The choice of the forms of Lyapunov functions for a positive 2D Roesser model

Tadeusz Kaczorek

International Journal of Applied Mathematics and Computer Science (2007)

  • Volume: 17, Issue: 4, page 471-475
  • ISSN: 1641-876X

Abstract

top
The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix P such that the matrix A^{T}PA-P is negative definite. The theoretical deliberations will be illustrated by numerical examples.

How to cite

top

Kaczorek, Tadeusz. "The choice of the forms of Lyapunov functions for a positive 2D Roesser model." International Journal of Applied Mathematics and Computer Science 17.4 (2007): 471-475. <http://eudml.org/doc/207852>.

@article{Kaczorek2007,
abstract = {The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix P such that the matrix A^\{T\}PA-P is negative definite. The theoretical deliberations will be illustrated by numerical examples.},
author = {Kaczorek, Tadeusz},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {asymptotic stability; positive 2D Roesser model; Lyapunov function},
language = {eng},
number = {4},
pages = {471-475},
title = {The choice of the forms of Lyapunov functions for a positive 2D Roesser model},
url = {http://eudml.org/doc/207852},
volume = {17},
year = {2007},
}

TY - JOUR
AU - Kaczorek, Tadeusz
TI - The choice of the forms of Lyapunov functions for a positive 2D Roesser model
JO - International Journal of Applied Mathematics and Computer Science
PY - 2007
VL - 17
IS - 4
SP - 471
EP - 475
AB - The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix P such that the matrix A^{T}PA-P is negative definite. The theoretical deliberations will be illustrated by numerical examples.
LA - eng
KW - asymptotic stability; positive 2D Roesser model; Lyapunov function
UR - http://eudml.org/doc/207852
ER -

References

top
  1. Benvenuti L. and Farina L. (2004): A tutorial on the positive realization problem. IEEE Transactions on Automatic Control, Vol.49, No.5, pp.651-664. 
  2. Bose N. K. (1985): Multidimensional Systems Theory Progress, Directions and Open Problems, Dordrecht: D. Reidel Publishing Co. Zbl0562.00017
  3. Farina L. and Rinaldi S. (2000): Positive Linear Systems. Theory and Applications. New York: Wiley. Zbl0988.93002
  4. Fornasini E. and Marchesini G. (1978): Double indexed dynamical systems. Mathematical Systems Theory, Vol.12, pp.59-72. Zbl0392.93034
  5. Fornasini E. and Marchesini G. (1976): State-space realization theory of two- dimensional filters. IEEE Transactions on Automatic Control, Vol.AC-21, pp.484-491. Zbl0332.93072
  6. Fornasini E. and Valcher M.E. (1996): On the spectral and combinatorial structure of 2D positive systems. Linear Algebra and Its Applications, Vol.245, pp.223-258. Zbl0857.93051
  7. Fornasini E. and Valcher M.E. (1997): Recent developments in 2D positive systems theory. International Journal of Applied Mathematics and Computer Science, Vol.7, No.4, pp.101-123. Zbl0913.93032
  8. Galkowski K. (1997): Elementary operation approach to state space realization of 2D systems. IEEE Transaction on Circuits and Systems, Vol.44, No.2, pp.120-129. Zbl0874.93028
  9. Kaczorek T. (1999): Externally positive 2D linear systems. Bulletin of the Polish Academy of Sciences: Technical Sciences, Vol.47, No.3, pp.227-234. Zbl0987.93044
  10. Kaczorek T. (1996): Reachability and controllability of non-negative 2D Roesser type models. Bulletin of the Polish Academy of Sciences: Technical Sciences, Vol.44, No.4, pp.405-410. Zbl0888.93009
  11. Kaczorek T. (2000): Positive 1D and 2D Systems. London: Springer. Zbl0978.93003
  12. Kaczorek T. (2002): When the equilibrium of positive 2D Roesser model are strictly positive. Bulletin of the Polish Academy of Sciences: Technical Sciences, Vol.50, No.3, pp.221-227. Zbl1138.93319
  13. Kaczorek T. (1985): Two-Dimensional Linear Systems. Berlin: Springer. Zbl0593.93031
  14. Klamka J. (1999): Controllability of 2D linear systems, In: Advances in Control Highlights of ECC 1999 (P.M. Frank, Ed.), Berlin: Springer, pp.319-326. 
  15. Klamka J. (1991): Controllability of dynamical systems. Dordrecht: Kluwer. Zbl0732.93008
  16. Kurek J. (1985): The general state-space model for a two-dimensional linear digital systems. IEEE Transactions on Automatic Control, Vol.-30, No.2, pp.600-602. Zbl0561.93034
  17. Kurek J. (2002): Stability of positive 2D systems described by the Roesser model. IEEE Transactions on Circuits and Systems I, Vol.49, No.4, pp.531-533. 
  18. Roesser R.P. (1975) A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control, Vol.AC-20, No.1, pp.1-10. Zbl0304.68099
  19. Valcher M.E. and Fornasini E. (1995): State models and asymptotic behaviour of 2D Roesser model. IMA Journal on Mathematical Control and Information, No.12, pp.17-36 Zbl0840.93047

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.