# Data-driven computation of invariant sets of discrete time-invariant black-box systems

## TL;DR

This paper introduces a data-driven method to compute almost-invariant sets for black-box nonlinear systems using trajectory observations, scenario optimization, and probabilistic guarantees, without requiring explicit models.

## Contribution

It presents a novel framework combining data-driven set identification with probabilistic invariance guarantees for black-box systems.

## Key findings

- Successfully computes almost-invariant sets with probabilistic guarantees.
- Provides explicit inner and outer set approximations.
- Demonstrates effectiveness through numerical examples.

## Abstract

We consider the problem of computing the maximal invariant set of discrete-time black-box nonlinear systems without analytic dynamical models. Under the assumption that the system is asymptotically stable, the maximal invariant set coincides with the domain of attraction. A data-driven framework relying on the observation of trajectories is proposed to compute almost-invariant sets, which are invariant almost everywhere except a small subset. Based on these observations, scenario optimization problems are formulated and solved. We show that probabilistic invariance guarantees on the almost-invariant sets can be established. To get explicit expressions of such sets, a set identification procedure is designed with a verification step that provides inner and outer approximations in a probabilistic sense. The proposed data-driven framework is illustrated by several numerical examples.