## Positivity and stabilization of fractional 2D linear systems described by the Roesser model.(English)Zbl 1300.93136

Summary: 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 $$\mathcal Z$$-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.

### MSC:

 93D15 Stabilization of systems by feedback 93C55 Discrete-time control/observation systems 26A33 Fractional derivatives and integrals
Full Text:

### References:

 [1] Bose, N. K. (1982). Applied Multidimensional Systems Theory, Van Nonstrand Reinhold Co., New York, NY. · Zbl 0574.93031 [2] Bose, N. K. (1985). Multidimensional Systems Theory Progress, Directions and Open Problems, D. Reidel Publishing Co., Dodrecht. · Zbl 0562.00017 [3] 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. [4] 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 [5] Farina, E. and Rinaldi, S. (2000). Positive Linear Systems; Theory and Applications, J. Wiley, New York, NY. · Zbl 0988.93002 [6] 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 [7] Fornasini, E. and Marchesini, G. (1978). Double indexed dynamical systems, Mathematical Systems Theory 12(1): 59-72. · Zbl 0392.93034 [8] Galkowski, K. (2001). State Space Realizations of Linear 2D Systems with Extensions to the General nD (n > 2) Case, Springer-Verlag, London. [9] Kaczorek, T. (1985). Two-Dimensional Linear Systems, Springer-Verlag, London. · Zbl 0593.93031 [10] 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 [11] Kaczorek, T. (2001). Positive 1D and 2D Systems, Springer-Verlag, London. · Zbl 1005.68175 [12] Kaczorek, T. (2005). Reachability and minimum energy control of positive 2D systems with delays, Control and Cybernetics 34(2): 411-423. · Zbl 1167.93359 [13] Kaczorek, T. (2007). Reachability and controllability to zero of positive fractional discrete-time systems, Machine Intelligence and Robotic Control 6(4): 139-143. [14] Kaczorek, T. (2008a). Asymptotic stability of positive 1D and 2D linear systems, in K. Malinowski and L. Rutkowski , Recent Advances in Control and Automation, Akademicka Oficyna Wydawnicza EXIT, Warsaw, pp. 41-52. [15] 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 [16] Kaczorek, T. (2008c). Fractional 2D linear systems, Journal of Automation, Mobile Robotics & Intelligent Systems 2(2): 5-9. · Zbl 1154.93017 [17] Kaczorek, T. (2008d). Positive different orders fractional 2D linear systems, Acta Mechanica et Automatica 2(2): 51-58. [18] Kaczorek, T. (2009a). LMI approach to stability of 2D positive systems, Multidimensional Systems and Signal Processing 20(1): 39-54. · Zbl 1169.93022 [19] 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 [20] Kaczorek, T. (2009c). Positivity and stabilization of 2D linear systems, Discussiones Mathematicae, Differential Inclusions, Control and Optimization 29(1): 43-52. · Zbl 1206.93085 [21] Kaczorek, T. (2009d). Stabilization of fractional discrete-time linear systems using state feedbacks, Proccedings of the LogiTrans Conference, Szczyrk, Poland, pp. 2-9. [22] 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 [23] Miller, K. S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Willey, New York, NY. · Zbl 0789.26002 [24] Nashimoto, K. (1984). Fractional Calculus, Descartes Press, Koriyama. [25] Oldham, K. B. and Spanier, J. (1974). The Fractional Calculus, Academic Press, New York, NY. · Zbl 0292.26011 [26] Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA. · Zbl 0924.34008 [27] 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 [28] 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. [29] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.