  We propose a stochastic MPC scheme using an optimization over the initial
state for the predicted trajectory. Considering linear discrete-time systems
under unbounded additive stochastic disturbances subject to chance constraints,
we use constraint tightening based on probabilistic reachable sets to design
the MPC. The scheme avoids the infeasibility issues arising from unbounded
disturbances by including the initial state as a decision variable. We show
that the stabilizing control scheme can guarantee constraint satisfaction in
closed loop, assuming unimodal disturbances. In addition to illustrating these
guarantees, the numerical example indicates further advantages of optimizing
over the initial state for the transient behavior.
