Table of Contents Author Guidelines Submit a Manuscript
Complexity
Volume 2017, Article ID 3452409, 11 pages
https://doi.org/10.1155/2017/3452409
Research Article

Generalized Fractional-Order Discrete-Time Integrator

1Faculty of Computer Science, Bialystok University of Technology, Białystok, Poland
2Institute of Applied Computer Science, Lodz University of Technology, Łódź, Poland

Correspondence should be addressed to Dorota Mozyrska; lp.ude.bp@aksryzom.d

Received 10 February 2017; Accepted 23 April 2017; Published 6 July 2017

Academic Editor: Ahmad T. Azar

Copyright © 2017 Dorota Mozyrska and Piotr Ostalczyk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. Kaczorek, Linear Control Systems: Analysis of Multivariable Systems, John Wiley & Sons, Inc, New York, NY, USA, 1992.
  2. T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, USA, 1980. View at MathSciNet
  3. R. Abu-Saris and Q. Al-Mdallal, “On the asymptotic stability of linear system of fractional-order difference equations,” Fractional Calculus and Applied Analysis. An International Journal for Theory and Applications, vol. 16, no. 3, pp. 613–629, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, Singapore, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  5. D. Baleanu, J. A. T. Machado, and A. C. J. Luo, Fractional Dynamics and Control, Springer-Verlag, New York, NY, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  6. R. Caponetto, G. Dongola, G. Fortuna, and I. Petras, Fractional Order Systems: Modeling and Control Applications, World Scientific, Singapore, 2010.
  7. S. Das, Functional Fractional Calculus for System Identification and Controls, Springer-Verlag, Berlin-Heidelberg, Germany, 2009.
  8. R. A. Ferreira and D. F. Torres, “Fractional h-difference equations arising from the calculus of variations,” Applicable Analysis and Discrete Mathematics, vol. 5, no. 1, pp. 110–121, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  9. D. Mozyrska and M. g. Wyrwas, “The Z-transform method and delta type fractional difference operators,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 852734, 12 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  10. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
  11. A. Halanay and J. Samuel, Differential Equations, Discrete Systems and Control, Kluwer Academic Publishers, Dordrecht, Netherlands, 1997.
  12. E. C. Ifeachor and B. W. Jervis, Digital Signal Processing, Addison-Wesley, Edinburgh Gate, UK, 1993.
  13. F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165–176, 2007. View at Google Scholar · View at MathSciNet
  14. M. A. Al-Alaoui, “Novel digital integrator and differentiator,” Electronics Letters, vol. 29, no. 4, pp. 376–378, 1993. View at Publisher · View at Google Scholar · View at Scopus
  15. N. R. O. Bastos, R. A. C. Ferreira, and D. F. M. Torres, “Discrete-time fractional variational problems,” Signal Processing, vol. 91, no. 3, pp. 513–524, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. D. Mozyrska, “Multiparameter fractional difference linear control systems,” Discrete Dynamics in Nature and Society, vol. 2014, Article ID 183782, 8 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  17. M. D. Ortigueira, F. J. V. Coito, and J. J. Trujillo, “Discrete-time differential systems,” Signal Processing, vol. 107, pp. 198–217, 2015. View at Publisher · View at Google Scholar · View at Scopus
  18. P. Ostalczyk, Discrete Fractional Calculus: Applications in Control and Image Processing, vol. 14 of Series in Computer Vision, World Scientific Publishing Co Pte Ltd, Singapore, 2016.
  19. R. Stanisławski and K. J. Latawiec, “Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: New necessary and sufficient conditions for the asymptotic stability,” Bulletin of the Polish Academy of Sciences: Technical Sciences, vol. 61, no. 2, pp. 353–361, 2013. View at Publisher · View at Google Scholar · View at Scopus
  20. R. Stanisławski and K. J. Latawiec, “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, vol. 61, no. 2, pp. 363–370, 2013. View at Google Scholar · View at Scopus
  21. J. Cervera, A. Baños, C. A. Monje, and B. M. Vinagre, “Tuning of fractional PID controllers by using QFT,” in Proceeding of the 32nd Annual Conference on IEEE Industrial Electronics (IECON '06), pp. 5402–5407, Paris, France, November 2006. View at Publisher · View at Google Scholar · View at Scopus
  22. P. Ostalczyk and D. Mozyrska, “The second form of the variable-, fractional-order discrete-time integrator,” in Proceedings of the 21st International Conference on Methods and Models in Automation and Robotics, MMAR 2016, pp. 859–864, pol, September 2016. View at Publisher · View at Google Scholar · View at Scopus
  23. D. Sierociuk, W. Malesza, and M. Macias, “On a new symmetric fractional variable order derivative,” in Theoretical developments and applications of non-integer order systems, S. Domek and P. Dworak, Eds., vol. 357, pp. 29–39, Springer, Cham, Heidelberg, Germany, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  24. D. Mozyrska and P. Ostalczyk, “Variable-fractional-order Grünwald-Letnikov backward difference selected properties,” in Proceedings of the 39th International Conference on Telecommunications and Signal Processing, TSP 2016, June 2016. View at Publisher · View at Google Scholar · View at Scopus
  25. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, Netherlands, 2006. View at MathSciNet
  26. A. Oustaloup, La dérivation non entière: théorie, synthèse et applications, Hermes, Paris, France, 1995.