Advertisement

Nonlinear Dynamics

, Volume 94, Issue 4, pp 3041–3052 | Cite as

A novel PID control with fractional nonlinear integral

  • Aldo-Jonathan Muñoz-Vázquez
  • Anand Sánchez-Orta
  • Vicente Parra-Vega
Original Paper
  • 127 Downloads

Abstract

A nonlinear PID controller for robust tracking of second-order nonlinear systems is proposed, which consists in a classical PD structure plus a fractional-order nonlinear integral action of control (FONLI). This nonlinear integral action of fractional-order induces robustness properties to the closed-loop system in order to withstand for a general class of continuous but not necessarily differentiable disturbances, while inducing a stable sliding motion in finite time, such that, the tracking error converges with exponential rate. The resulting controller is uniformly continuous, preserving the regularity of the control signal. The proposed scheme is extended for higher-order dynamical systems by means of a state-feedback control plus a FONLI, inducing a stable motion in finite time. Numerical results based on simulation are discussed to show the reliability of the proposed scheme.

Keywords

PID control Fractional-order sliding mode Nonlinear systems Disturbance rejection 

Notes

Acknowledgements

We authors dearly do acknowledge to Editors for handling our paper, and to anonymous reviewers for a thorough review that allowed us to produce an improved paper. Aldo Jonathan also acknowledges CONACYT—Mexico for the project Catedras 1086 “Ambientes Inteligentes”.

Compliance with ethical standards

Conflict of interest

The authors declare they have no conflict of interest regarding the publication of this paper.

References

  1. 1.
    Li, S., Li, D., Lee, K.Y.: Optimal disturbance rejection for PI controller with constraints on relative delay margin. ISA Trans. 63, 103–111 (2016)CrossRefGoogle Scholar
  2. 2.
    Barbosa, R.S., Tenreiro-Machado, J.A., Ferreira, I.M.: Tuning of PID controllers based on Bodes ideal transfer function. Nonlinear Dyn. 38(1–4), 305–321 (2004)CrossRefGoogle Scholar
  3. 3.
    Åström, K.J., Hägglund, T.: Advanced PID Control. The Instrumentation, Systems, and Automation Society, North Carolina (2006)Google Scholar
  4. 4.
    Monje, C.A., Calderon, A.J., Vinagre, B.M., Chen, Y., Feliu, V.: On fractional \(\text{ PI }^{\lambda }\) controllers: some tuning rules for robustness to plant uncertainties. Nonlinear Dyn. 38(1–4), 369–381 (2004)CrossRefGoogle Scholar
  5. 5.
    Podlubny, I.: Fractional-order systems and \(\text{ PI }^{\lambda }\text{ D }^{\mu }\)-controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)MathSciNetCrossRefGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. 7.
    Utkin, V.: Sliding Modes in Control and Optimization. Springer, Berlin (1992)CrossRefGoogle Scholar
  8. 8.
    Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76(9–10), 924–941 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Moreno, J., Osorio, M.: Strict Lyapunov functions for the super-twisting algorithm. IEEE Trans. Autom. Control 57(4), 1035–1040 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bartolini, G., Ferrara, A., Usai, E.: Chattering avoidance by second-order sliding mode control. IEEE Trans. Autom. Control 43(2), 241–246 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Davila, J., Fridman, L., Levant, A.: Second-order sliding-mode observer for mechanical systems. IEEE Trans. Autom. Control 50(11), 1785–1789 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pisano, A., Rapaić, M.R., Jelic̆ić, Z.D., Usai, E.: Sliding mode control approaches to the robust regulation of linear multivariable fractional-order dynamics. Int. J. Robust Nonlinear Control 20(18), 2045–2056 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jakovljević, B., Pisano, A., Rapaić, M.R., Usai, E.: On the sliding-mode control of fractional-order nonlinear uncertain dynamics. Int. J. Robust Nonlinear Control 26(4), 782–798 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Pisano, A., Rapaić, M.R., Usai, E., Jelic̆ić, Z.D.: Continuous finite-time stabilization for some classes of fractional order dynamics. In: IEEE International Workshop on Variable Structure Systems, pp. 16–21 (2012)Google Scholar
  16. 16.
    Kamal, S., Raman, A., Bandyopadhyay, B.: Finite-time stabilization of fractional order uncertain chain of integrator: an integral sliding mode approach. IEEE Trans. Autom. Control 58(6), 1597–1602 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: Uniformly continuous differintegral sliding mode control of nonlinear systems subject to Hölder disturbances. Automatica 66, 179–184 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: Fractional integral sliding modes for robust tracking of nonlinear systems. Nonlinear Dyn. 87(2), 895–901 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: Continuous fractional-order sliding PI control for nonlinear systems subject to non-differentiable disturbance. Asian J. Control 19(1), 279–288 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: A novel continuous fractional sliding mode control. Int. J. Syst. Sci. 48(13), 2901–2908 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A., Romero-Galván, R.: Output feedback finite-time stabilization of systems subject to Hölder disturbances via continuous fractional sliding modes. Math. Prbl. Eng. (2017).  https://doi.org/10.1155/2017/3146231 CrossRefGoogle Scholar
  22. 22.
    Izaguirre-Espinosa, C., Muñoz-Vázquez, A.J., Sánchez-Orta, A., Parra-Vega, V., Sanahuja, G.: Fractional attitude-reactive control for robust quadrotor position stabilization without resolving underactuation. Control Eng. Pract. 53, 47–56 (2016)CrossRefGoogle Scholar
  23. 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)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Muñoz-Vázquez, A.J., Ramírez-Rodríguez, H., Parra-Vega, V., Sánchez-Orta, A.: Fractional sliding mode control of underwater ROVs subject to non-differentiable disturbances. Int. J. Control Autom. Syst. 15(3), 1314–1321 (2017)CrossRefGoogle Scholar
  25. 25.
    Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A., Romero-Galván, G.: Finite-time disturbance observer via continuous fractional sliding modes. Trans. Inst. Measur. Control. (2017).  https://doi.org/10.1177/0142331217737833 CrossRefGoogle Scholar
  26. 26.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  27. 27.
    Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993)zbMATHGoogle Scholar
  28. 28.
    Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A., Romero-Galván, G.: Quadratic Lyapunov functions for stability analysis in fractional-order systems with not necessarily differentiable solutions. Syst. Control Lett. 116, 15–19 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Isidori, I.: Nonlinear Control Systems. Communications and Control Engineering. Springer, London (1995)CrossRefGoogle Scholar
  30. 30.
    Khalil, H.K.: Nonlinear Systems. Prentice-Hall, New Jersey (1996)Google Scholar
  31. 31.
    Spong, M.W., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control. Wiley, New York (2006)Google Scholar
  32. 32.
    Goldstein, H.: Classical Mechanics. Pearson Education, Bangalore (1965)zbMATHGoogle Scholar
  33. 33.
    Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Aldo-Jonathan Muñoz-Vázquez
    • 1
  • Anand Sánchez-Orta
    • 2
  • Vicente Parra-Vega
    • 2
  1. 1.CONACYT–School of Engineering, Autonomous University of Chihuahua (UACH), Campus IIChihuahuaMexico
  2. 2.Research Center for Advanced StudiesRamos ArizpeMexico

Personalised recommendations