Stability Of Switched Linear System With Delay

Switched systems are a special class of hybrid systems, which are controlled by a family of continuous time subsystems or discrete time subsystems and a switching law. At present, the switched systems research has become a hot problem in the modern control theory and some results were obtained. But in real life, there are a lot of the state of the systems is negative. For example, the concentration of the chemistry, the density of physics,the species and quantity of animals in biology and so on, this leads to the concept of positive systems. The so-called positive systems refers to a dynamic system if the input and initial condition are negative, its state trajectory and output are also negative. In this paper, we study a special class of systems by Lyapunov function, that is, the stability of switched system with delay. Content and organization of this paper are as follows:In the first chapter, we introduce the definition, development and existing three basic problems of stability for switching systems, as well as introduce the development of non-negative matrix theory and on witch the theory of positive switching system is well developed. Detailed introductions of the development of the system are presented on positive systems with delay, switched positive systems and positive observer of system. In the second chapter, we discuss the robust exponential stability of a class of switched systems with time varying delay and nonlinear perturbation. By constructing a new Lyapunov-Krasovskii function, using Schur complement lemma and the method of the free weighting matrix we study the stability of switched systems, taking into account the effects of varying delay on system stability. We get the switching control strategy for robust stability of the system which improving the solvability of the problem and the results are less conservative. In the third chapter, the stability of continuous time switched linear systems with time delay are discussed, mainly considering a special class of continuous time delay systems. Finally, we conclude that the stability of the positive switched systems with time delay is independent of time delay and the stability of the system is mainly dependent on the system matrix. In the fourth chapter, we study the stability of discrete-time linear switched systems with time delays, mainly extend the above conclusion to the discrete ones, the proof is given.