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group synchronization; coupled harmonic oscillators; directed topology; acyclic partition
This paper considers group synchronization issue of diffusively directed coupled harmonic oscillators for two cases with nonidentical and identical agent dynamics. For the case of coupled nonidentical harmonic oscillators with positive coupling, it is demonstrated that distributed group synchronization can always be achieved under two kinds of network structures, i. e., the strongly connected graph and the acyclic partition topology with a directed spanning tree. It is interesting to find that the group synchronization states under acyclic partition are some periodic orbits with the same frequency and are simply related with the initial values of certain group regardless of ones of the other groups. For the case of coupled identical harmonic oscillators with positive and negative coupling, some generic algebraic criteria on group synchronization with both local continuous and instantaneous interaction are established respectively. In particular, an explicit expression of group synchronization states in terms of initial values of the agents can be obtained by the property of acyclic partition topology, and so it is very convenient to yield the desired group synchronization in practical application. Finally, numerical examples illustrate and visualize the effectiveness and feasibility of theoretical results.
[1] Ballard, L., Cao, C. Y., Ren, W.: Distributed discrete-time coupled harmonic oscillators with application to synchronised motion coordination. IET Control Theory Appl. 4 (2010), 806-816. DOI 10.1049/iet-cta.2009.0053 | MR 2758795
[2] Chen, Y., L$\ddot{\mathrm{u}}$, J. H., Yu, X. H., Lin, Z. L.: Consensus of discrete-time second-order multiagent systems based on infinite products of general stochastic matrices. SIAM J. Control Optim. 51 (2013), 3274-3301. DOI 10.1137/110850116 | MR 3090151
[3] Cheng, S., Ji, C. J., Zhou, J.: Infinite-time and finite-time synchronization of coupled harmonic oscillators. Physica Scripta 84 (2011), 035006. DOI 10.1088/0031-8949/84/03/035006 | Zbl 1262.34057
[4] Desoer, C., Vidyasagar, M.: Feedback Systems: Input-output Properties. Academic, New York 1975. MR 0490289 | Zbl 1153.93015
[5] Godsil, C., Royle, G.: Algebraic Graph Theory. Springer-Verlag, London 2001. DOI 10.1007/978-1-4613-0163-9 | MR 1829620 | Zbl 0968.05002
[6] He, W. L., Qian, F., Lam, J., Chen, G. R., Han, Q. L., Kurths, J.: Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control: error estimation, optimization and design. Automatica 62 (2015), 249-262. DOI 10.1016/j.automatica.2015.09.028 | MR 3423996 | Zbl 1330.93011
[7] He, W. L., Zhang, B., Han, Q. L., Qian, F., Kurths, J., Cao, J. D.: Leader-following consensus of nonlinear multi-agent systems with stochastic sampling. IEEE Trans. Cybernetics (2016), 1-12. DOI 10.1109/tcyb.2015.2514119
[8] Hong, Y. G., Hu, J. P., Gao, L. X.: Tracking control for multi-agent consensus with an active leader and variable topology. Automatica 42 (2006), 1177-1182. DOI 10.1016/j.automatica.2006.02.013 | MR 2230987 | Zbl 1117.93300
[9] Horn, R., Johnson, C. R.: Matrix Analysis. Cambridge University Press, Cambridge 1990. DOI 10.1002/zamm.19870670330 | MR 1084815 | Zbl 0801.15001
[10] Liu, J., Zhou, J.: Distributed impulsive group consensus in second-order multi-agent systems under directed topology. Int. J. Control 88 (2015), 910-919. DOI 10.1080/00207179.2014.985717 | MR 3325386 | Zbl 1316.93005
[11] Lu, S. J., Chen, L.: A general synchronization method of chaotic communication system via kalman filtering. Kybernetika 44 (2008), 43-52. MR 2405054
[12] Lu, W. L., Liu, B., Chen, T. P.: Cluster synchronization in networks of coupled nonidentical dynamical systems. Chaos 20 (2010), 013120. DOI 10.1063/1.3329367 | MR 2730167 | Zbl 1311.34117
[13] Ma, M. H., Zhang, H., Cai, J. P., Zhou, J.: Impulsive practical synchronization of n-dimensional nonautonomous systems with parameter mismatch. Kybernetika 49 (2013), 539-553. MR 3117913 | Zbl 1274.70039
[14] Qin, J. H., Yu, C. B.: Cluster consensus control of generic linear multi-agent systems under directed topology with acyclic partition. Automatica 49 (2013), 2898-2905. DOI 10.1016/j.automatica.2013.06.017 | MR 3084481
[15] Ren, W.: Synchronization of coupled harmonic oscillators with local interaction. Automatica 44 (2008), 3195-3200. DOI 10.1016/j.automatica.2008.05.027 | MR 2531426 | Zbl 1153.93421
[16] Ren, W., Cao, Y. C.: Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues. Springer-Verlag, London 2011. Zbl 1225.93003
[17] Slotine, J. J. E., Li, W. P.: Applied Nonlinear Control. Prentice Hall, N.J. 1991. Zbl 0753.93036
[18] Su, H. S., Wang, X. F., Lin, Z. L.: Synchronization of coupled harmonic oscillators in a dynamic proximity network. Automatica 45 (2009), 2286-2291. DOI 10.1016/j.automatica.2009.05.026 | MR 2890789 | Zbl 1179.93102
[19] Su, H. S., Chen, M., Wang, X. F., Wang, H. W., Valeyev, N. V.: Adaptive cluster synchronisation of coupled harmonic oscillators with multiple leaders. IET Control Theory Appl. 7 (2013), 765-772. DOI 10.1049/iet-cta.2012.0910 | MR 3100186
[20] Wang, K. H., Fu, X. C., Li, K. Z.: Cluster synchronization in community networks with nonidentical nodes. Chaos 19 (2009), 023106. DOI 10.1063/1.3125714 | MR 2548747 | Zbl 1309.34107
[21] Wu, W., Zhou, W. J., Chen, T. P.: Cluster synchronization of linearly coupled complex networks under pinning control. IEEE Trans. Circuits Syst. I. Reg. Pap. 56 (2009), 819-839. DOI 10.1109/tcsi.2008.2003373 | MR 2724977
[22] Xia, W. G., Cao, M.: Clustering in diffusively coupled networks. Automatica 47 (2011), 2395-2405. DOI 10.1016/j.automatica.2011.08.043 | MR 2886867 | Zbl 1228.93015
[23] Yang, T.: Impulsive Control Theory. Springer 2001. MR 1850661 | Zbl 0996.93003
[24] Yu, W. W., Chen, G. R., Cao, M., Kurths, J.: Second-order consensus for multioscillator systems with directed topologies and nonlinear dynamics. IEEE T. Syst. Man Cy. B 40 (2010), 881-891. DOI 10.1109/tsmcb.2009.2031624
[25] Yu, C. B., Qin, J. H., Gao, H. J.: Cluster synchronization in directed networks of partial-state coupled linear systems under pinning control. Automatica 50 (2014), 2341-2349. DOI 10.1016/j.automatica.2014.07.013 | MR 3256724 | Zbl 1297.93019
[26] Yu, J. Y., Wang, L.: Group consensus of multi-agent systems with undirected communication graphs. In: Proc. 7th Asian Control Conference 2009, pp. 105-110.
[27] Zhang, H., Zhou, J.: Synchronization of sampled-data coupled harmonic oscillators with control inputs missing. Syst. Control Lett. 61 (2012), 1277-1285. DOI 10.1016/j.sysconle.2012.10.001 | MR 2998215 | Zbl 1256.93064
[28] Zhao, L. Y., Wu, Q. J., Zhou, J.: Impulsive sampled-data synchronization of directed coupled harmonic oscillators. In: Proc. 33rd Chinese Control Conference 2014, pp. 3950-3954. DOI 10.1109/chicc.2014.6895598
[29] Zhou, J., Zhang, H., Xiang, L., Wu, Q. J.: Synchronization of coupled harmonic oscillators with local instantaneous interaction. Automatica 48 (2012), 1715-1721. DOI 10.1016/j.automatica.2012.05.022 | MR 2950421 | Zbl 1267.93008
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