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.

翻译：由于技术系统日益复杂，往往难以获得精确的机理模型。监督式机器学习可通过从测量数据中推断模型来缓解这一问题。高斯过程回归凭借其高数据效率及明确的非确定性表示（可推导预测误差界）而特别适用于此。这类误差界已被用于证明多种控制方法的跟踪精度保证，但其对训练数据的直接依赖关系通常尚不明确。我们通过推导一种适用于高斯过程回归的贝叶斯预测误差界来解决该问题，证明该误差界会随着一种基于核的新型数据密度度量值的增长而衰减。基于该预测误差界，我们证明了当使用学习到的高斯过程模型作为未知非线性的反馈补偿时，跟踪精度具有时变保证，并表明随着数据密度增加可实现跟踪误差趋零。这使我们能够发展一种用于学习高斯过程模型的情景式方法，从而保证任意精度的跟踪。通过多项仿真验证了所推导理论的有效性。