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Impulse response of a generalized fractional second order filter

Zhuang Jiao
  • Department of Automation, Tsinghua University, Beijing, 100084, P.R. China
  • Center for Self-Organizing and Intelligent Systems (CSOIS), Electrical and Computer Engineering Department, Utah State University, Logan, UT, 84322, USA
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/ YangQuan Chen
  • Center for Self-Organizing and Intelligent Systems (CSOIS), Electrical and Computer Engineering Department, Utah State University, Logan, UT, 84322, USA
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Published Online: 2011-12-29 | DOI: https://doi.org/10.2478/s13540-012-0007-2


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.

MSC: Primary 26A33; Secondary 33E12, 34A08, 34K37, 35R11, 60G22

Keywords: generalized fractional second order filter; fractional-order signal processing; impulse response; critically stable

  • [1] K. Oldham, J. Spanier, The Fractional Calculus. Academic Press, New York, 1974. Google Scholar

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

  • [3] I. Podlubny, Fractional Differential Equations. Academic Press, New York, 1999. Google Scholar

  • [4] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science Inc., New York, 2006. 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. http://dx.doi.org/10.1115/1.3167615CrossrefGoogle Scholar

  • [6] B. Mandelbrot, The Fractal Geometry of Nature. W.H. Freeman and Co., San Francisco, 1982. 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/ http://dx.doi.org/10.2478/s13540-011-0031-7CrossrefGoogle 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. http://dx.doi.org/10.1122/1.549724CrossrefGoogle Scholar

  • [9] I. Podlubny, Fractional-order systems and PI λD µ controllers. IEEE Trans. on Automatic Control 44, No 1 (1999), 208–214. http://dx.doi.org/10.1109/9.739144CrossrefGoogle Scholar

  • [10] N. Laskin, Fractional Schrodinger equation. Physical Review E, 66, No 5 (2002), 7 p. Google Scholar

  • [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/ http://dx.doi.org/10.2478/s13540-011-0027-3CrossrefGoogle Scholar

  • [12] J. Sabatier, O. Agrawal, J. Tenreiro Machado, Advances in Fractional Calculus-Theoretical Developments and Applications in Physics and Engineering. Springer, Netherlands, 2007. 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. http://dx.doi.org/10.1007/s11433-006-0257-2CrossrefGoogle 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. http://dx.doi.org/10.1023/A:1016591006562CrossrefGoogle 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. http://dx.doi.org/10.1016/0956-5663(95)96795-ZCrossrefGoogle Scholar

  • [17] L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional operators. Mechanical Systems and Signal Processing 5, No 2 (1991), 81–88. http://dx.doi.org/10.1016/0888-3270(91)90016-XCrossrefGoogle Scholar

  • [18] N. Makris, Fractional-derivative Maxwell model for viscous dampers. J. of Structural Engineering 117, No 9 (1991), 2708–2724. http://dx.doi.org/10.1061/(ASCE)0733-9445(1991)117:9(2708)CrossrefGoogle 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. http://dx.doi.org/10.2514/3.20641CrossrefGoogle 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. http://dx.doi.org/10.1051/jphyslet:019840045019091300CrossrefGoogle 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/ http://dx.doi.org/10.2478/s13540-011-0037-1CrossrefGoogle Scholar

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

  • [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. http://dx.doi.org/10.1109/81.989172CrossrefGoogle Scholar

  • [25] Y. Q. Chen, B. Vinagre, A new IIR-type digital fractional-order differentiator. Signal Processing 83, No 11 (2003), 2359–2365. http://dx.doi.org/10.1016/S0165-1684(03)00188-9CrossrefGoogle Scholar

  • [26] C. Lubich, Discretized fractional calculus. SIAM J. on Mathematical Analysis 17, No 3 (1986), 704–719. http://dx.doi.org/10.1137/0517050CrossrefGoogle 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/ http://dx.doi.org/10.2478/s13540-011-0029-1CrossrefGoogle 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. http://dx.doi.org/10.1007/s11071-004-3752-xCrossrefGoogle 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. http://dx.doi.org/10.1016/j.sigpro.2010.01.017Web of ScienceCrossrefGoogle 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. http://dx.doi.org/10.1016/j.jfranklin.2010.11.009Web of ScienceCrossrefGoogle Scholar

  • [31] B. Davies, Integral Transforms and Their Applications, 3rd Ed., Springer, New York, 2002. 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. http://dx.doi.org/10.1080/10652460310001600672CrossrefGoogle Scholar

  • [33] R. Saxena, A. Mathai, H. Haubold, On generalized fractional kinetic equations. Physica A: Stat. Mechanics and its Applications 344 (2004), 657–664. http://dx.doi.org/10.1016/j.physa.2004.06.048CrossrefWeb of ScienceGoogle 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. http://dx.doi.org/10.1007/978-1-84996-335-0CrossrefGoogle Scholar

About the article

Published Online: 2011-12-29

Published in Print: 2012-03-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 15, Issue 1, Pages 97–116, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-012-0007-2.

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