LMI Approach to Suboptimal Guaranteed Cost Control for 2-D Discrete Uncertain Systems

Amit Dhawan, Haranath Kar

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DOI: 10.4236/jsip.2011.24042                

Abstract

This paper studies the problem of the guaranteed cost control via static-state feedback controllers for a class of two-dimensional (2-D) discrete systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model with norm bounded uncertainties. A convex optimization problem with linear matrix inequality (LMI) constraints is formulated to design the suboptimal guaranteed cost controller which ensures the quadratic stability of the closed-loop system and minimizes the associated closed-loop cost function. Application of the proposed controller design method is illustrated with the help of one example.

Keywords

Linear Matrix Inequality, Lyapunov Methods, Robust Stability, 2-D Discrete Systems, Uncertain Systems, Fornasini-Marchesini Second Local State-Space Model

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	N. K. Bose, “Applied Multidimensional System Theory,” Van Nostrand Reinhold, New York, 1982.
[2]	R. N. Bracewell, “Two-Dimensional Imaging,” Prentice-Hall, Englewood, 1995, pp. 505-537.
[3]	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
[4]	S. Foda and P. Agathoklis, “Control of the Metal Rolling Process: A Multidimensional System Approach,” Journal of the Franklin Institute, Vol. 329, No. 2, 1992, pp. 317-332. doi:10.1016/0016-0032(92)90037-H
[5]	T. Kaczorek, “Two-Dimensional Linear Systems,” Springer-Verlag, Berlin, 1985.
[6]	W.-S. Lu and A. Antoniou, “Two-Dimensional Digital Filters,” Marcel Dekker, New York, 1992.
[7]	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
[8]	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. doi:10.1109/81.974886
[9]	M. Yamada, L. Xu and O. Saito, “2-D Model Following Servo System,” Multidimensional Systems and Signal Processing, Vol. 10, No. 1, 1999, pp. 71-91. doi:10.1023/A:1008461019087
[10]	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
[11]	R. P. Roesser, “A Discrete State-Space Model for Linear Image Processing,” IEEE Transactions on Automatic Control, Vol. 20, No. 1, 1975, pp. 1-10. doi:10.1109/TAC.1975.1100844
[12]	S. Attasi, “Modeling and Recursive Estimation for Double Indexed Sequences,” In: R. K. Mehra and D. G. Lainiotis, Eds., System Identification: Advances and Case Studies, Academic, New York, 1976, pp. 289-348. doi:10.1016/S0076-5392(08)60875-9
[13]	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
[14]	E. Fornasini and G. Marchesini, “Doubly Indexed Dynamical Systems: State-Space Models and Structural Properties,” Theory of Computing Systems, Vol. 12, No. 1, 1978, pp. 59-72. doi:10.1007/BF01776566
[15]	N. G. El-Agizi and M. M. Fahmy, “Two-Dimensional Digital Filters with No Overflow Oscillations,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 27, No. 5, 1979, pp. 465-469. doi:10.1109/TASSP.1979.1163285
[16]	J. H. Lodge and M. M. Fahmy, “Stability and Overflow Oscillations in 2-D State-Space Digital Filters,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 29, No. 6, 1981, pp. 1161-1167. doi:10.1109/TASSP.1981.1163712
[17]	B. D. O. Anderson, P. Agathoklis, E. I. Jury and M. Mansour, “Stability and the Matrix Lypunov Equation for Discrete 2-Dimensional Systems,” IEEE Transactions on Circuits and Systems, Vol. 33, No. 3, 1986, pp. 261-267. doi:10.1109/TCS.1986.1085912
[18]	P. Agathoklis, E. I. Jury and M. Mansour, “The Discrete-Time Strictly Bounded-Real Lemma and the Computation of Positive Definite Solutions to the 2-D Lyapunov Equation,” IEEE Transactions on Circuits and Systems, Vol. 36, No. 6, 1989, pp. 830-837. doi:10.1109/31.90402
[19]	D. Liu and A. N. Michel, “Stability Analysis of State- Space Realizations for Two-Dimensional Filters with Overflow Nonlinearities,” IEEE Transactions on Circuits and Systems I, Vol. 41, No. 2, 1994, pp. 127-137. doi:10.1109/81.269049
[20]	H. Kar and V. Singh, “Stability Analysis of 2-D State- Space Digital Filters Using Lyapunov Function: A Caution,” IEEE Transactions on Signal Processing, Vol. 45, No. 10, 1997, pp. 2620-2621. doi:10.1109/78.640734
[21]	H. Kar and V. Singh, “Stability Analysis of 2-D State- Space Digital Filters with Overflow Nonlinearities,” IEEE Transactions on Circuits and Systems I, Vol. 47, No. 4, 2000, pp. 598-601. doi:10.1109/81.841865
[22]	H. Kar and V. Singh, “Stability Analysis of 1-D and 2-D Fixed-Point State-Space Digital Filters Using Any Combination of Overflow and Quantization Nonlinearities,” IEEE Transactions on Signal Processing, Vol. 49, No. 5, 2001, pp. 1097-1105. doi:10.1109/78.917812
[23]	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
[24]	T. Bose, “Stability of 2-D State-Space System with Overflow and Quantization,” IEEE Transactions on Circuits and Systems II, Vol. 42, No. 6, 1995, pp. 432-434. doi:10.1109/82.392319
[25]	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
[26]	A. Bhaya, E. Kaszkurewicz and Y. Su, “Stability of Asynchronous Two-Dimensional Fornasini-Marchesini Dynamical Systems,” Linear Algebra and Its Applications, Vol. 332, 2001, pp. 257-263.
[27]	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
[28]	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
[29]	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
[30]	T. Hinamoto, “2-D Lyapunov Equation and Filter Design Based on the Fornasini-Marchesini Second Model,” IEEE Transactions on Circuits and Systems I, Vol. 40, No. 2, 1993, pp. 102-110. doi:10.1109/81.219824
[31]	W.-S. Lu, “On a Lyapunov Approach to Stability Analysis of 2-D Digital Filters,” IEEE Transactions on Circuits and Systems I, Vol. 41, No. 10, 1994, pp. 665-669. doi:10.1109/81.329727
[32]	T. Ooba, “On Stability Analysis of 2-D Systems Based on 2-D Lyapunov Matrix Inequalities,” IEEE Transactions on Circuits and Systems I, Vol. 47, No. 8, 2000, pp. 1263-1265. doi:10.1109/81.873883
[33]	T. Hinamoto, “Stability of 2-D Discrete Systems Described by the Fornasini-Marchesini Second Model,” IEEE Transactions on Circuits and Systems I, Vol. 44, No. 3, 1997, pp. 254-257. doi:10.1109/81.557373
[34]	D. Liu, “Lyapunov Stability of Two-Dimensional Digital Filters with Overflow Nonlinearities,” IEEE Transactions on Circuits and Systems I, Vol. 45, No. 5, 1998, pp. 574-577. doi:10.1109/81.668870
[35]	H. Kar and V. Singh, “An Improved Criterion for the Asymptotic Stability of 2-D Digital Filters Described by the Fornasini-Marchesini Second Model Using Saturation Arithmetic,” IEEE Transactions on Circuits and Systems I, Vol. 46, No. 11, 1999, pp. 1412-1413. doi:10.1109/81.802847
[36]	H. Kar and V. Singh, “Stability Analysis of 2-D Digital Filters Described by the Fornasini-Marchesini Second Model Using Overflow Nonlinearities,” IEEE Transactions on Circuits and Systems I, Vol. 48, 2001, pp. 612-617.
[37]	C. Du, L. Xie and Y. C. Soh, “HooFiltering of 2-D Discrete Systems,” IEEE Transactions on Signal Processing, Vol. 48, No. 6, 2000, pp. 1760-1768. doi:10.1109/78.845933
[38]	H. Kar and V. Singh, “Robust Stability of 2-D Discrete Systems Described by the Fornasini-Marchesini Second Model Employing Quantization/Overflow Nonlinearities,” IEEE Transactions on Circuits and Systems II, Vol. 51, No. 11, 2004, pp. 598-602. doi:10.1109/TCSII.2004.836880
[39]	C. Du and L. Xie, “Stability Analysis and Stabilization of Uncertain Two-Dimensional Discrete Systems: An LMI Approach,” IEEE Transactions on Circuits and Systems I, Vol. 46, No. 11, 1999, pp. 1371-1374. doi:10.1109/81.802835
[40]	C. Du and L. Xie, “LMI Approach to Output Feedback Stabilization of 2-D Discrete Systems,” International Journal of Control, Vol. 72, No. 2, 1999, pp. 97-106. doi:10.1080/002071799221262
[41]	T. Bose, M. Q. Chen, K. S. Joo and G. F. Xu, “Stability of Two-Dimensional Discrete Systems with Periodic Coefficients,” IEEE Transactions on Circuits and Systems II, Vol. 45, No. 7, 1998, pp. 839-847. doi:10.1109/82.700930
[42]	X. Guan, C. Long and G. Duan, “Robust Optimal Guaranteed Cost Control for 2-D Discrete Systems,” IET Control Theory & Applications, Vol. 148, 2001, pp. 355-361.
[43]	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
[44]	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
[45]	S. S. L. Chang and T. K. C. Peng, “Adaptive Guaranteed Cost Control Systems with Uncertain Parameters,” IEEE Transactions on Automatic Control, Vol. 17, No. 4, 1972, pp. 474-483. doi:10.1109/TAC.1972.1100037
[46]	I. R. Petersen and D. C. Mcfarlane, “Optimal Guaranteed Cost Control and Filtering for Uncertain Linear Systems,” IEEE Transactions on Automatic Control, Vol. 39, No. 9, 1994, pp. 1971-1977. doi:10.1109/9.317138
[47]	I. R Petersen, D. C. Mcfarlane and M. A. Rotea, “Optimal Guaranteed Cost Control of Discrete-Time Uncertain Linear Systems,” International Journal of Robust and Nonlinear Control, Vol. 8, No. 8, 1998, pp. 649-657. doi:10.1002/(SICI)1099-1239(19980715)8:8<649::AID-RNC334>3.0.CO;2-6
[48]	M. S. Mahmoud and L. Xie, “Guaranteed Cost Control of Uncertain Discrete Systems with Delays,” International Journal of Control, Vol. 73, No. 2, 2000, pp. 105-114. doi:10.1080/002071700219812
[49]	H. Mukaidani, “An LMI Approach to Guaranteed Cost Control for Uncertain Delay Systems,” IEEE Transactions on Circuits and Systems I, Vol. 50, No. 6, 2003, pp. 795-800. doi:10.1109/TCSI.2003.812620
[50]	X. Nian and J. Feng, “Guaranteed Cost Control of a Linear Uncertain System with Multiple Time-Varying Delays: An LMI Approach,” IET Control Theory & Applications, Vol. 150, 2003, pp. 17-22.
[51]	L. Xie and Y. C. Soh, “Guaranteed Cost-Control of Uncertain Discrete-Time Systems,” Control Theory and Advanced Technology, Vol. 10, 1995, pp. 1235-1251.
[52]	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
[53]	P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, “LMI Control Toolbox—for Use with Matlab,” The MATH Works Inc., Natick, 1995.
[54]	V. Balakrishnan and E. Feron, “Linear Matrix Inequalities in Control Theory and Applications,” International Journal of Robust Nonlinear Control, Special Issue, 1996, pp. 896-1099.
[55]	L. El Ghaoui and S. I. Niculescu, “Advances in Linear Matrix Inequality Methods in Control, Advances in Design and Control,” SIAM, Philadelphia, 2000.
[56]	T. Iwasaki, “LMI and Control,” Shokodo, Japan, 1997.
[57]	Y. Nesterov and A. Nemirovskii, “Interior-Point Polynomial Algorithms in Convex Programing,” SIAM, Philadelphia, 1994.
[58]	L. Vandenberghe and V. Balakrishnan, “Algorithms and Software for LMI Problems in Control,” IEEE Control Systems, Vol. 17, No. 5, 1997, pp. 89-95. doi:10.1109/37.621480
[59]	S. Xu, J. Lam, Z. Lin and K. Galkowski, “Positive Real Control for Uncertain Two-Dimensional Systems,” IEEE Transactions on Circuits and Systems I, Vol. 49, No. 11, 2002, pp. 1659-1666. doi:10.1109/TCSI.2002.804531
[60]	P. P. Khargonekar, I. R. Petersen and K. Zhou, “Robust Stabilization of Uncertain Linear Systems: Quadratic Stability and Hoo Control Theory,” IEEE Transactions on Automatic Control, Vol. 35, 1991, pp. 356-361.
[61]	L. Xie, M. Fu and C. E. De Souza, “Hoo Control and Quadratic Stabilization of Systems with Parameter Uncertainty via Output Feedback,” IEEE Transactions on Automatic Control, Vol. 37, No. 8, 1992, pp. 1253-1256. doi:10.1109/9.151120
[62]	F. Yang and Y. S. Hung, “Robust Mixed H2/Hoo Filtering with Regional Pole Assignment for Uncertain Discrete-Time Systems,” IEEE Transactions on Circuits and Systems I, Vol. 49, No. 8, 2002, pp. 1236-1241. doi:10.1109/TCSI.2002.801267
[63]	M. S. Mahmoud, “Robust Control and Filtering for Time-Delay Systems,” Marcel-Dekker, New York, 2000.