When first principle models cannot be derived due to the complexity of the real system, data-driven methods allow us to build models from system observations. As these models are employed in learning-based control, the quality of the data plays a crucial role for the performance of the resulting control law. Nevertheless, there hardly exist measures for assessing training data sets, and the impact of the distribution of the data on the closed-loop system properties is largely unknown. This paper derives - based on Gaussian process models - an analytical relationship between the density of the training data and the control performance. We formulate a quality measure for the data set, which we refer to as $\rho$-gap, and derive the ultimate bound for the tracking error under consideration of the model uncertainty. We show how the $\rho$-gap can be applied to a feedback linearizing control law and provide numerical illustrations for our approach.

中文翻译：

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训练数据如何影响基于学习的控制中的性能

当由于真实系统的复杂性而无法推导出第一原理模型时，数据驱动的方法允许我们从系统观察中构建模型。由于这些模型用于基于学习的控制，因此数据质量对于最终控制律的性能起着至关重要的作用。然而，几乎没有评估训练数据集的措施，并且数据分布对闭环系统特性的影响在很大程度上是未知的。本文基于高斯过程模型推导出训练数据密度与控制性能之间的分析关系。我们为数据集制定了一个质量度量，我们将其称为 $\rho$-gap，并在考虑模型不确定性的情况下推导出跟踪误差的最终界限。