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Fixed-time Synchronization of Fractional-order Hopfield Neural Networks

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  • Control Theory and Applications
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Abstract

This paper investigates the fixed-time synchronization of fractional-order Hopfield neural networks (FHNNs). The aim of this paper is to design a state-feedback controller to make the synchronization error convergent to zero within bounded time. Based on the Lyapunov function and fractional calculus theory, we derived some criteria of synchronization for delay-free FHNNs and delayed FHNNs, respectively. At the same time, the upper bound of settling time for synchronization are given. Numerical simulations demonstrate the effectiveness of the theoretical analysis.

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References

  1. A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of linear control systems,” IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2106–2110, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. C. Maria and C. Andrea, “Nonsingular terminal sliding-mode control of nonlinear planar systems with global fixed-time stability guarantees,” Automatica, vol. 95, pp. 561–565, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

    MATH  Google Scholar 

  4. P. M. Aghababa, “Design of a chatter-free terminal sliding mode controller for nonlinear fractional-order dynamical systems,” International Journal of Control, vol. 86, no. 10, pp. 1744–1756, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Hu, J. Yu, Z. Chen, H. Jiang, and T. Huang, “Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks,” Neural Networks, vol. 89, pp. 74–83, 2017.

    Article  MATH  Google Scholar 

  6. H. Deng and H. B. Bao, “Fixed-time synchronization of quaternion-valued neural networks,” Physica A, vol. 527, p. 121351, 2019.

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Hayman, “The McCulloch-Pitts model,” Proc. of International Joint Conference on Neural Networks, IEEE, pp. 4438–4439, 1999.

  8. Y. Wang, C. Lu, G. Ji, and L. Wang, “Global exponential stability of high-order Hopfield-type neural networks with s-type distributed time delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 8, pp. 3319–3325, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. D. Wang, X. H. Hu, K. B. Shi, X. N. Song, and H. Shen, “Network-based passive estimation for switched complex dynamical networks under persistent dwell-time with limited signals,” Journal of the Franklin Institute, vol. 357, no. 15, pp. 10921–10936, 2020.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Zhao, “Global asymptotic stability of Hopfield neural network involving distributed delays,” Neural Networks, vol. 17, no. 1, pp. 47–53, 2004.

    Article  MATH  Google Scholar 

  11. L. Wang, “Comments on ‘Robust stability for interval Hop-field neural networks with time delay’ by X. F. Liao,” IEEE Transactions on Neural Networks, vol. 13, no. 1, pp. 250–251, 2002.

    Article  Google Scholar 

  12. X. Li, J. Fang, and H. Li, “Exponential synchronization of stochastic memristive recurrent neural networks under alternate state feedback control,” International Journal of Control, Automation, and Systems, vol. 16, pp. 2859–2869, 2018.

    Article  Google Scholar 

  13. X. Q. Feng and K. Shen, “Controlling hyperchaos and periodic synchronization in DOPO with parameter modulated by an external periodic signal,” Chaos, Solitons Fractals, vol. 35, no. 3, pp. 506–511, 2008.

    Article  Google Scholar 

  14. G. Wen and D. Xu, “Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems,” Chaos Solitons Fractals, vol. 26, no. 1, pp. 71–77, 2005.

    Article  MATH  Google Scholar 

  15. Y. Wu and L. Guo, “Enhancement of intercellular electrical synchronization by conductive materials in cardiac tissue engineering,” IEEE Transactions on Biomedical Engineering, vol. 65, no. 2, pp. 264–272, 2017.

    Article  Google Scholar 

  16. Y. A. Liu, J. W. Xia, B. Meng, X. N. Song, and H. Shen, “Extended dissipative synchronization for semi-Markov jump complex dynamic networks via memory sampled-data control scheme,” Journal of the Franklin Institute, vol. 357, no. 15, pp. 10900–10920, 2020.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Nijmeijer and A. Rodriguez-Angeles, Synchronization of Mechanical Systems, World Scientific, 2003.

  18. S. Lakshmanan, M. Prakash, C. P. Lim, R. Rakkiyappan, P. Balasubramaniam, and S. Nahavandi, “Synchronization of an inertial neural network with time-varying delays and its application to secure communication,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 1, pp. 195–207, 2016.

    Article  MathSciNet  Google Scholar 

  19. F. C. Hoppensteadt and E. M. Izhikevich, “Pattern recognition via synchronization in phase-locked loop neural networks,” IEEE Transactions on Neural Networks, vol. 11, no. 3, pp. 734–738, 2000.

    Article  Google Scholar 

  20. D. Li, L. Wei, T. Song, and O. Jin “Study on asymptotic stability of fractional singular systems with time delay,” International Journal of Control, Automation, and Systems, vol. 18, pp. 1002–1011, 2020.

    Article  Google Scholar 

  21. S. X. Liu, Y. G. Yu, S. Zhang, and Y. T. Zhang, “Robust stability of fractional-order memristor-based Hopfield neural networks with parameter disturbances,” Physica A, vol. 509, pp. 845–854, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  22. X. L. Yuan, L. P. Mo, Y. G. Yu, and G. J. Ren, “Distributed containment control of fractional-order multi-agent systems with double-integrator and nonconvex control input constraints,” International Journal of Control, Automation, and Systems, vol. 18, pp. 1728–1742, 2020.

    Article  Google Scholar 

  23. S. Marir and M. Chadli, “Robust admissibility and stabilization of uncertain singular fractional-order linear time-invariant systems,” IEEE/CAA Journal of Automatica Sinica, vol. 6, no. 3, pp. 685–692, 2019.

    Article  MathSciNet  Google Scholar 

  24. S. Marir, M. Chadli, and D. Bouagada, “A novel approach of admissibility for singular linear continuous-time fractional-order systems,” International Journal of Control, Automation, and Systems, vol. 15, no. 2, pp. 959–964, 2017.

    Article  MATH  Google Scholar 

  25. M. A. Ghezzar, D. Bouagada, and M. Chadli, “Influence of discretization step on positivity of a certain class of two-dimensional continuous-discrete fractional linear systems,” IMA Journal of Mathematical Control and Information, vol. 35, no. 3, pp. 845–860, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  26. F. H. Liu, Z. Gao, C. Yang, and R. C. Ma, “Extended Kalman filters for continuous-time nonlinear fractional-order systems involving correlated and uncorrelated process and measurement noises,” International Journal of Control, Automation, and Systems, vol. 18, pp. 2229–2241, 2020.

    Article  Google Scholar 

  27. H. L. Li, H. Jiang, and J. D. Cao, “Global synchronization of fractional-order quaternion-valued neural networks with leakage and discrete delays,” Neurocomputing, vol. 385, pp. 211–219, 2020.

    Article  Google Scholar 

  28. Q. Wang and D. L. Qi, “Synchronization for fractional order chaotic systems with uncertain parameters,” International Journal of Control, Automation, and Systems, vol. 14, pp. 211–216, 2016.

    Article  Google Scholar 

  29. N. Liu, J. Fang, W. Deng, Z.-J. Wu, and G.-Q. Ding, “Synchronization for a class of fractional-order linear complex networks via impulsive control,” International Journal of Control, Automation, and Systems, vol. 16, pp. 2839–2844, 2018.

    Article  Google Scholar 

  30. J. Ni, L. Liu, C. Liu, and X. Hu, “Fractional order fixed-time nonsingular terminal sliding mode synchronization and control of fractional order chaotic systems,” Nonlinear Dynamics, vol. 89, no. 3, pp. 2065–2083, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  31. W. Wang, X. Jia, Z. Wang, X. Luo, L. Li, J. Kurths, and M. Yuan, “Fixed-time synchronization of fractional order memristive MAM neural networks by sliding mode control,” Neurocomputing, vol. 401, pp. 364–376, 2020.

    Article  Google Scholar 

  32. S. Huang, B. Zhou, S. Bu, C. Li, C. Zhang, H. Wang, and T. Wang, “Robust fixed-time sliding mode control for fractional-order nonlinear hydro-turbine governing system,” Renewable Energy, vol. 139, pp. 447–458, 2019.

    Article  Google Scholar 

  33. A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of linear control systems,” IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2106–2110, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  34. G. H. Hardy and J. E. Littlewood, Inequalities, Cambridge University Press, 1952.

  35. Z. Zuo and L. Tie, “Distributed robust finite-time nonlinear consensus protocols for multi-agent systems,” International Journal of Systems Science, vol. 47, no. 6, pp. 1366–1375, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  36. H. K. Khalil and J. W. Grizzle, Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ, 2002.

    Google Scholar 

  37. A. Khanzadeh and I. Mohammadzaman, “Comment on ‘Fractional-order fixed-time nonsingular terminal sliding mode synchronization and control of fractional-order chaotic systems’,” Nonlinear Dynamics, vol. 94, no. 4, pp. 3145–3153, 2018.

    Article  MATH  Google Scholar 

  38. S. E. Parsegov, A. E. Polyakov, and P. S. Shcherbakov, “Nonlinear fixed-time control protocol for uniform allocation of agents on a segment,” Doklady Mathematics, vol. 87, no. 1, pp. 133–136, 2013.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Science, Southwest University of Science and Technology, Mianyang, 621010, China

    Xu Mei & Yucai Ding

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  1. Xu Mei
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Correspondence to Yucai Ding.

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Xu Mei received her B.S. degree in applied mathematics in 2020 and is working toward an M.S. degree in physics, both from Southwest University of Science and Technology, Mianyang, China. Her current research interests include stability of the fractional order systems.

Yucai Ding received his B.S. degree in applied mathematics from Dezhou University, Shandong, China, in 2006, and a Ph.D. degree in measuring and testing technologies and instruments from University of Electronic Science and Technology of China, Chengdu, China, in 2013. He is currently an associate professor with the School of Science, Southwest University of Science and Technology. His current research interests include fractional order systems, neural networks, and Markov jump systems.

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Mei, X., Ding, Y. Fixed-time Synchronization of Fractional-order Hopfield Neural Networks. Int. J. Control Autom. Syst. 20, 3584–3591 (2022). https://doi.org/10.1007/s12555-021-0529-7

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  • DOI: https://doi.org/10.1007/s12555-021-0529-7

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