  This paper discusses a novel probabilistic approach for the design of robust
model predictive control (MPC) laws for discrete-time linear systems affected
by parametric uncertainty and additive disturbances. The proposed technique is
based on the iterated solution, at each step, of a finite-horizon optimal
control problem (FHOCP) that takes into account a suitable number of randomly
extracted scenarios of uncertainty and disturbances, followed by a specific
command selection rule implemented in a receding horizon fashion. The scenario
FHOCP is always convex, also when the uncertain parameters and disturbance
belong to non-convex sets, and irrespective of how the model uncertainty
influences the system's matrices. Moreover, the computational complexity of the
proposed approach does not depend on the uncertainty/disturbance dimensions,
and scales quadratically with the control horizon. The main result in this
paper is related to the analysis of the closed loop system under
receding-horizon implementation of the scenario FHOCP, and essentially states
that the devised control law guarantees constraint satisfaction at each step
with some a-priori assigned probability p, while the system's state reaches the
target set either asymptotically, or in finite time with probability at least
p. The proposed method may be a valid alternative when other existing
techniques, either deterministic or stochastic, are not directly usable due to
excessive conservatism or to numerical intractability caused by lack of
convexity of the robust or chance-constrained optimization problem.
