The paper presents the results of the study of behavior theory as developed by J.C. Willems from the point of view of polynomial and rational models. Considering behaviors, in the discrete time case, to be generalizations of rational models, a natural focal point becomes the concept of a behavior homomorphism. We give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices. These results, which are of intrinsic interest, are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations. To a certain extent, these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of Rosenbrock. A study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory.
Fuhrmann, P. A. (2002). A study of behaviors. Linear Algebra and Its Applications, 351–352, 303–380. https://doi.org/10.1016/S0024-3795(02)00282-3