Consensus for double-integrator dynamics with velocity constraints

  • Tales A. Jesus
  • Luciano C. A. Pimenta
  • Leonardo A. B. Tôrres
  • Eduardo M. A. M. Mendes
Regular Papers Control Theory

Abstract

The problem of consensus for double-integrator dynamics with velocity constraints and a constant group reference velocity is addressed such that: (i) the control law of an agent does not depend on the local neighbors’ velocities or accelerations, but only on the neighbors’ positions and on the own agent velocity; (ii) the constraints are non-symmetric; (iii) the class of nonlinear functions used to account for the velocity constraints is more general than the ones that are normally considered in the literature. We propose a decentralized control strategy with the neighboring topology described by an undirected interaction graph that is connected. Mathematical guarantees of convergence without violating the constraints are given. A numerical experiment is provided to illustrate the effectiveness of our approach.

Keywords

Consensus multi-agent systems second-order dynamics velocity constraints 

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References

  1. [1]
    M. A. Hsieh, V. Kumar, and L. Chaimowicz, “Decentralized controllers for shape generation with robotic systems,” Robotica, vol. 26, no. 5, pp. 691–701, September 2008.CrossRefGoogle Scholar
  2. [2]
    D. W. Casbeer, D. B. Kingston, R. W. Beard, T. W. McLain, S.-M. Li, and R. Mehra, “Cooperative forest fire surveillance using a team of small unmanned air vehicles,” International Journal of Systems Science, vol. 37, no. 6, pp. 351–360, May 2006.CrossRefMATHGoogle Scholar
  3. [3]
    X. C. Ding, A. R. Rahmani, and M. Egerstedt, “Multi-UAV convoy protection: an optimal approach to path planning and coordination,” IEEE Trans. on Robotics, vol. 26, no. 2, pp. 256–268, April 2010.CrossRefGoogle Scholar
  4. [4]
    A. Franchi, P. Stegagno, M. D. Rocco, and G. Oriolo, “Distributed target localization and encircling with a multi-robot system,” Proc. of the 7th IFAC Symposium on Intelligent Autonomous Vehicles, pp. 1–6, 2010.Google Scholar
  5. [5]
    R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. on Automatic Control, vol. 49, no. 9, pp. 1520–1533, September 2004.CrossRefMathSciNetGoogle Scholar
  6. [6]
    M. Arcak, “Passivity as a design tool for group coordination,” IEEE Trans. on Automatic Control, vol. 52, no. 9, pp. 1380–1390, August 2007.CrossRefMathSciNetGoogle Scholar
  7. [7]
    W. Ren, R. W. Beard, and E. M. Atkins, “A survey of consensus problems in multi-agent coordination,” Proc. of the American Control Conference, pp. 1859–1864, 2005.Google Scholar
  8. [8]
    W. Ren and R. W. Beard, “Consensus seeking in multiagent systems under dynamically changing interaction topologies,” IEEE Trans. on Automatic Control, vol. 50, no. 5, pp. 655–661, May 2005.CrossRefMathSciNetGoogle Scholar
  9. [9]
    R. Olfati-Saber, “Flocking for multi-agent dynamic systems: algorithms and theory,” IEEE Trans. on Automatic Control, vol. 51, no. 3, pp. 401–420, March 2006.CrossRefMathSciNetGoogle Scholar
  10. [10]
    G. Xie and L. Wang, “Consensus control for a class of networks of dynamic agents,” International Journal of Robust and Nonlinear Control, vol. 17, pp. 941–949, November 2007.CrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    W. Ren and E. Atkins, “Distributed multi-vehicle coordinated control via local information exchange,” International Journal of Robust and Nonlinear Control, vol. 17, pp. 1002–1033, November 2007.CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    F. Xiao, L. Wang, J. Chen, and Y. Gao, “Finitetime formation control for multi-agent systems,” Automatica, vol. 45, no. 11, pp. 2605–2611, November 2009.CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    Y. Li, J. Xiang, and W. Wei, “Consensus problems for linear time-invariant multi-agent systems with saturation constraints,” IET Control Theory & Applications, vol. 5, no. 6, pp. 823–829, April 2011.CrossRefMathSciNetGoogle Scholar
  14. [14]
    L. Gao, J. Zhang, and W. Chen, “Second-order consensus for multiagent systems under directed and switching topologies,” Mathematical Problems in Engineering, pp. 1–21, 2012.Google Scholar
  15. [15]
    Z. Wang and J. Cao, “Quasi-consensus of secondorder leader-following multi-agent systems,” IET Control Theory & Applications, vol. 6, no. 4, pp. 545–551, March 2012.CrossRefMathSciNetGoogle Scholar
  16. [16]
    H. Zhao, S. Xu, and D. Yuan, “An LMI approach to consensus in second-order multi-agent systems,” International Journal of Control, Automation, and Systems, vol. 9, no. 6, pp. 1111–1115, December 2011.CrossRefGoogle Scholar
  17. [17]
    L. Gao, X. Zhu, and W. Chen, “Leader-following consensus problem with an accelerated motion leader,” International Journal of Control, Automation, and Systems, vol. 10, no. 5, pp. 931–939, October 2012.CrossRefGoogle Scholar
  18. [18]
    X. Luo, D. Liu, X. Guan, and S. Li, “Flocking in target pursuit for multi-agent systems with partial informed agents,” IET Control Theory & Applications, vol. 6, no. 4, pp. 560–569, March 2012.CrossRefMathSciNetGoogle Scholar
  19. [19]
    G. Wen, Z. Duan, W. Yu, and G. Chen, “Consensus in multi-agent systems with communication constraints,” International Journal of Robust and Nonlinear Control, vol. 22, pp. 170–182, January 2012.CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    G. Wen, Z. Duan, Z. Lu, and G. Chen, “Consensus and its L2 gain performance of multi-agent systems with intermittent information transmissions,” International Journal Control, vol. 85, no. 4, pp. 384–396, January 2012.CrossRefMATHGoogle Scholar
  21. [21]
    P. Lin and Y. Jia, “Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies,” Automatica, vol. 45, no. 9, pp. 2154–2158, September 2009.CrossRefMATHMathSciNetGoogle Scholar
  22. [22]
    Q. Zhang, Y. Niu, L. Wang, L. Shen, and H. Zhu, “Average consensus seeking of high-order continuous-time multi-agent systems with multiple timevarying communication delays,” International Journal of Control, Automation, and Systems, vol. 9, no. 6, pp. 1209–1218, December 2011.CrossRefGoogle Scholar
  23. [23]
    W. Yu, G. Chen, M. Cao, and J. Kurths, “Secondorder consensus for multiagent systems with directed topologies and nonlinear dynamics,” IEEE Trans. on Systems, Man, and Cybernetics — Part B: Cybernetics, vol. 40, no. 3, pp. 881–891, June 2010.CrossRefGoogle Scholar
  24. [24]
    W. Yu, G. Chen, and M. Cao, “Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems,” Automatica, vol. 46, no. 6, pp. 1089–1095, June 2010.CrossRefMATHMathSciNetGoogle Scholar
  25. [25]
    S. Li, H. Du, and X. Lin, “Finite-time consensus algorithm for multi-agent systems with doubleintegrator dynamics,” Automatica, vol. 47, no. 8, pp. 1706–1712, August 2011.CrossRefMATHMathSciNetGoogle Scholar
  26. [26]
    H. Du, S. Li, and C. Qian, “Finite-time attitude tracking control of spacecraft with application to attitude synchronization,” IEEE Trans. on Automatic Control, vol. 56, no. 11, pp. 2711–2717, November 2011.CrossRefMathSciNetGoogle Scholar
  27. [27]
    H. Li, X. Liao, and G. Chen, “Leader-following finite-time consensus in second-order multi-agent networks with nonlinear dynamics,” International Journal of Control, Automation, and Systems, vol. 11, no. 2, pp. 422–426, April 2013.CrossRefGoogle Scholar
  28. [28]
    D. Zhao, T. Zou, S. Li, and Q. Zhu, “Adaptive backstepping sliding mode control for leaderfollower multi-agent systems,” IET Control Theory & Applications, vol. 6, no. 8, pp. 1109–1117, May 2012.CrossRefMathSciNetGoogle Scholar
  29. [29]
    H. Du, S. Li, and P. Shi, “Robust consensus algorithm for second-order multi-agent systems with external disturbances,” International Journal of Control, vol. 85, no. 12, pp. 1913–1928, December 2012.CrossRefMATHMathSciNetGoogle Scholar
  30. [30]
    H. Hu, G. Chen, and G. Xie, “Second-order consensus of multi-agent systems with unknown but bounded disturbance,” International Journal of Control, Automation, and Systems, vol. 1, no. 2, pp. 258–267, April 2013.CrossRefGoogle Scholar
  31. [31]
    G. Ferrari-Trecate, L. Galbusera, M. P. E. Marciandi, and R. Scattolini, “Contractive distributed MPC for consensus in networks of single- and doubleintegrators,” Proc. of the 17th World Congress of the International Federation of Automatic Control, pp. 9033–9038, 2008.Google Scholar
  32. [32]
    A. Nedic, A. Ozdaglar, and P. A. Parrilo, “Constrained consensus and optimization in multi-agent networks,” IEEE Trans. on Automatic Control, vol. 55, no. 4, pp. 922–938, April 2010.CrossRefMathSciNetGoogle Scholar
  33. [33]
    J. Lee, J.-S. Kim, H. Song, and H. Shim, “A constrained consensus problem using MPC,” International Journal of Control, Automation and Systems, vol. 9, no. 5, pp. 952–957, October 2011.CrossRefGoogle Scholar
  34. [34]
    U. Lee and M. Mesbahi, “Constrained consensus via logarithmic barrier functions,” Proc. of the 5th Conf. on Decision and Control, pp. 3608–3613, 2011.Google Scholar
  35. [35]
    W. Ren, “On consensus algorithms for doubleintegrator dynamics,” IEEE Trans. on Automatic Control, vol. 53, no. 6, pp. 1503–1509, July 2008.CrossRefGoogle Scholar
  36. [36]
    K. Peng, H.-S. Su, and Y.-P. Yang, “Coordinated control of multi-agent systems with a varyingvelocity leader and input saturation,” Communications in Theoretical Physics, vol. 52, no. 3, pp. 449–456, September 2009.CrossRefMATHGoogle Scholar
  37. [37]
    A. Abdessameud and A. Tayebi, “On consensus algorithms for double-integrator dynamics without velocity measurements and with input constraints,” Systems & Control Letters, vol. 59, no. 12, pp. 812–821, December 2010.CrossRefMATHMathSciNetGoogle Scholar
  38. [38]
    W. Ren, “Distributed leaderless consensus algorithms for networked Euler-Lagrange systems,” Systems & Control Letters, vol. 82, no. 11, pp. 2137–2149, November 2009.MATHGoogle Scholar
  39. [39]
    J. R. T. Lawton, R. W. Beard, and B. J. Young, “A decentralized approach to formation maneuvers,” IEEE Trans. on Robotics and Automation, vol. 19, no. 6, pp. 933–941, December 2003.CrossRefGoogle Scholar
  40. [40]
    A. Abdessameud and A. Tayebi, “Synchronization of networked Lagrangian systems with input constraints,” Proc. of the 18h IFAC World Congress, pp. 2382–2387, 2011.Google Scholar
  41. [41]
    D. V. Dimarogonas and K. J. Kyriakopoulos, “On the rendezvous problem for multiple nonholonomic agents,” IEEE Trans. on Automatic Control, vol. 52, no. 5, pp. 916–922, May 2007.CrossRefMathSciNetGoogle Scholar
  42. [42]
    W. Ren, “Collective motion from consensus with Cartesian coordinate coupling,” IEEE Trans. on Automatic Control, vol. 54, no. 6, pp. 1330–1335, June 2009.CrossRefGoogle Scholar
  43. [43]
    P. Lin, K. Qin, Z. Lin, and W. Ren, “Collective rotating motions of second-order multi-agent systems in three-dimensional space,” Systems & Control Letters, vol. 60, no. 6, pp. 365–372, June 2011.CrossRefMATHMathSciNetGoogle Scholar
  44. [44]
    P. Lin and Y. Jia, “Distributed rotating formation control of multi-agent systems,” Systems & Control Letters, vol. 59, no. 10, pp. 587–595, October 2010.CrossRefMATHMathSciNetGoogle Scholar
  45. [45]
    X.-P Chen, H.-B. Xu, and Y.-X. Ban, “Rotating consensus of multi-agent systems without relative velocity measurement,” Chinese Physics B, vol. 20, no. 9, 090515, September 2011.CrossRefGoogle Scholar
  46. [46]
    S. Bayraktar, J. E. Fainekos, and G. J. Pappas, “Experimental cooperative control of fixed-wing unmanned aerial vehicles,” Proc. of the 43rd Conf. Decision and Control, pp. 4292–4298, 2004.Google Scholar
  47. [47]
    F. Bullo, J. Cortés, and S. Martínez, Distributed Control of Robotic Networks, Princeton University Press, New Jersey, 2009.MATHGoogle Scholar
  48. [48]
    D. S. Bernstein, Matrix Mathematics: Theory, Facts and Formulas, Princeton University Press, New Jersey, 2009.Google Scholar
  49. [49]
    J.-J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, NJ, 1991.MATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Tales A. Jesus
    • 1
  • Luciano C. A. Pimenta
    • 2
  • Leonardo A. B. Tôrres
    • 2
  • Eduardo M. A. M. Mendes
    • 2
  1. 1.Computation DepartmentCentro Federal de Educação Tecnológica de Minas GeraisBelo HorizonteBrazil
  2. 2.School of EngineeringUniversidade Federal de Minas GeraisBelo HorizonteBrazil

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