Notes on minimal realizations of multidimensional systems

Abstract

In this paper, we formalize two related but different notions for state-space realization of multidimensional ($$n\hbox {D}$$nD) single-input---single-output discrete systems in $$n\hbox {D}$$nD Roesser model, namely the "absolutely minimal realization" and the "minimal realization". We then focus our study mainly on first-degree 2D and 3D causal systems. A necessary and sufficient condition for absolutely minimal realizations is given for first-degree 2D systems. It is then shown that first-degree 2D systems that do not admit absolutely minimal realizations always admit minimal realizations of order 3. A Grobner basis approach is also proposed which leads to a sufficient condition for the absolutely minimal realizations of some higher-degree 2D systems. We then present a symbolic method that gives simple necessary conditions for the existence of absolutely minimal realizations for first-degree 3D systems. A two-step approach to absolutely minimal realizations for first-degree 3D systems is then presented, followed by techniques for minimal realizations of first-degree 3D systems. Throughout the paper, several non-trivial examples are illustrated with the aim of helping the reader to apply the realization methods proposed in this paper.