Abstract
The problem of synthesizing stabilizing state-feedback controllers is solved when the closed-loop system is required to remain positive, for the class of 2-D linear systems described by the Fornasini-Marchesini second model. First, a constructive necessary and sufficient condition expressed as a Linear Programming problem is provided for stabilization of these systems when the states must be nonnegative (assuming that the boundary conditions are nonnegative). It is shown how it is simple to include additional constraints (such as positive controls). Moreover, this result is also extended to include uncertainty in the model, making possible to synthesize robust state-feedback controllers, solving Linear Programming problems. Some numerical examples are included to illustrate the proposed approach for different design problems.
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References
Ait Rami, M., Tadeo, F.: Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circuits and Systems-II 54(2), 151–155 (2007)
Alfidi, M., Hmamed, A.: Control for stability and positivity of 2D linear discrete-time systems. WSEAS Transactions on Systems and Control 12(2), 546–556 (2007)
Anderson, B.D.O., Agathoklis, P., Jury, E.I., Mansour, M.: Stability and the Matrix Lyapunov Equation for Discrete 2-Dimensional Systems. IEEE Transactions on Circuits and Systems 33, 261–266 (1986)
Bailo, E., Bru, R., Gelonch, J., Romero, S.: On the Reachability Index of Positive 2-D Systems. IEEE Transactions on Circuits and Systems-II 53(19), 997–1001 (2006)
Dua, C., Xie, L., Zhang, C.: H  ∞  control and robust stabilization of two-dimensional systems in Roesser models. Automatica 37(2), 205–211 (2001)
Farina, L., Rinaldi, S.: Positive Linear Systems. Theory and Applications. Pure and Applied mathematics. John Wiley & Sons, Inc., New York (2000)
Fornasini, E., Marchesini, G.: State-space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control 21(4), 484–492 (1976)
Gao, H., Lam, J., Wang, C., Xu, S.: H  ∞  model reduction for uncertain two-dimensional discrete systems. Optimal Control Appl. and Methods 26(4), 199–227 (2005)
Galkowski, K., Rogers, E., Xu, S., Lam, J., Owens, D.H.: LMIs-A Fundamental Tool in Analysis and Controller Design for Discrete Linear Repetitive Process. IEEE Transactions on Circuits and Systems I 49(6), 768–778 (2002)
Givone, D.D., Roesser, R.P.: Multidimensional linear iterative circuits - General properties. IEEE Transactions on Computers 21(10), 1067–1073 (1972)
Hmamed, A., Ait Rami, M., Alfidi, M.: Controller synthesis for positive 2D systems described by the Roesser model. In: Proceedings of the 47th Conference on Decision and Control, Cancun, Mexico, pp. 387–391 (2008)
Kaczorek, T.: Realization problem, reachability and minimum energy control of positive 2D Roesser model. In: Proceedings of the 6th Annual International Conference on Advances in Communication and Control, pp. 765–776 (1997)
Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002)
Lancaster, P., Tismenetsky, M.: The Theory of Matrices. Academic Press, London (1985)
Lee, E.B., Lu, W.S.: Stabilization of Two-Dimensional Systems. IEEE Transactions on Automatic Control 30, 409–411 (1985)
Lu, W.S.: Some New Results on Stability Robustness of Two-Dimensional Discrete Systems. Multidimensional Systems and Signal Processing 5, 345–361 (1994)
Marszalek, W.: Two dimensional state-space discrete models for hyperbolic partial differential equations. Applied Mathematical Modelling 8, 11–14 (1984)
Roesser, R.: A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control 20, 1–10 (1975)
Twardy, M.: An LMI approach to checking stability of 2D positive systems. Bulletin of the Polish Academy of Sciences: Technical Sciences 55(4), 385–395 (2007)
Valcher, M.E.: On the Internal Stability and Asymptotic Behavior of 2-D Positive Systems. IEEE Transactions on Circuits and Systems I 44(7), 602–613 (1997)
Valcher, M.E., Fornasini, E.: State models and asymptotic behaviour of 2D positive systems. IMA Journal of Mathematical Control and Information 12(1), 17–36 (1995)
Wu-Sheng, L., Lee, E.B.: Stability analysis for two-dimensional systems via a Lyapunov Approach. IEEE Transactions on Circuits and Systems 32, 61–68 (1985)
Yaz, E.: On State-feedback Stabilization of Two-Dimensional Digital Systems. IEEE Transactions on Circuits and Systems 32, 1069–1070 (1985)
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Alfidi, M., Hmamed, A., Tadeo, F. (2009). Linear Programming Approach for 2-D Stabilization and Positivity. In: Bru, R., Romero-Vivó, S. (eds) Positive Systems. Lecture Notes in Control and Information Sciences, vol 389. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02894-6_21
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DOI: https://doi.org/10.1007/978-3-642-02894-6_21
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