Parametric solutions to fully-actuated generalized Sylvester equations—The homogeneous case

Inspired by the concept of fully-actuation for second-order mechanical systems, in this paper we introduce the definition of fully-actuated generalized Sylvester equations, which are closely related with the various control designs of fully-actuated systems. It is shown that when a general high-order generalized Sylvester equation is fully-actuated, its complete parametric solution can be obtained extremely simply, yet which possesses a very neat explicit closed form. 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, 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 second-order dynamical systems.