Abstract
This paper investigates the synchronization problem of neural networks with time-varying delay under sampled-data control in the presence of a constant input delay. Based on the extended Wirtinger inequality, a discontinuous Lyapunov functional is introduced, which makes full use of the sawtooth structure characteristic of sampling input delay. A simple and less conservative synchronization criterion is given to ensure the master systems synchronize with the slave systems by using the linear matrix inequality (LMI) approach. The design method of the desired sampled-data controller is also proposed. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.
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Zhang, H., Ma, T., Huang, G., Wang, Z.: Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40, 831–844 (2010)
Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)
Cao, J., Li, H., Ho, D.W.C.: Synchronization criteria of Lur’e systems with time-delay feedback control. Chaos Solitons Fractals 23, 1285–1298 (2005)
Xiong, W., Xie, W., Cao, J.: Adaptive exponential synchronization of delayed chaotic networks. Physica A 370, 832–842 (2006)
Karimi, H.R.: Robust synchronization and fault detection of uncertain master-slave systems with mixed time-varying delays and nonlinear perturbations. Int. J. Control. Autom. Syst. 9, 671–680 (2011)
Li, T., Wang, T., Song, A., Fei, S.: Exponential synchronization for arrays of coupled neural networks with time-delay couplings. Int. J. Control. Autom. Syst. 9, 187–196 (2011)
Lee, S.M., Kwon, O.M., Park, J.H.: Regional asymptotic stability analysis for discrete-time delayed systems with saturation nonlinearity. Nonlinear Dyn. 67, 885–892 (2012)
Kwon, O.M., Park, J.H., Lee, S.M.: Secure communication based on chaotic synchronization via interval time-varying delay feedback control. Nonlinear Dyn. 63, 239–252 (2011)
Lee, S.M., Choi, S.J., Ji, D.H., Park, J.H., Won, S.C.: Synchronization for chaotic Lur’e systems with sector restricted nonlinearities via delayed feedback control. Nonlinear Dyn. 59, 277–288 (2010)
Han, Q.L.: On designing time-varying delay feedback controllers for master-slave synchronization of Lur’e systems. IEEE Trans. Circuits Syst. I, Regul. Pap. 54, 1573–1583 (2007)
Lee, S.M., Ji, D.H., Park, J.H., Won, S.C.: \(\mathcal{H}_{\infty}\) synchronization of chaotic systems via dynamic feedback approach. Phys. Lett. A 372, 4905–4912 (2008)
Huang, H., Feng, G., Cao, J.: Exponential synchronization of chaotic Lur’e systems with delayed feedback control. Nonlinear Dyn. 57, 441–453 (2009)
Miao, Q., Tang, Y., Lu, S., Fang, J.: Lag synchronization of a class of chaotic systems with unknown parameters. Nonlinear Dyn. 57, 107–112 (2009)
Zhang, J., Tang, W.: Control and synchronization for a class of new chaotic systems via linear feedback. Nonlinear Dyn. 58, 675–686 (2009)
Park, J.H., Kwon, O.M.: Synchronization of cellular neural networks of neutral type via dynamic feedback controller. Chaos Solitons Fractals 42, 1299–1304 (2009)
Zhang, H., Huang, W., Wang, Z., Chai, T.: Adaptive synchronization between two different chaotic systems with unknown parameters. Phys. Lett. A 350, 363–366 (2006)
Zhang, C., He, Y., Wu, M.: Improved global asymptotical synchronization of chaotic Lur’e systems with sampled-data control. IEEE Trans. Circuits Syst. II, Express Briefs 56, 320–324 (2009)
Li, P., Cao, J., Wang, Z.: Robust impulsive synchronization of coupled delayed neural networks with uncertainties. Physica A 373, 261–272 (2007)
Lu, J., Cao, J., Ho, D.W.C.: Adaptive stabilization and synchronization for chaotic Lur’e systems with time-varying delay. IEEE Trans. Circuits Syst. I, Regul. Pap. 55, 1347–1356 (2008)
Balasubramaniam, P., Chandran, R., Theesar, S.J.S.: Synchronization of chaotic nonlinear continuous neural networks with time-varying delay. Cogn. Neurodyn. 5, 361–371 (2011)
Gupta, M.M., Jin, L., Homma, N.: Static and Dynamic Neural Networks: From Fundamentals to Advanced Theory. Wiley/IEEE Press, New York (2003)
Wang, Z., Zhang, H.: Global asymptotic stability of reaction–diffusion Cohen–Grossberg neural networks with continuously distributed delays. IEEE Trans. Neural Netw. 20, 39–49 (2010)
Kwon, O.M., Park, J.H., Lee, S.M.: Augmented Lyapunov functional approach to stability of uncertain neutral systems with time-varying delays. Appl. Math. Comput. 207, 202–212 (2009)
Park, J.H., Kwon, O.M.: Further results on state estimation for neural networks of neutral-type with time-varying delay. Appl. Math. Comput. 208, 69–75 (2009)
Park, M.J., Kwon, O.M., Park, J.H., Lee, S.M.: A new augmented Lyapunov–Krasovskii functional approach for stability of linear systems with time-varying delays. Appl. Math. Comput. 217, 7197–7209 (2011)
Ji, D., Koo, J.H., Won, S.C., Lee, S.M., Park, J.H.: Passivity-based control for Hopfield neural networks using convex representation. Appl. Math. Comput. 217, 6168–6175 (2011)
Kwon, O.M., Lee, S., Park, J.H.: Improved delay-dependent exponential stability for uncertain stochastic neural networks with time-varying delays. Phys. Lett. A 374, 1232–1241 (2010)
He, Y., Liu, G., Rees, D.: New delay-dependent stability criteria for neural networks with time-varying delay. IEEE Trans. Neural Netw. 18, 310–314 (2007)
Wu, L., Feng, Z., Zheng, W.: Exponential stability analysis for delayed neural networks with switching parameters: average dwell time approach. IEEE Trans. Neural Netw. 21, 1396–1407 (2010)
Liu, Y., Wang, Z., Serrano, A., Liu, X.: Discrete-time recurrent neural networks with time-varying delays: exponential stability analysis. Phys. Lett. A 362, 480–488 (2007)
Xu, S., Lam, J., Ho, D.W.C., Zou, Y.: Improved global robust asymptotic stability criteria for delayed cellular neural networks. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 35, 1317–1321 (2005)
Zhang, H., Liu, Z., Huang, G., Wang, Z.: Novel weighting-delay-based stability criteria for recurrent neural networks with time-varying delay. IEEE Trans. Neural Netw. 21, 91–106 (2010)
Balasubramaniam, P., Lakshmanan, S., Theesar, S.J.S.: State estimation for Markovian jumping recurrent neural networks with interval time-varying delays. Nonlinear Dyn. 60, 661–675 (2010)
Feng, Z., Lam, J.: Stability and dissipativity analysis of distributed delay cellular neural networks. IEEE Trans. Neural Netw. 22, 976–981 (2011)
Liu, X., Chen, T., Cao, J., Lu, W.: Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches. Neural Netw. 24, 1013–1021 (2011)
Liu, X., Cao, J.: Local synchronization of one-to-one coupled neural networks with discontinuous activations. Cogn. Neurodyn. 5, 13–20 (2011)
Yu, W., Cao, J.: Synchronization control of stochastic delayed neural networks. Physica A 373, 252–260 (2007)
Karimi, H.R., Maass, P.: Delay-range-dependent exponential \(\mathcal{H}_{\infty}\) synchronization of a class of delayed neural networks. Chaos Solitons Fractals 41, 1125–1135 (2009)
Qi, D., Liu, M., Qiu, M., Zhang, S.: Exponential \(\mathcal{H}_{\infty}\) synchronization of general discrete-time chaotic neural networks with or without time delays. IEEE Trans. Neural Netw. 21, 1358–1365 (2010)
Zhang, C., He, Y., Wu, M.: Exponential synchronization of neural networks with time-varying mixed delays and sampled-data. Neurocomputing 74, 265–273 (2010)
Liu, K., Fridman, E.: Wirtinger’s inequality and Lyapunov-based sampled-data stabilization. Automatica 48, 102–108 (2012)
Gao, H., Chen, T., Lam, J.: A new delay system approach to network-based control. Automatica 44, 39–52 (2008)
Wang, Y., Zhang, H., Wang, X., Yang, D.: Networked synchronization control of coupled dynamic networks with time-varying delay. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40, 1468–1479 (2010)
Yue, D., Han, Q., Lam, J.: Network-based robust \(\mathcal{H}_{\infty}\) control of systems with uncertainty. Automatica 41, 999–1007 (2005)
Liu, Y., Wang, Z., Liu, X.: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Netw. 19, 667–675 (2006)
Gu, K., Kharitonov, V.K., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003)
Liu, K., Suplin, V., Fridman, E.: Stability of linear systems with general sawtooth delay. IMA J. Math. Control Inf. 27, 419–436 (2011)
Wang, Z., Liu, Y., Yu, L., Liu, X.: Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Phys. Lett. A 356, 346–352 (2006)
Acknowledgements
The work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2010-0009373). This work was also supported in part by the National Natural Science Foundation of China under Grant 61174029.
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Wu, ZG., Park, J.H., Su, H. et al. Discontinuous Lyapunov functional approach to synchronization of time-delay neural networks using sampled-data. Nonlinear Dyn 69, 2021–2030 (2012). https://doi.org/10.1007/s11071-012-0404-4
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DOI: https://doi.org/10.1007/s11071-012-0404-4