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 <img src="/fulltext-image.asp?format=htmlnonpaginated&src=M0662Q177RV07K71_html\11045_2004_Article_1680_TeX2GIFIEq1.gif" border="0" alt="$${\cal B}$$" /> and one of its sub-behaviors <img src="/fulltext-image.asp?format=htmlnonpaginated&src=M0662Q177RV07K71_html\11045_2004_Article_1680_TeX2GIFIEq2.gif" border="0" alt="$${\cal B}_B$$" />, under what conditions a third complete behavior <img src="/fulltext-image.asp?format=htmlnonpaginated&src=M0662Q177RV07K71_html\11045_2004_Article_1680_TeX2GIFIEq3.gif" border="0" alt="$${\cal B}_A$$" /> can be found, such that <img src="/fulltext-image.asp?format=htmlnonpaginated&src=M0662Q177RV07K71_html\11045_2004_Article_1680_TeX2GIFIEq4.gif" border="0" alt="$${\cal B} = {\cal B}_A + {\cal B}_B$$" /> and <img src="/fulltext-image.asp?format=htmlnonpaginated&src=M0662Q177RV07K71_html\11045_2004_Article_1680_TeX2GIFIEq5.gif" border="0" alt="$${\cal B}_A \cap {\cal B}_B$$" /> 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].