Time-Average Constraints in Stochastic Model Predictive Control

This paper presents two alternatives to using chance constraints in stochastic MPC, motivated by the observation that many stochastic constrained control algorithms aim to impose a bound on the time-average of constraint violations. We consider imposing a robust constraint on the time-average of constraint violations over a finite period. By allowing the controller to respond to the effects of past violations, two algorithms are presented that solve this problem, both requiring a single convex optimization after a preprocessing step. Stochastic MPC formulations that 'remember' previous violations and react accordingly were given previously in [1], [2], but in those works the focus was on asymptotic guarantees on the average number of violations. In contrast we give stronger robust bounds on the violation permissible in any time period of a specified length. The method is also applied to a bound on the sum of convex loss functions of the amount of constraint violation, thus allowing controllers to place greater importance on avoiding large violations.