Distributed control for synchronization on the circle

Abstract

In this paper, the novel consensus protocol on the unit circle is proposed. The consensus problem on the unit circle refers to the synchronization of coupled oscillators. We consider the states of the agents on the unit circle as the position vector in the vector space. We add an auxiliary variable which is assumed to be communicated by agents. By using the auxiliary variables, we design the control law defined in the vector space for the synchronization. From the convex nature of the vector space, we guarantee the convergence of the auxiliary variables under the proposed consensus protocol for almost all initial conditions. With the proposed control input, the dynamics of agents on the circle achieves the synchronization exponentially. By the projection of the proposed algorithm onto the circle, it is not necessary to analyze the dynamics of the agent directly on the circle. Instead, the global convergence property of the agent is analyzed in the vector space.