Positive 2D discrete-time linear Lyapunov systems

Przemysław Przyborowski; Tadeusz Kaczorek

International Journal of Applied Mathematics and Computer Science (2009)

  • Volume: 19, Issue: 1, page 95-105
  • ISSN: 1641-876X

Abstract

top
Two models of positive 2D discrete-time linear Lyapunov systems are introduced. For both the models necessary and sufficient conditions for positivity, asymptotic stability, reachability and observability are established. The discussion is illustrated with numerical examples.

How to cite

top

Przemysław Przyborowski, and Tadeusz Kaczorek. "Positive 2D discrete-time linear Lyapunov systems." International Journal of Applied Mathematics and Computer Science 19.1 (2009): 95-105. <http://eudml.org/doc/207926>.

@article{PrzemysławPrzyborowski2009,
abstract = {Two models of positive 2D discrete-time linear Lyapunov systems are introduced. For both the models necessary and sufficient conditions for positivity, asymptotic stability, reachability and observability are established. The discussion is illustrated with numerical examples.},
author = {Przemysław Przyborowski, Tadeusz Kaczorek},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {positivity; Lyapunov systems; reachability; observability},
language = {eng},
number = {1},
pages = {95-105},
title = {Positive 2D discrete-time linear Lyapunov systems},
url = {http://eudml.org/doc/207926},
volume = {19},
year = {2009},
}

TY - JOUR
AU - Przemysław Przyborowski
AU - Tadeusz Kaczorek
TI - Positive 2D discrete-time linear Lyapunov systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2009
VL - 19
IS - 1
SP - 95
EP - 105
AB - Two models of positive 2D discrete-time linear Lyapunov systems are introduced. For both the models necessary and sufficient conditions for positivity, asymptotic stability, reachability and observability are established. The discussion is illustrated with numerical examples.
LA - eng
KW - positivity; Lyapunov systems; reachability; observability
UR - http://eudml.org/doc/207926
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, Buchberger, B. and Guiver, J.P. (2003). Multidimensional Systems Theory and Applications, Kluwer Academic Publishers, Dordrecht. 
  3. Busłowicz, M. (2006), Stability of positive linear discrete-time systems with unit delay with canonical forms of state matrices, Proceedings of 12-th IEEE International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland. 
  4. Farina, L. and Rinaldi, S. (2000). Positive Linear Systems Theory and Applications, Wiley, New York, NY. Zbl0988.93002
  5. Fornasini, E. and Marchesini, G. (1976) State-space realization theory of two-dimensional filters, IEEE Transactions on Automatic Control 21(4): 481-491. Zbl0332.93072
  6. Fornasini, E. and Marchesini, G. (1978). Double indexed dynamical systems, Mathematical Systems Theory 12: 59-72. Zbl0392.93034
  7. Gałkowski, K. (2001). State Space Realizations of Linear 2D Systems with Extensions to the General nD (n > 2) Case, Springer, London. Zbl1007.93001
  8. Kaczorek, T. (1985). Two-Dimensional Linear Systems, Springer, Berlin. Zbl0593.93031
  9. 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
  10. Kaczorek, T. (1998). Vectors and Matrices in Automation and Electrotechnics, Wydawnictwo Naukowo-Techniczne, Warsaw (in Polish). 
  11. Kaczorek, T. (2001). Positive 1D and 2D Systems, Springer-Verlag, London. Zbl1005.68175
  12. Kaczorek, T. (2003). Realizations problem for positive discretetime systems with delays, Systems Science 29(1): 15-29. 
  13. Kaczorek, T. (2004). Realization problem for positive 2D systems with delays, Machine Intelligence and Robotic Control 6(2): 61-68. 
  14. Kaczorek, T. (2005). Reachability and minimum energy control of positive 2D systems with delays, Control and Cybernetics 34(2): 411-423. Zbl1167.93359
  15. Kaczorek, T. (2006a). Minimal positive realizations for discretetime systems with state time-delays, The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, COMPEL 25(4): 812-826. Zbl1125.93409
  16. Kaczorek, T. (2006b). Positive 2D systems with delays, Proceedings of the 12-th IEEE|IFAC International Conference on Methods in Automation and Robotics, Międzyzdroje, Poland. Zbl1122.93318
  17. Kaczorek, T. (2007). Positive discrete-time linear Lyapunov systems, Proceedings of the 15-th Mediterranean Conference of Control and Automation, MED, Athens, Greece. Zbl1119.93370
  18. Kaczorek, T. (2008a). Asymptotic stability of positive 2D linear systems, Proceedings of the 13-th Scientific Conference on Computer Applications in Electrical Engineering, Poznań, Poland. Zbl1154.93017
  19. Kaczorek, T. (2008b). LMI approach to stability of 2D positive systems, Multidimensional Systems and Signal Processing, (in press). Zbl1169.93022
  20. Kaczorek, T. (2008c). Asymptotic stability of positive 2D linear systems with delays, Lecture Notes in Electrical Engineering: Numerical Linear Algebra in Signals, Systems and Control, Springer-Verlag. 
  21. Kaczorek, T. and Przyborowski, P. (2007a). Positive continuoustime linear Lyapunov systems, Proceedings of the International Conference on Computer as a Tool, EUROCON 2007, Warsaw, Poland, pp. 731-737. Zbl1176.93067
  22. Kaczorek, T. and Przyborowski, P. (2007b). Positive continuoustime linear time-varying Lyapunov systems, Proceedings of the 16-th International Conference on Systems Science, Wrocław, Poland, Vol. I, pp. 140-149. Zbl1176.93067
  23. Kaczorek, T. and Przyborowski, P. (2007c). Continuoustime linear Lyapunov cone-systems, Proceedings of the 13-th IEEE IFAC International Conference on Methods and Models in Automation and Robotics, Szczecin, Poland, pp. 225-229. 
  24. Kaczorek, T. and Przyborowski, P. (2007d). Positive discretetime linear Lyapunov systems with delays, Przegląd Elektrotechniczny (2): 12-15. Zbl1176.93067
  25. Kaczorek, T. and Przyborowski, P. (2007e). Positive linear Lyapunov systems, FNA-ANS International Journal - Problems of Nonlinear Analysis in Engineering Systems 13(2): 35-60. Zbl1176.93067
  26. Kaczorek, T. and Przyborowski, P. (2008). Reachability, controllability to zero and observability of the positive discretetime Lyapunov systems, Control and Cybernetics Journal, (submitted). Zbl1301.93025
  27. Klamka, J. (1991). Controllability of Dynamical Systems, Kluwer, Dordrecht. Zbl0732.93008
  28. Klamka, J. (1996a). Controllability of 2-D systems, Proceedings of the 3-rd Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 207-212. 
  29. Klamka, J. (1996b). Controllability and minimum energy control of 2-D linear systems, Proceedings of the International Conference on Circuits Systems and Computers, Athens, Greece, Vol. 1, pp. 45-50. 
  30. Klamka, J. (1997a). Controllability of infinite-dimensional 2-D linear systems, Advances in Systems Science and Applications 1(1): 537-543. 
  31. Klamka, J. (1997b). Controllability of nonlinear 2-D systems, Nonlinear Analysis, Theory, Methods and Applications 30(5): 2963-2968. Zbl0896.93005
  32. Klamka, J. (1997c). Controllability of 2-D systems systems: A survey, Applied Mathematics and Computer Science 7(4): 101-120. 
  33. Klamka, J. (1997d). Controllability and minimum energy control of 2-D linear systems, Proceedings of the American Control Conference ACC'97, Albuquerque, NM, USA, Vol. 5, pp. 3141-3143. 
  34. Klamka, J. (1998a). Constrained controllability of positive 2-D systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 46(1): 95-104. Zbl1039.93003
  35. Klamka, J. (1998b). Constrained controllability of 2-D systems, Proceedings of the Symposium on Modelling Analysis and Control, Hammamet, Tunisia. Zbl1039.93003
  36. Klamka, J. (1998c). Constrained controllability of linear positive 2-D systems, Proceedings of the 9-th Symposium on Systems, Modelling, Control, SMC-9, Zakopane, Poland. Zbl1039.93003
  37. Klamka, J. (1999a). Local controllability of 2-D nonlinear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 47(2): 153-161. Zbl0941.93005
  38. Klamka, J. (1999b). Controllability of 2-D linear systems, in P.M. Frank (Ed.), Advances in Control. Highlights of ECC'99, Springer, Berlin, pp. 319-326. 
  39. Klamka, J. (1999c). Controllability of 2-D nonlinear systems, Proceedings of the European Control Conference, Karlsruhe, Germany, pp. 1121-1127. 
  40. Klamka, J. (2002). Positive controllability of positive dynamical systems, Proceedings of the American Control Conference, Anchorage, AK, USA, (on CD-ROM). Zbl1017.93014
  41. Klamka, J. (2005). Approximate constrained controllability of mechanical systems, Journal of Theoretical and Applied Mechanics 43(3): 539-554. 
  42. Kurek, J. (1985). The general state-space model for a twodimensional linear digital systems, IEEE Transactions on Automatic Control 30(6): 600-602. Zbl0561.93034
  43. Murty, M.S.N. and Apparao, B.V. (2005). Controllability and observability of Lyapunov systems, Ranchi University Mathematical Journal 32: 55-65. Zbl1103.93035
  44. Przyborowski, P. (2008a). Positive fractional discrete-time Lyapunov systems, Archives of Control Sciences 18(LIV)(1): 5-18. Zbl1187.93088
  45. Przyborowski, P. (2008b). Fractional discrete-time Lyapunov cone-systems, Przegląd Elektrotechniczny (5): 47-52. Zbl1187.93088
  46. Przyborowski, P. and Kaczorek, T. (2008). Linear Lyapunov cone-systems, in J. M. Ramos Arreguin (Ed.), Automation and Robotics-New Challenges, I-Tech Education and Publishing, Vienna, (in press). 
  47. Roesser, R.P. (1975). A discrete state-space model for linear image processing, IEEE Transactions on Automatic Control 20(1): 1-10. Zbl0304.68099
  48. Twardy, M. (2007). An LMI approach to checking stability of 2D positive system, Bulletin of the Polish Academy of Sciences: Technical Sciences 54(4): 385-395 
  49. 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.