First–Order Representations of Discrete Linear Multidimensional Systems

The classical localstate–space models for discrete multidimensional linearsystems, as proposed by Roesser or Fornasini and Marchesini,require causality of the resulting transfer matrices. We considera generalization comprising non-causal systems, based on Willems'state-space behaviors. A vector of manifest and a vector of latentvariables are supposed to be linked via a first–order dynamicequation and a static equation. Any system of linear constant–coefficientpartial difference equations gives rise to such an ``output–nulling''(ON) representation. Controllable systems possess driving–variablerepresentations, which are the dual counterpart of ON repsin many aspects. We study these representations with respectto their minimality, observability and controllability, and wederive conditions for their reducibility to the standard input–state–outputsetting.