Abstract
The problem of observer-based robust control design is studied for discrete-time singular systems with norm-bounded uncertainties and a time-varying delay. More precisely, a delay-dependent criterion is established that guarantees the admissibility of the considered systems, without resorting to its decomposition. Based on the proposed criterion and without the assumption that the considered systems are admissible, robust observer-based controllers are designed for discrete-time singular time-delay systems such that the closed-loop systems have the characteristics of regularity, causality and asymptotic stability. Seeking computational convenience, all the developed results are cast in the format of strict linear matrix inequalities (LMIs). Finally, some numerical examples are presented to show the feasibility of the proposed approach.
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References
Y. Xia, M. Fu, and P. Shi, Analysis and Synthesis of Dynamical Systems with Time-Delays, Springer-Verlag, Berlin, 2009.
K. Gu, V. Kharitonov, and J. Chen, Stability of Time Delay Systems, Springer-Verlag, Berlin, 2003.
J. Chen, C. Yang, C. Lien, and J. Horng, “New delay-dependent non-fragile H∞ observer-based control for continuous time-delay systems,” Information Sciences, vol. 178, pp. 4699–4706, 2008.
M. Kchaou, A. Toumi, and M. Souissi, “Robust reliable guaranteed cost piecewise fuzzy control for discrete-time nonlinear systems with time-varying delay and actuator failures,” International Journal of General Systems, vol. 40, no. 5, pp. 531–558, 2011.
J. Sun, G. Liu, J. Chena, and D. Rees, “Improved delay-range-dependent stability criteria for linear systems with time-varying delays,” Automatica, vol. 46, pp. 466–470, 2010.
H. Shao, “Improved delay-dependent stability criteria for systems with a delay varying in a range,” Automatica, vol. 44, no. 12, pp. 3215–3218, 2008.
L. Dai, Singular Control Systems, vol. 118 of Lecture Notes in Control and Information Sciences, Springer, New York, 1989.
P. Muller, “Linear mechanical descriptor systems: identification, analysis and design,” Proc. of Conference on Control of Independent Systems, pp. 501–506, IFAC, Belfort, France, 1997.
S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, Springer-Verlag, 2006.
G. Duan, Analysis and Design of Descriptor Linear Systems, Springer Verlag, New York, 2010.
E. Boukas, “Singular linear systems with delay: H∞ stabilization,” Optimal Control Applications and Methods, vol. 28, pp. 259–274, 2007.
R. Zhong and Z. Yang, “Delay-dependent robust control of descriptor systems with time delay,” Asian Journal of Control, vol. 8, no. 1, pp. 36–44, 2008.
M. Kchaou, M. Souissi, and A. Toumi, “Delay dependent stability and robust L2-L∞ control for a class of fuzzy descriptor systems with time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 23, pp. 284–304, 2013.
S. Ma, C. Zhang, and Z. Wu, “Delay-dependent stability and H∞ control for uncertain discrete switched singular systems with time-delay,” Applied Mathematics and Computation, vol. 206, pp. 413–424, 2008.
S. Chen and J. Chou, “Robust D-stability analysis for linear uncertain discrete singular systems with state delay,” Applied Mathematics Letters, vol. 19, pp. 197–205, 2006.
C. Huang, “Stability analysis of discrete singular fuzzy systems,” Fuzzy Sets and Systems, vol. 151, pp. 155–165, 2005.
J. Kim, “New reduced-order H∞ filter design method for discrete-time singular systems with lossy measurements by strict LMI approach,” International Journal of Control, Automation, and Systems, vol. 9, pp. 1095–1102, 2011.
J. Zhao, Z. Hu, and L. Zhang, “Stability and stabilization for discrete-time singular systems with infinite distributed delays and actuator failures,” International Journal of Control, Automation, and Systems, vol. 10, pp. 721–726, 2012.
J. Feng, P. Cui, and Z. Cheng, “H∞ output feedback control for descriptor systems with delayed states,” Journal of Control Theory and Applications, vol. 4, pp. 341–347, 2005.
J. Feng, S. Zhu, and Z. Cheng, “Observer design for linear singular time-delay systems,” Proc. of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii USA, December 2003.
Z. Wu, H. Su, and J. Chu, “Output feedback stabilization for discrete singular systems with random abrupt changes,” International Journal of Robust and Nonlinear Control, vol. 20, pp. 1945–1957, 2010.
Z. Wang and H. Zhang, “Design of a bilinear fault detection observer for singular bilinear systems,” Journal of Control Theory and Applications, vol. 5, no. 1, pp. 28–36, 2007.
D. Koenig, B. Marx, and O. Sename, “Unknown inputs proportional integral observers for descriptor systems with multiple delays and unknown inputs,” Proc. of the American Control Conference, Boston. Massachusetts, June 2004.
Z. Wang, Y. Shen, X. Zhang, and Q. Wang, “Observer design for discrete-time descriptor systems: an LMI approach,” Systems & Control Letters, vol. 61, pp. 683–687, 2012.
S. Ma, Z. Cheng, and C. Zhang, “Delay-dependent robust stability and stabilization for uncertain discrete singular systems with time-varying delay,” IET Control Theory & Applications, vol. 1, no. 4, pp. 1086–1095, 2007.
I. Petersen, “A stabilization algorithm for a class of uncertain linear systems,” Systems & Control Letters, vol. 8, pp. 35–357, 1987.
X. Zhu, Y. Wang, and G. Yang, “New delay dependent stability results for discrete-time recurrent neural networks with time-varying delay,” Neurocomputing, vol. 72, pp. 3376–3383, 2009.
S. Ma, E. Boukas, and Y. Chinniah, “Stability and stabilization of discrete-time singular Markov jump systems with time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 20, pp. 531–543, 2010.
H. Wang, X. Zhao, A. Xue, and R. Lu, “Delay dependent robust control for uncertain discrete singular systems with time-varying delay,” Journal of Zhejiang University Science A, vol. 9, no. 8, pp. 1034–1042, 2008.
M. Fang, “Delay-dependent stability analysis for discrete singular systems with time-varying delays,” Acta Automatica Sinica, vol. 36, no. 5, pp. 751–755, 2010.
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Recommended by Associate Editor Guang-Hong Yang under the direction of Editor Hyungbo Shim.
Mourad Kchaou received his Mastery of Sciences from (ENSET/Tunisia) in 1993. In 2004, he obtained his Master degree and the Doctoral Thesis in 2009 in Automatic and Industrial Computing from Engineering National School of Sfax (ENIS). In 2013, he has obtained the University Habilitation (HDR) from (ENIS). He is now an Associate Professor in the Electronic Engineering department of High Institute of Applied Sciences and Technology of Sousse. His main research interests include fuzzy control, time-delay systems, descriptor systems, with particular attention paid to nonlinear systems represented by multiple-models.
Fernando Tadeo is a Professor of the School of Engineering at the University of Valladolid. He graduated from the same university, in Physics in 1992, and in Electronic Engineering in 1994. After completing an M.Sc. in Control Engineering in the University of Bradford, U.K., he went back to Valladolid, where he got his Ph.D. degree. His main interest area is Feedback Systems with Constraints, in particular Positive Systems, collaborating with teams in Morocco, Tunisia, France and Germany. This research is focused on applications in Process Control (Desalination and Electrolyzation Processes) and Renewable Energies (Wind, Solar and Salinity Gradient).
Mohamed Chaabane was born in Sfax, Tunisia, on August 26, 1961. He received the Ph.D. degree in Electrical Engineering from the University of Nancy, French in 1991. He was associate professor at the University of Nancy and is a researcher at Center of Automatic Control of Nancy (CRAN) from 1988–1992. Actually he is a professor in ENI-Sfax and editor in chief of the International Journal on Sciences and Techniques of Automatic Control and Computer Engineering IJSTA. Since 1997, he is holding a research position at Automatic Control Unit, ENIS. The main research interests are in the filed of robust control, delay systems, descriptor systems and applications of theses techniques to fed-batch processes and agriculture systems. Currently, he is an associate editor of International Journal on Sciences and Techniques of Automatic Control & Computer Engineering, IJ-STA (www.sta-tn.com).
Ahmed Toumi received his Electrical Engineering Diploma from the Engineering National School of Sfax/Tunisia (ENIS) and the DEA (Master) in Instrumentation and Measurement from University of Bordeaux-1/France in 1981. He obtained in 1985 and 2000, respectively, the Doctoral Thesis in Physical Sciences from the University of Tunis, and the University Habilitation (HDR) in Electrical Engineering (Automatic Control) from (ENIS). He is now a Professor on Automatic Control, and the Team Leader of Research in Control of Industrial Processes at ENIS. Since 2002, he was the co-President of the international conference on Sciences and Techniques of Automatic control and computer engineering (STA) which has taken place in a number of tourist cities of Tunisia. His main research area concerns Process modelling, Stability, Delay systems, Singular systems, Fuzzy logic control, Robust control.
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Kchaou, M., Tadeo, F., Chaabane, M. et al. Delay-dependent robust observer-based control for discrete-time uncertain singular systems with interval time-varying state delay. Int. J. Control Autom. Syst. 12, 12–22 (2014). https://doi.org/10.1007/s12555-012-9311-1
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DOI: https://doi.org/10.1007/s12555-012-9311-1