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Traffic Flow Theory

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Feedback Control Theory for Dynamic Traffic Assignment

Part of the book series: Advances in Industrial Control ((AIC))

Abstract

This chapter presents the basic traffic flow theory which is used in the following chapters for control problem formulations. The theory develops the Lighthill–Whitham–Richards (LWR) model that uses the conservation law for traffic. Additionally, a density-dependent speed formula is used. There are many relationships available for this fundamental diagram, the chapter uses Greenshields’ formula for further analysis. Elementary partial differential equations (PDE) theory is also presented including the method of characteristics needed for the analysis of the traffic model. Shockwaves and weak solutions are discussed followed by a brief discussion of traffic measurements.

Section 3.3.8 “Diffusion Model” is adapted from the paper by Pushkin Kachroo, Kaan Özbay, Sungkwon Kang, and John A. Burns, “System Dynamics and Feedback Control Problem Formulations for Real Time Dynamic Traffic Routing,” Mathl. Comput. Modelling Vol. 27, No. 9–11, pp. 27–49, DOI: https://doi.org/10.1016/S0895-7177(98)00050-8, ©1998, with permission from Elsevier.

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Notes

  1. 1.

    Implicit Function Theorem: Let F be a function of three variables of class \(C^1\) in an open set \(\mathcal {O}\) given by \(F(x,y,z)=C\). Then z can be solved in terms of x and y for (x, y, z) near the point \((x_{0},y_{0},z_{0})\) if \(F_z(x_{0},y_{0},z_{0})\ne 0\). Writing z as a function of x and y in the equation gives \(F(x,y,z(x,y))=C\). Differentiating with respect to x gives \(F_x+F_zz_x=0\) and differentiating with respect to y gives \(F_y+F_zz_y=0\). Therefore, we have \(z_x=-F_x/F_z\) and \(z_y=-F_y/F_z\). Hence, \(z(x,y)\approx z_0+z_xdx+z_ydy\).

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Nevada, Las Vegas, NV, USA

    Pushkin Kachroo

  2. Department of Civil and Urban Engineering, New York University, Brooklyn, NY, USA

    Kaan M. A. Özbay

Authors
  1. Pushkin Kachroo
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  2. Kaan M. A. Özbay
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Correspondence to Pushkin Kachroo .

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Kachroo, P., Özbay, K.M.A. (2018). Traffic Flow Theory. In: Feedback Control Theory for Dynamic Traffic Assignment. Advances in Industrial Control. Springer, Cham. https://doi.org/10.1007/978-3-319-69231-9_3

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  • DOI: https://doi.org/10.1007/978-3-319-69231-9_3

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