Least-square Fitting on the Circle using Optimal Control

Filtering of circular data from noisy measurements is well known to be a hard problem because of the ambiguity of the wrapped phase and further complicated by the constraint that filtered states need to be on the circle. Probabilistic models focus on specifying the dynamics of the random phase and then estimate these states recursively using Bayes' formula, while deterministic approaches normally define a cost function containing the error between the states and measurements and then minimize it over all allowed state paths. In this paper, we construct a deterministic filter on the circle by minimizing the least square error based on Pontryagin's minimum principle, where the optimal state trajectory is described by a bilinear differential equation with deterministic optimal control input. The effectiveness of the proposed filter is shown through numerical experiments.