Robust H_∞ Control And Positive Real Control For Time Delay Systems

In process of actual industries, time delays and uncertainties al-ways exist, which make the analysis and synthesis of systems more difficult andcomplicated, and are the sources of instability and poor performances of systems.There are many stability criterias of time delay systems, which can be gerneallyclassified into two categories: delay-independent and delay-dependent ones. Sincedelay-dependent criteria make use of exact information of delays, they are less con-servative than delay-independent ones. So studies of the delay-dependent stabilitycreteria are necessary.H_∞performance is an pivotal index of many performances of systems, thusthe analysis of H_∞performance and H_∞control for systems are important parts ofsystem control. Recently, much attention is paid on the studies of delay-dependentstability conditions and applications in the H_∞problem. And the notion of posi-tive realness is a fundamental property of theory of circuits and systems, which iswidely applied in the areas of robustness of systems, dissipativeness and nonlinearcontrol. There are close connections between positive-real control and H_∞control.The problem of positive-real control sometimes can be changed into H_∞control.Because of the existence of time delays and parameter uncertainties, the analysisand synthesis of robust positive realness of systems with delays and uncertaintiesare of practical importance.By using the tool of linear matrix inequality(LMI), this paper considers robustH_∞stability and H_∞control for several kinds of uncertain systems with time delays.The main conclusions in this paper are(1) By using the tool of LMI, the sufficient delay-dependent condition of robuststability for this system is obtained; Based on the condition of robust stability,the delay-dependent condition of the system with H_∞performance index(boundedreal lemma) is presented, Consequently, the condition of existence of robust H_∞controller such that the resulting closed-loop systems is robust stable and H_∞normof it is smller than a given scalar is discussed. Furthermore, the design of the controller is ginven. Finally, several examples illustrate that these conclusions areeffective and less conservative.(2) By introducing appropriate Lyapunov function, free matrices and zero-equalities, the delay-dependent sufficient condition of robust stability and bounedreal lemma for a class of discrete systems with norm-bounded uncertainties andmultiple time delays is given. The conclusions of some existed literatures are gener-alized.(3) The condition for the existence of memoryless state feedback controller isgotten, in order to guarantee the stability and extended strict positive realness ofthe closed-loop system for all admissible parameter uncertainties. The method ofthe design of state feedback controller is derived. An example result shows that thefeasibility of the mehtod of the design.