Stability Analysis Of Switched Systems With Discrete-time Subsystems

Switched systems are an important class of hybrid systems which consists of several continuous-time subsystems or discrete-time subsystems or continuous-time and discrete-time subsystems and a rule that orchestrates among them. Switched systems have broad applications and show complicated behaviours because of the interaction between the continuous dynamics and the switching signals. Hence switched systems attract a lot of authors' much attention.The BZ reaction is currently one of the most arresting chemical experiments and objects of theoretic research in nonlinear chemistry. It has received much attention because the BZ reaction is the simplest reaction, and can show rich dynamical behaviours.In this dissertation we mainly study stability of switched systems with discrete-time subsytems. The main results are summarized as follows:1. For a general nonlinear nonautonomous discrete-time switched system, we first use multiple Lyapunov functions and the comparison principle of discrete systems to give a result on stability in terms of two measures. When the number of subsystems is finite, based on the result we present a detailed result on stability in terms of two measures. Subsequently we study several input-to-state stable(ISS)-type properties (input-to-state stable,1/λ-weighted input-to-state stable,1/λ-weighted sum input-tostate stable) of discrete-time switched systems with inputs, and we give some new results on several input-to-state stable(ISS)-type properties under average dwell-time switching. As an application of the result on input-to-state stability, sufficient conditions for the global asymptotic stability of cascade discrete-time switched systems are easily obtained.2. For a class of discrete-time switched systems-cascade switched systems, by means of the Brower fixed point theorem we first obtain a result on the existence of periodic orbits of this class of switched systems, and then we present a result on stability of periodic orbits. Furthermore we consider a particular case of this class of switched systems that this class of switched systems has one subsystem, and give some results on the number and the minimum period of peridic orbits. Some examples show that the results are unique for the particular case . 3. For a new type of switched systems-switched systems composed of continuous-time subsystems and discrete-time subsystems proposed in the literature, by combining the method of myultiple Lyapunov functions with the comparison principle of continuous-time systems and discrete-time systems we first present a result on stability in terms of two measures for such type of switched systems. When the numbers of continuous-time and discrete-time sunsystems are finite, on the basis of the above result we obtain a detailed result on stability in terms of two measures. Then we give a result on global asymptotic stability of such type of switched systems under average dwell-time switching. Moreover, for a special case of such type of switched systems-switched sytems composed of continuous-time LTI(linear time invariant)and discrete-time LTI subsystems, a result on global asymptotic stability of the switched systems under periodic switching signals is obtained.4. For switched systems composed of continuous-time subsystems and discrete-time subsystems, we discuss their robust stability. First we give a sufficient and necessary condition for quadratic stability of the linear case of the switched systems with polytopic pertubations, and then for a nonlinear case of the switched systems in which there are nonlinear pertubations in discrete-time subsystems we present an estimate of state under arbitrary switching laws, and from the estimate a sufficient condition for global asymptotic stability under arbitrary switching laws is easily obtained.Moreover, we investigate stability of the typical BZ reaction system and estimates ofthe ultimate bounded set, by means of the small gain theorem we first present sufficient conditions for global asymptotic stability of an equilibrium point of the chemical system . Then we give an estimate of the ultimate bounded set of the chemical system.