The controllability test for behaviors revisited

Jan W. Polderman, P. Piztek (Editor)

    Research output: Contribution to conferencePaperAcademicpeer-review

    1 Citation (Scopus)


    Let B = {w 2 Lloc 1 (R,Rq) | R( d/dt )w = 0}. It is well-know that B is controllable if and only if R(λ) has the same rank for all complex λ. We want to re-examen the proof of this fundamental result. Denote by B1 the behavior B intersected with set of smooth functions. For B1 the proof is easy and one would expect that the fact that B1 is dense in B would provide a quick and easy proof for the controllability test for B. This, unfortunately, is not true. For the smooth case the proof uses a differential transformation of the behavior. This transformation corresponds to a right unimodular transformation V(E) of R(E) yielding its Smith form. Generally, such a transformation cannot be extended to B since B contains non-smooth trajectories. In this contribution we argue that, despite this fact, the unimodular matrix V(E) defines an injection from B into the behavior defined by the Smith form. With this observation, that is interesting in its own right, the controllability test can be proved relatively easy.
    Original languageUndefined
    Number of pages5
    Publication statusPublished - 2005
    Event16th IFAC World Congress 2005 - Prague, Czech Republic
    Duration: 3 Jul 20058 Jul 2005
    Conference number: 16


    Conference16th IFAC World Congress 2005
    Country/TerritoryCzech Republic
    Internet address


    • IR-68681
    • EWI-16603

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