Adaptively constrained Stochastic Model Predictive Control for closed-loop constraint satisfaction

Stochastic Model Predictive Control (SMPC) for discrete-time linear systems subject to additive disturbances with chance constraints on the states and hard constraints on the inputs is considered. Current chance constrained MPC methods-based on analytic reformulations or on sampling approaches-tend to be conservative partly because they fail to exploit the predefined violation level in closed-loop. For many practical applications, this conservatism can lead to a loss in performance. We propose an adaptive SMPC scheme that starts with a standard conservative chance constrained formulation and then on-line adapts the formulation of constraints based on the experienced violation frequency. Using martingale theory we establish guarantees of convergence to the desired level of constraint violation in closed-loop for a special class of linear systems. Comments are given on how to extend this to a broader class of (non-)linear systems. The developed methodology is demonstrated with an illustrative example.