Probabilistic Data-Driven Invariance for Constrained Control of Nonlinear Systems

We present a novel direct data-driven method for computing constraint-admissible positive invariant sets for general nonlinear systems with compact constraint sets. Our approach employs machine learning techniques to lift the state space and approximate invariant sets using finite data. The invariant sets are parameterized as sub-level-sets of scalar linear functions in the lifted space, which is suitable for control applications. We provide probabilistic guarantees of invariance through scenario optimization, with probability bounds on robustness against the uncertainty inherent in the data-driven framework. As the amount of data increases, these probability bounds approach 1. We use our invariant sets to switch between a collection of controllers to select a controller which enforces constraints. We demonstrate the practicality of our method by applying it to a nonlinear autonomous driving lane-keeping scenario.