  In this paper, we consider a stochastic Model Predictive Control able to
account for effects of additive stochastic disturbance with unbounded support,
and requiring no restrictive assumption on either independence nor Gaussianity.
We revisit the rather classical approach based on penalty functions, with the
aim of designing a control scheme that meets some given probabilistic
specifications. The main difference with previous approaches is that we do not
recur to the notion of probabilistic recursive feasibility, and hence we do not
consider separately the unfeasible case. In particular, two probabilistic
design problems are envisioned. The first randomization problem aims to design
\textit{offline} the constraint set tightening, following an approach inherited
from tube-based MPC. For the second probabilistic scheme, a specific
probabilistic validation approach is exploited for tuning the penalty
parameter, to be selected \textit{offline} among a finite-family of possible
values. The simple algorithm here proposed allows designing a \textit{single}
controller, always guaranteeing feasibility of the online optimization problem.
The proposed method is shown to be more computationally tractable than previous
schemes. This is due to the fact that the sample complexity for both
probabilistic design problems depends on the prediction horizon in a
logarithmic way, unlike scenario-based approaches which exhibit linear
dependence. The efficacy of the proposed approach is demonstrated with a
numerical example.
