Improved Results on Stability Analysis for Delayed Neural Network

Abstract

This paper deals with the delay-dependent stability analysis problem for neural network with a time-varying delay. A proper Lyapunov-Krasovskii functional (LKF) is established by revealing the features of the improved Jensen integral inequality and considering two complementary integral couples with more cross information. Based on the improved Jensen inequality, a generalized integral inequality involving more free matrices is developed. With the help of the new LKF and integral inequality, some improved stability conditions with less conservatism are derived in terms of linear matrix inequality (LMI). The efficiency of theoretical results is verified by three typical numerical examples.

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References

  1. [1]

    L. O. Chua and L. Yang, “Cellular neural networks: applications,” IEEE Transactions on Circuits and Systems. vol. 35, no. 10, pp. 1273–1290, 1988.

    MathSciNet  Article  Google Scholar 

  2. [2]

    G. Joya, M. A. Atencia and F. Sandoval, “Hopfleld neural networks for optimization: study of the different dynamics,” Neurocomputing, vol. 43, no. 1–4, pp. 219–237, 2002.

    MATH  Article  Google Scholar 

  3. [3]

    Y. H. Chen and S. C. Fang, “Neurocomputing with time delay analysis for solving convex quadratic programming problems,” IEEE Transactions on Neural Networks, vol. 11, no. 1, pp. 230–240, 2000.

    Article  Google Scholar 

  4. [4]

    Q. Liu and J. Wang, “A one-layer recurrent neural network with a discontinuous hard-limiting activation function for quadratic programming,” IEEE Transactions on Neural Networks, vol. 19, no. 4, pp. 558–570, 2008.

    Article  Google Scholar 

  5. [5]

    K. Liu, A. Selivanov, and E. Fridman, “Survey on time-delay approach to networked control,” Annual Reviews in Control, vol. 49, pp. 57–79, 2019.

    MathSciNet  Article  Google Scholar 

  6. [6]

    Y. M. Liu, J. K. Tian and Z. R. Ren, “New stability analysis for generalized neural networks with interval time-varying delays,” International Journal of Control, Automation and Systems, vol. 15, no. 4, pp. 1600–1610, 2017.

    Article  Google Scholar 

  7. [7]

    R-L. Liu, “Further improvement on delay-derivative-dependent stochastic stability criteria for Markovian jumping neutral-type interval time-varying delay systems with mixed delays,” International Journal of Control, Automation and Systems, vol. 16, no. 3, pp. 1186–1193, 2018.

    MathSciNet  Article  Google Scholar 

  8. [8]

    X.-M. Zhang and Q. L. Han, “Global asymptotic stability for a class of generalized neural networks with interval time-varying delays,” IEEE Transactions on Neural Networks, vol. 22, no. 8, pp. 1180–1192, 2011.

    Article  Google Scholar 

  9. [9]

    O. M. Kwon, J. H. Park, S. M. Lee and E. J. Cha, “New augmented Lyapunov-Krasovskii functional approach to stability analysis of neural networks with time-varying delay,” Nonlinear dynamics, vol. 76, no. 1, pp. 221–236, 2014.

    MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    C. C. Hua, Y. B. Wang, and S. S. Wu, “Stability analysis of neural networks with time-varying delay using a new augmented Lyapunov-Krasovskii functional,” Neurocomputing, vol. 332, pp. 1–9, 2019.

    Article  Google Scholar 

  11. [11]

    H.-B. Zeng, Y. He, M. Wu, and C.-K. Zhang, “Complete delay-decomposing approach to asymptotic stability for neural networks with time-varying delays,” IEEE Transactions on Neural Networks, vol. 22, no. 2, pp. 806–812, 2011.

    Article  Google Scholar 

  12. [12]

    C. Ge, C. C. Hua, and X. P. Guan, “New delay-dependent stability criteria for neural networks with time-varying delay using delay-decomposition approach,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 7, pp. 1378–1383, 2014.

    Article  Google Scholar 

  13. [13]

    J. Chen, J. H. Park, and S. Y. Xu, “Stability analysis for neural networks with time-varying delay via improved techniques”, IEEE Transactions on Cybernetics, vol. 49, no. 12, pp. 4495–4500, Dec. 2019.

    Article  Google Scholar 

  14. [14]

    C.-K. Zhang, Y. He, L. Jiang, W. J. Lin, and M. Wu, “Delay-dependent stability of neural networks with time-varying delay: a generalized free-weighting-matrix approach,” Applied Mathematics and Computation, vol. 294, pp. 102–120, 2017.

    MathSciNet  MATH  Article  Google Scholar 

  15. [15]

    C.-K. Zhang, Y. He, L. Jiang, and M. Wu, “Stability analysis for delayed neural networks considering both conser-vativeness and complexity,” IEEE Transactions on Neural Networks and Learning Systems, vol. 27, no. 7, pp. 1486–1501, 2016.

    MathSciNet  Article  Google Scholar 

  16. [16]

    T. H. Lee and J. H. Park, “Improved stability conditions of time-varying delay systems based on new Lyapunov functionals,” Journal of the Franklin Institute, vol. 355, no. 3, pp. 1176–1191, 2018.

    MathSciNet  MATH  Article  Google Scholar 

  17. [17]

    T. H. Lee, H. M. Trinh, and J. H. Park, “Stability analysis of neural networks with time-varying delay by constructing novel Lyapunov functionals,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 9, pp. 4238–4247, 2018.

    Article  Google Scholar 

  18. [18]

    T. H. Lee and J. H. Park, “A novel Lyapunov functional for stability of time-varying delay systems via matrix-refined-function,” Automatica, vol. 80, pp. 239–242, 2017.

    MathSciNet  MATH  Article  Google Scholar 

  19. [19]

    Y. He, M. Wu, and J. H. She, “Delay-dependent exponential stability of delayed neural networks with time-varying delay,” IEEE Transactions on Circuits Systems II: Express Briefs, vol. 53, no.7, pp. 553–557, 2006.

    Google Scholar 

  20. [20]

    Y. He, G. P. Liu, and D. Rees, “New delay-dependent stability criteria for neural networks with time-varying delay,” IEEE Transactions on Neural Networks, vol. 18, no. 1, pp. 310–314, 2007.

    Article  Google Scholar 

  21. [21]

    P. Park, J. W. Ko, and C. K. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, no. 1, pp. 235–238, 2011.

    MathSciNet  MATH  Article  Google Scholar 

  22. [22]

    H. G. Zhang, F. S. Yang, X. D. Liu, and Q. L. Zhang, “Stability analysis for neural networks with time-varying delay based on quadratic convex combination,” IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 5, pp. 513–521, 2013.

    Article  Google Scholar 

  23. [23]

    A. Seuret and F. Gouaisbaut, “Wirtinger-based integral inequality: application to time-delay systems,” Automatica, vol. 49, no. 9, pp. 2860–2866, 2013.

    MathSciNet  MATH  Article  Google Scholar 

  24. [24]

    P. G. Park, W. I. Lee, and S. Y Lee, “Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems,” Journal of the Franklin Institute, vol. 352, no. 4, pp. 1378–1396, 2015.

    MathSciNet  MATH  Article  Google Scholar 

  25. [25]

    A. Seuret and F. Gouaisbaut, “Hierarchy of LMI conditions for the stability analysis of time-delay systems,” Systems & Control Letters, vol. 81, pp. 1–7, 2015.

    MathSciNet  MATH  Article  Google Scholar 

  26. [26]

    K. Liu, A. Seuret, Y. Q. Xia, F. Gouaisbaut, and Y. Ariba, “Bessel-Laguerre inequality and its application to systems with infinite distributed delays,” Automatica, vol. 109, 108562, 2019.

    MathSciNet  MATH  Article  Google Scholar 

  27. [27]

    X.-M. Zhang, Q.-L. Han, and Z. G Zeng, “Hierarchical type stability criteria for delayed neural networks via canonical Bessel-Legendre inequalities,” IEEE Transactions on Cybernetics, vol. 48, no. 5, pp. 1660–1671, 1671.

    Article  Google Scholar 

  28. [28]

    H.-B. Zeng, Y. He, M. Wu, and J. H. She, “Free-matrix-based integral inequality for stability analysis of systems with time-varying delay,” IEEE Transactions on Automatic Control vol. 60, no. 10, pp. 2768–2772, 2772.

    MathSciNet  MATH  Article  Google Scholar 

  29. [29]

    M. J. Park, S. H. Lee, O. M. Kwon, and J. H. Ryu, “Enhanced stability criteria of neural networks with time-varying delays via a generalized free-weighting matrix integral inequality,” Journal of the Franklin Institute, vol. 355, pp. 6531–6548, 2018.

    MathSciNet  MATH  Article  Google Scholar 

  30. [30]

    M. J. Park, O. M. Kwon, and J. H. Ryu, “Passivity and stability analysis of neural networks with time-varying delays via extended free-weighting matrices integral inequality,” Neural Networks, vol. 106, pp. 67–78, 2018.

    MATH  Article  Google Scholar 

  31. [31]

    J. H. Kim, “Further improvement of Jensen inequality and application to stability of time-delayed systems,” Automatica, vol. 64, pp. 121–125, 2016.

    MathSciNet  MATH  Article  Google Scholar 

  32. [32]

    Z. C. Li, Y. Bai, C. Z. Huang, H. C. Yan, and S. C. Mu, “Improved stability analysis for delayed neural networks,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 9, pp. 4535–4541, 2018.

    Article  Google Scholar 

  33. [33]

    B. Yang, J. Wang, and X. Liu, “Improved delay-dependent stability criteria for generalized neural networks with time-varying delays,” Information Science, vol. 420, pp. 299–312, 2017.

    Article  Google Scholar 

  34. [34]

    B. Yang, J. Wang, and X. Liu, “Stability analysis of delayed neural networks via a new integral inequality,” Neural Networks, vol. 88, pp. 49–57, 2017.

    MATH  Article  Google Scholar 

  35. [35]

    X.-M. Zhang, W.-J. Lin, Q.-L. Han, Y. He, and M. Wu, “Global asymptotic stability for delayed neural networks using an integral inequality based on nonorthogonal polynomials,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 9, pp. 4487–4493, 2018.

    Article  Google Scholar 

  36. [36]

    S. H. Kim, P. Park, and C. Jeong, “Robust H∞ stabilization of networked control systems with packet analyzer,” IET Control Theory & Applications, vol. 4, no. 9, pp. 1828–1837, 2010.

    Article  Google Scholar 

  37. [37]

    R. Saravanakumar, G. Rajchakit, C. K. Ahn, and H. R. Karimi, “Exponential stability, passivity, and dissipativity analysis of generalized neural networks with mixed time-varying delays,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, no. 2, pp. 395–405, 2019.

    Article  Google Scholar 

  38. [38]

    R. M. Zhang, D. Q. Zeng, X. Z. Liu, S. M. Zhong, and J. Cheng, “New results on stability analysis for delayed markovian generalized neural networks with partly unknown transition rates,” IEEE Transactions on Neural Networks and Learning Systems, vol. 30, no. 11, pp. 3384–3395, 2019.

    MathSciNet  Article  Google Scholar 

  39. [39]

    R. M. Zhang, D. Q. Zeng, J. H. Park, Y. J. Liu, and S. M. Zhong, “Pinning event-triggered sampling control for synchronization of T-S fuzzy complex networks with partial and discrete-time couplings,” IEEE Transactions on Fuzzy Systems, vol. 27, no. 12, pp. 2368–2380, Dec. 2019.

    Article  Google Scholar 

  40. [40]

    R. M. Zhang, D. Q. Zeng, J. H. Park, Y.J. Liu, and S. M. Zhong, “Quantized sampled-data control for synchronization of inertial neural networks with heterogeneous time-varying delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 12, pp. 6385–6395, 2019.

    Article  Google Scholar 

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Correspondence to Jian-An Wang.

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Recommended by Editor Jessie (Ju H.) Park. The work is supported by Shanxi Province Science Foundation for Youths (Grant No. 201701D221107), Natural Science Foundation of Shanxi Province (Grant No. 201801D121132), Key R&D program of Shanxi Province (International Cooperation, 201903D421045), Open project of coal mine electrical equipment and intelligent control key laboratory of Shanxi province (Grant No. MEI201604).

Jian-An Wan received his B.S. degree from Jiangxi Normal University, Nanchang, China, in 2005, and a Ph.D. degree from University of Science and Technology Beijing, China, in 2011. He is currently an associate professor with the School of Electronics Information Engineering, Taiyuan University of Science and Technology. His research interests include the time-delay system, complex network and multi-agent system.

Li Fan received her Bachelor degree from Taiyuan University of Science and Technology, Taiyuan, China, in {dy2018}. She is currently pursuing an M.S. degree with Taiyuan University of Science and Technology. Her current research interests include the time-delay system and multiagent system.

Xin-Yu Wen received his Ph.D. degree from Southeast University, Nanjing, China, in 2011. He is currently an associate professor in Taiyuan University of Science and Technology. His research interests include nonlinear system, robust control, and disturbance observer.

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Wang, J., Fan, L. & Wen, X. Improved Results on Stability Analysis for Delayed Neural Network. Int. J. Control Autom. Syst. 18, 1853–1862 (2020). https://doi.org/10.1007/s12555-019-0536-0

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Keywords

  • Delay-dependent stability
  • improved Jensen inequality
  • neural network
  • time-varying delay