Scenario-based MPC with gradual relaxation of output constraints

Handling of uncertainty in Model Predictive Control (MPC) has received increasing attention the last decade. The robust open-loop approach often leads to overly conservative solutions, because constraints have to be satisfied for all possible realizations of the stochastic variables, over the entire prediction horizon. In this paper, we present a novel scenario-based approach, where the constraints are gradually relaxed over the prediction horizon. The concept of Conditional Value at Risk (CVaR) is employed for the relaxation, resulting in a computational tractable non-conservative open-loop problem formulation. The formulation is illustrated with a simple numerical example, and compared to a more traditional robust approach.