Skip to main content

Independence of Asymptotic Stability of Positive 2D Linear Systems with Delays of Their Delays Cover

.blurhash-client-img { display: none !important; }

Independence of Asymptotic Stability of Positive 2D Linear Systems with Delays of Their Delays

International Journal of Applied Mathematics and Computer Science

Volume 19 (2009): Issue 2 (June 2009)

By: Tadeusz Kaczorek

Open Access

|Jul 2009

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.

References

Bose, N.K. (1982). Applied Multidimensional System Theory, Van Nostrand Reinhold Co., New York, NY.
Search in Google Scholar
Bose, N.K. (1985). Multidimensional Systems Theory Progress: Directions and Open Problems, D. Reineld Publishing Co., Dordrecht.
Search in Google Scholar
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.
Search in Google Scholar
Busłowicz, M. (2008a). Robust stability of convex combination of two fractional degree characteristic polynomials, Acta Mechanica et Automatica 2(2): 5-10.
Search in Google Scholar
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.
Search in Google Scholar
Farina, L. and Rinaldi, S. (2000). Positive Linear Systems: Theory and Applications, J. Wiley, New York, NY.10.1002/9781118033029
Search in Google Scholar
Fornasini, E. and Marchesini, G. (1976). State-space realization theory of two-dimensional filters, IEEE Transactions on Automatic Control AC-21: 481-491.10.1109/TAC.1976.1101305
Search in Google Scholar
Fornasini, E. and Marchesini, G. (1978). Double indexed dynamical systems, Mathematical Systems Theory 12: 59-72.10.1007/BF01776566
Search in Google Scholar
Gałkowski, K. (1997). Elementary operation approach to state space realization of 2D systems, IEEE Transactions on Circuit and Systems 44: 120-129.10.1109/81.554329
Search in Google Scholar
Gałkowski, K. (2001). State Space Realizations of Linear 2D Systems with Extensions to the General nD (n>2) Case, Springer-Verlag, London.
Search in Google Scholar
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).10.1109/CDC.2008.4739294
Search in Google Scholar
Kaczorek, T. (1985). Two-Dimensional Linear Systems, Springer-Verlag, Berlin.10.1007/BFb0005617
Search in Google Scholar
Kaczorek, T. (2001). Positive 1D and 2D Systems, Springer-Verlag, London.10.1007/978-1-4471-0221-2
Search in Google Scholar
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).10.2478/v10175-010-0113-4
Search in Google Scholar
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.
Search in Google Scholar
Kaczorek, T. (2009c). LMI approach to stability of 2D positive systems, Multidimensional Systems and Signal Processing 20: 39-54.10.1007/s11045-008-0050-7
Search in Google Scholar
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.
Search in Google Scholar
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.
Search in Google Scholar
Kaczorek, T. (2007). Choice of the forms of Lyapunov functions for positive 2D Roesser model, International Journal Applied Mathematics and Computer Science 17(4): 471-475.10.2478/v10006-007-0039-7
Search in Google Scholar
Kaczorek, T. (2004). Realization problem for positive 2D systems with delays, Machine Intelligence and Robotic Control 6(2): 61-68.
Search in Google Scholar
Kaczorek, T. (1996). Reachability and controllability of non-negative 2D Roesser type models, Bulletin of the Polish Academy of Sciences: Technical Sciences 44(4): 405-410.
Search in Google Scholar
Kaczorek, T. (2005). Reachability and minimum energy control of positive 2D systems with delays, Control and Cybernetics 34(2): 411-423.
Search in Google Scholar
Kaczorek, T. (2006a). Minimal positive realizations for discrete-time systems with state time-delays, International Journal for Computation and Mathematics in Electrical and Electronic Engineering, COMPEL 25(4): 812-826.10.1108/03321640610684015
Search in Google Scholar
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.
Search in Google Scholar
Kaczorek, T. (2003). Realizations problem for positive discrete-time systems with delays, Systems Science 29(1): 15-29.
Search in Google Scholar
Klamka, J. (1991). Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht.
Search in Google Scholar
Kurek, J. (1985). The general state-space model for a two-dimensional linear digital systems, IEEE Transactions on Automatic Control AC-30: 600-602.10.1109/TAC.1985.1103998
Search in Google Scholar
Roesser, R.P. (1975). A discrete state-space model for linear image processing, IEEE Transactions on Automatic Control AC-20(1): 1-1010.1109/TAC.1975.1100844
Search in Google Scholar
Valcher, M.E. (1997). On the internal stability and asymptotic behavior of 2D positive systems, IEEE Transactions on Circuits and Systems—I 44(7): 602-613.10.1109/81.596941
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/v10006-009-0021-7 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X

Journal RSS Feed

Language: English

Page range: 255 - 261

Published on: Jul 8, 2009

Published by: University of Zielona Góra

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

systems with delays,

asymptotic stability,

positive systems

Related subjects:

Applied mathematics

© 2009 Tadeusz Kaczorek, published by University of Zielona Góra
This work is licensed under the Creative Commons License.

Volume 19 (2009): Issue 2 (June 2009)