Least-squares integration of one-dimensional codistributions with application to approximate feedback linearization

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 (L2) 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 anL2 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 inL2 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 (L2) 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.

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