On the characteristic polynomial of the dynamic matrix of linear time-invariant multivariable systems in Luenberger's canonical forms

This article presents a general representation of the characteristic polynomial of the dynamic matrix for multivariable systems in Luenberger's canonical forms. The characteristic polynomial is given by means of the determinant of a polynomial matrix of substantially lower order. Therein, the polynomial coefficients of the single elements are the coefficients of the corresponding blocks of the dynamic matrix. The proposed representation of the characteristic polynomial can be helpful for the design of state-feedback controllers and state observers which is demonstrated by a numerical example.