Exponential Stability of Uncertain Takagi-Sugeno Fuzzy Hopfield Neural Networks with Time Delays
In this paper, based on linear matrix inequality (LMI), by using Lyapunov functional theory, the exponential stability criterion is obtained for a class of uncertain Takagi-Sugeno fuzzy Hopfield neural networks (TSFHNNs) with time delays. Here we choose a generalized Lyapunov functional and introduce a parameterized model transformation with free weighting matrices to it, these techniques lead to generalized and less conservative stability condition that guarantee the wide stability region. Finally, an example is given to illustrate our results by using MATLAB LMI toolbox.
[1] J. Hopfield, Neural networks and physical systems with emergent collect
computational abilities. Proc. Natl. Acad. Sci. USA; 1982; 2554-2558 .
[2] J. Hopfield, Neurons with graded response have collective computational
proporties like those of two-state neurons. Proc. Natl. Acad. Sci. USA;
1984; 3088-3092.
[3] S. Chen, Q. Zhang, C. Wang, Existence and stability of equilibria of the
continuous-time Hopfield neural network. Journal of Computational and
Applied Mathematics 2004; 169; 117-125.
[4] S. Hu, X. Liao, X. Mao, Stochastic Hopfield neural networks. J. Phys.
A: Math. Gen. 2004; 9; 47-53.
[5] H. Huang, D. Ho, J. Lam, Stochastic stability analysis of fuzzy Hopfield
neural networks with Time-Varying Delays. IEEE Trans. Circuits Syst.
II, Exp. Briefs 2005; 52; 251- 255.
[6] Z. Wang, H. Shu, J. Fang, X. Liu. Robust stability for stochastic Hopfield
neurarl networks with time delays. Nonlinear Anal. Real World. Appl.
2006, 7: 1119-1128.
[7] X. Mao, N. Koroleva, A. Rodkina, Robust stability of uncertain stochastic
delay differential equations. Systems Control lett. 1998; 35; 325-336.
[8] K. Tanaka, T. Ikede, H. Wang, Robust stabilization of a class of uncertain
nonlinear systems via fuzzy control: Quadratic stabilizability, H control
theory, andlinear matrix inequalities. IEEE Trans. Fuzzy Systems 1996;
4; 1-13.
[9] Z. Wang, S. Lauria, J. Fang, X. Liu, Exponential stability of uncertain
stochastic neural networks with mixed time delays. Chaos Solitons
Fractals 2007; 32; 62-72.
[10] Z. Wang, H. Qiao, Robust filtering for bilinear uncertain stochastic
discrete-time systems. IEEE Trans. Signal Process. 2002; 50(3); 560-567.
[11] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications
to modeling and control. IEEE Trans. Syst. Man, Cybern. 1985; 15;
116 - 132.
[12] Y. Cao, P. Frank, Stability analysis and synthesis of nonlinear time-delay
systems via linear Takagi-Sugeno fuzzy models. Fuzzy Sets and Systems
2001; 124; 213-229.
[13] T. Takagi, M. Sugeno, Stability analysis andd esign of fuzzy control
systems. Fuzzy Sets and Systems 1993; 45; 135-156.
[14] K. Tanaka, T. Ikede, H. Wang, An LMI approach to fuzzy controller
designs basedon the relaxed stability conditions Proceedings of the IEEE
international conference on fuzzy systems. Barcelona Spain; 1997; 171-
176.
[15] K. Tanaka, T. Ikede, H. Wang, Fuzzy regulators and fuzzy observers:
Relaxedstability conditions and LMI-based design. IEEE Trans. Fuzzy
Systems 1998; 6; 250-265.
[16] P. Gahinet, Nemirovski A, Laub A, Chilali M. LMI control toolbox
user-s guide. The Mathworks; Massachusetts; 1995.
[17] B. Boyd, L. Ghoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities
in System and Control Theory. SIAM; philadephia; 1994.
[18] H. Zhao, Global asymptotic stability of Hopfield neural network involving
distributed delays. Neural Networks. 2004; 17; 4753.
[19] L. Xie, Output feedback H control of systems with parameter uncertainty.
Internat. J. control 1996; 63; 741-759.
[20] K. Gu, V. Kharitonov, J. Chen, Stability of time delay systems.
Birkhuser; Boston; 2003.
[1] J. Hopfield, Neural networks and physical systems with emergent collect
computational abilities. Proc. Natl. Acad. Sci. USA; 1982; 2554-2558 .
[2] J. Hopfield, Neurons with graded response have collective computational
proporties like those of two-state neurons. Proc. Natl. Acad. Sci. USA;
1984; 3088-3092.
[3] S. Chen, Q. Zhang, C. Wang, Existence and stability of equilibria of the
continuous-time Hopfield neural network. Journal of Computational and
Applied Mathematics 2004; 169; 117-125.
[4] S. Hu, X. Liao, X. Mao, Stochastic Hopfield neural networks. J. Phys.
A: Math. Gen. 2004; 9; 47-53.
[5] H. Huang, D. Ho, J. Lam, Stochastic stability analysis of fuzzy Hopfield
neural networks with Time-Varying Delays. IEEE Trans. Circuits Syst.
II, Exp. Briefs 2005; 52; 251- 255.
[6] Z. Wang, H. Shu, J. Fang, X. Liu. Robust stability for stochastic Hopfield
neurarl networks with time delays. Nonlinear Anal. Real World. Appl.
2006, 7: 1119-1128.
[7] X. Mao, N. Koroleva, A. Rodkina, Robust stability of uncertain stochastic
delay differential equations. Systems Control lett. 1998; 35; 325-336.
[8] K. Tanaka, T. Ikede, H. Wang, Robust stabilization of a class of uncertain
nonlinear systems via fuzzy control: Quadratic stabilizability, H control
theory, andlinear matrix inequalities. IEEE Trans. Fuzzy Systems 1996;
4; 1-13.
[9] Z. Wang, S. Lauria, J. Fang, X. Liu, Exponential stability of uncertain
stochastic neural networks with mixed time delays. Chaos Solitons
Fractals 2007; 32; 62-72.
[10] Z. Wang, H. Qiao, Robust filtering for bilinear uncertain stochastic
discrete-time systems. IEEE Trans. Signal Process. 2002; 50(3); 560-567.
[11] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications
to modeling and control. IEEE Trans. Syst. Man, Cybern. 1985; 15;
116 - 132.
[12] Y. Cao, P. Frank, Stability analysis and synthesis of nonlinear time-delay
systems via linear Takagi-Sugeno fuzzy models. Fuzzy Sets and Systems
2001; 124; 213-229.
[13] T. Takagi, M. Sugeno, Stability analysis andd esign of fuzzy control
systems. Fuzzy Sets and Systems 1993; 45; 135-156.
[14] K. Tanaka, T. Ikede, H. Wang, An LMI approach to fuzzy controller
designs basedon the relaxed stability conditions Proceedings of the IEEE
international conference on fuzzy systems. Barcelona Spain; 1997; 171-
176.
[15] K. Tanaka, T. Ikede, H. Wang, Fuzzy regulators and fuzzy observers:
Relaxedstability conditions and LMI-based design. IEEE Trans. Fuzzy
Systems 1998; 6; 250-265.
[16] P. Gahinet, Nemirovski A, Laub A, Chilali M. LMI control toolbox
user-s guide. The Mathworks; Massachusetts; 1995.
[17] B. Boyd, L. Ghoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities
in System and Control Theory. SIAM; philadephia; 1994.
[18] H. Zhao, Global asymptotic stability of Hopfield neural network involving
distributed delays. Neural Networks. 2004; 17; 4753.
[19] L. Xie, Output feedback H control of systems with parameter uncertainty.
Internat. J. control 1996; 63; 741-759.
[20] K. Gu, V. Kharitonov, J. Chen, Stability of time delay systems.
Birkhuser; Boston; 2003.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51758", author = "Meng Hu and Lili Wang", title = "Exponential Stability of Uncertain Takagi-Sugeno Fuzzy Hopfield Neural Networks with Time Delays", abstract = "In this paper, based on linear matrix inequality (LMI), by using Lyapunov functional theory, the exponential stability criterion is obtained for a class of uncertain Takagi-Sugeno fuzzy Hopfield neural networks (TSFHNNs) with time delays. Here we choose a generalized Lyapunov functional and introduce a parameterized model transformation with free weighting matrices to it, these techniques lead to generalized and less conservative stability condition that guarantee the wide stability region. Finally, an example is given to illustrate our results by using MATLAB LMI toolbox.
", keywords = "Hopfield neural network, linear matrix inequality, exponential stability, time delay, T-S fuzzy model.", volume = "5", number = "12", pages = "1841-5", }
