  A learning-based safety filter is developed for discrete-time linear
time-invariant systems with unknown models subject to Gaussian noises with
unknown covariance. Safety is characterized using polytopic constraints on the
states and control inputs. The empirically learned model and process noise
covariance with their confidence bounds are used to construct a robust
optimization problem for minimally modifying nominal control actions to ensure
safety with high probability. The optimization problem relies on tightening the
original safety constraints. The magnitude of the tightening is larger at the
beginning since there is little information to construct reliable models, but
shrinks with time as more data becomes available.
