Study On The Perturbed Discrete Matrix Equations And The Robust Control Of Uncertain Discrete Systems

In this paper, the estimation of the solutions to discrete matrix Lyapunov equations and the design of robust controller of uncertain discrete-time systems are both investigated. Design problem of robust controller is one of the main problem in robust control theory, and applications of the estimations of solutions to discrete-time matrix Lyapunov equations can be found in stability analysis, the design of optimal controllers and filters, and the estimation of the transient behavior. Both the problems mentioned above are studied in this paper. The present paper is organized as follows:First, the problems of characteristic estimation for the solution to the perturbed discrete matrix Lyapunov equations are studied. The estimation of the minimum and the maximum eigenvalues and trace of the solution to the perturbed discrete matrix Lyapunov equation are given by applying the properties of matrix eigenvalues and traces using the related matrix inequality respectively. At the same time, the existence condition and the supper and lower bounds estimation of the solution to the equation under a certain uncertainty assumption are presented, and the estimation results are given by a linear matrix inequality (LMI) and two matrix algebra Riccati equations. Combined to uncertainty assumptions for uncertain matrices, the supper and lower bounds estimation of the solution to the equation and the concrete form of matrix algebra Riccati equations are both given for four uncertainty assumptions. The validity of the results is illustrated by numerical examples.Second, the estimation of lower bounds of the symmetric positive definite matrix solution for the perturbation generalized Lyapunov matrix equations are presented. The concept of perturbation generalized matrix Lyapunov equation (PGMLE) is introduced according to the definition of generalized matrix Lyapunov equation (GMLE), the solution bounds of PGMLE in a certain perturbation structure are obtained by means of the approach for GMLE, and the existence conditions of the solution for the equation are presented. Numerical examples prove the validity of our proposed approach.Third, the robust controller design problem is studied for a class of uncertain discrete time-delay systems based on the state observer. By using LMI and Schur supplement lemma, the design method of robust controllers are developed. Furthermore, we present the method of controller design with a smaller feedback gain parameter by building and solving a problem of convex optimization. A numerical example is given to demonstrate the feasibility and efficiency of the method.Fourth, a new control scheme for the linear uncertain discrete system is presented. First a high performance controller based on the system's normal model is given, then based on the state observer, another control input to compensate the uncertainties and/or disturbances is obtained.