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Research Papers

Synchronization of Networked Harmonic Oscillators With Communication Delays Under Local Instantaneous Interaction

[+] Author and Article Information
Hua Zhang

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China; Department of Mathematics and Computer Science,  Tongren College, Tongren 554300, Chinazhanghwua@163.com

Jin Zhou1

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering,  Shanghai University, Shanghai 200072, Chinajinzhousu@yahoo.com.cn

Zengrong Liu

Institute of System Biology School of Science,  Shanghai University, Shanghai 200444, Chinazrongliu@126.com

1

Corresponding author.

J. Dyn. Sys., Meas., Control 134(6), 061009 (Sep 13, 2012) (9 pages) doi:10.1115/1.4006365 History: Received July 18, 2011; Revised February 16, 2012; Published September 13, 2012; Online September 13, 2012

The primary objective of this paper is to propose a distributed synchronization algorithm in undirected networks of coupled harmonic oscillators having communication delays under local instantaneous interaction. Some generic criteria on exponential convergence for such algorithm over, respectively, undirected fixed and switching network topologies are derived analytically. Different from the existing pure continuous or discrete-time algorithms, a distinctive feature of this work is to solve synchronization problem in undirected networks even if each oscillator instantaneously exchanges the information of the velocity with its neighbors only at some discrete moments. It is shown that the networked harmonic oscillators can be synchronized under instantaneous network connectivity. Subsequently, numerical examples illustrate and visualize the effectiveness and feasibility of the theoretical results.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Network topologies {G1,G2,G3,G4} for five oscillators. The line between agent i and j denotes that they two can exchange information with each other.

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Figure 2

Evolution of synchronization errors ri (t) − γ(t) and vi(t)-ν(t)(i=1,2,…,5) over fixed topology G1

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Figure 3

Time responses of E(t) in [10,20] over fixed topology G1 with different sampling periods Tk ∈ (0,0.07]

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Figure 4

Time responses of E(t) in [10,20] over fixed topology G1 with different sampling periods Tk∈(0.510,π4]

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Figure 5

Evolution of five harmonic oscillators states over randomly switching topologies {G1,G2,G3,G4}

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Figure 6

Trajectories of five agents. Circles denote the snapshot at t = 0 while the diamonds denote the snapshots at t = 10, 20, 30, and 40.

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