Realization Theory of Nonlinear Hybrid Systems

Mihály Petreczky 1 — Jean-Baptiste Pomet 2

1 Centrum voor Wiskunde en Informatica (CWI) P.O.Box 94079, 1090GB Amsterdam, The Netherlands M.Petreczky@cwi.nl

2 INRIA Sophia Antipolis P.O.Box 93, 2004 Route des Lucioles, 6902 Jean-Baptiste.Pomet@sophia.inria.fr

ABSTRACT. The paper investigates the realization problem for a class of analytic nonlinear hybrid systems without autonomous switching. Similarly to the classical nonlinear realization theory the realization problem for hybrid systems is translated to a formal realization problem of a class of abstract systems defined on rings of formal power series. Necessary conditions are presented for existence of a realization by such an abstract system and thus by a hybrid system. A notion analogous to the Lie-rank of nonlinear input-output maps is defined and the presented necessary condition involves a requirement that this generalised Lie-rank should be finite. We will also introduce the notion of strong Lie-rank and we will show that finiteness of the strong Lie-rank implies existence of a realization which is very close to the required hybrid system realization. Thus, finiteness of the strong Lie-rank can be seen as an "almost" sufficient condition. In the special case of nonlinear analytic systems both the finite Lie-rank and the finite strong Lie-rank condition presented in the paper reduces to the well-known finite Lie-rank condition. We will use theory of Sweedler-type coalgebras ...

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