Model-free Learning of Regions of Attraction via Recurrent Sets

TL;DR

This paper introduces a model-free method to approximate the region of attraction of a stable equilibrium by learning recurrent sets from sampled trajectories, without relying on explicit system models.

Contribution

It proposes a novel approach to estimate the ROA using recurrence properties and develops algorithms with convergence guarantees that operate sequentially from trajectory samples.

Findings

Algorithms effectively approximate the ROA using recurrence-based sets.

Provides theoretical guarantees and bounds on the number of samples needed.

Demonstrates the method's applicability without explicit system models.

Abstract

We consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. Rather than leveraging approximate models with bounded uncertainty to find a (robust) invariant set contained in the ROA, we propose to learn sets that satisfy a more relaxed notion of containment known as recurrence. We define a set to be $\tau$-recurrent (resp. $k$-recurrent) if every trajectory that starts within the set, returns to it after at most $\tau$ seconds (resp. $k$ steps). We show that under mild assumptions a $\tau$-recurrent set containing a stable equilibrium must be a subset of its ROA. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allow them to continue being executed even after an initial offline training stage. We further provide an upper bound on the number of counter-examples used by the algorithm, and almost sure convergence guarantees.