Solution to a type of nonhomogeneous generalized Sylvester equations

Abstract

In this paper, a new type of nonhomogeneous generalized Sylvester equations (GSEs) are proposed. A complete general parametric solution in a neat explicit closed form is established using the F -coprimeness condition. The primary feature of this solution is that the matrix F does not need to be in any canonical form, or may be even unknown a priori. The matrix R, together with the matrix F, may be both set undetermined and used as a part of the degree of freedom beyond the completely free parameter matrix Z. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many control systems analysis and design problems involving dynamical systems.

Parametric solutions to fully-actuated generalized Sylvester equations—The nonhomogeneous case

“…Particularly, in [21] a type of high-order Generalized Sylvester equations in the following nonhomogeneous form is investigated:…”

“…This property will allow us to simplify the general solution to fully-actuated GSEs. The purpose of this paper is to show that the complete parametric solution to the high-order GSE (1) obtained in [21] can be greatly simplified and easily obtained when the following fully-actuation assumption is met:…”