Advertisement

Nonlinear Dynamics

, Volume 73, Issue 4, pp 2175–2189 | Cite as

Passivity analysis of uncertain neural networks with mixed time-varying delays

  • O. M. Kwon
  • M. J. Park
  • Ju H. Park
  • S. M. Lee
  • E. J. Cha
Original Paper

Abstract

This paper addresses the passivity problem for uncertain neural networks with both discrete and distributed time-varying delays. It is assumed that the parameter uncertainties are norm-bounded. By construction of an augmented Lyapunov–Krasovskii functional and utilization of zero equalities, improved passivity criteria for the networks are derived in terms of linear matrix inequalities (LMIs) via new approaches. Through three numerical examples, the effectiveness to enhance the feasible region of the proposed criteria is demonstrated.

Keywords

Neural networks Time-varying delays Passivity Lyapunov method 

Notes

Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2012-0000479), and by a grant of the Korea Healthcare Technology R & D Project, Ministry of Health & Welfare, Republic of Korea (A100054).

References

  1. 1.
    Ensari, T., Arik, S.: Global stability of a class of neural networks with time-varying delay. IEEE Trans. Circuits Syst. II 52, 126–130 (2005) CrossRefGoogle Scholar
  2. 2.
    Xu, S., Lam, J., Ho, D.W.C.: Novel global robust stability criteria for interval neural networks with multiple time-varying delays. Phys. Lett. A 342, 322–330 (2005) MATHCrossRefGoogle Scholar
  3. 3.
    Ma, Q., Xu, S., Zou, Y., Shi, G.: Synchronization of stochastic chaotic neural networks with reaction–diffusion terms. Nonlinear Dyn. 67, 2183–2196 (2012) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Balasubramaniam, P., Vembarasan, V.: Synchronization of recurrent neural networks with mixed time-delays via output coupling with delayed feedback. Nonlinear Dyn. 70, 677–691 (2012) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Faydasicok, O., Arik, S.: Robust stability analysis of a class of neural networks with discrete time delays. Neural Netw. 29–30, 52–59 (2012) CrossRefGoogle Scholar
  6. 6.
    Kwon, O.M., Park Ju, H.: New delay-dependent robust stability criterion for uncertain neural networks with time-varying delays. Appl. Math. Comput. 205, 417–427 (2008) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Xu, S., Lam, J.: A survey of linear matrix inequality techniques in stability analysis of delay systems. Int. J. Syst. Sci. 39, 1095–1113 (2008) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Balasubramaniam, P., Lakshmanan, S.: Delay-range dependent stability criteria for neural networks with Markovian jumping parameters. Nonlinear Anal. Hybrid Syst. 3, 749–756 (2009) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Wang, G., Cao, J., Liang, J.: Exponential stability in the mean square for stochastic neural networks with mixed time-delays and Markovian jumping parameters. Nonlinear Dyn. 57, 209–218 (2009) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Balasubramaniam, P., Lakshmanan, S., Rakkiyappan, R.: Delay-interval dependent robust stability criteria for stochastic neural networks with linear fractional uncertainties. Neurocomputing 72, 3675–3682 (2009) CrossRefGoogle Scholar
  11. 11.
    Kwon, O.M., Park, J.H.: Improved delay-dependent stability criterion for neural networks with time-varying delays. Phys. Lett. A 373, 529–535 (2009) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Tian, J., Xie, X.: New asymptotic stability criteria for neural networks with time-varying delay. Phys. Lett. A 374, 938–943 (2010) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Tian, J., Zhong, S.: Improved delay-dependent stability criterion for neural networks with time-varying delay. Appl. Math. Comput. 217, 10278–10288 (2011) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Li, T., Zheng, W.X., Lin, C.: Delay-slope-dependent stability results of recurrent neural networks. IEEE Trans. Neural Netw. 22, 2138–2143 (2011) CrossRefGoogle Scholar
  15. 15.
    Mathiyalagan, K., Sakthivel, R., Marshal Anthoni, S.: Exponential stability result for discrete-time stochastic fuzzy uncertain neural networks. Phys. Lett. A 376, 901–912 (2012) MATHCrossRefGoogle Scholar
  16. 16.
    Mathiyalagan, K., Sakthivel, R., Marshal Anthoni, S.: New robust exponential stability results for discrete-time switched fuzzy neural networks with time delays. Comput. Math. Appl. 64, 2926–2938 (2012) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Sakthivel, R., Mathiyalagan, K., Marshal Anthoni, S.: Design of a passification controller for uncertain fuzzy Hopfield neural networks with time-varying delays. Phys. Scr. 84, 045024 (2011) CrossRefGoogle Scholar
  18. 18.
    Sakthivela, R., Arunkumarb, A., Mathiyalaganb, K., Marshal Anthoni, S.: Robust passivity analysis of fuzzy Cohen–Grossberg BAM neural networks with time-varying delays. Appl. Math. Comput. 218, 3799–3899 (2011) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mathiyalagan, K., Sakthivel, R., Marshal Anthoni, S.: New robust passivity criteria for stochastic fuzzy BAM neural networks with time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 17, 1392–1407 (2012) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Mathiyalagan, K., Sakthivel, R., Marshal Anthoni, S.: New robust passivity criteria for discrete-time genetic regulatory networks with Markovian jumping parameters. Can. J. Phys. 90, 107–118 (2012) CrossRefGoogle Scholar
  21. 21.
    Wu, Z.G., Shi, P., Su, H., Chu, J.: Stability and dissipativity analysis of static neural networks with time delay. IEEE Trans. Neural Netw. 23, 199–210 (2012) CrossRefGoogle Scholar
  22. 22.
    Chen, H.: Improved stability criteria for neural networks with two additive time-varying delay components. Circuits Syst. Signal Process. doi: 10.1007/s00034-013-9555-x
  23. 23.
    Chen, H., Zhu, C., Hu, P., Zhang, Y.: Delayed-state-feedback exponential stabilization for uncertain Markovian jump systems with mode-dependent time-varying state delays. Nonlinear Dyn. 69, 1023–1039 (2012) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Ruan, S., Filfil, R.S.: Dynamics of a two-neuron system with discrete and distributed delays. Physica D 191, 323–342 (2004) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Park, J.H.: A delay-dependent asymptotic stability criterion of cellular neural networks with time-varying discrete and distributed delays. Chaos Solitons Fractals 33, 436–442 (2007) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Park, J.H.: On global stability criterion for neural networks with discrete and distributed delays. Chaos Solitons Fractals 30, 897–902 (2006) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Lien, C.-H., Chung, L.-Y.: Global asymptotic stability for cellular neural networks with discrete and distributed time-varying delays. Chaos Solitons Fractals 34, 1213–1219 (2007) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Park, J.H.: An analysis of global robust stability of uncertain cellular neural networks with discrete and distributed delays. Chaos Solitons Fractals 32, 800–807 (2007) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Park, J.H.: Further results on passivity analysis of delayed cellular neural networks. Chaos Solitons Fractals 34, 1546–1551 (2007) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Willems, J.C.: Dissipative dynamical systems. Arch. Ration. Mech. Anal. 45, 321–393 (2008) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Chen, B., Li, H., Lin, C., Zhou, Q.: Passivity analysis for uncertain neural networks with discrete and distributed time-varying delays. Phys. Lett. A 373, 1242–1248 (2009) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Chen, Y., Li, W., Bi, W.: Improved results on passivity analysis of uncertain neural networks with time-varying discrete and distributed delays. Neural Process. Lett. 30, 155–169 (2009) CrossRefGoogle Scholar
  33. 33.
    Xu, S., Zheng, W.X., Zou, Y.: Passivity analysis of neural networks with time-varying delays. IEEE Trans. Circuits Syst. II 56, 325–329 (2009) CrossRefGoogle Scholar
  34. 34.
    Fu, J., Zhang, H., Ma, T., Zhang, Q.: On passivity analysis for stochastic neural networks with interval time-varying delay. Neurocomputing 73, 795–801 (2010) CrossRefGoogle Scholar
  35. 35.
    Zeng, H.-B., He, Y., Wu, M., Xiao, S.P.: Passivity analysis for neural networks with a time-varying delay. Neurocomputing 74, 730–734 (2011) CrossRefGoogle Scholar
  36. 36.
    Kwon, O.M., Lee, S.M., Park, J.H.: On improved passivity criteria of uncertain neural networks with time-varying delays. Nonlinear Dyn. 67, 1261–1271 (2012) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Song, Q., Cao, J.: Passivity of uncertain neural networks with both leakage delay and time-varying delay. Nonlinear Dyn. 2012, 1695–1707 (2012) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Li, H., Lam, J., Cheung, K.C.: Passivity criteria for continuous-time neural networks with mixed time-varying delays. Appl. Math. Comput. 218, 11062–11074 (2012) MathSciNetCrossRefGoogle Scholar
  39. 39.
    Ariba, Y., Gouaisbaut, F.: An augmented model for robust stability analysis of time-varying delay systems. Int. J. Control 82, 1616–1626 (2009) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Kim, S.H., Park, P., Jeong, C.K.: Robust H stabilisation of networks control systems with packet analyser. IET Control Theory Appl. 4, 1828–1837 (2010) CrossRefGoogle Scholar
  41. 41.
    Park, P., Ko, J.W., Jeong, C.K.: Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47, 235–238 (2011) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    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) MATHCrossRefGoogle Scholar
  43. 43.
    de Oliveira, M.C., Skelton, R.E.: Stability Tests for Constrained Linear Systems pp. 241–257. Springer, Berlin (2001) Google Scholar
  44. 44.
    Gu, K.: An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE Conference on Decision and Control, December, Sydney, Australia, pp. 2805–2810 (2000) Google Scholar
  45. 45.
    Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994) MATHCrossRefGoogle Scholar
  46. 46.
    Morita, M.: Associative memory with nonmonotone dynamics. Neural Netw. 6, 115–126 (1993) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • O. M. Kwon
    • 1
  • M. J. Park
    • 1
  • Ju H. Park
    • 2
  • S. M. Lee
    • 3
  • E. J. Cha
    • 4
  1. 1.School of Electrical EngineeringChungbuk National UniversityCheongjuRepublic of Korea
  2. 2.Department of Electrical EngineeringYeungnam UniversityKyongsanRepublic of Korea
  3. 3.School of Electronic EngineeringDaegu UniversityGyungsanRepublic of Korea
  4. 4.Department of Biomedical Engineering, School of MedicineChungbuk National UniversityCheongjuRepublic of Korea

Personalised recommendations