  In recent years, the increasing interest in Stochastic model predictive
control (SMPC) schemes has highlighted the limitation arising from their
inherent computational demand, which has restricted their applicability to
slow-dynamics and high-performing systems. To reduce the computational burden,
in this paper we extend the probabilistic scaling approach to obtain
low-complexity inner approximation of chance-constrained sets. This approach
provides probabilistic guarantees at a lower computational cost than other
schemes for which the sample complexity depends on the design space dimension.
To design candidate simple approximating sets, which approximate the shape of
the probabilistic set, we introduce two possibilities: i) fixed-complexity
polytopes, and ii) $\ell_p$-norm based sets. Once the candidate approximating
set is obtained, it is scaled around its center so to enforce the expected
probabilistic guarantees. The resulting scaled set is then exploited to enforce
constraints in the classical SMPC framework. The computational gain obtained
with the proposed approach with respect to the scenario one is demonstrated via
simulations, where the objective is the control of a fixed-wing UAV performing
a monitoring mission over a sloped vineyard.
