Solution to second-order nonhomogeneous generalized Sylvester equations

In this paper, a new type of nonhomogeneous second-order 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 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 second-order dynamical systems.