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**Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays.**
*(English)*
Zbl 1221.93254

Summary: This paper investigates delay-dependent robust exponential state estimation of Markovian jumping fuzzy neural networks with mixed random time-varying delay. In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the robust exponential state estimation of Markovian jumping Hopfield neural networks with mixed random time-varying delays. Moreover probabilistic delay satisfies a certain probability-distribution. By introducing a stochastic variable with a Bernoulli distribution, the neural networks with random time delays is transformed into one with deterministic delays and stochastic parameters. The main purpose is to estimate the neuron states, through available output measurements such that for all admissible time delays, the dynamics of the estimation error is globally exponentially stable in the mean square. Based on the Lyapunov–Krasovskii functional and stochastic analysis approach, several delay-dependent robust state estimators for such T–S fuzzy Markovian jumping Hopfield neural networks can be achieved by solving a linear matrix inequality (LMI), which can be easily facilitated by using some standard numerical packages. The unknown gain matrix is determined by solving a delay-dependent LMI. Finally some numerical examples are provided to demonstrate the effectiveness of the proposed method.

### MSC:

93E10 | Estimation and detection in stochastic control theory |

34K50 | Stochastic functional-differential equations |

60J28 | Applications of continuous-time Markov processes on discrete state spaces |

82C32 | Neural nets applied to problems in time-dependent statistical mechanics |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

### Keywords:

linear matrix inequality; Lyapunov; krasovskii functional; Hopfield neural networks; mixed random time-varying delays; state estimation
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\textit{P. Balasubramaniam} et al., Commun. Nonlinear Sci. Numer. Simul. 16, No. 4, 2109--2129 (2011; Zbl 1221.93254)

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### References:

[1] | Qiu, J.; Zhang, J.; Wang, J.; Xia, Y.; Shi, P., A new global robust stability criteria for uncertain neural networks with fast time-varying delays, Chaos Soliton Fract, 37, 360-368 (2008) · Zbl 1141.93046 |

[2] | Chua, L.; Yang, L., Cellular neural networks: theory and applications, IEEE Trans Circuits Syst I, 35, 1257-1290 (1988) |

[3] | Gopalsamy, K., Stability of artificial neural networks with impulses, Appl Math Comput, 154, 783-813 (2004) · Zbl 1058.34008 |

[4] | Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc Natl Acad Sci, 81, 3088-3092 (1984) · Zbl 1371.92015 |

[5] | Otawara, K.; Fan, L. T.; Tsutsumi, A.; Yano, T.; Kuramoto, K.; Yoshida, K., An artificial neural network as a model for chaotic behavior of a three-phase fluidized bed, Chaos Soliton Fract, 13, 353-362 (2002) · Zbl 1073.76656 |

[6] | Li, X., Global exponential stability for a class of neural networks, Appl Math Lett, 22, 1235-1239 (2009) · Zbl 1173.34345 |

[7] | Li, X.; Chen, Z., Stability properties for Hopfield neural networks with delays and impulsive perturbations, Nonlinear Anal: Real World Appl, 10, 3253-3265 (2009) · Zbl 1162.92003 |

[8] | Arik, S., An analysis of exponential stability of delayed neural networks with time varying delays, Neural Networks, 14, 1027-1031 (2004) · Zbl 1068.68118 |

[9] | Chen, T.; Wang, L., Global \(μ\)-stability of delayed neural networks with unbounded time-varying delays, IEEE Trans Neural Networks, 18, 1836-1840 (2007) |

[10] | Zeng, Z.; Wang, J., Improved conditions for global exponential stability of recurrent neural networks with time-varying delays, IEEE Trans Neural Networks, 17, 3, 623-635 (2006) |

[11] | Liao, X.; Wang, J.; Zeng, Z., Global asymptotic stability and global exponential stability of delayed cellular neural networks, IEEE Trans Circuits Syst-II: Express Briefs, 52, 403-409 (2005) |

[12] | Mou, S.; Gao, H.; Qiang, W.; Chen, K., New delay-dependent exponential stability for neural networks with time delay, IEEE Trans Syst Man Cybernet Part-B, 38, 571-576 (2008) |

[13] | Zhang, Q.; Ma, R.; Wang, C.; Xu, J., Delay-dependent exponential stability of cellular neural networks with time-varying delays, Chaos Soliton Fract, 23, 1363-1369 (2005) · Zbl 1094.34055 |

[14] | Yue, D.; Zhang, Y. J.; Tian, E. G., Delay-distribution-dependent exponential stability criteria for discrete-time recurrent neural networks with stochastic delay, IEEE Trans Neural Networks, 19, 1299-1306 (2008) |

[15] | Zhang, Y. J.; Yue, D.; Tian, E. G., Robust delay-distribution-dependent stability of discrete-time stochastic neural networks with time-varying delay, Neurocomputing, 72, 1265-1273 (2009) |

[16] | Fu, J.; Zhang, H. G.; Ma, T., Delay-probability-distribution-dependent robust stability analysis for stochastic neural networks with time-varying delay, Prog Natural Sci, 19, 1333-1340 (2009) |

[17] | Yang, R.; Gao, H.; Lam, J.; Shi, P., New stability criteria for neural networks with distributed and probabilistic delays, Circuits Syst Signal Process, 28, 505-522 (2009) · Zbl 1170.93027 |

[18] | Takagi, T.; Sugeno, M., Fuzzy identification of systems and its application to modeling and control, IEEE Trans Syst Man Cybernet, SMC-15, 116-132 (1985) · Zbl 0576.93021 |

[19] | Cao, Y. Y.; Frank, P. M., Analysis and synthesis of nonlinear timedelay systems via fuzzy control approach, IEEE Trans Fuzzy Syst, 8, 200-211 (2000) |

[20] | Takagi, T.; Sugeno, M., Stability analysis and design of fuzzy control systems, Fuzzy Set Syst, 45, 135-156 (1993) |

[22] | Huang, H.; Ho, D. W.C.; Lam, J., Stochastic stability analysis of fuzzy Hopfield neural networks with time-varying delays, IEEE Trans Circuits Syst-II: Express Briefs, 52, 251-255 (2005) |

[23] | Syed Ali, M.; Balasubramaniam, P., Stability analysis of uncertain fuzzy Hopfield neural networks with time delays, Commun Nonlinear Sci Numer Simulat, 14, 2776-2783 (2009) · Zbl 1221.34191 |

[24] | Sheng, L.; Gao, M.; Yang, H., Delay-dependent robust stability for uncertain stochastic fuzzy Hopfield neural networks with time-varying delays, Fuzzy Sets Syst, 160, 3503-3517 (2009) · Zbl 1185.93109 |

[25] | Li, H.; Chen, B.; Zhou, Q.; Qian, W., Robust stability for uncertain delayed fuzzy Hopfield Neural Networks with Markovian jumping parameters, IEEE Trans Syst Man Cybernet, 39, 94-102 (2009) |

[26] | Rakkiyappan, R.; Balasubramaniam, P., On exponential stability results for fuzzy impulsive neural networks, Fuzzy Sets Syst, 161, 1823-1835 (2010) · Zbl 1198.34160 |

[27] | Bolle, D.; Dupont, P.; Vinck, B., On the overlap dynamics of multi-state neural networks with a finite number of patterns, J Phys A, 25, 2859-2872 (1992) · Zbl 0800.82013 |

[28] | Cleeremans, A.; Servan-schreiber, D.; McClelland, J. L., Finite state automata and simple recurrent networks, Neural Comput, 1, 372-381 (1989) |

[29] | Wang, Z.; Liu, Y.; Liu, X., On global asymtotic stability of neural networks with discrete and distributed delays, Phys Lett A, 345, 299-308 (2005) · Zbl 1345.92017 |

[30] | Tino, P.; Cernansky, M.; Benuskova, L., Markovian architectural bias of recurrent neural networks, IEEE Trans Neural Networks, 15, 6-15 (2004) |

[31] | Balasubramaniam, P.; Lakshmanan, S., Delay-range dependent stability criteria for neural networks with Markovian jumping parameters, Nonlinear Anal: Hybrid Syst, 3, 749-756 (2009) · Zbl 1175.93206 |

[32] | He, Y.; Wang, Q.-G.; Wu, M.; Lin, C., Delay-dependent state estimation for delayed neural networks, IEEE Trans Neural Networks, 17, 1077-1081 (2006) |

[33] | Wang, Z.; Ho, D. W.C.; Liu, X., State estimation for delayed neural networks, IEEE Trans Neural Networks, 16, 279-284 (2005) |

[34] | Jin, L.; Nikiforuk, P. N.; Gupta, M. M., Adaptive control of discrete time nonlinear systems using recurrent neural networks, IEEE Proc Control Theory Appl, 141, 169-176 (1994) · Zbl 0803.93026 |

[35] | Huang, H.; Feng, G.; Cao, J., An LMI approach to delay-dependent state estimation for delayed neural networks, Neurocomputing, 71, 2857-2867 (2008) |

[36] | Park, J. H.; Kwon, O. M., Design of state estimator for neural networks of neutral-type, Appl Math Comput, 202, 1, 360-369 (2008) · Zbl 1142.93016 |

[37] | Li, T.; Fei, S-M.; Zhu, Q., Design of exponential state estimator for neural networks with distributed delays, Nonlinear Anal: Real World Appl, 10, 1229-1242 (2009) · Zbl 1167.93318 |

[38] | Lou, X.; Cui, B., Design of state estimator for uncertain neural networks via the integral-inequality method, Nonlinear Dynam, 53, 223-235 (2008) · Zbl 1402.92023 |

[39] | Liu, Y.; Wang, Z.; Liu, X., Design of exponential state estimators for neural networks with mixed time delays, Phys Lett A, 364, 401-412 (2007) |

[40] | Wang, H.; Song, Q., State estimation for neural networks with mixed interval time-varying delays, Neurocomputing, 73, 1281-1288 (2010) |

[41] | Mahmoud, M. S., New exponentially convergent state estimation method for delayed neural networks, Neurocomputing, 72, 3935-3942 (2009) |

[42] | Balasubramaniam, P.; Lakshmanan, S.; Jeeva Sathya Theesar, S., State estimation for Markovian jumping recurrent neural networks with interval time-varying delays, Nonlinear Dynam, 60, 661-675 (2010) · Zbl 1194.62109 |

[43] | Wang, Z.; Liu, Y.; Liu, X., State estimation for jumping recurrent neural networks with discrete and distributed delays, Neural Networks, 22, 41-48 (2009) · Zbl 1335.93125 |

[44] | Ahn, C. K., Delay-dependent state estimation for T-S fuzzy delayed Hopfield neural networks, Nonlinear Dynam, 61, 483-489 (2010) · Zbl 1204.93047 |

[45] | Boyd, B.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in systems and control theory (1994), SIAM: SIAM Philadelphia |

[46] | Xu, S. Y.; Chen, T. W., Robust \(H_∞\) control for uncertain stochastic systems with state delay, IEEE Trans Automatic Control, 47, 2089-2094 (2002) · Zbl 1364.93755 |

[47] | Gu, K.; Kharitonov, V. L.; Chen, J., Stability of time-delay systems (2003), Birkhäuser: Birkhäuser Boston · Zbl 1039.34067 |

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