An approach to multidimensional Fornasini–Marchesini state-space model realization w.r.t. columns of transfer matrices

Abstract

This paper proposes a new approach that can generate a low-order Fornasini–Marchesini (F–M) model realization for a given n-D system by taking into account the column structural properties of its transfer matrix. Specifically, a new necessary and sufficient realization condition is developed for the F–M model realization based on a resolvent invariant space associated with the Gleason’s problem specified by the given n-D transfer matrix. Then, a new constructive procedure with respect to the columns of a given transfer matrix is proposed for constructing a low-order F–M model realization. In order to apply this procedure to the multiple-input multiple-output case more effectively, an improved realization procedure based on a polynomial description is also proposed. It will be shown, by both algorithmic analysis and illustrative examples, that for a transfer matrix having more rows than columns the proposed realization approach may generate a much lower realization order than the existing one with respect to the rows of a given n-D transfer matrix recently developed by Cheng et al. Particularly, for a given scalar transfer function, our new approach always generates an F–M model realization with lower order than the existing methods.