JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (10): 64-73, 88.doi: 10.6040/j.issn.1671-9352.0.2023.328

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On ramp control of ARZ traffic flow model based on time-delay feedback

Yiyan WANG(),Dongxia ZHAO*(),Caixia GAO   

  1. School of Mathematics, North University of China, Taiyuan 030051, Shanxi, China
  • Received:2023-07-25 Online:2024-10-20 Published:2024-10-10
  • Contact: Dongxia ZHAO E-mail:3194967419@qq.com;zhaodongxia6@sina.com

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Abstract:

For the Aw-Rascle-Zhang (ARZ) non-equilibrium traffic flow model, if the traffic flow at the entrance is constant and the density of the traffic flow at the exit is constant, the system is in critical stability and there will be continuous oscillations near the equilibrium state. This article proposes the design of a time-delay feedback control strategy at the entrance ramp, and characterizes the time-delay term with the solution of the initial value problem of the first-order transportation equation, establishing the form of an infinite dimensional coupled closed-loop system for PDE-PDE. The operator semigroup theory is used to prove the well posedness of the system. The conclusion of exponential stability of the system is obtained by constructing a weighted strict Lyapunov function. The results indicate that when the feedback gain and delay values satisfy certain inequality constraints, the system energy reaches exponential decay. Finally, through numerical simulation, the effectiveness of the designed time-delay controller and the feasibility of parameter conditions are verified.

Key words: ARZ traffic flow model, time-delay feedback, Lyapunov function, exponential stability

CLC Number: 

  • O231.4

Table 1

Road traffic parameters"

道路长度L/km松弛时间τ0/s最大密度ρm/(辆·km-1)自由流速度vf/(km·h-1)稳态(ρ*, v*)
1150200150(120, 70)

Fig.1

Convergence of the state ω(x, t) of system (1.27)"

Fig.2

Convergence of the state ${\bar v}$(x, t) of system (1.27)"

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