Research On Input-to-State Stability For Several Classes Of Nonlinear Systems

Stability analysis of nonlinear systems is one of the most important problems in control theory and engineering.When studying stability of a system,it is vital to characterize effects of the external inputs,because they might cause loss of the stability for an otherwise stable one.In recent years,the input-to-state stability(ISS)and extensions of the ISS on different systems have attracted widespread attentions in the literatures due to their extensive usage in characterizing effects of the external inputs(such as sensor noise,actuator disturbances,parameter perturbations,or measurement errors)on the considered systems.The notion of ISS is formed to investigate how the external disturbance affects the system stability.Roughly speaking,the property of ISS means that no matter what the magnitude of the initial state is,the system state will ultimately enter into a neighborhood of the origin whose magnitude is proportional to the size of the input.Recently,various extensions of the ISS,such as integral ISS(iISS),stochastic ISS(SISS),and finite-time ISS,have been proposed and successfully applied to different kinds of dynamical systems,for instance,impulsive systems,switched systems,discrete-time dynamical networks,and nonlinear systems with delays.This dissertation is concerned with the ISS-related problems of several classes of nonlinear systems.The addressed systems cover stochastic impulsive time-delay systems,stochastic switched time-delay systems,discrete-time nonlinear systems and discrete-time impulsive nonlinear systems.The stability considered here includes input-to-state KB-stability,SISS,p-th moment exponential ISS and finite-time ISS.Furthermore,several new comparison principles in vector-version are established,and the finite-time ISS problems are extended to the discretetime nonlinear systems.On this basis,some novel concepts and methods are proposed to tackle the problem of finite-time ISS,for example,the concept and construction method of the settling-time function for discrete-time nonlinear systems,which greatly enrich the theoretical framework of ISS.More specifically,contents of this dissertation can be summarised from the following aspects.The second chapter addresses the ISS,the p-th moment exponential ISS,the input-tostate KB-stability and the SISS problems for stochastic impulsive systems with time delays via the comparison principle method.Firstly,several general comparison principles in vectorversion are proposed guaranteeing the existence,uniqueness and magnitude for solutions of the addressed system,which generalize the scalar version of the classical comparison principle for nonlinear systems that has been widely used in many literatures.Existence and uniqueness of the solution for a stochastic impulsive system can be deduced by comparing it with a lowerdimensional deterministic impulsive system,which is assumed to have a global solution so that the linear growth constraints are no longer necessary.Then,based on these established comparison principles,the ISS-related properties are investigated for the stochastic impulsive delayed model.Finally,two examples are given to illustrate effectiveness of the obtained results.The third chapter is concerned with the finite-time ISS-related(i.e.,the finite-time ISS in mean and the finite-time SISS)and the ISS-related(i.e.,the ISS in mean,the SISS and the mean-square ISS)problems for the discrete-time stochastic switched system with time-varying delay and external inputs by employing the vector-valued functions and the vector-version comparison principle.A deterministic impulsive system has been designed as a comparison system,where the impulsive time sequence is identified the same with the sequence of switching instants of the considered stochastic system.Based on these results,the finite-time ISS-related(ISS-related)issues concerning with the switched system can be deduced via the corresponding finite-time ISS(ISS)of its comparison model.As an application,the comparison principle has been used to investigate the mean-square ISS for the linear discrete-time stochastic switched system.Two numerical examples with simulations have also been given to illustrate feasibility of the main results.The fourth chapter considers the topic of finite-time ISS for discrete-time nonlinear systems.By using the Lyapunov method,sufficient criteria have been provided for the discretetime nonlinear system with external inputs to be finite-time input-to-state stable.In the framework of finite time,it has shown that the state trajectory of the considered system will enter into a neighborhood of the origin in finite time and will not exceed it thereafter,whose magnitude is proportional to that of the inputs.Moreover,the corresponding GKL functions and the settling-time functions capturing the finite settling time behavior of the addressed discrete-time system have also been considered/deduced.Two examples and the relating numerical simulations have also been given to illustrate effectiveness of the main results.The fifth chapter discusses the finite-time stability(FTS)for discrete-time impulsive systems without inputs and the finite-time ISS for discrete-time impulsive systems with inputs,respectively,by employing the Lyapunov method.In addition,the corresponding GKL functions as well as the settling-time functions capturing the finite settling time behavior of the addressed discrete-time systems have also been considered/deduced.Sufficient conditions for lower semi-continuity of the settling-time function are also presented.One example and the relating numerical simulations have been given to illustrate effectiveness of the main results.