  In this paper we propose a stochastic model predictive control (MPC)
algorithm for linear discrete-time systems affected by possibly unbounded
additive disturbances and subject to probabilistic constraints. Constraints are
treated in analogy to robust MPC using a constraint tightening based on the
concept of probabilistic reachable sets, which is shown to provide closed-loop
fulfillment of chance constraints under a unimodality assumption on the
disturbance distribution. A control scheme reverting to a backup solution from
a previous time step in case of infeasibility is proposed, for which an
asymptotic average performance bound is derived. Two examples illustrate the
approach, highlighting closed-loop chance constraint satisfaction and the
benefits of the proposed controller in the presence of unmodeled disturbances.
