On the connections between two classical notions of multidimensional system equivalence

In this paper, we study the connections between two classical notions of equivalence used in the literature of linear multidimensional (nD) systems. On one hand, we have the notion of zero coprime system equivalence which has been proved to be well suited for the preservation of input–output properties. On the other hand, we have the notion of equivalence induced by the algebraic analysis approach to linear systems theory which preserves algebraic properties of the associated module. We first prove that a zero coprime system equivalence yields an equivalence in the sense of algebraic analysis. Conversely, we show that, under some conditions, an equivalence in the sense of algebraic analysis provides a zero coprime system equivalence. In both cases, the results are completely explicit.