© The Institution of Engineering and Technology
This study is concerned with the globally exponential synchronisation problem for a class of non-linear singularly perturbed complex networks where all nodes possess the same structure and properties. The addressed global synchronisation problem is converted into the one for two lower-order sub-networks, namely, a non-linear slow sub-network and a linear fast sub-network, which are obtained by using the classical singular perturbation decomposition method. The network topology is directed and weighted, which means that the coupling configuration matrix is allowed to be asymmetric. By using the Lyapunov functional method and the Kronecker product technique, sufficient conditions are obtained under which the synchronisation is achieved, respectively, for the two sub-networks and the original complex network. These conditions can be easily verified by using the semi-definite programming method. A numerical example is finally simulated to validate the theoretical results and the effectiveness of the proposed synchronisation scheme.
References
-
-
1)
-
33. Horn, R., Johnson, C.: ‘Matrix analysis’ (Cambridge, UK, Cambridge Univ. Press, 1990).
-
2)
-
H.K. Khalil
.
Feedback control of nonstandard singularly perturbed systems.
IEEE Trans. Autom. Control
,
10 ,
1052 -
1060
-
3)
-
Z.S. Duan ,
G.R. Chen ,
L. Huang
.
Synchronization of weighted networks and complex synchronized regions.
Phys. Lett. A
,
21 ,
3741 -
3751
-
4)
-
34. Liu, Q., Wang, Z., He, X., Zhou, D.: ‘A survey of event-based strategies on control and estimation’, Syst. Sci. Control Eng., Open Access J., 2014, 2, (1), pp. 90–97 (doi: 10.1080/21642583.2014.880387).
-
5)
-
Z. Li ,
G. Chen
.
Global synchronization and asymptotic stability of complex dynamical networks.
IEEE Trans. Circuits Syst. II
,
1 ,
28 -
33
-
6)
-
J. Lu ,
D.W.C. Ho
.
Globally exponential synchronization and synchronizability for general dynamical networks.
IEEE Trans. Syst. Man Cybern., B
,
2 ,
350 -
361
-
7)
-
13. Wang, R., Zhou, T., Jing, Z., Chen, L.: ‘Modelling periodic oscillation of biological systems with multiple timescale networks’, Syst. Biol., 2004, 1, (1), pp. 71–84 (doi: 10.1049/sb:20045007).
-
8)
-
J. Liang ,
Z. Wang ,
Y. Liu ,
X. Liu
.
Global synchronization control of general delayed discrete-time networks with stochastic coupling and disturbances.
IEEE Trans. Syst. Man Cybern., B
,
4 ,
1073 -
1083
-
9)
-
18. Chow, J.H.: ‘Asymptotic stability of a class of non-linear singularly perturbed system’, J. Franklin Inst., 1978, 305, (5), pp. 275–281 (doi: 10.1016/S0016-0032(78)90015-7).
-
10)
-
30. Chen, X., Heidarinejad, M., Liu, J., de la Pena, D.M., Christofides, P.D.: ‘Model predictive control of nonlinear singularly perturbed systems: application to a large-scale process network’, J. Process Control, 2011, 21, (9), pp. 1296–1305 (doi: 10.1016/j.jprocont.2011.07.004).
-
11)
-
29. Lin, K.-J.: ‘Stabilisation of singularly perturbed nonlinear systems via neural network-based control and observer design’, Int. J. Syst. Sci., 2013, 44, (10), pp. 1925–1933 (doi: 10.1080/00207721.2012.670304).
-
12)
-
38. Sun, H.-Y., Li, N., Zhao, D.-P., Zhang, Q.-L.: ‘Synchronization of complex networks with coupling delays via adaptive pinning intermittent control’, Int. J. Autom. Comput., 2013, 10, (4), pp. 312–318 (doi: 10.1007/s11633-013-0726-9).
-
13)
-
24. Mostafa Asheghan, M., Miguez, J.: ‘Robust global synchronization of two complex dynamical networks’, Chaos, 2013, 23, (2), (doi: 10.1063/1.4803522).
-
14)
-
22. Shen, B., Wang, Z., Ding, D., Shu, H.: ‘H∞ state estimation for complex networks with uncertain inner coupling and incomplete measurements’, IEEE Trans. Neural Netw. Learn. Syst., 2013, 24, (12), pp. 2027–2037 (doi: 10.1109/TNNLS.2013.2271357).
-
15)
-
6. Wang, X.: ‘Complex networks: topology, dynamics and synchronization’, Int. J. Bifurcation Chaos, 2002, 12, (5), pp. 885–916 (doi: 10.1142/S0218127402004802).
-
16)
-
Y. Liu ,
Z. Wang ,
X. Liu
.
Global exponential stability of generalized recurrent neural networks with discrete and distributed delays.
Neural Netw.
,
5 ,
667 -
675
-
17)
-
19. Wang, L., Wei, G., Shu, H.: ‘State estimation for complex networks with randomly occurring coupling delays’, Neurocomputing, 2013, 122, pp. 513–520 (doi: 10.1016/j.neucom.2013.05.010).
-
18)
-
Y. Wang ,
Z. Wang ,
J. Liang
.
A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances.
Phys. Lett. A
,
39 ,
6066 -
6073
-
19)
-
X.F. Wang ,
G.R. Chen
.
Synchronization in scale-free dynamical networks: robustness and fragility.
IEEE Trans. Cir. Syst. I, Fundam. Theory Appl.
,
1 ,
54 -
62
-
20)
-
21. Hu, J., Chen, D., Du, J.: ‘State estimation for a class of discrete nonlinear systems with randomly occurring uncertainties and distributed sensor delays’, Int. J. General Syst., 2014, 43, (3–4), pp. 387–401 (doi: 10.1080/03081079.2014.892251).
-
21)
-
7. Shen, B., Wang, Z., Liu, X.: ‘Bounded H∞ synchronization and state estimation for discrete time-varying stochastic complex networks over a finite horizon’, IEEE Trans. Neural Netw., 2011, 22, (1), pp. 145–157 (doi: 10.1109/TNN.2010.2090669).
-
22)
-
10. Liu, Y., Wang, Z., Liu, X.: ‘Exponential synchronization of complex networks with Markovian jump and mixed delays’, Phys. Lett. A, 2008, 372, pp. 3986–3998 (doi: 10.1016/j.physleta.2008.02.085).
-
23)
-
16. Liang, J., Wang, Z., Liu, X., Louvieris, P.: ‘Robust synchronization for 2D discrete-time coupled dynamical networks’, IEEE Trans. Neural Netw. Learn. Syst., 2012, 23, (6), pp. 942–953 (doi: 10.1109/TNNLS.2012.2193414).
-
24)
-
37. Pan, W.-F., Jiang, B., Li, B.: ‘Refactoring software packages via community detection in complex software networks’, Int. J. Autom. Comput., 2013, 10, (2), pp. 157–166 (doi: 10.1007/s11633-013-0708-y).
-
25)
-
35. Qin, L., He, X., Zhou, D.: ‘A survey of fault diagnosis for swarm systems’, Syst. Sci. Control Eng., Open Access J., 2014, 2, (1), pp. 13–23 (doi: 10.1080/21642583.2013.873745).
-
26)
-
11. Ma, F., Fu, L.: ‘Principle of multi-time scale order reduction and its application in AC/DC hybrid power systems’. Proc. Int. Conf. on Electrical Machines and Systems, Wuhan, China, 2008, pp. 3951–3956.
-
27)
-
15. Liu, H., Sun, F., He, K., Sun, Z.: ‘Survey of singularly perturbed control systems: theory and applications’, Control Theory Appl., 2003, 20, (1), pp. 1–7.
-
28)
-
2. Wu, C.W., Chua, L.O.: ‘Synchronization in an array of linearly coupled dynamical systems’, IEEE Trans. Circuits Syst. I, 1995, 42, pp. 430–447 (doi: 10.1109/81.404047).
-
29)
-
36. Zhang, X., Yang, L.: ‘A fiber Bragg grating quasi-distributed sensing network with a wavelength-tunable chaotic fiber laser’, Syst. Sci. Control Eng., Open Access J., 2014, 2, (1), pp. 268–274 (doi: 10.1080/21642583.2014.888962).
-
30)
-
1. Dangalchv, C.: ‘Satistical mechanics and its applications’, Phys. A, 2004, 338, (3–4), pp. 659–671 (doi: 10.1016/j.physa.2004.01.056).
-
31)
-
31. Cai, C., Wang, Z., Hu, J.: ‘Synchronization of a linear singularly perturbed complex network with potential application in electric power systems’. Proc. 18th Int. Conf. on Automation & Computing, Loughborough, UK, pp. 1–6, 2012.
-
32)
-
W. Lu ,
T. Chen
.
New approach to synchronization analysis of linearly coupled ordinary differential systems.
Physica D
,
2 ,
214 -
230
-
33)
-
5. Gao, H., Lam, J., Chen, G.: ‘New criteria for synchronization stability of general complex dynamical networks with coupling delays’, Phys. Lett. A, 2006, 360, pp. 263–273 (doi: 10.1016/j.physleta.2006.08.033).
-
34)
-
14. Kokotovic, P., Khalil, H., O’Reilly, J.: ‘Singular perturbation methods in control: analysis and design’ (Academic Press, New York, 1986).
-
35)
-
7. Ding, D., Wang, Z., Shen, B., Shu, H.: ‘H∞ state estimation for discrete-time complex networks with randomly occurring sensor saturations and randomly varying sensor delays’, IEEE Trans. Neural Netw. Learn. Syst., 2012, 23, (5), pp. 725–736 (doi: 10.1109/TNNLS.2012.2187926).
-
36)
-
23. Tang, Y., Gao, H., Zou, W., Kurths, J.: ‘Distributed synchronization in networks of agent systems with nonlinearities and random switchings’, IEEE Trans. Cybern., 2013, 43, (1), pp. 358–370 (doi: 10.1109/TSMCB.2012.2207718).
-
37)
-
12. Karimi, H.R.: ‘Robust synchronization and fault detection of uncertain master-slave systems with mixed time-varying delays and nonlinear perturbations’, Int. J. Control Autom. Syst., 2011, 9, (4), pp. 671–680 (doi: 10.1007/s12555-011-0408-8).
-
38)
-
39. Zhao, Y.-H., Wang, J.-L.: ‘Exponential synchronization of impulsive complex networks with output coupling’, Int. J. Autom. Comput., 2013, 10, (4), pp. 350–359 (doi: 10.1007/s11633-013-0731-z).
-
39)
-
17. Winkelman, J., Chow, J., Allemong, J., Kokotovic, P.V.: ‘Multi-time-scale analysis of a power system’, Automatica, 1980, 16, pp. 35–43 (doi: 10.1016/0005-1098(80)90084-9).
-
40)
-
28. Barany, E., Schaffer, S., Wedeward, K., Ball, S.: ‘Nonlinear controllability of singularly perturbed models of power flow networks’. Proc. 43rd IEEE Conf. on Decision and Control, Atlantis, Paradise Island, Bahamas, 2004, pp. 4826–4832.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2014.0102
Related content
content/journals/10.1049/iet-cta.2014.0102
pub_keyword,iet_inspecKeyword,pub_concept
6
6