Scalable Estimation of Invariant Sets for Mixed-Integer Nonlinear Systems using Active Deep Learning

Set invariance is a crucial property for ensuring safe and feasible performance of closed-loop systems under state and input constraints. Classical set-theoretic methods for constructing reachable and invariant sets are generally inadequate in handling complex system dynamics and may not be scalable to high-dimensional systems. This paper presents a sample-efficient approach for data-driven estimation of invariant sets for constrained nonlinear systems that can exhibit a mixture of continuous, discrete, and/or switching-mode behavior. The approach relies on learning an oracle that verifies if a given system state is feasible. Thus, the invariant set construction problem is converted to a classification problem that can be effectively solved with deep learning. We also present an active learning algorithm to improve the sample efficiency of deep learning-based estimation of the feasibility oracle. Randomized verification is then used to provide probabilistic guarantees for set invariance. The proposed approach does not impose any assumptions on the structure of system dynamics, and is particularly suitable when the feasibility test for control invariance requires solving (expensive) mixed-integer nonlinear programs. The approach is illustrated on a benchmark problem.