Input-state Stability Study Of Pulse-switched Nonlinear Systems

Dynamical systems that are described by an interaction between continuous and discrete dynamic are usually called hybrid systems,which is the most important topic in the current time.As a subset of hybrid systems,impulsive signals have crucial applications in various fields.Another important subset of hybrid systems is switched systems.In order to study these two types of hybrid systems in the same framework,we connected them to shape a more complex model,i.e.impulsive switched systems.The notion of input-to-state stability,presented by Sontag,is testified very valuable in this region and has been promoted to nonlinear systems by various work.The input-to-state stability implies that no matter what the size of the initial state is,the state will eventually be small if the external inputs are uniformly small.In this paper,we introduce different impulses.By means of Lyapunov function and under condition of average impulsive interval,and get input-to-state stability and stochastic input-to-state stability respectively.The main conclusions of this paper can be summarized as the following two parts:1)Input-to-state stability of impulsive switched nonlinear system.By means of the multiple-Lyapunov function and the average impulsive interval approach,the conditions for input-to-state stability properties are derived under all subsystems stable,all subsystems unstable and some subsystems unstable.It is shown that if the continuous subsystems are all stable and though the impulsive effects are destabilizing,the system is input-to-state stability with respect to a lower bound of the average impulsive switched interval.Moreover,if all the subsystems are not stable,the impulsive effects can still successfully stabilize the system but for an upper bound of the average impulsive switched interval.However,it is unveiled that if some continuous subsystems are not stable,the system is still input-to-state stability under specific conditions.Finally,examples are illustrated with simulations to manifest the validity of the main results.2)Stochastic input-to-state stability of stochastic impulsive switched nonlinear system.A group of stochastic impulsive switched nonlinear systems are studied to estimate the stochastic input-to-state stability.By means of Lyapunov functional,some sufficient results are given to ensure the stochastic input-to-state stability.Then,based on average impulsive interval,we also discuss its property in three cases.With impulse effects,we still get stochastic input-to-state stability.Finally,examples are illustrated with simulations to manifest the validity of the main results.