Analog Data-Driven Theory and Estimation of the Region of Attraction Using Sampled-Data

TL;DR

This paper introduces a data-driven method for estimating the region of attraction of nonlinear systems using sampled data, involving convex optimization and geometric flow algorithms, validated through simulations.

Contribution

It formulates a convex optimization approach for linearizing nonlinear vector fields from trajectory data and proposes a geometric flow algorithm to accurately estimate the region of attraction boundary.

Findings

Convex cost function guarantees stability of the solution.

Algorithm converges to the exact boundary under certain conditions.

Validated through simulations on nonlinear autonomous systems.

Abstract

The contributions of this technical note are twofold. Firstly, we formulate an optimization problem to obtain a linear representation of a nonlinear vector field based on a system's trajectory. We also prove that its cost function is strictly convex, given the trajectory is persistently exciting. Under certain observability conditions, we provide results that guarantee the Hurwitz stability of the global minimizer. Secondly, we present a novel algorithm based on point-wise geometric flows to estimate the boundary of the region of attraction. We show that the algorithm converges to the exact boundary of the region of attraction under certain assumptions on the system dynamics. Finally, we validate the results using simulations on various nonlinear autonomous systems.