PARAMETRIC SOLUTIONS TO THE GENERALIZED SYLVESTER MATRIX EQUATION AX ‐ XF = BY AND THE REGULATOR EQUATION AX ‐ XF = BY + R

Abstract

ABSTRACTIn this paper, explicit parametric solutions to the generalized Sylvester matrix equation AX ‐ XF = BY and the regulator matrix equation AX ‐ XF = BY + R are proposed without any transformation and factorization. The proposed solutions are presented in terms of the Krylov matrix of matrix pair (A, B), a symmetric operator and the generalized observability matrix of matrix pair (Z, F) where Z is an arbitrary matrix and is used to denote the degree of freedom in the solution. Due to its elegant form and convenient computation, these proposed solutions will play an important role in solving and analyzing these types of equations in control systems theory.