Least Squares Approximate Feedback Linearization
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We study the least squares approximate feedback linearization problem: given a single input nonlinear system, find a linearizable nonlinear system 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 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. One can use least squares approximate integrating factors in approximate feedback linearization of nonlinearizable single input affine systems. Moreover, least squares approximate integrating factors allow a unified approach to both least squares approximate and exact feedback linearization.