Episodic Gaussian Process-Based Learning Control with Vanishing Tracking Errors

Due to the increasing complexity of technical systems, accurate first principle models can often not be obtained. Supervised machine learning can mitigate this issue by inferring models from measurement data. Gaussian process regression is particularly well suited for this purpose due to its high data-efficiency and its explicit uncertainty representation, which allows the derivation of prediction error bounds. These error bounds have been exploited to show tracking accuracy guarantees for a variety of control approaches, but their direct dependency on the training data is generally unclear. We address this issue by deriving a Bayesian prediction error bound for GP regression, which we show to decay with the growth of a novel, kernel-based measure of data density. Based on the prediction error bound, we prove time-varying tracking accuracy guarantees for learned GP models used as feedback compensation of unknown nonlinearities, and show to achieve vanishing tracking error with increasing data density. This enables us to develop an episodic approach for learning Gaussian process models, such that an arbitrary tracking accuracy can be guaranteed. The effectiveness of the derived theory is demonstrated in several simulations.