Synchronization of Coupled Harmonic Oscillators via Sampled Position Data Control

This paper proposes two distributed synchronization protocols for a network of continuous-time coupled harmonic oscillators by utilizing current and past relative sampled position data, respectively. Some necessary and sufficient conditions in terms of coupling strength and sampling period are established to achieve network synchronization. By designing the coupling strength based on the nonzero eigenvalues of the Laplacian matrix of the network, it is shown that the synchronization in the network can be reached if and only if the sampling period is taken from a sequence of disjoint open intervals. In particular, when the Laplacian matrix has some complex eigenvalues, it is found that the sampling period should be larger than a positive threshold, that is, any small sampling period less than this threshold will not lead to network synchronization. Numerical examples are given to demonstrate the effectiveness of the theoretical analysis.