Solution to High-order Generalized Sylvester Matrix Equations

This paper provides a complete general parametric solution (V,W) to the high-order generalized Sylvester matrix equation ∑i=0^mAiVFⁱ=BW, with F being an arbitrary square matrix. 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 analysis and design problems involving high-order dynamical systems.