Kybernetika 52 no. 6, 914-928, 2016

Perfect observers for fractional discrete-time linear systems

Ewa PawluszewiczDOI: 10.14736/kyb-2016-6-0914

Abstract:

A perfect (exact) fractional observer of discrete-time singular linear control system of fractional order is studied. Conditions for its existence are given. The obtained results are applied to the detectability problem of the class of systems under consideration.

Keywords:

perfect observer, $h$-difference fractional operator, linear control system, singular system

Classification:

93C05, 39A70

References:

  1. T. Abdeljawad and D. Baleanu: Fractional differences and integration by parts. J. Comput. Analysis Appl. 13 (2011), 3, 574-582.   CrossRef
  2. F. M. Atici and P. W. Eloe: A transform method in discrete fractional calculus. Int. J. Difference Equations 2 (2007), 165-176.   CrossRef
  3. N. R. O. Bastos, R. A. C. Ferreira and D. F. M. Torres: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin. Dyn. Syst. 29 (2011), 2, 417-437.   DOI:10.3934/dcds.2011.29.417
  4. L. Dai: Observers for discrete singular systems. IEEE Trans. Automat. Control 33 (1988), 2, 187-191.   DOI:10.1109/9.387
  5. M. Darouach and L. Boutat-Baddas: Observers for a class of nonlinear singular systems. IEEE Trans. Automat. Control 53 (2008), 11, 2627-2633.   DOI:10.1109/tac.2008.2007868
  6. M. A. Duarte-Mermoud, M. J. Mira, I. S. Pelissier and J. C. Travieso-Torres: Evaluation of a fractional order PI controller applied to induction moror speed control. In: Proc. 8th IEEE Int. Conf. on Control and Automation, Xiamen 2010, pp. 573-577.   DOI:10.1109/icca.2010.5524496
  7. A. Dzielinski, D. Sierociuk and G. Sarwas: Some applications of fractional order calculus. Bull. Pol. Acad. Sci. Tech. Sci. 58 (2010), 4, 583-59.   DOI:10.2478/v10175-010-0059-6
  8. R. A. C. Ferreira and D. F. M. Torres: Fractional h-difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 5 (2011), 1, 110-121.   DOI:10.2298/aadm110131002f
  9. M. Fiacchini and G. Millerioux: Deat-beat functional observers for discrete-time LVP systems with unknown inputs. IEEE Trans. Automat. Control 58 (2013), 12, 3230-3235.   DOI:10.1109/tac.2013.2261712
  10. E. Girejko, D. Mozyrska and M. Wyrwas: Advances in the theory and applications of non-integer order systems. In: Comparison of $h$-difference fractional operators (W. Mitkowski, J. Kacprzyk, and J. Baranowski, eds.), Springer 257 (2013), pp. 191-197.   DOI:10.1007/978-3-319-00933-9_17
  11. A. Isidori: Nonlinear Control Theory. Springer, 1991.   CrossRef
  12. T. Kaczorek: Full-order perfect observers for continuous-time linear systems. Pomiary, Automatyka, Kontrola 1 (2001), 3-6.   CrossRef
  13. T. Kaczorek: Advances in Modelling and control of non-integer-order systems. In: Perfect Observers of Fractional Descriptor Continuous-Time Linear System (K. J. Latawiec, M. Lukaniszyn and R. Stanislawski, eds.), Lecture Notes in Electrical Engineering, Springer International Publishing 320 (2015), pp. 3-12.   DOI:10.1007/978-3-319-09900-2_1
  14. K. S. Miller, B and Ross: Fractional difference calculus. In: Proc. Int. Symp. on Univalent Functions, Fractional Calculus and their Applications, Nihon University, K\=oriyama 1988, pp. 139-152.   CrossRef
  15. D. Mozyrska and E. Girejko: Advances in Harmonic Analysis and Operator Theory: The Stefan Samko Anniversary. In: Overview of the fractional h-difference operators, Springer 229 (2013), pp. 253-267.   DOI:10.1007/978-3-0348-0516-2_14
  16. D. Mozyrska, E. Pawluszewicz and M. Wyrwas: Local observability and controllability of nonlinear discrete-time fractional order systems based on their linearization. Int. J. Syst. Sci. 48 (2017), 4, 788-794.   CrossRef
  17. D. Mozyrska and M. Wyrwas: The $\mathcal{Z}$-transform method and delta-type fractional difference operators. Discrete Dynamics in Nature and Society 2015, pp. 47-58.   DOI:10.1007/978-3-319-09900-2_5
  18. D. Mozyrska, M. Wyrwas and E. Pawluszewicz: Stabilization of linear multi-parameter fractional difference control systems. In: Proc. 20th Int. Conf. on Methods and Models in Automation and Robotics MMAR'2015, Miedzyzdroje 2915, pp. 315-319.   DOI:10.1109/mmar.2015.7283894
  19. I. N'Doye, M. Darouach, M. Zasadzinski and N.-E. Radhy: Observers design for singular fractional-order system. In: Proc. 50th Int. Conf. on Decision and Control and European Control Conference CDC-ECC'2011, Orlando 2011, pp. 4017-4022.   DOI:10.1109/cdc.2011.6161336
  20. M. Slawinski and T. Kaczorek: Perfect observers for continuous time linear systems. Pomiary, Automatyka, Kontrola 1 (2004), 39-44.   CrossRef
  21. E. D. Sontag: Mathematical Control Theory. Springer 1998.   DOI:10.1007/978-1-4612-0577-7
  22. J. C. Trigeassou, T. Poinot, J. Lin, A. Oustaloup and F. Levron: Modelling and identification of a non integer order system. In: Proc. European Control Conference ECC'1999, Karlsruhe 1999, pp. 2453-2458.   CrossRef
  23. W. A. Wolowich: Linear Multivariable Systems. Springer-Verlag, 1974.   DOI:10.1007/978-1-4612-6392-0
  24. M. Wyrwas, E. Pawluszewicz and E. Girejko: Stability of nonlinear $h$- difference systems with $n$ fractional orders. Kybernetika 51 (2015), 1, 112-136.   DOI:10.14736/kyb-2015-1-0112