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A switching fractional calculus-based controller for normal non-linear dynamical systems

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Abstract

In this paper, a fractional calculus-based terminal sliding mode controller is introduced for finite-time control of non-autonomous non-linear dynamical systems in the canonical form. A fractional terminal switching manifold which is appropriate for canonical integer-order systems is firstly designed. Then some conditions are provided to avoid the inherent singularities of the conventional terminal sliding manifolds. A non-smooth Lyapunov function is adopted to prove the finite time stability and convergence of the sliding mode dynamics. Afterward, based on the sliding mode control theory, an equivalent control and a discontinuous control law are designed to guarantee the occurrence of the sliding motion in finite time. The proposed control scheme uses only one control input to stabilize the system. The proposed controller is also robust against system uncertainties and external disturbances. Two illustrative examples show the effectiveness and applicability of the proposed fractional finite-time control strategy. It is worth noting that the proposed sliding mode controller can be applied for control and stabilization of a large class of non-autonomous non-linear uncertain canonical systems.

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References

  1. Utkin, V.I.: Sliding Modes in Control Optimization. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  2. Zak, M.: Terminal attractors in neural networks. Neural Netw. 2, 259–274 (1989)

    Article  Google Scholar 

  3. Man, Z., Paplinski, A.P., Wu, H.R.: A robust MIMO terminal sliding mode control for rigid robotic manipulators. IEEE Trans. Autom. Control 39, 2464–2469 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Man, Z., Yu, X.: Terminal sliding mode control of MIMO linear systems. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 44, 1065–1070 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Yu, X., Man, Z.: Multi-input uncertain linear systems with terminal sliding-mode control. Automatica 34, 389–392 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Tang, Y.: Terminal sliding mode control for rigid robots. Automatica 34, 51–56 (1998)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Li, T.-H.S., Huang, Y.-C.: MIMO adaptive fuzzy terminal sliding-mode controller for robotic manipulators. Inf. Sci. 180, 4641–4660 (2010)

    Article  MATH  Google Scholar 

  10. Liu, C.-H., Hsiao, M.-Y.: A finite time synergetic control scheme for robot manipulators. Comput. Math. Appl. 64, 1163–1169 (2012)

    Article  Google Scholar 

  11. Hui, L., Li, J.: Terminal sliding mode control for spacecraft formation flying. IEEE Trans. Aerosp. Electron. Syst. 45, 835–846 (2009)

    Article  Google Scholar 

  12. 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, Cybern. 41, 950–963 (2011)

    Article  Google Scholar 

  13. Aghababa, M.P., Aghababa, H.P.: Chaos suppression of a class of unknown uncertain chaotic systems via single input. Commun. Nonlinear Sci. Numer. Simul. 17, 3533–3538 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Aghababa, M.P., Khanmohammadi, S., Alizadeh, G.: Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique. Appl. Math. Model. 35, 3080–3091 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Aghababa, M.P., Feizi, H.: Nonsingular terminal sliding mode approach applied to synchronize chaotic systems with unknown parameters and nonlinear inputs. Chin. Phys. B 21, 060506 (2012)

    Article  Google Scholar 

  16. Aghababa, M.P., Feizi, H.: Design of a sliding mode controller for synchronizing chaotic systems with parameter and model uncertainties and external disturbances. Trans. Inst. Meas. Control 34, 990–997 (2012)

    Article  Google Scholar 

  17. Aghababa, M.P., Aghababa, H.P.: A novel finite-time sliding mode controller for synchronization of chaotic systems with input nonlinearity. Arab. J. Sci. Eng. (2012). doi:10.1007/s13369-012-0459-z

    Google Scholar 

  18. Aghababa, M.P., Akbari, M.E.: A chattering-free robust adaptive sliding mode controller for synchronization of two different chaotic systems with unknown uncertainties and external disturbances. Appl. Math. Comput. 218, 5757–5768 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Aghababa, M.P., Heydari, A.: Chaos synchronization between two different chaotic systems with uncertainties, external disturbances, unknown parameters and input nonlinearities. Appl. Math. Model. 36, 1639–1652 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Chen, S.-Y., Lin, F.-J.: Robust nonsingular terminal sliding-mode control for nonlinear magnetic bearing system. IEEE Trans. Control Syst. Technol. 19, 636–643 (2011)

    Article  Google Scholar 

  21. Aghababa, M.P., Aghababa, H.P.: The rich dynamics of fractional-order gyros applying a fractional controller. J. Syst. Control Eng. 227, 588–601 (2013)

    Google Scholar 

  22. Luo, Y., Chen, Y.Q., Pi, Y.: Experimental study of fractional order proportional derivative controller synthesis for fractional order systems. Mechatronics 21, 204–214 (2011)

    Article  Google Scholar 

  23. Sabatier, J., Cugnet, M., Laruelle, S., Grugeon, S., Sahut, B., Oustaloup, A., Tarascon, J.M.: A fractional order model for lead-acid battery crankability estimation. Commun. Nonlinear Sci. Numer. Simul. 15, 1308–1317 (2010)

    Article  Google Scholar 

  24. Aghababa, M.P.: Chaos in a fractional-order micro-electro-mechanical resonator and its suppression. Chin. Phys. B 21, 100505 (2012)

    Article  Google Scholar 

  25. Rivero, M., Trujillo, J.J., Vàzquez, L., Velasco, M.P.: Fractional dynamics of populations. Appl. Math. Comput. 218, 1089–1095 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Jesus, I.S., Machado, J.A.T.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54, 263–282 (2008)

    Article  MATH  Google Scholar 

  27. Rapaić, M.R., Jeličić, Z.D.: Optimal control of a class of fractional heat diffusion systems. Nonlinear Dyn. 62, 39–51 (2010)

    Article  MATH  Google Scholar 

  28. Aghababa, M.P.: A novel terminal sliding mode controller for a class of non-autonomous fractional-order systems. Nonlinear Dyn. 73, 679–688 (2013)

    Article  MathSciNet  Google Scholar 

  29. Aghababa, M.P.: No-chatter variable structure control for fractional nonlinear complex systems. Nonlinear Dyn. 73, 2329–2342 (2013)

    Article  MathSciNet  Google Scholar 

  30. Aghababa, M.P.: Design of a chatter-free terminal sliding mode controller for nonlinear fractional-order dynamical systems. Int. J. Control 86, 1744–1756 (2013)

    Article  Google Scholar 

  31. Aghababa, M.P.: Robust finite-time stabilization of fractional-order chaotic systems based on fractional Lyapunov stability theory. J. Comput. Nonlinear Dyn. 7, 021010 (2012)

    Article  Google Scholar 

  32. Aghababa, M.P.: Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller. Commun. Nonlinear Sci. Numer. Simul. 17, 2670–2681 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  33. Aghababa, M.P.: Finite-time chaos control and synchronization of fractional-order chaotic (hyperchaotic) systems via fractional nonsingular terminal sliding mode technique. Nonlinear Dyn. 69, 247–261 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  34. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  35. Li, C., Qian, D., Chen, Y.Q.: On Riemann–Liouville and Caputo derivatives. Discrete Dyn. Nat. Soc. 2011, 562494 (2011)

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  37. Fliess, M.: Generalized controller canonical forms for linear and nonlinear dynamics. IEEE Trans. Autom. Control 35, 994–1001 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  38. Yu, X., Man, Z.: Fast terminal sliding-mode control design for nonlinear dynamical systems. IEEE Trans. Circuits Syst. 49, 261–264 (2002)

    Article  MathSciNet  Google Scholar 

  39. Wang, H., Han, Z., Xie, Q., Zhang, W.: Finite-time chaos control via nonsingular terminal sliding mode control. Commun. Nonlinear Sci. Numer. Simul. 14, 2728–2733 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  40. 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)

    Article  MATH  MathSciNet  Google Scholar 

  41. Hui, L., Li, J.: Terminal sliding mode control for spacecraft formation flying. IEEE Trans. Aerosp. Electron. Syst. 45, 835–846 (2009)

    Article  Google Scholar 

  42. Xiang, W., Huangpu, Y.: Second-order terminal sliding mode controller for a class of chaotic systems with unmatched uncertainties. Commun. Nonlinear Sci. Numer. Simul. 15, 3241–3247 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  43. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  44. Genesio, R., Tesi, A.: A harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28, 531–548 (1992)

    Article  MATH  Google Scholar 

  45. Chen, H.K.: Chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping. J. Sound Vib. 255, 719–740 (2002)

    Article  MATH  Google Scholar 

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  1. Electrical Engineering Department, Urmia University of Technology, Urmia, Iran

    Mohammad Pourmahmood Aghababa

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  1. Mohammad Pourmahmood Aghababa
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Correspondence to Mohammad Pourmahmood Aghababa.

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Aghababa, M.P. A switching fractional calculus-based controller for normal non-linear dynamical systems. Nonlinear Dyn 75, 577–588 (2014). https://doi.org/10.1007/s11071-013-1087-1

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