Nonlinear Dynamics

, Volume 95, Issue 3, pp 2181–2195 | Cite as

Observer-based adaptive consensus tracking control for nonlinear multi-agent systems with actuator hysteresis

  • Junwei WangEmail author
  • Kairui Chen
  • Qiuli Liu
  • Qinghua Ma
Original Paper


This paper addresses the consensus tracking problem of a class of nonlinear multi-agent systems by using observer-based control. The systems are in output-feedback form with both actuator hysteresis and external disturbances. Radial basis function neural networks are used to approximate unknown nonlinear functions. By constructing a state observer and using the backstepping technique, a distributed adaptive neural output-feedback control scheme is proposed to solve the consensus tracking problem. Approximation errors of neural networks together with external disturbances are adaptively estimated and counteracted. For a communication graph containing a spanning tree, we show that the proposed controller guarantees all signals of the closed-loop system are semi-globally uniformly ultimately bounded, and the consensus tracking error and the observer error converge to an adjustable neighborhood of the origin. Finally, two simulation examples are provided to verify the performance of the control design.


Consensus Nonlinear observer Adaptive control Actuator hysteresis 



This work is supported by the National Natural Science Foundation of China (11771102, U1501251), the Characteristic Innovation Project of Education Department of Guangdong Province (2015KTSCX034), the Zhujiang New Star (201506010056), the Guangdong Province Outstanding Young Teacher Training Plan (YQ2015050) and the Natural Science Foundation of Guangdong Province (2017A030313397, 2018A030313738).

Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflict of interest.


  1. 1.
    Ren, W., Cao, Y.: Distributed Coordination of Multi-agent Networks. Springer, London, U.K. (2011)CrossRefzbMATHGoogle Scholar
  2. 2.
    Li, Z., Duan, Z.: Cooperative Control of Multi-agent Systems: A Consensus Region Approach. CRC, Boca Raton (2014)Google Scholar
  3. 3.
    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)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Olfati-Saber, R., Murray, R.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520–1533 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Münz, U., Papachristodoulou, A., Allgöwer, F.: Consensus in multi-agent systems with coupling delays and switching topology. IEEE Trans. Autom. Control 56(12), 2976–2982 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Komareji, M., Shang, Y., Bouffanais, R.: Consensus in topologically interacting swarms under communication constraints and time-delays. Nonlinear Dyn. 93(3), 1287–1300 (2018)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ren, W.: On consensus algorithms for double-integrator dynamics. IEEE Trans. Autom. Control 53(64), 1503–1509 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Hu, J., Hong, Y.: Leader-following coordination of multi-agent systems with coupling time delays. Phys. A 374(2), 853–863 (2007)CrossRefGoogle Scholar
  9. 9.
    Abdessameud, A., Tayebi, A.: On consensus algorithms design for double integrator dynamics. Automatica 49(1), 253–260 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Wang, J., Chen, K., Lewis, F.L.: Coordination of multi-agent systems on interacting physical and communication topologies. Syst. Control Lett. 100, 56–65 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Abdessameud, A., Tayebi, A.: Distributed output regulation of heterogeneous linear multi-agent systems with communication constraints. Automatica 91, 152–158 (2018)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Yu, W., Wen, G., Chen, G., Cao, J.: Distributed Cooperative Control of Multi-agent Systems. Wiley, Newark (2016)CrossRefGoogle Scholar
  13. 13.
    Ding, L., Han, Q.-L., Ge, X., Zhang, X.-M.: An overview of recent advances in event-triggered consensus of multiagent systems. IEEE Trans. Cybern. 48(4), 1110–1123 (2018)CrossRefGoogle Scholar
  14. 14.
    Yu, H., Xia, X.: Adaptive consensus of multi-agents in networks with jointly connected topologies. Automatica 48(8), 1783–1790 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Li, Z., Ren, W., Liu, X., Fu, M.: Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols. IEEE Trans. Autom. Control 58(7), 1786–1791 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Liu, K., Xie, G., Ren, W., Wang, L.: Consensus for multi-agent systems with inherent nonlinear dynamics under directed topologies. Syst. Control Lett. 62(2), 152–162 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Chen, W., Li, X., Ren, W., Wen, C.: Adaptive consensus of multi-agent systems with unknown identical control directions based on a novel nussbaum-type function. IEEE Trans. Autom. Control 59(7), 1887–1892 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Netw. 2(5), 359–366 (1989)CrossRefzbMATHGoogle Scholar
  19. 19.
    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. B Cybern. 39(3), 636–647 (2009)CrossRefGoogle Scholar
  20. 20.
    Zhang, H., Lewis, F.L., Qu, L.: Lyapunov, adaptive, and optimal design techniques for cooperative systems on directed communication graphs. IEEE Trans. Ind. Electron. 59(7), 3026–3041 (2012)CrossRefGoogle Scholar
  21. 21.
    El-Ferik, S., Qureshi, A., Lewis, F.L.: Neuro-adaptive cooperative tracking control of unknown higher-order affine nonlinear systems. Automatica 50(3), 798–808 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Peng, Z., Wang, D., Zhang, H., Sun, G.: Distributed neural network control for adaptive synchronization of uncertain dynamical multiagent systems. IEEE Trans. Neural Netw. Learn. Syst. 25(8), 1508–1519 (2014)CrossRefGoogle Scholar
  23. 23.
    Rezaee, H., Abdollahi, F.: Consensus problem over high-order multiagent systems with uncertain nonlinearities under deterministic and stochastic topologies. IEEE Trans. Cybern. 47(8), 2079–2088 (2017)CrossRefGoogle Scholar
  24. 24.
    Krstić, M., Kanellakopoulos, I., Kokotović, P.: Nonlinear and Adaptive Control Design. Wiley, New York (1995)zbMATHGoogle Scholar
  25. 25.
    Polycarpou, M.M.: Stable adaptive neural scheme for nonlinear systems. IEEE Trans. Autom. Control 41(3), 447–451 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Ge, S.S., Wang, C.: Direct adaptive NN control for a class of nonlinear systems. IEEE Trans. Neural Netw. 13(1), 214–221 (2002)CrossRefGoogle Scholar
  27. 27.
    Tong, S.-C., Li, Y.-M., Feng, G., Li, T.-S.: Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems. IEEE Trans. Syst. Man Cybern. B Cybern. 41(4), 1124–1135 (2011)CrossRefGoogle Scholar
  28. 28.
    Edalati, L., Sedigh, A.K., Shooredeli, M.A., Moarefianpour, A.: Asymptotic tracking control of strict-feedback nonlinear systems with output constraints in the presence of input saturation. IET Control Theory Appl. 12(6), 778–785 (2018)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Yoo, S.J.: Distributed consensus tracking for multiple uncertain nonlinear strict-feedback systems under a directed graph. IEEE Trans. Neural Netw. Learn. Syst. 24(4), 666–672 (2013)CrossRefGoogle Scholar
  30. 30.
    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)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Yoo, S.J.: Synchronised tracking control for multiple strict-feedback non-linear systems under switching network. IET Control Theory Appl. 8(8), 546–553 (2014)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Chen, K., Wang, J., Zhang, Y., Liu, Z.: Leader-following consensus for a class of nonlinear strick-feedback multiagent systems with state time-delays. IEEE Trans. Syst. Man Cybern. Syst. (2018).
  33. 33.
    Liu, Z., Su, L., Ji, Z.: Neural network observer-based leader-following consensus of heterogenous nonlinear uncertain systems. Int. J. Mach. Learn. Cybern. 9(9), 1435–1443 (2018)CrossRefGoogle Scholar
  34. 34.
    Wang, X., Li, S., Chen, M.: Composite backstepping consensus algorithms of leader-follower higher-order nonlinear multiagent systems subject to mismatched disturbances. IEEE Trans. Cybern. 48(6), 1935–1946 (2018)CrossRefGoogle Scholar
  35. 35.
    Chen, C.L.P., Ren, C.E., Du, T.: Fuzzy observed-based adaptive consensus tracking control for second-order multiagent systems with heterogeneous nonlinear dynamics. IEEE Trans. Fuzzy Syst. 24(4), 906–915 (2016)CrossRefGoogle Scholar
  36. 36.
    Chen, C.L.P., Wen, G.-X., Liu, Y.-J., Liu, Z.: Observer-based adaptive backstepping consensus tracking control for high-order nonlinear semi-strict-feedback multiagent systems. IEEE Trans. Cybern. 62(7), 3423–3429 (2016)Google Scholar
  37. 37.
    Tao, G., Kokotović, P.V.: Adaptive control of plants with unknown hystereses. IEEE Trans. Autom. Control. 40(2), 200–212 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Li, H., Bai, L., Wang, L., Zhou, Q., Wang, H.: Adaptive neural control of uncertain nonstrict-feedback stochastic nonlinear systems with output constraint and unknown dead zone. IEEE Trans. Syst. Man Cybern. Syst. 47(8), 2048–2059 (2017)CrossRefGoogle Scholar
  39. 39.
    Edardar, M., Tan, X., Khalil, H.K.: Design and analysis of sliding mode controller under approximate hysteresis compensation. IEEE Trans. Control Syst. Technol. 23(2), 598–608 (2015)CrossRefGoogle Scholar
  40. 40.
    Macki, J.W., Nistri, P., Zecca, P.: Mathematical models for hysteresis. SIAM Rev. 35, 94–123 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Zhou, J., Wen, C.Y., Li, T.S.: Adaptive output feedback control of uncertain nonlinear systems with hysteresis nonlinearity. IEEE Trans. Autom. Control 57(10), 2627–2633 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  42. 42.
    Zhang, Z., Xu, S., Zhang, B.: Asymptotic tracking control of uncertain nonlinear systems with unknown actuator nonlinearity. IEEE Trans. Autom. Control 59(5), 1336–1341 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Liu, Z., Chen, C., Zhang, Y., Chen, C.L.P.: Adaptive neural control for dual-arm coordination of humanoid robot with unknown nonlinearities in output mechanism. IEEE Trans. Cybern. 45(3), 521–532 (2015)CrossRefGoogle Scholar
  44. 44.
    Marino, R., Tomei, P., Verrelli, C.M.: Learning control for nonlinear systems in output feedback form. Syst. Control Lett. 61(12), 1242–1247 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  45. 45.
    Wang, C., Wen, C., Wang, W., Hu, Q.: Output-feedback adaptive consensus tracking control for a class of high-order nonlinear multi-agent systems. Int. J. Robust Nonlinear Control 27, 4931–4948 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  46. 46.
    Chen, K., Wang, J., Zhang, Y., Liu, Z.: Adaptive consensus of nonlinear multi-agent systems with unknown backlash-like hysteresis. Neurocomputing 175, 698–703 (2016)CrossRefGoogle Scholar
  47. 47.
    Marino, R., Tomei, P.: Nonlinear Control Design-Geometric. Adaptive and Robust. Prentice Hall, London (1995)zbMATHGoogle Scholar
  48. 48.
    Krishnamurthy, P., Khorrami, F.: Robust adaptive control for nonlinear systems in generalized output-feedback canonical form. Int. J. Adapt. Control Signal Process. 17(4), 285–311 (2003)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsGuangdong University of Foreign StudiesGuangzhouChina
  2. 2.School of AutomationGuangdong University of TechnologyGuangzhouChina
  3. 3.School of Mathematical SciencesSouth China Normal UniversityGuangzhouChina

Personalised recommendations