Advertisement

Synchronization of Networked Lagrangian Systems

  • Mark W. Spong
  • Nikhil Chopra
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 366)

Abstract

In this paper we study the problem of synchronization of networked Lagrangian systems. This problem arises in applications such as the control of multiple robots, formation control of UAV’s, mobile sensor networks, Kuramoto oscillators, and teleoperation. We assume that the agents exchange position and velocity information over a network described by an interconnection graph. We consider only linear interconnections on balanced, directed graphs but similar results can also be derived for nonlinear interconnections on undirected graphs. Our results exploit the well-known passivity property of Lagrangian mechanical systems. We demonstrate synchronization on both fixed and switching graphs. In the case of fixed graphs, the passivity property allows us, in addition, to treat the practically important problem of time delay in communication.

Keywords

IEEE Transaction Communication Delay Lagrangian System Consensus Problem State Synchronization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Murat Arcak. Passivity as a design tool for group coordination. In Proc. ACC, Minneapolis, MN, June 2006.Google Scholar
  2. 2.
    N. Chopra. Output Synchronization of Networked Passive Systems. Ph.d. thesis, University of Illinois at Urbana-Champaign, Department of Industrial and Enterprise Systems Engineering, 2006.Google Scholar
  3. 3.
    N. Chopra and M. W. Spong. On synchronization of kuramoto oscillators. In Proceedings of the Joint 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC’05), pages 3916–3922, Seville, Spain, December 2005.Google Scholar
  4. 4.
    N. Chopra and M. W. Spong. Output synchronization of networked passive systems. Preprint, 2006.Google Scholar
  5. 5.
    N. Chopra and M. W. Spong. Output synchronization of nonlinear systems with time delay in communication. To Appear, IEEE Conference on Decision and Control, 2006.Google Scholar
  6. 6.
    J. Cortés, S. Martínez, T. Karatas, and F. Bullo. Coverage control for mobile sensing networks. IEEE Transactions on Robotics and Automation, 20(2):243–255, 2004.CrossRefGoogle Scholar
  7. 7.
    J.A. Fax and R.M. Murray. Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, 49(4):1465–1476, 2004.CrossRefMathSciNetGoogle Scholar
  8. 8.
    C. Godsil and G. Royle. Algebraic graph theory. In Springer Graduate Texts in Mathematics No. 207. Springer, 2001.Google Scholar
  9. 9.
    A. Jadbabaie, J. Lin, and A. S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48:988–1001, June 2003.CrossRefMathSciNetGoogle Scholar
  10. 10.
    A. Jadbabaie, N. Motee, and M. Barahona. On the stability of the kuramoto model of coupled nonlinear oscillators. In Proceedings of the American Control Conference, pages 4296–4301, 2004.Google Scholar
  11. 11.
    E. W. Justh and P. S. Krishnaprasad. A simple control law for UAV formation flying. Technical 2002–38, Institute for Systems Research, 2002.Google Scholar
  12. 12.
    H. K. Khalil. Nonlinear systems. In Upper Saddle River. Prentice Hall, New Jersey, 2002.Google Scholar
  13. 13.
    Y. Kuramoto. In international symposium on mathematical problems in theoretical physics. In Lecture Notes in Physics, volume 39. Springer, New York, 1975.Google Scholar
  14. 14.
    D. J. Lee and M. W. Spong. Stable flocking of multiple inertial, agents on balanced graphs. IEEE Transactions on Automatic Control, submitted, 2005.Google Scholar
  15. 15.
    N. E. Leonard and E. Fiorelli. Virtual leaders, artificial potentials and coordinated control of groups. In Proceedings of the 40th IEEE Conference on Decision and Control, pages 2968–2973, Orlando, Florida, 2001.Google Scholar
  16. 16.
    J. A. Marshall, M. E. Broucke, and B. A. Francis. Formation of vehicles in cyclic pursuit. IEEE Transactions on Automatic Control, 49(11):1963–1974, 2004.CrossRefMathSciNetGoogle Scholar
  17. 17.
    L. Moreau. Stability of multi-agent systems with time-dependent communication links. IEEE Transactions on Automatic Control, 50(2):169–182, 2005.CrossRefMathSciNetGoogle Scholar
  18. 18.
    P. J. Moylan. Implications of passivity for a class of nonlinear systems. IEEE Tran. Automat. Contr., 19:373–381, August 1974.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    R. Olfati-Saber and R. M. Murray. Consensus problems in networks of dynamic agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49:1520–1533, September 2004.CrossRefMathSciNetGoogle Scholar
  20. 20.
    R. Ortega and M. W. Spong. Adaptive motion control of rigid robots: a tutorial. Automatica, 25:877–888, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    A. Y. Pogromsky. Passivity based design of synchronizing systems. Int. J. of Bifurcation and Chaos, 8:295–319, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    A. Y. Pogromsky and H. Nijmeijer. Cooperative oscillatory behavior of mutually coupled dynamical systems. IEEE Transactions on Circuits and Systems 1, 48:152–162, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    W. Ren and R. W. Beard. Consensus seeking in multi-agent systems using dynamically changing interaction topologies. IEEE Transactions on Automatic Control, 5(5):655–661, May 2005.CrossRefMathSciNetGoogle Scholar
  24. 24.
    R. Sepulchre, M. Janković, and P. Kokotović. Constructive Nonlinear Control. Springer-Verlag, London, 1997.zbMATHGoogle Scholar
  25. 25.
    R. Sepulchre, D. Paley, and N. Leonard. Collection motion and oscillator synchronization. In V. Kumar, N. Leonard, and A. S. Morse, editors, Cooperative Control, Lecture Notes in Control and Information Sciences, volume 309. Springer Verlag, 2004.Google Scholar
  26. 26.
    J.-J. E. Slotine and W. Wang. A study of synchronization and group cooperation using partial contraction theory. In V. Kumar, N. E. Leonard, and A. S. Morse, editors, Cooperative Control, volume 309 of Lecture Notes in Control and Information Sciences, 2004.Google Scholar
  27. 27.
    M. W. Spong, S. Hutchinson, and M. Vidyasagar. Robot Modeling and Control. John Wiley & Sons, Inc., New York, 2006.Google Scholar
  28. 28.
    S. H. Strogatz. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143:1–20, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    H. G. Tanner and D. K. Christodoulakis. State synchronization in local-interaction networks is robust with respect to time delays, December 2005.Google Scholar
  30. 30.
    H. G. Tanner, G. J. Pappas, and V. Kumar. Leader-to-formation stability. IEEE Transactions on Robotics and Automation, 20(3):443–455, 2004.CrossRefGoogle Scholar
  31. 31.
    J. N. Tsitsiklis, D. P. Bertsekas, and M. Athans. Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Transactions on Automatic Control, 31(9):803–812, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    W. Wang and J.-J. E. Slotine. Contraction analysis of time-delayed communications and group cooperation. IEEE Transactions on Automatic Control, 51(4):712–717, 2006.CrossRefMathSciNetGoogle Scholar
  33. 33.
    J. Zhao and David J. Hill. Dissipative theory for switched systems. Im Proceedings of the Joint 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC’05), pages 7003–7008, Seville, Spain, December 2005.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Mark W. Spong
    • 1
  • Nikhil Chopra
    • 1
  1. 1.Coordinated Science LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations