  In this paper, we propose a novel approach for computing robust backward
reachable sets from noisy data for unknown constrained linear systems subject
to bounded disturbances. In particular, we develop an algorithm for obtaining
zonotopic inner approximations that can be used for control purposes. It is
shown that such sets, if built on an extended space including states and
inputs, can be used to embed the system's one-step evolution in the computed
extended regions. Such a result is then exploited to build a set-theoretic
model predictive controller that, offline, builds a recursive family of robust
data-driven reachable sets, and online, computes recursively admissible control
actions without explicitly resorting to either a model of the system or the
available data.The validity of the proposed data-driven solution is verified by
means of a numerical simulation and its performance is contrasted with the
model-based counterpart.
