Abstract
This paper deals with the delay-dependent stabilization problem for singular systems with Markovian jump parameters and time delays. A delay-dependent condition is established for the considered system to be regular, impulse free and stochastically stable. Based on the condition, a design algorithm of the desired state feedback controller which guarantees the resultant closed-loop system to be regular, impulse free and stochastically stable is proposed in terms of a set of strict linear matrix inequalities (LMIs). Numerical examples show the effectiveness of the proposed methods.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Xu, J. Lam. Robust Control and Filtering of Singular Systems[M]. Berlin: Springer-Verlag, 2007.
L. Dai. Singular Control Systems[M]. Berlin: Springer-Verlag, 1989.
C. Lin, Q. Wang, T. H. Lee. Stability and stabilization of a class of fuzzy time-delay descriptor systems[J]. IEEE Transsactions on Fuzzy Systems, 2006, 14(4): 542–551.
S. Zhu, C. Zhang, Z. Cheng, et al. Delay-dependent robust stability criteria for two classes of uncertain singular time-delay systems[J]. IEEE Transsactions on Automatic Control, 2007, 52(5): 880–885.
S. Xu, J. Lam. Robust stability for uncertain discrete singular systems with state delay[J]. Asian Journal of Control, 2003, 5(3): 399–405.
S. Ma, Z. Cheng, C. Zhang. Delay-dependent robust stability and stabilisation for uncertain discrete singular systems with time-varying delays[J]. IET Control Theory & Applications, 2007, 1(4): 1086–1095.
W. Chen, Z. Guan, X. Lu. Delay-dependent output feedback stabilisation of Markovian jump system with time-delay[J]. IEEE Proceedings-Control Theory & Applications, 2004, 151(5): 561–566.
J. Wu, T. Chen, L. Wang. Delay-dependent robust stability and H ∞ control for jump linear systems with delays[J]. Systems & Control Letters, 2006, 55(11): 939–948.
L. Hu, P. Shi, Y. Cao. Delay-dependent filtering design for time-delay systems with Markovian jumping parameters[J]. International Journal of Adaptive Control and Signal Processing, 2007, 21(5):434–448.
E. K. Boukas. On state feedback stabilization of singular systems with random abrupt changes[J]. Journal of Optimization Theory and Applications, 2008, 137(2): 335–345.
J. Lam, Z. Shu, S. Xu, et al. Robust H∞ control of descriptor discrete-time Markovian jump systems[J]. International Journal of Control, 2007, 80(3): 374–385.
E. K. Boukas, S. Xu, J. Lam. On stability and stabilizability of singular stochastic systems with delays[J]. Journal of Optimization Theory and Applications, 2005, 127(2): 249–262.
E. K. Boukas, Z. Liu. Delay-dependent stabilization of singularly perturbed jump linear systems[J]. International Journal of Control, 2004, 77(3): 310–319.
K. Gu, V. L. Kharitonov, J. Chen. Stability of Time-Delay Systems[M]. Boston: Birkhäuser, 2003.
Y. He, M. Wu, J. She, et al. Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays[J]. Systems & Control Letters, 2004, 51(1): 57–65.
M. Wu, Y. He, J. She, et al. Delay-dependent criteria for robust stability of time-varying delay systems[J]. Automatica, 2004, 40(8):1435–1439.
Y. He, M. Wu, J. She, et al. Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties[J]. IEEE Transactions Automatic Control, 2004, 49(5):828–832.
H. Gao, J. Lam, T. Chen, et al. Stability analysis of uncertain discrete-time systems with time-varying state delay: a parameter-dependent Lyapunov function approach[J]. Asian Journal of Control, 2006, 8(4):433–440.
X. Zhang, Q. Han. A new finite sum inequality approach to delay-dependent H∞ control of discrete-time systems with time-varying delay[J]. International Journal of Robust and Nonlinear Control, 2008, 18(6): 630–647.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Creative Research Groups Science Foundation of China (No.60721062), the National High Technology Research and Development Program of China (863 Program) (2006AA04 Z182) and the National Natural Science Foundation of China (No.60736021).
Zhengguang WU was born in 1982. He received the M.E. degree from Zhejiang Normal University in 2007. Now he is a doctoral student in Institute of Cyber-Systems and Control, Zhejiang University. His research interests include robust control and system theory.
Hongye SU was born in 1969. He received a first class B.S. degree in Industrial Automation from Nanjin University of Chemical Technology in 1990. He obtained the M.S. and Ph.D. degrees from Zhejiang University in 1993 and 1995 respectively. From 1995 to 1997, he was a lecturer in Department of Chemical Engineering, Zhejiang University. From 1998 to 2000, he was an associate professor in Institute of Advanced Process Control, Zhejiang University. Now he is a professor in Institute of Cyber-Systems and Control, Zhejiang University. His recent research interests include robust control, time-delay systems, advanced process control theory and application.
Jian CHU was born in 1963. He received the B.E. and M.S. degree from Zhejiang University in 1982 and 1984, respectively. From 1986 to 1989, he studied in Kyoto University, Japan, and finished his Ph.D. thesis there. Then, he came back to Zhejiang University for postdoctoral research in Institute of Advanced Process Control, Zhejiang University. In 1991, he was promoted as an associate professor, and promoted as a full professor in 1993. In 1994 he was approved to be a doctorial advisor. He is interested in control theory and application, R & D of computer control systems and advanced process control software.
Rights and permissions
About this article
Cite this article
Wu, Z., Su, H. & Chu, J. Delay-dependent stabilization of singular Markovian jump systems with state delay. J. Control Theory Appl. 7, 231–236 (2009). https://doi.org/10.1007/s11768-009-8145-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11768-009-8145-8