The Matrix Series Solutions For Some Kinds Linear Matrix Equations And Numeric Algorithms

The matrix theory is widely used in the areas of statistics, ladder networks,transport theory, dynamic programming, control theory and stochastic filtering.Several problems in the stability analysis of continuous time linear systems and inthe design of optimal control systems are usually transformed to solve the linearmatrix equations.In this paper, we present the matrix series solutions for some linear matrixequations, such as equation P=AP B+C, discrete and continuous Lyapunovmatrix equations, and give some corresponding numeric algorithms. This papermainly contains three chapters:In chapter one, we present some background knowledge of these linear ma-trix equations, and introduce some symbols and definition used in this paper.In chapter two, we obtain the matrix series solution for the linear matrixequation P=AP B+C, by using the property of the matrix series convergence.We present several matrix series solutions to discrete Lyapunov matrix equationP=AP AT+Q, while the module of the eigenvalues of the coefcient matrixA belong to diferent ranges. And design the corresponding numeric algorithmsto the discrete Lyapunov matrix equation. Finally the superiority of the newnumeric algorithms will be showed by numerical examples.In chapter three, based on the results of chapter two, and taking in a pa-rameter, we can change Sylvester matrix equation and the continuous Lyapunovmatrix equation into the linear matrix equations, which were studied in chaptertwo, and present the matrix series solutions to them. At the same time we designthe numeric algorithms to the continuous Lyapunov matrix equation. Finally thesuperiority will be showed by numerical examples.