Robust stochastic stability of uncertain discrete-time impulsive Markovian jump delay systems with multiplicative noises

In this paper, the robust stochastic stability is investigated for a class of uncertain discrete-time impulsive Markovian jump delay systems with multiplicative noises. Using the method of stochastic Lyapunov functionals construction, it is shown that impulses can stabilise the original impulse-free unstable systems. Moreover, the stability property of the impulse-free systems can be retained in the cases of appropriately large impulsive time interval. Some numerical examples are exploited to demonstrate the effectiveness and the superiority of the proposed results.     

Robust stability of impulsive Takagi-Sugeno fuzzy systems with parametric uncertainties

This paper presents a kind of time-varying impulsive Takagi-Sugeno (T-S) fuzzy model with parametric uncertainties in which each subsystem of the model is time-varying. Several robust stabilities of time-varying systems with parametric uncertainties, such as general robust stability, robustly asymptotical stability and exponential stability, are studied using uniformly positive definite matrix functions and the Lyapunov method. Specifically, robust stability conditions of time-invariant impulsive T-S fuzzy systems are also derived in the formulation of quasi-linear matrix inequalities (QLMIs) and an iterative LMIs algorithm is designed for solving QLMIs. Finally, a unified chaotic system with continuous periodic switch and a unified time-invariant chaotic system are used for demonstrating the effectiveness of our respective results.     

Robust exponential stability of uncertain impulsive neural networks with time-varying delays and delayed impulses

In this paper, the robust exponential stability of uncertain impulsive neural networks with time-varying delays and delayed impulses is considered. It is assumed that the considered impulsive neural networks have norm-bounded parametric uncertainties and time-varying delays and the state variables on the impulses may relate to the time-varying delays. By using Lyapunov functions together with Razumikhin technique or with differential inequalities, some new robust exponential stability criteria are provided. Some examples and their simulations, including examples that the stability of which can not be tackled by the existing results, are also presented to illustrate the effectiveness and the advantage of the obtained results.     

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.     

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.     

Input/output‐to‐state stability of nonlinear impulsive delay systems based on a new impulsive inequality

In this paper, input/output‐to‐state stability (IOSS) and integral IOSS (iIOSS) are investigated for nonlinear impulsive systems with delay. Based on a new impulsive inequality, we propose some sufficient criteria for IOSS and iIOSS of impulsive delay systems. It is shown that the obtained results for IOSS and iIOSS are regardless of the length of the impulsive interval and the size of time delay if the impulsive gain satisfies a given condition. In addition, based on the average impulsive interval method, some more useful sufficient conditions are derived for IOSS and iIOSS of impulsive delay systems with persistent large‐scale destabilizing impulses. Furthermore, a relationship is established among the average impulsive interval, impulses, time delay, and the decay of the system without impulses such that the impulsive delay system is input/output‐to‐state stable and integral input/output‐to‐state stable, respectively. Two examples are given to show the validity of the obtained results.     

Enhancement of robust performance for fuzzy parametric uncertain time‐delay systems using differential evolution algorithm

This paper proposes a systematic methodology for the enhancement of robust stability and performance of a fuzzy parametric uncertain time‐delay system. A fuzzy parametric uncertain time‐delay system is an example for a linear time‐invariant uncertain time‐delay system with fuzzy coefficients. By using the nearest approximation, these fuzzy coefficients are approximated into crisp sets called intervals to get an interval system. The proposed approach develops the necessary and sufficient stability conditions of interval polynomials for determining the robust stability. Then, by using these developed stability conditions, a set of inequalities in terms of controller parameters are obtained from the closed‐loop characteristic polynomial of fuzzy parametric uncertain time‐delay system. Finally, these inequalities are solved to obtain robust controller with the help of a differential evolution algorithm for an unstable fuzzy parametric uncertain time‐delay system. Consequently, a lead‐lag compensator is constructed based on the frequency domain approach to improve the performance of the fuzzy parametric uncertain time‐delay system. The proposed method has the advantage of less computational complexity and easy to implement on a digital computer. The viability of the proposed methodology is illustrated through a numerical example for its successful implementation. The efficacy of the proposed methodology is also evaluated against the available approach in the literature and the simulation results are successfully implemented for robust stability and performance of fuzzy parametric uncertain time‐delay systems.     

Impulsive controller design for singular networked control systems with packet dropouts

This paper considers the problem of impulsive time-delay control for singular networked impulsive control systems(SNICSs) and uncertain SNICSs both with network-induced delay and packet dropouts. The parameter uncertainty is assumed to be norm bounded. The problem to be addressed is the design of robust impulsive time-delay feedback controllers such that the exponential stability of the resulting closed-loop system is guaranteed for admissible uncertainties. By applying Lyapunov function theory and Halanay Lemma, impulsive time-delay controller is derived through solving LMIs. Numerical examples are provided to demonstrate the application of the proposed method.     

Finite-time annular domain stability of Itô stochastic impulsive systems with markovian jumping under asynchronous switching

ABSTRACT

This study examines the finite time annular domain stability (FTADS) and stabilisation of a class of Itô stochastic impulsive systems with asynchronous switching controller. The asynchronous switching means that the controller switching does not accurately coincide with system switching in delayed time interval. The design of the controller depends on the observed jumping parameters, which cannot be precisely measured in real-time because of switching delay. Our results apply to cases where some subsystems of the switched systems are not necessarily stable under the influence of input delay. When the subsystem is stable in the synchronous switching interval and unstable in the asynchronous case, a compromise among the average impulsive interval, the upper bound of delay, and the decay/increasing rate of Lyapunov function in the synchronous/asynchronous switching interval respectively is given. By the mode-dependent parameter approach (MDPA) and allowing the increase of the impulses on all the switching times, the extended FTADS criteria for Itô stochastic impulsive systems in generally nonlinear setting are derived first. Then, we focus on the case when the system in both synchronous and asynchronous switching intervals are unstable. By reaching a tradeoff among average impulsive interval, the upper bound of delay, the magnitude of impulses and the difference between the increasing rate of Lyapunov function in the synchronous and asynchronous switching interval, new sufficient conditions for existence of the state feedback controller are also developed by MDPA. In addition, we consider the effect of different impulsive strengths (harmful and beneficial impulses) and obtained less conservative results because the Lyapunov function may be non-decreasing during switching interval. Moreover, we extend the conclusion from nonlinear stochastic impulsive switching systems to linear case. Finally, we present two examples to illustrate the effectiveness of the results obtained in this study.     

Delay‐dependent robust stability analysis for switched time‐delay systems with polytopic uncertainties

In this paper, sufficient conditions are provided for the stability of switched retarded and neutral time‐delay systems with polytopic‐type uncertainties. It is assumed that the delay in the system dynamics is time‐varying and bounded. Parameter‐dependent Lyapunov functionals are employed to obtain criteria for the exponential stability of the system in the form of linear matrix inequality (LMI). Free‐weighting matrices are then provided to express the relationship between the system variables and the terms in the Leibniz–Newton formula. Numerical examples are presented to show the effectiveness of the results. Copyright © 2014 John Wiley & Sons, Ltd.     

Exponential estimates for neutral time delay systems with multiple delays

Exponential estimates and sufficient conditions for the exponential stability of linear neutral time delay for systems with multiple delays are given. The case of systems with uncertainties, including uncertainties in the difference operator, is considered. The proofs follows from new results on non‐homogeneous difference equations evolving in continuous time combined with the Lyapunov–Krasovskii functionals approach. The conditions are expressed in terms of linear matrix inequalities. The particular case of neutral time delay systems with commensurate delays, which leads to less restrictive exponential estimates, is also addressed. Copyright © 2005 John Wiley & Sons, Ltd.     

Global robust asymptotic stability analysis of uncertain switched Hopfield neural networks with time delay in the leakage term

This paper deals with the problem of delay-dependent global robust asymptotic stability of uncertain switched Hopfield neural networks (USHNNs) with discrete interval and distributed time-varying delays and time delay in the leakage term. Some Lyapunov––Krasovskii functionals are constructed and the linear matrix inequality (LMI) approach are employed to derive some delay-dependent global robust stability criteria which guarantee the global robust asymptotic stability of the equilibrium point for all admissible parametric uncertainties. The proposed results that do not require the boundedness, differentiability, and monotonicity of the activation functions. Moreover, the stability behavior of USHNNs is very sensitive to the time delay in the leakage term. It can be easily checked via the LMI control toolbox in Matlab. In the absence of leakage delay, the results obtained are also new results. Finally, nine numerical examples are given to show the effectiveness of the proposed results.     

Robust H∞ control of uncertain linear impulsive stochastic systems

This paper develops robust stability theorems and robust H_∞ control theory for uncertain impulsive stochastic systems. The parametric uncertainties are assumed to be time varying and norm bounded. Impulsive stochastic systems can be divided into three cases, namely, the systems with stable/stabilizable continuous‐time stochastic dynamics and unstable/unstabilizable discrete‐time dynamics, the systems with unstable/unstabilizable continuous dynamics and stable/stabilizable discrete‐time dynamics, and the systems in which both the continuous‐time stochastic dynamics and the discrete‐time dynamics are stable/stabilizable. Sufficient conditions for robust exponential stability and robust stabilization for uncertain impulsive stochastic systems are derived in terms of an average dwell‐time condition. Then, a linear matrix inequality‐based approach to the design of a robust H_∞ controller for each system is presented. Finally, the numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Robust Absolute Stability Analysis of Multiple Time‐Delay Lur'e Systems With Parametric Uncertainties

The problem of robust absolute stability for time‐delay Lur'e systems with parametric uncertainties is investigated in this paper. The nonlinear part of the Lur'e system is assumed to be both time‐invariant and time‐varying. The structure of uncertainty is a general case that includes norm‐bounded uncertainty. Based on the Lyapunov–Krasovskii stability theory, some delay‐dependent sufficient conditions for the robust absolute stability of the Lur'e system will be derived and expressed in the form of linear matrix inequalities (LMIs). These conditions reduce the conservativeness in computing the upper bound of the maximum allowed delay in many cases. Numerical examples are given to show that the proposed stability criteria are less conservative than those reported in the established literatures.     

DELAY‐DEPENDENT ROBUST CONTROL OF DESCRIPTOR SYSTEMS WITH TIME DELAY

This paper deals with the problem of stability and robust control for both certain and uncertain continuous‐time singular systems with state delay. Systems with norm‐bounded parameter uncertainties are considered. Robust delay‐dependent stability criteria and linear memoryless state feedback controllers based on linear matrix inequality are obtained. By choosing some Lyapunov‐Krasovskii functionals, neither model transformation nor bounding for cross terms is required in the derivation of our delay‐dependent results. Finally, numerical example is provided to illustrate the effectiveness of the proposed method.     

Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems

Exponential stability and robust exponential stability relating to switched systems consisting of stable and unstable nonlinear subsystems are considered in this study. At each switching time instant, the impulsive increments which are nonlinear functions of the states are extended from switched linear systems to switched nonlinear systems. Using the average dwell time method and piecewise Lyapunov function approach, when the total active time of unstable subsystems compared to the total active time of stable subsystems is less than a certain proportion, the exponential stability of the switched system is guaranteed. The switching law is designed which includes the average dwell time of the switched system. Switched systems with uncertainties are also studied. Sufficient conditions of the exponential stability and robust exponential stability are provided for switched nonlinear systems. Finally, simulations show the effectiveness of the result.     

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.     

具有时变不确定性的脉冲时滞大系统的鲁棒稳定性

提出并研究了具有时变不确定性的脉冲时滞大系统的鲁棒稳定性,借助于向量比较原理和矩阵不等式等方法,对该系统建立了若干大范围指数鲁棒稳定的判据,并举例说明了结论的有效性。     

Uncertain impulsive differential systems of fractional order: almost periodic solutions

In this paper, we investigate the existence and stability of almost periodic solutions of impulsive fractional-order differential systems with uncertain parameters. The impulses are realised at fixed moments of time. For the first time, we determine the impact of the uncertainties on the qualitative behaviour of such systems. The main criteria for the existence of almost periodic solutions are proved by employing the fractional Lyapunov method. The global perfect robust uniform-asymptotic stability of such solutions is also considered. We apply our results to uncertain impulsive neural network systems of fractional order.     

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.