Approximate Sampled-Data Models for Fractional Order Systems

  • Juan I. Yuz
  • Graham C. Goodwin
Part of the Communications and Control Engineering book series (CCE)


Fractional-order systems have received considerable attention by the engineering community in the last years. This chapter extends the ideas presented in Chaps.  5 and  8 to systems having fractional-order dynamics. Specifically, fractional-order Euler–Frobenius polynomials are defined and used to characterize the asymptotic sampling zeros for fractional systems as the sampling period tends to zero.


Fractional Order Fractional Derivative Fractional Calculus Fractional Order System Fractional Order Integrator 
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Further Reading

Background on fractional calculus can be found in

  1. Kilbas A, Srivastava M, Trujillo J (2006) Theory and application of fractional differential equations. Elsevier, Amsterdam Google Scholar
  2. Miller KS, Ross B (1993) An introduction to the fractional calculus and differential equations. Wiley, New York zbMATHGoogle Scholar
  3. Oldham K, Spanier J (1974) The fractional calculus. Academic Press, New York zbMATHGoogle Scholar
  4. Podlubny I (1999) Fractional differential equations. Academic Press, New York zbMATHGoogle Scholar

Applications of fractional calculus in engineering can be found in

  1. Baleanu D, Guvenc ZB, Tenreiro-Machado JA (eds) (2009) New trends in nanotechnology and fractional calculus applications. Springer, New York Google Scholar
  2. Caponetto R, Dongola G, Fortuna L, Petras I (2010) Fractional order systems: modeling and control applications. World Scientific, Singapore Google Scholar
  3. Monje CA, Chen Y, Vinagre BM, Xue D, Feliu V (2010) Fractional-order systems and controls: fundamentals and applications. Springer, Berlin CrossRefGoogle Scholar
  4. Sabatier J, Agrawal OP, Tenreiro-Machado JA (2007) Advances in fractional calculus: theoretical developments and applications in physics and engineering. Springer, Berlin CrossRefGoogle Scholar

Other approaches for obtaining approximate sampled-data models for fractional order systems can be found in

  1. Aoun M, Malti R, Levron F, Oustaloup A (2004) Numerical simulations of fractional systems: an overview of existing methods and improvements. In: Nonlinear dynamics, pp 117–131 Google Scholar
  2. Chen YQ, Moore KL (2002) Discretization schemes for fractional-order differentiators and integrators. IEEE Trans Circuits Syst I, Fundam Theory Appl 49(3):363–367 MathSciNetCrossRefGoogle Scholar
  3. Chen YQ, Vinagre BM, Podlubny I (2004) Continued fraction expansion approaches to discretizing fractional order derivatives—an expository review. Nonlinear Dyn 38(1):155–170 MathSciNetCrossRefzbMATHGoogle Scholar
  4. Lubich C (1986) Discretized fractional calculus. SIAM J Math Anal 17(3):704–719 MathSciNetCrossRefzbMATHGoogle Scholar
  5. Maione G (2008) Continued fractions approximation of the impulse response of fractional-order dynamic systems. IET Control Theory Appl 2(7):564–572 MathSciNetCrossRefGoogle Scholar
  6. Maione G (2011) High-speed digital realizations of fractional operators in the delta domain. IEEE Trans Autom Control 56(3):697–702 MathSciNetCrossRefGoogle Scholar
  7. Valério D, da Costa JS (2005) Time-domain implementation of fractional order controllers. In: Control theory and applications. IEE proceedings, pp 539–552 Google Scholar
  8. Yucra E, Yuz JI (2013) Sampling zeros of discrete models for fractional order systems. IEEE Trans Autom Control 58(9):2383–2388. doi: 10.1109/TAC.2013.2254000 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Juan I. Yuz
    • 1
  • Graham C. Goodwin
    • 2
  1. 1.Departamento de ElectrónicaUniversidad Técnica Federico Santa MaríaValparaísoChile
  2. 2.School of Electrical Engineering & Computer ScienceUniversity of NewcastleCallaghanAustralia

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