In this paper, the synchronization of complex dynamical networks (CDNs) is investigated, where coupling connections are expressed in terms of state-space equations. As it is shown in simulation results, such links can greatly affect the synchronization and cause synchronization loss, while many real-world networks have these types of connections. With or without time-delay, two different models of the CDNs are presented. Then, by introducing a distributed adaptive controller, the synchronization conditions are derived by utilizing the Lyapunov(–Krasovskii) theorem. These conditions are provided in the form of linear matrix inequalities (LMIs), which can be easily solved by standard LMI solvers even for large networks due to a few numbers of scalar decision variables. At the end, illustrative numerical examples are given to specify the effectiveness of the proposed methods.

References

References
1.
Kazemy
,
A.
,
2018
, “
Global Synchronization of Neural Networks With Hybrid Coupling: A Delay Interval Segmentation Approach
,”
Neural Comput. Appl.
,
30
(
2
), pp.
627
637
.
2.
Dahlem
,
M.
,
Rode
,
S.
,
May
,
A.
,
Fujiwara
,
N.
,
Hirata
,
Y.
,
Aihara
,
K.
, and
Kurths
,
J.
,
2013
, “
Towards Dynamical Network Biomarkers in Neuromodulation of Episodic Migraine
,”
Transl. Neurosci.
,
4
(
3
), pp.
282
294
.https://link.springer.com/article/10.2478/s13380-013-0127-0
3.
Liu
,
K.
, and
Fridman
,
E.
,
2012
, “
Networked-Based Stabilization Via Discontinuous Lyapunov Functionals
,”
Int. J. Robust Nonlinear Control
,
22
(
4
), pp.
420
436
.
4.
Pagani
,
G. A.
, and
Aiello
,
M.
,
2013
, “
The Power Grid as a Complex Network: A Survey
,”
Phys. A
,
392
(
11
), pp.
2688
2700
.
5.
Manaffam
,
S.
, and
Seyedi
,
A.
,
2013
, “
Synchronization Probability in Large Complex Networks
,”
IEEE Trans. Circuits Syst. II
,
60
(
10
), pp.
697
701
.
6.
Wang
,
J. A.
,
2017
, “
New Synchronization Stability Criteria for General Complex Dynamical Networks With Interval Time-Varying Delays
,”
Neural Comput. Appl.
,
28
(
4
), pp.
805
815
.
7.
Kazemy
,
A.
,
2017
, “
Synchronization Criteria for Complex Dynamical Networks With State and Coupling Time-Delays
,”
Asian J. Control
,
19
(
1
), pp.
131
138
.
8.
Li
,
X.
, and
Fu
,
X.
,
2012
, “
Lag Synchronization of Chaotic Delayed Neural Networks Via Impulsive Control
,”
IMA J. Math. Control Inf.
,
29
(
1
), pp.
133
145
.
9.
Wang
,
J.
,
Zhang
,
H.
,
Wang
,
Z.
, and
Wang
,
B.
,
2013
, “
Local Exponential Synchronization in Complex Dynamical Networks With Time-Varying Delay and Hybrid Coupling
,”
Appl. Math. Comput.
,
225
(1), pp.
16
32
.https://www.sciencedirect.com/science/article/abs/pii/S0096300313009946
10.
Tang
,
Z.
,
Park
,
J. H.
,
Lee
,
T. H.
, and
Feng
,
J.
,
2016
, “
Random Adaptive Control for Cluster Synchronization of Complex Networks With Distinct Communities
,”
Int. J. Adapt. Control Signal Process.
,
30
(
3
), pp.
534
549
.
11.
Yang
,
X.
,
Feng
,
Y.
,
Yiu
,
K. F. C.
,
Song
,
Q.
, and
Alsaadi
,
F. E.
,
2018
, “
Synchronization of Coupled Neural Networks With Infinite-Time Distributed Delays Via Quantized Intermittent Pinning Control
,”
Nonlinear Dyn.
,
94
(
3
), pp.
2289
2303
.
12.
,
L.
,
Li
,
C.
,
Li
,
G.
, and
Zhao
,
G.
,
2017
, “
Projective Synchronization for Uncertain Network Based on Modified Sliding Mode Control Technique
,”
Int. J. Adapt. Control Signal Process.
,
31
(
3
), pp.
429
440
.
13.
Wang
,
A.
,
Dong
,
T.
, and
Liao
,
X.
,
2016
, “
Event-Triggered Synchronization Strategy for Complex Dynamical Networks With the Markovian Switching Topologies
,”
Neural Networks
,
74
, pp.
52
57
.
14.
Xu
,
Y.
,
Zhou
,
W.
,
Fang
,
J.
,
Xie
,
C.
, and
Tong
,
D.
,
2016
, “
Finite-Time Synchronization of the Complex Dynamical Network With Non-Derivative and Derivative Coupling
,”
Neurocomputing
,
173
, pp.
1356
1361
.
15.
Wang
,
Z.
,
Cao
,
J.
,
Chen
,
G.
, and
Liu
,
X.
,
2013
, “
Synchronization in an Array of Nonidentical Neural Networks With Leakage Delays and Impulsive Coupling
,”
Neurocomputing
,
111
, pp.
177
183
.
16.
Sun
,
Y.
,
Ma
,
Z.
,
Liu
,
F.
, and
Wu
,
J.
,
2016
, “
Theoretical Analysis of Synchronization in Delayed Complex Dynamical Networks With Discontinuous Coupling
,”
Nonlinear Dyn.
,
86
(
1
), pp.
489
499
.
17.
Li
,
X.
,
Rakkiyappan
,
R.
, and
Sakthivel
,
N.
,
2015
, “
Non-Fragile Synchronization Control for Markovian Jumping Complex Dynamical Networks With Probabilistic Time-Varying Coupling Delays
,”
Asian J. Control
,
17
(
5
), pp.
1678
1695
.
18.
Wang
,
J. L.
,
Wu
,
H. N.
, and
Huang
,
T.
,
2015
, “
Passivity-Based Synchronization of a Class of Complex Dynamical Networks With Time-Varying Delay
,”
Automatica
,
56
, pp.
105
112
.
19.
Yang
,
X.
,
Song
,
Q.
,
Cao
,
J.
, and
Lu
,
J.
,
2019
, “
Synchronization of Coupled Markovian Reaction-Diffusion Neural Networks With Proportional Delays Via Quantized Control
,”
IEEE Trans. Neural Networks Learn. Syst.
,
30
(3), pp. 951–958.
20.
Yang
,
X.
,
Feng
,
Z.
,
Feng
,
J.
, and
Cao
,
J.
,
2017
, “
Synchronization of Discrete-Time Neural Networks With Delays and Markov Jump Topologies Based on Tracker Information
,”
Neural Networks
,
85
, pp.
157
164
.
21.
Wei
,
T.
,
Yao
,
Q.
,
Lin
,
P.
, and
Wang
,
L.
,
2018
, “
Adaptive Synchronization of Stochastic Complex Dynamical Networks and Its Application
,”
Neural Comput. Appl.
(epub).
22.
Kazemy
,
A.
, and
Cao
,
J.
,
2018
, “
Consecutive Synchronization of a Delayed Complex Dynamical Network Via Distributed Adaptive Control Approach
,”
Int. J. Control, Autom. Syst.
,
16
(
6
), pp.
2656
2664
.
23.
Kazemy
,
A.
, and
Gyurkovics
,
E.
, “
Sliding Mode Synchronization of a Delayed Complex Dynamical Network in the Presence of Uncertainties and External Disturbances
,”
Trans. Inst. Meas. Control.
(epub).
24.
Gyurkovics
,
E.
,
Kiss
,
K.
, and
Kazemy
,
A.
,
2018
, “
Non-Fragile Exponential Synchronization of Delayed Complex Dynamical Networks With Transmission Delay Via Sampled-Data Control
,”
J. Franklin Inst.
,
355
(
17
), pp.
8934
8956
.
25.
Chen
,
Z.
,
Shi
,
K.
, and
Zhong
,
S.
,
2016
, “
New Synchronization Criteria for Complex Delayed Dynamical Networks With Sampled-Data Feedback Control
,”
ISA Trans.
,
63
, pp.
154
169
.
26.
Cai
,
S.
,
Li
,
X.
,
Jia
,
Q.
, and
Liu
,
Z.
,
2016
, “
Exponential Cluster Synchronization of Hybrid-Coupled Impulsive Delayed Dynamical Networks: Average Impulsive Interval Approach
,”
Nonlinear Dyn.
,
85
(
4
), pp.
2405
2423
.
27.
Yang
,
X.
,
Lu
,
J.
,
Ho
,
D. W. C.
, and
Song
,
Q.
,
2018
, “
Synchronization of Uncertain Hybrid Switching and Impulsive Complex Networks
,”
Appl. Math. Modell.
,
59
, pp.
379
392
.
28.
Lee
,
T. H.
,
Park
,
J. H.
,
Ji
,
D. H.
,
Kwon
,
O. M.
, and
Lee
,
S. M.
,
2012
, “
Guaranteed Cost Synchronization of a Complex Dynamical Network Via Dynamic Feedback Control
,”
Appl. Math. Comput.
,
218
(
11
), pp.
6469
6481
.https://www.sciencedirect.com/science/article/abs/pii/S0096300311014767
29.
Xiong
,
X.
,
Tang
,
R.
, and
Yang
,
X.
,
2018
, “
Finite-Time Synchronization of Memristive Neural Networks With Proportional Delay
,”
Neural Process. Lett.
(epub).
30.
Yang
,
X.
,
Cao
,
J.
,
Xu
,
C.
, and
Feng
,
J.
,
2018
, “
Finite-Time Stabilization of Switched Dynamical Networks With Quantized Couplings Via Quantized Controller
,”
Sci. China Technol. Sci.
,
61
(
2
), pp.
299
308
.
31.
Yang
,
X.
,
Lam
,
J.
,
Ho
,
D. W. C.
, and
Feng
,
Z.
,
2017
, “
Fixed-Time Synchronization of Complex Networks With Impulsive Effects Via Non-Chattering Control
,”
IEEE Trans. Autom. Control
,
62
(
11
), pp.
5511
5521
.
32.
Fan
,
Y.
,
Liu
,
H.
,
Zhu
,
Y.
, and
Mei
,
J.
,
2016
, “
Fast Synchronization of Complex Dynamical Networks With Time-Varying Delay Via Periodically Intermittent Control
,”
Neurocomputing
,
205
, pp.
182
194
.
33.
Feng
,
Y.
,
Yang
,
X.
,
Song
,
Q.
, and
Cao
,
J.
,
2018
, “
Synchronization of Memristive Neural Networks With Mixed Delays Via Quantized Intermittent Control
,”
Appl. Math. Comput.
,
339
, pp.
874
887
.https://www.sciencedirect.com/science/article/abs/pii/S0096300318306489
34.
Guo
,
W.
,
Chen
,
S.
, and
Francis
,
A.
,
2015
, “
Pinning Synchronization of Complex Networks With Delayed Nodes
,”
Int. J. Adapt. Control Signal Process.
,
29
(
5
), pp.
603
613
.
35.
Xu
,
C.
,
Yang
,
X.
,
Lu
,
J.
,
Feng
,
J.
,
Alsaadi
,
F. E.
, and
Hayat
,
T.
,
2018
, “
Finite-Time Synchronization of Networks Via Quantized Intermittent Pinning Control
,”
IEEE Trans. Cybern.
,
48
(
10
), pp.
3021
3027
.
36.
Li
,
X. J.
, and
Yang
,
G. H.
,
2016
, “
FLS-Based Adaptive Synchronization Control of Complex Dynamical Networks With Nonlinear Couplings and State-Dependent Uncertainties
,”
IEEE Trans. Cybern.
,
46
(
1
), pp.
171
180
.
37.
Zheng
,
S.
,
2017
, “
Pinning and Impulsive Synchronization Control of Complex Dynamical Networks With Non-Derivative and Derivative Coupling
,”
J. Franklin Inst.
,
354
(
14
), pp.
6341
6363
.
38.
Kazemy
,
A.
, and
Shojaei
,
K.
,
2018
, “
Synchronization of Complex Dynamical Networks With Dynamical Behavior Links
,”
Asian J. Control
(epub).
39.
Hu
,
C.
,
Yu
,
J.
,
Jiang
,
H.
, and
Teng
,
Z.
,
2012
, “
Pinning Synchronization of Weighted Complex Networks With Variable Delays and Adaptive Coupling Weights
,”
Nonlinear Dyn.
,
67
(
2
), pp.
1373
1385
.
40.
Feng
,
J.
,
Sun
,
S.
,
Xu
,
C.
,
Zhao
,
Y.
, and
Wang
,
J.
,
2012
, “
The Synchronization of General Complex Dynamical Network Via Pinning Control
,”
Nonlinear Dyn.
,
67
(
2
), pp.
1623
1633
.
41.
Cai
,
S.
,
Hao
,
J.
,
He
,
Q.
, and
Liu
,
Z.
,
2011
, “
Exponential Synchronization of Complex Delayed Dynamical Networks Via Pinning Periodically Intermittent Control
,”
Phys. Lett. A
,
375
(
19
), pp.
1965
1971
.
You do not currently have access to this content.