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**Synchronization of sampled-data coupled harmonic oscillators with control inputs missing.**
*(English)*
Zbl 1256.93064

Summary: This paper is concerned with the synchronization problem of sampled-data coupled harmonic oscillators with control inputs missing. A distributed algorithm for synchronization in undirected networks of coupled harmonic oscillators is proposed based on the sampled-data measurement with controller failures. Some generic criteria for such algorithm over, respectively, undirected fixed and switching network topologies are derived analytically. Compared with some existing works, a distinctive feature of this work is to solve synchronization problems in undirected networks even if each oscillator intermittently exchanges the velocity information with its neighbors in terms of sampled-data setting only at discrete moments. Subsequently, numerical examples illustrate and visualize the effectiveness and feasibility of the theoretical results.

### MSC:

93C57 | Sampled-data control/observation systems |

93B40 | Computational methods in systems theory (MSC2010) |

### Keywords:

coordinated behavior; coupled harmonic oscillators; sampled-data control; control inputs missing; intermittent measurement; undirected networks; sampled-data measurement with controller; switching network topologies; velocity information; discrete moments; synchronization problem
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\textit{H. Zhang} and \textit{J. Zhou}, Syst. Control Lett. 61, No. 12, 1277--1285 (2012; Zbl 1256.93064)

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### References:

[1] | Jadbabaie, A.; Lin, J.; Morse, A. S., Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, 48, 6, 988-1001 (2003) · Zbl 1364.93514 |

[2] | Fax, J.; Murray, R., Information flow and cooperative control of vehicle formations, IEEE Transactions on Automatic Control, 49, 9, 1465-1476 (2004) · Zbl 1365.90056 |

[3] | Olfati-Saber, R., Flocking for multi-agent dynamic systems: algorithms and theory, IEEE Transactions on Automatic Control, 51, 3, 401-420 (2006) · Zbl 1366.93391 |

[4] | Xie, G.; Wang, L., Consensus control for a class of networks of dynamic agents, International Journal of Robust and Nonlinear Control, 17, 10-11, 941-959 (2007) · Zbl 1266.93013 |

[5] | Chen, T.; Liu, X.; Lu, W., Pinning complex networks by a single controller, IEEE Transactions on Circuits and Systems I: Regular Papers, 54, 6, 1317-1326 (2007) · Zbl 1374.93297 |

[6] | Nair, S.; Leonard, N., Stable synchronization of mechanical system networks, SIAM Journal on Control and Optimization, 47, 2, 661-683 (2008) · Zbl 1158.70013 |

[7] | Ren, W., On consensus algorithms for double-integrator dynamics, IEEE Transactions on Automatic Control, 53, 6, 1503-1509 (2008) · Zbl 1367.93567 |

[8] | Song, Q.; Cao, J.; Yu, W., Second-order leader-following consensus of nonlinear multi-agent systems via pinning control, Systems & Control Letters, 59, 9, 553-562 (2010) · Zbl 1213.37131 |

[9] | Yu, W.; Chen, G.; Cao, M., Some necessary and suffient conditions for second-order consensus in multi-agent dynamical systems, Automatica, 46, 6, 1089-1095 (2010) · Zbl 1192.93019 |

[10] | Yu, W.; Chen, G.; Cao, W.; Kurths, J., Second-Order consensus for multiagent systems with directed topologies and nonlinear dynamics, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 40, 3, 881-891 (2010) |

[11] | Qin, J.; Gao, H.; Zheng, W., Second-order consensus for multi-agent systems with switching topology and communication delay, Systems & Control Letters, 60, 6, 390-397 (2011) · Zbl 1225.93020 |

[12] | Wu, Q.; Zhou, J.; Xiang, L., Impulsive consensus seeking in directed networks of multi-agent systems with communication time delays, International Journal of Systems Science, 43, 8, 1479-1491 (2012) · Zbl 1417.93290 |

[13] | Zhou, J.; Wu, Q.; Xiang, L., Pinning complex delayed dynamical networks by a single impulsive controller, IEEE Transactions on Circuits and Systems I: Regular Papers, 58, 12, 2882-2893 (2011) · Zbl 1468.93008 |

[14] | Ren, W.; Cao, Y., Distributed Coordination of Multi-agent Networks (2011), Springer · Zbl 1225.93003 |

[15] | Wang, L.; Wang, X., New conditions for synchronization in dynamical communication networks, Systems & Control Letters, 60, 4, 219-225 (2011) · Zbl 1216.93009 |

[16] | Ballard, L.; Cao, Y.; Ren, W., Distributed discrete-time coupled harmonic oscillators with application to synchronised motion coordination, IET Control Theory & Applications, 4, 5, 806-816 (2010) |

[17] | Ren, W., Synchronization of coupled harmonic oscillators with local interaction, Automatica, 44, 12, 3195-3200 (2008) · Zbl 1153.93421 |

[18] | Su, H.; Wang, X.; Lin, Z., Synchronization of coupled harmonic oscillators in a dynamic proximity network, Automatica, 45, 10, 2286-2291 (2009) · Zbl 1179.93102 |

[19] | C. Cai, S.E. Tuna, Synchronization of nonlinearly coupled harmonic oscillators. in: Proceedings of American Control Conference 2010, pp. 1767-1771.; C. Cai, S.E. Tuna, Synchronization of nonlinearly coupled harmonic oscillators. in: Proceedings of American Control Conference 2010, pp. 1767-1771. |

[20] | Cheng, S.; Ji, J.; Zhou, J., Infinite-time and finite-time synchronization of coupled harmonic oscillators, Physica Scripta, 84, 3, 035006 (2011) · Zbl 1262.34057 |

[21] | T. Hayakawa, T. Matsuzawa, S. Hara, Formation control of multi-agent systems with sampled information, in: Proceedings of the 45th IEEE Conference on Decision Control, 2006, pp. 4333-4338.; T. Hayakawa, T. Matsuzawa, S. Hara, Formation control of multi-agent systems with sampled information, in: Proceedings of the 45th IEEE Conference on Decision Control, 2006, pp. 4333-4338. |

[22] | G. Xie, H. Liu, L. Wang, Y. Jia, Consensus in networked multi-agent systems via sampled control: switching topologies case, in: Proceedings of American Control Conference, 2009, pp. 4525-4530.; G. Xie, H. Liu, L. Wang, Y. Jia, Consensus in networked multi-agent systems via sampled control: switching topologies case, in: Proceedings of American Control Conference, 2009, pp. 4525-4530. |

[23] | Gao, Y.; Wang, L.; Xie, G.; Wu, B., Consensus of multi-agent systems based on sampled-data control, International Journal of Control, 82, 12, 2193-2205 (2009) · Zbl 1178.93091 |

[24] | Gao, Y.; Wang, L., Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology, IEEE Transactions on Automatic Control, 56, 5, 1226-1231 (2011) · Zbl 1368.93424 |

[25] | Yu, W.; Zheng, W.; Chen, G.; Ren, W.; Cao, J., Second-order consensus in multi-agent dynamical systems with sampled position data, Automatica, 47, 7, 1496-1503 (2011) · Zbl 1220.93005 |

[26] | Zhang, Y.; Tian, Y., Consensus of data-sampled multi-agent systems with random communication delay and packet loss, IEEE Transactions on Automatic Control, 55, 2, 939-943 (2010) · Zbl 1368.94067 |

[27] | Zhang, W.; Yu, L., Stabilization of sampled-data control systems with control inputs missing, IEEE Transactions on Automatic Control, 55, 2, 447-452 (2010) · Zbl 1368.93586 |

[28] | Z¨ochowski, M., Intermittent dynamical control, Physica D, 145, 3-4, 181-190 (2000) · Zbl 0963.34030 |

[29] | Huang, T.; Li, C.; Liu, X., Synchronization of chaotic systems with delay using intermittent linear state feedback, Chaos, 18, 033122 (2008) · Zbl 1309.34096 |

[30] | Xia, W.; Cao, J., Pinning synchronization of delayed dynamical networks via periodically intermittent control, Chaos, 19, 013120 (2009) · Zbl 1311.93061 |

[31] | G. Wen, Z. Duan, G. Chen, W. Yu, Second-order consensus for nonlinear multi-agent systems with intermittent measurements, in: Proceedings of the 30th Chinese Control and Decision Conference, 2011, pp. 3710-3714.; G. Wen, Z. Duan, G. Chen, W. Yu, Second-order consensus for nonlinear multi-agent systems with intermittent measurements, in: Proceedings of the 30th Chinese Control and Decision Conference, 2011, pp. 3710-3714. |

[32] | Sun, X.; Liu, G.; Rees, D.; Wang, W., Stability of systems with controller failure and time-varying delay, IEEE Transactions on Automatic Control, 53, 10, 2391-2396 (2008) · Zbl 1367.93573 |

[33] | Liu, X.; Chen, T., Cluster synchronization in directed networks via intermittent pinning control, IEEE Transactions on Neural Networks, 22, 7, 1009-1020 (2011) |

[34] | Clarke, F. H.; Ledyave, Y. S.; Sontag, E. D., Asymptotic controllability implies feedback stabilization, IEEE Transactions on Automatic Control, 42, 10, 1394-1407 (1997) · Zbl 0892.93053 |

[35] | Cortes, J., Discontinuous dynamical systems, IEEE Control Systems Magazine, 28, 3, 36-73 (2008) · Zbl 1395.34023 |

[36] | R.A. Horn, C.R. Johnson, Matrix Analysis. Cambridge, 1985.; R.A. Horn, C.R. Johnson, Matrix Analysis. Cambridge, 1985. · Zbl 0576.15001 |

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