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.
Spatial Gaussian process regression models typically contain finite dimensional covariance parameters that need to be estimated from the data. We study the Bayesian estimation of covariance parameters including the nugget parameter in a general class of stationary covariance functions under fixed-domain asymptotics, which is theoretically challenging due to the increasingly strong dependence among spatial observations. We propose a novel adaptation of the Schwartz's consistency theorem for showing posterior contraction rates of the covariance parameters including the nugget. We derive a new polynomial evidence lower bound, and propose new consistent higher-order quadratic variation estimators that satisfy concentration inequalities with exponentially small tails. Our Bayesian fixed-domain asymptotics theory leads to explicit posterior contraction rates for the microergodic and nugget parameters in the isotropic Matérn covariance function under a general stratified sampling design. We verify our theory and the Bayesian predictive performance in simulation studies and an application to sea surface temperature data.
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