Non-Fragile Controller Design for 2-D Discrete Uncertain Systems Described by the Roesser Model

Affiliation(s)

Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology, Allahabad, India..

Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology, Allahabad, India..

ABSTRACT

This paper is concerned with the design problem of non-fragile controller for a class of two-dimensional (2-D) discrete uncertain systems described by the Roesser model. The parametric uncertainties are assumed to be norm-bounded. The aim of this paper is to design a memoryless non-fragile state feedback control law such that the closed-loop system is asymptotically stable for all admissible parameter uncertainties and controller gain variations. A new linear matrix inequality (LMI) based sufficient condition for the existence of such controllers is established. Finally, a numerical example is provided to illustrate the applicability of the proposed method.

This paper is concerned with the design problem of non-fragile controller for a class of two-dimensional (2-D) discrete uncertain systems described by the Roesser model. The parametric uncertainties are assumed to be norm-bounded. The aim of this paper is to design a memoryless non-fragile state feedback control law such that the closed-loop system is asymptotically stable for all admissible parameter uncertainties and controller gain variations. A new linear matrix inequality (LMI) based sufficient condition for the existence of such controllers is established. Finally, a numerical example is provided to illustrate the applicability of the proposed method.

KEYWORDS

2-D Discrete Systems; Non-Fragile Control; Roesser Model; Linear Matrix Inequality; Lyapunov Methods

2-D Discrete Systems; Non-Fragile Control; Roesser Model; Linear Matrix Inequality; Lyapunov Methods

Cite this paper

A. Dhawan, "Non-Fragile Controller Design for 2-D Discrete Uncertain Systems Described by the Roesser Model,"*Journal of Signal and Information Processing*, Vol. 3 No. 2, 2012, pp. 248-251. doi: 10.4236/jsip.2012.32033.

A. Dhawan, "Non-Fragile Controller Design for 2-D Discrete Uncertain Systems Described by the Roesser Model,"

References

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[16] D. Yue and J. Lam, “Non-Fragile Guaranteed Cost Control for Uncertain Descriptor Systems with Time-Varying State and Input Delays,” Optimal Control Applications & Methods, Vol. 26, 2005, pp. 85-105. doi:10.1002/oca.753

[17] L. H. Keel and S. P. Bhattacharya, “Robust, Fragile or Optimal,” IEEE Transactions on Automatic Control, Vol. 42, No. 8, 1997, pp. 1098-1105. doi:10.1109/9.618239

[18] 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, No. 15, 2000, pp. 1405-1423. doi:10.1080/002071700445424

[19] 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

[20] S. Xu, J. Lam, G. Yang and J. Wang, “Stabilization and*H*_{oo} Control for Uncertain Stochastic Time-Delay Systems via Non-Fragile Control,” Asian Journal of Control, Vol. 8, No. 2, 2006, pp. 197-200.
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[21] 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

[22] C. Lien, “*H*_{oo} 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] C. Lien, “Non-Fragile Guaranteed Cost Control for Uncertain Neutral Dynamic Systems with Time-Varying Delays in State and Control Input,” Chaos, Solitons and Fractals, Vol. 31, No. 4, 2007, pp. 889-899. doi:10.1016/j.chaos.2005.10.080

[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, MA, 1995.

[1] 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] 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.

[6] 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, 1986, pp. 261-267. doi:10.1109/TCS.1986.1085912

[7] H. Kar and V. Singh, “Stability Analysis of 2-D State-Space Digital Filters Using Lyapunov Function: A Caution,” IEEE Transactions on Signal Process, Vol. 45, No. 10, 1997, pp. 2620-2621. doi:10.1109/78.640734

[8] 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 Process, Vol. 49, 2001, pp. 1097-1105. doi:10.1109/78.917812

[9] D. Liu and A. N. Michel, “Stability Analysis of State-Space Realizations for Two-Dimensional Filters with Over-flow Nonlinearities,” IEEE Transactions on Circuits and Systems I, Vol. 41, No. 2, 1994, pp. 127-137. doi:10.1109/81.269049

[10] 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

[11] V. Singh, “New LMI Condition for the Nonexistence of Overflow Oscillations in 2-D State-Space Digital Filters Using Saturation Arithmetic,” Digital Signal Process, Vol. 17, No. 1, 2007, pp. 345-352. doi:10.1016/j.dsp.2006.01.003

[12] H. Kar, “Comments on ‘New LMI Condition for the Nonexistence of Overflow Oscillations in 2-D State-Space Digital Filters Using Saturation Arithmetic’,” Digital Signal Process, Vol. 18, No. 2, 2008, pp. 148-150. doi:10.1016/j.dsp.2007.02.001

[13] C. Du, L. Xie and C. Zhang, “

[14] R. Yang, L. Xie and C. Zhang, “

[15] 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 Process, Vol. 90, No. 9, 2010, pp. 2648-2654. doi:10.1016/j.sigpro.2010.03.008

[16] D. Yue and J. Lam, “Non-Fragile Guaranteed Cost Control for Uncertain Descriptor Systems with Time-Varying State and Input Delays,” Optimal Control Applications & Methods, Vol. 26, 2005, pp. 85-105. doi:10.1002/oca.753

[17] L. H. Keel and S. P. Bhattacharya, “Robust, Fragile or Optimal,” IEEE Transactions on Automatic Control, Vol. 42, No. 8, 1997, pp. 1098-1105. doi:10.1109/9.618239

[18] 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, No. 15, 2000, pp. 1405-1423. doi:10.1080/002071700445424

[19] 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

[20] S. Xu, J. Lam, G. Yang and J. Wang, “Stabilization and

[21] 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

[22] C. Lien, “

[23] C. Lien, “Non-Fragile Guaranteed Cost Control for Uncertain Neutral Dynamic Systems with Time-Varying Delays in State and Control Input,” Chaos, Solitons and Fractals, Vol. 31, No. 4, 2007, pp. 889-899. doi:10.1016/j.chaos.2005.10.080

[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, MA, 1995.