Almost direct sum decomposition and implementation of 2D behaviors

Abstract

This paper deals with multidimensional systems, for example, systems described by linear, constant coefficient partial differential/difference equations. In the behavioral approach, the notion of interconnection is the basis of control. In this setting, feedback interconnection of systems is based on the still more fundamental concept of regular interconnection, which has been introduced by J.C. Willems. The dual problem of regular interconnection is the one of direct sum decomposition. The following two problems are addressed: given a behavior ${\mathfrak {B}}$ and one of its sub-behaviors ${\mathfrak {B}_1}$ , under what conditions does there exist another sub-behavior ${\mathfrak {B}_2}$ such that ${\mathfrak {B}_1 \cap \mathfrak {B}_2}$ has finite dimension and ${\mathfrak {B}_1 + \mathfrak {B}_2}$ has finite codimension with respect to ${\mathfrak {B}}$ i.e. we treat the direct sum decomposition of ${\mathfrak {B}}$ up to finite-dimensional behaviors, which, in this context, are considered negligible. The second related problem concerns regular interconnections and reads as follows: given a plant behavior together with a desired behavior, find, if possible, another behavior (a controller) such that the interconnection is regular and has finite codimension with respect to the given desired behavior. A constructive solution to the problems is provided for two-dimensional behaviors.