Advertisement

Approximation of Fractional-Order Operators

  • Reyad El-KhazaliEmail author
  • Iqbal M. Batiha
  • Shaher Momani
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 303)

Abstract

In order to deal with some difficult problems in fractional-order systems, like computing analytical time responses such as unit impulse and step responses; some rational approximations for the fractional-order operators are presented with satisfying results in simulation and realization. In this chapter, several comparisons in the time response and Bode results between four well-known methods; Oustaloup’s method, Matsuda’s method, AbdelAty’s method, and El-Khazali’s method are made for the rational approximation of fractional-order operator (fractional Laplace operator). The various methods along with their advantages and limitations are described in this chapter. Simulation results are shown for different orders of the fractional operator. It has been shown in several numerical examples that the El-Khazali’s method is very successful in comparison with Oustaloup’s, Matsuda’s, and AbdelAty’s methods.

Keywords

Fractional-Order models Oustaloup’s approximation Matsuda’s approximation AbdelAty’s approximation El-Khazali’s approximation 

Notes

Acknowledgements

We thank the sponsors of the International Conference on Fractional Differentiation and its Applications (ICFDA 2018), who provided insight and expertise that greatly assisted the research to be in hand. We would also like to thank the international editors, Praveen Agarwal, Dumitru Baleanu, YangQuan Chen, and Tenreiro Machado, for their valuable comments and remarks.

References

  1. 1.
    Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuit. Syst. I, Fund. Theory and Appl. 47, 25–39 (2000)CrossRefGoogle Scholar
  2. 2.
    Vinagre, B.M., Podlubny, I., Hernandez, A., Feliu, V.: Some approximations of fractional order operators used in control theory and applications. Fract. Calculus Appl. Anal. 3, 231–248 (2000)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Yce, A., Deniz, F.N., Tan, N.: A new integer-order approximation table for fractional order derivative operators. Int. Federation Autom. Control 50, 9736–9741 (2017)Google Scholar
  4. 4.
    AbdelAty, A.M., AbdelAty, A.S., Radwan, A.G., Psychalinos, C., Maundy, B.J.: Approximation of the fractional-order Laplacian \(s^\alpha \) As a weighted sum of first-order high-pass filters. IEEE Trans. Circuit. Sys. II: Express Briefs 65 (2018)Google Scholar
  5. 5.
    El-Khazali, R.: Discretization of Fractional-Order Laplacian Operators. 19th IFAC Congress, Cape Town, 25/8/2014Google Scholar
  6. 6.
    Atherton, D.P., Tan, N., Yce, A.: Methods for computing the time response of fractional-order systems. IET Control Theory Appl. 9, 817–830 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Podlubny, I.: Fractional-order systems and \(PI^\lambda D^\mu \) controllers. IEEE Trans. Automat. Control 44, 208–214 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lin, W., Chongquan, Z.: Design of optimal fractional-order PID controllers using particle swarm optimization algorithm for DC motor system. In: IEEE Advanced Information Technology, Electronic and Automation Control Conference (IAEAC), pp. 175–179 (2015)Google Scholar
  9. 9.
    El-Khazali, R.: Fractional-order \(PI^\lambda D^\mu \) controller design. Comp. Math. App. 66, 639–646 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    El-Khazali, R.: Discretization of Fractional-Order Differentiators and Integrators. 19th IFAC Congress (2014)CrossRefGoogle Scholar
  11. 11.
    Tepljakov, A.: Fractional-order Modeling and Control of Dynamic Systems. Tallin University of Technology, Tallinn (2015)zbMATHGoogle Scholar
  12. 12.
    Petras, I., Podlubny, I., OLeary, P.: Analogue Realization of Fractional Order Controllers. Fakulta BERG, TU Kosice (2002)Google Scholar
  13. 13.
    Anis Kharisovich Gilmutdinov: Pyotr Arkhipovich Ushakov, Reyad El-Khazali: Fractal Elements and their Applications. Springer, Switzerland (2017)CrossRefGoogle Scholar
  14. 14.
    Matsuda, K., Fujii, H.: Optimized wave-absorbing control: analytical and experimental results. J. Guidance Control Dyn. 16, 1153–1164 (1993)CrossRefGoogle Scholar
  15. 15.
    Ogata, K.: Modern Control Engineering. Prentice Hall, New Jersey (2010)zbMATHGoogle Scholar
  16. 16.
    Yang, X.S.: Flower pollination algorithm for global optimization in Unconventional computation and natural computation, pp. 240–249. Springer (2012)Google Scholar
  17. 17.
    El-Khazali, R.: On the biquadratic approximation of fractional-order Laplacian operators. Analog Int. Circuits and Sig. Proc. 82, 503–517 (2015)CrossRefGoogle Scholar
  18. 18.
    Astrom, K., Hagglund, T.: PID Controllers; Theory. Design and Tuning. Instrument Society of America, Research Triangle Park (1995)Google Scholar
  19. 19.
    Saleh, K.: Fractional Order PID Controller Tuning by Frequency Loop-Shaping: Analysis and Applications. Arizona State University, Arizona (2017)Google Scholar
  20. 20.
    Podlubny, I.: Fractional-order systems and \(PI^\lambda D^\mu \)-controllers. IEEE Trans. Autom. Control 44, 208–214 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Cech, M., Schlegel, M.: The Fractional-order PID Controller Outperforms the Classical One. Process control 2006: Pardubice Technical University, pp. 1–6 (2006)Google Scholar
  22. 22.
    Podlubny, I., Dorcak, L., Kostial, I.: On fractional derivatives, fractional-order dynamic systems and \(PI^\lambda D^\mu \) controllers. In: Proceedings of the 36th Conference on Decision and Control, San Diego, California, USA (1997)Google Scholar
  23. 23.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)zbMATHGoogle Scholar
  24. 24.
    Rinu, P.S.: Design and analysis of optimal fractional-order PID controller. Int. J. Appl. Eng. Res. 10, 23000–23002 (2015)Google Scholar
  25. 25.
    Xue, D., Chen, Y.Q., Atherton, D.P.: Linear Feedback Control: Analysis and Design with MATLAB. Society for Industrial and Applied Mathematics, Philadelphia (2007)CrossRefGoogle Scholar
  26. 26.
    Luo, Y., Chen, Y.Q.: Fractional-order [proportional derivative] controller for robust motion control: tuning procedure and validation. Am. Control Conf., 1412–1417 (2009)Google Scholar
  27. 27.
    Kumar, A.: Controller Design for Fractional Order Systems. National Institute of Technology- Rourkela, Odisha (2013)Google Scholar
  28. 28.
    Maione, G., Lino, P.: New tuning rules for fractional \(PI^\alpha \) controllers. Nonlinear Dyn. 49, 251–257 (2007)CrossRefGoogle Scholar
  29. 29.
    Luo, Y., Chen, Y.Q.: Fractional Order Motion Controls. Wiley, Odisha (2012)CrossRefGoogle Scholar
  30. 30.
    Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, London, New York (2010)CrossRefGoogle Scholar
  31. 31.
    Dimeas, I.: Design of an Integrated Fractional-Order Controller. University of Patras, Patras (2017)Google Scholar
  32. 32.
    Oldham, K.B., Spanier, J.: The fractional calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)zbMATHGoogle Scholar
  33. 33.
    Caponetto, R., Dongola, G., Fortuna, L., Petras, I.: Fractional order Systems: Modelling and Control Applications. World Scientific, Singapore (2010)CrossRefGoogle Scholar
  34. 34.
    Chen, Y.Q.: Ubiquitous fractional order controls. In: Proceedings of the Second IFAC Symposium on Fractional Derivatives and Applications. Porto (2006)Google Scholar
  35. 35.
    Monje, C.A., Vinagre, B.M., Feliu, V., Chen, Y.Q.: Tuning and auto-tuning of fractional order controllers for industry application. Control Eng. Pract. 16, 798–812 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Reyad El-Khazali
    • 1
    Email author
  • Iqbal M. Batiha
    • 2
  • Shaher Momani
    • 2
    • 3
  1. 1.ECSE DepartmentKhalifa UniversityAbu DhabiUAE
  2. 2.Department of MathematicsThe University of JordanAmmanJordan
  3. 3.Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of ScienceKing Abdulaziz UniversityJeddahKingdom of Saudi Arabia

Personalised recommendations