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Synchronization and stabilization of fractional order nonlinear systems with adaptive fuzzy controller and compensation signal

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Abstract

In this paper, a direct adaptive fuzzy controller with compensation signal is presented to control and stabilize a class of fractional order systems with unknown nonlinearities. Based on a Lyapunov function candidate the global Mittag–Leffler stability is proved and a new fractional order adaptation law is derived. The adaptation law adjusts free parameters of the fuzzy controller and bounds them by utilizing a novel fractional order projection algorithm. Furthermore, due to the use of compensation term, the proposed approach does not demand suitable membership functions in the fuzzy system. In addition, the stability of the closed-loop system is guaranteed by utilizing a supervisory controller. Numerical simulations show the validity and effectiveness of the introduced scheme for various fractional order nonlinear models that perturbed by disturbance and uncertainty.

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References

  1. Petras, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations. Academic press, San Diego, CA (1999)

    MATH  Google Scholar 

  3. Gabano, J.D., Poinot, T.: Fractional modelling and identification of thermal systems. Signal Process. 91(3), 531–541 (2011)

    Article  MATH  Google Scholar 

  4. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  5. Magin, R.L.: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32(1), 1–104 (2004)

    Article  Google Scholar 

  6. Sabatier, J., Aoun, M., Oustaloup, A., Grégoire, G., Ragot, F., Roy, P.: Fractional system identification for lead acid battery state of charge estimation. Signal Process. 86(10), 2645–2657 (2006)

    Article  MATH  Google Scholar 

  7. Vinagre, B., Feliu, V.: Modeling and control of dynamic system using fractional calculus: Application to electrochemical processes and flexible structures. In: Proceedings 41st IEEE Conference Decision and Control 2002, pp. 214–239

  8. Oustaloup, A., Moreau, X., Nouillant, M.: The CRONE suspension. Control Eng. Pract. 4(8), 1101–1108 (1996)

    Article  Google Scholar 

  9. Podlubny, I.: Fractional-order systems and PI/sup/spl lambda//D/sup/spl mu//-controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)

    Article  MATH  Google Scholar 

  10. Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-order Systems and Controls: Fundamentals and Applications. Springer, London (2010)

    Book  MATH  Google Scholar 

  11. Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Phys. A: Stat. Mech. Appl. 387(1), 57–70 (2008)

    Article  Google Scholar 

  12. Valerio, D., Da Costa, J.S.: An introduction to fractional control, vol. 91. IET, (2013)

  13. Wei, Y., 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)

    Article  MATH  MathSciNet  Google Scholar 

  14. Shao, S., Chen, M., Yan, X.: Adaptive sliding mode synchronization for a class of fractional-order chaotic systems with disturbance. Nonlinear Dyn. 83(4), 1855–1866 (2016)

  15. 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 

  16. Gallegos, J.A., Duarte-Mermoud, M.A., Aguila-Camacho, N., Castro-Linares, R.: On fractional extensions of Barbalat Lemma. Syst. Control Lett. 84, 7–12 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  17. Li, Y., Chen, Y., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965–1969 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. 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(5), 1810–1821 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Momani, S., Hadid, S.: Lyapunov stability solutions of fractional integrodifferential equations. Int. J. Math. Math. Sci. 2004(47), 2503–2507 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhang, F., Li, C., Chen, Y.: Asymptotical stability of nonlinear fractional differential system with Caputo derivative. Int. J. Differ. Equ. 2011 (2011)

  21. Jun-Guo, L.: Chaotic dynamics and synchronization of fractional-order Genesio-Tesi systems. Chin. Phys. 14(8), 1517 (2005)

    Article  Google Scholar 

  22. Li, C., Chen, G.: Chaos and hyperchaos in the fractional-order Rössler equations. Phys. A: Stat. Mech. Appl. 341, 55–61 (2004)

    Article  Google Scholar 

  23. Lu, J.G.: Chaotic dynamics of the fractional-order Lü system and its synchronization. Phys. Lette. A 354(4), 305–311 (2006)

    Article  MathSciNet  Google Scholar 

  24. Lu, J.G., Chen, G.: A note on the fractional-order Chen system. Chaos, Solitons and Fractals 27(3), 685–688 (2006)

    Article  MATH  Google Scholar 

  25. Wu, X., Li, J., Chen, G.: Chaos in the fractional order unified system and its synchronization. J. Franklin Inst. 345(4), 392–401 (2008)

    Article  MATH  Google Scholar 

  26. Yang, Q., Zeng, C.: Chaos in fractional conjugate Lorenz system and its scaling attractors. Commun. Nonlinear Sci. Numer. Simul. 15(12), 4041–4051 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. Aghababa, M.P.: Synchronization and stabilization of fractional second-order nonlinear complex systems. Nonlinear Dyn. 80(4), 1731–1744 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  28. Hosseinnia, S.H., Ghaderi, R., Ranjbar, A., Sadati, J., Momani, S.: Synchronization of gyro systems via fractional-order adaptive controller. New Trends in Nanotechnology and Fractional Calculus Applications, pp. 495–502. Springer, Berlin. (2010)

  29. Aghababa, M.P.: Design of hierarchical terminal sliding mode control scheme for fractional-order systems. IET Sci., Meas. Technol. 9(1), 122–133 (2014). doi:10.1049/iet-smt.2014.0039

    Article  Google Scholar 

  30. Passino, K.M., Yurkovich, S., Reinfrank, M.: Fuzzy control, vol. 42. Citeseer, (1998)

  31. Su, C.-Y., Stepanenko, Y.: Adaptive control of a class of nonlinear systems with fuzzy logic. Fuzzy Syst., IEEE Trans. 2(4), 285–294 (1994)

    Article  Google Scholar 

  32. Ying, H.: Fuzzy Control and Modeling: Analytical Foundations and Applications, 1st edn. Wiley-IEEE Press, New York (2000)

  33. Golea, N., Golea, A., Benmahammed, K.: Stable indirect fuzzy adaptive control. Fuzzy Sets Syst. 137(3), 353–366 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. Tang, Y., Zhang, N., Li, Y.: Stable fuzzy adaptive control for a class of nonlinear systems. Fuzzy Sets Syst. 104(2), 279–288 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wang, C.-H., Lin, T.-C., Lee, T.-T., Liu, H.-L.: Adaptive hybrid intelligent control for uncertain nonlinear dynamical systems. IEEE Trans. Syst., Man, Cybern., Part B: Cybern. 32(5), 583–597 (2002)

    Article  Google Scholar 

  36. Wang, C.-H., Liu, H.-L., Lin, T.-C.: Direct adaptive fuzzy-neural control with state observer and supervisory controller for unknown nonlinear dynamical systems. IEEE Trans. Fuzzy Syst., 10(1), 39–49 (2002)

    Article  Google Scholar 

  37. Chen, B.-S., Lee, C.-H., Chang, Y.-C.: H\(\infty \) tracking design of uncertain nonlinear SISO systems: adaptive fuzzy approach. IEEE Trans. Fuzzy Syst., 4(1), 32–43 (1996)

    Article  Google Scholar 

  38. Hsueh, Y.-C., Su, S.-F., Chen, M.-C.: Decomposed fuzzy systems and their application in direct adaptive fuzzy control. IEEE Trans. Cybern. 44(10), 1772–1783 (2014)

    Article  Google Scholar 

  39. Efe, M.Ö.: Fractional fuzzy adaptive sliding-mode control of a 2-DOF direct-drive robot arm. IEEE Trans. Syst., Man, Cybern., Part B: Cybern., 38(6), 1561–1570 (2008)

    Article  Google Scholar 

  40. Lin, T.-C., Lee, T.-Y., Balas, V.E.: Adaptive fuzzy sliding mode control for synchronization of uncertain fractional order chaotic systems. Chaos, Solitons and Fractals 44(10), 791–801 (2011)

    Article  MATH  Google Scholar 

  41. Lin, T.-C., Lee, T.-Y.: Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control. IEEE Trans. Fuzzy Syst., 19(4), 623–635 (2011)

    Article  Google Scholar 

  42. Lin, T.-C., Kuo, C.-H., Lee, T.-Y., Balas, V.E.: Adaptive fuzzy H \(\infty \) tracking design of SISO uncertain nonlinear fractional order time-delay systems. Nonlinear Dyn. 69(4), 1639–1650 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  43. Aghababa, M.P.: Comments on Adaptive fuzzy H\(\infty \) tracking design of SISO uncertain nonlinear fractional order time-delay systems. Nonlinear Dyn. 69(4), 1639–1650 (2012)

    Article  MathSciNet  Google Scholar 

  44. Lin, T.-C., Kuo, C.-H., Balas, V.E.: Uncertain fractional order chaotic systems tracking design via adaptive hybrid fuzzy sliding mode control. Int. J. Comput., Commun. Control, ISSN 9836, 400–409 (2011)

    Google Scholar 

  45. Ullah, N., Han, S., Khattak, M.: Adaptive fuzzy fractional-order sliding mode controller for a class of dynamical systems with uncertainty. Trans. Inst. Measure. Control 38(4), 402–413 (2016)

    Article  Google Scholar 

  46. Ullah, N., Shaoping, W., Khattak, M.I., Shafi, M.: Fractional order adaptive fuzzy sliding mode controller for a position servo system subjected to aerodynamic loading and nonlinearities. Aerosp. Sci. Technol. 43, 381–387 (2015)

    Article  Google Scholar 

  47. Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., Castro-Linares, R.: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22(1), 650–659 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  48. Ioannou, P.A., Sun, J.: Robust Adaptive Control. Prentice Hall, New York (1996)

    MATH  Google Scholar 

  49. Wang, L.-X.: Stable adaptive fuzzy control of nonlinear systems. IEEE Trans. Fuzzy Syst., 1(2), 146–155 (1993)

    Article  MathSciNet  Google Scholar 

  50. Wang, L.-X., Mendel, J.M.: Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Trans. Neural Networks 3(5), 807–814 (1992)

    Article  Google Scholar 

  51. Kratmüller, M., Murgaš, J.: B-spline Fuzzy adaptive system. Acta Electrotechnica et Informatica No 6(1), 3 (2006)

    Google Scholar 

  52. Ge, S., Hang, C., Lee, T.: Stable Adaptive Neural Network Control. Springer, Amsterdam (2002)

  53. Slotine, J.-J.E., Li, W.: Applied nonlinear control, vol. 199. vol. 1. prentice-Hall Englewood Cliffs, NJ, (1991)

  54. Delavari, H.: A novel fractional adaptive active sliding mode controller for synchronization of non-identical chaotic systems with disturbance and uncertainty. Int. J. Dyn. Control 5(1), 1–13 (2015)

    MathSciNet  Google Scholar 

  55. Aghababa, M.P., Aghababa, H.P.: The rich dynamics of fractional-order gyros applying a fractional controller. In: Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 0959651813492326 (2013)

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Authors and Affiliations

  1. Intelligent Systems Laboratory (ISLab.), Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, Iran

    Pouria Jafari & Mohammad Teshnehlab

  2. Industrial Control Center of Excellence, Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, Iran

    Mohammad Teshnehlab & Mahsan Tavakoli-Kakhki

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  1. Pouria Jafari
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  2. Mohammad Teshnehlab
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  3. Mahsan Tavakoli-Kakhki
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Correspondence to Mohammad Teshnehlab.

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Jafari, P., Teshnehlab, M. & Tavakoli-Kakhki, M. Synchronization and stabilization of fractional order nonlinear systems with adaptive fuzzy controller and compensation signal. Nonlinear Dyn 90, 1037–1052 (2017). https://doi.org/10.1007/s11071-017-3709-5

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  • DOI: https://doi.org/10.1007/s11071-017-3709-5

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