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Synchronization in Coupled Harmonic Oscillator Systems Based on Sampled Position Data

  • Qiang Song
  • Fang Liu
  • Guanghui Wen
  • Jinde Cao
  • Yang Tang
Living reference work entry

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.

Keywords

Coupled harmonic oscillators Multi-agent system Synchronization Sampled position data Sampling period 

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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Qiang Song
    • 1
  • Fang Liu
    • 2
  • Guanghui Wen
    • 3
  • Jinde Cao
    • 3
  • Yang Tang
    • 4
  1. 1.College of Electrical EngineeringHenan University of TechnologyZhengzhouChina
  2. 2.School of Information EngineeringHenan International Joint Laboratory of Behavior Optimization Control for Smart Robots, Huanghuai UniversityHenanChina
  3. 3.School of MathematicsSoutheast UniversityNanjingChina
  4. 4.Key Laboratory of Advanced Control and Optimization for Chemical ProcessesMinistry of Education, East China University of Science and TechnologyShanghaiChina

Section editors and affiliations

  • Yang Tang
    • 1
  1. 1.Department of AutomationEast China University of Science and TechnologyShanghaiChina

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