Gaussian Process-based Min-norm Stabilizing Controller for Control-Affine Systems with Uncertain Input Effects

This paper presents a method to design a min-norm Control Lyapunov Function (CLF)-based stabilizing controller for a control-affine system with uncertain dynamics using Gaussian Process (GP) regression. We propose a novel compound kernel that captures the control-affine nature of the problem, which permits the estimation of both state and input-dependent model uncertainty in a single GP regression problem. Furthermore, we provide probabilistic guarantees of convergence by the use of GP Upper Confidence Bound analysis and the formulation of a CLF-based stability chance constraint which can be incorporated in a min-norm optimization problem. We show that this resulting optimization problem is convex, and we call it Gaussian Process-based Control Lyapunov Function Second-Order Cone Program (GP-CLF-SOCP). The data-collection process and the training of the GP regression model are carried out in an episodic learning fashion. We validate the proposed algorithm and controller in numerical simulations of an inverted pendulum and a kinematic bicycle model, resulting in stable trajectories which are very similar to the ones obtained if we actually knew the true plant dynamics.