Observability of linear discrete-time systems with different fractional orders

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wyślij Rafał Kociszewski Faculty of Electrical Engineering, Białystok University of Technology, Poland

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Abstract

In the paper the observability problem for the linear discrete-time positive systems with different fractional orders is presented. Necessary and sufficient conditions for observability of this class of dynamical systems are given. A method for computing the initial state is proposed. Considerations are illustrated by theoretical example. Numerical calculations have been performed in the MATLAB program environment.

Keywords

discrete-time, fractional order, observability, positive, system

Obserwowalność liniowych układów dyskretnych różnych niecałkowitych rzędów

Streszczenie

W pracy rozpatrzono problem obserwowalności układów dyskretnych dodatnich przy różnych niecałkowitych rzędach w równaniu stanu. Podano warunki konieczne i wystarczające obserwowalności rozpatrywanej klasy układów dynamicznych. Zaproponowano prostą metodę wyznaczania nieujemnego stanu początkowego takiego układu. Rozważania zilustrowano przykładem teoretycznym, zaś niezbędne obliczenia wykonano w środowisku programowym MATLAB.

Słowa kluczowe

dodatni, obserwowalność, rząd niecałkowity, układ dyskretny

Bibliografia

  1. Kalman R.E., On the general theory of control system., Proc. Of the 1st IFAC Congr., Butterworth, London 1960.
  2. Kaczorek T., Control theory and systems, PWN, Warsaw 1996.
  3. Kaczorek T., Positive 1D and 2D systems, Springerverlag, London 2002.
  4. Debnath L., Recent applications of fractional calculus to science and engineering, “Int. Journal of Mathematics and Mathematical Sciences”, Vol. 54, 2003, 3413-3442 [on-line: www.ijmms.hindawi.com].
  5. Dzieliński A., Sierociuk D., Sarwas G., Some applications of fractional order calculus, “Bull. of the Polish Acad. Of Sciences, Technical Sciences”, Vol. 58, No. 4, 2010, 583-592.
  6. Kilbas A.A., Srivastava H. M., Trujillo J. J., Theory and applications of fractional differential equations. Elsevier, Amsterdam 2006.
  7. Lino P., Maione G., Loop-shaping annd easy tuning of fractional-order proportional integral controllers for position servo systems, “Asian Journal of Control”, 2012.
  8. Podlubny I., Fractional differential equations, Acad. Press, San Diego 1999.
  9. Sabatier J., Agraval O. P., Machado, Advances in fractional calculus. Theoretical developments and applications in physics and engineering, Springer, London 2007.
  10. Sierociuk D., Estimation and control of discrete-time dybnamical fractional systems described in state space, Ph.D. thesis. Warsaw University of Technology, Warsaw 2007.
  11. Kaczorek T., Selected problems of fractional systems theory, Springer, Berlin 2011.
  12. Kaczorek T., Vectors and matrices in automatics and electrotechnics, WNT, Warsaw 1998.
  13. Bettayeb M., Djennoune S., A note on the controllability and the observability of fractional dynamical systems, [in:] Proc. of the 2nd IFAC Workshop on Fractional Differentation and its Applications, Porto, Portugal 2006, 506-511.
  14. Mantignon D., d’Andrea-Novel B., Some results on controllability and observability of finite-dimensional fractional differential systems, [in:] Proc. of the IMACS. IEEE SMC Conf., France 1996, 952-956.
  15. Mozyrska D., Pawłuszewicz E., Observability of linear q-difference fractional order systems with finite initial memory, “Bull. of the Polish Acad. of Sci.”, Vol. 58, No. 4, 2010, 601-605.
  16. Kociszewski R., Controllability and observability of linear time-invariant positive discrete-time systems with delays, Ph.D. thesis, Białystok University of Technology, Białystok 2008.
  17. Busłowicz M., Simple analytic conditions for stability of fractional discrete-time linear systems with diagonal state matrix, “Bull. of the Polish Acad. of Sciences, Technical Sciences” (in press).