantage of the new criterion lies in its less conservatism and simplicity. Numerical examples are given to illustrate the effectiveness and the less conservatism of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Exponential stability and exponential stabilization of singularly perturbed stochastic systems with time‐varying delay

In this paper, the problems of exponential stability and exponential stabilization for linear singularly perturbed stochastic systems with time‐varying delay are investigated. First, an appropriate Lyapunov functional is introduced to establish an improved delay‐dependent stability criterion. By applying free‐weighting matrix technique and by equivalently eliminating time‐varying delay through the idea of convex combination, a less conservative sufficient condition for exponential stability in mean square is obtained in terms of ε‐dependent linear matrix inequalities (LMIs). It is shown that if this set of LMIs for ε=0 are feasible then the system is exponentially stable in mean square for sufficiently small ε?0. Furthermore, it is shown that if a certain matrix variable in this set of LMIs is chosen to be a special form and the resulting LMIs are feasible for ε=0, then the system is ε‐uniformly exponentially stable for all sufficiently small ε?0. Based on the stability criteria, an ε‐independent state‐feedback controller that stabilizes the system for sufficiently small ε?0 is derived. Finally, numerical examples are presented, which show our results are effective and useful. Copyright © 2010 John Wiley & Sons, Ltd.     

Krasovskii and Razumikhin criteria on input‐to‐state stability of discrete‐time time‐varying switched delayed systems

In this article, we are concerned with the problem on input‐to‐state stability (ISS) for discrete‐time time‐varying switched delayed systems. Some Krasovskii and Razumikhin ISS criteria are provided by using the notions of uniformly asymptotically stable (UAS) function and mode‐dependent average dwell time (MDADT). With the help of the concept of UAS function, the advantage of our results in this article is that the coefficients of the first‐order difference inequalities for the mode‐dependent Krasovskii functionals and mode‐dependent Razumikhin functions are allowed to be time‐varying, mode‐dependent, and can even take both positive and negative values, and the whole switched system can be allowed to have both ISS subsystems and non‐ISS subsystems. With the aid of the notion of MDADT, each subsystem can have its own average dwell time. As an application, we also provide an ISS criterion for discrete‐time time‐varying switched delayed Hopfield neural networks with disturbance inputs. Numerical simulations verify the effectiveness of the established criteria.     

A new result on the delay‐dependent stability of discrete systems with time‐varying delays

This paper proposes an improvement to the delay‐dependent stability of discrete systems with time‐varying delays. The approach is based on the observation that the positive definiteness of a chosen Lyapunov–Krasovskii functional does not necessarily require all the involved symmetric matrices to be positive definite, which has been overlooked in the literature. The derived delay‐dependent stability conditions are in terms of linear matrix inequalities. It is theoretically proved that our results are less conservative than the corresponding ones obtained by requiring the positive definiteness of all the symmetric matrices in a chosen Lyapunov–Krasovskii functional. The importance of the present approach is that a great number of delay‐dependent analysis and synthesis results obtained by the aforementioned requirement in the literature can be improved by the present approach without introducing any new decision variables. Copyright © 2013 John Wiley & Sons, Ltd.     

Delay‐dependent conditions for finite time stability of linear time‐varying systems with delay

This paper investigates the problem of finite time stability of linear time‐varying system with delay. By constructing an augmented time‐varying Lyapunov functional and using the Wirtinger‐type inequality deductively, delay‐dependent finite time stability conditions are derived and presented in terms of differential linear matrix inequalities (DLMIs). Then, the DLMIs are transformed into a series of recursive linear matrix inequalities (RLMIs) by discretizing the time interval into equally spaced time distances, and an algorithm is given to solve the RLMIs. Examples illustrate the feasibility and effectiveness of the proposed method.