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Exponential Stabilization Control of Delayed Quaternion-Valued Memristive Neural Networks: Vector Ordering Approach

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Abstract

The stabilization control of the quaternion-valued memristive system is investigated in this paper. By starting from the basic quaternion-valued algorithms, the memristive system described by quaternion-valued connection weights is derived. Subsequently, a comprehensive set of results to ensure the existence of the equilibrium point and its stability analysis have been developed. Particularly, vector ordering approach is proposed in this paper, which can be employed to determine the “magnitude” of two different quaternion-valued, and thus the closed convex hull derived by two different quaternion-valued connections can be obtained correspondingly. In the end, the proposed method is substantiated with two numerical examples.

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References

  1. A. Ascoli, F. Corinto, Memristor models in a chaotic neural circuit. Int. J. Bifurc. Chaos 23(3), 1350052 (2013)

    Article  MathSciNet  Google Scholar 

  2. J. Aubin, A. Cellina, Differential Inclusions (Springer, Berlin, 1984)

    Book  Google Scholar 

  3. X. Chen, Q. Song, Z. Li, Z. Zhao, Y. Liu, Stability analysis of continuous-time and discrete-time quaternion-valued neural networks with linear threshold neurons. IEEE Trans. Neural Netw. Learn. Syst. 29, 2769–2781 (2018)

    MathSciNet  Google Scholar 

  4. S. Choe, J. Faraway, Modeling head and hand orientation during motion using quaternions. J. Aerosp. 113, 186–192 (2004)

    Google Scholar 

  5. J. Chou, Quaternions kinematic and dynamic differential equations. IEEE Trans. Robot. Autom. 8, 53–64 (1992)

    Article  Google Scholar 

  6. L. Chua, Memristor—the missing circuit element. IEEE Trans. Circuit Theory 18, 507–519 (1971)

    Article  Google Scholar 

  7. S. Ding, Z. Wang, H. Niu, H. Zhang, Stop and go adaptive strategy for synchronization of delayed memristive recurrent neural networks with unknown synaptic weights. J. Frankl. Inst. 354, 4989–5010 (2017)

    Article  MathSciNet  Google Scholar 

  8. Z. Guo, J. Wang, Z. Yan, Attractivity analysis of memristor-based cellular neural networks with time-varying delays. IEEE Trans. Neural Netw. Learn. Syst. 25, 704–717 (2014)

    Article  Google Scholar 

  9. W. Hamilton, Lectures on Quaternions (Hodges and Smith, Dublin, 1853)

    Google Scholar 

  10. T. Huang, C. Li, X. Liao, Synchronization of a class of coupled chaotic delayed systems with parameter mismatch. Chaos 17, 821–439 (2017)

    MathSciNet  Google Scholar 

  11. T. Isokawa, N. Matsui, H. Nishimura, Quaternionic neural networks: fundamental properties and applications, in Complex-Valued Neural Networks: Utilizing High-Dimensional Parameters, ed. by T. Nitta (Pennsylvania, IGI global, 2009), pp. 411–439

    Chapter  Google Scholar 

  12. M. Itoh, L.O. Chua, Memristor cellular automata and memristor discrete-time cellular neural networks. Int. J. Bifurc. Chaos 19(11), 3605–3656 (2009)

    Article  Google Scholar 

  13. A. Khan, C. Tammer, C. Zalinescu, Set-Valued Optimization: An Introduction with Applications (Springer, Berlin, 2015)

    Book  Google Scholar 

  14. R. Li, J. Cao, Finite-time stability analysis for Markovian jump memristive neural networks with partly unknown transition probabilities. IEEE Trans. Neural Netw. Learn. Syst. 28, 2924–2935 (2017)

    Article  MathSciNet  Google Scholar 

  15. Y. Liu, D. Zhang, J. Lou, J. Lu, J. Cao, Stability analysis of quaternion-valued neural networks: decomposition and direct approaches. IEEE Trans. Neural Netw. Learn. Syst. 29(9), 4201–4211 (2018)

    Article  Google Scholar 

  16. Y. Liu, Y. Zheng, J. Lu, J. Cao, L. Rutkowski, Constrained quaternion-variable convex optimization: a quaternion-valued recurrent neural network approach. IEEE Trans. Neural Netw. Learn. Syst. (2019). https://doi.org/10.1109/TNNLS.2019.2916597

    Article  Google Scholar 

  17. N. Matsui, T. Isokawa, H. Kusamichi, F. Peper, H. Nishimura, Quaternion neural network with geometrical operators. J. Intell. Fuzzy Syst. Appl. Eng. Technol. 15, 149–164 (2004)

    MATH  Google Scholar 

  18. R. Mukundan, Quaternions: from classical mechanics to computer graphics and beyond, in Proceedings of the 7th Asian Technology Conference in Mathematics (2002), pp. 97–105

  19. H. Pan, X. Jing, W. Sun, H. Gao, A bioinspired dynamics-based adaptive tracking control for nonlinear suspension systems. IEEE Trans. Control Syst. Technol. 26(3), 903–914 (2018)

    Article  Google Scholar 

  20. H. Pan, W. Sun, H. Gao, X. Jing, Disturbance observer-based adaptive tracking control with actuator saturation and its application. IEEE Trans. Autom. Sci. Eng. 13(2), 868–875 (2016)

    Article  Google Scholar 

  21. M. Scarabello, M. Messias, Bifurcations leading to nonlinear oscillations in a 3D piecewise linear memristor oscillator. Int. J. Bifurc. Chaos 24(1), 1430001 (2014)

    Article  MathSciNet  Google Scholar 

  22. Q. Song, X. Chen, Multistability analysis of quaternion-valued neural networks with time delays. IEEE Trans. Neural Netw. Learn. Syst. 29, 5430–5440 (2018)

    Article  MathSciNet  Google Scholar 

  23. D. Strukov, G. Snider, D. Stewart, R. Williams, The missing memristor found. Nature 453, 80–83 (2008)

    Article  Google Scholar 

  24. Z. Tu, J. Cao, A. Alsaedi, T. Hayat, Global dissipativity analysis for delayed quaternion-valued neural networks. Neural Netw. 89, 97–104 (2017)

    Article  Google Scholar 

  25. B. Ujang, C. Took, D. Mandic, Quaternion-valued nonlinear adaptive filtering. IEEE Trans. Neural Netw. 22, 1193–1206 (2011)

    Article  Google Scholar 

  26. L. Wang, E. Drakakis, S. Duan, P. He, X. Liao, Memristor model and its application for chaos generation. Int. J. Bifurc. Chaos 22(8), 1250205 (2012)

    Article  Google Scholar 

  27. Q. Wang, B. Du, J. Lam, M. Chen, Stability analysis of Markovian jump systems with multiple delay components and polytopic uncertainties. Circuits Syst. Signal Process. 31, 143–162 (2012)

    Article  MathSciNet  Google Scholar 

  28. H. Wei, R. Li, C. Chen, Z. Tu, Stability analysis of fractional order complex-valued memristive neural networks with time delays. Neural Process. Lett. 45, 379–399 (2017)

    Article  Google Scholar 

  29. S. Wen, T. Huang, Z. Zeng, Y. Chen, P. Li, Circuit design and exponential stabilization of memristive neural networks. Neural Netw. 63, 48–56 (2015)

    Article  Google Scholar 

  30. A. Wu, Z. Zeng, Exponential stabilization of memristive neural networks with time delays. IEEE Trans. Neural Netw. Learn. Syst. 23(12), 1919–1929 (2012)

    Article  Google Scholar 

  31. X. Yang, D.W.C. Ho, Synchronization of delayed memristive neural networks: robust analysis approach. IEEE Trans. Cybern. 46, 3377–3387 (2015)

    Article  Google Scholar 

  32. S. Yang, C. Li, T. Huang, Exponential stabilization and synchronization for fuzzy model of memristive neural networks by periodically intermittent control. Neural Netw. 75, 162–172 (2016)

    Article  Google Scholar 

  33. S. Zhu, W. Luo, Y. Shen, Robustness analysis for connection weight matrices of global exponential stability of stochastic delayed recurrent neural networks. Circuits Syst. Signal Process. 33, 2065–2083 (2014)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, China

    Ruoxia Li, Xingbao Gao & Kai Zhang

  2. The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing, 211189, China

    Jinde Cao

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  1. Ruoxia Li
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  2. Xingbao Gao
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  3. Jinde Cao
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  4. Kai Zhang
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Correspondence to Ruoxia Li.

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This work was supported by National Natural Science Foundation of China under Grant Nos. 61803247, 61273311 and 61173094, Project funded by Young Scientists Fund 61802243, China Postdoctoral Science Foundation 2018M640948, the Fundamental Research Funds for the Central Universities under Grant No. GK201903003, Shaanxi Postdoctoral Science Foundation under Grant No. 2018BSHEDZZ129.

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Li, R., Gao, X., Cao, J. et al. Exponential Stabilization Control of Delayed Quaternion-Valued Memristive Neural Networks: Vector Ordering Approach. Circuits Syst Signal Process 39, 1353–1371 (2020). https://doi.org/10.1007/s00034-019-01225-8

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  • DOI: https://doi.org/10.1007/s00034-019-01225-8

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