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Synchronization in Coupled Harmonic Oscillator Systems Based on Sampled Position Data

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Handbook of Real-Time Computing

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Abstract

A harmonic oscillator is a typical second-order spring-mass system exhibiting periodic motions. In the last decade, much effort has been devoted to the study on the synchronization in networks composed by a set of identical harmonic oscillators. Most of existing synchronization algorithms for coupled harmonic oscillators are developed based on relative velocity measurements. This chapter proposes two distributed synchronization protocols to solve the synchronization problem for a network of harmonic oscillators in continuous-time setting by utilizing current and past relative sampled position data between neighboring nodes, respectively. Some necessary and sufficient conditions in terms of coupling strength and sampling period are established to achieve synchronization in the network. By designing the coupling strength according to the nonzero eigenvalues of the Laplacian matrix of the network, it is shown that the synchronization problem of coupled harmonic oscillators can be solved if and only if the sampling period is taken from a sequence of disjoint open intervals. Interestingly, 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 illustrate the feasibility and the effectiveness of the theoretical analysis.

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Author information

Authors and Affiliations

  1. College of Electrical Engineering, Henan University of Technology, Zhengzhou, China

    Qiang Song

  2. School of Information Engineering, Henan International Joint Laboratory of Behavior Optimization Control for Smart Robots, Huanghuai University, Henan, China

    Fang Liu

  3. School of Mathematics, Southeast University, Nanjing, P. R. China

    Guanghui Wen

  4. School of Mathematics, Southeast University, Nanjing, China

    Jinde Cao

  5. The Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai, China

    Yang Tang

Authors
  1. Qiang Song
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  2. Fang Liu
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  3. Guanghui Wen
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  4. Jinde Cao
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  5. Yang Tang
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Editor information

Editors and Affiliations

  1. School of Computer Science, Queensland University of Technology, Brisbane, QLD, Australia

    Yu-Chu Tian

  2. School of Electrical and Information Engineering, The University of Sydney, Sydney, NSW, Australia

    David Charles Levy

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© 2022 Springer Nature Singapore Pte Ltd

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Song, Q., Liu, F., Wen, G., Cao, J., Tang, Y. (2022). Synchronization in Coupled Harmonic Oscillator Systems Based on Sampled Position Data. In: Tian, YC., Levy, D.C. (eds) Handbook of Real-Time Computing. Springer, Singapore. https://doi.org/10.1007/978-981-287-251-7_21

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