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A pencil approach to high gain feedback and generalized state space systems. (English) Zbl 0862.93033
In this article, limits of generalized state space systems under high-gain feedback are characterized. More precisely, for systems of the kind $E\dot x(t)=Ax(t)+Bu(t)$ conditions are given under which a given system can be obtained as a limit of a sequence of systems which are feedback equivalent to some system of the same structure.
Here the limit and the feedback equivalence (either strong or state restricted) are defined in terms of the pencil $$[sE-A,B]$$ of the system matrices.
After defining these concepts, a theorem on orbit closures of singular matrix pencils proved by the authors [J. Pure Appl. Algebra 81, No. 2, 117–137 (1992, Zbl 0754.15007)] is used in order to characterize these limits. Thus an if-and-only-if characterization in terms of closures of orbits of pencils under the action of full feedback groups is obtained.
Finally, some system theoretic applications of this result are given.

##### MSC:
 93C05 Linear systems in control theory 93B52 Feedback control 93B25 Algebraic methods 15A22 Matrix pencils
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