Selected Implementation Issues in Computation of the Grünwald-Letnikov Fractional-Order Difference by Means of Embedded System

  • Kamil KoziołEmail author
  • Rafał Stanisławski
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 559)


This paper presents practical aspects of the implementation of discrete-time fractional-order models in embedded systems, which use single floating-point operations. To improve the numerical performance of the modeling process for fractional-order difference and discrete-time fractional-order systems the ‘error-free transformation’ in the calculation process is proposed. Simulation examples present that the methodology proposed in the paper significantly improves modeling accuracy.


Discrete-time fractional-order system Numerical accuracy 


  1. 1.
    Barbosa, R.S., Machado, J.A.T.: Implementation of discrete-time fractional-order controllers based on LS approximations. Acta Polytechnica Hungarica 3(4), 5–22 (2006)Google Scholar
  2. 2.
    Dekker, T.J.: A floating-point technique for extending the available precision. Numerische Mathematik 18(3), 224–242 (1971). Scholar
  3. 3.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002). Scholar
  4. 4.
    Knuth, D.E.: The Art of Computer Programming: Seminumerical Algorithms, vol. 2, 3rd edn. Addison-Wesley, Reading (1998). Scholar
  5. 5.
    Kootsookos, P.J., Williamson, R.C.: FIR approximation of fractional sample delay systems. IEEE Trans. Circ. Syst. II: Analog Digit. Sig. Process. 43(3), 269–271 (1996). Scholar
  6. 6.
    Latawiec, K.J., Stanisławski, R., Łukaniszyn, M., Czuczwara, W., Rydel, M.: Fractional-order modeling of electric circuits: modem empiricism vs. classical science. In: Progress in Applied Electrical Engineering (PAEE) (2017).
  7. 7.
    Ogita, T., Rump, S.M., Oishi, S.: Accurate sum and dot product. SIAM J. Sci. Comput. (SISC) 26(6), 1955–1988 (2005). Scholar
  8. 8.
    Rump, S.M.: Error-free transformations and ill-conditioned problems. In: International Workshop on Verified Computations and Related Topics, University of Karlsruhe, Germany, 7–10 March 2009Google Scholar
  9. 9.
    Rump, S.M., Ogita, T., Oishi, S.: Accurate floating-point summation part I: faithful rounding. SIAM J. Sci. Comput. (SISC) 31(1), 189–224 (2008). Scholar
  10. 10.
    Rump, S.M., Ogita, T., Oishi, S.: Accurate floating-point summation part II: sign, K-fold faithful and rounding to nearest. SIAM J. Sci. Comput. (SISC) 31(2), 1269–1302 (2008). Scholar
  11. 11.
    Stanisławski, R., Hunek, W.P., Latawiec, K.J.: Finite approximations of a discrete-time fractional derivative. In: Proceedings of the 16th International Conference on Methods and Models in Automation and Robotics, MMAR 2011, Miedzyzdroje, Poland, pp. 142–145, August 2011.
  12. 12.
    Stanisławski, R., Latawiec, K.J., Gałek, M., Łukaniszyn, M.: Modeling and identification of fractional-order discrete-time Laguerre-based feedback-nonlinear systems. Lecture Notes in Electrical Engineering, vol. 320, pp. 101–112 (2015). Scholar
  13. 13.
    Stanisławski, R., Rydel, M., Latawiec, K.J.: Modeling of discrete-time fractional-order state space systems using the balanced truncation method. J. Franklin Inst. 354(7), 3008–3020 (2017). Scholar
  14. 14.
    Vinagre, B.M., Podlubny, I., Hernandez, A., Feliu, V.: Some approximations of fractional order operators used in control theory and applications. Fract. Calc. Appl. Anal. 3(3), 945–950 (2000)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Yamanaka, N., Ogita, T., Rump, S.M., Oishi, S.: A parallel algorithm for accurate dot product. Parallel Comput. 34(6–8), 392–410 (2008). Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Departament of Electrial, Control and Computer EngineeringOpole University of TechnologyOpolePoland

Personalised recommendations