Fractional order chattering-free robust adaptive backstepping control technique


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.

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  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)

    Article  MATH  Google Scholar 

  2. 2.

    Krsti, M., Kanellakopoulos, I., Kokotovi, P.V.: Adaptive nonlinear control without overparametrization. Syst. Control Lett. 19(3), 177–185 (1992)

    Article  MathSciNet  Google Scholar 

  3. 3.

    Kokotovic, P.V.: The joy of feedback: nonlinear and adaptive. IEEE Control Syst. 12(3), 7–17 (1992)

    Article  Google Scholar 

  4. 4.

    Zhou, J., Wen, C.Y.: Adaptive Backstepping Control of Uncertain Systems: Nonsmooth Nonlinearities, Interactions or Time-variations. Springer, Berlin (2008)

    Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  MATH  Google 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)

    Article  MATH  Google 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)

  10. 10.

    Efe, M.Ö.: Fractional order systems in industrial automation-a survey. IEEE Trans. Ind. Inform. 7(4), 582–591 (2011)

    Article  Google 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)

    Google 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)

  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)

    Article  Google 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)

    Article  MathSciNet  Google 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)

    Article  MathSciNet  MATH  Google 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)

    Article  MathSciNet  MATH  Google 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)

    Article  MathSciNet  MATH  Google 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)

    Article  MathSciNet  MATH  Google 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)

    Article  MathSciNet  MATH  Google 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)

    Article  Google 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)

  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)

    Article  Google 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)

    Article  Google 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)

    Article  MATH  Google 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)

    Article  MATH  Google 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)

    Article  Google 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)

    Article  MathSciNet  MATH  Google 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)

    Article  MathSciNet  MATH  Google 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)

    Article  Google 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)

    Article  MathSciNet  Google 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)

  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)

    Article  MathSciNet  MATH  Google 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)

    Article  Google 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)

    Article  MATH  Google Scholar 

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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).

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Correspondence to Yong Wang.

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Wei, Y., Sheng, D., Chen, Y. et al. Fractional order chattering-free robust adaptive backstepping control technique. Nonlinear Dyn 95, 2383–2394 (2019).

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