  In this paper, we address the problem of designing stochastic model
predictive control (MPC) schemes for linear systems affected by unbounded
disturbances. The contribution of the paper is twofold. First, motivated by the
difficulty of guaranteeing recursive feasibility in this framework, due to the
nonzero probability of violating chance-constraints in the case of unbounded
noise, we introduce the novel definition of measured-state conditioned
recursive feasibility in expectation. Second, we construct a stochastic MPC
scheme, based on the introduction of ellipsoidal probabilistic reachable sets,
which implements a closed-loop initialization strategy, i.e., the current
measured-state is employed for initializing the optimization problem. This new
scheme is proven to satisfy the novel definition of recursive feasibility, and
its superiority with respect to open-loop initialization schemes, arising from
the fact that one never neglects the information brought by the current
measurement, is shown through numerical examples.
