Predictive control for nonlinear stochastic systems: Closed-loop guarantees with unbounded noise

We present a stochastic model predictive control framework for nonlinear systems subject to unbounded process noise with closed-loop guarantees. First, we provide a conceptual shrinking-horizon framework that utilizes general probabilistic reachable sets and minimizes the expected cost. Then, we provide a tractable receding-horizon formulation that uses a nominal state to minimize a deterministic quadratic cost and satisfy tightened constraints. Our theoretical analysis demonstrates recursive feasibility, satisfaction of chance constraints, and bounds on the expected cost for the resulting closed-loop system. We provide a constructive design for probabilistic reachable sets of nonlinear continuously differentiable systems using stochastic contraction metrics and an assumed bound on the covariance matrices. Numerical simulations highlight the computational efficiency and theoretical guarantees of the proposed method. Overall, this paper provides a framework for computationally tractable stochastic predictive control with closed-loop guarantees for nonlinear systems with unbounded noise.