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Group synchronization of coupled harmonic oscillators without velocity measurements

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Abstract

This paper investigates the group synchronization of coupled harmonic oscillators over a directed network topology in the absence of velocity measurements. Each harmonic oscillator can only obtain the sampled position states relative to its neighbors at a series of discrete-time instants. Two distributed control protocols are proposed based on the impulsive control and sampled-data control strategies. Theoretical analysis shows that the desired sampling period is determined by the position gain and the eigenvalues of the Laplacian matrix associated with the network topology. Some necessary and sufficient conditions for group synchronization are analytically established in virtue of matrix analysis, graph theory and polynomial Schur stability theory. Different to the synchronization criteria presented in the form of linear matrix inequality or general inequality, which may need to be verified, this paper explicitly gives the ranges for all feasible sampling periods. A significant feature of the synchronization criteria is that certain functional relationships between the feasible sampling period, the largest real part of the eigenvalues of the Laplacian matrix, the largest ratio of the imaginary part to the real part of the eigenvalues of the Laplacian matrix (if there exist complex eigenvalues) and the position gain are analytically established. Some effective iterative methods are then derived to calculate the endpoints of the feasible range of the sampling periods for achieving group synchronization. Finally, numerical experiments further verify the correctness of the theoretical results.

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References

  1. Jadbabaie, A., Lin, J., Morse, A.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Jain, A., Ghose, D.: Synchronization of multi-agent systems with heterogeneous controllers. Nonlinear Dyn. 89(2), 1433–1451 (2017)

    Article  Google Scholar 

  3. Moreau, L.: Stability of multiagent systems with time-dependent communication links. IEEE Trans. Autom. Control 50(2), 169–182 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226–1229 (1995)

    Article  MathSciNet  Google Scholar 

  5. Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)

    Article  MATH  Google Scholar 

  6. Blekhman, I., Fradkov, A., Nijmeijer, H., Pogromsky, A.: On self-synchronization and controlled synchronization. Syst. Control Lett. 31(5), 299–305 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: a universal concept in nonlinear sciences, Cambridge Nonlinear Science Series 12. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  8. Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)

    Article  MathSciNet  Google Scholar 

  9. Lawton, J.R., Beard, R.W.: Synchronized multiple spacecraft rotations. Automatica 38(8), 1359–1364 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ma, M., Cai, J.: Synchronization of master-slave lagrangian systems via intermittent control. Nonlinear Dyn. 89(1), 39–48 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Montbrió, E., Kurths, J., Blasius, B.: Synchronization of two interacting populations of oscillators. Phys. Rev. E 70, 056125 (2004)

    Article  Google Scholar 

  12. Fax, J., Murray, R.: Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 49(9), 1465–1476 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, Z., Guan, Z., Shen, X., Feng, G.: Consensus of multi-agent networks with aperiodic sampled communication via impulsive algorithms using position-only measurements. IEEE Trans. Autom. Control 57(10), 2639–2643 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sun, Y.G., Wang, L.: Consensus of multi-agent systems in directed networks with nonuniform time-varying delays. IEEE Trans. Autom. Control 54(7), 1607–1613 (2009)

    Article  MATH  Google Scholar 

  15. Wu, Y., Su, H., Shi, P., Shu, Z., Wu, Z.: Consensus of multiagent systems using aperiodic sampled-data control. IEEE Trans. Cybern. 46(9), 2132–2143 (2016)

    Article  Google Scholar 

  16. Xie, G., Wang, L.: Consensus control for a class of networks of dynamic agents. Int. J. Robust Nonlinear Control 17(10–11), 941–959 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, X., Su, H., Chen, M.Z.Q.: Flocking of networked euler-lagrange systems with uncertain parameters and time-delays under directed graphs. Nonlinear Dyn. 85(1), 415–424 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Olfati-Saber, R.: Flocking for multi-agent dynamic systems: algorithms and theory. IEEE Trans. Autom. Control 51(3), 401–420 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Toner, J., Tu, Y.: Flocks, herds, and schools: a quantitative theory of flocking. Phys. Rev. E 58, 4828–4858 (1998)

    Article  MathSciNet  Google Scholar 

  20. Gazi, V., Passino, K.M.: Stability analysis of swarms. IEEE Trans. Autom. Control 48(4), 692–697 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, Y., Passino, K.M., Polycarpou, M.: Stability analysis of one-dimensional asynchronous swarms. IEEE Trans. Autom. Control 48(10), 1848–1854 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, X., Chen, Z.: Robust sampled-data output synchronization of nonlinear heterogeneous multi-agents. IEEE Trans. Autom. Control 62(3), 1458–1464 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Grip, H.F., Yang, T., Saberi, A., Stoorvogel, A.A.: Output synchronization for heterogeneous networks of non-introspective agents. Automatica 48(10), 2444–2453 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wu, W., Zhou, W., Chen, T.: Cluster synchronization of linearly coupled complex networks under pinning control. IEEE Trans. Circuits Syst. I Regul. Pap. 56(4), 829–839 (2009)

    Article  MathSciNet  Google Scholar 

  25. Yu, J., Wang, L.: Group consensus in multi-agent systems with switching topologies and communication delays. Syst. Control Lett. 59(6), 340–348 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xia, W., Cao, M.: Clustering in diffusively coupled networks. Automatica 47(11), 2395–2405 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Qin, J., Yu, C.: Cluster consensus control of generic linear multi-agent systems under directed topology with acyclic partition. Automatica 49(9), 2898–2905 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pan, L., Voos, H., N’Doye, I., Darouach, M.: Group synchronization and control of a new class of adaptive complex network with brownian motion and time-varying delay. In: 2014 European Control Conference (ECC), pp. 1771–1776 (2014)

  29. Hou, H., Zhang, Q., Zheng, M.: Cluster synchronization in nonlinear complex networks under sliding mode control. Nonlinear Dyn. 83(1), 739–749 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xia, H., Huang, T.Z., Shao, J.L., Yu, J.Y.: Group consensus of multi-agent systems with communication delays. Neurocomputing 171, 1666–1673 (2016)

    Article  Google Scholar 

  31. Park, J.H., Tang, Z., Feng, J.: Pinning cluster synchronization of delay-coupled lur’e dynamical networks in a convex domain. Nonlinear Dyn. 89(1), 623–638 (2017)

    Article  MATH  Google Scholar 

  32. Shi, L., Zhu, H., Zhong, S., Shi, K., Cheng, J.: Cluster synchronization of linearly coupled complex networks via linear and adaptive feedback pinning controls. Nonlinear Dyn. 88(2), 859–870 (2017)

    Article  MATH  Google Scholar 

  33. Wang, X., She, K., Zhong, S., Yang, H.: Pinning cluster synchronization of delayed complex dynamical networks with nonidentical nodes and impulsive effects. Nonlinear Dyn. 88(4), 2771–2782 (2017)

    Article  Google Scholar 

  34. Chopra, N., Spong, M.: On exponential synchronization of kuramoto oscillators. IEEE Trans. Autom. Control 54(2), 353–357 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Scardovi, L., Sepulchre, R.: Synchronization in networks of identical linear systems. Automatica 45(11), 2557–2562 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tuna, S.E.: Synchronizing linear systems via partial-state coupling. Automatica 44(8), 2179–2184 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ballard, L., Cao, Y., Ren, W.: Distributed discrete-time coupled harmonic oscillators with application to synchronised motion coordination. IET Control Theory Appl. 4(5), 806–816 (2010)

    Article  MathSciNet  Google Scholar 

  38. Tuna, S.E.: Synchronization of harmonic oscillators under restorative coupling with applications in electrical networks. Automatica 75, 236–243 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ren, W.: Synchronization of coupled harmonic oscillators with local interaction. Automatica 44(12), 3195–3200 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Su, H., Wang, X., Lin, Z.: Synchronization of coupled harmonic oscillators in a dynamic proximity network. Automatica 45(10), 2286–2291 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, H., Zhou, J.: Synchronization of sampled-data coupled harmonic oscillators with control inputs missing. Syst. Control Lett. 61(12), 1277–1285 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhou, J., Zhang, H., Xiang, L., Wu, Q.: Synchronization of coupled harmonic oscillators with local instantaneous interaction. Automatica 48(8), 1715–1721 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Sun, W., Lü, J., Chen, S., Yu, X.: Synchronisation of directed coupled harmonic oscillators with sampled-data. IET Control Theory Appl. 8(11), 937–947 (2014)

    Article  MathSciNet  Google Scholar 

  44. Wang, J., Feng, J., Xu, C., Chen, M.Z., Zhao, Y., Feng, J.: The synchronization of instantaneously coupled harmonic oscillators using sampled data with measurement noise. Automatica 66, 155–162 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang, H., Yan, Q., Wu, Q., Wan, M.: Synchronization of impulsive coupled harmonic oscillators based on sampled position data. In: 35th Chinese Control Conference (CCC), 2016, pp. 7867–7872 (2016)

  46. Zhao, L., Zhou, J., Wu, Q.: Sampled-data synchronisation of coupled harmonic oscillators with communication and input delays subject to controller failure. Int. J. Syst. Sci. 47(1), 235–248 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Gao, Y., Wang, L., Xie, G., Wu, B.: Consensus of multi-agent systems based on sampled-data control. Int. J. Control 82(12), 2193–2205 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rakkiyappan, R., Sivaranjani, K.: Sampled-data synchronization and state estimation for nonlinear singularly perturbed complex networks with time-delays. Nonlinear Dyn. 84(3), 1623–1636 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wang, Y., Yi, J.: Consensus in second-order multi-agent systems via impulsive control using position-only information with heterogeneous delays. IET Control Theory Appl. 9(3), 336–345 (2015)

    Article  MathSciNet  Google Scholar 

  50. Zhang, B., Jia, Y., Matsuno, F.: Finite-time observers for multi-agent systems without velocity measurements and with input saturations. Syst. Control Lett. 68, 86–94 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Abdessameud, A., Tayebi, A.: Attitude synchronization of a group of spacecraft without velocity measurements. IEEE Trans. Autom. Control 54(11), 2642–2648 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Lawton, J., Beard, R., Young, B.: A decentralized approach to formation maneuvers. IEEE Trans. Robot. Autom. 19(6), 933–941 (2003)

    Article  Google Scholar 

  53. Gudino-Lau, J., Arteaga, M., Munoz, L., Parra-Vega, V.: On the control of cooperative robots without velocity measurements. IEEE Trans. Control Syst. Technol. 12(4), 600–608 (2004)

    Article  Google Scholar 

  54. Do, K.D., Pan, J.: Underactuated ships follow smooth paths with integral actions and without velocity measurements for feedback: theory and experiments. IEEE Trans. Control Syst. Technol. 14(2), 308–322 (2006)

    Article  Google Scholar 

  55. Zou, A.M., Kumar, K.D., Hou, Z.G.: Attitude coordination control for a group of spacecraft without velocity measurements. IEEE Trans. Control Syst. Technol. 20(5), 1160–1174 (2012)

    Article  Google Scholar 

  56. Zheng, Y., Wang, L.: Consensus of heterogeneous multi-agent systems without velocity measurements. Int. J. Control 85(7), 906–914 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  57. Altafini, C.: Consensus problems on networks with antagonistic interactions. IEEE Trans. Autom. Control 58(4), 935–946 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  58. Guan, Z., Liu, Z., Feng, G., Jian, M.: Impulsive consensus algorithms for second-order multi-agent networks with sampled information. Automatica 48(7), 1397–1404 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  59. Liu, H., Xie, G., Wang, L.: Necessary and sufficient conditions for solving consensus problems of double-integrator dynamics via sampled control. Int. J. Robust Nonlinear Control 20(15), 1706–1722 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  60. Kang, Y., Qin, J., Ma, Q., Song, W., Fu, W.: On group synchronization for clusters of agents with collectively acyclic intercluster couplings. IEEE Trans. Ind. Electron. 64(12), 9560–9568 (2017)

    Google Scholar 

  61. Liu, J., Zhou, J.: Distributed impulsive group consensus in second-order multi-agent systems under directed topology. Int. J. Control 88(5), 910–919 (2015)

    MathSciNet  MATH  Google Scholar 

  62. Ren, W., Beard, R.: Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control 50(5), 655–661 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  63. Huang, L., Wand, L., Hollot, C.: On robust stability of polynomials and related topics. J. Syst. Sci. Complexity 5(1), 42–54 (1992)

    MathSciNet  MATH  Google Scholar 

  64. Zhao, L., Liu, J., Xiang, L., Zhou, J.: Group synchronization of diffusively coupled harmonic oscillators. Kybernetika 52(4), 629–647 (2016)

    MathSciNet  MATH  Google Scholar 

  65. Gautschi, W.: Numerical Analysis. Birkhäuser, Boston (2012)

    Book  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the Editor, the Associate Editor and the reviewers for their helpful comments and suggestions on which the quality of this paper has been greatly improved. This work was partially supported by the National Science Foundation of China (Grant No. 61364003), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ1500915).

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Authors and Affiliations

  1. School of Science, Chongqing University of Technology, Chongqing, 400054, People’s Republic of China

    Hua Zhang

  2. School of Data Science, Tongren University, Tongren, 5231004, Guizhou, People’s Republic of China

    Hua Zhang

  3. School of Mechanical and Mechatronic Engineering, University of Technology Sydney, Sydney, NSW, 2007, Australia

    Hua Zhang & Jinchen Ji

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  1. Hua Zhang
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  2. Jinchen Ji
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Correspondence to Hua Zhang.

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This is to certify that all authors have approved the revised manuscript. We warrant that the article is the Authors’ original research work. We warrant that this manuscript has not been submitted to other journals and proceedings and is not under consideration for publication elsewhere. We declare that there is no conflict of interest in authorial ascription to organization of financial and personal relationships with other people. On behalf of the co-author Jinchen Ji, the corresponding author Hua Zhang would like to bear full responsibility for the submission.

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Zhang, H., Ji, J. Group synchronization of coupled harmonic oscillators without velocity measurements. Nonlinear Dyn 91, 2773–2788 (2018). https://doi.org/10.1007/s11071-017-4045-5

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