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Stabilization for T-S model based uncertain stochastic systems. (English) Zbl 1209.93163
Summary: This paper considers the stabilization problem of a class of uncertain Itô stochastic fuzzy systems driven by a multidimensional Wiener process. The uncertainty modeled in the systems is of linear fractional type which includes the norm-bounded uncertainty as a special case. The objective is to design a state-feedback fuzzy controller such that the closed-loop system is robustly asymptotically stable under a stochastic setting. By using a stochastic Lyapunov approach, sufficiency conditions for the stability and stabilization of this class of systems are established based on a novel matrix decomposition technique. The derived stability conditions are then employed to design controllers which stabilize the uncertain Itô stochastic fuzzy systems. Two simulation examples are given to illustrate the effectiveness of the approaches proposed.

93E15 Stochastic stability in control theory
93C42 Fuzzy control/observation systems
93C41 Control/observation systems with incomplete information
Full Text: DOI
[1] Balachandran, K.; Karthikeyan, S., Controllability of nonlinear Itô type stochastic integrodifferential systems, Journal of the franklin institute, 345, 382-391, (2008) · Zbl 1151.93005
[2] Dong, J.; Yang, G., State feedback control of continuous-time T-S fuzzy systems via switched fuzzy controllers, Information sciences, 178, 1680-1695, (2008) · Zbl 1139.93018
[3] El Ghaoui, L.; Scorletti, G., Control of rational systems using linear fractional representations and linear matrix inequalities, Automatica, 32, 9, 1273-1284, (1996) · Zbl 0857.93040
[4] Handel, R.V.; Stockton, J.K.; Mabuchi, H., Feedback control of quantum state reduction, IEEE transactions on automatic control, 50, 6, 768-780, (2005) · Zbl 1365.81068
[5] Higham, D., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM review, 43, 525-546, (2001) · Zbl 0979.65007
[6] Hu, L.; Yang, A., Fuzzy model-based control of nonlinear stochastic systems with time-delay, Nonlinear analysis, 71, 2855-2865, (2009)
[7] Lee, H.; Park, J.; Chen, G., Robust fuzzy control of nonlinear systems with parametric uncertainties, IEEE transactions on fuzzy systems, 9, 369-379, (2001)
[8] Lee, H.J.; Kim, D.W., Robust stabilization of T-S fuzzy systems: fuzzy static output feedback under parametric uncertainty, International journal of control, automation, and systems, 7, 5, 731-736, (2009)
[9] Lee, K.R.; Jeung, E.T.; Park, H.B., Robust fuzzy H∞ control for uncertain nonlinear systems via state feedback: an LMI approach, Fuzzy sets and systems, 120, 123-134, (2001) · Zbl 0974.93546
[10] Li, L.; Liu, X., New results on delay-dependent robust stability criteria of uncertain fuzzy systems with state and input delays, Information sciences, 179, 1134-1148, (2009) · Zbl 1156.93354
[11] Liu, P.; Li, H., Approximation of stochastic processes by T-S fuzzy systems, Fuzzy sets and systems, 155, 215-235, (2005) · Zbl 1140.93414
[12] Liu, X.; Zhang, Q., Approaches to quadratic stability conditions and H∞ control designs for T-S fuzzy systems, IEEE transactions on fuzzy systems, 11, 6, 830-839, (2003)
[13] Liu, Y.; Zhang, J.-F., Reduced-order observer-based control design nonlinear stochastic systems, Systems & control letters, 52, 123-135, (2004) · Zbl 1157.93538
[14] Mansouri, B.; Manamanni, N.; Guelton, K.; Kruszewski, A.; Guerra, T.M., Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed T-S models, Information sciences, 179, 446-457, (2009) · Zbl 1158.93024
[15] Mao, X., Stochastic differential equations and applications, (1997), Horwood Publication Chichester · Zbl 0874.60050
[16] Niu, Y.; Ho, D.W.C.; Lam, J., Robust integral sliding mode control for uncertain stochastic systems with time-varying delay, Automatica, 41, 873-880, (2005) · Zbl 1093.93027
[17] Ting, C., Stability analysis and design of takagi – sugeno fuzzy systems, Information sciences, 176, 2817-2845, (2006) · Zbl 1097.93023
[18] Ugrinovskii, V.A.; Petersen, I.R., Robust stability and performance of stochastic uncertain systems on an infinite time interval, Systems & control letters, 44, 291-308, (2001) · Zbl 0986.93052
[19] Wang, W.; Sun, C.H., Relaxed stability and stabilization conditions for a T-S fuzzy discrete system, Fuzzy sets and systems, 156, 2, 208-225, (2005) · Zbl 1082.93035
[20] Xie, L., Output feedback H∞ control of systems with parameter uncertainty, International journal of control, 63, 741-750, (1996) · Zbl 0841.93014
[21] Xu, S.; Chen, T., Robust H∞ control for uncertain stochastic systems with state delay, IEEE transactions automatic control, 47, 2089-2094, (2002) · Zbl 1364.93755
[22] Xu, S.; Lam, J.; Mao, X., Delay-dependent H∞ control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE transactions on circuit and systems-I: regular papers, 9, 2070-2077, (2007) · Zbl 1374.93134
[23] Yoneyama, J., Robust stability and stabilizing controller design of fuzzy systems with discrete and distributed delays, Information sciences, 178, 1935-1947, (2008) · Zbl 1144.93018
[24] Zhang, B.; Lam, J.; Xu, S.; Shu, Z., Robust stabilization of uncertain T-S fuzzy time-delay systems with exponential estimates, Fuzzy sets and systems, 160, 1720-1737, (2009) · Zbl 1175.93200
[25] Zhang, B.; Xu, S.; Zong, G.; Zou, Y., Delay-dependent stabilization for stochastic fuzzy systems with time delays, Fuzzy sets and systems, 158, 2238-2250, (2007) · Zbl 1122.93051
[26] Zhou, S.; Feng, G.; Lam, J.; Xu, S., Robust H∞ control for discrete-time fuzzy systems via basis-dependent Lyapunov functions, Information sciences, 174, 3-4, 197-217, (2005) · Zbl 1113.93038
[27] Zhou, S.; Lam, J., Control design for fuzzy systems based on relaxed nonquadratic stability and H∞ performance conditions, IEEE transactions on fuzzy systems, 15, 2, 188-199, (2007)
[28] Zhou, S.; Lam, J., Robust stabilization of delayed singular systems with linear fractional parametric uncertainties, Circuits systems signal precessing, 22, 6, 579-588, (2003) · Zbl 1045.93042
[29] Zhou, S.; Li, T., Robust stabilization for delayed discrete-time fuzzy systems via basis-dependent lyapunov – krasovskii function, Fuzzy sets and systems, 151, 139-153, (2005) · Zbl 1142.93379
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