The Stability And Stabilization Of Nonlinear Singular Systems

Study on singular system comes from practical problems, which are an accurate model description of a set of problems exist in practical applications. With constantly innovative system theory and increasingly mature computer technology, singular control system has been rapidly developing. Since nonlinear singular systems can be applied to generally describe objective systems, it’s very necessary to conduct study on nonlinear singular system. In order to research this kind of systems, we first need to start from basic properties and system states to study its general quantitative theory. Major results of this article are as follows:In the first part, we give an introduction to the background of nonlinear singular systems as well as the development of the theory. Meanwhile, the study of relevant subjects of nonlinear singular system is mentioned. At the end of the first part, the research content and main tasks were concluded. In the second part, problems about the stability of nonlinear time-varying singular system and nonlinear system were concerned. Here we provide sufficient conditions for impulse-free and stable singular system, and then judge the variation of fast system through the changes of fast-low subsystem, obtain sufficient criterion for the asymptotic stability of nonlinear time-varying singular system. In the third part, the design of state observer of the nonlinear singular system is proposed. Under the assumption that nonlinear singular system can be detected, we obtain the sufficient condition for the existence of state observer in nonlinear singular system by using Lyapunov function. At the same time, we utilize Riccati equation to get the concrete form of gain matrix in state observer. The fourth part analyzes the robust stabilization of nonlinear singular system with uncertain parameters. Under appropriate assumptions, we chose Lyapunov function and use liner matrix inequality to obtain robust stabilization condition of uncertain nonlinear singular system. In the fifth part, the stability of a special nonlinear singular system——the singular bilinear system is investigated. Under the admissible condition, we design the control input and chose suitable Lyapunov function, then we obtained the conclusion of asymptotic stability of closed-loop system. Subsequently, we provide the stable region of the closed-loop system when the singular bilinear system is not regular or impulse.