  Providing safety guarantees for learning-based controllers is important for
real-world applications. One approach to realizing safety for arbitrary control
policies is safety filtering. If necessary, the filter modifies control inputs
to ensure that the trajectories of a closed-loop system stay within a given
state constraint set for all future time, referred to as the set being positive
invariant or the system being safe. Under the assumption of fully known
dynamics, safety can be certified using control barrier functions (CBFs).
However, the dynamics model is often either unknown or only partially known in
practice. Learning-based methods have been proposed to approximate the CBF
condition for unknown or uncertain systems from data; however, these techniques
do not account for input constraints and, as a result, may not yield a valid
CBF condition to render the safe set invariant. In this work, we study
conditions that guarantee control invariance of the system under input
constraints and propose an optimization problem to reduce the conservativeness
of CBF-based safety filters. Building on these theoretical insights, we further
develop a probabilistic learning approach that allows us to build a safety
filter that guarantees safety for uncertain, input-constrained systems with
high probability. We demonstrate the efficacy of our proposed approach in
simulation and real-world experiments on a quadrotor and show that we can
achieve safe closed-loop behavior for a learned system while satisfying state
and input constraints.
