On the Krylov subspace solutions of matrix equations in control theory.

摘要

This dissertation deals with numerical solutions of large and sparse matrix equations, such as the Lyapunov, Sylvester, and algebraic Riccati equations (both continuous and discrete-time) arising in control systems design and applications.Research on large-scale matrix computations is still at the developing stage and, in particular, only a very small number of methods have been developed for the large-scale solutions of the above matrix equations. Furthermore, most of the existing methods have some computational limitations.In this dissertation, we propose three new methods for these matrix equations: an Arnoldi-based divide-and-conquer method for the discrete Sylvester equation, a block Arnoldi method for the continuous-time Lyapunov and Sylvester equations, and a block Arnoldi method for the continuous-time algebraic Riccati equation.The divide-and-conquer method is based on exploitation of the sparsity pattern of one of the system matrices, and the other Arnoldi methods for the continuous-time Lyapunov, Sylvester, and algebraic Riccati equations are completely general purpose in the sense that they work with large matrices with arbitrary sparsity patterns.All these new methods are practical for restarting because these methods are such that the residuals after m fixed number of steps can be cheaply computed using information available after m steps.The results of our numerical experiments on practical data demonstrate that these methods are more efficient and accurate than their existing counterparts.