Abstract
This chapter drops the assumption of exactly realizable trajectories and allows for arbitrary desired trajectories \(\varvec{x}_{d}\left( t\right) \). The regularization parameter of optimal control is assumed to be small and used for a perturbation expansion. Rearranging the necessary optimality conditions leads to a reinterpretation of unregularized optimal control problems as singularly perturbed differential equations. For systems satisfying a linearizing assumption, the leading order equations become linear. The linearity allows the derivation of closed form expressions for optimal trajectory tracking in a general class of nonlinear systems affine in control. The perturbative approach yields exact results for vanishing regularization parameter. However, this exact result comes at a price in form of a diverging control signal and a discontinuous state trajectory.
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Löber, J. (2017). Analytical Approximations for Optimal Trajectory Tracking. In: Optimal Trajectory Tracking of Nonlinear Dynamical Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-46574-6_4
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DOI: https://doi.org/10.1007/978-3-319-46574-6_4
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