Positive fractional linear systems

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send Tadeusz Kaczorek Białystok University of Technology Faculty of Electrical Engineering

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Abstract

An overview of some recent published and unpublished results on positive fractional continuous-time and discrete-time linear systems is given. The first part of the paper is devoted to the positive continuous-time fractional systems. For those systems the solutions to the fractional state equations are proposed. Necessary and sufficient conditions for the positivity, reachability and stability are established. In the second part similar problems are considered for positive discrete-time fractional systems.

Keywords

discrete-time systems, linear system

Dodatnie układy liniowe niecałkowitego rzędu

Streszczenie

W pracy dokonano syntetycznego przeglądu nowych publikowanych i niepublikowanych wyników dotyczących dodatnich ciągłych i dyskretnych układów liniowych niecałkowitego rzędu. W części pierwszej poświęconej układom ciągłym podano rozwiązanie układu równań stanu, warunki konieczne i wystarczające dodatniości, osiągalności i stabilności układów dodatnich. W części drugiej przedstawiono podobne wyniki dla układów dyskretnych.

Słowa kluczowe

układ dyskretny, układ liniowy

Bibliography

  1. Busłowicz M., Stability of linear continuous time fractional order systems with delay of the retarder type, Bull. Pol. Acad. Sci. Tech., vol. 56, no. 4, 2008, pp. 319-324.
  2. Dzieliński A., Sierociuk D. and Sarwas G., Ultracapacitor parameters identification based on fractional order model, Proc ECC’09, Budapest 2009.
  3. Dzieliński A. and Sierociuk D., Stability of discrete fractional order state-space systems, Journal of Vibrations and Control, vol. 14, no. 9/10, 2008, pp. 1543-1556.
  4. Farina E. and Rinaldi S., Positive Linear Systems. Theory and Applications, J. Wiley New York 200.
  5. Kaczorek T., Positive 1D and 2D Systems, Springer-Verlag, London 2002.
  6. Kaczorek T., Fractional positive continuous-time systems and their Reachability, Int. J. Appl. Math. Comput. Sci., vol. 18, no. 2, 2008, pp. 223-228.
  7. Kaczorek T., Selected Problems in Fractional Systems Theory, Springer-Verlag 2011.
  8. Kaczorek T., Stability of positive continuous-time systems with delays, Bull. Pol. Acad. Sci. Tech., vol. 57, no. 4, 2009, pp. 395-398.
  9. Kaczorek T., Asymptotic stability of positive fractional 2D linear systems, Bull. Pol. Acad. Sci. Tech., vol. 57, no. 3, 2009, pp. 287-292.
  10. Kaczorek T., Practical stability of positive fractional discrete-time linear systems, Bull. Pol. Acad. Sci. Tech., vol. 56, no. 4, 2008, pp. 313-318.
  11. Kaczorek T., Positive linear systems with different fractional orders, Bull. Pol. Acad. Sci. Tech., vol. 58, no. 3, 2010, pp. 453-458.
  12. Kaczorek T., Decomposition of the pairs (A,B) and (A,C) of positive discrete-time linear systems, Archives of Control Sciences, vol. 20, no. 3, 2010, pp. 341-361.
  13. Kaczorek T., Positive linear systems consisting of n subsystems with different fractional orders, IEEE Trana. Circuits and Systems, 2011 (in Press).
  14. Oldham K. B. and Squier J., The Fractional Calculus, Academic Press, New York 1974.
  15. Ostalczyk P., Epitome of the fractional calculus: Theory and its Applications in Automatics, Wydawnictwo Politechniki Łódzkiej, Łódź 2008 (in Polish).
  16. Podlubny I., Fractional Differential Equations, Academic Press, San Diego 1999.
  17. Radwan A. G., Soliman A. M., Elwakil A. S. and Sedeek A., On the stability of linear systems with fractional-order elements, Chaos, Solitons and Fractals, vol. 40, no. 5, 2009, pp. 2317-2328.
  18. Ruszewski A., Stability regions of closed-loop system with time delay inertial plant of fractional order and fractional order PI controller, Bull. Pol. Acad. Sci. Tech., vol. 56, no. 4, 2008, pp. 329-332.
  19. Tenreiro Machado J. A. and Ramiro Barbosa S., Editors of special issue on fractional differentiation and its application, Journal of Vibration and Control, vol. 14, no. 9/10, 2008, pp. 1543-1556.
  20. Vinager B. M., Monje C. A. and Calderon A. J., Fractional order systems and fractional order control actions, 41th IEEE Conf. on Decision and Control, Las Vegas NV, December 2002.