Fixed-Domain Posterior Contraction Rates for Spatial Gaussian Process Model with Nugget

Abstract

A combination of control Lyapunov functions (CLFs) and control barrier functions (CBFs) forms an efficient framework for addressing control challenges in safe stabilization. In our previous research, we developed an analytical control strategy, namely the universal formula, that incorporates CLF and CBF conditions for safe stabilization. However, successful implementation of this universal formula relies on an accurate model, as any mismatch between the model and the actual system can compromise stability and safety. In this paper, we propose a new universal formula that leverages Gaussian processes (GPs) learning to address safe stabilization in the presence of model uncertainty. By utilizing the results related to bounded learning errors, we achieve a high probability of stability and safety guarantees with the proposed universal formula. Additionally, we introduce a probabilistic compatibility condition to evaluate conflicts between the modified CLF and CBF conditions with GP learning results. In cases where compatibility assumptions fail and control system limits are present, we propose a modified universal formula that relaxes stability constraints and a projection-based method accommodating control limits. We illustrate the effectiveness of our approach through a simulation of adaptive cruise control (ACC), highlighting its potential for practical applications in real-world scenarios.