Positivity and stabilization of fractional 2D linear systems described by the Roesser model

Tadeusz Kaczorek; Krzysztof Rogowski

International Journal of Applied Mathematics and Computer Science (2010)

  • Volume: 20, Issue: 1, page 85-92
  • ISSN: 1641-876X

Abstract

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A new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definition is applied to each dimension of a 2D Roesser model. Solutions of these systems are derived using a 2D Z-transform. The classical Cayley-Hamilton theorem is extended to 2D fractional systems described by the Roesser model. Necessary and sufficient conditions for the positivity and stabilization by the state-feedback of fractional 2D linear systems are established. A procedure for the computation of a gain matrix is proposed and illustrated by a numerical example.

How to cite

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Tadeusz Kaczorek, and Krzysztof Rogowski. "Positivity and stabilization of fractional 2D linear systems described by the Roesser model." International Journal of Applied Mathematics and Computer Science 20.1 (2010): 85-92. <http://eudml.org/doc/207980>.

@article{TadeuszKaczorek2010,
abstract = {A new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definition is applied to each dimension of a 2D Roesser model. Solutions of these systems are derived using a 2D Z-transform. The classical Cayley-Hamilton theorem is extended to 2D fractional systems described by the Roesser model. Necessary and sufficient conditions for the positivity and stabilization by the state-feedback of fractional 2D linear systems are established. A procedure for the computation of a gain matrix is proposed and illustrated by a numerical example.},
author = {Tadeusz Kaczorek, Krzysztof Rogowski},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {positivity; stabilization; fractional systems; Roesser model; 2D systems},
language = {eng},
number = {1},
pages = {85-92},
title = {Positivity and stabilization of fractional 2D linear systems described by the Roesser model},
url = {http://eudml.org/doc/207980},
volume = {20},
year = {2010},
}

TY - JOUR
AU - Tadeusz Kaczorek
AU - Krzysztof Rogowski
TI - Positivity and stabilization of fractional 2D linear systems described by the Roesser model
JO - International Journal of Applied Mathematics and Computer Science
PY - 2010
VL - 20
IS - 1
SP - 85
EP - 92
AB - A new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definition is applied to each dimension of a 2D Roesser model. Solutions of these systems are derived using a 2D Z-transform. The classical Cayley-Hamilton theorem is extended to 2D fractional systems described by the Roesser model. Necessary and sufficient conditions for the positivity and stabilization by the state-feedback of fractional 2D linear systems are established. A procedure for the computation of a gain matrix is proposed and illustrated by a numerical example.
LA - eng
KW - positivity; stabilization; fractional systems; Roesser model; 2D systems
UR - http://eudml.org/doc/207980
ER -

References

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  1. Bose, N. K. (1982). Applied Multidimensional Systems Theory, Van Nonstrand Reinhold Co., New York, NY. Zbl0574.93031
  2. Bose, N. K. (1985). Multidimensional Systems Theory Progress, Directions and Open Problems, D. Reidel Publishing Co., Dodrecht. Zbl0562.00017
  3. Busłowicz, M. (2008). Simple stability conditions for linear positive discrete-time systems with delays, Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4): 325-328. 
  4. Busłowicz, M. and Kaczorek, T. (2009). Simple conditions for practical stability of positive fractional discrete-time linear systems, International Journal of Applied Mathematics and Computer Science 19(2): 263-269, DOI: 10.2478/v10006-009-0022-6. Zbl1167.93019
  5. Farina, E. and Rinaldi, S. (2000). Positive Linear Systems; Theory and Applications, J. Wiley, New York, NY. Zbl0988.93002
  6. Fornasini, E. and Marchesini, G. (1976). State-space realization theory of two-dimensional filters, IEEE Transactions on Automatic Control AC-21(4): 484-491. Zbl0332.93072
  7. Fornasini, E. and Marchesini, G. (1978). Double indexed dynamical systems, Mathematical Systems Theory 12(1): 59-72. Zbl0392.93034
  8. Galkowski, K. (2001). State Space Realizations of Linear 2D Systems with Extensions to the General nD (n > 2) Case, Springer-Verlag, London. Zbl1007.93001
  9. Kaczorek, T. (1985). Two-Dimensional Linear Systems, Springer-Verlag, London. Zbl0593.93031
  10. Kaczorek, T. (1996). Reachability and controllability of nonnegative 2D Roesser type models, Bulletin of the Polish Academy of Sciences: Technical Sciences 44(4): 405-410. Zbl0888.93009
  11. Kaczorek, T. (2001). Positive 1D and 2D Systems, Springer-Verlag, London. Zbl1005.68175
  12. Kaczorek, T. (2005). Reachability and minimum energy control of positive 2D systems with delays, Control and Cybernetics 34(2): 411-423. Zbl1167.93359
  13. Kaczorek, T. (2007). Reachability and controllability to zero of positive fractional discrete-time systems, Machine Intelligence and Robotic Control 6(4): 139-143. 
  14. Kaczorek, T. (2008a). Asymptotic stability of positive 1D and 2D linear systems, in K. Malinowski and L. Rutkowski (Eds), Recent Advances in Control and Automation, Akademicka Oficyna Wydawnicza EXIT, Warsaw, pp. 41-52. 
  15. Kaczorek, T. (2008b). Asymptotic stability of positive 2D linear systems, Proceedings of the 13th Scientific Conference on Computer Applications in Electrical Engineering, Poznań, Poland, pp. 1-5. Zbl1154.93017
  16. Kaczorek, T. (2008c). Fractional 2D linear systems, Journal of Automation, Mobile Robotics & Intelligent Systems 2(2): 5-9. Zbl1154.93017
  17. Kaczorek, T. (2008d). Positive different orders fractional 2D linear systems, Acta Mechanica et Automatica 2(2): 51-58. 
  18. Kaczorek, T. (2009a). LMI approach to stability of 2D positive systems, Multidimensional Systems and Signal Processing 20(1): 39-54. Zbl1169.93022
  19. Kaczorek, T. (2009b). Positive 2D fractional linear systems, International Journal for Computation and Mathematics in Electrical and Electronic Engineering, COMPEL 28(2): 341-352. Zbl1173.93017
  20. Kaczorek, T. (2009c). Positivity and stabilization of 2D linear systems, Discussiones Mathematicae, Differential Inclusions, Control and Optimization 29(1): 43-52. Zbl1206.93085
  21. Kaczorek, T. (2009d). Stabilization of fractional discrete-time linear systems using state feedbacks, Proccedings of the LogiTrans Conference, Szczyrk, Poland, pp. 2-9. 
  22. Kurek, J. (1985). The general state-space model for a twodimensional linear digital systems, IEEE Transactions on Automatic Control AC-30(2): 600-602. Zbl0561.93034
  23. Miller, K. S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Willey, New York, NY. Zbl0789.26002
  24. Nashimoto, K. (1984). Fractional Calculus, Descartes Press, Koriyama. 
  25. Oldham, K. B. and Spanier, J. (1974). The Fractional Calculus, Academic Press, New York, NY. Zbl0292.26011
  26. Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA. Zbl0924.34008
  27. Roesser, R. (1975). A discrete state-space model for linear image processing, IEEE Transactions on Automatic Control AC-20(1): 1-10. Zbl0304.68099
  28. Twardy, M. (2007). An LMI approach to checking stability of 2D positive systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 55(4): 385-395. 
  29. Valcher, M. E. (1997). On the internal stability and asymptotic behavior of 2D positive systems, IEEE Transactions on Circuits and Systems-I 44(7): 602-613. Zbl0891.93046

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