Synchronization of Coupled Discrete-Time Harmonic Oscillators With Rational Frequency

This technical note studies the synchronization of coupled discrete-time harmonic oscillators with rational frequency under switching topology. The remarkable feature of this problem is that the conditions that merely involve the connectivity structure of topology does not suffice for synchronizing the oscillators. We propose a frequency dependent topology condition that indicates in what way the topology switches, and introduce firmly nonexpansive mapping (FNM) from functional analysis. Under the condition, the synchronization of coupled oscillators is related to an infinite product of FNMs, which share only one zero common fixed point. By a convergence result on infinite product of a finite number of FNM, we present a synchronization result for the coupled oscillators. Finally, a simulation example is given to illustrate the effectiveness of the result.