  In this paper we present a multi-rate control architecture for safety
critical systems. We consider a high level planner and a low level controller
which operate at different frequencies. This multi-rate behavior is described
by a piecewise nonlinear model which evolves on a continuous and a discrete
level. First, we present sufficient conditions which guarantee recursive
constraint satisfaction for the closed-loop system. Afterwards, we propose a
control design methodology which leverages Control Barrier Functions (CBFs) for
low level control and Model Predictive Control (MPC) policies for high level
planning. The control barrier function is designed using the full nonlinear
dynamical model and the MPC is based on a simplified planning model. When the
nonlinear system is control affine and the high level planning model is linear,
the control actions are computed by solving convex optimization problems at
each level of the hierarchy. Finally, we show the effectiveness of the proposed
strategy on a simulation example, where the low level control action is updated
at a higher frequency than the high level command.
