Fractional order chattering-free robust adaptive backstepping control technique

  • Yiheng Wei
  • Dian Sheng
  • Yuquan Chen
  • Yong WangEmail author
Original Paper


This paper proposes an observer-based fractional order robust adaptive backstepping control scheme for incommensurate fractional order systems with partial measurable state. The chattering phenomenon is carefully analyzed, and then a class of chattering-free controllers are proposed. To handle the time-varying disturbance, a robust adaptive control scheme is developed via the backstepping procedure. The method to generate the required fractional order differential signals online is provided. After designing the controller, the stability of the resulting closed-loop system is analyzed systematically. To highlight the efficiency of our findings, one illustrative example is provided at last.


Robust adaptive control Fractional order systems Incommensurate case Indirect Lyapunov method Chattering free 



The authors would like to thank the Associate Editor and the anonymous reviewers for their keen and insightful comments which greatly improved the contents and the presentation. The work described in this paper was fully supported by the National Natural Science Foundation of China (61601431, 61573332), the Anhui Provincial Natural Science Foundation (1708085QF141), the Fundamental Research Funds for the Central Universities (WK2100100028), and the General Financial Grant from the China Postdoctoral Science Foundation (2016M602032).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


  1. 1.
    Balachandra, M., Sethna, P.R.: Adaptive backstepping control of a dual-manipulator cooperative system handling a flexible payload. Arch. Ration. Mech. Anal. 58, 261–283 (1975)CrossRefGoogle Scholar
  2. 2.
    Krsti, M., Kanellakopoulos, I., Kokotovi, P.V.: Adaptive nonlinear control without overparametrization. Syst. Control Lett. 19(3), 177–185 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Kokotovic, P.V.: The joy of feedback: nonlinear and adaptive. IEEE Control Syst. 12(3), 7–17 (1992)CrossRefGoogle Scholar
  4. 4.
    Zhou, J., Wen, C.Y.: Adaptive Backstepping Control of Uncertain Systems: Nonsmooth Nonlinearities, Interactions or Time-variations. Springer, Berlin (2008)zbMATHGoogle Scholar
  5. 5.
    Guo, Q., Zhang, Y., Celler, B.G., Su, S.W.: Backstepping control of electro-hydraulic system based on extended-state-observer with plant dynamics largely unknown. IEEE Trans. Ind. Electron. 63(11), 6909–6920 (2016)CrossRefGoogle Scholar
  6. 6.
    Chen, C.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. 46(7), 1591–1601 (2016)CrossRefGoogle Scholar
  7. 7.
    Chen, F.Y., Lei, W., Zhang, K.K., Tao, G., Jiang, B.: A novel nonlinear resilient control for a quadrotor uav via backstepping control and nonlinear disturbance observer. Nonlinear Dyn. 85(2), 1281–1295 (2016)CrossRefGoogle Scholar
  8. 8.
    Liu, S., Liu, Y., Wang, N.: Nonlinear disturbance observer-based backstepping finite-time sliding mode tracking control of underwater vehicles with system uncertainties and external disturbances. Nonlinear Dyn. 88(1), 465–476 (2017)CrossRefGoogle Scholar
  9. 9.
    Efe, M.Ö.: Backstepping control technique for fractional order systems. In: The 3rd Conference on Nonlinear Science and Complexity. No. Paper 105, Ankara, Turkey (2010)Google Scholar
  10. 10.
    Efe, M.Ö.: Fractional order systems in industrial automation-a survey. IEEE Trans. Ind. Inform. 7(4), 582–591 (2011)CrossRefGoogle Scholar
  11. 11.
    Efe, M.Ö.: Application of backstepping control technique to fractional order dynamic systems. Fractional Dynamics and Control, vol. 3, pp. 33–47. Springer, New York (2012)CrossRefGoogle Scholar
  12. 12.
    Shahiri, T.M., Ranjbar, A., Ghaderi, R., Karami, M., Hosseinnia, S.H.: Adaptive backstepping chaos synchronization of fractional order coullet systems with mismatched parameters. In: The 4th IFAC Workshop Fractional Differentiation and its Applications, No. FDA10-104. Badajoz, Spain (2010)Google Scholar
  13. 13.
    Sahab, A.R., Ziabari, M.T., Modabbernia, M.R.: A novel fractional-order hyperchaotic system with a quadratic exponential nonlinear term and its synchronization. Adv. Differ. Equ. (2012).
  14. 14.
    Takamatsu, T., Ohmori, H.: Sliding mode controller design based on backstepping technique for fractional order system. SICE J. Control, Meas. Syst. Integr. 9(4), 151–157 (2016)CrossRefGoogle Scholar
  15. 15.
    Ding, D.S., Qi, D.L., Wang, Q.: Non-linear Mittag-Leffler stabilisation of commensurate fractional-order non-linear systems. IET Control Theory Appl. 9(5), 681–690 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shukla, M.K., Sharma, B.B.: Stabilization of a class of fractional order chaotic systems via backstepping approach. Chaos, Solitons Fractals 98, 56–62 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Shukla, M.K., Sharma, B.B.: Control and synchronization of a class of uncertain fractional order chaotic systems via adaptive backstepping control. Asian J. Control 20(2), 707–720 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ding, D.S., Qi, D.L., Peng, J.M., Wang, Q.: Asymptotic pseudo-state stabilization of commensurate fractional-order nonlinear systems with additive disturbance. Nonlinear Dyn. 81(1), 667–677 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wang, Q., Zhang, J.L., Ding, D.S., Qi, D.L.: Adaptive Mittag-Leffler stabilization of a class of fractional order uncertain nonlinear systems. Asian J. Control 18(6), 2343–2351 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Bigdeli, N., Ziazi, H.A.: Finite-time fractional-order adaptive intelligent backstepping sliding mode control of uncertain fractional-order chaotic systems. J. Frankl. Inst. 354(1), 160–183 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Liu, H., Pan, Y., Li, S., Chen, Y.: Adaptive fuzzy backstepping control of fractional-order nonlinear systems. IEEE Trans. Syst. Man Cybern: Syst. 47(8), 2209–2217 (2017)CrossRefGoogle Scholar
  22. 22.
    Zhao, Y.H., Chen, N., Tai, Y.P.: Trajectory tracking control of wheeled mobile robot based on fractional order backstepping. In: The 28th Chinese Control and Decision Conference, pp. 6730–6734. Yinchuan, China (2016)Google Scholar
  23. 23.
    Liang, Z.H., Gao, J.F.: Chaos in a fractional-order single-machine infinite-bus power system and its adaptive backstepping control. Int. J. Mod. Nonlinear Theory Appl. 5(3), 122–131 (2016)CrossRefGoogle Scholar
  24. 24.
    Nikdel, N., Badamchizadeh, M., Azimirad, V., Nazari, M.A.: Fractional-order adaptive backstepping control of robotic manipulators in the presence of model uncertainties and external disturbances. IEEE Trans. Ind. Electron. 63(10), 6249–6256 (2016)CrossRefGoogle Scholar
  25. 25.
    Luo, S.H., Li, S.B., Tajaddodianfar, F., Hu, J.J.: Observer-based adaptive stabilization of the fractional-order chaotic MEMS resonator. Nonlinear Dyn. 92(3), 1079–1089 (2018)CrossRefGoogle Scholar
  26. 26.
    Trigeassou, J.C., Maamri, N., Sabatier, J., Oustaloup, A.: A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91(3), 437–445 (2011)CrossRefGoogle Scholar
  27. 27.
    Wei, Y.H., Chen, Y.Q., Liang, S., Wang, Y.: A novel algorithm on adaptive backstepping control of fractional order systems. Neurocomputing 165, 395–402 (2015)CrossRefGoogle Scholar
  28. 28.
    Wei, Y.H., Tse, P.W., Yao, Z., Wang, Y.: Adaptive backstepping output feedback control for a class of nonlinear fractional order systems. Nonlinear Dyn. 86(2), 1047–1056 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sheng, D., Wei, Y.H., Cheng, S.S., Shuai, J.M.: Adaptive backstepping control for fractional order systems with input saturation. J. Frankl. Inst. 354(5), 2245–2268 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhou, X., Wei, Y.H., Liang, S., Wang, Y.: Robust fast controller design via nonlinear fractional differential equations. ISA Trans. 69, 20–30 (2017)CrossRefGoogle Scholar
  31. 31.
    Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2951–2957 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    La Salle, J.P.: An invariance principle in the theory of stability. In: International Symposium on Differential Equations and Dynamical Systems, pp. 277–286. Puerto Rico, USA (1965)Google Scholar
  33. 33.
    Wei, Y.H., Du, B., Cheng, S.S., Wang, Y.: Fractional order systems time-optimal control and its application. J. Optim. Theory Appl. 174(1), 122–138 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Wei, Y.H., Tse, P.W., Du, B., Wang, Y.: An innovative fixed-pole numerical approximation for fractional order systems. ISA Trans. 62, 94–102 (2016)CrossRefGoogle Scholar
  35. 35.
    Chen, Y.Q., Wei, Y.H., Zhou, X., Wang, Y.: Stability for nonlinear fractional order systems: an indirect approach. Nonlinear Dyn. 89(2), 1011–1018 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of AutomationUniversity of Science and Technology of ChinaHefeiChina

Personalised recommendations