Stability Study Of Discrete Impulsive Switching Systems Under Random Constraint

Hybrid systems are dynamic systems that represent continuous and discrete dynamics.Typical systems include switched systems,impulsive systems,and so on.The switched system is composed of a set of subsystems,and the mode switching between subsystems is expressed by switching rules.Impulsive system is a control system that combines continuous evolution of state with instantaneous jump.These systems are widely used in many fields,such as power electronics,stirred tank reactors,and flight control systems.In fact,switching behavior and impulsive behavior often occur in the system synchronously,thus forming impulsive switched system.At present,numerous achievements have been made in the in-depth exploration of impulsive switched system.This paper mainly summarizes the research on impulsive switched systems into the following three parts:1.Input-to-state stability for time-delay impulsive switched systems based on a class of generalized impulsive and switching signalsThe input-to-state stability(ISS)of time-delay impulsive switched systems with a class of generalized impulsive and switched signals is studied,where the system consists of partially stable subsystems and partially unstable subsystems.By using multiple Lyapunov-Krasovskii functional(MLKFs),the delay effect of switching signal and impulsive signal is divided into delayindependent part and delay-dependent part.On the premise that switching behavior and impulsive behavior occur infrequently and the activation time of unstable subsystem is relatively short,sufficient conditions for the system to reach ISS are given.Meanwhile,a relationship is established between the admissible edge-dependent average dwell time(AED-ADT),admissible edge-dependent average impulsive interval(AED-AII),the impulse jump amplitude and the decay/increase rate of Lyapunov function.Compared with previous work,the influence of delay in impulsive signal and switching signal is fully explored,and the results obtained are applicable to systems with different impulsive behaviors.In addition,the innovation lies in putting forward the concept of AED-AII for the first time to process impulsive signals,and combining it with AED-ADT to analyze the stability of the system,and giving corresponding numerical examples to illustrate the effectiveness of the results.2.Input-to-state stability in probability for constrained impulsive switched systems with stochastic impulsesThe input-to-state stability in probability(ISSi P)of constrained impulsive switched systems with stochastic impulses is studied.Here,“constrained” means to impose constraints on switching mode conversion,impulse mode jump and impulse characteristics.Under asynchronous switching and impulse,the ISSi P characteristics of constrained impulsive switching system are studied,in which the impulsive signal is described from two aspects of intensity and density.The first case is to investigate the stability of the constrained impulsive switched system with the impulsive intensity assumed to be stochastic and the impulsive density constrained by the determined impulsive interval.The second case explores the stability of the system with stochastic impulsive intensity and density,where the impulsive interval is triggered by the update process.By applying AEDADT,AED-AII under the intensity and density of stochastic impulses,and the decay/increase rate of Lyapunov function,a sufficient condition for the whole constrained impulsive switching system to achieve ISSi P is established.Finally,the validity of the proposed stability criteria is illustrated by numerical examples.3.Input-to-state stability in probability of constrained switched delayed systems subject to constrained and stochastic impulsesThe ISSi P characteristics of switched time-delay systems subject to constrained stochastic impulses are studied.The switched signals and impulsive signals are constrained in a subset of all modes,and the switching modes and impulse modes jump are stochastic.Considering the asynchrony of switching signal and impulsive signal,AED-AII is used to describe impulsive signal comprehensively.Under switching impulsive signals with stochastic constraints,expand the case that all subsystems are stable,and explore the stability problems of systems with different subsystems,such as all subsystems are unstable,or some subsystems are stable,and some subsystems are unstable.Among them,the impulsive signal is also divided into two situations,one is the stochastic impulsive intensity and the impulsive density constrained by the determined impulsive interval,and the other is the stochastic impulsive intensity and density constrained by the update process.The relationship between AED-ADT,AED-AII under stochastic impulses,and the decay/increase rate of Lyapunov function is built,so that the switched delay system subject to constrained stochastic impulses can achieve ISSi P characteristics.Finally,two numerical examples are given to prove the generality of the obtained conditions.