Distributed cooperative adaptive tracking control for heterogeneous systems with hybrid nonlinear dynamics

  • Xiaojie Li
  • Peng ShiEmail author
  • Yiguang Wang
Original Paper


The cooperative leader-following tracking for a group of heterogeneous mechanical systems with nonlinear hybrid order dynamics is studied. The controlled systems are considered to be composed of followers (agents) with hybrid first- and second-order time-varying dynamics. The leader is an unknown nonautonomous nonlinear system and can only give the state information of position and velocity to its neighboring followers. The followers are linked by the directed graph with fixed communication topology. And, not all of them have the information path to the leader directly. The directed information topology graph is required to have at least one spanning tree for position and velocity, respectively. Distributed cooperative adaptive control protocols are developed for all followers with first- or second-order dynamics to achieve the ultimate synchronization to the leader. The control protocols are designed based on the neural networks and the adaptive estimation algorithm for unknown time-varying functions and control coefficients. The convergence and boundedness of the synchronization error is proved by the Lyapunov theory. The simulation example verifies the correctness of the developed distributed control protocols.


Control synchronization Consensus tracking Neural networks Heterogeneous multi-agent systems Unknown nonlinear dynamics 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Lewis, F.L., Zhang, H., Hengster-Movric, K., et al.: Cooperative Control of Multi-agent Systems: Optimal and Adaptive Design Approaches. Springer, Berlin (2013)zbMATHGoogle Scholar
  2. 2.
    Ren, W., Beard, R.W., Atkins, E.M.: A survey of consensus problems in multi-agent coordination. In: Proceedings of the American Control Conference, pp. 1859–1864 (2005)Google Scholar
  3. 3.
    Zhang, H., Lewis, F.L.: Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics. Automatica 48(7), 1432–1439 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ren, W., Beard, R.W., Atkins, E.M.: Information consensus in multivehicle cooperative control. IEEE Trans. Control Syst. Technol. 27(2), 71–82 (2007)CrossRefGoogle Scholar
  5. 5.
    Song, J.: Observer-based consensus control for networked multi-agent systems with delays and packet-dropouts. Int. J. Innov. Comput. Inf. Control 12(4), 1287–1302 (2016)Google Scholar
  6. 6.
    Wang, W., Yu, Y.: Fuzzy adaptive consensus of second-order nonlinear multi-agent systems in the presence of input saturation. Int. J. Innov. Comput. Inf. Control 12(2), 533–542 (2016)Google Scholar
  7. 7.
    Shen, Q., Shi, P.: Output consensus control of multiagent systems with unknown nonlinear dead zone. IEEE Trans. Syst. Man Cybern. Syst. 46(10), 1329–1337 (2016)CrossRefGoogle Scholar
  8. 8.
    Lee, T.H., Wu, Z.G., Park, J.H.: Synchronization of a complex dynamical network with coupling time-varying delays via sampled-data control. Appl. Math. Comput. 219(3), 1354–1366 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Shen, Q., Shi, P.: Distributed command filtered backstepping consensus tracking control of nonlinear multiple-agent systems in strict-feedback form. Automatica 53, 120–124 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lee, T.H., Park, J.H.: Improved criteria for sampled-data synchronization of chaotic Lur’e systems using two new approaches. Nonlinear Anal. Hybrid Syst. 24, 132–145 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Meng, Z., Ren, W., Cao, Y., et al.: Leaderless and leader-following consensus with communication and input delays under a directed network topology. IEEE Trans. Syst. Man Cybern. Part B Cybern. 41(1), 75–88 (2011)CrossRefGoogle Scholar
  12. 12.
    Shen, Q., Jiang, B., Shi, P., et al.: Cooperative adaptive fuzzy tracking control for networked unknown nonlinear multiagent systems with time-varying actuator faults. IEEE Trans. Fuzzy Syst. 22(3), 494–504 (2014)CrossRefGoogle Scholar
  13. 13.
    Shi, P., Shen, Q.: Cooperative control of multi-agent systems with unknown state-dependent controlling effects. IEEE Trans. Autom. Sci. Eng. 12(3), 827–834 (2015)CrossRefGoogle Scholar
  14. 14.
    Cao, Y., Yu, W., Ren, W., et al.: An overview of recent progress in the study of distributed multi-agent coordination. IEEE Trans. Ind. Inform. 9(1), 427–438 (2013)CrossRefGoogle Scholar
  15. 15.
    Ren, W., Beard, R.W.: Distributed Consensus in Multi-vehicle Cooperative Control. Springer, London (2008)CrossRefGoogle Scholar
  16. 16.
    Ren, W., Cao, Y.: Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues. Springer, Berlin (2010)zbMATHGoogle Scholar
  17. 17.
    Qu, Z.: Cooperative Control of Dynamical Systems: Applications to Autonomous Vehicles. Springer, Berlin (2009)zbMATHGoogle Scholar
  18. 18.
    Zheng, Y., Zhu, Y., Wang, L.: Consensus of heterogeneous multi-agent systems. IET Control Theory Appl. 5(16), 1881–1888 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Liu, C.L., Liu, F.: Stationary consensus of heterogeneous multi-agent systems with bounded communication delays. Automatica 47(9), 2130–2133 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zheng, Y., Wang, L.: Consensus of heterogeneous multi-agent systems without velocity measurements. Int. J. Control 85(7), 906–914 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Liu, C.L., Liu, F.: Dynamical consensus seeking of heterogeneous multic-systems under input delays. Int. J. Commun. Syst. 26(10), 1243–1258 (2013)Google Scholar
  22. 22.
    Feng, Y., Xu, S., Lewis, F.L., et al.: Consensus of heterogeneous first-second-order multic-systems with directed communication topologies. Int. J. Robust Nonlinear Control 25(3), 362–375 (2015)CrossRefGoogle Scholar
  23. 23.
    Zheng, Y., Wang, L.: Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements. Syst. Control Lett. 61(8), 871–878 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Das, A., Lewis, F.L.: Distributed adaptive control for synchronization of unknown nonlinear networked systems. Automatica 46(12), 2014–2021 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Das, A., Lewis, F.L.: Cooperative adaptive control for synchronization of second-order systems with unknown nonlinearities. Int. J. Robust Nonlinear Control 21(13), 1509–1524 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lewis, F.W., Jagannathan, S., Yesildirak, A.: Neural Network Control of Robot Manipulators and Non-linear Systems. CRC Press, Boca Raton (1998)Google Scholar
  27. 27.
    Khoo, S., Xie, L., Man, Z.: Robust finite-time consensus tracking algorithm for multirobot systems. IEEE/ASME Trans. Mechatron. 14(2), 219–228 (2009)CrossRefGoogle Scholar
  28. 28.
    Shivakumar, P.N., Chew, K.H.: A sufficient condition for nonvanishing of determinants. In: Proceedings of the American Mathematical Society, pp. 63–66 (1974)Google Scholar
  29. 29.
    Stone, M.H.: The generalized Weierstrass approximation theorem. Math. Mag. 21(5), 237–254 (1948)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Khalil, H.K.: Nonlinear Systems, vol. 9(4.2), 3rd edn. Prentice Hall, New Jersey (2002)zbMATHGoogle Scholar
  31. 31.
    Ge, S.S., Wang, C.: Adaptive neural control of uncertain MIMO nonlinear systems. IEEE Trans. Neural Netw. 15(3), 674–692 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.College of AutomationHarbin Engineering UniversityHarbinChina
  2. 2.School of Electrical and Electronic EngineeringThe University of AdelaideAdelaideAustralia
  3. 3.College of Electronic EngineeringHeilongjiang UniversityHarbinChina

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