On a type of generalized sylvester equations

In this paper a new type of generalized Sylvester equations (GSEs) associated with the generalized eigen-structure assignment in a type of descriptor linear systems are proposed. Based on the concept of F-coprimeness, degrees of freedom existing in the general solution to this type of equations are first given, and then a general complete parametric solution in explicit closed form is established based on generalized right factorization. The primary feature of this solution is that the parameter matrix F, which corresponds to the finite closed-loop Jordan matrix in the generalized eigenstructure assignment problem, does not need to be in any canonical form, and 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.