Learning Region of Attraction for Nonlinear Systems

TL;DR

This paper introduces a neural network-based, counterexample-guided approach to estimate the region of attraction for nonlinear systems, providing guarantees under certain approximation conditions.

Contribution

It presents a novel method combining neural network approximation and counterexample-guided search to reliably estimate the ROA of nonlinear systems.

Findings

Method guarantees to find a robust Lyapunov function if it exists.

Uses Mixed-Integer Quadratic Programming to generate counterexamples.

Demonstrates effectiveness through numerical examples.

Abstract

Estimating the region of attraction (ROA) of general nonlinear autonomous systems remains a challenging problem and requires a case-by-case analysis. Leveraging the universal approximation property of neural networks, in this paper, we propose a counterexample-guided method to estimate the ROA of general nonlinear dynamical systems provided that they can be approximated by piecewise linear neural networks and that the approximation error can be bounded. Specifically, our method searches for robust Lyapunov functions using counterexamples, i.e., the states at which the Lyapunov conditions fail. We generate the counterexamples using Mixed-Integer Quadratic Programming. Our method is guaranteed to find a robust Lyapunov function in the parameterized function class, if exists, after collecting a finite number of counterexamples. We illustrate our method through numerical examples.