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

In this paper, a type of fully-actuated nonhomogeneous high-order generalized Sylvester equations (GSEs) are proposed. It is shown that the complete general parametric solution in neat explicit closed form to a nonhomogeneous high-order generalized Sylvester equations can be obtained in a extremely easy and simple way once the full-actuation assumption is met. 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 degrees 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 high-order dynamical systems.