  Modern control systems must operate in increasingly complex environments
subject to safety constraints and input limits, and are often implemented in a
hierarchical fashion with different controllers running at multiple time
scales. Yet traditional constructive methods for nonlinear controller synthesis
typically "flatten" this hierarchy, focusing on a single time scale, and
thereby limited the ability to make rigorous guarantees on constraint
satisfaction that hold for the entire system. In this work we seek to address
the stabilization of constrained nonlinear systems through a
\textit{multi-rate} control architecture. This is accomplished by iteratively
planning continuous reference trajectories for a nonlinear system using a
linearized model and Model Predictive Control (MPC), and tracking said
trajectories using the full-order nonlinear model and Control Lyapunov
Functions (CLFs). Connecting these two levels of control design in a way that
ensures constraint satisfaction is achieved through the use of
\textit{B\'{e}zier curves}, which enable planning continuous trajectories
respecting constraints by planning a sequence of discrete points. Our framework
is encoded via convex optimization problems which may be efficiently solved, as
demonstrated in simulation.
