Data driven synthesis of provable invariant sets via stochastically sampled data

TL;DR

This paper presents a data-driven method for synthesizing positive invariant sets for dynamical systems using stochastically sampled data, leveraging Lipschitz continuity and probabilistic bounds to ensure safety constraints.

Contribution

It introduces a novel approach that synthesizes PI sets from arbitrary sampled data without deterministic sampling, using a tree structure and Lipschitz continuity for guarantees.

Findings

Successfully synthesizes PI sets from stochastic data samples.

Provides probabilistic bounds on sample complexity.

Ensures safety constraints with minimal system information.

Abstract

Positive invariant (PI) sets are essential for ensuring safety, i.e. constraint adherence, of dynamical systems. With the increasing availability of sampled data from complex (and often unmodeled) systems, it is advantageous to leverage these data sets for PI set synthesis. This paper uses data driven geometric conditions of invariance to synthesize PI sets from data. Where previous data driven, set-based approaches to PI set synthesis used deterministic sampling schemes, this work instead synthesizes PI sets from any pre-collected data sets. Beyond a data set and Lipschitz continuity, no additional information about the system is needed. A tree data structure is used to partition the space and select samples used to construct the PI set, while Lipschitz continuity is used to provide deterministic guarantees of invariance. Finally, probabilistic bounds are given on the number of samples needed for the algorithm to determine of a certain volume.