Equivalence conditions for behaviors and the Kronecker canonical form

P. A. Fuhrmann, U. Helmke

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In this paper we explore equivalence conditions and invariants for behaviors given in kernel representations. In case the kernel representation is given in terms of a linear matrix pencil, the invariants for strict equivalence are given by theKronecker canonical form which, in turn, we interpret in geometric control terms. If the behavior is given in a kernel representation by a higher order rectangular polynomial matrix, the natural equivalence concept is behavior equivalence. These notions are closely related to the Morse group that incorporates state space similarity transformations, state feedback, and output injection. A simple canonical form for behavioral equivalence is given that clearly exhibits the reachable and autonomous parts of the behavior. Using polynomial models we also present a unified approach to pencil equivalence that elucidates the close connections between classification problems from linear algebra, geometric control theory, and behavior theory.We also indicate how to derive the invariants under behavior equivalence from the Kronecker invariants.

Original languageEnglish
Pages (from-to)267-293
Number of pages27
JournalMathematics of Control, Signals, and Systems
Issue number4
StatePublished - 1 Aug 2011


  • Behaviors
  • Kronecker canonical form
  • Matrix pencils
  • Strict equivalence

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Signal Processing
  • Control and Optimization
  • Applied Mathematics


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