Stability and Hopf Bifurcation of PD-Controlled XCP Network Congestion Model

Na Han; Yanhui Zhai

doi:https://doi.org/10.14445/22315373/IJMTT-V67I9P516

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 67 | Issue 9 | Year 2021 | Article Id. IJMTT-V67I9P516 | DOI : https://doi.org/10.14445/22315373/IJMTT-V67I9P516

Stability and Hopf Bifurcation of PD-Controlled XCP Network Congestion Model

Na Han, Yanhui Zhai

Abstract

Based on feedback control and bifurcation theory, a PD controller is proposed to solve the Hopf branch problem of XCP network congested power system. Firstly, α is selected as the branching parameter to obtain the critical value that keeps the original system and the controlled system stable. When the delay value passes the critical value, the system will lose stability at the equilibrium point and Hopf branch will be generated. Then, the addition of PD controller increases the critical value of system branch parameters, expands the stability region, and effectively alleviates the generation of Hopf branches. Finally, the feasibility of theoretical analysis is verified by numerical simulation with mathematical software.

Keywords

Hopf bifurcation, PD controller, Stability, network congestion

References

[1] Wu Zh W,Yuan D Ch, Han K.Congestion control model of TCP network based on hybrid automaton[J].Chinese Journal of Scientific Instrument. 2008(soppl.3) 2001-2003.
[2] Sun W,Wang J P,Li Q Y,et al.Research on MAC layer communication performance model of wireless sensor network[J].Journal of Electronic Measurement and Instrument, 26(2) (2012) 144-149.
[3] L. Massoulie, Stability of distributed congestion control with heterogeneous feedback delays,IEEE Trans. Automat. Control 47(6)(2002) 895-902.
[4] P. Ranian, E.H. Abed, Nonlinear instabilities in TCP-RED,IEEE/ACM Trans. Automat. Control 49(6)(2004) 1006-1011.
[5] Z. Chen, P. Yu. Hopf bifurcation control for an internet congestion model, Int. J. Bifur. Chaos 15(2005) 2643-2651.
[6] Liu F, Wang H O, Guan Z H.Hopf bifurcation control in the XCP for the Internet congestion control system[J].Nonlinear Analysis: Real World Applications, 13(3) (2012) 1466-1477.
[7] T. T. Wu. Research on Network Congestion Control System Based on Hybrid Control [D]. Anhui University, (2014).
[8] Liu F, Wang H O , Guan Z H. Hopf bifurcation control in the XCP for the Internet congestion control system[J]. Nonlinear Analysis: Real World Applications, 13(3) (2011) 1466-1479.
[9] J.Y. Zhang, B.Y. Feng. Geometric theory and Bifurcating problems of ordinary differential equations [M]. Beijing: Peking University Press, (2000).
[10] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Applications of Hopf Bifurcation[M], Cambridge University Press, Cambridge, (1981).

Citation :

Na Han, Yanhui Zhai, "Stability and Hopf Bifurcation of PD-Controlled XCP Network Congestion Model," International Journal of Mathematics Trends and Technology (IJMTT), vol. 67, no. 9, pp. 138-145, 2021. Crossref, https://doi.org/10.14445/22315373/IJMTT-V67I9P516