Common eigenvector approach to exact order reduction for Roesser state-space models of multidimensional systems

Abstract The well-known Popov–Belevitch–Hautus (PBH) tests play an important role in the Kalman decomposition of 1-D systems and reveal the relationship among the eigenvalues, the eigenvectors and the reducibility of a given 1-D state-space model. This paper is to try to generalize the PBH tests to the n -D case for the exact reducibility of n -D Roesser models by exploiting the so-called common eigenvectors. Specifically, the notion of constrained common eigenvectors is introduced, for the first time, which provides insight into the relationship between reducibility and multiple eigenvalues. Based on this result, new reducibility conditions and the corresponding reduction procedure are developed for n -D Roesser models. It will be shown that this common eigenvector approach is applicable to a larger class of Roesser models for which the existing approaches may not be applied to do further order reduction. A Grobner basis approach is proposed to compute such a constrained common eigenvector, which also leads to an equivalent reducibility condition. Moreover, a generalization to the state delay case is also given so that the eigenvalues of both the system matrix and the state-delay system matrix can be treated simultaneously.