基于时滞反馈的ARZ交通流模型的入口匝道控制

doi:10.6040/j.issn.1671-9352.0.2023.328

《山东大学学报(理学版)》 ›› 2024, Vol. 59 ›› Issue (10): 64-73.doi: 10.6040/j.issn.1671-9352.0.2023.328

• • 上一篇

基于时滞反馈的ARZ交通流模型的入口匝道控制

王一言,赵东霞^*,高彩霞

中北大学数学学院, 山西太原 030051

发布日期:2024-10-10
通讯作者: 赵东霞(1981— ),女,副教授,硕士生导师,博士,研究方向为微分方程控制理论与应用. E-mail:zhaodongxia6@sina.com
基金资助:
山西省基础研究计划资助项目(20210302123046)

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

WANG Yiyan, ZHAO Dongxia^*, GAO Caixia

School of Mathematics, North University of China, Taiyuan 030051, Shanxi, China

Published:2024-10-10

摘要/Abstract

摘要： 对于Aw-Rascle-Zhang(ARZ)非平衡交通流模型,若入口处的交通流量恒定,出口处交通流的密度恒定,则系统处于临界稳定,在平衡状态附近会有持续的振荡。提出在入口匝道处设计时滞反馈控制策略,并将时滞项用一阶运输方程初值问题的解进行刻画,建立了PDE-PDE无穷维耦合闭环系统的形式。采用算子半群理论证明系统的适定性。构造加权严格Lyapunov函数得到系统指数稳定的结论。结果表明,当反馈增益和时滞取值满足某个不等式约束条件时,系统能量达到指数衰减。最后,通过数值仿真,验证所设计时滞控制器的有效性和参数条件的可行性。

关键词: ARZ交通流模型, 时滞反馈, 李雅普诺夫函数, 指数稳定性

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

中图分类号:

O231.4

王一言,赵东霞,高彩霞. 基于时滞反馈的ARZ交通流模型的入口匝道控制[J]. 《山东大学学报(理学版)》, 2024, 59(10): 64-73.

WANG Yiyan, ZHAO Dongxia, GAO Caixia. On ramp control of ARZ traffic flow model based on time-delay feedback[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(10): 64-73.

参考文献

[1] GREENSHIELDS B D, BIBBINS J R, CHANNING W S, et al. A study of traffic capacity[C] //Highway Research Board Proceedings. Washington, D. C.: Highway Research Board, 1935, 14(2):448-477.
[2] AW A, RASCLE M. Resurrection of “second order” models of traffic flow[J]. SIAM Journal on Applied Mathematics, 2000, 60(3):916-938.
[3] ZHANG H M. A non-equilibrium traffic model devoid of gas-like behavior[J]. Transportation Research Part B: Methodological, 2002, 36(3):275-290.
[4] BASTIN G, CORON J M. Stability and boundary stabilization of 1-D hyperbolic systems[M]. Cham, Switzerland: Springer, 2016:31-33.
[5] LI Zhibin, XU Chengheng, LI Dawei, et al. Comparing the effects of ramp metering and variable speed limit on reducing travel time and crash risk at bottlenecks[J]. IET Intelligent Transport Systems, 2018, 12(2):120-126.
[6] YU H, KRSTIC M. Traffic congestion control by PDE backstepping[M]. Cham, Switzerland: Springer, 2022:87-111.
[7] BURKHARDT M, YU H, KRTIC M. Stop-and-go suppression in two-class congested traffic[J]. Automatica, 2021, 125:109381.
[8] FRANCOIS B, MANDAY H, XAVIER L, et al. Prediction of traffic convective instability with spectral analysis of the Aw-Rascle-Zhang model[J]. Physics Letters A, 2015, 379(38):2319-2330.
[9] ZHANG L G, PRIEUR C, QIAO J F. PI boundary control of linear hyperbolic balance laws with stabilization of ARZ traffic flow models[J]. Systems & Control Letters, 2019, 123:85-91.
[10] ZHAO Dongxia, WANG Junmin. Exponential stability and spectral analysis of the inverted pendulum system under two delayed position feedbacks[J]. Journal of Dynamical and Control Systems, 2012, 18(2):269-295.
[11] CHENTOUF B, SMAOUI N. Time-delayed feedback control of a hydraulic model governed by a diffusive wave system[J/OL]. Complexity, 2020[2023-07-10]. https://doi.org/10.1155/2020/4986026.
[12] 范东霞,赵东霞,史娜,等. 一类扩散波方程的PDP反馈控制和稳定性分析[J]. 数学物理学报, 2021, 41(4):1088-1096. FAN Dongxia, ZHAO Dongxia, SHI Na, et al. PDP feedback control and stability analysis for a class of diffusion wave equations[J]. Journal of Mathematical Physics, 2021, 41(4):1088-1096.
[13] YU H, KRSTIC M. Traffic congestion control of Aw-Rascle-Zhang model[J]. Automatica, 2019, 100:38-51.
[14] DIAGNE A, BASTIN G, CORON J M. Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws[J]. Automatica, 2012, 48(1):109-114.

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed

基于时滞反馈的ARZ交通流模型的入口匝道控制

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

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 6

多维度评价

本文评价

推荐阅读 0

[1]	黄祖达,彭乐群,徐敏. 论变时滞高阶细胞神经网络模型的反周期解[J]. J4, 2012, 47(10): 121-126.
[2]	赵永昌1,王林山1,2. 具有不同时间尺度的分布时滞竞争神经网络概周期解的全局指数稳定性[J]. J4, 2010, 45(6): 60-64.
[3]	刘静1,2 . 一类延迟细胞神经网络全局渐近稳定性条件[J]. J4, 2009, 44(4): 61-65 .
[4]	王顺康,王林山 . 具有马尔可夫跳跃参数的变时滞静态神经网络的全局指数稳定性[J]. J4, 2008, 43(4): 81-84 .
[5]	考永贵,高存臣,孟波 . 非自治分布时滞BAM神经网络的绝对指数稳定性[J]. J4, 2008, 43(4): 9-13 .
[6]	张丽娟,斯力更 . 变时滞细胞神经网络的全局指数稳定性[J]. J4, 2007, 42(4): 58-62 .