The Estimation For Solution Matrix Of Perturbed Lyapunov / Riccati Matrix Equation

In this paper, the estimations for the solution matrix of perturbed generalized Lyapunov matrix equation and perturbed continuous Riccati matrix equation are both researched. Algebraic Lyapunov matrix equation and algebraic Riccati matrix equation are widely used in different systems for many control problems such as stability analysis, controller design of time-delay systems, maximum cost estimation, convergence of numerical algorithm and behavior of Riccati differential equation. Furthermore, the two types of equations mentioned above have important theoretical significance and practical value. Particularly, perturbed generalized Lyapunov matrix equation and perturbed Riccati matrix equation have important research significance. Thus these two matrix equations are discussed in the paper. The specific contents are included as follows:First of all, the estimations of solution matrix boundary and eigenvalue for solution matrix of perturbed generalized Lyapunov matrix equation are studied. The estimation for solution matrix of perturbed generalized Lyapunov equation defined in a special complex plane region is researched. The perturbed parameter is norm bounded uncertain. Firstly, by using of matrix inequalities and character of matrix eigenvalue and quadratic equation with one unknown of calculus, the general form of perturbed generalized Lyapunov equation is studied. Then, a new lower bound for eigenvalue of solution matrix is obtained. Next, the general form of perturbed generalized Lyapunov matrix equation is transformed to that of perturbed continuous Lyapunov matrix equation by the variable substitution method of a variable. Moreover, by constructing two semi-definition matrices and using matrix inequalities and character of eigenvalue, four forms lower bound for solution matrix and two forms lower bound for eigenvalue are derived.Second, the estimations for solution matrix and eigenvalue of perturbed continuous Riccati matrix equation are studied. Under the condition that the perturbed parameter is norm bounded uncertain, the estimation of solution matrix is discussed. Then, by constructing two semi-definition matrices and using perturbed parameter inequalities and character of eigenvalue, three upper and lower bounds for solution matrix and two upper and lower bounds of eigenvalue for perturbed continuous Riccati matrix equation are all derived.Finally, the main research results obtained above is verified by some numerical examples, meanwhile the feasibility of research results is illustrated.