Multi‐Step probabilistic sets in model predictive control for stochastic systems with multiplicative uncertainty

This study designs a model predictive controller for linear, discrete‐time, stochastic systems with multiplicative noise and probabilistic constraints. The probabilistic invariance has shown its advantage in characterising the stochastic dynamics of the controlled state. Here multi‐step probabilistic sets strengthen probabilistic invariance to further satisfy infinite‐horizon probabilistic constraints. In addition, multi‐step probabilistic sets offer some degrees of freedom to enlarge the feasible region ensured by probabilistic invariance. The controller satisfies given constraints and guarantees closed‐loop mean‐square stability. Moreover, a simplified controller with lower on‐line computational burden is presented. Numerical examples show the performance of the proposed approach.