Enforcing synchronization in oscillators with Huygens’ coupling via feed-forward control

This paper considers the problem of synchronizing a pair of oscillators with Huygens’ coupling to a desired periodic trajectory, by using feed-forward control. Two particular examples are considered: a modified version of the classical Huygens’ system of pendulum clocks coupled through a common bar and a pair of self-sustained mass–spring–damper oscillators mounted on a suspended bar. For both cases, it is demonstrated that if the coupling bar is excited by a suitably designed feed-forward control, then it is possible to indirectly force the oscillators to track a desired periodic trajectory. Additionally, sufficient and necessary conditions for the stability of the desired synchronous solution are derived by using the Poincaré method of perturbation and a numerical study is conducted in order to illustrate the theoretical findings. Ultimately, it is demonstrated that the proposed feed-forward control induces the desired synchronous motion provided that, besides the coupling bar, there exists a diffusive coupling between the oscillators and the intrinsic parameters of the system satisfy certain constraints.