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Stability analysis of delayed Takagi-Sugeno fuzzy systems: a new integral inequality approach. (English) Zbl 1412.37017
Summary: This paper is concerned with the problem of the stability analysis for Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delay. The delay is assumed to be differential with interval bounds, and has both the lower and upper bounds of the delay derivatives, in which the upper bound of delay derivative may be greater than one. By constructing some delaydependent Lyapunov functions, some stability criteria are derived by using the convex optimization method and new integral inequality techniques. Utilizing integral inequalities for quadratic functions plays a key role in the field of stability analysis for delayed T-S fuzzy systems, and some integral inequalities for quadratic functions are derived and employed in order to produce tighter bounds than what the Jensen inequality and Wirtinger-based inequality produce. Then, less conservative stability criteria are derived by using convex combination method and improved integral inequalities based on appropriate Lyapunov-Krasovskii (LK) functional. Finally, several examples are given to show the advantages of the proposed results.
MSC:
37B25 Stability of topological dynamical systems
93C42 Fuzzy control/observation systems
93D20 Asymptotic stability in control theory
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[1] An, J.-Y.; Li, T.; Wen, G.-L.; Li, R.-F., New stability conditions for uncertain TS fuzzy systems with interval time-varying delay, Int. J. Control Autom. Syst., 10, 490-497, (2012)
[2] An, J.-Y.; Li, T.; Wen, G.-L.; Li, R.-F., Improved stability criteria for time-varying delayed T-S fuzzy systems via delay partitioning approach, Fuzzy Sets and Systems, 185, 83-94, (2011) · Zbl 1237.93156
[3] An, J.-Y.; Wen, G.-L.; Lin, C.; Li, R.-F., New results on delay-derivative- dependent fuzzy \(H^\infty\) filter design for T-S fuzzy systems, IEEE Trans. Fuzzy Syst., 19, 770-779, (2011)
[4] An, J.-Y.; Wen, G.-L.; Xu, W., Improved results on fuzzy \(H_\infty\) filter design for T-S fuzzy systems, Discrete Dyn. Nat. Soc., 2010, 1-21, (2010) · Zbl 1205.93040
[5] Briat, C., Convergence and equivalence results for the Jensen’s inequality—application to time-delay and sampled-data systems, IEEE Trans. Automat. Control, 56, 1660-1665, (2011) · Zbl 1368.26020
[6] 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
[7] Faria, F. A.; Silva, G. N.; Oliverira, V. A., Reducing the conservatism of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy Lyapunov functions, Internat. J. Systems Sci., 44, 1956-1969, (2013) · Zbl 1307.93218
[8] Gu, K.; Kharitonov, V. L.; Chen, J., Stability of time-delay systems, Control Engineering, Birkhäuser Boston, Inc., Boston, MA, (2003) · Zbl 1039.34067
[9] Hale, J. K.; Lunel, S. M. Verduyn, Introduction to functional-differential equations, Applied Mathematical Sciences, Springer-Verlag, New York, (1993) · Zbl 0787.34002
[10] Kwon, O. M.; Park, M. J.; Lee, S. M.; Park, J. H., Augmented Lyapunov-Krasovskii functional approaches to robust stability criteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delays, Fuzzy Sets and Systems, 201, (2012) · Zbl 1251.93072
[11] Lee, D. H., Relaxed LMI conditions for local stability and local stabilization of continuous-time TakagiSugeno fuzzy systems, IEEE Trans. Cybern., 44, 394-405, (2014)
[12] Lien, C.-H.; Yu, K.-W.; Chen, W.-D.; Wan, Z.-L.; Chung, Y.-J., Stability criteria for uncertain Takagi-Sugeno fuzzy systems with interval time-varying delay, IET Control Theory Appl., 1, 764-769, (2007)
[13] Liu, F.; Liu, M.; He, Y.; Yokoyama, R., New delay-dependent stability criteria for T-S fuzzy systems with time-varying delay, Fuzzy Sets and Systems., 161, 2033-2042, (2010) · Zbl 1194.93117
[14] Moon, Y. S.; Park, P. G.; Kwon, W. H.; Lee, Y. S., Delay-dependent robust stabilization of uncertain state-delayed systems, Internat. J. Control, 74, 1447-1455, (2001) · Zbl 1023.93055
[15] Narimani, M.; Lam, H.-K.; Dilmaghani, R.; Wolfe, C., LMI-based stability analysis of fuzzy-model-based control systems using approximated polynomial membership functions, IEEE Trans. Syst., Man, Cybern. B, 41, 713-724, (2011)
[16] Park, P. G.; Ko, J. W.; Jeong, C.-K., Reciprocally convex approach to stability of systems with time-varying delays, Automatica J. IFAC, 47, 235-238, (2011) · Zbl 1209.93076
[17] Park, P. G.; Lee, W. I.; Lee, S. Y., Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Franklin Inst., 352, 1378-1396, (2015) · Zbl 1395.93450
[18] Peng, C.; M.-R. Fei, An improved result on the stability of uncertain T-S fuzzy systems with interval time-varying delay, Fuzzy Sets and Systems, 212, 97-109, (2013) · Zbl 1285.93054
[19] Peng, C.; Fei, M.-R.; Tian, E.-G., Networked control for a class of T-S fuzzy systems with stochastic sensor faults, Fuzzy Sets and Systems, 212, 62-77, (2013) · Zbl 1285.93055
[20] Peng, C.; Han, Q.-L., Delay-range-dependent robust stabilization for uncertain T-S fuzzy control systems with interval time-varying delays, Inform. Sci., 181, 4287-4299, (2011) · Zbl 1242.93067
[21] Peng, C.; Wen, L.-Y.; Yang, J.-Q., On delay-dependent robust stability criteria for uncertain T-S fuzzy systems with interval time-varying delay, Int. J. Fuzzy Syst., 13, 35-44, (2011)
[22] Qiu, J.-B.; Feng, G.; Gao, H.-J., Fuzzy-model-based piecewise \(H_\infty\) static-output-feedback controller design for networked nonlinear systems, IEEE Trans. Fuzzy Syst., 18, 919-934, (2010)
[23] Seuret, A.; Gouaisbaut, F., Wirtinger-based integral inequality: application to time-delay systems, Automatica J. IFAC, 49, 2860-2866, (2013) · Zbl 1364.93740
[24] Seuret, A.; Gouaisbaut, F., Complete quadratic Lyapunov functionals using Bessel-Legendre inequality, Proceedings of European Control Conference, 448-453, (2014)
[25] Souza, F. O.; Campos, V. C. S.; Palhares, R. M., On delay-dependent stability conditions for Takagi-Sugeno fuzzy systems, J. Franklin Inst., 351, 3707-3718, (2014) · Zbl 1290.93130
[26] Takagi, T.; Sugeno, M., Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst., Man, Cybern., 1, 116-132, (1985) · Zbl 0576.93021
[27] Tian, E.-G.; Yue, D.; Zhang, Y.-J., Delay-dependent robust \(H_\infty\) control for T-S fuzzy system with interval time-varying delay, Fuzzy Sets and Systems, 160, 1708-1719, (2009) · Zbl 1175.93134
[28] Wu, L.-G.; Su, X.-J.; Shi, P.; Qiu, J.-B., A new approach to stability analysis and stabilization of discrete-time TS fuzzy time-varying delay systems, IEEE Trans. Syst., Man, Cybern. B, 41, 273-286, (2011)
[29] Xie, X.-P.; Hu, S.-L., Relaxed stability criteria for discrete-time TakagiSugeno fuzzy systems via new augmented nonquadratic Lyapunov functions, Neurocomputing, 166, 416-421, (2015)
[30] Xie, X.-P.; Weng, S.-X.; Zhang, H.-F., Reducing the conservatism of stability analysis for discrete-time TS fuzzy systems based on a delayed Lyapunov function, Neurocomputing, 171, 1139-1145, (2016)
[31] Yang, J.; Luo, W.-P.; Shi, K.-B.; Zhao, X., Robust stability analysis of uncertain T-S fuzzy systems with time-varying delay by improved delay-partitioning approach, J. Nonlinear Sci. Appl., 9, 171-185, (2016) · Zbl 1327.93254
[32] Zeng, H.-B.; Park, J. H.; Xia, J.-W.; Xiao, S.-P., Improved delay-dependent stability criteria for T-S fuzzy systems with time-varying delay, Appl. Math. Comput., 235, 492-501, (2014) · Zbl 1334.93110
[33] Zhang, X.-M.; Han, Q.-L., Novel delay-derivative-dependent stability criteria using new bounding techniques, Internat. J. Robust Nonlinear Control, 23, 1419-1432, (2013) · Zbl 1278.93230
[34] Zhang, Z.-Y.; Lin, C.; Chen, B., New stability and stabilization conditions for T-S fuzzy systems with time delay, Fuzzy Sets and Systems, 263, 82-91, (2015) · Zbl 1361.93031
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