Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processes

TL;DR

This paper introduces a safe learning method for estimating the region of attraction in uncertain nonlinear systems using Gaussian processes, ensuring safety during exploration and improving stability guarantees.

Contribution

It presents a novel approach combining Gaussian process models with Lyapunov functions to safely learn and expand the ROA without risking unsafe states.

Findings

Successfully estimates ROA with high probability

Safely explores state space to expand ROA estimates

Validated effectiveness through simulation experiments

Abstract

Control theory can provide useful insights into the properties of controlled, dynamic systems. One important property of nonlinear systems is the region of attraction (ROA), a safe subset of the state space in which a given controller renders an equilibrium point asymptotically stable. The ROA is typically estimated based on a model of the system. However, since models are only an approximation of the real world, the resulting estimated safe region can contain states outside the ROA of the real system. This is not acceptable in safety-critical applications. In this paper, we consider an approach that learns the ROA from experiments on a real system, without ever leaving the true ROA and, thus, without risking safety-critical failures. Based on regularity assumptions on the model errors in terms of a Gaussian process prior, we use an underlying Lyapunov function in order to determine a region in which an equilibrium point is asymptotically stable with high probability. Moreover, we provide an algorithm to actively and safely explore the state space in order to expand the ROA estimate. We demonstrate the effectiveness of this method in simulation.