An Extended Kuramoto Model for Frequency and Phase Synchronization in Delay-Free Networks with Finite Number of Agents

Due to its description of a synchronization between oscillators, the Kuramoto model is an ideal choice for a synchronisation algorithm in networked systems. This requires to achieve not only a frequency synchronization but also a phase synchronization - something the standard Kuramoto model can not provide for a finite number of agents. In this case, a remaining phase difference is necessary to offset differences of the natural frequencies. Setting the Kuramoto model into the context of dynamic consensus and making use of the $n$th order discrete average consensus algorithm, this paper extends the standard Kuramoto model in such a way that frequency and phase synchronization are separated. This in turn leads to an algorithm achieve the required frequency and phase synchronization also for a finite number of agents. Simulations show the viability of this extended Kuramoto model.