Skip to main content
SpringerLink
Log in

Fractional integro-differential sliding mode control of a class of distributed-order nonlinear systems

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

Distributed-order systems arise as a natural generalization of fractional- and integer-order systems, and these are commonly associated with slow and ultra-slow dynamics, which motivates designing robust schemes to enforce fast stabilization. This paper proposes a robust sliding mode controller that induces a stable motion in a finite time, relying on a dynamic extension to produce an integer-order reaching phase. Thus, a continuous fractional sliding mode controller is designed to compensate not necessarily integer-order differentiable disturbances. The theoretical results demonstrate that both the disturbance is exactly compensated and the pseudo-state converges asymptotically to the origin. A numerical simulation is carried out to demonstrate the reliability and efficacy of the proposed method, where a comparison to a conventional sliding mode scheme reveals a superior performance for the proposed fractional sliding mode approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Alkasassbeh, M., Omar, Z., Mebarek-Oudina, F., Raza, J., Chamkha, A.: Heat transfer study of convective fin with temperature-dependent internal heat generation by hybrid block method. Heat Transf. Asian Res. 48(4), 1225–1244 (2019)

    Article  Google Scholar 

  2. Atanackovic, T., Budincevic, M., Pilipovic, S.: On a fractional distributed-order oscillator. J. Phys. A Math. Gen. 38(30), 6703 (2005)

    Article  MathSciNet  Google Scholar 

  3. Atanackovic, T.M.: On a distributed derivative model of a viscoelastic body. C.R. Mec. 331(10), 687–692 (2003)

    Article  Google Scholar 

  4. Atanacković, T.M., Oparnica, L., Pilipović, S.: On a nonlinear distributed order fractional differential equation. J. Math. Anal. Appl. 328(1), 590–608 (2007)

    Article  MathSciNet  Google Scholar 

  5. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory, vol. 178. Springer Science & Business Media, Berlin (2008)

    MATH  Google Scholar 

  6. Dadkhah, E., Shiri, B., Ghaffarzadeh, H., Baleanu, D.: Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods. J. Appl. Math. Comput. 63(1), 29–57 (2020)

    Article  MathSciNet  Google Scholar 

  7. Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225(1), 96–104 (2009)

    Article  MathSciNet  Google Scholar 

  8. Efe, M.Ö.: Integral sliding mode control of a quadrotor with fractional order reaching dynamics. Trans. Inst. Meas. Control. 33(8), 985–1003 (2011)

    Article  Google Scholar 

  9. Farhan, M., Omar, Z., Mebarek-Oudina, F., Raza, J., Shah, Z., Choudhari, R., Makinde, O.: Implementation of the one-step one-hybrid block method on the nonlinear equation of a circular sector oscillator. Comput. Math. Model. 31(1), 116–132 (2020)

    Article  MathSciNet  Google Scholar 

  10. Fernández-Anaya, G., Nava-Antonio, G., Jamous-Galante, J., Muñoz-Vega, R., Hernández-Martínez, E.: Asymptotic stability of distributed order nonlinear dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 48, 541–549 (2017)

    Article  MathSciNet  Google Scholar 

  11. 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)

    Article  MathSciNet  Google Scholar 

  12. Jensen, J.L.W.V.: Om konvekse funktioner og uligheder imellem middelvaerdier. Nyt Tidsskrift for Matematik 30, 49–69 (1905) https://www.jstor.org/stable/24528332

  13. Jensen, J.L.W.V.: Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 30, 175–193 (1906). https://doi.org/10.1007/BF02418571

    Article  MathSciNet  MATH  Google Scholar 

  14. Jiao, Z., Chen, Y.Q.: Stability of fractional-order linear time-invariant systems with multiple noncommensurate orders. Comput. Math. Appl. 64(10), 3053–3058 (2012)

    Article  MathSciNet  Google Scholar 

  15. Jiao, Z., Chen, Y.Q., Podlubny, I.: Distributed-order dynamic systems: stability. Simulation, Applications and Perspectives, London (2012)

  16. Kamal, S., Chalanga, A., Moreno, J.A., Fridman, L., Bandyopadhyay, B.: Higher order super-twisting algorithm. In: 2014 13th International Workshop on Variable Structure Systems (VSS), IEEE, pp 1–5 (2014)

  17. Kochubei, A.N.: Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl. 340(1), 252–281 (2008)

    Article  MathSciNet  Google Scholar 

  18. Lazović, G., Vosika, Z., Lazarević, M., Simić-Krstić, J., Koruga, D.: Modeling of bioimpedance for human skin based on fractional distributed-order modified cole model. FME Transact. 42(1), 74–81 (2014)

    Article  Google Scholar 

  19. Li, A., Liu, G., Luo, Y., Yang, X.: An indirect lyapunov approach to robust stabilization for a class of linear fractional-order system with positive real uncertainty. J. Appl. Math. Comput. 57(1), 39–55 (2018)

    Article  MathSciNet  Google Scholar 

  20. Li, Y., Chen, Y.: Theory and implementation of distributed-order element networks. Int. Des. Eng. Tech. Conf. Comput. Inf. Eng. Conf. 54808, 361–367 (2011)

    Google Scholar 

  21. Li, Y., Chen, Y.Q.: Theory and implementation of weighted distributed order integrator. In: IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, IEEE, pp 119–124 (2012)

  22. Liu, L., Feng, L., Xu, Q., Zheng, L., Liu, F.: Flow and heat transfer of generalized maxwell fluid over a moving plate with distributed order time fractional constitutive models. Int. Commun. Heat Mass Transf. 116, 104679 (2020)

    Article  Google Scholar 

  23. Mahmoud, G.M., Farghaly, A.A., Abed-Elhameed, T.M., Aly, S.A., Arafa, A.A.: Dynamics of distributed-order hyperchaotic complex van der pol oscillators and their synchronization and control. Eur. Phys. J. Plus 135(1), 1–16 (2020)

    Article  Google Scholar 

  24. Mebarek-Oudina, F.: Numerical modeling of the hydrodynamic stability in vertical annulus with heat source of different lengths. Eng. Sci. Technol. Int. J. 20(4), 1324–1333 (2017)

    Google Scholar 

  25. Morgado, M.L., Rebelo, M.: Numerical approximation of distributed order reaction-diffusion equations. J. Comput. Appl. Math. 275, 216–227 (2015)

    Article  MathSciNet  Google Scholar 

  26. 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)

    Article  MathSciNet  Google Scholar 

  27. Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: Non-smooth convex Lyapunov functions for stability analysis of fractional-order systems. Trans. Inst. Meas. Control. 41(6), 1627–1639 (2019)

    Article  Google Scholar 

  28. Muñoz-Vázquez, A.J., Fernández-Anaya, G., Sánchez-Torres, J.D., Meléndez-Vázquez, F.: Predefined-time control of distributed-order systems. Nonlinear Dyn. 103(3), 2689–2700 (2021)

    Article  Google Scholar 

  29. Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course, vol. 87. Springer Science & Business Media, Berlin (2013)

    MATH  Google Scholar 

  30. Patnaik, S., Semperlotti, F.: Application of variable-and distributed-order fractional operators to the dynamic analysis of nonlinear oscillators. Nonlinear Dyn. 100(1), 561–580 (2020)

    Article  Google Scholar 

  31. Pérez-Ventura, U., Fridman, L.: Design of super-twisting control gains: a describing function based methodology. Automatica 99, 175–180 (2019)

    Article  MathSciNet  Google Scholar 

  32. Pisano, A., Rapaić, M.R., Jeličić, 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)

    Article  MathSciNet  Google Scholar 

  33. Pisano, A., Rapaić, M.R., Usai, E., Jeličić, Z.D.: Continuous finite-time stabilization for some classes of fractional order dynamics. In: International Workshop on Variable Structure Systems, IEEE, pp 16–21 (2012)

  34. Podlubny, I.: Fractional Differential Equations. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  35. Rayal, A., Verma, S.R.: An approximate wavelets solution to the class of variational problems with fractional order. J. Appl. Math. Comput. 65(1), 735–769 (2021)

    Article  MathSciNet  Google Scholar 

  36. Ross, B., Samko, S.G., Love, E.R.: Functions that have no first order derivative might have fractional derivatives of all orders less than one. Real Anal. Exch. 20(1), 140–157 (1994)

    Article  MathSciNet  Google Scholar 

  37. Taghavian, H., Tavazoei, M.S.: Stability analysis of distributed-order nonlinear dynamic systems. Int. J. Syst. Sci. 49(3), 523–536 (2018)

    Article  MathSciNet  Google Scholar 

  38. Toaldo, B.: Lévy mixing related to distributed order calculus, subordinators and slow diffusions. J. Math. Anal. Appl. 430(2), 1009–1036 (2015)

    Article  MathSciNet  Google Scholar 

  39. Wang, F., Liu, X.: Pseudo-state estimation for fractional order neural networks. Neural Process. Lett. pp 1–14 (2021)

  40. Wei, L., Liu, L., Sun, H.: Stability and convergence of a local discontinuous galerkin method for the fractional diffusion equation with distributed order. J. Appl. Math. Comput. 59(1), 323–341 (2019)

    Article  MathSciNet  Google Scholar 

  41. Yang, W., Chen, X., Zhang, X., Zheng, L., Liu, F.: Flow and heat transfer of viscoelastic fluid with a novel space distributed-order constitution relationship. Comput. Math. Appl. 94, 94–103 (2021)

    Article  MathSciNet  Google Scholar 

  42. Yang, Z., Zhang, J., Niu, Y.: Finite-time stability of fractional-order bidirectional associative memory neural networks with mixed time-varying delays. J. Appl. Math. Comput. 63(1), 501–522 (2020)

    Article  MathSciNet  Google Scholar 

  43. Zaky, M.A., Machado, J.T.: On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 52, 177–189 (2017)

    Article  MathSciNet  Google Scholar 

  44. Želi, V., Zorica, D.: Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law. Phys. A 492, 2316–2335 (2018)

    Article  MathSciNet  Google Scholar 

  45. Zhou, F., Zhao, Y., Li, Y., Chen, Y.: Design, implementation and application of distributed order pi control. ISA Trans. 52(3), 429–437 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Multidisciplinary Engineering, Texas A&M University, 6200 Tres Lagos Blvd, Higher Education Center at McAllen, Texas, 78504, USA

    Aldo Jonathan Muñoz-Vázquez

  2. Department of Physics and Mathematics, Universidad Iberoamericana, 880 Prolongacion Paseo de la Reforma, Lomas de Santa Fe, Alvaro Obregón, Mexico City, 01219, Mexico

    Guillermo Fernández-Anaya

  3. Department of Mathematics and Physics, ITESO University, Anillo Perif. Sur Manuel Gómez Morin 8585, Tlaquepaque, Jalisco, 45604, Mexico

    Juan Diego Sánchez-Torres

Authors
  1. Aldo Jonathan Muñoz-Vázquez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guillermo Fernández-Anaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Juan Diego Sánchez-Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guillermo Fernández-Anaya.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Muñoz-Vázquez, A.J., Fernández-Anaya, G. & Sánchez-Torres, J.D. Fractional integro-differential sliding mode control of a class of distributed-order nonlinear systems. J. Appl. Math. Comput. 68, 2743–2760 (2022). https://doi.org/10.1007/s12190-021-01632-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01632-8

Keywords

Mathematics Subject Classification

Search

Navigation