The Estimation Of The Solutions Of The Algebraic Lyapunov Equation And Riccati Equation

The algebraic Lyapunov equation and Riccati equation are usually utilized to perform the stability analysis of control systems and solve the problem of controller design. Therefore, both solving and estimating the solutions about the two kinds equations are very important. Furthermore, the bounds of the solution matrix of these equations can also be applied to deal with many control problems such as stability analysis, the design of optimal controllers and filters, and the estimates of the transient behavior . In this paper, the estimation of the solutions of the algebraic Riccati equation and Lyapunov equation are both investigated, and the present paper is organized as follows:First, the estimation problem of the trace of solution matrix of the general discrete time algebraic Riccati equation is discussed. The equivalent form of the general discrete time algebraic Riccati equation is given by using the formula of matrix inversion. Then, three new lower bounds for the trace of the solution matrix are derived by means of the properties of eigenvaules and traces of matrices; compared to the majority of the approach proposed in the literature, the present results are less conservative under certain conditions.Second, the estimation problem of the solution matrix of the general discrete time algebraic Riccati equation is discussed. Bounds of the solution matrix about it are given by using the formula of matrix inversion and the properties of eigenvalue of the matrix, and the results are improved by establishing a formula of iteration.Third, the estimation problem of the solution matrix of the perturbed discrete time algebraic Riccati equation is discussed. The estimation of upper and lower bounds of the solution matrix to the equation under a certain uncertainty assumption is presented by applying the matrix calculation property, and the estimation results are given by a matrix inequality and discrete time algebraic Riccati equations.Fourth, the estimation problem of the solution matrix of the perturbed continuous time algebraic Lyapunov equation is discussed. The estimation of upper and lower bounds of the solution matrix to the equation under a certain uncertainty assumption are presented by applying some calculation properties of matrix and inequality, and the estimation results are given by a continuous time algebra Riccati equations or given by some inequalities. At the same time, some existence conditions of symmetric positive definite matrix solution are shown for the equation.Finally, the estimation problem of the solution matrix of the perturbed continuous time algebraic Riccati equation is introduced. An estimation of upper bound and existence condition of the solution matrix to the equation under a certain uncertainty assumption is presented by applying the Schur complement and some calculation properties of matrix inequality. At the same time, two lower bounds of the solution matrix to the equation are shown by some properties of matrix inequality.