Stability and stabilization of delayed fuzzy systems via a novel quadratic polynomial inequality. (English) Zbl 1501.93087
Summary: The stability and stabilization problems of a class of Takagi-Sugeno (T-S) fuzzy systems with time-varying delay are concerned in this paper. By introducing the tunable parameter \(l\), a novel quadratic polynomial inequality is proposed to reduce the estimation gap with previous work. Sequentially, an appropriate Lyapunov-Krasovskii functional (LKF) containing some new s-dependent integral terms is constructed, the delay-dependent stability condition is further improved by combining the generalized free-matrix-based integral inequality (GFMBII) with the proposed inequality. Finally, four typical numerical examples are carried out to demonstrate the validity and feasibility of the stability criterion and controller design method.
MSC:
| 93C42 | Fuzzy control/observation systems |
| 93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |
| 93C43 | Delay control/observation systems |
References:
| [1] | Zeng, K.; Zhang, N. Y.; Xu, W. L., A comparative study on sufficient conditions for Takagi-Sugeno fuzzy systems as universal approximators, IEEE Trans. Fuzzy Syst., 8, 6, 773-780 (2000) |
| [2] | Johansen, T. A.; Shorten, R.; Murray-Smith, R., On the interpretation and identification of dynamic Takagi-Sugeno fuzzy models, IEEE Trans. Fuzzy Syst., 8, 3, 297-313 (2000) |
| [3] | Liu, F.; Wu, M.; He, Y.; Yokoyama, R., New delay-dependent stability criteria for T-S fuzzy systems with time-varying delay, Fuzzy Sets Syst., 161, 15, 2033-2042 (2010) · Zbl 1194.93117 |
| [4] | 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 |
| [5] | Ku, C. C.; Chang, W. J.; Tsai, M. H.; Lee, Y. C., Observer-based proportional derivative fuzzy control for singular Takagi-Sugeno fuzzy systems, Inf. Sci. (Ny), 570, 815-830 (2021) · Zbl 1530.93239 |
| [6] | Cheng, J.; Huang, W.; Park, J. H.; Cao, J., A hierarchical structure approach to finite-time filter design for fuzzy Markov switching systems with deception attacks, IEEE Trans. Cybern. (2021) |
| [7] | Cheng, J.; Wang, Y.; Park, J. H.; Cao, J.; Shi, K., Static output feedback quantized control for fuzzy Markovian switching singularly perturbed systems with deception attacks, IEEE Trans. Fuzzy Syst., 30, 4, 1036-1047 (2021) |
| [8] | Zhao, Y.; Gao, H.; Lam, J.; Du, B., Stability and stabilization of delayed T-S fuzzy systems: a delay partitioning approach, IEEE Trans. Fuzzy Syst., 17, 4, 750-762 (2008) |
| [9] | Lian, Z.; He, Y.; Zhang, C. K.; Wu, M., Further robust stability analysis for uncertain Takagi-Sugeno fuzzy systems with time-varying delay via relaxed integral inequality, Inf. Sci. (Ny), 409, 139-150 (2017) · Zbl 1432.93269 |
| [10] | Zhang, X. M.; Han, Q. L.; Ge, X.; Zhang, B. L., Delay-variation-dependent criteria on extended dissipativity for discrete-time neural networks with time-varying delay, IEEE Trans. Neural Netw. Learn. Syst. (2021) |
| [11] | Zhou, W.; Fu, J.; Yan, H.; Du, X.; Wang, Y.; Zhou, H., Event-triggered approximate optimal path-following control for unmanned surface vehicles with state constraints, IEEE Trans. Neural Netw. Learn. Syst. (2021) |
| [12] | Cheng, J.; Liang, L.; Park, J. H.; Yan, H.; Li, K., A dynamic event-triggered approach to state estimation for switched memristive neural networks with nonhomogeneous sojourn probabilities, IEEE Trans. Circuits Syst. I Regul. Pap., 68, 12, 4924-4934 (2021) |
| [13] | Zeng, H. B.; Zhai, Z. L.; He, Y.; Teo, K. L.; Wang, W., New insights on stability of sampled-data systems with time-delay, Appl. Math. Comput., 374, 125041 (2020) · Zbl 1433.93110 |
| [14] | Tian, E.; Peng, C., Delay-dependent stability analysis and synthesis of uncertain T-S fuzzy systems with time-varying delay, Fuzzy Sets Syst., 157, 4, 544-559 (2006) · Zbl 1082.93031 |
| [15] | Lien, C.; Yu, K.; Chen, W.; Wan, Z.; Chung, Y., Stability criteria for uncertain Takagi-Sugeno fuzzy systems with interval time-varying delay, IET Control Theory Appl., 1, 3, 764-769 (2007) |
| [16] | Yang, Z.; Yang, Y. P., New delay-dependent stability analysis and synthesis of T-S fuzzy systems with time-varying delay, Int. J. Robust Nonlinear Control, 20, 3, 313-322 (2010) · Zbl 1185.93071 |
| [17] | Peng, C.; Fei, M. R., An improved result on the stability of uncertain T-S fuzzy systems with interval time-varying delay, Fuzzy Sets Syst., 212, 97-109 (2013) · Zbl 1285.93054 |
| [18] | Lian, Z.; He, Y.; Zhang, C. K.; Wu, M., Stability and stabilization of T-S fuzzy systems with time-varying delays via delay-product-type functional method, IEEE Trans. Cybern., 50, 6, 2580-2589 (2019) |
| [19] | Peng, C.; Tian, Y.; Tian, E., Improved delay-dependent robust stabilization conditions of uncertain T-S fuzzy systems with time-varying delay, Fuzzy Sets Syst., 159, 20, 2713-2729 (2008) · Zbl 1170.93344 |
| [20] | Tian, E.; Yue, D.; Zhang, Y., Delay-dependent robust \(h \infty\) control for T-S fuzzy system with interval time-varying delay, Fuzzy Sets Syst., 160, 12, 1708-1719 (2009) · Zbl 1175.93134 |
| [21] | Peng, C.; Han, Q. L., Delay-range-dependent robust stabilization for uncertain T-S fuzzy control systems with interval time-varying delays, Inf. Sci. (Ny), 181, 19, 4287-4299 (2011) · Zbl 1242.93067 |
| [22] | Souza, F. O.; Campos, V. C.S.; Palhares, R. M., On delay-dependent stability conditions for Takagi-Sugeno fuzzy systems, J. Frankl. Inst., 351, 7, 3707-3718 (2014) · Zbl 1290.93130 |
| [23] | Cheng, J.; Zhang, D.; Qi, W.; Cao, J.; Shi, K., Finite-time stabilization of T-S fuzzy semi-Markov switching systems: a coupling memory sampled-data control approach, J. Frankl. Inst., 357, 16, 11265-11280 (2020) · Zbl 1450.93054 |
| [24] | Xie, X.; Yue, D.; Peng, C., Relaxed real-time scheduling stabilization of discrete-time Takagi-Sugeno fuzzy systems via an alterable-weights-based ranking switching mechanism, IEEE Trans. Fuzzy Syst., 26, 6, 3808-3819 (2018) |
| [25] | Hu, M. J.; Park, J. H.; Wang, Y. W., Stabilization of positive systems with time delay via the Takagi-Sugeno fuzzy impulsive control, IEEE Trans. Cybern. (2020) |
| [26] | Xie, X.; Yue, D.; Park, J. H., Enhanced switching stabilization of discrete-time Takagi-Sugeno fuzzy systems: reducing the conservatism and alleviating the online computational burden, IEEE Trans. Fuzzy Syst., 29, 8, 2419-2424 (2020) |
| [27] | Lian, Z.; He, Y.; Zhang, C. K.; Wu, M., Stability analysis for T-S fuzzy systems with time-varying delay via free-matrix-based integral inequality, Int. J. Control Autom. Syst., 14, 1, 21-28 (2016) |
| [28] | Li, Z.; Yan, H.; Zhang, H.; Sun, J.; Lam, H. K., Stability and stabilization with additive freedom for delayed Takagi-Sugeno fuzzy systems by intermediary-polynomial-based functions, IEEE Trans. Fuzzy Syst., 28, 4, 692-705 (2019) |
| [29] | Xie, X.; Yue, D.; Park, J. H.; Liu, J., Enhanced stabilization of discrete-time Takagi-Sugeno fuzzy systems based on a comprehensive real-time scheduling model, IEEE Trans. Syst. Man Cybern. Syst. (2020) |
| [30] | Chaibi, R.; El Aiss, H.; El Hajjaji, A.; Hmamed, A., Stability analysis and robust \(H \infty\) controller synthesis with derivatives of membership functions for T-S fuzzy systems with time-varying delay: input-output stability approach, Int. J. Control Autom. Syst., 18, 7, 1872-1884 (2020) |
| [31] | Lian, Z.; He, Y.; Wu, M., Stability and stabilization for delayed fuzzy systems via reciprocally convex matrix inequality, Fuzzy Sets Syst., 402, 124-141 (2021) · Zbl 1464.93039 |
| [32] | Huang, W., Generalization of Lypaunov’s theorem in a linear delay system, J. Math. Anal. Appl., 142, 1, 83-94 (1989) · Zbl 0705.34084 |
| [33] | Ge, X.; Ding, D., an overview of recent developments in Lyapunov-Krasovskii functionals and stability criteria for recurrent neural networks with time-varying delays, Neurocomputing, 313, 392-401 (2018) |
| [34] | Feng, Z.; Zheng, W. X., Improved stability condition for takagi-Sugeno fuzzy systems with time-varying delay, IEEE Trans. Cybern., 47, 3, 661-670 (2016) |
| [35] | 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 Syst., 201, 1-19 (2012) · Zbl 1251.93072 |
| [36] | Zhang, X. M.; Han, Q. L.; Wang, J., Admissible delay upper bounds for global asymptotic stability of neural networks with time-varying delays, IEEE Trans. Neural Netw. Learn. Syst., 29, 11, 5319-5329 (2018) |
| [37] | Shanmugam, L.; Joo, Y. H., Stability and stabilization for T-S fuzzy large-scale interconnected power system with wind farm via sampled-data control, IEEE Trans. Neural Netw. Learn. Syst., 51, 4, 2134-2144 (2020) |
| [38] | Pan, X. J.; Yang, B.; Cao, J. J.; Zhao, X. D., Improved stability analysis of Takagi-Sugeno fuzzy systems with time-varying delays via an extended delay-dependent reciprocally convex inequality, Inf. Sci. (Ny), 571, 24-37 (2021) · Zbl 1535.93017 |
| [39] | Seuret, A.; Gouaisbaut, F., Wirtinger-based integral inequality: application to time-delay systems, Automatica, 49, 9, 2860-2866 (2013) · Zbl 1364.93740 |
| [40] | 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. Frankl. Inst., 352, 4, 1378-1396 (2015) · Zbl 1395.93450 |
| [41] | Zeng, H. B.; He, Y.; Wu, M.; She, J., New results on stability analysis for systems with discrete distributed delay, Automatica, 60, 189-192 (2015) · Zbl 1331.93166 |
| [42] | Seuret, A.; Gouaisbaut, F., Stability of linear systems with time-varying delays using Bessel-Legendre inequalities, IEEE Trans. Autom. Control, 63, 1, 225-232 (2017) · Zbl 1390.34213 |
| [43] | Ma, X.; Liu, B.; Jia, X. C., On estimating neuronal states of delayed neural networks based on canonical Bessel-Legendre inequalities, J. Frankl. Inst., 357, 13, 9025-9044 (2020) · Zbl 1448.93122 |
| [44] | Zeng, H. B.; Liu, X. G.; Wang, W., A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems, Appl. Math. Comput., 354, 1-8 (2019) · Zbl 1428.34112 |
| [45] | Zhang, C. K.; Long, F.; He, Y.; Yao, W.; Jiang, L.; Wu, M., A relaxed quadratic function negative-determination lemma and its application to time-delay systems, Automatica, 113, 108764 (2020) · Zbl 1440.93144 |
| [46] | Lin, H. C.; Zeng, H. B.; Zhang, X. M.; Wang, W., Stability analysis for delayed neural networks via a generalized reciprocally convex inequality, IEEE Trans. Neural Netw. Learn. Syst., early access (2022) |
| [47] | Kim, J. H., Further improvement of Jensen inequality and application to stability of time-delayed systems, Automatica, 64, 121-125 (2016) · Zbl 1329.93123 |
| [48] | Zeng, H. B.; Lin, H. C.; He, Y.; Zhang, C. K.; Teo, K. L., Improved negativity condition for a quadratic function and its application to systems with time-varying delay, IET Control Theory Appl., 14, 18, 2989-2993 (2020) · Zbl 1542.93298 |
| [49] | Zeng, H. B.; He, Y.; Teo, K. L., Monotone-delay-interval-based Lyapunov functionals for stability analysis of systems with a periodically varying delay, Automatica, 138, 110030 (2022) · Zbl 1489.93093 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
