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Feedback invariants, critical trajectories and hamiltonian formalism

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Nonlinear control in the Year 2000

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 258))

Abstract

We show that the Hamiltonian formalism, as used for optimal control problems, is a natural tool for studying the feedback equivalence problem and for constructing invariants. The most important invariants (covariants) are the critical curves or trajectories of the system, also called extremals or singular curves. They are obtained as the curves satisfying formally the necessary conditions of Pontryagin maximum principle. In particular, we show that for arbitrary scalar control systems the set of critical trajectories (if nonempty), together with a canonical involutive distribution, determine the system in neighbourhood of a regular point, up to feedback equivalence. We describe the structure of critical trajectories around regular points and show that the system can be decomposed into a feedback linearizable part and a fully nonlinear part. A complete set of invariants is given for fully nonlinear analytic systems.

Supported by Polish KBN grant 2P03A 035 16

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Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950, Warsaw, Poland

    Bronisław Jakubczyk

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  1. Bronisław Jakubczyk
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Editor information

Alberto Isidori (Professor)Françoise Lamnabhi-Lagarrigue (Docteur D’état)Witold Respondek (Professor)

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© 2001 Springer-Verlag London Limited

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Jakubczyk, B. (2001). Feedback invariants, critical trajectories and hamiltonian formalism. In: Isidori, A., Lamnabhi-Lagarrigue, F., Respondek, W. (eds) Nonlinear control in the Year 2000. Lecture Notes in Control and Information Sciences, vol 258. Springer, London. https://doi.org/10.1007/BFb0110240

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  • DOI: https://doi.org/10.1007/BFb0110240

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-363-8

  • Online ISBN: 978-1-84628-568-4

  • eBook Packages: Springer Book Archive

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