Synchronization of two inverted pendula via dynamic coupling

This work presents a controlled synchronization strategy for a pair of underactuated inverted pendula. The ultimate aim is to synchronize the pendula to a stable periodic orbit. This goal is achieved in two steps. First, a suitable control input is designed such that each inverted pendulum converges to a stable limit cycle, i.e. each pendulum is converted into a self-sustained oscillatory system. Then, once the system exhibits the desired oscillatory motion, a dynamic coupling is constructed in order to synchronize the pendula. In the proposed synchronization scheme, two types of controlled synchronous motion are possible, namely in-phase and anti-phase synchronization. Interestingly, these synchronous modes can be achieved without changing neither the structure nor the parameters of the dynamical coupling system, but rather changing the sign of the coupling signal. The resulting closed-loop dynamics is analyzed by using the Poincaré method of perturbation, which allows to derive analytic conditions for achieving stable synchronous motion in the inverted and underactuated pendula. The analytic results are illustrated by means of computer simulations and supported by experiments.