The controllability test for behaviors revisited


Polderman, Jan Willem (2005) The controllability test for behaviors revisited. In: 16th IFAC World Congress, July 3-8, 2005, Prague, Czech Republic (pp. pp. 1204-1208).

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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.
Item Type:Conference or Workshop Item
Copyright:© 2005 Elsevier
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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