Perturbation feedback control: A geometric interpretation

Heinz Schättler; Urszula Ledzewicz

doi:10.3934/naco.2012.2.631

Article Contents

2012, Volume 2, Issue 3: 631-654. Doi: 10.3934/naco.2012.2.631

This issue Previous Article Noether's symmetry Theorem for variational and optimal control problems with time delay Next Article

Perturbation feedback control: A geometric interpretation

Heinz Schättler^1, and
Urszula Ledzewicz^2,

1.
Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899
2.
Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026

Received: August 31, 2011

Revised: February 29, 2012

Published: July 2012

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Abstract

Perturbation feedback control is a classical procedure in control engineering that is based on linearizing a nonlinear system around some locally optimal nominal trajectory. In the presence of terminal constraints defined by a $k$-dimensional embedded submanifold, the corresponding flow of extremals for the underlying system gives rise to a canonical foliation in the $(t,x)$-space consisting of $(n-k+1)$-dimensional leaves and $k$-dimensional cross sections. In this paper, the connections between the formal computations in the engineering literature and the geometric meaning underlying these constructions are described.

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Mathematics Subject Classification: Primary: 49K15, 49L99; Secondary: 93C10.

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