Data-driven certificates of constraint enforcement and stability for unmodeled, discrete dynamical systems using tree data structures

TL;DR

This paper introduces a data-driven method for certifying stability and safety of unmodeled dynamical systems using tree data structures, Lipschitz bounds, and invariant set synthesis without prior system knowledge.

Contribution

It proposes a novel approach that prunes constraint sets to synthesize invariant sets and guarantees stability via a piecewise affine Lyapunov function, eliminating the need for an initial invariant set.

Findings

Successfully synthesizes invariant sets for unmodeled systems.

Provides stability guarantees through Lyapunov functions.

Requires only Lipschitz continuity, no prior system knowledge.

Abstract

This paper addresses the critical challenge of developing data-driven certificates for the stability and safety of unmodeled dynamical systems by leveraging a tree data structure and an upper bound of the system's Lipschitz constant. Previously, an invariant set was synthesized by iteratively expanding an initial invariant set. In contrast, this work iteratively prunes the constraint set to synthesize an invariant set -- eliminating the need for a known, initial invariant set. Furthermore, we provide stability assurances by characterizing the asymptotic stability of the system relative to an invariant approximation of the minimal positive invariant set through synthesis of a discontinuous piecewise affine Lyapunov function over the computed invariant set. The proposed method takes inspiration from subdivision techniques and requires no prior system knowledge beyond Lipschitz continuity.