Nonlinear Set Membership Regression with Adaptive Hyper-Parameter Estimation for Online Learning and Control

Jan-Peter Calliess, Stephen J. Roberts, Carl E. Rasmussen, Jan M. Maciejowski

Abstract: Methods known as Lipschitz Interpolation or Nonlinear Set Membership regression have become established tools for nonparametric system-identification and data-based control. They utilise presupposed Lipschitz properties to compute inferences over unobserved function values. Unfortunately, they rely on the a priori knowledge of a Lipschitz constant of the underlying target function which serves as a hyper-parameter. We propose a closed-form estimator of the Lipschitz constant that is robust to bounded observational noise in the data. The merger of Lipschitz Interpolation with the new hyper-parameter estimator gives a new nonparametric machine learning method for which we derive online learning convergence guarantees. Furthermore, we apply our learning method to model-reference adaptive control and provide a convergence guarantee on the closed-loop dynamics. In a simulated flight manoeuvre control scenario, we compare the performance of our approach to recently proposed alternative learning-based controllers.