Model-free Learning of Regions of Attraction via Recurrent Sets
This addresses the challenge of model-free control for stability analysis, but it is incremental as it builds on existing notions of recurrence.
The paper tackles the problem of approximating the region of attraction for stable equilibrium points without a model, by learning sets that satisfy recurrence conditions, and shows that these sets are subsets of the region of attraction with convergence guarantees.
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 $τ$-recurrent (resp. $k$-recurrent) if every trajectory that starts within the set, returns to it after at most $τ$ seconds (resp. $k$ steps). We show that under mild assumptions a $τ$-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.