Skip to main content
SpringerLink
Log in

Design of An Arcak-Type Generalized \(H_2\) Filter for Delayed Static Neural Networks

  • Short Paper
  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

In this paper, an Arcak-type generalized \(H_2\) filter is designed for a class of static neural networks with time-varying delay. By employing some inequalities and constructing a suitable Lyapunov functional, a delay-dependent condition is derived by means of linear matrix inequalities such that the filtering error system is globally asymptotically stable and a prescribed generalized \(H_2\) performance is achieved. It is shown that the design of such a desired filter for a delayed static neural network is successfully transformed into solving a convex optimization problem subject to some linear matrix inequalities. It is thus facilitated readily by some standard algorithms. A numerical example is finally provided to show the effectiveness of the developed approach. A comparison on the generalized \(H_2\) performance for different gain parameters of the activation function is also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Arcak, P. Kokotovic, Nonlinear observers: A circle criterion design and robustness analysis. Automatica 37(12), 1923–1930 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Balasubramaniam, S. Lakshmanan, LMI conditions for robust stability analysis of stochastic Hopfield neural networks with interval time-varying delays and linear fractional uncertainties. Circuits Sys. Signal Process. 30(5), 1011–1028 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Bao, J. Cao, Delay-distribution-dependent state estimation for discrete-time stochastic neural networks with random delay. Neural Netw. 24, 19–28 (2011)

    Article  MATH  Google Scholar 

  4. S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, PA, 1994)

    Book  MATH  Google Scholar 

  5. Q. Duan, H. Su, Z. Wu, \(H_\infty \) state estimation of static neural networks with time-varying delay. Neurocomputing 97, 16–21 (2012)

    Article  Google Scholar 

  6. O. Faydasicok, S. Arik, Robust stability analysis of a class of neural networks with discrete time delays. Neural Netw. 29–30, 52–59 (2012)

    Article  Google Scholar 

  7. K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems (Birkhauser, Boston, MA, 2003)

    Book  MATH  Google Scholar 

  8. Y. He, Q.-G. Wang, M. Wu, C. Lin, Delay-dependent state estimation for delayed neural networks. IEEE Trans. Neural Netw. 17(4), 1077–1081 (2006)

    Article  MATH  Google Scholar 

  9. H. Huang, G. Feng, Delay-dependent \(H_\infty \) and generalized \(H_2\) filtering for delayed neural networks. IEEE Trans. Circuits Syst. I 56(4), 846–857 (2009)

    Article  MathSciNet  Google Scholar 

  10. H. Huang, G. Feng, J. Cao, Guaranteed performance state estimation of static neural networks with time-varying delay. Neurocomputing 74(4), 606–616 (2011)

    Article  Google Scholar 

  11. H. Huang, G. Feng, J. Cao, State estimation for static neural networks with time-varying delay. Neural Netw. 23, 1202–1207 (2010)

    Article  Google Scholar 

  12. H. Huang, T. Huang, X. Chen, Guaranteed \(H_\infty \) performance state estimation of delayed static neural networks. IEEE Trans. Circuits Syst. II 60(6), 371–375 (2013)

    Article  Google Scholar 

  13. H. Huang, T. Huang, X. Chen, C. Qian, Exponential stabilization of delayed recurrent neural networks: a state estimation based approach. Neural Netw. 48, 153–157 (2013)

    Article  MATH  Google Scholar 

  14. L. Jin, P.N. Nikiforuk, M.M. Gupta, Adaptive control of discrete-time nonlinear systems using recurrent neural networks. IEE Proc. Control Theory Appl. 141, 169–176 (1994)

    Article  MATH  Google Scholar 

  15. H. Li, B. Chen, C. Lin, Q. Zhou, Mean square exponential stability of stochastic fuzzy Hopfield neural networks with discrete and distributed time-varying delays. Neurocomputing 72(7–9), 2017–2023 (2009)

    Article  Google Scholar 

  16. H. Li, J. Lam, K.C. Cheung, Passivity criteria for continuous-time neural networks with mixed time-varying delays. Appl. Math. Comput. 218(22), 11062–11074 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Li, J. Cao, Stability in static delayed neural networks: a nonlinear measure approach. Neurocomputing 69, 1776–1781 (2006)

    Article  Google Scholar 

  18. X. Li, H. Gao, X. Yu, A unified approach to the stability of generalized static neural networks with linear fractional uncertainties and delays. IEEE Trans. Sys. Man Cybern. B 41, 1275–1286 (2011)

    Article  Google Scholar 

  19. S.S. Li, T. Okada, X.M. Chen, Z.H. Tang, An individual adaptive gain parameter backpropagation algorithm for complex-valued neural networks, Advance in Neural networks-ISNN 2006. vol 3971 (Springer Berlin, 2006), pp. 551–557

  20. J. Lian, Z. Feng, P. Shi, Observer design for switched recurrent neural networks: an average dwell time approach. IEEE Trans. Neural Netw. 22(10), 1547–1556 (2011)

    Article  Google Scholar 

  21. M. Liu, S. Zhang, Z. Fan, S. Zheng, W. Sheng, Exponential \(H_\infty \) synchronization and state estimation for chaotic systems via a unified model. IEEE Trans. Neural Netw. Learn. Sys. 24(7), 1114–1126 (2013)

    Article  Google Scholar 

  22. Y. Liu, Z. Wang, X. Liu, Design of exponential state estimators for neural networks with mixed time delays. Phy. Lett. A 364, 401–412 (2007)

    Article  Google Scholar 

  23. M. Di Marco, M. Forti, M. Grazzini, L. Pancioni, Limit set dichotomy and multistability for a class of cooperative neural networks with delays. IEEE Trans. Neural Netw. Learn. Sys. 23(9), 1473–1485 (2013)

    Article  Google Scholar 

  24. P. Park, J.W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47, 235–238 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. H.S. Seung, How the brain keeps the eye still. Proc. Natl. Acad. Sci. USA 93, 13339–13344 (1996)

    Article  Google Scholar 

  26. B. Shen, Z. Wang, D. Ding, H. Shu, \(H_\infty \) state estimation for complex networks with uncertain inner coupling and incomplete measurements. IEEE Trans. Neural Netw. Learn. Sys. 24(12), 2027–2037 (2013)

    Article  Google Scholar 

  27. I. Varga, G. Elek, H. Zak, On the brain-state-in-a-convex-domain neural models. Neural Netw. 9, 1173–1184 (1996)

    Article  Google Scholar 

  28. G. Wang, Z. Mu, C. Wen, Y. Li, A new global stability criteria for neural network with two time-varying delays. Circuits Sys. Signal Process. 31(1), 177–187 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Z. Wang, D.W.C. Ho, X. Liu, State estimation for delayed neural networks. IEEE Trans. Neural Netw. 16(1), 279–284 (2005)

    Article  Google Scholar 

  30. Z. Wang, Y. Liu, X. Liu, State estimation for jumping recurrent neural networks with discrete and distributed delays. Neural Netw. 22, 41–48 (2009)

    Article  Google Scholar 

  31. Y. Xia, An extended projection neural network for constrained optimization. Neural Comput. 16, 863–883 (2004)

    Article  MATH  Google Scholar 

  32. Z.-B. Xu, H. Qiao, J. Peng, B. Zhang, A comparative study of two modeling approaches in neural networks. Neural Netw. 17, 73–85 (2004)

    Article  MATH  Google Scholar 

  33. A. Zemouche, M. Boutayeb, A note on observers for discrete-time Lipschitz nonlinear systems. IEEE Trans. Circuits Sys. II 60(1), 56–60 (2013)

    Article  MathSciNet  Google Scholar 

  34. A. Zemouche, M. Boutayeb, Nonlinear-observer-based \(\cal H_\infty \) synchronization and unknown input recovery. IEEE Trans. Circuits Sys. I 56(8), 1720–1731 (2009)

    Article  MathSciNet  Google Scholar 

  35. Z. Zeng, W.X. Zheng, Multistability of neural networks with time-varying delays and concave-convex characteristics. IEEE Trans. Neural Netw. Learn. Sys. 23(2), 293–305 (2012)

    Article  Google Scholar 

  36. D. Zhang, L. Yu, Q.-G. Wang, C.-J. Ong, Estimator design for discrete-time switched neural networks with asynchronous switching and time-varying delay. IEEE Trans. Neural Netw. Learn. Sys. 23(5), 827–834 (2012)

    Article  Google Scholar 

  37. H. Zhang, F. Yang, X. Liu, Q. Zhang, Stability analysis for neural networks with time-varying delay based on quadratic convex combination. IEEE Trans. Neural Netw. Learn. Sys. 24(4), 513–521 (2013)

    Article  Google Scholar 

  38. C.-D. Zheng, M. Ma, Z. Wang, Less conservative results of state estimation for delayed neural networks with fewer LMI variables. Neurocomputing 74, 974–982 (2011)

    Article  Google Scholar 

  39. C.-D. Zheng, H. Zhang, Z. Wang, Delay-dependent globally exponential stability criteria for static neural networks: an LMI approach. IEEE Trans. Circuits Sys. II 56(7), 605–609 (2009)

    Article  Google Scholar 

  40. S. Zhu, W. Luo, Y. Shen, Robustness analysis for connection weight matrices of global exponential stability of stochastic delayed recurrent neural networks. Circuits Sys. Signal Process (2014). doi:10.1007/s00034-013-9735-8

Download references

Acknowledgments

The authors would like to thank the associate editor and anonymous reviewers for their valuable comments that have greatly improved the quality of this paper. The work was supported by the National Natural Science Foundation of China under Grant No. 61005047, and the Natural Science Foundation of Jiangsu Province of China under Grant No. BK2010214. Also, this publication was made possible by NPRP grant #4-1162-1-181 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the author[s].

Author information

Authors and Affiliations

  1. School of Electronics and Information Engineering, Soochow University, Suzhou , 215006, People’s Republic of China

    Danfeng Hu & He Huang

  2. Texas A&M University at Qatar, 5825 , Doha, Qatar

    Tingwen Huang

Authors
  1. Danfeng Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. He Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tingwen Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to He Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, D., Huang, H. & Huang, T. Design of An Arcak-Type Generalized \(H_2\) Filter for Delayed Static Neural Networks. Circuits Syst Signal Process 33, 3635–3648 (2014). https://doi.org/10.1007/s00034-014-9814-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-014-9814-5

Keywords

Search

Navigation