Nonlinear Dynamics

, Volume 73, Issue 3, pp 1367–1383 | Cite as

Design of sampled data state estimator for Markovian jumping neural networks with leakage time-varying delays and discontinuous Lyapunov functional approach

  • R. Rakkiyappan
  • Quanxin Zhu
  • T. Radhika
Original Paper


This paper is concerned with the sampled-data state estimation problem for neural networks with both Markovian jumping parameters and leakage time-varying delays. Instead of the continuous measurement, the sampled measurement is used to estimate the neuron states, and a sampled-data estimator is constructed. In order to make full use of the sawtooth structure characteristic of the sampling input delay, a discontinuous Lyapunov functional is proposed based on the extended Wirtinger inequality. A less conservative delay dependent stability criterion is derived via constructing a new triple-integral Lyapunov–Krasovskii functional and the famous Jenson integral inequality. Based on the Lyapunov–Krasovskii functional approach, a state estimator of the considered neural networks has been achieved by solving some linear matrix inequalities, which can be easily facilitated by using the standard numerical software. Finally, two numerical examples are provided to show the effectiveness of the proposed methods.


Sampled data state estimator neural network Lyapunov–Krasovskii functional leakage time-varying delay 



The work of R. Rakkiyappan was supported by NBHM Research Project under the sanctioned No: 2/48(7)/2012/NBHM(R.P.)/R and D II/12669 and Quanxin Zhu’s work was jointly supported by the National Natural Science Foundation of China (10801056), the Natural Science Foundation of Zhejiang Province (LY12F03010) and the Natural Science Foundation of Ningbo (2012A610032).


  1. 1.
    Hagan, M.T., Demuth, H.B., Beale, M.: Neural Network Design. PWS Publishing Company, Boston (1996) Google Scholar
  2. 2.
    Gupta, M.M., Jin, L., Homma, N.: Static and Dynamic Neural Networks: from Fundamentals to Advanced Theory. Wiley, New York (2003) CrossRefGoogle Scholar
  3. 3.
    Cichoki, A., Unbehauen, R.: Neural Networks for Optimization and Signal Processing. Wiley, Chichester (1993) Google Scholar
  4. 4.
    Haykin, S.: Neural Networks: a Comprehensive Foundation. Prentice Hall, New York (1998) Google Scholar
  5. 5.
    Ahn, C.K.: Robust stability of recurrent neural networks with ISS learning algorithm. Nonlinear Dyn. 65, 413–419 (2011) CrossRefGoogle Scholar
  6. 6.
    Haken, H.: Pattern recognition and synchronization in pulse-coupled neural networks. Nonlinear Dyn. 44, 269–276 (2006) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Hendzel, Z.: An adaptive critic neural network for motion control of a wheeled mobile robot. Nonlinear Dyn. 50, 849–855 (2007) MATHCrossRefGoogle Scholar
  8. 8.
    Civalleri, P.P., Gilli, M., Pandolfi, L.: On stability of cellular neural networks with delay. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 40, 157–165 (1993) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Marcus, C.M., Westervelt, R.M.: Stability of analog neural networks with delay. Phys. Rev. A 39, 347–359 (1989) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cao, J.: Global stability conditions for delayed CNNs. IEEE Trans. Circuits Syst. I 48, 1330–1333 (2001) MATHCrossRefGoogle Scholar
  11. 11.
    Liao, T.L., Wang, F.C.: Global stability for cellular neural networks with time delay. IEEE Trans. Neural Netw. 11, 1481–1484 (2000) CrossRefGoogle Scholar
  12. 12.
    Li, X., Cao, J.: Delay-dependent stability of neural networks of neutral-type with time delay in the leakage term. Nonlinearity 23, 1709–1726 (2010) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Fu, X., Li, X., Akca, H.: Exponential state estimation for impulsive neural networks with time delay in the leakage term. Arab. J. Math. (2012). doi: 10.1007/s40065-012-0045-y Google Scholar
  14. 14.
    Zhu, Q., Cao, J.: Robust exponential stability of Markovian jump impulsive stochastic Cohen–Grossberg neural networks with mixed time delays. IEEE Trans. Neural Netw. 21, 1314–1325 (2010) CrossRefGoogle Scholar
  15. 15.
    Zhu, Q., Cao, J.: Stability analysis of Markovian jump stochastic BAM neural networks with impulsive control and mixed time delays. IEEE Trans. Neural Netw. Learn. Syst. 23, 467–479 (2012) CrossRefGoogle Scholar
  16. 16.
    Van Den Driessche, P., Zou, X.: Global attractivity in delayed Hopfield neural network models. SIAM J. Appl. Math. 6, 1878–1890 (1998) CrossRefGoogle Scholar
  17. 17.
    Park, J., Kwon, O.: Design of state estimator for neural networks of neutral-type. Appl. Math. Comput. 202, 360–369 (2008) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Park, J., Kwon, O.: Further results on state estimation for neural networks of neutral-type with time-varying delay. Appl. Math. Comput. 208, 69–75 (2009) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Liu, Y., Wang, Z., Liu, X.: State estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delays. Phys. Lett. A 372, 7147–7155 (2008) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Balasubramaniam, P., Lakshmanan, S., Jeeva, S.: Sathya theesar, state estimation for Markovian jumping recurrent neural networks with interval time-varying delays. Nonlinear Dyn. 60, 661–675 (2010) MATHCrossRefGoogle Scholar
  21. 21.
    Liu, Y., Wang, Z., Liang, J., Liu, X.: Synchronization and state estimation for discrete-time complex networks with distributed delays. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 38, 1314–1325 (2008) CrossRefGoogle Scholar
  22. 22.
    Shen, B., Wang, Z., Liu, X.: Bounded synchronization and state estimation for discrete time-varying stochastic complex networks over a finite horizon. IEEE Trans. Neural Netw. 22, 145–157 (2011) CrossRefGoogle Scholar
  23. 23.
    Huang, H., Feng, G., Cao, J.: Robust state estimation for uncertain neural networks with time-varying delay. IEEE Trans. Neural Netw. 19, 1329–1339 (2008) CrossRefGoogle Scholar
  24. 24.
    Liu, X., Cao, J.: Robust state estimation for neural networks with discontinuous activations. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40, 1425–1437 (2010) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Bao, H., Cao, J.: Delay-distribution-dependent state estimation for discrete-time stochastic neural networks with random delay. Neural Netw. 24, 19–28 (2011) MATHCrossRefGoogle Scholar
  26. 26.
    Huang, H., Feng, G., Cao, J.: Guaranteed performance state estimation of static neural networks with time-varying delay. Neurocomputing 74, 606–616 (2011) CrossRefGoogle Scholar
  27. 27.
    Lakshmanan, S., Park, J.H., Ji, D.H., Jung, H.Y., Nagamani, G.: State estimation of neural networks with time-varying delays and Markovian jumping parameter based on passivity theory. Nonlinear Dyn. 70, 1421–1434 (2012) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Fridman, E., Shaked, U., Suplin, V.: Input/output delay approach to robust sampled-data H control. Syst. Control Lett. 54, 271–282 (2005) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Li, N., Hu, J., Hu, J., Li, L.: Exponential state estimation for delayed recurrent neural networks with sampled-data. Nonlinear Dyn. 69, 555–564 (2012) MATHCrossRefGoogle Scholar
  30. 30.
    Mikheev, Y., Sobolev, V., Fridman, E.: Asymptotic analysis of digital control systems. Autom. Remote Control 49, 1175–1180 (1988) MathSciNetMATHGoogle Scholar
  31. 31.
    Astrom, K., Wittenmark, B.: Adaptive Control. Addison-Wesley, Reading (1989) Google Scholar
  32. 32.
    Fridman, E., Seuret, A., Richard, J.P.: Robust sampled-data stabilization of linear systems: an input delay approach. Automatica 40, 1441–1446 (2004) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Wang, Z., Ho, D.W.C., Liu, X.: State estimation for delayed neural networks. IEEE Trans. Neural Netw. 16, 279–284 (2005) CrossRefGoogle Scholar
  34. 34.
    Naghshtabrizi, P., Hespanha, J., Teel, A.: Exponential stability of impulsive systems with application to uncertain sampled-data systems. Syst. Control Lett. 57, 378–385 (2008) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Zhu, X.: Stabilization for sampled-data neural network based control systems. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 41, 210–221 (2011) MATHCrossRefGoogle Scholar
  36. 36.
    Naghshtabrizi, P., Hespanha, J., Teel, A.: Stability of delay impulsive systems with application to networked control systems. In: Proceedings of the 26th American Control Conference, New York, USA, July 2007 Google Scholar
  37. 37.
    Fridman, E.: A refined input delay approach to sampled-data control. Automatica 46, 421–427 (2010) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Liu, K., Fridman, E.: Stability analysis of networked sontrol systems: a discontinuous Lyapunov functional approach. In: Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China, December 2009 Google Scholar
  39. 39.
    Hardy, G., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1934) Google Scholar
  40. 40.
    Mirkin, L.: Some remarks on the use of time-varying delay to model sample and hold circuits. IEEE Trans. Autom. Control 52, 1109–1112 (2007) MathSciNetCrossRefGoogle Scholar
  41. 41.
    Fridman, E.: A refined input delay approach to sampled-data control. Automatica 46, 421–427 (2010) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Kharitonov, V., Niculescu, S.I., Moreno, J., Michiels, M.: Static output feedback stabilization: necessary conditions for multiple delay controllers. IEEE Trans. Autom. Control 52, 1109–1112 (2007) CrossRefGoogle Scholar
  43. 43.
    Krasovskii, N.N., Lidskii, E.A.: Analysis and design of controllers in systems with random attributes. Autom. Remote Control 22, 1021–1025 (1961) MathSciNetGoogle Scholar
  44. 44.
    Kim, S., Li, H., Dougherty, E.R., Chao, N., Chen, Y., Bittner, M.L., Suh, E.B.: Can Markov chain models mimic biological regulation? J. Biol. Syst. 10, 337–357 (2002) MATHCrossRefGoogle Scholar
  45. 45.
    Wang, Z., Liu, Y., Liu, X.: State estimation for jumping recurrent neural networks with discrete and distributed delays. Neural Netw. 22, 41–48 (2009) CrossRefGoogle Scholar
  46. 46.
    Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht (1992) MATHCrossRefGoogle Scholar
  47. 47.
    Li, X., Fu, X.: Effects of leakage time-varying delay on stability of nonlinear differential systems. Journal of Franklin Institute. doi: 10.2016/j.jfranklin.2012.04.007
  48. 48.
    Gahinet, P., Nemirovski, A., Laub, A.J., Chilali, M.: LMI Control Toolbox. The Mathworks, Natick (1995) Google Scholar
  49. 49.
    Liu, Z., Yu, J., Xu, D., Peng, D.: Triple-integral method for the stability analysis of delayed neural networks. Neurocomputing 99, 283–289 (2013) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsBharathiar UniversityCoimbatoreIndia
  2. 2.School of Mathematical Sciences and Institute of MathematicsNanjing Normal UniversityNanjingChina

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