On a type of second-order generalized Sylvester equations

In this paper, a new type of second-order generalized Sylvester equations (GSEs) associated with the general eigenstructure assignment of a type of second-order linear systems are proposed. Degrees of freedom is first investigated using the concept of F-coprimeness, and a complete general parametric solution in a neat explicit closed form is then established using a generalized matrix fraction right factorization. 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 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.