  In this paper, we introduce the notion of periodic safety, which requires
that the system trajectories periodically visit a subset of a forward-invariant
safe set, and utilize it in a multi-rate framework where a high-level planner
generates a reference trajectory that is tracked by a low-level controller
under input constraints. We introduce the notion of fixed-time barrier
functions which is leveraged by the proposed low-level controller in a
quadratic programming framework. Then, we design a model predictive control
policy for high-level planning with a bound on the rate of change for the
reference trajectory to guarantee that periodic safety is achieved. We
demonstrate the effectiveness of the proposed strategy on a simulation example,
where the proposed fixed-time stabilizing low-level controller shows successful
satisfaction of control objectives, whereas an exponentially stabilizing
low-level controller fails.
