Skip to main content
SpringerLink
Log in

Improved Digital Rational Approximation of the Operator \(S^{\alpha }\) Using Second-Order s-to-z Transform and Signal Modeling

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

In this paper, an improved stable digital rational approximation of the fractional-order operator \(s^\alpha ,\alpha \in \,R\) is developed. First, a novel efficient second-order digital differentiator is derived from the transfer function of the digital integrator proposed by Tseng. Then, the fractional power of the new s-to-z transform is expanded using power series expansion (PSE)-signal-modeling technique to obtain stable rational approximation of \(s^\alpha \). Simulation results show that the proposed rational approximation has better frequency characteristics in almost the whole frequency range than that of existing first-order s-to-z transforms based approximations for different values of the fractional-order \(\alpha \). This paper also shows the benefit of using PSE-signal-modeling approach with first- or second-order mapping functions over PSE-truncation approach that is used in recent works for rational approximation of the operator \(s^\alpha \), and highlights the major disadvantage of the latter approach that leads to undesirable rational models with complex conjugate poles and zeros.

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

Similar content being viewed by others

References

  1. M.A. Al-Alaoui, Novel stable higher order s-to-z transforms. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 48(11), 1326–1329 (2001)

    Article  Google Scholar 

  2. M.A. Al-Alaoui, Novel class of digital integrators and differentiators. IET Signal Process 5(2), 251–260 (2011)

    Article  Google Scholar 

  3. M. Benmalek, A. Charef, Digital fractional order operators for R-wave detection in electrocardiogram signal. IET signal process 3(5), 381–391 (2009)

    Article  Google Scholar 

  4. T. Blaszczyk, M. Ciesielski, M. Klimek, J. Leszczynski, Numerical solution of fractional oscillator equation. Appl. Math. Comput. 218(6), 2480–2488 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. O. Cem, S. Melin, Y. Yavuz, An experimental performance evaluation for the suppression of vibrations of the second mode of a smart beam. in Proceedings of Ankara International Aerospace Conference, METU, Ankara, Turkey, 14–16 September, 2011, pp. 1–9

  6. A. Charef, H.H. Sun, Y.Y. Tsao, B. Onaral, Fractal system as represented by singularity function. IEEE Trans. Autom. Control. 37(9), 1465–1470 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Y.Q. Chen, K.L. Moore, Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49(3), 363–367 (2002)

    Article  MathSciNet  Google Scholar 

  8. Y.Q. Chen, B.M. Vinagre, A new IIR-type digital fractional order differentiator. Signal Process 83(11), 2359–2365 (2003)

    Article  MATH  Google Scholar 

  9. Y. Ferdi, Computation of fractional order derivative and integral via power series expansion and signal modeling. Nonlinear Dyn. 46(1–2), 1–15 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Y. Ferdi, Impulse invariance-based method for the computation of fractional integral of order 0\(<\alpha <\)1. Comput. Electr. Eng. 35(5), 722–729 (2009)

    Article  MATH  Google Scholar 

  11. Y. Ferdi, Improved lowpass differentiator for physiological signal processing. in Proceedings of IEEE 7th International Conference on Communication Systems Networks and Digital Signal Process. (CSNDSP), Newcastle upon Tyne, England, 21–23, July, 2010, pp. 747–750

  12. Y. Ferdi, Fractional order calculus-based filters for biomedical signal processing. in Proceedings of IEEE 1st Middle East Conference on Biomedical Engineering (MECBME), Sharjah, UAE, 21–24, February, 2011, pp. 73–76

  13. Y. Ferdi, A. Taleb-Ahmed, M.R. Lakehal, Efficient generation of \(1/f^\beta \) noise using signal modeling techniques. IEEE Trans. Circuits Syst. I 55(6), 1704–1710 (2008)

    Article  MathSciNet  Google Scholar 

  14. M. Gupta, P. Varshney, G.S. Visweswaran, Digital fractional order differentiator and integrator models based on first and higher order operators. Int. J. Circ. Theor. Appl. (2010). doi:10.1002/cta.650

    Google Scholar 

  15. B.T. Krishna, Binary phase coded sequence generation using fractional order logistic equation. Circuits. Syst. Signal Process 31(1), 401–411 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. B.T. Krishna, K.V.V.S. Reddy, Design of fractional order digital differentiators and integrators using indirect discretization. Int. J. Circ. Theor. Appl. 11(2), 143–151 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. P. Kumar, O.P. Agrawal, An approximate method for numerical solution of fractional differential equations. Signal Process 86, 2602–2610 (2006)

    Article  MATH  Google Scholar 

  18. P. Lanusse, H. Benlaoukli, D. Nelson-Gruel, A. Oustaloup, Fractional-order control and interval analysis of SISO systems with time-delayed state. IET Cont. Theory Appl. 2(1), 16–23 (2008)

    Article  MathSciNet  Google Scholar 

  19. F. Leulmi, Y. Ferdi, An improvement of the rational approximation of the fractional operator \(s^\alpha \). in Proceedings of IEEE International Conference on Electronics, Communications and Photonics Conference (SIECPC), Riyadh, Saudi Arabia, 24–26, April, 2011, pp. 1–6

  20. Y. Li, H. Tang, H. Chen, Fractional order derivative spectroscopy for resolving simulated overlapped Lorenzian peaks. Chemom. Intell. Lab. Syst. 107(1), 83–89 (2011)

    Article  Google Scholar 

  21. J. Lu, M. Xie, Use fractional calculus in iris localization. in Proceedings of International Conference Communications, Circuits and Systems (ICCCAS), Fujian, China, 25–27, May, 2008, pp. 946–949

  22. G. Maione, High-speed digital realizations of fractional operators in delta domain. IEEE Trans. Autom. Control 56(3), 697–702 (2011)

    Article  MathSciNet  Google Scholar 

  23. B. Mathieu, P. Melchior, A. Oustaloup, C. Ceyral, Fractional differentiation for edge detection. Signal Process 83(11), 2421–2432 (2003)

    Article  MATH  Google Scholar 

  24. C.A. Monje, F. Ramos, V. Feliu, B.M. Vinagre, Tip position control of a lightweight flexible manipulator using a fractional order controller. IET Cont. Theory Appl. 1(5), 1451–1460 (2007)

    Article  Google Scholar 

  25. I. Petras, Fractional-order feedback control of DC motor. J. Electr. Eng. 60(3), 117–128 (2009)

    MathSciNet  Google Scholar 

  26. A.G. Radwan, K.N. Salama, Fractional-order RC and RL circuits. Circuits Syst. Signal Process 31(6), 1901–1915 (2012)

    Article  MathSciNet  Google Scholar 

  27. A. Razminia, A.F. Dizaji, V.J. Majd, Solution existence for non-autonomous variable-order fractional differential equations. Math. Comput. Model. 55, 1106–1117 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  28. C.C. Tseng, Digital integrator design using Simpson rule and fractional delay filter. IEE Proc. Vis. Image Signal Process 153(1), 79–85 (2006)

    Article  Google Scholar 

  29. C.C. Tseng, Design of FIR and IIR fractional order Simpson digital integrators. Signal Process 87(5), 1045–1057 (2007)

    Article  MATH  Google Scholar 

  30. P. Varshney, G.S. Visweswaran, First and higher order operator based fractional order differentiator and integrator models. in Proceedings of 2009 IEEE Region 10 Conference, TENCON 2009, pp. 1–6

  31. B.M. Vinagre, I. Podlubny, A. Hernandez, V. Feliu, Some approximations of fractional order operators used in control theory and applications. Fract. Calc. Appl. Anal. 3(3), 231–248 (2000)

    MATH  MathSciNet  Google Scholar 

  32. G.S. Visweswaran, P. Varshney, M. Gupta, New approach to realize fractional power in z-domain at low frequency. IEEE Trans. Circuits syst. II Express. Briefs 58(3), 179–183 (2011)

    Article  Google Scholar 

  33. R. Yadav, M. Gupta, Design of fractional order differentiators and integrators using indirect discretization scheme IICPE, New Delhi, India, January. in Proceedings of IEEE Indian International Conference on Power Electronics 2011

Download references

Acknowledgments

The authors would like to thank the anonymous reviewers for their useful comments and suggestions to improve the quality of the paper.

Author information

Authors and Affiliations

  1. LRES Laboratory, Electrical Engineering Department, Skikda University, BP. 26, Route d’El-Hadaîek, 21000, Skikda, Algeria

    Fouzia Leulmi & Youcef Ferdi

Authors
  1. Fouzia Leulmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Youcef Ferdi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fouzia Leulmi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leulmi, F., Ferdi, Y. Improved Digital Rational Approximation of the Operator \(S^{\alpha }\) Using Second-Order s-to-z Transform and Signal Modeling. Circuits Syst Signal Process 34, 1869–1891 (2015). https://doi.org/10.1007/s00034-014-9928-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-014-9928-9

Keywords

Search

Navigation