Delay-dependent stabilization for stochastic fuzzy systems with time delays. (English) Zbl 1122.93051

Summary: This paper is concerned with the delay-dependent stabilization problem for a class of time-delay stochastic fuzzy systems. The time delays are assumed to appear in both the state and the control input. The purpose is the design of a state-feedback fuzzy controller such that the resulting closed-loop system is asymptotically stable in the mean square. A delay-dependent condition for the solvability of this problem is obtained in terms of relaxed linear matrix inequalities (LMIs). By solving these LMIs, a desired controller can be obtained. Finally, a numerical example is given to demonstrate the effectiveness of the present results.


93C42 Fuzzy control/observation systems
93E03 Stochastic systems in control theory (general)
93E15 Stochastic stability in control theory


LMI toolbox
Full Text: DOI


[1] Cao, S.G.; Rees, N.W.; Feng, G., Analysis and design of a class of continuous time fuzzy control systems, Int. J. control, 64, 1069-1087, (1996) · Zbl 0867.93053
[2] Cao, Y.Y.; Frank, P.M., Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach, IEEE trans. fuzzy systems, 8, 2, 200-211, (2000)
[3] Cao, Y.Y.; Frank, P.M., Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-sugeno fuzzy models, Fuzzy sets and systems, 124, 213-229, (2001) · Zbl 1002.93051
[4] Cao, Y.Y.; Lin, Z., Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation, IEEE trans. fuzzy systems, 11, 1, 57-67, (2003)
[5] Chen, B.; Zhang, W., Stochastic \(H_2 / H_\infty\) control with state-dependent noise, IEEE trans. automat. control, 49, 1, 45-57, (2004) · Zbl 1365.93539
[6] Chen, W.-H.; Guan, Z.-H.; Lu, X., Delay-dependent robust stabilization and \(H_\infty\)-control of uncertain stochastic systems with time-varying delay, IMA J. math. control inform., 21, 3, 345-358, (2004) · Zbl 1053.93012
[7] Gahinet, P.; Nemirovskii, A.; Laub, A.J.; Chilali, M., LMI control toolbox User’s guide, (1995), The Math. Works Inc. Natick, MA
[8] H. Huang, D.W.C. Ho, Y. Niu, Sliding mode based fuzzy control of stochastic systems with time delay, in: Proc. 6th World Congress on Intelligent Control and Automation, Dalian, China, 2006, pp. 295-299.
[9] Kim, E.; Lee, H., New approaches to relaxed quadratic stability condition of fuzzy control systems, IEEE trans. fuzzy systems, 8, 5, 523-534, (2000)
[10] Li, C.; Wang, H.; Liao, X., Delay-dependent robust stability of uncertain fuzzy systems with time-varying delays, IEE proc. control theory appl., 151, 417-421, (2004)
[11] Lien, C.H., Further results on delay-dependent robust stability of uncertain fuzzy systems with time-varying delay, Chaos solitons fractals, 28, 422-427, (2006) · Zbl 1091.93022
[12] Lin, C.; Wang, Q.G.; Lee, T.H., Improvement on observer-based \(H_\infty\) control for T-S fuzzy systems, Automatica, 41, 1651-1656, (2005) · Zbl 1093.93016
[13] Lin, C.; Wang, Q.G.; Lee, T.H., Delay-dependent LMI conditions for stability and stabilization of T-S fuzzy systems with bounded time delay, Fuzzy sets systems, 157, 1229-1247, (2006) · Zbl 1090.93024
[14] H. Liu, K. He, F. Sun, Z. Sun, Analysis and synthesis of fuzzy stochastic systems via LMI approach, in: Proc. 17th IEEE Region 10 Internat. Conf. on Computers, Communications, Control and Power Engineering, Beijing, China, 2002, pp. 1700-1703
[15] Liu, X.; Zhang, Q.L., New approaches to controller designs based on fuzzy observers for T-S fuzzy systems via LMI, Automatica, 39, 1571-1582, (2003) · Zbl 1029.93042
[16] Mao, X., Robustness of exponential stability of stochastic differential delay equations, IEEE trans. automat. control, 41, 442-447, (1996) · Zbl 0851.93074
[17] Mao, X., Stochastic differential equations and applications, (1997), Horwood West Sussex, UK · Zbl 0874.60050
[18] Mao, X.; Koroleva, N.; Rodkina, A., Robust stability of uncertain stochastic differential delay equations, Systems control lett., 35, 325-336, (1998) · Zbl 0909.93054
[19] Takagi, T.; Sugeno, M., Fuzzy identification of systems and its application to modeling and control, IEEE trans. systems man cybernet, 15, 1, 116-132, (1985) · Zbl 0576.93021
[20] Tanaka, K.; Ikeda, T.; Wang, H.O., Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs, IEEE trans. fuzzy systems, 6, 2, 250-265, (1998)
[21] Tanaka, K.; Sugeno, M., Stability analysis and design of fuzzy control systems, Fuzzy sets systems, 45, 135-156, (1992) · Zbl 0758.93042
[22] Tian, E.; Peng, C., Delay-dependent stability analysis and synthesis of uncertain T-S fuzzy systems with time-varying delay, Fuzzy sets systems, 157, 544-559, (2006) · Zbl 1082.93031
[23] Wang, Z.; Ho, D.W.C.; Liu, X., A note on the robust stability of uncertain stochastic fuzzy systems with time-delays, IEEE trans. systems man cybernet A, 34, 4, 570-576, (2004)
[24] Xu, S.; Chen, T., Reduced-order \(H_\infty\) filtering for stochastic systems, IEEE trans. signal processing, 50, 12, 2998-3007, (2002) · Zbl 1369.94325
[25] Xu, S.; Chen, T., Robust \(H_\infty\) control for uncertain stochastic systems with state delay, IEEE trans. automat. control, 47, 12, 2089-2094, (2002) · Zbl 1364.93755
[26] Xu, S.; Chen, T., \(H_\infty\) output feedback control for uncertain stochastic systems with time-varying delays, Automatica, 40, 2091-2098, (2004) · Zbl 1073.93022
[27] Xu, S.; Lam, J.; Mao, X.; Zou, Y., A new LMI condition for delay-dependent robust stability of stochastic time-delay systems, Asian J. control, 7, 4, 419-423, (2005)
[28] Yi, Z.; Heng, P.A., Stability of fuzzy control systems with bounded uncertain delays, IEEE trans. fuzzy systems, 10, 92-97, (2002) · Zbl 1142.93377
[29] Yuan, C.; Mao, X., Robust stability and controllability of stochastic differential delay equations with Markovian switching, Automatica, 40, 3, 343-354, (2004) · Zbl 1040.93069
[30] Zhou, S.; Feng, G.; Lam, J.; Xu, S., Robust \(H_\infty\) control for discrete-time fuzzy systems via basis-dependent Lyapunov functions, Inform. sci., 174, 3-4, 197-217, (2005) · Zbl 1113.93038
[31] Zhou, S.; Li, T., Robust stabilization for delayed discrete-time fuzzy systems via basis-dependent lyapunov – krasovskii function, Fuzzy sets systems, 151, 139-153, (2005) · Zbl 1142.93379
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.