Mean-square exponential stability of impulsive stochastic time-delay systems with delayed impulse effects

This paper is concerned with the mean-square exponential stability problem for a class of impulsive stochastic systems with delayed impulses. The delays exhibit in both continuous subsystem and discrete subsystem. By constructing piecewise time-varying Lyapunov functions and Razumikhin technique, sufficient conditions are derived which guarantee the mean-square exponential stability for impulsive stochastic delay system. It is shown that the obtained stability conditions depend both on the lower bound and the upper bound of impulsive intervals, and the stability of system is robust with regard to sufficiently small impulse input delays. Finally, two examples are proposed to verify the efficiency of the proposed results.     

Exponential stability of impulsive discrete-time systems with infinite delays

This paper investigates the exponential stability of impulsive discrete-time systems with infinite delays. Some sufficient conditions are obtained to guarantee the exponential stability of the considered systems by employing Lyapunov functions together with Razumikhin technique. Finally, three numerical examples are given to illustrate the effectiveness and superiority of the obtained results. It is the first time that the exponential stability of impulsive discrete-time systems with infinite delays is studied, the obtained results complement some recent works.     

非线性采样系统指数稳定的新条件

研究了非线性采样系统的稳定性问题. 对以采样周期为参数的离散时间系统族, 证明了全局指数稳定的Lyapunov定理和逆定理. 分别基于系统的一般近似模型和Euler近似模型, 给出了闭环系统全局指数稳定的新条件. 与现有结果相比, 取消了Lyapunov函数全局Lipschitz连续的假设, 减弱了闭环系统全局指数稳定的充分条件.     

Direct and converse Lyapunov theorems for functional difference systems

New Lyapunov criteria for asymptotic stability and input-to-state stability of infinite dimensional systems described by functional difference equations are provided. Conditions in terms of both Lyapunov–Razumikhin functions defined on Euclidean spaces and of Lyapunov–Krasovskii functionals defined on infinite dimensional spaces are found. For the case of Lyapunov–Krasovskii functionals, necessary and sufficient conditions are provided for the asymptotic stability, in both the local and the global case, and for the input-to-state stability. This is the first time in the literature that converse Lyapunov theorems are provided for the class of nonlinear systems here studied.     

Memoryless H ∞ controller design for switched non-linear systems with mixed time-varying delays

This article deals with the H _∞ control problem for a class of switched non-linear systems with mixed time-varying delays. The novel features here are that the system in consideration is non-linear perturbation with discrete and distributed delays, the time-varying delay is also involved in the observation output, and the controllers to be designed satisfy some exponential stability constraints on the closed-loop poles. By using Lyapunov–Razumikhin functional approach, new sufficient conditions for the H _∞ control with exponential stability constraint are derived in terms of the solution of Riccati-type equations. The approach allows for simultaneous computation of the two bounds that characterise the stability rate of the solution.     

Razumikhin stability theorems for a general class of stochastic impulsive switched time‐delay systems

This paper studies stability of a general class of impulsive switched systems under time delays and random disturbances using multiple Lyapunov functions and fixed dwell‐time. In the studied system model, the impulses and switches are allowed to occur asynchronously. As a result, the switching may occur in the impulsive intervals and the impulses can occur in the switching intervals, which have great effects on system stability. Since the switches do not bring about the change of the system state, we study two cases in terms of the impulses, ie, the stable continuous dynamics case and the stable impulsive dynamics case. According to multiple Lyapunov‐Razumikhin functions and the fixed dwell‐time, Razumikhin‐type stability conditions are established. Finally, the obtained results are illustrated via a numerical example from the synchronization problem of chaotic systems.     

带有时滞状态的不确定Lur''e-Postnikov系统的鲁棒镇定

由Lyapunov和一种Razumikhin型的方法, 讨论了带有时滞状态的不确定Lur’e_Postnikov系统的鲁棒镇定. 证明对带有状态时滞和范数有界扰动的不确定Lur’e_Postnikov系统, 若其系统矩阵满足某个代数Riccati不等式, 则可通过某 (静态 )线性状态反馈或 (动态 )状态反馈使其闭环系统是二次稳定的. 同样, 应用一Razumikhin型方法, 对带有时变时滞的一类不确定非线性系统的能稳定性问题, 也给出一个充分条件.     

Discrete-time stochastic control systems: A continuous Lyapunov function implies robustness to strictly causal perturbations

Discrete-time stochastic systems employing possibly discontinuous state-feedback control laws are addressed. Allowing discontinuous feedbacks is fundamental for stochastic systems regulated, for instance, by optimization-based control laws. We introduce generalized random solutions for discontinuous stochastic systems to guarantee the existence of solutions and to generate enough solutions to get an accurate picture of robustness with respect to strictly causal perturbations. Under basic regularity conditions, the existence of a continuous stochastic Lyapunov function is sufficient to establish that asymptotic stability in probability for the closed-loop system is robust to sufficiently small, state-dependent, strictly causal, worst-case perturbations. Robustness of a weaker stochastic stability property called recurrence is also shown in a global sense in the case of state-dependent perturbations, and in a semiglobal practical sense in the case of persistent perturbations. An example shows that a continuous stochastic Lyapunov function is not sufficient for robustness to arbitrarily small worst-case disturbances that are not strictly causal. Our positive results are also illustrated by examples.     

Stability of linear continuous‐time difference equations with distributed delay: Constructive exponential estimates

This paper is concerned with the construction of exponential estimates for a class of systems governed by continuous‐time difference equations with distributed delay. With the Lyapunov–Krasovskii approach, we propose sufficient conditions for exponential stability, with numerical constructive estimates. A conservatism analysis is made to illustrate the improvement of these stability conditions with respect to conditions already presented in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Exponential Stability Analysis for Stochastic Delayed Differential Systems with Impulsive Effects: Average Impulsive Interval Approach

This paper is concerned with the exponential stability analysis of stochastic delayed systems with impulsive effects. By using the average impulsive interval approach, and together with comparison lemma and Razumikhin techniques, sufficient conditions ensuring the moment exponential stability of the systems under consideration are established. A stability criterion for non‐delayed stochastic systems with impulses is also derived as a corollary. Compared with the existing stability results in the literature, which are usually based on the supremum or infimum of impulsive intervals, the results reported in this paper are less conservative. Two illustrative examples are provided to validate the effectiveness and advantages of our theoretical results.     

Robust model predictive control of discrete nonlinear systems with time delays and disturbances via T–S fuzzy approach

In this paper, robust fuzzy model predictive control of a class of nonlinear discrete systems subjected to time delays and persistent disturbances is investigated. Based on the modeling method of delay difference inclusions, nonlinear discrete time-delay systems can be represented by T–S fuzzy systems comprised of piecewise linear delay difference equations. Moreover, Lyapunov–Razumikhin function (LRF), instead of Lyapunov–Krasovskii functional (LKF), is employed for time-delay systems due to its ability to reflect system original state space and its advantages in controller synthesis and computation. The robust positive invariance and input-to-state stability with respect to disturbance under such circumstances are investigated. A robust constraint set is adopted that the system state is converged to this set round the desired point. In addition, the controller synthesis conditions are derived by solving a set of matrix inequalities. Simulation results show that the proposed approach can be successfully applied to the well-known continuous stirred tank reactor (CSTR) systems subjected to time delay.     

Exponentially incremental dissipativity for nonlinear stochastic switched systems

ABSTRACT

In this paper, we investigate the exponentially incremental dissipativity for nonlinear stochastic switched systems by using the designed state-dependent switching law and multiple Lyapunov functions approach. Specifically, using incremental supply rate as well as a state dissipation inequality in expectation, a stochastic version of exponentially incremental dissipativity is presented. The sufficient conditions for nonlinear stochastic switched systems to be exponentially incrementally dissipative are given by the designed state-dependent switching law. Furthermore, the extended Kalman–Yakubovich–Popov conditions are derived by using two times continuously differentiable storage functions. Moreover, the incremental stability conditions in probability for nonlinear stochastic switched systems are derived based on exponentially incremental dissipativity. The exponentially incremental dissipativity is preserved for the feedback-interconnected nonlinear stochastic switched systems with the composite state-dependent switching law; meanwhile, the incremental stability in probability is preserved under some certain conditions. A numerical example is given to illustrate the validity of our results.     

Impulsive observers with variable update intervals for Lipschitz nonlinear time-delay systems

This article is concerned with the design of impulsive observers with variable update intervals for Lipschitz nonlinear systems with delays in state. Discontinuous Lyapunov function/funtional approaches are developed to analyse the stability of error dynamics. Delay-independent sufficient conditions for uniform exponential stability of the error dynamics over variable update intervals are derived in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, the observer gain matrix can be solved numerically with an LMI-based optimisation algorithm. Numerical examples are provided to show the efficiency of the proposed approach.     

Razumikhin-type stability theorems for discrete delay systems

In this paper, by employing the Razumikhin technique and Lyapunov functions, Razumikhin-type theorems that guarantee the uniform stability, uniformly asymptotic stability and uniformly exponential stability for the general discrete delay systems are established, respectively. Moreover, Razumikhin-type uniformly exponential stability theorem gives the estimation of the convergence speed. As theoretic application, the Razumikhin-type uniformly exponential stability result is further studied and used to show some well-known stability results for some kinds of discrete delay systems. Finally, examples are also worked through to illustrate our results.     

Sampling‐interval‐dependent stability for sampled‐data systems with state quantization

This paper is concerned with the stability of sampled‐data systems with state quantization. A new piecewise differentiable Lyapunov functional is first constructed by fully utilizing information about sampling instants. This functional has two features: one is that it is of the second order in time t and of every term being dependent on time t explicitly and the other is that it is discontinuous and is only required to be definite positive at sampling instants. Then, on the basis of this piecewise differentiable Lyapunov functional, a sampling‐interval‐dependent exponential stability criterion is derived by applying the technique of a convex quadratic function with respect to the time t to check the negative definiteness for the derivative of the piecewise differentiable Lyapunov functional. In the case of no quantization, a new sampling‐interval‐dependent stability criterion is also obtained. It is shown that the new stability criterion is less conservative than some existing one in the literature. Finally, two examples are given to illustrate the effectiveness of the stability criterion. Copyright © 2013 John Wiley & Sons, Ltd.     

New results concerning the exponential stability of delayed neural networks with impulses

Employing the matrix measure approach and Lyapunov function, the author studies the global exponential stability of delayed neural networks with impulses in this paper. Some novel and sufficient conditions are given to guarantee the global exponential stability of the equilibrium point for such delayed neural networks with impulses. Finally, three examples are given to show the effectiveness of the results obtained here.     

Mean square stability of linear stochastic neutral‐type time‐delay systems with multiple delays

This paper studies mean square exponential stability of linear stochastic neutral‐type time‐delay systems with multiple point delays by using an augmented Lyapunov‐Krasovskii functional (LKF) approach. To build a suitable augmented LKF, a method is proposed to find an augmented state vector whose elements are linearly independent. With the help of the linearly independent augmented state vector, the constructed LKF, and properties of the stochastic integral, sufficient delay‐dependent stability conditions expressed by linear matrix inequalities are established to guarantee the mean square exponential stability of the system. Differently from previous results where the difference operator associated with the system needs to satisfy a condition in terms of matrix norms, in the current paper, the difference operator only needs to satisfy a less restrictive condition in terms of matrix spectral radius. The effectiveness of the proposed approach is illustrated by two numerical examples.     

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.     

Multiple-mode adaptive state estimator for nonlinear switched systems

This paper deals with the issue of state estimator design for nonlinear switched systems. A multiplemode adaptive estimator is proposed under mode-dependent average dwell time (MDADT) switching, and the switching signal with MDADT constraint is also obtained to guarantee the exponential stability of estimation error dynamics, where the Lipschitz constant may be unknown since it is adaptively adjusted by designing an adaptation law. Based on both Lyapunov stable theory and the feasible solution of an optimization problem with linear matrix inequality constraint, the gain matrices and switching signals are provided, respectively. The sufficient conditions of the existence of multiple-mode adaptive switched estimator are also derived. Meanwhile, the above methods are also extended to the case of the average dwell time (ADT) switching, and an algorithm is given to summarize the implementation of the proposed estimators. Finally, the effectiveness of the designed methods is illustrated by simulation examples.