Gaussian Process-Based Predictive Control for Periodic Error Correction

Many controlled systems suffer from unmodeled nonlinear effects that recur periodically over time. Model-free controllers generally cannot compensate these effects, and good physical models for such periodic dynamics are challenging to construct. We investigate nonparametric system identification for periodically recurring nonlinear effects. Within a Gaussian process (GP) regression framework, we use a locally periodic covariance function to shape the hypothesis space, which allows for a structured extrapolation that is not possible with more widely used covariance functions. We show that hyperparameter estimation can be performed online using the maximum a posteriori point estimate, which provides an accuracy comparable with sampling methods as soon as enough data to cover the periodic structure has been collected. It is also shown how the periodic structure can be exploited in the hyperparameter optimization. The predictions obtained from the GP model are then used in a model predictive control framework to correct the external effect. The availability of good continuous predictions allows control at a higher rate than that of the measurements. We show that the proposed approach is particularly beneficial for sampling times that are smaller than, but of the same order of magnitude as, the period length of the external effect. In experiments on a physical system, an electrically actuated telescope mount, this approach achieves a reduction of about 20% in root mean square tracking error.