Two-dimensional behavior decompositions with Finite-Dimensional Intersection: A Complete Characterization

Abstract

In this paper, the following problem is addressed: given a two-dimensional complete behavior $${\cal B}$$ and one of its sub-behaviors $${\cal B}_B$$ , under what conditions a third complete behavior $${\cal B}_A$$ can be found, such that $${\cal B} = {\cal B}_A + {\cal B}_B$$ and $${\cal B}_A \cap {\cal B}_B$$ is finite-dimensional autonomous? This constitutes a complete generalization of the decomposition theorem, as it represents a decomposition with "minimal intersection", in which one of the two terms has been a priori fixed. The analysis carried on here completes the preliminary results reported in [Bisiacco and Valcher, Multidimensional Systems and Signal Processing, vol. 13,2002, pp. 289--315]. and completely generalizes the direct sum decomposition problem presented in [Bisiacco and Valcher, IEEE Transactions on Circuits and Systems Part I, CAS-I-48, no-4, 2001, pp. 490--494].