Advertisement

Modeling and Identification of Fractional-Order Discrete-Time Laguerre-Based Feedback-Nonlinear Systems

  • Rafał Stanisławski
  • Krzysztof J. Latawiec
  • Marcin Gałek
  • Marian Łukaniszyn
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 320)

Abstract

This paper presents a new implementable strategy for modeling and identification of a fractional-order discrete-time block-oriented feedback-nonlinear system. Two different concepts of orthonormal basis functions (OBF) are used to model a linear dynamic part, namely ”regular” OBF and inverse IOBF. It is shown that the IOBF concept enables to separate linear and nonlinear submodels, which leads to a linear regression formulation of the parameter estimation problem, with the detrimental bilinearity effect totally eliminated. Finally, Laguerre filters are uniquely embedded in modeling of the fractional-order dynamics. Unlike for regular OBF, simulation experiments show a very good identification performance for an IOBF-structured, fractional-order Laguerre-based feedback-nonlinear model, both in terms of low prediction errors and accurate reconstruction of the actual system characteristics.

Keywords

Recursive Little Square Recursive Little Square Algorithm Hammerstein System Orthonormal Basis Function Linear Dynamic Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Astrom, K.J., Bell, R.D.: A nonlinear model for steam generation process. In: Preprints IFAC 12th World Congress, Sydney, Australia (1993)Google Scholar
  2. 2.
    Hasiewicz, Z.: Hammerstein system identification by the haar multiresolution approximation. Int J. Adaptive Control and Signal Processing 74, 191–217 (1999)MathSciNetGoogle Scholar
  3. 3.
    Greblicki, W.: Stochastic approximation in nonparametric identification of Hammerstein system. IEEE Transactions on Automatic Control 47, 1800–1810 (2002)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Ninness, B., Gibson, S., Weller, S.: Practical aspects of using orthonormal system parameterisations in estimation problems. In: Proc. 12th IFAC Symp. on System Identification (SYSID 2000), Santa Barbara, CA, pp. 463–468 (2000)Google Scholar
  5. 5.
    Hanczewski, S., Stasiak, M., Weissenberg, J.: The queueing model of a multiservice system with dynamic resource sharing for each class of calls. In: Kwiecień, A., Gaj, P., Stera, P. (eds.) CN 2013. CCIS, vol. 370, pp. 436–445. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Latawiec, K., Marciak, C., Hunek, W., Stanisławski, R.: A new analytical design methodology for adaptive control of nonlinear block-oriented systems. In: Proceedings the 7th World Multi-Conference on Systemics, Cybernetics and Informatics, Orlando, Florida, vol. 11, pp. 215–220 (2003)Google Scholar
  7. 7.
    Latawiec, K.J., Marciak, C., Stanislawski, R., Oliveira, G.H.C.: The mode separability principle in modeling of linear and nonlinear block-oriented systems. In: Proc. 10th IEEE MMAR Conference (MMAR 2004), Miedzyzdroje, Poland, pp. 479–484 (2004)Google Scholar
  8. 8.
    Heuberger, P.S.C., Van den Hof, P.M.J., Wahlberg, B.: Modeling and identification with rational orthogonal basis functions. Springer, London (2005)CrossRefGoogle Scholar
  9. 9.
    Stanisławski, R.: Hammerstein system identification by means of orthonormal basis functions and radial basis functions. In: Pennacchio, S. (ed.) Emerging Technologies, Robotics and Control Systems, Internationalsar, Italy, pp. 69–73 (2007)Google Scholar
  10. 10.
    Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  11. 11.
    Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems and Applications Multiconference, Lille, France, vol. 2, pp. 963–968 (1996)Google Scholar
  12. 12.
    Miller, K., Ross, B.: An Introduction to the fractional calculus and fractional differential equations. Willey, New York (1993)zbMATHGoogle Scholar
  13. 13.
    Monje, C., Chen, Y., Vinagre, B., Xue, D., Feliu, V.: Fractional-order Systems and Controls. Springer, London (2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    Ostalczyk, P.: Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains. International Journal of Applied Mathematics and Computer Science 22(3), 533–538 (2012)MathSciNetGoogle Scholar
  15. 15.
    Oldham, K., Spanier, J.: The fractional calculus. Academic Press, Orlando (1974)zbMATHGoogle Scholar
  16. 16.
    Podlubny, I.: Fractional differential equations. Academic Press, Orlando (1999)zbMATHGoogle Scholar
  17. 17.
    Stanisławski, R., Latawiec, K.J.: Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: New necessary and sufficient conditions for asymptotic stability. Bulletin of the Polish Academy of Sciences, Technical Sciences 61(2), 353–361 (2013)Google Scholar
  18. 18.
    Stanisławski, R., Latawiec, K.J.: Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: New stability criterion for FD-based systems. Bulletin of the Polish Academy of Sciences, Technical Sciences 61(2), 362–370 (2013)Google Scholar
  19. 19.
    Stanisławski, R.: New Laguerre filter approximators to the Grünwald-Letnikov fractional difference. Mathematical Problems in Engineering 2012 (2012), Paper ID: 732917Google Scholar
  20. 20.
    Stanisławski, R.: Advances in Modeling of Fractional Difference Systems - New Accuracy, Stability and Computational Results. Opole University of Technology Press, Opole (2013)Google Scholar
  21. 21.
    Stanisławski, R., Hunek, W.P., Latawiec, K.J.: Normalized finite fractional discrete-time derivative a new concept and its application to obf modeling. Measurements, Automation and Monitoring 57(3), 241–243 (2011)Google Scholar
  22. 22.
    Stanisławski, R., Latawiec, K.J.: Modeling of open-loop stable linear systems using a combination of a finite fractional derivative and orthonormal basis functions. In: Proceedings of the 15th International Conference on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, pp. 411–414 (2010)Google Scholar
  23. 23.
    Latawiec, K.J., Stanisławski, R., Hunek, W.P., Łukaniszyn, M.: Laguerre-based modeling of fractional-order LTI SISO systems. In: Proceedings of the 18th International Conference on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, pp. 64–69 (2013)Google Scholar
  24. 24.
    Sierociuk, D., Dzieliński, A.: Fractional Kalman filter algorithm for states, parameters and order of fractional system estimation. International Journal of Applied Matchematics and Computer Science 16(1), 101–112 (2006)Google Scholar
  25. 25.
    Stanisławski, R.: Identification of open-loop stable linear systems using fractional orthonormal basis functions. In: Proceedings of the 14th International Conference on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, pp. 935–985 (2009)Google Scholar
  26. 26.
    Ljung, L.: System Identification. Prentice-Hall, Englewood Cliffs (1987)zbMATHGoogle Scholar
  27. 27.
    Boukis, C., Mandic, D.P., Constantinides, A.G., Polymenakos, L.C.: A novel algorithm for the adaptation of the pole of Laguerre filters. IEEE Signal Processing Letters 13(7), 429–432 (2006)CrossRefGoogle Scholar
  28. 28.
    Oliveira, S.T.: Optimal pole conditions for Laguerre and two-parameter Kautz models: A survey of known results. In: 12th IFAC Symposium on System Identification (SYSID 2000), Santa Barbara, CA, pp. 457–462 (2000)Google Scholar
  29. 29.
    Stanisławski, R., Hunek, W.P., Latawiec, K.J.: Modeling of non-linear block-oriented systems using orthonormal basis and radial basis functions. Systems Science 35(2), 11–18 (2009)MathSciNetGoogle Scholar
  30. 30.
    Stanisławski, R., Latawiec, K.J.: Regular vs. inverse orthonormal basis functions concepts in identification of feedback-nonlinear systens by means of radial basis functions. International Journal of Factory Automation, Robotics and Soft Computing 2007(4), 64–68 (2007)Google Scholar
  31. 31.
    Latawiec, K.J.: The Power of Inverse Systems in Linear and Nonlinear Modeling and Control. Opole University of Technology Press, Opole (2004)Google Scholar
  32. 32.
    Stanisławski, R., Latawiec, K.J.: Normalized finite fractional differences: the computational and accuracy breakthroughs. International Journal of Applied Mathematics and Computer Science 22(4), 907–919 (2012)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Rafał Stanisławski
    • 1
  • Krzysztof J. Latawiec
    • 1
  • Marcin Gałek
    • 1
  • Marian Łukaniszyn
    • 1
  1. 1.Department of Electrical, Control and Computer EngineeringOpole University of TechnologyOpolePoland

Personalised recommendations