Synchronisation of directed coupled harmonic oscillators with sampled-data

This study investigates the synchronisation problem of instantaneously coupled harmonic oscillators with directed topology. Two distributed algorithms, in which only sampled data are utilised, are proposed for directed coupled harmonic oscillators without and with communication delay, respectively. Necessary and sufficient conditions for achieving synchronised oscillatory motions are established. It is shown that instantaneously coupled harmonic oscillators can be synchronised if and only if each oscillator exchanges the velocity information with its neighbours in an appropriate sampling period which relates to the positive gain, the coupling strength, time delay and the non-zero eigenvalues of the Laplacian matrix. Simulations are given to illustrate the feasibility of the proposed algorithms.