Enhanced FPGA realization of the fractional-order derivative and application to a variable-order chaotic system


The efficiency of the hardware implementations of fractional-order systems heavily relies on the efficiency of realizing the fractional-order derivative operator. In this work, a generic hardware implementation of the fractional-order derivative based on the Grünwald–Letnikov’s approximation is proposed and verified on a field-programmable gate array. The main advantage of this particular realization is its flexibility in applications which enable easy real-time configuration of the values of the fractional orders, step sizes, and/or other system parameters without changing the hardware architecture. Different approximation techniques are used to improve the hardware performance including piece-wise linear/quadratic methods. As an application, a variable-order chaotic oscillator is implemented and verified using fractional orders that vary in time.

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15


  1. 1.

    Alvarez, G., Li, S.: Some basic cryptographic requirements for chaos-based cryptosystems. Int. J. Bifurc. Chaos 16(08), 2129–2151 (2006)

  2. 2.

    Assadi, I., Charef, A., Bensouici, T., Belgacem, N.: Arrhythmias discrimination based on fractional order system and KNN classifier. In: IET Conference Proceedings, pp. 6–6 (2015)

  3. 3.

    Barakat, M.L.: Generalized hardware post-processing technique for chaos-based pseudorandom number generators. ETRI J. 35(3), 448–458 (2013)

  4. 4.

    Bayasi, N., Tekeste, T., Saleh, H., Mohammad, B., Khandoker, A., Ismail, M.: Low-power ECG-based processor for predicting ventricular arrhythmia. IEEE Trans. Very Large Scale Integr. VLSI Syst. 24(5), 1962–1974 (2015)

  5. 5.

    Bhrawy, A., Zaky, M.: Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Comput. Math. Appl. 73(6), 1100–1117 (2017)

  6. 6.

    Caponetto, R., Dongola, G., Maione, G., Pisano, A.: Integrated technology fractional order proportional-integral-derivative design. J. Vib. Control 20(7), 1066–1075 (2014)

  7. 7.

    Clemente-López, D., Muñoz-Pacheco, J., Félix-Beltrán, O., Volos, C.: Efficient computation of the Grünwald–Letnikov method for ARM-based implementations of fractional-order chaotic systems. In: 8th International Conference on Modern Circuits and Systems Technologies (MOCAST), pp. 1–4. IEEE (2019)

  8. 8.

    Ferdi, Y.: Computation of fractional order derivative and integral via power series expansion and signal modelling. Nonlinear Dyn. 46(1), 1–15 (2006)

  9. 9.

    Hartley, T.T., Lorenzo, C.F.: Dynamics and control of initialized fractional-order systems. Nonlinear Dyn. 29(1–4), 201–233 (2002)

  10. 10.

    Howard, R.M.: Principles of Random Signal Analysis and Low Noise Design. Wiley, Hoboken (2002)

  11. 11.

    Hsiao, S.F., Ko, H.J., Tseng, Y.L., Huang, W.L., Lin, S.H., Wen, C.S.: Design of hardware function evaluators using low-overhead nonuniform segmentation with address remapping. IEEE Trans. Very Large Scale Integr. VLSI Syst. 21(5), 875–886 (2012)

  12. 12.

    Huang, X., Zhang, B., Qin, H., An, W.: Closed-form design of variable fractional-delay fir filters with low or middle cutoff frequencies. IEEE Trans. Circuits Syst. I Regul. Pap. 65(2), 628–637 (2018)

  13. 13.

    Jiang, C., Adams, J., Carletta, J., Hartley, T.: Hardware implementation of fractional-order systems as infinite impulse response filters. IFAC Proc. 39(11), 408–413 (2006)

  14. 14.

    Jiang, C.X., Carletta, J.E., Hartley, T.T.: Implementation of fractional-order operators on field programmable gate arrays. In: Advances in Fractional Calculus, pp. 333–346. Springer, Netherlands (2007)

  15. 15.

    Jiang, C.X., Carletta, J.E., Hartley, T.T., Veillette, R.J.: A systematic approach for implementing fractional-order operators and systems. IEEE J. Emerg. Sel. Top. Circuits Syst. 3(3), 301–312 (2013)

  16. 16.

    Munoz-Pacheco, J., Zambrano-Serrano, E., Volos, C., Tacha, O., Stouboulos, I., Pham, V.T.: A fractional order chaotic system with a 3d grid of variable attractors. Chaos Solitons Fractals 113, 69–78 (2018)

  17. 17.

    Pano-Azucena, A.D., Tlelo-Cuautle, E., Muñoz-Pacheco, J.M., de la Fraga, L.G.: FPGA-based implementation of different families of fractional-order chaotic oscillators applying Grünwald–Letnikov method. Commun. Nonlinear Sci. Numer. Simul. 72, 516–527 (2019)

  18. 18.

    Petráš, I.: Method for simulation of the fractional order chaotic systems. Acta Montan. Slovaca 11(4), 273–277 (2006)

  19. 19.

    Petráš, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, London (2011)

  20. 20.

    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Amsterdam (1998)

  21. 21.

    Rajagopal, K., Akgul, A., Jafari, S., Aricioglu, B.: A chaotic memcapacitor oscillator with two unstable equilibriums and its fractional form with engineering applications. Nonlinear Dyn. 91(2), 957–974 (2018)

  22. 22.

    Rajagopal, K., Karthikeyan, A., Srinivasan, A.: Dynamical analysis and FPGA implementation of a chaotic oscillator with fractional-order memristor components. Nonlinear Dyn. 91(3), 1491–1512 (2018)

  23. 23.

    Rajagopal, K., Karthikeyan, A., Srinivasan, A.K.: FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization. Nonlinear Dyn. 87(4), 2281–2304 (2017)

  24. 24.

    Rana, K., Kumar, V., Mittra, N., Pramanik, N.: Implementation of fractional order integrator/differentiator on field programmable gate array. Alex. Eng. J. 55(2), 1765–1773 (2016)

  25. 25.

    Sabzalian, M.H., Mohammadzadeh, A., Lin, S., et al.: Robust fuzzy control for fractional-order systems with estimated fraction-order. Nonlinear Dyn. 98, 2375–2385 (2019)

  26. 26.

    Sayed, W.S., Tolba, M.F., Radwan, A.G., Abd-El-Hafiz, S.K.: FPGA realization of a speech encryption system based on a generalized modified chaotic transition map and bit permutation. Multimed. Tools Appl. 78(12), 16097–16127 (2019)

  27. 27.

    Tavakoli-Kakhki, M.: Implementation of fractional-order transfer functions in the viewpoint of the required fractional-order capacitors. Int. J. Syst. Sci. 48(1), 63–73 (2017)

  28. 28.

    Tekeste, T., Bayasi, N., Saleh, H., Khandoker, A., Mohammad, B., Al-Qutayri, M., Ismail, M.: Adaptive ECG interval extraction. In: IEEE International Symposium on Circuits and Systems (ISCAS), pp. 998–1001. IEEE (2015)

  29. 29.

    Tolba, M.F., et al.: FPGA implementation of two fractional order chaotic systems. AEU Int. J. Electron. Commun. 78, 162–172 (2017)

  30. 30.

    Tolba, M.F., et al.: FPGA realization of caputo and Grünwald–Letnikov operators. In: 6th International Conference on Modern Circuits and Systems Technologies (MOCAST), pp. 1–4. IEEE (2017)

  31. 31.

    Tolba, M.F., et al.: Fractional order integrator/differentiator: FPGA implementation and fopid controller application. AEU Int. J. Electron. Commun. 98, 220–229 (2019)

  32. 32.

    Tolba, M.F., Said, L.A., Madian, A.H., Radwan, A.G.: FPGA implementation of fractional-order integrator and differentiator based on Grünwald–Letnikov definition. In: 29th International Conference on Microelectronics (ICM), pp. 1–4 (2017).

  33. 33.

    Tolba, M.F., Said, L.A., Madian, A.H., Radwan, A.G.: FPGA implementation of the fractional order integrator/differentiator: two approaches and applications. IEEE Trans. Circuits Syst. I Regul. Pap. 66(4), 1484–1495 (2018)

  34. 34.

    Yasin, M., Tekeste, T., Saleh, H., Mohammad, B., Sinanoglu, O., Ismail, M.: Ultra-low power, secure iot platform for predicting cardiovascular diseases. IEEE Trans. Circuits Syst. I Regul. Pap. 64(9), 2624–2637 (2017)

  35. 35.

    Zambrano-Serrano, E., Munoz-Pacheco, J., Campos-Cantón, E.: Chaos generation in fractional-order switched systems and its digital implementation. AEU Int. J. Electron. Commun. 79, 43–52 (2017)

Download references


This work is supported by the System-On-Chip center at the Khalifa University of Science and Technology under Award No. (RC2-2018-020).

Author information

Correspondence to Baker Mohammad.

Ethics declarations

Conflict of Interest

The authors declare no competing interests.

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

Verify currency and authenticity via CrossMark

Cite this article

Tolba, M.F., Saleh, H., Mohammad, B. et al. Enhanced FPGA realization of the fractional-order derivative and application to a variable-order chaotic system. Nonlinear Dyn (2020).

Download citation


  • Fractional-order systems
  • Chaotic oscillators
  • FPGA