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Improved Results on Guaranteed Generalized \({\mathcal {H}}_{2}\) Performance State Estimation for Delayed Static Neural Networks

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Abstract

This paper is concerned with the guaranteed generalized \({\mathcal {H}}_{2}\) performance state estimation for a class of static neural networks with a time-varying delay. A more general Arcak-type state estimator rather than the Luenberger-type state estimator is adopted to deal with this problem. Based on the Lyapunov stability theory, the inequality techniques and the delay-partitioning approach, some novel delay-dependent design criteria in terms of linear matrix inequalities (LMIs) are proposed ensuring that the resulting error system is globally asymptotically stable and a prescribed generalized \({\mathcal {H}}_{2}\) performance is guaranteed. The estimator gain matrices can be derived by solving the LMIs. Compared with the existing results, the sufficient conditions presented in this paper are with less conservatism. Numerical examples are given to illustrate the effectiveness and superiority of the developed method over the existing approaches. A comparison between the Arcak-type state estimator and Luenberger-type state estimator is given simultaneously.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. K.A. Choon, Switched exponential state estimation of neural networks based on passivity theory. Nonlinear Dyn. 67, 573–586 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. X.H. Chang, G.H. Yang, Nonfragile \(\cal{H}_{\infty }\) filtering of continuous-time fuzzy systems. IEEE Trans. Signal Process. 59(4), 1528–1538 (2011)

    Article  MathSciNet  Google Scholar 

  4. X.H. Chang, Robust nonfragile \({\cal{H}}_{\infty }\) filtering of fuzzy systems with linear fractional parametric uncertainties. IEEE Trans. Fuzzy Syst. 20(6), 1001–1011 (2012)

    Article  Google Scholar 

  5. X.H. Chang, G.H. Yang, New results on output feedback control for linear discrete-time systems. IEEE Trans. Automat. Contr. 59(5), 1355–1359 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Q.H. Duan, H.Y. Su, Z.G. Wu, \({\cal{H}}_{\infty }\) state estimation of static neural networks with time-varying delay. Neurocomputing 97, 16–21 (2012)

    Article  Google Scholar 

  7. Z.G. Feng, J. Lam, Stability and dissipativity analysis of distributed delay cellular neural networks. IEEE Trans. Neural Netw. 22(6), 976–981 (2011)

    Article  Google Scholar 

  8. K. Gopalsamy, X.Z. He, Delay-independent stability in bidirectional associative memory networks. IEEE Trans. Neural Netw. 5(6), 998–1002 (1994)

    Article  Google Scholar 

  9. Y. He, M.D. Ji, C.K. Zhang, M. Wu, Global exponential stability of neural networks with time-varying delay based on free-matrix-based integral inequality. Neural Netw. 77, 80–86 (2016)

    Article  Google Scholar 

  10. L.V. Hien, H. Trinh, Refined Jensen-based inequality approach to stability analysis of time-delays systems. IET Control Theory Appl. 9, 2188–2194 (2015)

    Article  MathSciNet  Google Scholar 

  11. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. U.S.A. 81, 3088–3092 (1984)

    Article  Google Scholar 

  12. M. Hua, H. Tan, J. Chen, Delay-dependent \({\cal{H}}_{\infty }\) and generalized \({\cal{H}}_{2}\) filtering for stochastic neural networks with time-varying delay and noise disturbance. Neural Comput. Appl. 25, 613–624 (2014)

    Article  Google Scholar 

  13. H. Huang, G. Feng, Delay-dependent \({\cal{H}}_{\infty }\) and generalized \({\cal{H}}_{2}\) filtering for delayed neural networks. IEEE Trans. Circuits Syst. 56(4), 846–857 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  15. D. Hu, H. Huang, T. Huang, Design of Arcak-type generalized \({\cal{H}}_{2}\) filter for delayed static neural networks. Circuits. Syst. Signal Process. 33, 3635–3648 (2014)

    Article  MathSciNet  Google Scholar 

  16. H. Huang, T.W. Huang, X.P. Chen, Guaranteed \({\cal{H}}_{\infty }\) performance state estimation of delayed static neural networks. IEEE Trans. Circuits Syst. 60(6), 371–375 (2013)

    Google Scholar 

  17. H. Huang, T. Huang, X. Chen, Further result on guaranteed \({\cal{H}}_{\infty }\) performance state estimation of delayed static neural networks. IEEE Trans. Neural Netw. Learn. Syst. 26(6), 1335–1341 (2015)

    Article  MathSciNet  Google Scholar 

  18. S. Lakshmanan, K. Mathiyalagan, J.H. Park, R. Sakthivel, F.A. Rihan, Delay-dependent \({\cal{H}}_{\infty }\) state estimation of neural networks with mixed time-varying delays. Neurocomputing 129, 392–400 (2014)

    Article  Google Scholar 

  19. T.H. Lee, J.H. Park, M.J. Park, O.M. Kwon, H.Y. Jung, On stability criteria for neural networks with time-varying delay using Wirtinger-based multiple integral inequality. J. Franklin Inst. 352, 5627–5645 (2015)

  20. N. Li, J.W. Hu, L. Li, Exponential state estimation for delayed recurrent neural networks with sampled-data. Nonlinear Dyn. 69, 555–564 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. X.G. Liu, R.R. Martin, M. Wu, M.L. Tang, Global exponential stability of bidirectional associative memory neural networks with time delays. IEEE Trans. Neural Netw. 19(3), 397–407 (2008)

    Article  Google Scholar 

  22. M.Q. Liu, S.L. Zhang, Z. Fan, S.Y. Zheng, W.H. Sheng, Exponential \({\cal{H}}_{\infty }\) synchronization and state estimation for chaotic systems via a unified model. IEEE Trans. Neural Netw. Learn. Syst. 24(7), 1114–1126 (2013)

    Article  Google Scholar 

  23. Y.J. Liu, S.M. Lee, H.G. Lee, Robust delay-dependent stability criteria for uncertain neural networks with two additive time-varying delay components. Neurocomputing 151, 770–775 (2015)

    Article  Google Scholar 

  24. Y.J. Liu, S.M. Lee, O.M. Kwon, J.H. Park, A study on \({\cal{H}}_{\infty }\) state estimation of static neural networks with time-varying delays. Appl. Math. Comput. 226, 589–597 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Z.W. Liu, H.G. Zhang, Q.L. Zhang, Novel stability analysis for recurrent neural networks with multiple delays via line integral-type L–K functional. IEEE Trans. Neural Netw. 21(11), 1710–1718 (2010)

    Article  Google Scholar 

  26. B. Niu, J. Zhao, Barrier Lyapunov functions for the output tracking control of constrained nonlinear switched systems. Syst. Control Lett. 62(10), 963–971 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. B. Niu, X.D. Zhao, L.X. Zhang, H.Y. Li, \({p}\)-times differentiable unbounded functions for robust control of uncertain switched nonlinear systems with tracking constraints. Int. J. Robust Nonlin. 25(16), 2965–2983 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. J.H. Park, O.M. Kwon, Delay-dependent stability criterion for bidirectional associative memory neural networks with interval time-varying delays. Mod. Phys. Lett. B 23, 35–46 (2009)

    Article  MATH  Google Scholar 

  29. 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 

  30. P.G. Park, WIl Lee, S.Y. Lee, Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems. J. Franklin Inst. 352(4), 1378–1396 (2015)

    Article  MathSciNet  Google Scholar 

  31. A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application to time delay systems. Automatica 49, 2860–2866 (2013)

    Article  MathSciNet  Google Scholar 

  32. H. Shen, L. Su, J.H. Park, Further results on stochastic admissibility for singular Markov jump systems using a dissipative constrained condition. ISA Trans. 59, 65–71 (2015)

  33. X.M. Song, X.H. Yan, Linear quadratic Gaussian control for linear time-delay systems. IET Control Theory Appl. 8(6), 375–383 (2014)

    Article  MathSciNet  Google Scholar 

  34. W.C. Sun, H.J. Gao, O. Kaynak, Finite frequency \({\cal{H}}_{\infty }\) control for vehicle active suspension systems. IEEE Trans. Control Syst. Technol. 19(2), 416–422 (2011)

    Article  Google Scholar 

  35. M. Syed Ali, R. Saravanakumar, S. Arik, Novel \({\cal{H}}_{\infty }\) state estimation of static neural networks with interval time-varying delays via augmented Lyapunov–Krasovskii functional. Neurocomputing 171, 949–954 (2016)

    Article  Google Scholar 

  36. Z.S. Wang, L. Liu, Q.H. Shan, H.G. Zhang, Stability criteria for recurrent neural networks with time-varying delay based on secondary delay partitioning method. IEEE Trans. Neural Netw. Learn. Syst. 26(10), 2589–2595 (2015)

    Article  MathSciNet  Google Scholar 

  37. Z.S. Wang, H.G. Zhang, B. Jiang, LMI-based approach for global asymptotic stability analysis of recurrent neural networks with various delays and structures. IEEE Trans. Neural Netw. 22(7), 1032–1045 (2011)

    Article  Google Scholar 

  38. T.B. Wang, S.W. Zhao, W.N. Zhou, W.Q. Yu, Finite-time state estimation for delayed Hopfield neural networks with Markovian jump. Neurocomputing 156, 193–198 (2015)

    Article  Google Scholar 

  39. Z.D. Wang, WCHo Daniel, X.H. Liu, State estimation for delayed neural networks. IEEE Trans. Neural Netw. 16(1), 279–284 (2005)

    Article  Google Scholar 

  40. J.Y. Xiao, S.M. Zhong, Y.T. Li, New passivity criteria for memristive uncertain neural networks with leakage and time-varying delays. ISA Trans. 59, 133–148 (2015)

    Article  Google Scholar 

  41. 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 

  42. D.S. Yang, X.R. Liu, Y.K. Xu, Y.C. Wang, Z.B. Liu, State estimation of recurrent neural networks with interval time-varying delay: an improved delay-dependent approach. Neural Comput. Appl. 23, 1149–1158 (2013)

    Article  Google Scholar 

  43. X.D. Zhao, P. Shi, X.L. Zheng, L.X. Zhang, Adaptive tracking control for switched stochastic nonlinear systems with unknown actuator dead-zone. Automatica 60, 193–200 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. X.D. Zhao, X.L. Zheng, B. Niu, L. Liu, Adaptive tracking control for a class of uncertain switched nonlinear systems. Automatica 52, 185–191 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. H.G. Zhang, Z.S. Wang, D.R. Liu, Global asymptotic stability of recurrent neural networks with multiple time-varying delays. IEEE Trans. Neural Netw. 19(5), 855–873 (2008)

    Article  Google Scholar 

  46. H.G. Zhang, Z.W. Liu, G.B. Huang, Z.S. Wang, Novel weighting-delay-based stability criteria for recurrent neural networks with time-varying delay. IEEE Trans. Neural Netw. 21(1), 91–106 (2010)

    Article  Google Scholar 

  47. H.G. Zhang, Z.S. Wang, D.R. Liu, A comprehensive review of stability analysis of continuous-time recurrent neural networks. IEEE Trans. Neural Netw. Learn. Syst. 25(7), 1229–1262 (2014)

    Article  Google Scholar 

  48. H.G. Zhang, J.Y. Wang, Z.S. Wang, H.J. Liang, Mode-dependent stochastic synchronization for Markovian coupled neural networks with time-varying mode-delays. IEEE Trans. Neural Netw. Learn. Syst. 26(11), 2621–2634 (2015)

    Article  MathSciNet  Google Scholar 

  49. C.K. Zhang, Y. He, L. Jiang, M. Wu, H.B. Zeng, Delay-variation-dependent stability of delayed discrete-time systems. IEEE Trans. Automat. Contr. 61(9), 2663–2669 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  50. C.K. Zhang, Y. He, L. Jiang, M. Wu, Stability analysis for delayed neural networks considering both conservativeness and complexity. IEEE Trans. Neural Netw. Learn. Syst. 27(7), 1486–1501 (2016)

    Article  MathSciNet  Google Scholar 

  51. C.K. Zhang, Y. He, L. Jiang, M. Wu, H.B. Zeng, Stability analysis of systems with time-varying delay via relaxed integral inequalities. Syst. Control Lett. 92, 52–61 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  52. X.M. Zhang, Q.L. Han, Global asymptotic stability for a class of generalized neural networks with interval time-varying delays. IEEE Trans. Neural Netw. 22(8), 1180–1192 (2011)

    Article  Google Scholar 

  53. X.M. Zhang, Q.L. Han, Global asymptotic stability analysis for delayed neural networks using a matrix quadratic convex approach. Neural Netw. 54, 57–69 (2014)

    Article  MATH  Google Scholar 

  54. H.B. Zeng, Y. He, M. Wu, S.P. Xiao, Passivity analysis for neural networks with a time-varying delay. Neurocomputing 74(5), 730–734 (2011)

    Article  Google Scholar 

  55. H.B. Zeng, Y. He, M. Wu, J.H. She, New results on stability analysis for systems with discrete distributed delay. Automatica 60, 189–192 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  56. H.B. Zeng, Y. He, M. Wu, J.H. She, Free-matrix-based integral inequality for stability analysis of systems with time-varying delay. IEEE Trans. Automat. Contr. 60(10), 2768–2772 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  57. A. Zemouche, M. Boutayeb, Comments on “A note on observers for discrete-time Lipschitz nonlinear systems”. IEEE Trans. Circuits Syst. 60(1), 56–60 (2013)

    MATH  Google Scholar 

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Acknowledgements

This work is partly supported by National Nature Science Foundation of China under Grant Numbers 61271355 and 61375063. The authors would like to thank the anonymous reviewers and the editor for their constructive comments that have greatly improved the quality of this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Yanjun Shu & Xin-Ge Liu

  2. Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan, 38541, Republic of Korea

    Yajuan Liu & Ju H. Park

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  1. Yanjun Shu
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  2. Xin-Ge Liu
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  3. Yajuan Liu
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  4. Ju H. Park
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Correspondence to Xin-Ge Liu or Ju H. Park.

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Shu, Y., Liu, XG., Liu, Y. et al. Improved Results on Guaranteed Generalized \({\mathcal {H}}_{2}\) Performance State Estimation for Delayed Static Neural Networks. Circuits Syst Signal Process 36, 3114–3142 (2017). https://doi.org/10.1007/s00034-016-0463-8

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