Data-Driven Abstractions for Control Systems via Random Exploration

Abstract: At the intersection of dynamical systems, control theory, and formal methods lies the construction of symbolic abstractions: these typically represent simpler, finite-state models whose behavior mimics that of an underlying concrete system but are easier to analyse. Building an abstraction usually requires an accurate knowledge of the underlying model: this knowledge may be costly to gather, especially in real-life applications. We aim to bridge this gap by building abstractions based on sampling finite length trajectories. To refine a controller built for the abstraction to one for the concrete system, we newly define a notion of probabilistic alternating simulation, and provide Probably Approximately Correct (PAC) guarantees that the constructed abstraction includes all behaviors of the concrete system and that it is suitable for control design, for arbitrarily long time horizons, leveraging scenario theory. Our method is then tested on several numerical benchmarks.