Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constrained Control of PDE Traffic Flow via Spatial Control Barrier Functions

Brian Block, Stephanie Stockar

TL;DR

The paper tackles safety-aware, in-domain control of PDE traffic models described by the Lighthill-Whitham-Richards equation. It extends control Lyapunov function (CLF) theory to PDEs and introduces spatially varying control barrier functions (sCBFs) that enforce density safety bounds within a quadratic programming framework that also accounts for CLF objectives. A unified CLF+sCBF controller is applied to in-domain variable speed limit (VSL) control, showing that CLF alone can violate safety while sCBF enforces the bounds and the combined controller tracks a target density profile. The approach offers a principled method for safety-critical PDE control in traffic networks with potential practical applications in congestion management.

Abstract

In this paper, a constrained control approach to variable speed limit (VSL) control for macroscopic partial differential equations (PDE) traffic models is developed. Control Lyapunov function (CLF) theory for ordinary differential equations (ODE) is extended to account for spatially and temporally varying states and control inputs. The stabilizing CLF is then unified with safety constraints through the introduction of spatially varying control barrier functions (sCBF). These methods are applied to in-domain VSL control of the Lighthill-Whitham-Richards (LWR) model to regulate traffic density to a desired profile while ensuring the density remains below prescribed limits enforced by the sCBF. Results show that incorporating constrained control minimally affects the stabilizing control input while successfully maintaining the density with the defined safe set.

Constrained Control of PDE Traffic Flow via Spatial Control Barrier Functions

TL;DR

The paper tackles safety-aware, in-domain control of PDE traffic models described by the Lighthill-Whitham-Richards equation. It extends control Lyapunov function (CLF) theory to PDEs and introduces spatially varying control barrier functions (sCBFs) that enforce density safety bounds within a quadratic programming framework that also accounts for CLF objectives. A unified CLF+sCBF controller is applied to in-domain variable speed limit (VSL) control, showing that CLF alone can violate safety while sCBF enforces the bounds and the combined controller tracks a target density profile. The approach offers a principled method for safety-critical PDE control in traffic networks with potential practical applications in congestion management.

Abstract

In this paper, a constrained control approach to variable speed limit (VSL) control for macroscopic partial differential equations (PDE) traffic models is developed. Control Lyapunov function (CLF) theory for ordinary differential equations (ODE) is extended to account for spatially and temporally varying states and control inputs. The stabilizing CLF is then unified with safety constraints through the introduction of spatially varying control barrier functions (sCBF). These methods are applied to in-domain VSL control of the Lighthill-Whitham-Richards (LWR) model to regulate traffic density to a desired profile while ensuring the density remains below prescribed limits enforced by the sCBF. Results show that incorporating constrained control minimally affects the stabilizing control input while successfully maintaining the density with the defined safe set.

Paper Structure

This paper contains 15 sections, 27 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. MODEL DESCRIPTION
  3. Lighthill-Whitham-Richards Model
  4. Variable Speed Limits
  5. Linearization of LWR Model
  6. STABILIZING CONTROL OF TRAFFIC MODELS
  7. Control Lyapunov Theory for ODEs
  8. Application of CLF Theory to Traffic PDEs
  9. SAFE CONTROL OF TRAFFIC MODELS
  10. Control Barrier Function Theory for ODEs
  11. Application of CBF Theory to Traffic PDEs
  12. SIMULATION RESULTS
  13. Case Study Definition
  14. Results
  15. CONCLUSION

Figures (7)

  • Figure 1: Triangular fundamental diagram.
  • Figure 2: Initial density profile (a) and $\lambda$ (b)
  • Figure 3: Desired density profiles for the CLF-based controller.
  • Figure 4: Uncontrolled density (a) and speed limit (b).
  • Figure 5: Density and VSL results for CLF control (a)(b), sCBF control (c)(d), and CLF+sCBF control (e)(f).
  • ...and 2 more figures