Analysis And Synthesis Of Linear Polytopic Uncertain Discrete-Time Systems

Polytopic uncertain systems, a class of the important systems are studied by robust control theory in time-domain. Many scholars have investigated the problems of analysis and synthesis for the systems with quadratic stability concept and Linear Matrix Inequality. Due to a single Lyapuonv function for all vertices of quadratic stability analysis in the entire uncertain domain, it has been well recognized to be conservative. It is a new approach of robust control theory that parameter-dependent Lyapunov stability concept is used to analyze and synthesize polytopic uncertain systems. The results obtained by this approach are less conservative than those obtained on quadratic stability concept. Moreover, majority of those methods concentrate on continuous-time systems rather than discrete-time systems.Based on previous research works of others, Linear Matrix Inequality and parameter-dependent Lyapunov stability, the problems of analysis and synthesis were systematically and deeply investigated for linear polytopic uncertain discrete-time systems in this thesis. On aspect of robust analysis, the robust stability problems for uncertain discrete-time systems without and with delays, the Gl₂ performance computational problems for uncertain discrete-time systems were addressed. While on robust synthesis, the delay-dependent state feedback robust stabilization problems, Gl₂ robust control problems for the systems with single- and multi- delay(s), Gl₂ robust filtering problems for uncertain discrete-time systems without and with delays were studied. In details, the aspects included in this thesis are mainly as follows:1. The main results based on parameter-dependent Lyapunov stability theory were summarized. The imitations of the Algebraic Structure Method used for achieving parameter-dependent Lyapunov functions were pointed out in reducing conservativeness. An improved Algebraic Structure Method was presented through increasing the degree of parameter-dependent Lyapunov functions on uncertain parameter. A new robust stability condition was proposed based on combining this method and Extra Matrix Variable Method. Compared with the main results in this area, the condition possesses less conservativeness.2. Based on parameter-dependent Lyapunov stability theory, the thesis studied the Gl₂ performance computational problems for polytopic uncertain discrete-time systems. According to a simplified Gl₂ analysis theorem and Bounded Real Lemma, a Gl₂ performance condition in terms of Linear Matrix Inequality was obtained. A series of sufficient conditions for Gl₂ performance were presented by using Extra Matrix Variable Method, Algebraic Structure Method, Integrating Method and New Integrating Method. By investigating the relationship of these conditions about conservativeness, the condition based on New Integrating Method possesses the lessconservativeness, which is used to verify the robust stability condition obtained based on New Integrating Method.3. The less conservative sufficient conditions of delay-independent and -dependent robust stability were proposed for polytopic uncertain discrete-time systems with features of engineering application. Based on Lyapunov-Krasovskii stability theory and parameter-dependent Lyapunov stability theory, the parameter-dependent stability conditions were presented, whereafter Extra Matrix Variable Method and Integrating Method were used to achieve parameter-dependent Lyapunov functions for evaluating the stability. At the same time, it is noted that delay-dependent robust stability conditions were derived without Park Inequality, so the conservativeness is furtherly reduced.4. The problems of delay-dependent state feedback stabilization and G/2 control were addressed for polytopic uncertain discrete-time systems with delay. On the basis of the new delay-dependent robust stability conditions, a new sufficient condition of admissible controllers existence was proposed. Compared with the existing methods to solve the controller design problems, which is by solving optimization problems with constraints, the methods proposed in this thesis are much simpler and less conservative. At the same time, G/2 performance was introduced into the controller design problems for the systems with delay. The methods solving delay-independent and delay-dependent G/2 controller design problems were presented, which are less conservative than the existing Hx controller design methods.5. G/2 robust filtering problem, which is dual to G/2 control, was investigated for polytopic uncertain discrete-time systems without and with delay. Based on quadratic stability concept and parameter-dependent Lyapunov stability theory, the sufficient conditions of filter existence for uncertain systems and for delay-independent and delay-dependent for uncertain systems with single and multi- delay(s) were presented. In addition, based on the new delay-dependent robust stability conditions, the delay-dependent G/2 robust filter design methods were proposed. The filter design methods based on parameter-dependent was less conservative than those based on quadratic stability concept.