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Fuzzy hyperbolic neural network with time-varying delays. (English) Zbl 1195.93073
Summary: The formula of fuzzy basis functions with time-varying delays (DFBF) is firstly given. Based on the DFBF, a novel fuzzy hyperbolic neural network with time-varying delays (DFHNN) and its approximation properties are presented, respectively. By constructing an appropriate Lyapunov-Krasovskii functional, the stability of DFHNN is discussed, and some new criteria concerning the global exponential stability of DFHNN are proposed. Finally, three numerical examples are given to demonstrate the effectiveness of the obtained results.
##### MSC:
 93C42 Fuzzy control systems 92B20 General theory of neural networks (mathematical biology) 93D20 Asymptotic stability of control systems
##### References:
 [1] Driankov, D.; Palm, R.: Advances in fuzzy control, (1998) · Zbl 0888.00014 [2] Pedrycz, W.: An identification algorithm in fuzzy relational systems, Fuzzy sets and systems 13, No. 2, 153-167 (1984) · Zbl 0554.93070 · doi:10.1016/0165-0114(84)90015-0 [3] Takagi, T.; Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control, IEEE trans. Syst. man cybern. 15, No. 1, 116-132 (1985) · Zbl 0576.93021 [4] Quan, L. D.: TS fuzzy realization of chaotic Lü system, Phys. lett. A 356, 51-58 (2006) · Zbl 1160.37347 · doi:10.1016/j.physleta.2006.03.020 [5] Lou, X. Y.; Cui, B. T.: Robust asymptotic stability of uncertain fuzzy BAM neural networks with time-varying delays, Fuzzy sets and systems 158, 2746-2756 (2007) · Zbl 1133.93366 · doi:10.1016/j.fss.2007.07.015 [6] Nguyen, H. T.; Sugeno, M.: Fuzzy systems: modeling and control, (1998) · Zbl 0911.00016 [7] Kim, J. H.; Park, C. W.; Kim, E.; Park, M.: Adaptive synchronization of T – S fuzzy chaotic system with unknown parameters, Chaos solitions fractals 24, 1353-1361 (2005) · Zbl 1092.37512 · doi:10.1016/j.chaos.2004.09.082 [8] Wu, S. J.; Chiang, H. H.; Lin, H. T.; Lee, T. T.: Neural-network-based optimal fuzzy controller design for nonlinear systems, Fuzzy sets and systems 154, 182-207 (2005) · Zbl 1068.93036 · doi:10.1016/j.fss.2005.03.011 [9] Novák, V.: Are fuzzy sets a reasonable tool for modeling vague phenomena?, Fuzzy sets and systems 156, 341-348 (2005) · Zbl 1084.03043 · doi:10.1016/j.fss.2005.05.029 [10] Hu, S. S.; Liu, Y.: Robust H$\infty$ control of multiple time-delay uncertain nonlinear system using fuzzy model and adaptive neural network, Fuzzy sets and systems 146, 403-420 (2004) · Zbl 1053.93023 · doi:10.1016/j.fss.2003.09.009 [11] Margaliot, M.; Langholz, G.: Hyperbolic optimal control and fuzzy control, IEEE trans. Syst. man cybern. Part A 29, No. 1, 1-10 (1999) [12] Margaliot, M.; Langholz, G.: New approaches to fuzzy modeling and control: design and analysis, (2000) [13] Zhang, H. G.; Quan, Y. B.: Modeling and control based on fuzzy hyperbolic model, Acta autom. Sin. 26, No. 6, 729-735 (2000) [14] Zhang, H. G.; Quan, Y. B.: Modeling, identification and control of a class of nonlinear system, IEEE trans. Fuzzy syst. 49, 349-354 (2001) [15] Margaliot, M.; Langholz, G.: A new approach to fuzzy modeling and control of discrete-time system, IEEE trans. Fuzzy syst. 11, No. 4, 486-494 (2003) [16] Zhang, H. G.; Wang, Z. L.: Generalized fuzzy hyperbolic model: a universal approximator, Acta autom. Sin. 30, No. 3, 416-422 (2004) [17] Zhang, H. G.; Wang, Z. L.: Chaotifying fuzzy hyperbolic model using adaptive inverse optimal control approach, Int. J. Bifurcation chaos 14, No. 10, 3505-3517 (2004) · Zbl 1129.93431 · doi:10.1142/S0218127404011442 [18] Zhang, H. G.; Wang, Z. L.: Chaotifying fuzzy hyperbolic model using impulsive and nonlinear feedback control approaches, Int. J. Bifurcation chaos 15, No. 8, 2603-2610 (2005) · Zbl 1092.93553 · doi:10.1142/S021812740501354X [19] Zhang, H. G.; Zhang, M. J.: Robust direct adaptive fuzzy control for nonlinear MIMO systems, Prog. nat. Sci. 16, No. 10, 1098-1105 (2006) · Zbl 1129.93028 · doi:10.1080/10020070612330116 [20] Lun, S. X.; Zhang, H. G.: Delayed-independent fuzzy hyperbolic guaranteed cost control design for a class of nonlinear continuous-time systems with uncertainties, Acta autom. Sin. 31, No. 5, 711-726 (2005) [21] Zhang, H. G.; Liu, D. R.: Fuzzy modeling and fuzzy control, (2006) [22] Gao, D. X.; Xue, D. Y.: Fuzzy adaptive control of manipulators based on the generalized hyperbolic model, Control decision 21, No. 10, 1124-1128 (2006) · Zbl 1117.93339 [23] Zhou, S. S.; Li, T.; Shao, H. Y.; Zheng, W. X.: Output feedback H$\infty$ control for uncertain discrete-time hyperbolic fuzzy systems, Eng. appl. Artif. intell. 19, No. 5, 487-499 (2006) [24] Zeng, X. J.; Singh, M. G.: Approximation theory of fuzzy systems — SISO case, IEEE trans. Fuzzy syst. 2, No. 2, 162-176 (1994) [25] Wang, L. X.; Mendel, J. M.: Fuzzy basic function, universal approximation, and orthogonal least-squares learning, IEEE trans. Neural networks 3, No. 5, 807-814 (1992) [26] Kolman, E.; Margaliot, M.: Knowledge-based neurocomputing: A fuzzy logic approach, (2009) [27] Zeng, X. J.; Singh, M. G.: Approximation theory of fuzzy systems — MIMO case, IEEE trans. Fuzzy syst. 3, No. 2, 219-235 (1995) [28] Liu, P. I.: Universal approximations of continuous fuzzy-valued functions by multi-layer regular fuzzy neural networks, Fuzzy sets and systems 119, 313-320 (2001) · Zbl 0976.93055 · doi:10.1016/S0165-0114(99)00132-3 [29] Liu, P. I.; Li, H. X.: Approximation analysis of feedforward regular fuzzy neural network with two hidden layers, Fuzzy sets and systems 150, 373-396 (2005) · Zbl 1087.41018 · doi:10.1016/j.fss.2004.02.013 [30] Kolmanovskii, V. B.; Nosov, V. R.: Stability of functional differential equations, (1986) [31] Niculescu, S. I.; Losano, R.: On passivity of linear delay systems, IEEE trans. Autom. control 46, No. 3, 460-464 (2001) · Zbl 1056.93610 · doi:10.1109/9.911424 [32] Kharitonov, V. L.; Niculescu, S. I.: On the stability of linear systems with uncertain delay, IEEE trans. Autom. control 48, No. 1, 127-132 (2003) [33] Marcus, C. M.; Westervelt, R. M.: Stability of analog neural networks with delay, Phys. rev. A 39, No. 1, 347-359 (1989) [34] Huang, H.; Ho, D.; Lam, J.: Stochastic stability analysis of fuzzy Hopfield neural networks with time-varying delays, IEEE trans. Circuits syst. II 52, No. 5, 251-255 (2005) [35] Arik, S.: An improved global stability result for delayed cellular neural networks, IEEE trans. Circuits syst. I 49, No. 8, 1211-1214 (2002) [36] Zhang, J. Y.; Yang, Y. R.: Global stability analysis of bidirectional associative memory neural networks with time delay, Int. J. Circuit theory appl. 29, No. 2, 185-196 (2001) · Zbl 1001.34066 · doi:10.1002/cta.144 [37] Liao, X.; Chen, G.; Sanchez, E. N.: LMI-based approach for asymptotic stability analysis of delayed neural networks, IEEE trans. Circuits syst. I 49, No. 7, 1033-1039 (2002) [38] Li, C. G.; Liao, X. F.: Passivity analysis of neural networks with time delay, IEEE trans. Circuits syst. II 52, No. 8, 471-475 (2005) [39] Ensari, T.; Arik, S.: Global stability of a class of neural networks with time-varying delays, IEEE trans. Circuits syst. II 52, No. 3, 126-130 (2005) [40] Xu, S.; Lam, J.; Ho, D.; Zou, Y.: Novel global asymptotic stability criteria for delayed cellular neural networks, IEEE trans. Circuits syst. II 52, No. 6, 349-353 (2005) [41] Yi, Z.; Heng, P. A.; Vadakkepat, P.: Absolute periodicity and absolute stability of delayed neural networks, IEEE trans. Circuits syst. I 49, No. 2, 256-261 (2002) [42] Senan, S.; Arik, S.: New results for exponential stability of delayed cellular neural networks, IEEE trans. Circuits syst. II 52, No. 3, 154-158 (2005) [43] Michel, A. N.; Liu, D.: Qualitative analysis and synthesis of recurrent neural networks, (2002) [44] Zhou, D. M.; Cao, J. D.: Globally exponential stability conditions for cellular neural networks with time-varying delays, Appl. math. Comput. 131, No. 2 – 3, 487-496 (2002) · Zbl 1034.34093 · doi:10.1016/S0096-3003(01)00162-X [45] Yang, T.; Yang, L. B.; Wu, C. W.; Chau, L. O.: Fuzzy cellular neural networks: theory, , 181-186 (1996) [46] Yang, T.; Yang, L. B.; Wu, C. W.; Chau, L. O.: Fuzzy cellular neural networks: applications, , 225-230 (1996) [47] Liu, Y. Q.; Tang, W. S.: Exponential stability of fuzzy cellular neural networks with constant and time-varying delays, Phys. lett. A 323, 224-233 (2004) · Zbl 1118.81400 · doi:10.1016/j.physleta.2004.01.064 [48] Kun, Y.; Cao, J. D.; Deng, J. M.: Exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying delays, Neurocomputing 69, 1619-1627 (2006) [49] Lehman, B.; Verriest, E.: Stability of a continuous stirred reactor with delay in the recycle streams, , 1875-1876 (1991) [50] Cao, Y. Y.; Frank, P. M.: Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach, IEEE trans. Fuzzy syst. 8, No. 2, 200-211 (2000)