Several Problems Of The Viability On The Switched System

The relationship between the system and its environment is an important issue in the system science.In the evolution of the system,it is necessary to constantly adjust its structure or state to adapt to environmental changes,so as to achieve sustainable and coordinated development of the system and its environment.Viability theory is a method to study the evolution of a system in the state constraint region.By describing the possible paths,we try to find the proper control input,such that the trajectory can evolve in the state constraint region forever.This provides the basic guarantee for the security evolution of the system.Therefore,the research on the viability theory has important theoretical value and practical significance.In this paper,the problem of the viability on a switched system is deeply studied.Methods of determining the viability on the polytope and algorithms for computing the viability kernel are proposed.Based on the theory of nonsmooth analysis,the problem of determining the viability for the linear switched system is studied.A sufficient viability condition for a switched system with arbitrary switching on a polytope,which is expressed by a convex hull of a finite number of points,is proposed.The condition transforms the viability of boundary points into determining the viability condition on the vertices;The problem of determining the viability for convex cone expressed by non-negative linear combinations of a finite number of directions is discussed,and the method of determining the viability is proposed;Furthermore,considering the unbounded polyhedron represented by a convex hull of a finite number of points and non-negative linear combinations of a finite number of directions,a sufficient viability condition for the linear switched system is presented and the method of determining the viability is proposed.The problems of the viability and the attraction for nonlinear switched system are studied by using the directional derivative of the Lyapunov function.For a switched system with a piecewise smooth Lyapunov function,a sufficient condition is derived to guarantee the level set of the piecewise smooth function is a viable region and attractive region,and the switching laws are constructed by using the directional derivatives of the piecewise smooth Lyapunov function along the trajectories of the subsystems;For a switched system with sliding motions,a sufficient condition to guarantee the level set is a viable region and attractive region is proposed,and the switching laws are given;The specific viable region and attraction region are constructed by using the level sets of two classes of piecewise smooth functions.The problems of computing the viability kernel for the discrete time and the continuous time switched system are studied by using the set theory and reachability theory.Firstly,the computation of the viability kernel is transformed into the iteration of the reachable sets by establishing the relationship between the reachable set and the viability kernel for a switched system,two algorithms of computing the viability kernel for the discrete time switched system on the polyhedral region and the ellipsoidal region are proposed,respectively;Secondly,taking into account the special structure of the linear switched system,a simple algorithm is developed based on the set theory;Finally,by discretizing the continuous time switched system,the method of computing the viability kernel for the discrete time switched system is extended to the continuous time switched system,and a method of computing the approximation viability kernel is proposed.The approximation viability kernel obtained by the method is closer to the real viability kernel as the time step decreases.Based on the set theory and the robust one step set,the problem of computing the robust viability kernel for the switched system is studied.For the different forms of the disturbance,algorithms of computing the robust viability kernel for the discrete time switched system are presented;For the linear switched system,a simple algorithm that is easy to implement has been developed;In addition,an algorithm of computing an inner approximation of the robust viability kernel for the switched system is proposed.The convergence of the algorithm is proved by using the null controllable set.