Least-squares integration of one-dimensional codistributions with application to approximate feedback linearization
نویسندگان
چکیده
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 one to nd a linearizable nonlinear system that is close to the given system in a least squares (L 2) sense. A linearly controllable single input aane 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 nding (least squares approximate) integrating factors that make a xed characteristic one-form close to being exact in an L 2 sense. One can decompose a given one-form 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 in L 2 together with some derivatives and derive similar bounds for the inexact part. This allows one to nd a linearizable non-linear system that is close to the given system in a least squares (L 2) sense together with some derivatives. The Sobolev embedding techniques allow to obtain an upper bound on the uniform (L 1) distance between the nonlinear system and its linearizable approximation.
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عنوان ژورنال:
- MCSS
دوره 9 شماره
صفحات -
تاریخ انتشار 1996