Robust Non-Fragile Control of 2-D Discrete Uncertain Systems: An LMI Approach

Paramanand Sharma, Amit Dhawan

Motilal Nehru National Institute of Technology.

DOI: 10.4236/jsip.2012.33049                

Abstract

This paper considers the problem of robust non-fragile control for a class of two-dimensional (2-D) discrete uncertain systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model under controller gain variations. The parameter uncertainty is assumed to be norm-bounded. The problem to be addressed is the design of non-fragile robust controllers via state feedback such that the resulting closed-loop system is asymptotically stable for all admissible parameter uncertainties and controller gain variations. A sufficient condition for the existence of such controllers is derived based on the linear matrix inequality (LMI) approach combined with the Lyapunov method. Finally, a numerical example is illustrated to show the contribution of the main result.

Keywords

2-D Discrete Systems; Fornasini-Marchesini Second Local State-Space Model; Non-Fragile Control; Linear Matrix Inequality; Lyapunov Methods

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	T. Kaczorek, “Two-Dimensional Linear Systems,” Springer-Verlag, Berlin, 1985.
[2]	R. N. Bracewell, “Two-Dimensional Imaging,” Prentice- Hall, Englewood, 1995, pp. 505-537.
[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]	E. Fornasini, “A 2-D System Approach to River Pollution Modeling,” Multidimensional Systems and Signal Processing, Vol. 2, No. 3, 1991, pp. 233-265. doi:10.1007/BF01952235
[6]	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.
[7]	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.
[8]	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
[9]	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
[10]	T. Hinamoto, “Stability of 2-D Discrete Systems Described by the Fornasini-Marchesini Second Model,” IEEE Transactions on Circuits and Systems I, Vol. 34, No. 2, 1997, pp. 254-257. doi:10.1109/81.557373
[11]	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
[12]	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
[13]	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, No. 5, 2001, pp. 612-617. doi:10.1109/81.922464
[14]	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
[15]	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
[16]	W. M. Haddad and J. R. Corrado, “Robust Resilient Dynamic Controllers for Systems with Parameter Uncertainty and Controller Gain Variations,” International Journal of Control, Vol. 73, 2000, pp. 1405-1423. doi:10.1080/002071700445424
[17]	G.-H. Yang, J. L. Wang and C. Lin, “ Control for Linear Systems with Additive Controller Gain Variations,” International Journal of Control, Vol. 73, No. 16, 2000, pp. 1500-1506. doi:10.1080/00207170050163369
[18]	G.-H. Yang and J. L. Wang, “Non-Fragile Control for Linear Systems with Multiplicative Controller Gain Variations,” Automatica, Vol. 37, No. 5, 2001, pp. 727- 737.
[19]	J. H. Park, “Robust Non-Fragile Control for Uncertain Discrete-Delay Large-Scale Systems with a Class of Controller Gain Variations,” Applied Mathematics and Computation, Vol. 149, No. 1, 2004, pp. 147-164. doi:10.1016/S0096-3003(02)00962-1
[20]	S. Xu, J. Lam, J. Wang and G. Yang, “Non-Fragile Positive Real Control for Uncertain Linear Neutral Delay Systems,” Systems and Control Letters, Vol. 52, No. 1, 2004, pp. 59-74. doi:10.1016/j.sysconle.2003.11.001
[21]	C. Lien, W. Cheng, C. Tsai and K. Yu, “Non-Fragile Observer-Based Controls of Linear System via LMI Approach,” Chaos, Solitons and Fractals, Vol. 32, No. 4 2007, pp. 1530-1537. doi:10.1016/j.chaos.2005.11.092
[22]	C. Lien, “ Non-Fragile Observer-Based Controls of Dynamical Systems via LMI Optimization Approach,” Chaos, Solitons and Fractals, Vol. 34, No. 2, 2007, pp. 428-436. doi:10.1016/j.chaos.2006.03.050
[23]	L. Xie, “Output Feedback Control of Systems with Parameter Uncertainty,” International Journal of Control, Vol. 63, No. 4, 1996, pp. 741-750. doi:10.1080/00207179608921866
[24]	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
[25]	P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, “LMI Control Toolbox-for Use with Matlab,” The MATH Works Inc., Natick, 1995.