  Safety for control systems is often posed as an invariance constraint; the
system is said to be safe if state trajectories avoid some unsafe region of the
statespace for all time. An assured controller is one that enforces safety
online by filtering a desired control input at runtime, and control barrier
functions (CBFs) provide an assured controller that renders a safe subset of
the state-space forward invariant. Recent extensions propose CBF-based assured
controllers that allow the system to leave a known safe set so long as a given
backup control strategy eventually returns to the safe set, however, these
methods have yet to be extended to consider systems subjected to unknown
disturbance inputs.
  In this work, we present a problem formulation for CBF-based runtime
assurance for systems with disturbances, and controllers which solve this
problem must, in some way, incorporate the online computation of reachable
sets. In general, computing reachable sets in the presence of disturbances is
computationally costly and cannot be directly incorporated in a CBF framework.
To that end, we present a particular solution to the problem, whereby reachable
sets are approximated via the mixed-monotonicity property. Efficient algorithms
exist for overapproximating reachable sets for mixed-monotone systems with
hyperrectangles, and we show that such approximations are suitable for
incorporating into a CBF-based runtime assurance framework.
