Design of Dynamical Compensators for Matrix Second-order Linear Systems: A Parametric Approach

In this paper, the issue of designing a type of dynamical compensators for matrix second-order linear (MSOL) systems is addressed in the matrix second-order framework by using a complete parametric approach. Based on simple and complete parametric solutions of the generalized Sylvester matrix equation of MSOL systems, parametric expressions for the left and right eigenvector matrices of the closed-loop system are obtained. Also a pair of complete parametric expressions for the coefficient matrices of the dynamical compensator is presented. These expressions contain two groups of parameter vectors which represent the degrees of freedom existing in the solution of the closed-loop eigenvectors. The proposed approach provides all the degrees of design freedom, which can be further utilized to achieve additional system specifications. A spring-mass system is utilized to show the effect of the proposed approach.