Controlled and Conditioned Invariance for Polynomial and Rational Feedback Systems

Schilli, Christian; Zerz, Eva; Levandovskyy, Viktor

doi:10.1007/978-3-030-38356-5_10

Cited by 3 publications

(1 citation statement)

References 9 publications

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“…This includes the concept of a differential equation on a variety, see also [26], and the fact that the variety is forward invariant with respect to the differential equation. Controlled invariant hypersurfaces of polynomial control systems on varieties are considered in [27], [28].…”

Section: Rational Systemsmentioning

confidence: 99%

An Observable Canonical Form for a Rational System on a Variety

Nemcova,

van Schuppen

2018

Preprint

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An observable canonical form is formulated for the set of rational systems on a variety each of which is a single-input-single-output, affine in the input, and a minimal realization of its response map. The equivalence relation for the canonical form is defined by the condition that two equivalent systems have the same response map. A proof is provided that the defined form is well-defined canonical form. Special cases are discussed.The support of the University of Chemistry and Technology for the cooperation of the two authors is herewith gratefully acknowledged.

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Section: Rational Systemsmentioning

confidence: 99%

An Observable Canonical Form for a Rational System on a Variety

Nemcova,

van Schuppen

2018

Preprint

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Attracting and Natural Invariant Varieties for Polynomial Vector Fields and Control Systems

Kruff

Schilli

Walcher

et al. 2020

Qual. Theory Dyn. Syst.

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We discuss real and complex polynomial vector fields and polynomially nonlinear, input-affine control systems, with a focus on invariant algebraic varieties. For a given real variety we consider the construction of polynomial ordinary differential equationṡ x = f (x) such that the variety is invariant and locally attracting, and show that such a construction is possible for any compact connected component of a smooth variety satisfying a weak additional condition. Moreover we introduce and study natural controlled invariant varieties (NCIV) with respect to a given input matrix g, i.e. varieties which are controlled invariant sets ofẋ = f (x) + g(x)u for any choice of the drift vector f . We use basic tools from commutative algebra and algebraic geometry in order to characterize NCIV's, and we present a constructive method to decide whether a variety is a NCIV with respect to an input matrix. The results and the algorithmic approach are illustrated by examples.

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