Skip to main content
SpringerLink
Log in

Impulse response of a generalized fractional second order filter

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

The impulse response of a generalized fractional second order filter of the form (s 2α + as α + b)−γ is derived, where 0 < α ≤ 1, 0 < γ < 2. The asymptotic properties of the impulse responses are obtained for two cases, and within these two cases, the properties are shown when changing the value of γ. It is shown that only when (s 2α + as α + b)−1 has the critical stability property, the generalized fractional second order filter (s 2α + as α + b)−γ has different properties as we change the value of γ. Finally, numerical examples to illustrate the impulse response are provided to verify the obtained results.

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

Similar content being viewed by others

References

  1. K. Oldham, J. Spanier, The Fractional Calculus. Academic Press, New York, 1974.

    MATH  Google Scholar 

  2. K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.

    MATH  Google Scholar 

  3. I. Podlubny, Fractional Differential Equations. Academic Press, New York, 1999.

    MATH  Google Scholar 

  4. A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science Inc., New York, 2006.

    MATH  Google Scholar 

  5. P. Torvik, R. Bagley, On the appearance of the fractional derivative in the behavior of real materials. J. of Applied Mechanics ASME 51, No 22 (1984), 294–298.

    Article  MATH  Google Scholar 

  6. B. Mandelbrot, The Fractal Geometry of Nature. W.H. Freeman and Co., San Francisco, 1982.

    MATH  Google Scholar 

  7. P. Lanusse, J. Sabatier, PLC implementation of a CRONE controller. Fract. Calc. Appl. Anal. 14, No 4 (2011), 505–522; DOI: 10.2478/s13540-011-0031-7, http://www.springerlink.com/content/1311-0454/14/4/

    Article  Google Scholar 

  8. R. Bagley, P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology 27, No 3 (1983), 201–210.

    Article  MATH  Google Scholar 

  9. I. Podlubny, Fractional-order systems and PI λ D µ controllers. IEEE Trans. on Automatic Control 44, No 1 (1999), 208–214.

    Article  MathSciNet  MATH  Google Scholar 

  10. N. Laskin, Fractional Schrodinger equation. Physical Review E, 66, No 5 (2002), 7 p.

  11. Z. Jiao and YangQuan Chen, Stability analysis of fractional-order systems with double noncommensurate orders for matrix case. Fract. Calc. Appl. Anal. 14, No 3 (2011), 436–453; DOI: 10.2478/s13540-011-0027-3, http://www.springerlink.com/content/1311-0454/14/3/

    Article  Google Scholar 

  12. J. Sabatier, O. Agrawal, J. Tenreiro Machado, Advances in Fractional Calculus-Theoretical Developments and Applications in Physics and Engineering. Springer, Netherlands, 2007.

    MATH  Google Scholar 

  13. M. Xu, W. Tan, Intermediate processes and critical phenomena: theory, method and progress of fractional operators and their applications to modern mechanics. Science in China: Series G, Physics, Mechanics and Astronomy 49, No 3 (2006), 257–272.

    Article  MATH  Google Scholar 

  14. Y. Chen, K. Moore, Analytical stability bound for a class of delayed fractional order dynamic systems. Nonlinear Dynamics 29, No 1–4 (2002), 191–200.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Ichise, Y. Nagayanagi, T. Kojima, An analog simulation of noninteger order transfer functions for analysis of electrode. J. of Electro Analytical Chemistry 33, No 2 (1971), 253–265.

    Google Scholar 

  16. E. McAdams, A. Lackermeier, J. McLaughlin, D. Macken, J. Jossinet, The linear and non-linear electrical properties of the electrode-electrolyte interface. Biosensors and Bioelectronics 10, No 1 (1995), 67–74.

    Article  Google Scholar 

  17. L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional operators. Mechanical Systems and Signal Processing 5, No 2 (1991), 81–88.

    Article  Google Scholar 

  18. N. Makris, Fractional-derivative Maxwell model for viscous dampers. J. of Structural Engineering 117, No 9 (1991), 2708–2724.

    Article  Google Scholar 

  19. R. Bagley, R. Calico, Fractional order state equations for the control of viscoelastically damped structures. J. of Guidance, Control and Dynamics 14, No 2 (1991), 304–311.

    Article  Google Scholar 

  20. J. Clerc, A. Tremblay, G. Albinet, C. Mitescu, AC response of fractal networks. J. de Physique Lettres 45, No 19 (1984), 913–924.

    Article  Google Scholar 

  21. J. Tenreiro Machado, And I say to myself: “What a fractional world!”. Fract. Calc. Appl. Anal. 14, No 4 (2011), 635–654; DOI: 10.2478/s13540-011-0037-1, http://www.springerlink.com/content/1311-0454/14/4/

    Article  Google Scholar 

  22. J. Machado, Analysis and design of fractional-order digital control systems. Systems Analysis Modelling Simulation 27, No 2–3 (1997), 107–122.

    MATH  Google Scholar 

  23. B. Vinagre, I. Petras, P. Merchan, L. Dorcak, Two digital realization of fractional controllers: Application to temperature control of a solid. In: Proc. of the European Control Conference (2001), 1764–1767.

  24. Y. Q. Chen, K. L. Moore, Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications 49, No 3 (2002), 363–367.

    Article  MathSciNet  Google Scholar 

  25. Y. Q. Chen, B. Vinagre, A new IIR-type digital fractional-order differentiator. Signal Processing 83, No 11 (2003), 2359–2365.

    Article  MATH  Google Scholar 

  26. C. Lubich, Discretized fractional calculus. SIAM J. on Mathematical Analysis 17, No 3 (1986), 704–719.

    Article  MathSciNet  MATH  Google Scholar 

  27. K. Diethelm, An efficient parallel algorithm for the numerical solution of fractional differential equations. Fract. Calc. Appl. Anal. 14, No 3 (2011), 475–490; DOI:10.2478/s13540-011-0029-1, http://www.springerlink.com/content/1311-0454/14/3/

    Article  Google Scholar 

  28. Y. Q. Chen, B. Vinagre, I. Podlubny, Continued fraction expansion approaches to discretizing fractional order derivatives — An expository review. Nonlinear Dynamics 38, No 16 (2004), 155–170.

    Article  MathSciNet  MATH  Google Scholar 

  29. Y. Li, H. Sheng, Y. Q. Chen, Analytical impulse response of a fractional second order filter and its impulse response invariant discretization. Signal Processing 91, No 3 (2011), 498–507.

    Article  MATH  Google Scholar 

  30. Hu Sheng, Yan Li, YangQuan Chen, Application of numerical inverse Laplace transform algorithms in fractional calculus, Journal of the Franklin Institute 348, No 2 (2011), 315–330.

    Article  MathSciNet  MATH  Google Scholar 

  31. B. Davies, Integral Transforms and Their Applications, 3rd Ed., Springer, New York, 2002.

    MATH  Google Scholar 

  32. A. Kilbas, M. Saigo, R. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators. Integr. Transf. Spec. Funct. 15, No 1 (2004), 1–13.

    Article  MathSciNet  Google Scholar 

  33. R. Saxena, A. Mathai, H. Haubold, On generalized fractional kinetic equations. Physica A: Stat. Mechanics and its Applications 344 (2004), 657–664.

    Article  MathSciNet  Google Scholar 

  34. C. Monje, Y. Q. Chen, B. Vinagre, D. Xue, V. Feliu, Fractional-order Systems and Controls: Fundamentals and Applications. Springer-Verlag, London, 2010.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Automation, Tsinghua University, Beijing, 100084, P.R. China

    Zhuang Jiao

  2. Center for Self-Organizing and Intelligent Systems (CSOIS), Electrical and Computer Engineering Department, Utah State University, Logan, UT, 84322, USA

    Zhuang Jiao & YangQuan Chen

Authors
  1. Zhuang Jiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YangQuan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhuang Jiao.

About this article

Cite this article

Jiao, Z., Chen, Y. Impulse response of a generalized fractional second order filter. fcaa 15, 97–116 (2012). https://doi.org/10.2478/s13540-012-0007-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s13540-012-0007-2

MSC 2010

Key Words and Phrases

Search

Navigation