Distributed finite-time coordinated tracking control for multiple Euler–Lagrange systems with input nonlinearity

  • Yanchao Sun
  • Liangliang Chen
  • Hongde QinEmail author
  • Wenjia Wang
Original Paper


In this paper, distributed finite-time coordinated tracking control for multiple Euler–Lagrange systems with input nonlinearity is investigated by using backstepping design technique under directed topology. The controller is designed under the condition that the information of the dynamic leader is available to only a subset of the followers. We first design an auxiliary variable relating to trajectory errors among neighbor agents. Then a distributed finite-time tracking control algorithm is developed where two neural networks are used to approximate the nonlinear model uncertainties and input nonlinearity, respectively. When considering that there exists incomplete known state for each follower, a modified distributed finite-time tracking control strategy is designed by utilizing high-gain observers. Based on backstepping method, finite-time technique, and graph theory, both proposed control strategies guarantee that tracking errors between each follower and the leader could be ultimately bounded in finite time. Numerical simulations show the superiorities of the proposed protocols by comparisons with existing methods.


Euler–Lagrange systems Distributed tracking control Finite-time control Backstepping control Neural network Input nonlinearity 



This work was supported by the National Natural Science Foundation of China, under grant 61803119 and U1713205.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Hao, Y.Q., Duan, Z.S., Chen, G.R.: Further on the controllability of networked MIMO LTI systems. Int. J. Robust Nonlinear Control 28(5), 1778–1788 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Zhao, Q.L., Dong, X.W., Liang, Z.X., Ren, Z.: Distributed group cooperative guidance for multiple missiles with fixed and switching directed communication topologies. Nonlinear Dyn. 90(4), 2507–2523 (2017)CrossRefGoogle Scholar
  3. 3.
    Wang, F., Chen, B., Zhang, Z.Y., Lin, C.: Adaptive tracking control of uncertain switched stochastic nonlinear systems. Nonlinear Dyn. 84(4), 2099–2109 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Xu, C.J., Zheng, Y., Su, H.S., Chen, M.Z.Q., Zhang, C.F.: Cluster consensus for second-order mobile multi-agent systems via distributed adaptive pinning control under directed topology. Nonlinear Dyn. 83(4), 1975–1985 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hu, G.Q.: Robust consensus tracking for an integrator-type multi-agent system with disturbances and unmodelled dynamics. Int. J. Control 84(1), 1–8 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cao, Y.C., Ren, W.: Distributed coordinated tracking with reduced interaction via a variable structure approach. IEEE Trans. Autom. Control 57(1), 33–48 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li, Z.X., Ji, H.B.: Distributed consensus and tracking control of second-order time-varying nonlinear multi-agent systems. Int. J. Robust Nonlinear Control 27(17), 3549–3563 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Zhao, Y., Liu, Y.F., Wen, G.H., Alotaibi, N.D., Shi, Z.K.: Distributed finite-time tracking of second-order multi-agent systems: an edge-based approach. IET Control Theory Appl. 12(1), 149–154 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhao, D.Y., Li, S.Y., Zhu, Q.M.: Adaptive synchronised tracking control for multiple robotic manipulators with uncertain kinematics and dynamics. Int. J. Syst. Sci. 47(4), 791–804 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cheng, L., Cheng, M., Yu, H.N., Deng, L., Hou, Z.G.: Distributed tracking control of uncertain multiple manipulators under switching topologies using neural networks. In: Cheng, L., Liu, Q. S., Ronzhin, A. (eds.) Advances in Neural Networks—ISNN 2016: 13th International Symposium on Neural Networks, ISNN 2016, St. Petersburg, Russia, July 6–8, 2016, Proceedings, Springer International Publishing, Cham, pp. 233–241 (2016)Google Scholar
  11. 11.
    Hao, Y.Q., Duan, Z.S., Wen, G.H.: Controllability and observability of an n-link robot with multiple active links. Int. J. Robust Nonlinear Control 27(18), 4633–4647 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Yin, Y.F., Hu, L.L., Xu, N., Wang, X.Y.: Synchronization for non-uniform sampling networked rigid bodies. Neurocomputing 173, 1761–1767 (2016)CrossRefGoogle Scholar
  13. 13.
    Chen, G., Lewis, F.L.: Distributed tracking control for networked mechanical systems. Asian J. Control 14(6), 1459–1469 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Yang, Z.J., Shibuya, Y., Qin, P.: Distributed robust control for synchronised tracking of networked Euler–Lagrange systems. Int. J. Syst. Sci. 46(4), 720–732 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yang, Q.K., Fang, H., Chen, J., Wang, X.J.: Distributed observer-based coordination for multiple Lagrangian systems using only position measurements. IET Control Theory Appl. 8(17), 2102–2114 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yang, Q.K., Fang, H., Chen, J., Jiang, Z.P., Cao, M.: Distributed global output-feedback control for a class of Euler–Lagrange systems. IEEE Trans. Autom. Control 62(9), 4855–4861 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhao, Y., Duan, Z.S., Wen, G.H., Chen, G.R.: Distributed finite-time tracking for a multi-agent system under a leader with bounded unknown acceleration. Syst. Control Lett. 81, 8–13 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, S.H., Du, H.B., Lin, X.Z.: Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics. Automatica 47(8), 1706–1712 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Chen, G., Yue, Y., Song, Y.: Finite-time cooperative-tracking control for networked Euler–Lagrange systems. IET Control Theory Appl. 7(11), 1487–1497 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sun, Y.C., Ma, G.F., Liu, M.M., Li, C.J., Liang, J.B.: Distributed finite-time coordinated control for multi-robot systems. Trans. Inst. Meas. Control 40(9), 2912–2927 (2018)CrossRefGoogle Scholar
  21. 21.
    Sun, Y.C., Ma, G.E., Liu, M.M., Chen, L.M.: Distributed finite-time configuration containment control for satellite formation. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 231(9), 1609–1620 (2017)CrossRefGoogle Scholar
  22. 22.
    Jiang, Y.M., Yang, C.G., Na, J., Li, G., Li, Y.N., Zhong, J.P.: A brief review of neural networks based learning and control and their applications for robots. Complexity (2017).
  23. 23.
    Chen, G., Lewis, F.L.: Distributed adaptive tracking control for synchronization of unknown networked Lagrangian systems. IEEE Trans. Syst. Man Cybern. Part B Cybern. 41(3), 805–816 (2011)CrossRefGoogle Scholar
  24. 24.
    Sun, Y.C., Wang, W.J., Ma, G.F., Li, Z., Li, C.J.: Backstepping-based distributed coordinated tracking for multiple uncertain Euler–Lagrange systems. J. Syst. Eng. Electron. 27(5), 1083–1095 (2016)CrossRefGoogle Scholar
  25. 25.
    Li, D.Y., Ma, G.F., He, W., Zhang, W., Li, C.J., Ge, S.S.: Distributed coordinated tracking control of multiple Euler–Lagrange systems by state and output feedback. IET Control Theory Appl. 11(14), 2213–2221 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Su, H.S., Chen, M.Z.Q., Chen, G.R.: Robust semi-global coordinated tracking of linear multi-agent systems with input saturation. Int. J. Robust Nonlinear Control 25(14), 2375–2390 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Chen, M., Ge, S.S., How, B.V.E.: Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities. IEEE Trans. Neural Netw. 21(5), 796–812 (2010)CrossRefGoogle Scholar
  28. 28.
    Shen, Q.K., Shi, P., Shi, Y., Zhang, J.H.: Adaptive output consensus with saturation and dead-zone and its application. IEEE Trans. Ind. Electron. 64(6), 5025–5034 (2017)CrossRefGoogle Scholar
  29. 29.
    Li, Z.K., Wen, G.H., Duan, Z.S., Ren, W.: Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs. IEEE Trans. Autom. Control 60(4), 1152–1157 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Mei, J., Ren, W., Ma, G.F.: Distributed coordinated tracking with a dynamic leader for multiple Euler–Lagrange systems. IEEE Trans. Autom. Control 56(6), 1415–1421 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Hua, C.C., Chang, Y.F.: Decentralized adaptive neural network control for mechanical systems with dead-zone input. Nonlinear Dyn. 76(3), 1845–1855 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Hou, Z.G., Cheng, L., Tan, M.: Decentralized robust adaptive control for the multiagent system consensus problem using neural networks. IEEE Trans. Syst. Man Cybern. Part B Cybern. 39(3), 636–647 (2009)CrossRefGoogle Scholar
  33. 33.
    Ge, S.S., Jin, Z.: Neural-network control of nonaffine nonlinear system with zero dynamics by state and output feedback. IEEE Trans. Neural Netw. 14(4), 900–918 (2003)CrossRefGoogle Scholar
  34. 34.
    He, W., Ouyang, Y.C., Hong, J.: Vibration control of a flexible robotic manipulator in the presence of input deadzone. IEEE Trans. Ind. Inform. 13(1), 48–59 (2017)CrossRefGoogle Scholar
  35. 35.
    He, W., Chen, Y.H., Yin, Z.: Adaptive neural network control of an uncertain robot with full-state constraints. IEEE Trans. Cybern. 46(3), 620–629 (2016)CrossRefGoogle Scholar
  36. 36.
    He, W., Ge, S.S., Li, Y.A., Chew, E., Ng, Y.S.: Neural network control of a rehabilitation robot by state and output feedback. J. Intell. Robot. Syst. 80(1), 15–31 (2015)CrossRefGoogle Scholar
  37. 37.
    Mei, J., Zhang, H.B., Ma, G.F.: Adaptive coordinated tracking for networked Euler–Lagrange systems under a directed graph. Acta Auto. Sin. 37(5), 596–603 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Tee, K.P., Ge, S.S.: Control of fully actuated ocean surface vessels using a class of feedforward approximators. IEEE Trans. Control Syst. Technol. 14(4), 750–756 (2006)CrossRefGoogle Scholar
  39. 39.
    Ge, S.S., Hang, C.C., Lee, T.H., Zhang, T.: Stable Adaptive Neural Network Control, vol. 13. Springer, Berlin (2001)zbMATHGoogle Scholar
  40. 40.
    Xu, Y.S., Kanade, T.: Space Robotics: Dynamics and Control. Springer, Berlin (1992)Google Scholar
  41. 41.
    Wang, H.L.: Task-space synchronization of networked robotic systems with uncertain kinematics and dynamics. IEEE Trans. Autom. Control 58(12), 3169–3174 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Yanchao Sun
    • 1
  • Liangliang Chen
    • 2
  • Hongde Qin
    • 1
    Email author
  • Wenjia Wang
    • 2
  1. 1.Science and Technology on Underwater Vehicle LaboratoryHarbin Engineering UniversityHarbinPeople’s Republic of China
  2. 2.Department of Control Science and EngineeringHarbin Institute of TechnologyHarbinPeople’s Republic of China

Personalised recommendations