The controllability test for behaviors revisited

Jan W. Polderman, P. Piztek (Editor)

    Research output: Contribution to conferencePaper

    1 Citation (Scopus)

    Abstract

    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
    Pages1204-1208
    Number of pages5
    Publication statusPublished - 2005
    Event16th IFAC World Congress 2005 - Prague, Czech Republic
    Duration: 3 Jul 20058 Jul 2005
    Conference number: 16
    http://www.utia.cas.cz/news/608

    Conference

    Conference16th IFAC World Congress 2005
    CountryCzech Republic
    CityPrague
    Period3/07/058/07/05
    Internet address

    Keywords

    • IR-68681
    • EWI-16603

    Cite this

    Polderman, J. W., & Piztek, P. (Ed.) (2005). The controllability test for behaviors revisited. 1204-1208. Paper presented at 16th IFAC World Congress 2005, Prague, Czech Republic.