Nonparametric dynamics estimation for time periodic systems

Screws and gears are a source of periodically recurring nonlinear effects in mechanical dynamical systems. Unless the state sampling frequency is much higher than the periodic effect, model-free controllers cannot always compensate these effects, and good physical models for such periodic dynamics are challenging to construct. We investigate nonparametric system identification with an explicit focus on periodically recurring nonlinear effects. Within a Gaussian process regression framework, we design 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. These predictions are then used in model predictive control to construct a control signal correcting for the predicted external effect. We show that this approach is beneficial for state sampling times that are smaller than, but comparable to, 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 error.