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.

References

References
1.
Jadbabaie
,
A.
,
Lin
,
J.
, and
Morse
,
A.
,
2003
, “
Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules
,”
IEEE Trans. Autom. Control
,
48
(
6
),
pp.
988
1001
.10.1109/TAC.2003.812781
2.
Fax
,
J.
, and
Murray
,
R.
,
2004
, “
Information Flow and Cooperative Control of Vehicle Formations
,”
IEEE Trans. Autom. Control
,
49
(
9
),
pp.
1465
1476
.10.1109/TAC.2004.834433
3.
Olfati-Saber
,
R.
, and
Murray
,
R.
,
2004
, “
Consensus Problems in Networks of Agents With Switching Topology and Time-Delays
,”
IEEE Trans. Autom. Control
,
49
(
9
),
pp.
1520
1533
.10.1109/TAC.2004.834113
4.
Swaroop
,
D.
, and
Hedrick
,
J. K.
,
1999
, “
Constant Spacing Strategies for Platooning in Automated Highway Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
121
(
3
),
pp.
462
470
.10.1115/1.2802497
5.
Yu
,
W.
,
Chen
,
G.
, and
Cao
,
M.
,
2010
, “
Some Necessary and Sufficient Conditions for Second-Order Consensus in Multi-Agent Dynamical Systems
,”
Automatica
,
46
(
6
),
pp.
1089
1095
.10.1016/j.automatica.2010.03.006
6.
Ren
,
W.
, and
Cao
,
Y.
,
2011
,
Distributed Coordination of Multi-Agent Networks
,
Springer
,
New York
.
7.
Murray
,
R. M.
,
2007
, “
Recent Research in Cooperative Control of Multivehicle Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
129
(
5
),
pp.
571
583
.10.1115/1.2766721
8.
Kuramoto
,
Y.
,
1984
.
Chemical Oscillators, Waves and Turbulence
,
Springer
,
Berlin
.
9.
Yeung
,
M. K. S.
, and
Strogatz
,
S. H.
,
1999
, “
Time Delay in the Kuramoto Model of Coupled Oscillators
,”
Phys. Rev. Lett.
,
82
(
3
),
pp.
648
651
.10.1103/PhysRevLett.82.648
10.
Chopra
,
N.
, and
Spong
,
M.
,
2009
, “
On Exponential Synchronization of Kuramoto Oscillators
,”
IEEE Trans. Autom. Control
,
54
(
2
),
pp.
353
357
.10.1109/TAC.2008.2007884
11.
Arenas
,
A.
,
Diaz-Guilera
,
A.
,
Kurths
,
J.
,
Moreno
,
Y.
, and
Zhou
,
C.
,
2008
, “
Synchronization in Complex Networks
,”
Phys. Rep.
,
469
(
3
),
pp.
93
153
.10.1016/j.physrep.2008.09.002
12.
Ballard
,
L.
,
Cao
,
Y.
, and
Ren
,
W.
,
2010
, “
Distributed Discrete-Time Coupled Harmonic Oscillators With Application to Synchronised Motion Coordination
,”
IET Control Theory Appl.
,
4
(
5
),
pp.
806
816
.10.1049/iet-cta.2009.0053
13.
Ren
,
W.
,
2008
, “
Synchronization of Coupled Harmonic Oscillators With Local Interaction
,”
Automatica
,
44
(
12
),
pp.
3195
3200
.10.1016/j.automatica.2008.05.027
14.
Su
,
H.
,
Wang
,
X.
, and
Lin
,
Z.
,
2009
, “
Synchronization of Coupled Harmonic Oscillators in a Dynamic Proximity Network
,”
Automatica
,
45
(
10
),
pp.
2286
2291
.10.1016/j.automatica.2009.05.026
15.
Cai
,
C.
, and
Tuna
,
S.
,
2010
, “
Synchronization of Nonlinearly Coupled Harmonic Oscillators
,”
Proceedings of the American Control Conference
,
pp.
1767
1771
.
16.
Cheng
,
S.
,
Zhang
,
G.
,
Xiang
,
L.
, and
Zhou
,
J.
,
2010
, “
Synchronization of Networked Harmonic Oscillators Under Nonlinear Protocols
,”
Proceedings of the 11th International Conference on Control Automation Robotics
,
pp.
1693
1698
.
17.
Cheng
,
S.
,
Ji
,
J.
, and
Zhou
,
J.
, 2011, “
Infinite-Time and Finite-Time Synchronization of Coupled Harmonic Oscillators
,”
Phys. Scr.
,
84
(
3
), p.
035006
.10.1088/0031-8949/84/03/035006
18.
Imer
,
O. C.
,
Yüksel
,
S.
, and
Basar
,
T.
,
2006
, “
Optimal Control of LTI Systems Over Unreliable Communication Links
,”
Automatica
,
42
(
9
),
pp.
1429
1439
.10.1016/j.automatica.2006.03.011
19.
Nair
,
S.
, and
Leonard
,
N. E.
,
2008
, “
Stable Synchronization of Mechanical System Networks
,”
SIAM J. Control Optim.
,
47
(
2
),
pp.
661
683
.10.1137/050646639
20.
Zhang
,
W.
, and
Yu
,
L.
,
2010
, “
Stabilization of Sampled-Data Control Systems With Control Inputs Missing
,”
IEEE Trans. Autom. Control
,
55
(
2
),
pp.
447
452
.10.1109/TAC.2009.2036325
21.
Wu
,
Q.
,
Zhou
,
J.
,
Zhang
,
H.
, and
Xiang
,
L
,
2011
, “
Distributed δ-Consensus in Directed Delayed Networks of Multi-Agent Systems
,”
Int. J. Syst. Sci.
,
online
10.1080/00207721.2011.649365.
22.
Lakshmikantham
,
V.
,
Bainov
,
D. D.
, and
Simeonov
,
P. S.
,
1989
,
Theory of Impulsive Differential Equations
,
World Scientific
,
Singapore
.
23.
Yang
,
T.
,
2001
,
Impulsive Control Theory
,
Springer
,
New York
.
24.
Guan
,
Z.
,
Chen
,
G.
,
Yu
,
X.
, and
Qin
,
Y.
,
2002
, “
Robust Decentralized Stabilization for a Class of Large-Scale Time-Delay Uncertain Impulsive Dynamical Systems
,”
Automatica
,
38
(
12
),
pp.
2075
2084
.10.1016/S0005-1098(02)00104-8
25.
Yang
,
T.
, and
Chua
,
L. O.
,
1997
, “
Impulsive Stabilization for Control and Synchronization of Chaotic Systems: Theory and Application to Secure Communication
,”
IEEE Trans. Circuits Syst., I: Fundam. Theory Appl.
,
44
(
10
),
pp.
976
988
.10.1109/81.633887
26.
Wu
,
Q.
,
Zhou
,
J.
, and
Xiang
,
L.
,
2010
, “
Impulsive Consensus Seeking in Directed Networks of Multi-Agent Systems With Communication Timedelays
,”
Int. J. Syst. Sci.
,
online
10.1080/00207721.2010.547630.
27.
Zhou
,
J.
,
Wu
,
Q.
, and
Xiang
,
L.
,
2011
, “
Pinning Complex Delayed Dynamical Networks by a Single Impulsive Controller
,”
IEEE Trans. Circuits Syst., I: Regul. Pap.
,
58
(
12
),
pp.
2882
2893
.10.1109/TCSI.2011.2161363
28.
Lu
,
Q.
,
Gu
,
H.
,
Yang
,
Z.
,
Shi
,
X.
,
Duan
,
L.
, and
Zheng
,
Y.
,
2008
, “
Dynamics of Firing Patterns, Synchronization and Resonances in Neuronal Electrical Activities: Experiments and Analysis
,”
Acta Mech. Sin.
,
24
(
6
),
pp.
593
628
.10.1007/s10409-008-0204-8
29.
Gerstner
,
W.
, and
Kistler
,
W. M.
,
2002
,
Spiking Neuron Models
,
Cambridge University
,
Cambridge
.
30.
Naghshtabrizi
,
P.
,
Hespanha
,
J. P.
, and
Teel
,
A. R.
,
2008
, “
Exponential Stability of Impulsive Systems With Application to Uncertain Sampled-Data Systems
,”
Syst. Control Lett.
,
57
(
5
),
pp.
378
385
.10.1016/j.sysconle.2007.10.009
31.
Fridman
,
E.
,
2010
, “
A Refined Input Delay Approach to Sampled-Data Control
,”
Automatica
,
46
(
2
),
pp.
421
427
.10.1016/j.automatica.2009.11.017
32.
Sun
,
X.
,
Liu
,
G.
,
Rees
,
D.
, and
Wang
,
W.
,
2008
, “
Stability of Systems With Controller Failure and Time-Varying Delay
,”
IEEE Trans. Autom. Control
,
53
(
10
),
pp.
2391
2396
.10.1109/TAC.2008.2007528
33.
Hu
,
L.
,
Bai
,
T.
,
Shi
,
P.
, and
Wu
,
Z.
,
2007
, “
Sampled-Data Control of Networked Linear Control Systems
,”
Automatica
,
43
(
5
),
pp.
903
911
.10.1016/j.automatica.2006.11.015
34.
Qin
,
J.
,
Gao
,
H.
, and
Zheng
,
W. X.
,
2011
, “
Second-Order Consensus for Multi-Agent Systems With Switching Topology and Communication Delay
,”
Syst. Control Lett.
,
60
(
6
),
pp.
390
397
.10.1016/j.sysconle.2011.03.004
35.
Hale
,
J.
,
1977
,
Theory of Functional Differential Equations
,
Springer-Verlag
,
New York
.
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