Least-squares integration of one-dimensional codistributions with application to approximate feedback linearization |
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Authors: | Andrzej Banaszuk Andrzej ?wi?ch John Hauser |
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Affiliation: | (1) Department of Mathematics, University of California, 95616-8633 Davis, California, U.S.A.;(2) School of Mathematics, Georgia Institute of Technology, 30332 Atlanta, Georgia, U.S.A.;(3) Electrical and Computer Engineering, University of Colorado, 80309-0425 Boulder, Colorado, U.S.A. |
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Abstract: | We study the problem of approximating one-dimensional nonintegrable codistributions by integrable ones and apply the resulting
approximations to approximate feedback linearization of single-input systems. The approach derived in this paper allows a
linearizable nonlinear system to be found that is close to the given system in a least-squares (L
2) sense. A linearly controllable single-input affine nonlinear system is feedback linearizable if and only if its characteristic
distribution is involutive (hence integrable) or, equivalently, any characteristic one-form (a one-form that annihilates the
characteristic distribution) is integrable. We study the problem of finding (least-squares approximate) integrating factors
that make a fixed characteristic one-form close to being exact in anL
2 sense. A given one-form can be decomposed into exact and inexact parts using the Hodge decomposition. We derive an upper
bound on the size of the inexact part of a scaled characteristic one-form and show that a least-squares integrating factor
provides the minimum value for this upper bound. We also consider higher-order approximate integrating factors that scale
a nonintegrable one-form in a way that the scaled form is closer to being integrable inL
2 together with some derivatives and derive similar bounds for the inexact part. This allows a linearizable nonlinear system
that is close to the given system in a least-squares (L
2) sense together with some derivatives to be found. The Sobolev embedding techniques allow us to obtain an upper bound on
the uniform (L
∞) distance between the nonlinear system and its linearizable approximation.
This research was supported in part by NSF under Grant PYI ECS-9396296, by AFOSR under Grant AFOSR F49620-94-1-0183, and by
a grant from the Hughes Aircraft Company. |
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Keywords: | Nonlinear systems Feedback linearization Differential forms Calculus of variations Sobolev spaces Elliptic PDEs |
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