Fractional-order sliding-mode controller for semi-active vehicle MRD suspensions

Abstract

Due to the complexly natural attributes of technical systems, reality has been shown that many systems could be modeled more precisely if they are modeled by using fractional calculus and fractional-order differential equations. Inspired by this advantage, in this work a fractional-order derivative-based sliding mode controller (FD-SMC) for magneto-rheological damper based on semi-active vehicle suspensions (MRD-SAVSs) is proposed to make the states of the given system asymptotically stable in the finite time. To show this assertion, a new estimate result for fractional differential inequality is presented to derive an FD-SMC law for the systems of MRD-SAVS. Then, this corresponding fractional-order sliding mode controller is designed to provide robustness, high performance control, finite time convergence in the presence of uncertainties and external disturbances. Finally, numerical simulation results are presented to demonstrate the effectiveness of the proposed control method.

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References

  1. 1.

    Do, X.P., Choi, S.B., Lee, Y.S., Han, M.S.: Vibration control of a vehicle’s seat suspension featuring a magnetorheological damper based on a new adaptive fuzzy sliding-mode controller. Proc. Inst. Mech. Eng. Part D J. Aut. Eng. 230, 437–458 (2015)

    Google Scholar 

  2. 2.

    Nguyen, S.D., Choi, S.B., Seo, T.I.: Adaptive fuzzy sliding control enhanced by compensation for explicitly unidentified aspects. Int. J. Control Automat Syst. IJCAS 15, 2906–2920 (2017)

    Google Scholar 

  3. 3.

    Nguyen, S.D., Jung, D., Choi, S.B.: A robust vibration control of a magnetorheological damper based railway suspension using a novel adaptive type-2 fuzzy sliding mode controller. Shock Vib (2017)

  4. 4.

    Zhang, H., Wang, E., Min, F., Subash, R., Su, C.: Skyhook-based semi-active control of full-vehicle suspension with magneto-rheological dampers. Chin. J. Mech. Eng. 26(3), 498–505 (2013)

    Google Scholar 

  5. 5.

    Nguyen, S.D., Choi, S.B., Seo, T.I.: Recurrent mechanism and impulse noise filter for establishing ANFIS. IEEE Trans. Fuzzy Syst. 26(2), 985–997 (2017)

    Google Scholar 

  6. 6.

    Nguyen, S.D., Choi, S.B.: A new neuro-fuzzy training algorithm for identifying dynamic characteristics of smart dampers. Smart Mater. Struct. 21(8), 085021 (2012)

    Google Scholar 

  7. 7.

    Qin, Y., Zhao, F., Wang, Z., Gu, L., Dong, M.: Comprehensive analysis for influence of controllable damper time delay on semi-active suspension control strategies. J. Vib. Acoust. ASME 139, 3 (2017)

    Google Scholar 

  8. 8.

    Duc, T.M., Hoa, N.V., Dao, T.P.: Adaptive fuzzy fractional-order nonsingular terminal sliding mode control for a class of second-order nonlinear systems. J. Comput. Nonlinear Dyn. 13, 3 (2018)

    Google Scholar 

  9. 9.

    Nekoukar, V., Erfanian, A.: Adaptive fuzzy terminal sliding mode control for a class of MIMO uncertain nonlinear systems. Fuzzy Sets Syst. 179, 34–49 (2011)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Ren, H., Chen, S., Zhao, Y., Liu, G., Yang, L.: State observer-based sliding mode control for semi-active hydro-pneumatic suspension. Veh. Syst. Dyn. 54, 168–190 (2016)

    Google Scholar 

  11. 11.

    Nguyen, S.D., Ho, H.V., Nguyen, T.T., Truong, N.T., Seo, T.I.: Novel fuzzy sliding controller for MRD suspensions subjected to uncertainty and disturbance. Eng. Appl. Artif. Intell. 61, 65–76 (2017)

    Google Scholar 

  12. 12.

    Oveisi, A., Nestorovic, T.: Robust observer-based adaptive fuzzy sliding mode controller. Mech. Syst. Signal Proc. 76, 58–71 (2016)

    Google Scholar 

  13. 13.

    Nguyen, S.D., Vo, H.D., Seo, T.I.: Nonlinear adaptive control based on fuzzy sliding mode technique and fuzzy-based compensator. ISA Trans. 70, 309–321 (2017)

    Google Scholar 

  14. 14.

    Feng, Y., Yu, X., Man, Z.: Non-singular terminal sliding mode control of rigid manipulators. Automatica 38, 2159–2167 (2002)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Wu, Y., Yu, X., Man, Z.: Terminal sliding mode control design for uncertain dynamic systems. Syst. Control Lett. 34, 281–287 (1998)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Levant, A.: Chattering Anal. IEEE Trans. Autom. Control 55, 1380–1389 (2010)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Davila, J., Fridman, L., Levant, A.: Second-order sliding-mode observer for mechanical systems. IEEE Trans. Autom. Control 50, 1785–1789 (2005)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Rivera, J., Garcia, L., Mora, C., Raygoza, J.J., Ortega, S.: Super-twisting sliding mode in motion control systems. In: Bartoszewicz, A. (ed.) Sliding Mode Control. InTech, Rijeka, Croatia (2011)

  19. 19.

    Jiang, Y., Wang, Q., Dong, C.: A reaching law based neural network terminal sliding-mode guidance law design. In: 2013 IEEE International Conference of IEEE Region 10 (TENCON 2013). Xi’an, China (2013)

  20. 20.

    Li, H., Wang, J., Wu, L., Lam, H.K., Gao, Y.: Optimal guaranteed cost sliding mode control of interval type-2 fuzzy time-delay systems. IEEE Trans. Fuzzy Syst. 26, 246–257 (2018)

    Google Scholar 

  21. 21.

    Nguyen, S.D., Kim, W.H., Park, J.H., Choi, S.B.: A new fuzzy sliding mode controller for vibration control systems using integrated structure smart dampers. Smart Mater. Struct. 26, 045038 (2017)

    Google Scholar 

  22. 22.

    Suresh, T., Wen, Y.: Sliding mode control of wind-induced vibrations using fuzzy sliding surface and gain adaptation. Int. J. Syst. Sci. 47, 1258–1267 (2016)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Yu, X., Man, Z.: Variable structure systems with terminal sliding modes. In Lecture Notes in Control and Information Sciences 274, pp. 109–128, Springer, New York (2002)

  24. 24.

    Yu, S., Xinghuo, Y., Bijan, S., Zhihong, M.: Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica 41, 1957–1964 (2005)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Tiwari, P.M., Janardhanan, S., Un Nabi, M.: A finite-time convergent continuous time sliding mode controller for spacecraft attitude control. In: International Workshop on Variable Structure Systems. IEEE., Mexico City, Mexico, pp. 399-403 (2010)

  26. 26.

    Zou, A.M., Kumar, K.D., Hou, Z.G., Liu, X.: Finite-time attitude tracking control for spacecraft using terminal sliding mode and Chebyshev neural network. IEEE Trans. Syst. Man Cybern. Part B 41, 950–963 (2011)

    Google Scholar 

  27. 27.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. North-Holland Mathematics Studies, Elsevier, Amsterdam, 204 (2006)

  28. 28.

    Baleanu, D., Machado, J.A.T., Luo, A.C.J.: Fractional dynamics and control. Springer, Berlin (2011)

  29. 29.

    Martínez-Fuentes, O., Martínez-Guerra, R.: A novel Mittag-Leffler stable estimator for nonlinear fractional-order systems: a linear quadratic regulator approach. Nonlinear Dyn. 94, 1973–86 (2018)

    MATH  Google Scholar 

  30. 30.

    Sabzalian, M.H., Mohammadzaeh, A., Lin, S., Zhang, W.: Robust fuzzy control for fractional-order systems with estimated fraction-order. Nonlinear Dyn. 98(3), 2375–2385 (2019)

    MATH  Google Scholar 

  31. 31.

    Ladaci, S., Charef, A.: On fractional adaptive control. Nonlinear Dyn. 43, 365–78 (2006)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Muñoz-Vázquez, A.J., Sánchez-Orta, A., Parra-Vega, V.: A novel PID control with fractional nonlinear integral. Nonlinear Dyn. 94, 3041–52 (2018)

    MATH  Google Scholar 

  33. 33.

    Padula, F., Visioli, A.: Optimal tuning rules for proportional-integral-derivative and fractional-order proportional-integral-derivative controllers for integral and unstable processes. IET Control. Theory Appl. 6, 776–786 (2012)

    MathSciNet  Google Scholar 

  34. 34.

    Vinagre, B.M., Petras, I., Podlubny, I., Chen, Y.Q.: Using fractional-order adjustment rules and fractional-order reference models in model-reference adaptive control. Nonlinear Dyn. 29, 269–79 (2002)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Zhong, J., Li, L.: Tuning fractional-order \(P{I^\lambda }{D^\mu }\) controllers for a solid-core magnetic nearing system. IEEE Trans. Control Syst. 23, 1648–1656 (2015)

    Google Scholar 

  36. 36.

    Aghababa, M.P.: A Lyapunov-based control scheme for robust stabilization of fractional chaotic systems. Nonlinear Dyn. 78, 2129–2140 (2014)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Aghababa, M.P.: Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology. Nonlinear Dyn. 89, 1357–1370 (2017)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Aghababa, M.P.: Adaptive switching control of uncertain fractional systems: application to Chua’s circuit. Int. J. Adapt. Control Signal Process. 32, 1206–1221 (2018)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Dadras, S., Momeni, H.R.: Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty. Commun. Nonlinear Sci. Numer. Simul. 17, 367–377 (2012)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Nojavanzadeh, D., Badamchizadeh, M.A.: Adaptive fractional-order non-singular fast terminal sliding mode control for robot manipulators. IET Control. Theory Appl. 10(3), 1565–1572 (2016)

    MathSciNet  Google Scholar 

  41. 41.

    Aghababa, M.P.: A switching fractional calculus-based controller for normal non-linear dynamical systems. Nonlinear Dyn. 75, 577–588 (2014)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Wang, Y., Luo, G., Gu, L., Li, X.: Fractional-order nonsingular terminal sliding mode control of hydraulic manipulators using time delay estimation. J. Vibration Control. 22, 3998–4011 (2016)

    Google Scholar 

  43. 43.

    Chen, Y., Wei, Y., Zhong, H., Wang, Y.: Sliding mode control with a second-order switching law for a class of nonlinear fractional order systems. Nonlinear Dyn. 85, 633–43 (2016)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Cheng, C.C., Hsu, S.C.: Design of adaptive sliding mode controllers for a class of perturbed fractional-order nonlinear systems. Nonlinear Dyn. 98(2), 1355–1365 (2019)

    Google Scholar 

  45. 45.

    Wang, P., Wang, Q., Xu, X., Chen, N.: Fractional Critical Damping Theory and Its Application in Active Suspension Control, Shock and Vibration, Article ID 2738976 (2017)

  46. 46.

    You, H., Shen, Y., Xing, H., Yang, S.: Optimal control and parameters design for the fractional-order vehicle suspension system. J. Low Frequency Noise Vib. Active Control 37, 456–467 (2018)

    Google Scholar 

  47. 47.

    Zhang, C., Xiao, J.: Chaotic behavior and feedback control of magnetorheological suspension system with fractional-order derivative. J. Comput. Nonlinear Dyn. 13, 2 (2018)

    Google Scholar 

  48. 48.

    Al-Refai, M.: On the fractional derivatives at extreme points. Electron. J. Qualitat. Theory Differ. Eq. 55, 1–5 (2012)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Li, C., Deng, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187, 777–784 (2007)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19, 2951–2957 (2014)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Baleanu, D., Guo-Cheng, W., Sheng-Da, Z.: Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solitons Fractals 102, 99–105 (2017)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Dadras, S., Dadras, S., Malek, H., Chen, Y.: A note on the lyapunov stability of fractional-order nonlinear systems, In ASME 2017 International design engineering technical conferences and computers and information in engineering conference. American Society of Mechanical Engineers Digital Collection (2017)

  53. 53.

    Li, Y., Chen, Y., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)

    MathSciNet  MATH  Google Scholar 

  54. 54.

    Li, Y., Chen, Y., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

    MathSciNet  MATH  Google Scholar 

  55. 55.

    Liu, S., Jiang, W., Li, X., Zhou, X.F.: Lyapunov stability analysis of fractional nonlinear systems. Appl. Math. Lett. 51, 13–19 (2016)

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Hua, C., Chen, J., Li, Y., Li, L.: Adaptive prescribed performance control of half-car active suspension system with unknown dead-zone input. Mech. Syst. Signal Process. 111, 135–148 (2018)

    Google Scholar 

  57. 57.

    Weichao, S., Zhao, Z., Gao, H.: Saturated adaptive robust control for active suspension systems. IEEE Trans. Ind. Electron. 60, 498–505 (2013)

    Google Scholar 

  58. 58.

    Gong, W., Cai, Z.: Differential evolution with ranking-based mutation operators. IEEE Trans. Cybern. 43, 1 (2013). https://doi.org/10.1109/TCYB.2013.2239988

    Article  Google Scholar 

  59. 59.

    Bettayeb, M., Djennoune, S.: Design of sliding mode controllers for nonlinear fractional-order systems via diffusive representation. Nonlinear Dyn. 84, 593–605 (2016)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are very grateful to the referees for their valuable suggestions, which helped to improve the paper significantly. The authors would like to thank the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.01-2019.328

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Correspondence to Van Hoa Ngo.

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Nguyen, S.D., Lam, B.D. & Ngo, V.H. Fractional-order sliding-mode controller for semi-active vehicle MRD suspensions. Nonlinear Dyn 101, 795–821 (2020). https://doi.org/10.1007/s11071-020-05818-w

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Keywords

  • Fractional-oder sliding mode control
  • Fractional-oder control
  • Fractional-order Lyapunov direct method
  • Semi-active MRD suspension