The Stability And Stabilization Of Singular Bilinear Systems

The asymptotical stability and stabilization problem is one of the fundamental problems in the theory of singular systems.Its results have been attracting more and more attentions,and are becoming more perfect along the development of the control theory.Because more systems can only be described by nonlinear systems in practical systems rather than linear systems,it has great significance extraordinarily to study nonlinear systems.Singular bilinear system(SBS)is a class of important singular nonlinear systems,and there are two advantages of using SBS models.One advantage is that they provide a better approximation to singular nonlinear systems than those of singular linear systems,the other advantage is that many real physical process may be appropriately modeled as SBS when singular linear systems models are inadequate.Up to now,the study of SBS is scarcely reported in literatures,so to investigate SBS has great significance.In this dissertation,the asymptotical stability and stabilization problems of SBS are considered.Based on Lyapunov theory,variable structure control method,etc,the stability,the design of state observer,the feedback control of SBS based on state observer,optimal control,robust stabilization and variable structure control problems of SBS are studied,respectively,in this dissertation.The main results obtained in this dissertation are as follows:1.By citing the concept of E-stability and generalized Lyapunov function the stability of SBS is introduced,and the Lyapunov method is adopted at the same time. The sufficient conditions of stabilization of SBS are given in two differential situations.2.The design of state observer for SBS with bounded inputs are presented via equivalent decomposition and pole-assignment methods,simultaneously,the feedback control based on the state observer of SBS is discussed.3.The optimal control of SBS is studied by employing Lyapunov stability theory and the solution of generalized Lyapunov equations.At the same time,the result global asymptotically stabilizes the closed-loop system and minimizes the generalized quadratic performance index.In the last,the design method of optimal controller of SBS is presented.4.To deal with the problem of robust stabilization for uncertain SBS,the sufficient conditions of stabilization for SBS with uncertainties in two different conditions are presented separately via the concept of generated quadratic stability and Lyapunov method in this dissertation.5.In order to solve the problem of variable structure control of SBS,the sliding surface is chosen by employing sliding mode compensator,and the variable structure controller is designed,simultaneously,the stability of the closed-loop system is guaranteed.At the same time,the designing of variable structure controller of SBS with uncertainty is presented by the similar method.Further more,in this dissertation,the effectiveness and rationality of almost all of the design methods and results are illustrated by numerical examples.