Data-Driven Estimation of Backward Reachable and Invariant Sets for Unmodeled Systems via Active Learning

Ensuring control performance with state and input constraints is facilitated by the understanding of reachable and invariant sets. While exploiting dynamical models have provided many set-based algorithms for constructing these sets, set-based methods typically do not scale well, or rely heavily on model accuracy or structure. In contrast, it is relatively simple to generate state trajectories in a data-driven manner by numerically simulating complex systems from initial conditions sampled from within an admissible state space, even if the underlying dynamics are completely unknown. These samples can then be leveraged for reachable/invariant set estimation via machine learning, although the learning performance is strongly linked to the sampling pattern. In this paper, active learning is employed to intelligently select batches of samples that are most informative and least redundant to previously labeled samples via submodular maximization. Selective sampling reduces the number of numerical simulations required for constructing the invariant set estimator, thereby enhancing scalability to higher-dimensional state spaces. The potential of the proposed framework is illustrated via a numerical example.