  Safe control of constrained linear systems under both epistemic and aleatory
uncertainties is considered. The aleatory uncertainty characterizes random
noises and is modeled by a probability distribution function (PDF) and the
epistemic uncertainty characterizes the lack of knowledge on the system
dynamics. Data-based probabilistic safe controllers are designed for the cases
where the noise PDF is 1) zero-mean Gaussian with a known covariance, 2)
zero-mean Gaussian with an uncertain covariance, and 3) zero-mean non-Gaussian
with an unknown distribution. Easy-to-check model-based conditions for
guaranteeing probabilistic safety are provided for the first case by
introducing probabilistic contractive sets. These results are then extended to
the second and third cases by leveraging distributionally-robust probabilistic
safe control and conditional value-at-risk (CVaR) based probabilistic safe
control, respectively. Data-based implementations of these probabilistic safe
controllers are then considered. It is shown that data-richness requirements
for directly learning a safe controller is considerably weaker than
data-richness requirements for model-based safe control approaches that
undertake a model identification. Moreover, an upper bound on the minimal risk
level, under which the existence of a safe controller is guaranteed, is learned
using collected data. A simulation example is provided to show the
effectiveness of the proposed approach.
