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A new and simple proof of the equivalence theorem for behaviors

Summary: A new and simple proof of the equivalence
theorem for behaviors
Jan Willem Polderman
Two full row rank representations of the same behavior are related
through a left unimodular transformation. We present a new and ex-
tremely simple and insightful proof for this well-established fact.
Mathematics subject classification: 93A30
1 Introduction
One of the features of the behavioral approach to linear systems theory is the
emphasis on trajectories rather than on specific representations of the set of all
possible trajectories. If a behavior is specified through a set of linear, constant
coefficient differential or difference equations, then the set of all possible repre-
sentations is completely characterized by the equivalence theorem for behaviors.
This theorem states that two full row rank matrices of polynomials that define
the same behavior are related by a left unimodular transformation. Proofs of
this result may be found in [2, Corollary 2.5] and [1, Theorem 3.6.4]. Both
proofs use properties of the behavior, in particular the fact that every behavior
admits an input/output structure. A proof in the spirit of the behavioral philos-
ophy would rely on as few properties of the behavior as possible. In [1, Chapter


Source: Al Hanbali, Ahmad - Department of Applied Mathematics, Universiteit Twente


Collections: Engineering