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Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

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Abstract

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 61573194, 61374180, 61473158 and 61573096), the Six Talent Peaks High Level Project of Jiangsu Province of China (Grant No. 2014-ZNDW-004), and the 1311 Talents Project through the Nanjing University of Posts and Telecommunications.

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Authors and Affiliations

  1. College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China

    Yuhong Tang, Min Xiao, Guoping Jiang & Jinxing Lin

  2. School of Mathematics, Southeast University, Nanjing, 210096, China

    Jinde Cao

  3. School of Mathematics and Statistics, Shandong Normal University, Ji’nan, 250014, China

    Jinde Cao

  4. School of Computing, Engineering and Mathematics, University of Western Sydney, Sydney, NSW, 2751, Australia

    Wei Xing Zheng

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  1. Yuhong Tang
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Correspondence to Min Xiao.

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Tang, Y., Xiao, M., Jiang, G. et al. Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system. Nonlinear Dyn 90, 2185–2198 (2017). https://doi.org/10.1007/s11071-017-3794-5

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