Parametric solutions to rectangular high-order sylvester equations - Case of F Jordan

In an recent paper, we have proposed a general complete parametric solution in a simple and neat analytical closed form to a type of rectangular high-order Sylvester matrix equations with the parameter matrix F being an arbitrary matrix. Based on this result, in this paper we are presenting, for a type of rectangular high-order Sylvester matrix equations with the parameter matrix F being in a Jordan matrix form, a complete parametric solution which is expressed in terms of a free parameter matrix Z representing the degrees of freedom. The primary feature of this solution is that the matrix F, together with the parameter matrix R, does not need to be even known a priori, and thus may be 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.