Estimation And Control Of Reachable Set For Singular Systems

Singular systems have been widely used in various fields such as electronic network systems,decision-making systems and so on since they were put forward in the early 1960 s,and many valuable theoretical results have been obtained.Singular system is a kind of system described by differential equation and algebraic equation.It has more description parts of algebraic equation in structure form than regular system which is described by pure differential equation or difference equation only.Therefore,the applicability of singular systems is much wider than that of normal systems.The reachable set is the state range of the dynamic system under bounded external interference.If the reachable set of the system can avoid the unsafe state in the state space,the system is considered safe.Therefore,in order to improve the performance of the system and ensure the stability of the system,it is of great theoretical and practical significance to study the estimation of reachable for singular systems in the field of control science.Since the mid-1960 s,with the rapid development of computer and large-scale integrated circuits,computer control has been widely used in various fields.The research on discrete systems is deepening day by day.In view of the good reliability and flexibility of digital systems,this paper studies the estimation of reachable sets and the design of controllers based on discrete singular systems.The specific contents of the study are as follows:(1)For the discrete singular system,the problem of the estimation of reachable set and the design of controller is studied.Considering the new system decomposition method,the state of the system is decomposed into the form of x=x₁=x₂.The reachable set analysis of the two subsystems is carried out respectively.In this way,the result of the reachable set analysis can be easily applied to the feedback control.The result is given in the form of linear matrix inequality(LMI).The reachable set of the system is defined within the ellipsoid.Based on the results of reachable set analysis,a state feedback controller is designed to define the reachable set of closed-loop system in a given range.(2)For the discrete singular system with time delay,the problem of the estimation of reachable set and the design of controller is studied.On the basis of reachable set analysis and controller design of discrete singular systems,in order to obtain a larger upper limit of delay and a smaller ellipsoid defining reachable sets,this study used the properties of discrete Laguerre polynomials(DLP)to obtain a new summation inequality for processing delay terms.Combined with the new system state decomposition method,the discriminant criterion of delay correlation reachable set analysis is obtained.The results are given as linear matrix inequality.Finally,the result of reachable set for discrete singular systems with time delay is given.(3)For the discrete singular system,the problem of the estimation of output reachable set and the design of controller in nonzero initial state is studied.A suitable Lyapunov function is selected to give the inequality theorem for defining the output reachable set.Based on this theorem,the matrix inequality conditions for defining the output reachable set of discrete singular systems under non-zero initial conditions are given by using the method of state decomposition.Based on the sufficient condition of the reachable set analysis,the design of the state feedback controller is given,so that the output of the closed-loop system is defined within a given range.