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.
@article{bwmeta1.element.bwnjournal-article-amcv20i1p85bwm,
author = {Tadeusz Kaczorek and Krzysztof Rogowski},
title = {Positivity and stabilization of fractional 2D linear systems described by the Roesser model},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {20},
year = {2010},
pages = {85-92},
zbl = {1300.93136},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv20i1p85bwm}
}
Tadeusz Kaczorek; Krzysztof Rogowski. Positivity and stabilization of fractional 2D linear systems described by the Roesser model. International Journal of Applied Mathematics and Computer Science, Tome 20 (2010) pp. 85-92. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv20i1p85bwm/
[000] Bose, N. K. (1982). Applied Multidimensional Systems Theory, Van Nonstrand Reinhold Co., New York, NY. | Zbl 0574.93031
[001] Bose, N. K. (1985). Multidimensional Systems Theory Progress, Directions and Open Problems, D. Reidel Publishing Co., Dodrecht. | Zbl 0562.00017
[002] 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.
[003] 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. | Zbl 1167.93019
[004] Farina, E. and Rinaldi, S. (2000). Positive Linear Systems; Theory and Applications, J. Wiley, New York, NY. | Zbl 0988.93002
[005] Fornasini, E. and Marchesini, G. (1976). State-space realization theory of two-dimensional filters, IEEE Transactions on Automatic Control AC-21(4): 484-491. | Zbl 0332.93072
[006] Fornasini, E. and Marchesini, G. (1978). Double indexed dynamical systems, Mathematical Systems Theory 12(1): 59-72. | Zbl 0392.93034
[007] Galkowski, K. (2001). State Space Realizations of Linear 2D Systems with Extensions to the General nD (n > 2) Case, Springer-Verlag, London. | Zbl 1007.93001
[008] Kaczorek, T. (1985). Two-Dimensional Linear Systems, Springer-Verlag, London. | Zbl 0593.93031
[009] 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. | Zbl 0888.93009
[010] Kaczorek, T. (2001). Positive 1D and 2D Systems, Springer-Verlag, London. | Zbl 1005.68175
[011] Kaczorek, T. (2005). Reachability and minimum energy control of positive 2D systems with delays, Control and Cybernetics 34(2): 411-423. | Zbl 1167.93359
[012] Kaczorek, T. (2007). Reachability and controllability to zero of positive fractional discrete-time systems, Machine Intelligence and Robotic Control 6(4): 139-143.
[013] 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.
[014] 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. | Zbl 1154.93017
[015] Kaczorek, T. (2008c). Fractional 2D linear systems, Journal of Automation, Mobile Robotics & Intelligent Systems 2(2): 5-9. | Zbl 1154.93017
[016] Kaczorek, T. (2008d). Positive different orders fractional 2D linear systems, Acta Mechanica et Automatica 2(2): 51-58.
[017] Kaczorek, T. (2009a). LMI approach to stability of 2D positive systems, Multidimensional Systems and Signal Processing 20(1): 39-54. | Zbl 1169.93022
[018] Kaczorek, T. (2009b). Positive 2D fractional linear systems, International Journal for Computation and Mathematics in Electrical and Electronic Engineering, COMPEL 28(2): 341-352. | Zbl 1173.93017
[019] Kaczorek, T. (2009c). Positivity and stabilization of 2D linear systems, Discussiones Mathematicae, Differential Inclusions, Control and Optimization 29(1): 43-52. | Zbl 1206.93085
[020] Kaczorek, T. (2009d). Stabilization of fractional discrete-time linear systems using state feedbacks, Proccedings of the LogiTrans Conference, Szczyrk, Poland, pp. 2-9.
[021] Kurek, J. (1985). The general state-space model for a twodimensional linear digital systems, IEEE Transactions on Automatic Control AC-30(2): 600-602. | Zbl 0561.93034
[022] Miller, K. S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Willey, New York, NY. | Zbl 0789.26002
[023] Nashimoto, K. (1984). Fractional Calculus, Descartes Press, Koriyama.
[024] Oldham, K. B. and Spanier, J. (1974). The Fractional Calculus, Academic Press, New York, NY. | Zbl 0292.26011
[025] Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA. | Zbl 0924.34008
[026] Roesser, R. (1975). A discrete state-space model for linear image processing, IEEE Transactions on Automatic Control AC-20(1): 1-10. | Zbl 0304.68099
[027] 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.
[028] 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. | Zbl 0891.93046