Robust Suboptimal Guaranteed Cost Control for 2-D Discrete Systems Described by Fornasini-Marchesini First Model

Abstract

This paper considers the guaranteed cost control problem for a class of two-dimensional (2-D) uncertain discrete systems described by the Fornasini-Marchesini (FM) first model with norm-bounded uncertainties. New linear matrix inequality (LMI) based characterizations are presented for the existence of static-state feedback guaranteed cost controller which guarantees not only the asymptotic stability of closed loop systems, but also an adequate performance bound over all the admissible parameter uncertainties. Moreover, a convex optimization problem is formulated to select the suboptimal guaranteed cost controller which minimizes the upper bound of the closed-loop cost function.

Share and Cite:

Tiwari, M. and Dhawan, A. (2012) Robust Suboptimal Guaranteed Cost Control for 2-D Discrete Systems Described by Fornasini-Marchesini First Model. Journal of Signal and Information Processing, 3, 252-258. doi: 10.4236/jsip.2012.32034.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] A. T. Kaczorek, “Two-Dimensional Linear Systems,” Springer-Verlag: Berlin, 1985.
[2] R. N. Bracewell, “Two-Dimensional Imaging,” Prentice-Hall Signal Processing Series, Prentice-Hall, Englewood Cliffs, 1995.
[3] W.-S. Lu and A. Antoniou, “Two-Dimensional Digital Filters,” Marcel Dekker, New York, 1992.
[4] N. K. Bose, “Applied Multidimensional System Theory,” Van Nostrand Reinhold, New York, 1982.
[5] W. Marszalek, “Two-Dimensional State-Space Discrete Models for Hyperbolic Partial Differential Equations,” Applied Mathematical Modelling, Vol. 8, No. 1, 1984, pp. 11-14. doi:10.1016/0307-904X(84)90170-7
[6] C. Du, L. Xie and C. Zhang, “Hoo Control and Robust Stabilization of Two-Dimensional Systems in Roesser Models,” Automatica, Vol. 37, No. 2, 2001, pp. 205-211. doi:10.1016/S0005-1098(00)00155-2
[7] J. S.-H. Tsai, J. S. Li and L.-S. Shieh, “Discretized Quadratic Optimal Control for Continuous-Time Two-Dimensional Systems,” IEEE Transactions on Circuits and Systems I, Vol. 49, No. 1, 2002, pp. 116-125.
[8] R. Yang, L. Xie and C. Zhang, “H2 and Mixed H2/Hoo Control of Two-Dimensional Systems in Roesser Model,” Automatica, Vol. 42, No. 9, 2006, pp. 1507-1514. doi:10.1016/j.automatica.2006.04.002
[9] E. Fornasini and G. Marchesini, “State-Space Realization Theory of Two Dimensional Filters,” IEEE Transactions on Automatic Control, Vol. 21, No. 4, 1976, pp. 484-492. doi:10.1109/TAC.1976.1101305
[10] G.-D. Hu and M. Liu, “Simple Criteria for Stability of Two-Dimensional Linear Systems,” IEEE Transactions on Signal Processing, Vol. 53, No. 12, 2005, pp. 4720-4723. doi:10.1109/TSP.2005.859265
[11] T. Bose and D. A. Trautman, “Two’s Complement Quantization in Two-Dimensional State-Space Digital Filters,” IEEE Transactions on Signal Processing, Vol. 40, No. 10, 1992, pp. 2589-2592. doi:10.1109/78.157299
[12] Y. Su and A. Bhaya, “On the Bose-Trautman Condition for Stability of Two-Dimensional Linear Systems,” IEEE Transactions on Signal Processing, Vol. 46, No. 7, 1998, pp. 2069-2070. doi:10.1109/78.700987
[13] T. Bose, “Stability of 2-D State-Space System with Over-flow and Quantization,” IEEE Transactions on Circuits and Systems II, Vol. 42, No. 6, 1995, pp. 432-434. doi:10.1109/82.392319
[14] H. Kar and V. Singh, “Stability of 2-D Systems Described by Fornasini-Marchesini First Model,” IEEE Transactions on Signal Processing, Vol. 51, No. 6, 2003, pp. 1675-1676. doi:10.1109/TSP.2003.811237
[15] T. Zhou, “Stability and Stability Margin for a Two-Dimensional System,” IEEE Transactions on Signal Processing, Vol. 54, No. 9, 2006, pp. 3483-3488. doi:10.1109/TSP.2006.879300
[16] R. Thamvichai and T. Bose, “Stability of 2-D Periodically Shift Variant Filters,” IEEE Transactions on Circuits and Systems II, Vol. 49, No. 1, 2002, pp. 61-64. doi:10.1109/82.996060
[17] A. Bhaya, E. Kaszkurewicz and Y. Su, “Stability of Asynchronous Two-Dimensional Fornasini-Marchesini Dynamical Systems,” Linear Algebra and Its Application, Vol. 332, 2001, pp. 257-263. doi:10.1016/S0024-3795(00)00317-7
[18] D. Henrion, M. Sebek and O. Bachelier, “Rank-1 LMI Approach to Stability of 2-D Polynomial Matrices,” Multidimens Systems Signal Process, Vol. 12, No. 11, 2001, pp. 33-48. doi:10.1023/A:1008464726878
[19] B. Dumitrescu, “LMI Stability Tests for the Fornasini-Marchesini Model,” IEEE Transactions on. Signal Processing, Vol. 56, No. 8, 2008, pp. 4091-4095.
[20] T. Liu, “Stability Analysis of Linear 2-D Systems,” Signal Processing, Vol. 88, No. 8, 2008, pp. 2078-2084. doi:10.1016/j.sigpro.2008.02.007
[21] M. Tiwari and A. Dhawan, “A Survey on Stability of 2-D Discrete Systems Described by Fornasini-Marchesini First Model,” Proceedings of the International Conference on Power Control and Embedded Systems, Allahabad, 2010, pp. 1-4. doi:10.1109/ICPCES.2010.5698674
[22] X. Guan, C. Long and G. Duan, “Robust Optimal Guaranteed Cost Control for 2D Discrete Systems,” IET-Control Theory & Applications, Vol. 148, No. 5, 2001, pp. 355-361.
[23] A. Dhawan and H. Kar, “Comment on Robust Optimal Guaranteed Cost Control for 2-D Discrete Systems,” IET-Control Theory & Applications, Vol. 1, No. 4, 2007, pp. 1188-1190.
[24] A. Dhawan and H. Kar, “LMI-Based Criterion for the Robust Guaranteed Cost Control of 2-D Systems Described by the Fornasini-Marchesini Second Model,” Signal Processing, Vol. 87, No. 3, 2007, pp. 479-488. doi:10.1016/j.sigpro.2006.06.002
[25] A. Dhawan and H. Kar, “Optimal Guaranteed Cost Control of 2-D Discrete Uncertain Systems: An LMI Approach,” Signal Processing, Vol. 87, No. 12, 2007, pp. 3075-3085. doi:10.1016/j.sigpro.2007.06.001
[26] A. Dhawan and H. Kar, “LMI Approach to Suboptimal Guaranteed Cost Control for 2-D Discrete Uncertain Systems,” Journal of Signal and Information Processing, Vol. 2, No. 4, 2011, pp. 292-300. doi:10.4236/jsip.2011.24042
[27] A. Dhawan and H. Kar, “An LMI Approach to Robust Optimal Guaranteed Cost Control of 2-D Discrete Systems Described by the Roesser Model,” Signal Processing, Vol. 90, No. 9, 2010, pp. 2648-2654. doi:10.1016/j.sigpro.2010.03.008
[28] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, “Linear Matrix Inequalities in System and Control Theory,” SIAM, Philadelphia, 1994. doi:10.1137/1.9781611970777
[29] P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, “LMI Control Toolbox—For Use with Matlab,” The MATH Works Inc., Natick, 1995.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.