﻿ 考虑限速的最优速度模型的稳定性与孤波 The Stability and Soliton of the Optimal Velocity Model Considering Speed Limit

International Journal of Mechanics Research
Vol. 08  No. 03 ( 2019 ), Article ID: 32084 , 10 pages
10.12677/IJM.2019.83021

The Stability and Soliton of the Optimal Velocity Model Considering Speed Limit

Guangzhu He, Cuncai Hua

School of Mathematics, Yunnan Normal University, Kunming Yunnan

Received: Aug. 19th, 2019; accepted: Sep. 2nd, 2019; published: Sep. 9th, 2019

ABSTRACT

Based on the optimal velocity model, this paper designs a traffic flow model which takes into account the driver’s advance time to know the speed limit information. By using the linear stability analysis method, the stability condition of the model is obtained. It shows that the influence of the speed limit makes the stable region of traffic flow expand obviously. The density wave equations such as Burgers equation, KdV equation and mKdV equation are derived respectively from the reduced perturbation method in the stable region, metastable region and unstable region. The phenomena of traffic congestion under the speed limit are described by the solitary wave solution of Burgers and KdV equation, and by the kink-antikink solution of mKdV equation.

Keywords:The Optimal Velocity Model, Speed Limit, Stability, Density Wave

1. 引言

2. 模型的提出

1995年，Bando等 [9] 提出的最优速度模型很好地描述了交通流的一些非线性现象，其运动方程如下：

${\stackrel{¨}{x}}_{n}\left(t\right)=a\left[V\left(\Delta {x}_{n}\left(t\right)\right)-{v}_{n}\left(t\right)\right]$ (1)

$V\left(\Delta {x}_{n}\left(t\right)\right)=\frac{{v}_{\mathrm{max}}}{2}\left[\mathrm{tanh}\left(\Delta {x}_{n}\left(t\right)-{h}_{c}\right)+\mathrm{tanh}\left({h}_{c}\right)\right]$ (2)

2015年，Lie等 [19] 通过调整车辆自身的速度差(当前速度与历史速度)达到稳定交通流的目的，提出了如下自稳定控制驾驶的跟驰模型：

${\stackrel{¨}{x}}_{n}\left(t\right)=a\left[V\left(\Delta {x}_{n}\left(t\right)\right)-{v}_{n}\left(t\right)\right]+\rho \left[{v}_{n}\left(t\right)-{v}_{n}\left(t-{t}_{1}\right)\right]$ (3)

2017年，Chen等 [20] 考虑驾驶员对自身车速变化的连续记忆效应，将模型(2)拓展为具有连续记忆效应的如下交通流跟驰模型：

${\stackrel{¨}{x}}_{n}\left(t\right)=a\left[V\left(\Delta {x}_{n}\left(t\right)\right)-{v}_{n}\left(t\right)\right]+\gamma \left[{v}_{n}\left(t\right)-\frac{1}{{t}_{1}}\underset{t-{t}_{1}}{\overset{t}{\int }}{v}_{n}\left(t\right)\text{d}t\right]$ (4)

2018年，Sun等 [21] 将车头时距自稳控制因子加入OVM中，提出如下扩展的自稳定最优速度模型：

${\stackrel{¨}{x}}_{n}\left(t\right)=a\left[V\left(\Delta {x}_{n}+\kappa \left(h-\Delta {x}_{n}\right)\right)-{v}_{n}\right]$ (5)

${\stackrel{¨}{x}}_{n}\left(t\right)=a\left[V\left(\Delta {x}_{n}\left(t\right)\right)-{v}_{n}\left(t\right)\right]+\lambda \left[{v}_{n}\left(t+{t}_{0}\right)-{v}_{n}\left(t\right)\right]$ (6)

${\stackrel{¨}{x}}_{n}\left(t\right)=a\left[V\left(\Delta {x}_{n}\left(t\right)\right)-{\stackrel{˙}{x}}_{n}\left(t\right)\right]+\lambda \left[{\stackrel{˙}{x}}_{n}\left(t+{t}_{0}\right)-{\stackrel{˙}{x}}_{n}\left(t\right)\right]$ (7)

3. 模型(7)的稳定性分析

${x}_{n}^{0}\left(t\right)=hn+V\left(h\right)t,h=L/N$ (8)

${x}_{n}\left(t\right)={x}_{n}^{0}\left(t\right)+{y}_{n}\left(t\right)$ (9)

${x}_{n}\left(t\right)$ 为扰动影响下车辆的实际位移。

${\stackrel{¨}{y}}_{n}\left(t\right)=a\left[V\left(h+\Delta {y}_{n}\left(t\right)\right)-V\left(h\right)-{\stackrel{˙}{y}}_{n}\left(t\right)\right]+\lambda \left[{\stackrel{˙}{y}}_{n}\left(t+{t}_{0}\right)-{\stackrel{˙}{y}}_{n}\left(t\right)\right]$ (10)

${\stackrel{¨}{y}}_{n}\left(t\right)=a\left[{V}^{\prime }\left(h\right)\Delta {y}_{n}\left(t\right)-{\stackrel{˙}{y}}_{n}\left(t\right)\right]+\lambda \left[{\stackrel{˙}{y}}_{n}\left(t+{t}_{0}\right)-{\stackrel{˙}{y}}_{n}\left(t\right)\right]$ (11)

${z}^{2}=a\left[{V}^{\prime }\left(h\right)\left(\mathrm{exp}\left(ik\right)-1\right)-z\right]+\lambda z\left({\text{e}}^{{t}_{0}z}-1\right)$ (12)

${z}_{1}={V}^{\prime }\left(h\right)$${z}_{2}=\frac{{V}^{\prime }\left(h\right)}{2}+\frac{\left(\lambda {t}_{0}-1\right){V}^{\prime }{\left(h\right)}^{2}}{a}$ (13)

$a=2\left(1-\lambda {t}_{0}\right){V}^{\prime }\left(h\right)$ (14)

(a) (b)

Figure 1. The neutral stability lines in the headway-sensitivity space for different parameter combinations. (a) ${t}_{0}=1$ , (b) $\lambda =0.3$

4. 模型(7)的约化摄动分析和孤立波

$\begin{array}{c}\frac{{\text{d}}^{2}\left(\Delta {x}_{n}\left(t\right)\right)}{\text{d}{t}^{2}}=a\left[V\left(\Delta {x}_{n+1}\left(t\right)\right)-V\left(\Delta {x}_{n}\left(t\right)\right)-\frac{\text{d}\left(\Delta {x}_{n}\left(t\right)\right)}{\text{d}t}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\lambda \left[\frac{\text{d}\left(\Delta {x}_{n}\left(t+{t}_{0}\right)\right)}{\text{d}t}-\frac{\text{d}\left(\Delta {x}_{n}\left(t\right)\right)}{\text{d}t}\right]\end{array}$ (15)

4.1. 在稳定流区域导出Burgers方程

$X=\epsilon \left(n+bt\right)$$T={\epsilon }^{2}t$ (16)

$\Delta {x}_{n}\left(t\right)=h+\epsilon R\left(X,T\right)$ (17)

$a{\epsilon }^{2}\left(b-{V}^{\prime }\left(h\right)\right){\partial }_{X}R+{\epsilon }^{3}\left[\left({b}^{2}-\frac{a{V}^{\prime }\left(h\right)}{2}-\lambda {t}_{0}{b}^{2}\right){\partial }_{X}^{2}R-a{V}^{″}\left(h\right)R{\partial }_{X}R+a{\partial }_{T}R\right]=0$ (18)

${\partial }_{T}R=\frac{\partial R}{\partial T}$${\partial }_{X}R=\frac{\partial R}{\partial X}$${\partial }_{X}^{k}R=\frac{{\partial }^{k}R}{\partial {X}^{k}}$

$b={V}^{\prime }\left(h\right)$，消去 $\epsilon$ 的二次项后，将(18)简化为：

$a{\partial }_{T}R-a{V}^{″}\left(h\right)R{\partial }_{X}R+{V}^{\prime }\left(h\right)\left({V}^{\prime }\left(h\right)-\frac{a}{2}-\lambda {t}_{0}{V}^{\prime }\left(h\right)\right){\partial }_{X}^{2}R=0$ (19)

$\frac{1}{2}{V}^{\prime }\left(h\right)\left[a-2\left(1-\lambda {t}_{0}\right){V}^{\prime }\left(h\right)\right]>0$ (20)

$\begin{array}{c}R\left(X,T\right)=\frac{1}{|{V}^{″}\left({h}_{c}\right)|T}\left[X-\frac{1}{2}\left({\eta }_{n}+{\eta }_{n+1}\right)\right]-\frac{1}{2|{V}^{″}\left({h}_{c}\right)|T}\left({\eta }_{n+1}-{\eta }_{n}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\mathrm{tanh}\left[\frac{{c}_{1}}{4|{V}^{″}\left({h}_{c}\right)|T}\left({\eta }_{n+1}-{\eta }_{n}\right)\left(X-{\zeta }_{n}\right)\right]\end{array}$ (21)

4.2. 在不稳定区域导出mKdV方程

$X=\epsilon \left(n+bt\right)$$T={\epsilon }^{3}t$(22)

$\Delta {x}_{n}\left(t\right)={h}_{c}+\epsilon R\left(X,T\right)$ (23)

$\begin{array}{l}a{\epsilon }^{2}\left(b-{V}^{\prime }\right){\partial }_{X}R+{\epsilon }^{3}\left[\left({b}^{2}-\frac{a{V}^{\prime }}{2}-\lambda {t}_{0}{b}^{2}\right){\partial }_{X}^{2}R-{V}^{″}{\partial }_{X}{R}^{2}\right]\\ \text{ }+{\epsilon }^{4}\left[a{\partial }_{T}R-\frac{a{V}^{‴}}{2}{\partial }_{X}{R}^{3}-\left(\frac{a{V}^{\prime }}{6}+\frac{1}{2}\lambda {t}_{0}^{2}{b}^{3}\right){\partial }_{X}^{3}R\right]\\ \text{ }+{\epsilon }^{5}\left[-2b{t}_{0}\lambda {\partial }_{X}R{\partial }_{T}R-\frac{a{V}^{‴}}{2}{\partial }_{X}^{2}{R}^{3}-\left(\frac{a{V}^{\prime }}{24}+\frac{1}{6}\lambda {t}_{0}^{3}{b}^{4}\right){\partial }_{X}^{4}R-\frac{a{V}^{\left(4\right)}}{6}{\partial }_{X}{R}^{4}\right]=0\end{array}$ (24)

$b={V}^{\prime }\left({h}_{c}\right)$$a=\left(1-{\epsilon }^{2}\right){a}_{c}$，和 ${a}_{c}=2{V}^{\prime }\left(1-\lambda {t}_{0}\right)$。在临界点 $\left({h}_{c},{a}_{c}\right)$ 附近，忽略 ${\epsilon }^{2}$${\epsilon }^{3}$ 量级，方程(24)可简化为：

${\partial }_{T}R-{k}_{1}{\partial }_{X}^{3}R+{k}_{2}{\partial }_{X}{R}^{3}+\epsilon \left({k}_{3}{\partial }_{X}^{2}R+{k}_{4}{\partial }_{X}^{4}R+{k}_{5}{\partial }_{X}^{2}{R}^{3}\right)=0$ (25)

${k}_{4}=\left(\frac{{V}^{\prime }}{3{a}_{c}}\lambda {t}_{0}-\frac{1}{24}\right){V}^{\prime }-\frac{1}{6{a}_{c}^{2}}\left(1+6\lambda \right)\lambda {t}_{0}^{3}{{V}^{\prime }}^{4}$${k}_{5}=\left(\frac{{V}^{\prime }}{{a}_{c}}\lambda {t}_{0}-\frac{1}{2}\right){V}^{‴}$

${T}^{\prime }={k}_{1}T$$R=\sqrt{\frac{{k}_{1}}{{k}_{2}}}{R}^{\prime }$ (26)

${\partial }_{{T}^{\prime }}{R}^{\prime }-{\partial }_{X}^{3}{R}^{\prime }+{\partial }_{X}{{R}^{\prime }}^{3}+\epsilon M\left[{R}^{\prime }\right]=0$ (27)

$M\left[{R}^{\prime }\right]=\frac{1}{{k}_{1}}\left({k}_{3}{\partial }_{X}^{2}{R}^{\prime }+{k}_{4}{\partial }_{X}^{4}{R}^{\prime }+\frac{{k}_{1}{k}_{5}}{{k}_{2}}{\partial }_{X}^{2}{{R}^{\prime }}^{3}\right)$ (28)

${{R}^{\prime }}_{0}\left(X,T\right)=\sqrt{c}\mathrm{tanh}\left(\sqrt{\frac{c}{2}}\left(X-c{T}^{\prime }\right)\right)$ (29)

$\left({{R}^{\prime }}_{0},M\left[{{R}^{\prime }}_{0}\right]\right)={\int }_{-\infty }^{+\infty }\text{d}X{{R}^{\prime }}_{0}\left(X,{T}^{\prime }\right)M\left[{{R}^{\prime }}_{0}\left(X,{T}^{\prime }\right)\right]=0$ (30)

${\int }_{-\infty }^{+\infty }\sqrt{\frac{c}{{k}_{1}{k}_{2}}}\left({k}_{2}{k}_{3}{\partial }_{X}^{2}{R}^{\prime }+{k}_{2}{k}_{4}{\partial }_{X}^{4}{R}^{\prime }+{k}_{1}{k}_{5}{\partial }_{X}^{2}{{R}^{\prime }}^{3}\right)\mathrm{tanh}\left(\sqrt{\frac{c}{2}}\left(X-c{T}^{\prime }\right)\right)\text{d}X=0$ (31)

$c=\frac{5{k}_{2}{k}_{3}}{2{k}_{2}{k}_{4}-3{k}_{1}{k}_{5}}$ (32)

$\Delta {x}_{n}\left(t\right)={h}_{c}+\sqrt{\frac{c{k}_{1}}{{k}_{2}}\left(1-\frac{a}{{a}_{c}}\right)}\mathrm{tanh}\sqrt{\frac{c}{2}\left(1-\frac{a}{{a}_{c}}\right)}\left[n+\left(1-c{k}_{1}\left(1-\frac{a}{{a}_{c}}\right)\right)t\right]$ (33)

$A=\sqrt{\frac{c{k}_{1}}{{k}_{2}}\left(1-\frac{a}{{a}_{c}}\right)}$，其中， ${a}_{c}=2{V}^{\prime }\left({h}_{c}\right)\left(1-\lambda {t}_{0}\right)$ (34)

4.3. 在亚稳态区域导出KdV方程

$X=\epsilon \left(n+bt\right)$$T={\epsilon }^{3}t$(35)

$\Delta {x}_{n}\left(t\right)={h}_{c}+{\epsilon }^{2}R\left(X,T\right)$ (36)

${\partial }_{T}R-{m}_{1}{\partial }_{X}^{3}R-{m}_{2}{\partial }_{X}{R}^{2}+\epsilon \left[{m}_{3}{\partial }_{X}^{2}R-{m}_{4}{\partial }_{X}^{4}R-{m}_{5}{\partial }_{X}^{2}{R}^{2}\right]=0$ (37)

${m}_{4}=\frac{1}{2{a}_{s}}+\frac{2{V}^{\prime }}{{a}_{s}}\left(\lambda -1\right)$${m}_{5}=\frac{{t}_{0}}{{a}_{s}}\left(\frac{{t}_{0}^{2}}{6}+\lambda {t}_{0}^{2}-\frac{{t}_{0}}{{a}_{s}}\right)\lambda {{V}^{\prime }}^{4}+\frac{\lambda {t}_{0}-1}{3{a}_{s}}{{V}^{\prime }}^{2}+\frac{{V}^{\prime }}{24}$

$T=\sqrt{{m}_{1}}{T}^{\prime }$$X=-\sqrt{{m}_{1}}{X}^{\prime }$$R=\frac{1}{{m}_{2}}{R}^{\prime }$(38)

${\partial }_{{T}^{\prime }}{R}^{\prime }+{\partial }_{{X}^{\prime }}^{3}{R}^{\prime }+{R}^{\prime }{\partial }_{{X}^{\prime }}{R}^{\prime }+\epsilon \sqrt{\frac{1}{{m}_{1}}}\left({m}_{3}{\partial }_{{X}^{\prime }}^{2}{R}^{\prime }-\frac{{m}_{4}}{{m}_{1}}{\partial }_{{X}^{\prime }}^{4}{R}^{\prime }-\frac{{m}_{5}}{{m}_{2}}{\partial }_{{X}^{\prime }}^{2}{{R}^{\prime }}^{2}\right)=0$ (39)

${{R}^{\prime }}_{0}\left({X}^{\prime },{T}^{\prime }\right)=A{\mathrm{sech}}^{2}\left[\sqrt{\frac{A}{12}}\left({X}^{\prime }-\frac{A}{3}{T}^{\prime }\right)\right]$ (40)

$\left({R}_{0},M\left[{R}_{0}\right]\right)\equiv {\int }_{-\infty }^{\infty }\text{d}{X}^{\prime }{R}_{0}M\left[{R}_{0}\right]=0$ (41)

$A=\frac{21{m}_{1}{m}_{2}{m}_{3}}{5{m}_{2}{m}_{4}-24{m}_{1}{m}_{5}}$ (42)

$R\left(X,T\right)=\frac{1}{{m}_{2}}A{\mathrm{sech}}^{2}\sqrt{\frac{A}{12{m}_{1}}}\left(n+{V}^{\prime }+\frac{A{\epsilon }^{2}}{3}\right)t$ (43)

$\Delta {x}_{n}\left(t\right)={h}_{c}+\frac{A}{{m}_{2}}\left(1-\frac{a}{{a}_{s}}\right)×\mathrm{sech}\left\{\sqrt{\frac{A}{12{m}_{1}}\left(1-\frac{a}{{a}_{s}}\right)}×\left[n+\left({V}^{\prime }\left(h\right)+\frac{A}{3}\left(1-\frac{a}{{a}_{s}}\right)\right)t\right]\right\}$ (44)

5. 结论

The Stability and Soliton of the Optimal Velocity Model Considering Speed Limit[J]. 力学研究, 2019, 08(03): 187-196. https://doi.org/10.12677/IJM.2019.83021

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