Input-to-state stability of nonlinear systems with hybrid inputs and delayed impulses

This paper addresses input-to-state stability (ISS) and integral input-to-state stability (iISS) problem of impulsive systems with hybrid inputs and delayed impulses. By adopting Lyapunov function method, sufficient conditions for ISS/iISS are established, and the impact of time delay in hybrid impulses, that is, the stabilizing impulses and destabilizing impulses, are further studied. Moreover, several examples are given and numerical simulations are performed to illustrate their usefulness.     

Event-triggered delayed impulsive control for input-to-state stability of nonlinear impulsive systems

This paper studies the input-to-state stability (ISS) and integral input-to-state stability (iISS) of nonlinear impulsive systems in the framework of event-triggered impulsive control (ETIC), where the stabilizing effect of time delays in impulses is fully considered. Some sufficient conditions which can avoid Zeno behavior and guarantee the ISS/iISS property of impulsive systems are proposed, where external inputs are considered in both the continuous dynamics and impulsive dynamics. A novel event-triggered delayed impulsive control (ETDIC) strategy which establishes a relationship among event-triggered parameters, impulse strength and time delays in impulses is presented. It is shown that time delays in impulses can contribute to the stabilization of impulsive systems in ISS/iISS sense. Finally, the effectiveness of the proposed theoretical results is illustrated by two numerical examples.     

Impulsive systems with hybrid delayed impulses: Input-to-state stability

This paper investigates input-to-state stability (ISS) and integral-input-to-state stability (iISS) of nonlinear impulsive systems with hybrid delayed impulses. Based on Lyapunov method, some sufficient conditions ensuring ISS and iISS of impulsive systems are obtained, where the time derivative of Lyapunov function is indefinite, and the hybrid effects of delayed impulses are also fully considered. It is shown that the impulsive system is ISS provided that the combined action of time delay existing in impulses, continuous dynamic, and the cumulative strength of hybrid impulses satisfies some conditions, even if the hybrid delayed impulses play a destabilizing effect on ISS. Examples and their simulations are presented to illustrate the applicability of the proposed results.     

Input-to-state stability of impulsive systems and their networks

This paper considers input-to-state stability (ISS) characterization for a class of impulsive systems which jump map depends on time. We provide sufficient conditions in terms of exponential ISS-Lyapunov functions equipped with an appropriate dwell-time condition for establishing ISS property. Some modifications of dwell times which are more conservative, but easy to be verified are being introduced. We also show that impulsive system with multiple jump maps can naturally represent an interconnection of several impulsive systems with different impulse time sequences. Then we present a procedure to verify ISS of such networks.     

Uniform stability and ISS of discrete-time impulsive hybrid systems

This paper studies the uniform stability and ISS (input-to-state stability) properties for DIHS (discrete-time impulsive hybrid systems) via comparison approach. By employing the vector-value function, the comparison principle is established for DIHS with external inputs. Then the comparison principle is used to establish the uniform stability and ISS criteria for DIHS, respectively. Moreover, regions in which the uniform stability and ISS properties can be guaranteed are estimated. As applications, the comparison principle and the results of uniform stability and ISS are used to study the robustly globally uniformly exponential stability for uncertain DIHS and exponential ISS of DIHS. It is shown that impulses contribute to stability and ISS properties for a discrete-time system which has no such properties. Two examples with numerical simulations are worked out for illustration.     

Exponential stability of singularly perturbed systems with mixed impulses

This paper investigates the exponential stability problem for a class of singularly perturbed impulsive systems in which the flow dynamics is unstable and is affected at discrete time instants by impulses that have both destabilizing and stabilizing effects. More precisely the impulses have stabilizing effects on the slow variables but destabilizing effects on the fast ones. Thus, a first contribution of our work is related to stability analysis of singularly perturbed impulsive systems in the case when neither the flow dynamics nor the impulsive one is stable. In order to take full advantage of the jump matrix structure and its stabilizing effects on the slow dynamics, we introduce a new impulse-dependent vector Lyapunov function. This function allows us to better describe the behavior between two consecutive impulses as well as the jumps at impulse instants. Several numerically tractable criteria for stability of singularly perturbed impulsive systems are established based on vector comparison principle. Additionally, upper bounds on the singular perturbation parameter are derived. Finally, the validity of our results is verified by two numerical examples.     

Generalized average dwell time approach to stability and input-to-state stability of hybrid impulsive stochastic differential systems

This paper is concerned with the stability properties of a class of impulsive stochastic differential systems with Markovian switching. Employing the generalized average dwell time (gADT) approach, some criteria on the global asymptotic stability in probability and the stochastic input-to-state stability of the systems under consideration are established. Two numerical examples are given to illustrate the effectiveness of the theoretical results, as well as the effects of the impulses and the Markovian switching on the systems stability.     

Lyapunov formulation of the ISS cyclic-small-gain theorem for hybrid dynamical networks

This paper presents a Lyapunov-based cyclic-small-gain theorem for the hybrid dynamical networks composed of input-to-state stable (ISS) subsystems whose motions may be continuous, impulsive or piecewise constant on the time-line. On the one hand, it is shown that hybrid dynamic networks with interconnection gains less than the identity function are ISS by means of Lyapunov functions. Additionally, an ISS-Lyapunov function for the total network is constructed using the ISS-Lyapunov functions of the subsystems. On the other hand, a novel result of this paper shows that a hybrid dynamic network satisfying the cyclic-small-gain condition can be transformed into one with interconnection gains less than the identity. In sharp contrast with several previously known results, the impulses of the subsystems are time triggered and the impulsive times for different subsystems may be different.     

Lyapunov formulation of the ISS cyclic-small-gain theorem for hybrid dynamical networks

This paper presents a Lyapunov-based cyclic-small-gain theorem for the hybrid dynamical networks composed of input-to-state stable (ISS) subsystems whose motions may be continuous, impulsive or piecewise constant on the time-line. On the one hand, it is shown that hybrid dynamic networks with interconnection gains less than the identity function are ISS by means of Lyapunov functions. Additionally, an ISS-Lyapunov function for the total network is constructed using the ISS-Lyapunov functions of the subsystems. On the other hand, a novel result of this paper shows that a hybrid dynamic network satisfying the cyclic-small-gain condition can be transformed into one with interconnection gains less than the identity. In sharp contrast with several previously known results, the impulses of the subsystems are time triggered and the impulsive times for different subsystems may be different.     

A Notion of Stochastic Input-to-State Stability and Its Application to Stability of Cascaded Stochastic Nonlinear Systems

In this paper, the property of practical input-to-state stability and its application to stability of cascaded nonlinear systems are investigated in the stochastic framework. Firstly, the notion of （practical） stochastic input-to-state stability with respect to a stochastic input is introduced, and then by the method of changing supply functions, （a） an （practical） SISS-Lyapunov function for the overall system is obtained from the corresponding Lyapunov functions for cascaded （practical） SISS subsystems.     

Exponential stability analysis of impulsive stochastic functional differential systems with delayed impulses

This paper is concerned with the exponential stability analysis of impulsive stochastic functional differential systems with delayed impulses. Although the stability of impulsive stochastic functional differential systems have received considerable attention. However, relatively few works are concerned with the stability of systems with delayed impulses and our aim here is mainly to close the gap. Based on the Lyapunov functions and Razumikhin techniques, some exponential stability criteria are derived, which show that the system will stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous flows. The obtained results improve and complement ones from some recent works. Three examples are discussed to illustrate the effectiveness and the advantages of the results obtained.     

切换多项式非线性系统的输入-状态稳定性

利用耗散不等式研究了切换多项式非线性系统的输入-状态稳定性分析问题,在任意切换信号下,给出了使得切换多项式非线性系统输入-状态稳定的充分条件.采用平方和分解方法来寻找切换多项式非线性系统的输入-状态稳定共同Lyapunov函数.数值算例验证了所提方法的可行性.     

Exponential stability of nonlinear state-dependent delayed impulsive systems with applications

The exponential stability problem for impulsive systems subject to double state-dependent delays is studied in this paper, where state-dependent delay (SDD) is involved in both continuous dynamics and discrete dynamics and the boundedness of it with respect to states is prior unknown. According to impulsive control theory, we present some Lyapunov-based sufficient conditions for the exponential stability of the concerned system. It is shown that the stabilizing effect of SDD impulses on an unstable SDD system changes the stability and achieves desired performance. In addition, the destabilizing effect of SDD impulses is also fully considered and the corresponding sufficient conditions are derived, which reveals the fact that a stable SDD system can maintain its performance when it is subject to SDD impulsive disturbance. As an application, the proposed result can be employed to the stability analysis of impulsive genetic regulatory networks (GRNs) with SDD and the corresponding sufficient conditions are proposed in terms of the model transformation technique and the linear matrix inequalities (LMIs) technique. In order to demonstrate the effectiveness and applicability of the derived results, we give two examples including impulsive GRNs with SDD and the impulsive controller design for the nonlinear system with SDD.     

Exponential stability of hybrid stochastic functional differential systems with delayed impulsive effects: average impulsive interval approach

In this paper, we aim to investigate the exponential stability of general hybrid stochastic functional differential systems with delayed impulses. By using the average impulsive interval and the Lyapunov function method, we derive some sufficient conditions for exponential stability, which are less conservative than those existing results based on the supremum or infimum of impulsive interval and more convenient to be applied than those Razumikhin‐type conditions in the literature. Meanwhile, we show that unstable hybrid stochastic delay differential systems, both linear and nonlinear, can be stabilized by suitably impulsive sequence. Finally, two examples are discussed to illustrate the effectiveness and advantages of the obtained results. Copyright © 2017 John Wiley & Sons, Ltd.     

Input-to-state stability and sliding mode control of the nonlinear singularly perturbed systems via trajectory-based small-gain theorem

This paper presents the trajectory-based input-to-state stability (ISS) and input-to-output stability (IOS) small-gain theorem, and the finite-time ISS (FTISS) and finite-time IOS (FTIOS) of nonlinear singularly perturbed systems. The contribution of this paper is threefold. Firstly, a novel idea is proposed to analyze the stability of the nonlinear singularly perturbed system, which is regarded as an interconnected system by using two-time-scale decomposition. Secondly, the trajectory-based approach is applied to establish ISS and IOS small-gain theorem for singularly perturbed systems and the FTISS and FTIOS properties are proposed. Thirdly, a novel sliding mode controller is developed for a class of nonlinear singularly perturbed systems. Finally, the effectiveness of proposed method is illustrated by using a numerical example, a DC motor simulation and a multi-agent singularly perturbed system.     

Stability of nonlinear variable-time impulsive differential systems with delayed impulses

This paper investigates the stability of nonlinear variable-time impulsive differential systems with delayed impulses. By using the comparison principle, the Lyapunov method and inequality techniques, some sufficient conditions for quasi-uniformly asymptotic stability and quasi-exponential stability of a given solution of variable-time impulsive differential systems with delayed impulses are established.     

Multiple periodic solutions in impulsive hybrid neural networks with delays

The existence of multiple periodic solutions and their exponential stability are investigated for impulsive hybrid Hopfield-type neural networks with both time-dependent and distributed delays, using the Leray-Schauder fixed point theorem and Lyapunov functionals. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses, providing flexibility for the analysis and design of delayed neural networks with impulse effects. Examples are given.     

Stability of random impulsive coupled systems on networks with Markovian switching

Abstract

This article is intended to study global asymptotical stability in probability for random impulsive coupled systems on networks with Markovian switching. Two cases are considered. (1) Continuous dynamics are stable while impulses are unstable; (2) impulses are stable while continuous dynamics are unstable. To begin with, based on Lyapunov method as well as graph-theoretic technique, several new stability criteria in two cases are derived, that are, the Lyapunov-type criteria and the coefficients-type criteria. Then main results are used for a class of random impulsive coupled oscillators. Finally, the effectiveness of the obtained results is verified by numerical simulations.     

Stability properties of nonlinear stochastic impulsive systems with time delay

This article establishes mean square stability and stabilization for stochastic delay systems with impulses. Using Razumikhin methodology, two approaches, classical Lyapunov-based method and comparison principle, are proposed to develop sufficient conditions that guarantee the stability and stabilization properties. It is shown that if the continuous system is stable and the impulses are destabilizing, the impulses should not be applied frequently. On the other hand, if the continuous system is unstable, but the impulses are stabilizing, the impulses should occur frequently to compensate the continuous state growth. Numerical examples are also presented to clarify the proposed theoretical results.     

Stabilization of time-delay neural networks via delayed pinning impulses

This paper studies the pinning stabilization problem of time-delay neural networks. A new pinning delayed-impulsive controller is proposed to stabilize the neural networks with delays. First, we consider the general nonlinear time-delay systems with delayed impulses, and establish several global exponential stability criteria by employing the method of Lyapunov functionals. Our results are then applied to obtain sufficient conditions under which the proposed pinning controller can exponentially stabilize the time-delay neural networks. It is shown that the global exponential stabilization of delayed neural networks can be effectively realized by controlling a small portion of neurons in the networks via delayed impulses, and, for fixed impulsive control gain, increasing the impulse delay or decreasing the number of neurons to be pinned at the impulsive moments will lead to high frequency of impulses added the corresponding neurons. Numerical examples are provided to illustrate the theoretical results, which demonstrate that our results are less conservative than the results reported in the existing literatures when the proposed pinning controller reduces to the delayed impulsive controller.