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Fractional integral sliding modes for robust tracking of nonlinear systems

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Abstract

Memory and heritage of differintegral operators require knowledge of the error manifold derivative at the initial time to sustain a sliding motion for any initial condition. Moreover, when the system is subject to (unknown) disturbances, such initial condition is unknown; thus, the enforcement of an integral sliding motion has been elusive with a chatter-less controller. In this paper, a novel fractional-order integral sliding mode (FISM) is proposed to maintain an invariant sliding mode due to an exact estimation of disturbances at first step. Our scheme is continuous after initial condition, avoiding chattering effects thanks to the topological properties of differintegral operators. In contrast to other FISM approaches, the proposed scheme induces a fractional-order reaching dynamics of order \((1+\nu )\in (1,2)\) to enforce an integral sliding mode for any initial condition, even in the presence of Hölder (continuous but not necessarily differentiable) disturbances and model uncertainties. Simulations show the reliability of the proposed scheme.

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References

  1. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  2. Hartley, T.T., Lorenzo, C.F.: Dynamics and control of initialized fractional-order systems. Nonlinear Dyn. 29(1), 201–233 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barbosa, R.S., Tenreiro-Machado, J.A., Ferreira, I.M.: Tuning of PID controllers based on Bodes ideal transfer function. Nonlinear Dyn. 31(1), 305–321 (2004)

    Article  MATH  Google Scholar 

  4. Tenreiro-Machado, J.A.: Fractional order modelling of dynamic backlash. Mechatronics 23(7), 741–745 (2013)

    Article  Google Scholar 

  5. Barbosa, R.S., Tenreiro-Machado, J.A.: Describing function analysis of systems with impacts and backlash. Nonlinear Dyn. 29(1), 235–250 (2002)

    Article  MATH  Google Scholar 

  6. Silva, M.F., Tenreiro-Machado, J.A., Lopes, A.M.: Fractional order control of a hexapod robot. Nonlinear Dyn. 31(1), 417–433 (2004)

    Article  MATH  Google Scholar 

  7. Ross, B., Samko, S., Love, E.: Functions that have no first order derivative might have fractional derivatives of all order less than one. Real Anal. Exch. 20(2), 140–157 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Humphrey, J., Schuler, C., Rubinsky, B.: On the use of the Weierstrass-Mandelbrot function to describe the fractal component of turbulent velocity. Fluid Dyn. Res. 9, 81–95 (1992)

    Article  Google Scholar 

  9. Utkin, V.: Sliding Modes in Control and Optimization. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  10. Rubagotti, M., Estrada, A., Castanos, F., Ferrara, A., Fridman, L.: Integral sliding mode control for nonlinear systems with matched and unmatched perturbations. IEEE Trans. Autom. Control 56(11), 2699–2704 (2011)

    Article  MathSciNet  Google Scholar 

  11. Utkin, V., Shi, J.: Integral sliding mode in systems operating under uncertainty conditions. In: IEEE Conference on Decision and Control, pp. 4591–4596 (1996)

  12. Utkin, V.: On convergence time and disturbance rejection of super-twisting control. IEEE Trans. Autom. Control 58(8), 2013–2017 (2013)

    Article  MathSciNet  Google Scholar 

  13. Vinagre, B.M., Calderón, A.J.: On fractional sliding mode control. In: Portuguese Conference on Automatic Control (2006)

  14. 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. (2016). doi:10.1007/s11071-016-2712-6

    MathSciNet  MATH  Google Scholar 

  15. Chen, D.-Y., Liu, Y.-X., Ma, X.-Y., Zhang, R.-F.: Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn. 67(1), 893–901 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: Control of constrained robot manipulators based on fractional order error manifolds. In: Symposium on Robot Control, pp. 131–126 (2015)

  17. Muñoz-Vázquez, A.J., Parra-Vega, V.,Sánchez-Orta, A.: Free-model fractional-order absolutely continuous sliding mode control for Euler–Lagrange systems. In: IEEE Conference on Decision and Control, pp. 6933–6938 (2014)

  18. Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A., García, O., Izaguirre-Espinosa, I.: Attitude tracking control of a quadrotor based on absolutely continuous fractional integral sliding modes. In: IEEE Conference on Control Applications, pp. 717–722 (2014)

  19. Shen, J., James, L.: Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica 50(2), 547–551 (2014)

    Article  MathSciNet  Google Scholar 

  20. Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: Continuous fractional sliding mode-like control for exact rejection of non-differentiable Hölder disturbances. IMA J. Math. Control Inf. (2015). doi:10.1093/imamci/dnv064

    MATH  Google Scholar 

  21. Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: Uniformly continuous differintegral sliding mode control of nonlinear systems subject to Hölder disturbances. Automatica 52(6), 179–184 (2016)

    Article  MATH  Google Scholar 

  22. Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A., Castillo, P., Lozano, R.: Generalized order integral sliding mode control for non-differentiable disturbance rejection: A comparative study. In: IEEE Conference on Decision and Control, pp. 4092–4097 (2015)

  23. Izaguirre-Espinosa, C., Muñoz-Vázquez, A.J., Sánchez-Orta, A., Parra-Vega, V., Castillo, P.: Attitude control of quadrotors based on fractional sliding modes: theory and experiments. IET Control Theory Appl. 10(7), 825–832 (2016). doi:10.1049/iet-cta.2015.1048

    Article  MathSciNet  Google Scholar 

  24. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  25. Danca, M.: Numerical approximation of a class of discontinuous of fractional order. Nonlinear Dyn. 66(1), 133–139 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Garrapa, R.: On some generalizations of the implicit Euler method for discontinuous fractional differential equations. Math. Comput. Simul. 95(1), 213–228 (2013)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Research Center for Advanced Studies (Cinvestav), Saltillo Campus, Ramos Arizpe, Mexico

    Aldo-Jonathan Muñoz-Vázquez, Vicente Parra-Vega & Anand Sánchez-Orta

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  1. Aldo-Jonathan Muñoz-Vázquez
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  2. Vicente Parra-Vega
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  3. Anand Sánchez-Orta
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Correspondence to Aldo-Jonathan Muñoz-Vázquez.

Additional information

Authors acknowledge partial support form Conacyt Basic Research Grants 133346 and 133544, as well as Ph.D. Scholarship Grant 243206.

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Muñoz-Vázquez, AJ., Parra-Vega, V. & Sánchez-Orta, A. Fractional integral sliding modes for robust tracking of nonlinear systems. Nonlinear Dyn 87, 895–901 (2017). https://doi.org/10.1007/s11071-016-3086-5

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  • DOI: https://doi.org/10.1007/s11071-016-3086-5

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