Common eigenvector approach to exact order reduction for multidimensional Fornasini-Marchesini state-space models

This paper proposes an exact order reduction approach for the multidimensional (n-D) Fornasini-Marchesini (F-M) model by making use of the common eigenvector. Specifically, by introducing the concept of common eigenvectors, sufficient conditions of exact order reductions are developed for an n-D F-M model, which are able to simultaneously deal with n eigenvalues of the system matrices of the n-D F-M model. The obtained results reveal, for the first time, the internal connection between the multiple eigenvalues of the system matrices and the reducibility of the considered n-D F-M model. Then, a corresponding algorithm is proposed to exactly reduce the order of an n-D F-M model as much as possible. Examples are given to illustrate the details as well as the effectiveness of the proposed approach.