Numerical Algorithm For The Generalized Lyapunov Equation

The generalized Lyapunov equation has been widely used in the fields of controllability of bilinear systems,model simplification,stability analysis of linear stochastic systems and special linear stochastic differential equations.For example,some nonlinear control systems and bilinear control systems in boundary control dynamics can be transformed to study the properties and solutions of the generalized Lyapunov equation.Therefore,it is of great scientific significance to study the numerical algorithm for the solution of this kind of the generalized Lyapunov equation.In this paper,first of all,for the generalized Lyapunov equation in real number domain,the effective algorithms for solving linear system are used to consider the matrix equation,then the biconjugate residual(BCR)algorithm,bi-conjugate gradient stabilized(Bi-CGSTAB)algorithm and conjugate residual squared(CRS)algorithm are given,and the convergence of BCR algorithm is proved.Secondly,for the generalized Lyapunov equation in complex domain,the generalized modified HSS(GMHSS)algorithm,inexact generalized modified HSS(IGMHSS)algorithm and accelerated double-step scale splitting(ADSS)iterative algorithm are given based on the idea of matrix splitting,and the convergence of the corresponding algorithms are analyzed.The first chapter introduces the theoretical background and research status of the generalized Lyapunov equation,and gives the relevant notation,preliminary knowledge,definition and lemma used in this paper.The second chapter considers the numerical algorithm of the generalized Lyapunov equation in the real number domain,and introduces the biconjugate residual(BCR)algorithm,bi-conjugate gradient stabilized(Bi-CGSTAB)algorithm and conjugate residual squared(CRS)algorithm for solving the linear system.We use Kronecker product and straightening operator calculation process to consider the generalized Lyapunov equation,the matrix version of BCR algorithm,Bi-CGSTAB algorithm and CRS algorithm can be obtained.In this paper,the preprocessing is applied to the three matrix versions of the algorithm.By introducing parameters and Cayley transformation,the corresponding three new matrix versions are obtained,which can solve the generalized Lyapunov equation more effectively.Finally,several numerical examples are given to illustrate the effectiveness of the three algorithms.In chapter three,the numerical method of generalized Lyapunov equation in complex number domain is given,based on the generalized modified HSS(GMHSS),inexact generalized modified HSS(IGMHSS)algorithm and accelerated double-step scale splitting(ADSS)iterative algorithm for solving the standard Sylvester equation,they are generalized to obtain the generalized Lyapunov equation solution method,the convergence of the three algorithms is guaranteed under certain conditions.The main idea of the algorithm is to decompose the coefficient matrix into two matrices,one of which is positive definite matrix belongs to the real number field,and the other is semi-positive definite matrix with imaginary part,and then a new algorithm is obtained by using the idea of alternating iteration algorithm.Finally,three numerical examples are given to verify the effectiveness of the algorithm.