# Positivity and stabilization of 2D linear systems

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2009)

- Volume: 29, Issue: 1, page 43-52
- ISSN: 1509-9407

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topTadeusz Kaczorek. "Positivity and stabilization of 2D linear systems." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 29.1 (2009): 43-52. <http://eudml.org/doc/271147>.

@article{TadeuszKaczorek2009,

abstract = {The problem of finding a gain matrix of the state-feedback of 2D linear system such that the closed-loop system is positive and asymptotically stable is formulated and solved. Necessary and sufficient conditions for the solvability of the problem are established. It is shown that the problem can be reduced to suitable linear programming problem. The proposed approach can be extended to 2D linear system described by the 2D Roesser model.},

author = {Tadeusz Kaczorek},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {linear 2D systems; general model; positivity; stabilization; state-feedback},

language = {eng},

number = {1},

pages = {43-52},

title = {Positivity and stabilization of 2D linear systems},

url = {http://eudml.org/doc/271147},

volume = {29},

year = {2009},

}

TY - JOUR

AU - Tadeusz Kaczorek

TI - Positivity and stabilization of 2D linear systems

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2009

VL - 29

IS - 1

SP - 43

EP - 52

AB - The problem of finding a gain matrix of the state-feedback of 2D linear system such that the closed-loop system is positive and asymptotically stable is formulated and solved. Necessary and sufficient conditions for the solvability of the problem are established. It is shown that the problem can be reduced to suitable linear programming problem. The proposed approach can be extended to 2D linear system described by the 2D Roesser model.

LA - eng

KW - linear 2D systems; general model; positivity; stabilization; state-feedback

UR - http://eudml.org/doc/271147

ER -

## References

top- [1] N.K. Bose, Applied Multidimensional System Theory (Van Nostrand Reinhold Co, New York, 1982). Zbl0574.93031
- [2] N.K. Bose, Multidimensional Systems Theory Progress, Directions and Open Problems (D. Reidel Publishing Co., 1985). Zbl0562.00017
- [3] M. Busłowicz, Stability of linear continuous-time fractional order systems with delays of the retarded type, Bull. Pol. Acad. Sci. Techn. 56 (2008), 319-324.
- [4] M. Busłowicz, Simple stability conditions for linear positive discrete-time systems with delays, Bull. Pol. Acad. Sci. Techn. 56 (2008), 325-328.
- [5] H.A. Eiselt and C.L. Sandblom, Linear Programming and its Applications (Springer, Berlin, 2007). Zbl1154.90006
- [6] L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications (J. Wiley, New York, 2000). Zbl0988.93002
- [7] M.C. Ferris, O.L. Mangasarian and S.J. Wright, Linear Programming with Matlab (SIAM, 2007).
- [8] E. Fornasini and G. Marchesini, State-space realization theory of two-dimensional filters, IEEE Trans, Autom. Contr. AC-21 (1976), 481-491. Zbl0332.93072
- [9] E. Fornasini and G. Marchesini, Double indexed dynamical systems, Math. Sys. Theory 12 (1978), 59-72. Zbl0392.93034
- [10] K. Gałkowski, Elementary operation approach to state space realization of 2D systems, IEEE Trans. On Circuit and Systems 44 (1997), 120-129. Zbl0874.93028
- [11] K. Gałkowski, State Space Realizations of Linear 2D Systems with Extensions to the General nD (n > 2) Case (Springer-Verlag, London, 2001). Zbl1007.93001
- [12] A. Hmamed, M. Ait Rami and M. Alfidi, Controller synthesis for positive 2D systems described by the Roesser model, submitted to IEEE Trans. on Circuits and Systems.
- [13] T. Kaczorek, Two-dimensional Linear Systems (Springer Verlag, Berlin, 1985). Zbl0593.93031
- [14] T. Kaczorek, Positive 1D and 2D Systems (Springer-Verlag, London, 2001).
- [15] T. Kaczorek, Asymptotic stability of positive fractional 2D linear systems, Bull. Polish Acad. Sci. Technical Sci. 57 (3) (2009), 289-292.
- [16] T. Kaczorek, Asymptotic stability of positive 1D and 2D linear systems, (Recent Advances in Control and Automation, Acad. Publ. House EXIT 2008) 41-52.
- [17] T. Kaczorek, LMI approach to stability of 2D positive systems, Multidim. Syst. Sign. Process. 20 (1) (2009), 39-54. Zbl1169.93022
- [18] T. Kaczorek, Asymptotic stability of positive 2D linear systems with delays, Proc. of XII Scientific Conf. Computer Applications in Electr. Engin. 2008.
- [19] T. Kaczorek, Practical stability of positive fractional discrete-time systems, Bull. Pol. Acad. Sci. Techn. 56 (2008), 313-318.
- [20] T. Kaczorek, Choice of the forms of Lyapunov functions for positive 2D Roesser model, Intern. J. Applied Math. And Comp. Science 17 (2007), 471-475. Zbl1234.93089
- [21] T. Kaczorek, Reachability and controllability of non-negative 2D Roesser type models, Bull. Acad. Pol. Sci. Techn. 44 (1996), 405-410. Zbl0888.93009
- [22] T. Kaczorek, Reachability and minimum energy control of positive 2D systems with delays, Control and Cybernetics 34 (2005), 411-423. Zbl1167.93359
- [23] J. Klamka, Controllability of Dynamical Systems (Kluwer Academic Publ., Dordrecht, 1991). Zbl0732.93008
- [24] J. Kurek, The general state-space model for a two-dimensional linear digital systems, IEEE Trans. Autom. Contr. AC-30 (1985), 600-602. Zbl0561.93034
- [25] R.P. Roesser, A discrete state-space model for linear image processing, IEEE Trans. Autom. Contr. AC-20 (1975), 1-10. Zbl0304.68099
- [26] M. Twardy, An LMI approach to checking stability of 2D positive systems, Bull. Pol. Acad. Sci. Techn. 55 (2007), 386-395.
- [27] M.E. Valcher, On the internal stability and asymptotic behavior of 2D positive systems, IEEE Trans. On Circuits and Systems - I, 44 (1997), 602-613. Zbl0891.93046

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