ArticleOriginal scientific text

Title

Reduced-order fractional descriptor observers for a class of fractional descriptor continuous-time nonlinear systems

Authors 1

Affiliations

  1. Faculty of Electrical Engineering, Białystok University of Technology, ul. Wiejska 45D, 15-351 Białystok, Poland

Abstract

Fractional descriptor reduced-order nonlinear observers for a class of fractional descriptor continuous-time nonlinear systems are proposed. Sufficient conditions for the existence of the observers are established. The design procedure for the observers is given and demonstrated on a numerical example.

Keywords

fractional system, descriptor system, nonlinear system, reduced order observer

Bibliography

  1. Cuihong, W. (2012). New delay-dependent stability criteria for descriptor systems with interval time delay, Asian Journal of Control 14(1): 197–206.
  2. Dodig, M. and Stosic, M. (2009). Singular systems state feedbacks problems, Linear Algebra and Its Applications 431(8): 1267–1292.
  3. Dai, L. (1989). Singular Control Systems, Lecture Notes in Control and Information Sciences, Vol. 118, Springer-Verlag, Berlin.
  4. Fahmy, M.M. and O’Reill, J. (1989). Matrix pencil of closed-loop descriptor systems: Infinite-eigenvalue assignment, International Journal of Control 49(4): 1421–1431.
  5. Gantmacher, F.R. (1960). The Theory of Matrices, Chelsea Publishing Co., New York, NY.
  6. Guang-ren, D. (2010). Analysis and Design of Descriptor Linear Systems, Springer, New York, NY.
  7. Kaczorek, T. (1992). Linear Control Systems, Vol. 1, Research Studies Press, J. Wiley, New York, NY.
  8. Kaczorek, T. (2001). Full-order perfect observers for continuous-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 49(4).
  9. Kaczorek, T. (2004). Infinite eigenvalue assignment by an output feedback for singular systems, International Journal of Applied Mathematics and Computer Science 14(1): 19–23.
  10. Kaczorek, T. (2008). Fractional positive continuous-time linear systems and their reachability, International Journal of Applied Mathematics and Computer Science 18(2): 223–228, DOI: 10.2478/v10006-008-0020-0.
  11. Kaczorek, T. (2011a). Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions on Circuits and Systems 58(7): 1203–1210.
  12. Kaczorek, T. (2011b). Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin.
  13. Kaczorek, T. (2012a). Checking of the positivity of descriptor linear systems with singular pencils, Archive of Control Sciences 22(1): 77–86.
  14. Kaczorek, T. (2012b). Positive fractional continuous-time linear systems with singular pencils, Bulletin of the Polish Academy of Sciences: Technical Sciences 60(1): 9–12.
  15. Kaczorek, T. (2013). Descriptor fractional linear systems with regular pencils, Asian Journal of Control 15(4): 1051–1064.
  16. Kaczorek, T. (2014a). Fractional descriptor observers for fractional descriptor continuous-time linear system, Archives of Control Sciences 24(1): 5–15.
  17. Kaczorek, T. (2014b). Reduced-order fractional descriptor observers for fractional descriptor continuous-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 62(4): 889–895.
  18. Kaczorek, T. (2015). Prefect observers of fractional descriptor continuous-time linear systems, in K.J. Latawiec et al. (Eds.), Advances in Modeling and Control of Non-integer orders Systems, Lecture Notes in Electrical Engineering, Vol. 320, Springer, Berlin/Heidelberg, pp. 5–12.
  19. Kociszewski, R. (2013). Observer synthesis for linear discrete-time systems with different fractional orders, Pomiary Automatyka Robotyka (2): 376–381, (on CD-ROM).
  20. Kucera, V. and Zagalak, P. (1988). Fundamental theorem of state feedback for singular systems, Automatica 24(5): 653–658.
  21. Lewis, F.L. (1983). Descriptor systems, expanded descriptor equation and Markov parameters, IEEE Transactions on Automatic Control AC-28(5): 623–627.
  22. Luenberger, D.G. (1977). Dynamical equations in descriptor form, IEEE Transactions on Automatic Control AC-22(3): 312–321.
  23. Luenberger, D.G. (1978). Time-invariant descriptor systems, Automatica 14(5): 473–480.
  24. Matignon, D. (1996). Stability result on fractional differential equations with applications to control processing, IMACSSMC Proceedings, Lille, France, pp. 963–968.
  25. N’Doye I., Darouach M., Voos H. and Zasadzinski M. (2013). Design of unknown input fractional-order observers for fractional-order systems, International Journal of Applied Mathematics and Computer Science 23(3): 491–500, DOI: 10.2478/amcs-2013-0037.
  26. Oldham, K.B. and Spanier, J. (1974). The Fractional Calculus, Academic Press, New York, NY.
  27. Ostalczyk, P. (2008). Epitome of the Fractional Calculus: Theory and Its Applications in Automatics, Technical University of Łódź Press, Łódź, (in Polish).
  28. Podlubny, I. (1999). Fractional Differential Equations, Academic Press, New York, NY.
  29. Van Dooren, P. (1979). The computation of Kronecker’s canonical form of a singular pencil, Linear Algebra and Its Applications 27: 103–140.
  30. Vinagre, B.M., Monje, C.A. and Calderon, A.J. (2002). Fractional order systems and fractional order control actions, Lecture 3, IEEE CDC’02, Las Vegas, NV, USA.
  31. Virnik, E. (2008). Stability analysis of positive descriptor systems, Linear Algebra and Its Applications 429: 2640–2659.

Additional information

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.

Pages:
277-283
Main language of publication
English
Published
2016
Exact and natural sciences