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

TL;DR

This paper introduces a novel Gaussian Process-based control method for stabilizing control-affine systems with uncertain dynamics, providing probabilistic error bounds and convex optimization formulation for improved robustness.

Contribution

It develops a new compound kernel for GP regression capturing control-affine structure and formulates a convex stability optimization incorporating probabilistic error bounds.

Findings

Successfully stabilizes inverted pendulum and bicycle models.

Achieves trajectories similar to true dynamics with uncertainty.

Provides probabilistic guarantees for stability.

Abstract

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. In order to estimate both state and input-dependent model uncertainty, we propose a novel compound kernel that captures the control-affine nature of the problem. Furthermore, by the use of GP Upper Confidence Bound analysis, we provide probabilistic bounds of the regression error, leading to 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.