Stability Of Linear Singular System And H_∞ Control Problem

Stabilization is the premise of practical system normal operation, and stability is an important task for system control theory. It is a fundamental technique for stability study to analysis and synthesize system stability by considering the Lyapunov equation. For many years, numerous scholars have proposed many kinds of generalized Lyapunuv equation to study stability of singular system. This paper puts forward a kind of new generalized Lyapunuv equation, which is used to study the stability of discrete singular system.H_∞ control theory can successfully solve robust stability and disturb restrain, hence it has been abroadly regarded and fully developed in control domain. The disscussion of the H_∞ control problem stems from the linear system with extra perturbance, and its detail is to design dynamic compensator so that closed loop system has stated inhibitory action to extra perturbance, and if no such extra perturbance exists, the dynamic compensator still makes the closed loop system stable. Using Riccati equation and Riccati inequality, many perfect results have been obtained to design H_∞ controller. In singular system, earlier scholars design H_∞ controller using the generalized Riccati equation and generalized Riccati inequality, but the solving of Riccati equation is still a problem. As linear matrix inequality has ripe software package to use, linear matrix inequality theory has become a main technique to study H_∞ control problem in recent years. In this paper, we mainly discuss H_∞ performance of continuous singular system, H_∞ state feedback control, H_∞ output feedback control, H_∞ performance of discrete singular system, etc.The main conclusions in the paper are:(1)A generalized Lyapunov equation is given to verdict stability of discrete singular system. In common with construction of Lyapunov equation which can verdict stability of continuous singular system in paper[14], we construct a generalized Lyapunov equation to verdict stability of discrete singular system. Moreover, through it we get the generalized Lyapunov inequality condition to verdict stability of discrete singular system with that equation.