Independence of asymptotic stability of positive 2D linear systems with delays of their delays

Tadeusz Kaczorek

International Journal of Applied Mathematics and Computer Science (2009)

  • Volume: 19, Issue: 2, page 255-261
  • ISSN: 1641-876X

Abstract

top
It is shown that the asymptotic stability of positive 2D linear systems with delays is independent of the number and values of the delays and it depends only on the sum of the system matrices, and that the checking of the asymptotic stability of positive 2D linear systems with delays can be reduced to testing that of the corresponding positive 1D systems without delays. The effectiveness of the proposed approaches is demonstrated on numerical examples.

How to cite

top

Tadeusz Kaczorek. "Independence of asymptotic stability of positive 2D linear systems with delays of their delays." International Journal of Applied Mathematics and Computer Science 19.2 (2009): 255-261. <http://eudml.org/doc/207932>.

@article{TadeuszKaczorek2009,
abstract = {It is shown that the asymptotic stability of positive 2D linear systems with delays is independent of the number and values of the delays and it depends only on the sum of the system matrices, and that the checking of the asymptotic stability of positive 2D linear systems with delays can be reduced to testing that of the corresponding positive 1D systems without delays. The effectiveness of the proposed approaches is demonstrated on numerical examples.},
author = {Tadeusz Kaczorek},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {2D systems; systems with delays; asymptotic stability; positive systems},
language = {eng},
number = {2},
pages = {255-261},
title = {Independence of asymptotic stability of positive 2D linear systems with delays of their delays},
url = {http://eudml.org/doc/207932},
volume = {19},
year = {2009},
}

TY - JOUR
AU - Tadeusz Kaczorek
TI - Independence of asymptotic stability of positive 2D linear systems with delays of their delays
JO - International Journal of Applied Mathematics and Computer Science
PY - 2009
VL - 19
IS - 2
SP - 255
EP - 261
AB - It is shown that the asymptotic stability of positive 2D linear systems with delays is independent of the number and values of the delays and it depends only on the sum of the system matrices, and that the checking of the asymptotic stability of positive 2D linear systems with delays can be reduced to testing that of the corresponding positive 1D systems without delays. The effectiveness of the proposed approaches is demonstrated on numerical examples.
LA - eng
KW - 2D systems; systems with delays; asymptotic stability; positive systems
UR - http://eudml.org/doc/207932
ER -

References

top
  1. Bose, N.K. (1982). Applied Multidimensional System Theory, Van Nostrand Reinhold Co., New York, NY. Zbl0574.93031
  2. Bose, N.K. (1985). Multidimensional Systems Theory Progress: Directions and Open Problems, D. Reineld Publishing Co., Dordrecht. Zbl0562.00017
  3. Busłowicz, M. (2007). Robust stability of positive discrete-time linear systems with multiple delays with linear unity rank uncertainty structure or non-negative perturbation matrices, Bulletin of the Polish Academy of Sciences: Technical Sciences 55(1): 1-5. Zbl1203.93164
  4. Busłowicz, M. (2008a). Robust stability of convex combination of two fractional degree characteristic polynomials, Acta Mechanica et Automatica 2(2): 5-10. 
  5. Busłowicz, M. (2008b). Simple stability conditions for linear positive discrete-time systems with delays, Bulletin of the Polish Academy of Sciences: Technical Sciences 57(1): 325-328. 
  6. Farina, L. and Rinaldi, S. (2000). Positive Linear Systems: Theory and Applications, J. Wiley, New York, NY. Zbl0988.93002
  7. Fornasini, E. and Marchesini, G. (1976). State-space realization theory of two-dimensional filters, IEEE Transactions on Automatic Control AC-21: 481-491. Zbl0332.93072
  8. Fornasini, E. and Marchesini, G. (1978). Double indexed dynamical systems, Mathematical Systems Theory 12: 59-72. Zbl0392.93034
  9. Gałkowski, K. (1997). Elementary operation approach to state space realization of 2D systems, IEEE Transactions on Circuit and Systems 44: 120-129. Zbl0874.93028
  10. Gałkowski, K. (2001). State Space Realizations of Linear 2D Systems with Extensions to the General nD (n>2) Case, Springer-Verlag, London. Zbl1007.93001
  11. Hmamed, A., Ait, Rami, M. and Alfidi, M. (2008). Controller synthesis for positive 2D systems described by the Roesser model, IEEE Transactions on Circuits and Systems, (submitted). 
  12. Kaczorek, T. (1985). Two-Dimensional Linear Systems, Springer-Verlag, Berlin. Zbl0593.93031
  13. Kaczorek, T. (2001). Positive 1D and 2D Systems, Springer-Verlag, London. Zbl1005.68175
  14. Kaczorek, T. (2009a). Asymptotic stability of positive 2D linear systems with delays, Bulletin of the Polish Academy of Sciences: Technical Sciences 57(1), (in press). Zbl1167.93023
  15. Kaczorek, T. (2009b). Asymptotic stability of positive 2D linear systems, Proceedings of the 14-th Scientific Conference on Computer Applications in Electrical Engineering, Poznań, Poland, pp. 1-11. 
  16. Kaczorek, T. (2009c). LMI approach to stability of 2D positive systems, Multidimensional Systems and Signal Processing 20: 39-54. Zbl1169.93022
  17. Kaczorek, T. (2008a). Asymptotic stability of positive 1D and 2D linear systems, Recent Advances in Control and Automation, Academic Publishing House EXIT, pp. 41-52. 
  18. Kaczorek, T. (2008c). Checking of the asymptotic stability of positive 2D linear systems with delays, Proceedings of the Conference on Computer Systems Aided Science and Engineering Work in Transport, Mechanics and Electrical Engineering, TransComp, Zakopane, Poland, Monograph No. 122, pp. 235-250, Technical University of Radom, Radom. 
  19. Kaczorek, T. (2007). Choice of the forms of Lyapunov functions for positive 2D Roesser model, International Journal Applied Mathematics and Computer Science 17(4): 471-475. Zbl1234.93089
  20. Kaczorek, T. (2004). Realization problem for positive 2D systems with delays, Machine Intelligence and Robotic Control 6(2): 61-68. 
  21. 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
  22. Kaczorek, T. (2005). Reachability and minimum energy control of positive 2D systems with delays, Control and Cybernetics 34(2): 411-423. Zbl1167.93359
  23. Kaczorek, T. (2006a). Minimal positive realizations for discretetime systems with state time-delays, International Journal for Computation and Mathematics in Electrical and Electronic Engineering, COMPEL 25(4): 812-826. Zbl1125.93409
  24. Kaczorek, T. (2006b). Positive 2D systems with delays, Proceedings of the 12-th IEEE/IFAC International Conference on Methods in Automation and Robotics, MMAR 2006, Międzyzdroje, Poland. Zbl1122.93318
  25. Kaczorek, T. (2003). Realizations problem for positive discretetime systems with delays, Systems Science 29(1): 15-29. 
  26. Klamka, J. (1991). Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht. Zbl0732.93008
  27. Kurek, J. (1985). The general state-space model for a twodimensional linear digital systems, IEEE Transactions on Automatic Control AC-30: 600-602. Zbl0561.93034
  28. Roesser, R.P. (1975). A discrete state-space model for linear image processing, IEEE Transactions on Automatic Control AC-20(1): 1-10, Zbl0304.68099
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.