Synchronization of Coupled oscillators through Controlled Energy Transfer

We study the phenomena of synchronization in a system of two coupled oscillators. The oscillators, identical when decoupled, model forced pendula and are coupled with a linear torsional spring. Using the idea of energy transfer between the two oscillators, we derive nonlinear control laws which stabilize the in-phase motion. Physically, the additional control terms act through a controlled transfer the energy between the two oscillators. Thus the mechanical energy of the system remains constant as the oscillators are synchronized. Trajectories of the controlled system are required to remain on the constant energy surface of the Hamiltonian of the uncontrolled system, which represents the mechanical energy of the coupled oscillator system.