Sampled-data control of coupled harmonic oscillators using measured position States only

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Journal Article
IET Control Theory and Applications, 2018, 12 (7), pp. 985 - 991
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© The Institution of Engineering and Technology 2018. This brief study investigates the synchronisation of undirected coupled harmonic oscillators by using a sampled-data control strategy. In the absence of measured velocity information, each of the networked oscillators can only obtain the measurements of the position States relative to its neighbours at the sampling moments. A distributed protocol based on the sampled relative position measurements at a series of time instants is proposed for the coupled harmonic oscillators. Necessary and sufficient conditions for achieving synchronisation are analytically established in virtue of the graph theory, matrix analysis theory and Schur stability theory of cubic polynomials. Theoretical results indicate that the explicit range that contains all the desirable sampling periods is uniquely determined by the largest eigenvalue of the Laplacian matrix associated with the network graph and the position gain of the harmonic oscillators. An effective simple iterative method is then developed to calculate the range in which the desirable sampling periods fall. Finally, numerical simulations are performed to illustrate the correctness of the theoretical results and effectiveness of the proposed protocol.
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