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LQ Control of Traffic Flow Models via Variable Speed Limits

Brian Block, Stephanie Stockar

Abstract

In this paper, an extension of a linear control design for hyperbolic linear partial differential equations is presented for a first-order traffic flow model. Starting from the Lighthill-Whitham-Richards (LWR) model, variable speed limit control (VSL) is applied through a modification of Greenshield's equilibrium flow model. Then, an optimal linear quadratic (LQ) controller is designed on the linear LWR model. The LQ state feedback function is found via the solution of a Riccati differential equation. Unlike previous studies, the control input is the rate of change of the input, not the input itself. The proposed controller is then verified on both the linear and nonlinear models. In both cases, the controller is able to drive the system to a desired density profile. In the nonlinear application, a higher control gain is needed to achieve similar results as in the linear case.

LQ Control of Traffic Flow Models via Variable Speed Limits

Abstract

In this paper, an extension of a linear control design for hyperbolic linear partial differential equations is presented for a first-order traffic flow model. Starting from the Lighthill-Whitham-Richards (LWR) model, variable speed limit control (VSL) is applied through a modification of Greenshield's equilibrium flow model. Then, an optimal linear quadratic (LQ) controller is designed on the linear LWR model. The LQ state feedback function is found via the solution of a Riccati differential equation. Unlike previous studies, the control input is the rate of change of the input, not the input itself. The proposed controller is then verified on both the linear and nonlinear models. In both cases, the controller is able to drive the system to a desired density profile. In the nonlinear application, a higher control gain is needed to achieve similar results as in the linear case.
Paper Structure (14 sections, 39 equations, 7 figures, 1 table)

This paper contains 14 sections, 39 equations, 7 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. MODEL DESCRIPTION
  3. Lighthill-Whitham-Richards Model
  4. Linearized LWR Model
  5. CONTROL FORMULATION
  6. Variable Speed Limit Control
  7. Optimal Control Design for Hyperbolic PDE model
  8. LQ Control Deisgn for LWR Model with VSL
  9. SIMULATION RESULTS
  10. Uncontrolled Scenario
  11. Linear LQR Results
  12. Nonlinear LQR Results
  13. Comparison
  14. CONCLUSIONS

Figures (7)

  • Figure 1: Fundamental diagram showing free flow region (green) and congested region (red).
  • Figure 2: Influence of $Q_0$ on the state feedback function, $\Phi$.
  • Figure 3: Baseline results for linear model (a), (b) and nonlinear model (c), (d) given the initial and boundary conditions in \ref{['eq:IC']}, \ref{['eq:BC']}.
  • Figure 4: Total cars on highway section for varying values of $Q_0$ when control is applied to linear model.
  • Figure 5: Total cars on highway section for varying values of $Q_0$ when control is applied to nonlinear model.
  • ...and 2 more figures