Cascaded synchronization of two pendulums

We consider two pendulums in open loop perturbed by sinusoidals where it is assumed that one can tune the oscillating frequency of one of the two systems. The synchronization goal is then to make the first pendulum to oscillate at the frequency of the second. The controller is a simple linear proportional feedback of the frequency error. The stability proof is based on the key observation that with this linear dynamic extension we can rearrange the “error” dynamics in a cascade structure. Then we invoke some results on stability analysis of cascade non-autonomous nonlinear systems in order to prove global uniform asymptotic stability