Observer-based fractional-order adaptive type-2 fuzzy backstepping control of uncertain nonlinear MIMO systems with unknown dead-zone

  • Adeleh Arabzadeh Jafari
  • Seyed Mohammad Ali MohammadiEmail author
  • Maliheh Maghfoori Farsangi
  • Mohsen Hasanpour Naseriyeh
Original Paper


A new problem of observer-based fractional adaptive type-2 fuzzy backstepping control for a class of fractional-order MIMO nonlinear dynamic systems with dead-zone input nonlinearity is considered in the presence of model uncertainties and external disturbances where the control scheme is constructed by combining the backstepping dynamic surface control (DSC) and fractional adaptive type-2 fuzzy technique. First, a linear state observer estimates immeasurable states. Second, the unknown nonlinear functions of the uncertain system are approximated with interval type-2 fuzzy logic systems. Third, to avoid the complication of backstepping design process, the DSC is used. Fourth, by using the fractional adaptive backstepping, fractional adaptive laws are constructed, the proposed method is applied to a class of uncertain fractional-order nonlinear MIMO system. In order to have a better control performance in reducing tracking error, the controller parameters are tuned by using the PSO algorithm. Stability of the system is proven by the Mittag-Leffler method. It is presented that the proposed design guarantees the boundedness property for the system and also the tracking error can converge to a small neighborhood of the zero. The simulation examples are given to show the efficiency of the proposed controller.


MIMO nonlinear system Fractional-order Adaptive backstepping Dynamic surface control (DSC) Interval type-2 fuzzy logic system (IT2FLS) Unknown dead-zone 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.


  1. 1.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)zbMATHGoogle Scholar
  2. 2.
    Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls Fundamentals and Applications. Springer, London (2010)zbMATHGoogle Scholar
  3. 3.
    Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)zbMATHGoogle Scholar
  4. 4.
    Baleanu, D., Machado, J.A.T., Luo, A.C.J.: Fractional Dynamics and Control. Springer, New York (2012)Google Scholar
  5. 5.
    Petras, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, Berlin (2011)zbMATHGoogle Scholar
  6. 6.
    Wei, Y.H., Chen, Y.Q., Liang, S., Wang, Y.: A novel algorithm on adaptive backstepping control of fractional order systems. Neurocomputing 127, 395–402 (2015)Google Scholar
  7. 7.
    Oustaloup, A., Sabatier, J., Lanusse, P.: From fractal robustness to CRONE control. Fract. Calc. Appl. Anal. 2, 1–30 (1999)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Wei, Y., Sun, Z., Hu, Y., Wang, Y.: On fractional order composite model reference adaptive control. Int. J. Syst. Sci. 47(11), 1–11 (2015)MathSciNetGoogle Scholar
  9. 9.
    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)Google Scholar
  10. 10.
    Wu, Y., Lv, H.: Adaptive neural network backstepping control for a class of uncertain fractional order chaotic systems with unknown backlash-like hysteresis. AIP Adv. 6(8), 085–121 (2016)Google Scholar
  11. 11.
    Li, L., Sun, Y.: Adaptive fuzzy control for nonlinear fractional-order uncertain systems with unknown uncertainties and external disturbance. SOURCE Entropy 17(8), 5580–5592 (2015)Google Scholar
  12. 12.
    Khettab, K., Ladaci, S., Bensafia, Y.: Fuzzy adaptive control of a fractional-order chaotic system with unknown control gain sign using a fractional order Nussbaum gain. IEEE/CAA J. Autom. Sin. (2017). Google Scholar
  13. 13.
    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)MathSciNetGoogle Scholar
  14. 14.
    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)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wei, Y.H., Peter, W.T., Yao, Z., Wang, Y.: Adaptive backstepping output feedback control for a class of nonlinear fractional order systems. Nonlinear Dyn. 86(2), 1047–1056 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ding, D.S., Qi, D.L., Meng, Y.M., Xu, L.: Adaptive Mittag-Leffler stabilization of commensurate fractional-order nonlinear systems. In: The 53rd IEEE Conference on Decision and Control, Los Angeles, USA, pp. 6920–6926 (2014)Google Scholar
  17. 17.
    Hu, T., Zhang, X., Zhong, S.: Global asymptotic synchronization of nonidentical fractional-order neural networks. Neurocomputing 313(3), 39–46 (2018)Google Scholar
  18. 18.
    Mohammadzadeh, A., Ghaemi, S.: Robust synchronization of uncertain fractional-order chaotic systems with time-varying delay. Nonlinear Dyn. 93(4), 1809–1821 (2018)zbMATHGoogle Scholar
  19. 19.
    Yang, X., Li, C., Huang, T., Song, Q., Huang, J.: Synchronization of fractional-order memristor-based complex-valued neural networks with uncertain parameters and time delays. Chaos Solitons Fractals 110, 105–123 (2018)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zhu, G., Du, J., Kao, Y.: Command filtered robust adaptive NN control for a class of uncertain strict-feedback nonlinear systems under input saturation. J. Frankl. Inst. 355(15), 7548–7569 (2018)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Maa, J., Xu, S., Li, Y., Chu, Y., Zhang, Z.: Neural networks-based adaptive output feedback control for a class of uncertain nonlinear systems with input delay and disturbances. J. Frankl. Inst. 355(13), 5503–5519 (2018)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ma, H., Liang, H., Zhou, Q., Ahn, C.K.: Adaptive dynamic surface control design for uncertain nonlinear strict-feedback systems with unknown control direction and disturbances. Chaos Solitons Fractals PP(99), 1–10 (2018). Google Scholar
  23. 23.
    Liu, Z., Lai, G., Zhang, Y., Chen, C.: Adaptive fuzzy tracking control of nonlinear time-delay systems with dead-zone output mechanism based on a novel smooth model. IEEE Trans. Fuzzy Syst. 23(6), 1998–2011 (2015)Google Scholar
  24. 24.
    Wang, C., Guo, L., Wen, C., Hu, Q., Qiao, J.: Adaptive neural network control for a class of nonlinear systems with unknown control direction. IEEE Trans. Syst. Man Cybern. Syst. PP(99), 1–11 (2018). Google Scholar
  25. 25.
    Tong, S.C., Sui, S., Li, Y.M.: Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained. IEEE Trans. Fuzzy Syst. 23(4), 729–743 (2015)Google Scholar
  26. 26.
    Tong, S.C., Li, Y.M.: Adaptive fuzzy output feedback control of MIMO nonlinear systems with unknown dead-zone inputs. IEEE Trans. Fuzzy Syst. 21, 134–146 (2013)Google Scholar
  27. 27.
    Zhao, X., Wang, X., Zong, G., Li, H.: Fuzzy-approximation-based adaptive output-feedback control for uncertain non-smooth nonlinear systems. IEEE Trans. Fuzzy Syst. (2018). Google Scholar
  28. 28.
    Zhou, Q., Wu, C., Shi, P.: Observer-based adaptive fuzzy tracking control of nonlinear systems with time delay and input saturation. Fuzzy Sets Syst. 316, 49–68 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Wang, T., Tong, S.C.: Observer-based fuzzy adaptive optimal stabilization control for completely unknown nonlinear interconnected systems. Neurocomputing 313, 415–425 (2018)Google Scholar
  30. 30.
    Chen, B., Lin, C., Liu, X., Liu, K.: Observer-based adaptive fuzzy control for a class of nonlinear delayed systems. IEEE Trans. Syst. Man Cybern. Syst. 46(1), 27–36 (2016)Google Scholar
  31. 31.
    Liu, H., Pan, Y., Li, S., Chen, Y.: Adaptive fuzzy backstepping control of fractional-order nonlinear systems. IEEE Trans. Syst. Man Cybern. Syst. 47, 2209–2217 (2017)Google Scholar
  32. 32.
    Yip, P., Hedrick, J.: Adaptive dynamic surface control: a simplified algorithm for adaptive backstepping control of nonlinear systems. Int. J. Control 71(5), 959–979 (1998)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Zhou, Q., Wu, C., Jing, X., Wang, L.: Adaptive fuzzy backstepping dynamic surface control for nonlinear Input-delay systems. Neurocomputing 199, 58–65 (2016)Google Scholar
  34. 34.
    Shi, X., Lim, C.C., Shi, P., Xu, S.: Adaptive neural dynamic surface control for nonstrict-feedback systems with output dead zone. IEEE Trans. Neural Netw. Learn. Syst. 29(11), 5200–5213 (2018)MathSciNetGoogle Scholar
  35. 35.
    Lin, T.C.: Based on interval type-2 fuzzy-neural network direct adaptive sliding mode control for SISO nonlinear systems. Commun. Nonlinear Sci. Numer. Simul. 15, 4084–4099 (2010)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Shahriari, M., Rahmani, S.: Adaptive interval type-2 fuzzy dynamic surface control for uncertain nonlinear systems with unknown asymmetric dead-zone input. Trans. Inst. Meas. Control (2018). Google Scholar
  37. 37.
    Kumar, A., Kumar, V.: Performance analysis of optimal hybrid novel interval type-2 fractional order fuzzy logic controllers for fractional order systems. Expert Syst. Appl. 93, 435–455 (2018)Google Scholar
  38. 38.
    Shahnazi, R.: Observer-based adaptive interval type-2 fuzzy control of uncertain MIMO nonlinear systems with unknown asymmetric saturation actuators. Neurocomputing 171, 1053–1065 (2016)Google Scholar
  39. 39.
    Lin, X.L., Wu, C.F., Chen, B.S.: Robust H adaptive fuzzy tracking control for MIMO nonlinear stochastic poisson jump diffusion systems. IEEE Trans. Cybern. (2018). Google Scholar
  40. 40.
    Lv, W., Wang, F., Li, Y.: Finite-time adaptive fuzzy output-feedback control of MIMO nonlinear systems with hysteresis. Neurocomputing 296, 74–81 (2018)Google Scholar
  41. 41.
    Wang, N., Tong, S.C., Shi, P.: Observer-based adaptive fuzzy control of a class of MIMO non-strict feedback nonlinear systems. J. Frankl. Inst. 355(12), 4873–4896 (2018)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Hamdy, M., Ramadan, A., Abozalam, B.: A novel inverted fuzzy decoupling scheme for MIMO systems with disturbance: a case study of binary distillation column. J. Intell. Manuf. 29, 1–13 (2016)Google Scholar
  43. 43.
    Li, Y.M., Tong, S.C.: Adaptive fuzzy control with prescribed performance for block-triangular-structured nonlinear systems. IEEE Trans. Fuzzy Syst. 26(3), 1153–1163 (2017)Google Scholar
  44. 44.
    Chen, B., Lin, C., Liu, X., Liu, K.: Adaptive fuzzy tracking control for a class of MIMO nonlinear systems in nonstrict-feedback form. IEEE Trans. Cybern. 45(12), 2744–2755 (2015)Google Scholar
  45. 45.
    Li, Y.M., Tong, S.C., Li, T.: Observer-based adaptive fuzzy tracking control of MIMO stochastic nonlinear systems with unknown control directions and unknown dead zones. IEEE Trans. Fuzzy Syst. 23(4), 1228–1241 (2015)Google Scholar
  46. 46.
    Roohi, M., Aghababa, M.P., Haghighi, A.H.: Switching adaptive controllers to control fractional-order complex systems with unknown structure and input nonlinearities. Complexity 21(2), 211–223 (2015)MathSciNetGoogle Scholar
  47. 47.
    Shahnazi, R.: Output feedback adaptive fuzzy control of uncertain MIMO nonlinear systems with unknown input nonlinearities. ISA Trans. 54, 39–51 (2015)Google Scholar
  48. 48.
    Liu, Y.J., Gao, Y., Tong, S., Li, Y.: Fuzzy approximation-based adaptive backstepping optimal control for a class of nonlinear discrete-time systems with dead-zone. IEEE Trans. Fuzzy Syst. 24(1), 16–28 (2016)Google Scholar
  49. 49.
    Sheng, D., Wei, Y., Cheng, S., Shuai, J.: Adaptive backstepping control for fractional order systems with input saturation. J. Frankl. Inst. 354(5), 2245–2268 (2017)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Arabzadeh Jafari, A., Mohammadi, S.M.A., Hasanpour Naseriyeh, M.: Adaptive backstepping control of uncertain fractional order systems by fuzzy approximation approach. Iran. J. Fuzzy Syst. (2017). zbMATHGoogle Scholar
  51. 51.
    Zouari, F., Boulkroune, A., Ibeas, A., Arefi, M.M.: Observer-based adaptive neural network control for a class of MIMO uncertain nonlinear time-delay non-integer-order systems with asymmetric actuator saturation. Neural Comput. Appl. 28(1), 993–1010 (2017)Google Scholar
  52. 52.
    Hasanpour Naseriyeh, M., Arabzadeh Jafari, A., Mohammadi, S.M.A.: Adaptive fuzzy output feedback tracking control for a class of nonlinear time-varying delay systems with unknown backlash-like hysteresis. Iran. J. Fuzzy Syst. 14(5), 43–64 (2017)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Mendel, J.M.: Computing derivatives in interval type-2 fuzzy logic system. IEEE Trans. Fuzzy Syst. 12, 84–98 (2004)Google Scholar
  54. 54.
    Shen, J., Lam, J.: Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica 50, 547–551 (2014)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Bouzeriba, A., Boulkroune, A., Bouden, T.: Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control. Neural Comput. Appl. 27(5), 1349–1360 (2016)zbMATHGoogle Scholar
  56. 56.
    Zheng, Y., Nian, Y., Wang, D.: Controlling fractional order chaotic systems based on TakagiSugeno fuzzy model and adaptive adjustment mechanism. Phys. Lett. A 375, 125–129 (2010)zbMATHGoogle Scholar
  57. 57.
    Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948 (1995)Google Scholar
  58. 58.
    Majidabad, S.S., Shandiz, H.T., Hajizadeh, A.: Decentralized sliding mode control of fractional-order large-scale nonlinear systems. Nonlinear Dyn. 77, 119–134 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Electrical Engineering, College of EngineeringShahid Bahonar UniversityKermanIran

Personalised recommendations