Studies On The Discrete Algebraic Matrix Equations And The Stability Of Uncertain Discrete-time Systems

In this paper, the estimation of the solutions to discrete-time algebraic Riccati equations and Lyapunov equations and the analysis of stability of uncertain discrete-time systems are both investigated. Stability analysis of uncertain discrete-time system is a main theme in control theory, and applications of the estimations of solutions to discrete-time algebraic Riccati equations and Lyapunov equations can be found in the stability analysis, the design of optimal controllers and filters, and the estimates of the transient behavior. Both the problems mentioned above are studied as an organic combination in this paper. The present paper is organized as follows:First, the estimation of positive definite solutions to general discrete-time algebraic Riccati equations is studied. Equivalent form of the general discrete-time algebraic Riccati equations was obtained by applying the formula of matrix inversion. A tighter lower and upper bounds for positive definite matrix P of discrete-time algebraic Riccati equations are presented by combining the Rayleigh inequality and properties of eigenvalue. The validity of the results is illustrated by numerical examples.Second, we study the estimation of the positive definite solutions to standard discrete-time algebraic Riccati equations. The bound of positive definite solutions to perturbed standard discrete-time algebraic Riccati equations are obtained by constructing new matrix and applying correlative theory of discrete-time algebraic Riccati equations, under the condition that the perturbed parameter is norm bounded uncertain. The bound is presented by solving discrete-time algebraic Riccati equation, thus it is avoided to solving higher-order algebraic equation. Finally a numerical example is presented.Third, the estimation of the positive definite solutions to discrete-time algebraic...