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Performance-Barrier Event-Triggered PDE Control of Traffic Flow

Peihan Zhang, Bhathiya Rathnayake, Mamadou Diagne, Miroslav Krstic

TL;DR

The paper addresses STOP-and-GO traffic instabilities by applying boundary VSL control to the linearized Aw–Rascle–Zhang PDE model. It advances the control design with Performance-Barrier ETC (P-ETC), allowing Lyapunov function increases as long as a performance barrier is respected, and extends this framework to continuous-time, periodic, and self-triggered implementations (P-CETC, P-PETC, P-STC). These methods yield exponentially stable convergence in the spatial $L^2$ norm while ensuring Zeno-free behavior and enabling sparser VSL updates, which improves driver safety and comfort and reduces cognitive load. The paper also extends existing R-ETC methods to PETC/STC forms, providing explicit dwell-time guarantees and a robust mechanism to switch to R-ETC during disturbances. Simulation results on the linear ARZ model demonstrate substantial reductions in driver discomfort and notable safety improvements, with a tunable trade-off controlled by the resource-aware parameter $c$. The work lays groundwork for practical VSL implementations under event-based schemes and points to future extensions to nonlinear ARZ dynamics and quantization effects.

Abstract

For stabilizing stop-and-go oscillations in traffic flow by actuating a variable speed limit (VSL) at a downstream boundary of a freeway segment, we introduce event-triggered PDE backstepping designs employing the recent concept of performance-barrier event-triggered control (P-ETC). Our design is for linearized hyperbolic Aw-Rascle-Zhang (ARZ) PDEs governing traffic velocity and density. Compared to continuous feedback, ETC provides a piecewise-constant VSL commands-more likely to be obeyed by human drivers. Unlike the existing regular ETC (R-ETC), which enforces conservatively a strict decrease of a Lyapunov function, our performance-barrier (P-ETC) approach permits an increase, as long as the Lyapunov function remains below a performance barrier, resulting in fewer control updates than R-ETC. To relieve VSL from continuously monitoring the triggering function, we also develop periodic event-triggered (PETC) and self-triggered (STC) versions of both R-ETC and P-ETC. These are referred to as R/P-PETC and R/P-STC, respectively, and we show that they both guarantee Zeno-free behavior and exponential convergence in the spatial $L^2$ norm. With comparative simulations, we illustrate the benefits of the performance-barrier designs through traffic metrics (driver comfort, safety, travel time, fuel consumption). The proposed algorithms reduce discomfort nearly in half relative to driver behavior without VSL, while tripling the driver safety, measured by the average dwell time, relative to the R-ETC frequent-switching VSL schedule.

Performance-Barrier Event-Triggered PDE Control of Traffic Flow

TL;DR

The paper addresses STOP-and-GO traffic instabilities by applying boundary VSL control to the linearized Aw–Rascle–Zhang PDE model. It advances the control design with Performance-Barrier ETC (P-ETC), allowing Lyapunov function increases as long as a performance barrier is respected, and extends this framework to continuous-time, periodic, and self-triggered implementations (P-CETC, P-PETC, P-STC). These methods yield exponentially stable convergence in the spatial norm while ensuring Zeno-free behavior and enabling sparser VSL updates, which improves driver safety and comfort and reduces cognitive load. The paper also extends existing R-ETC methods to PETC/STC forms, providing explicit dwell-time guarantees and a robust mechanism to switch to R-ETC during disturbances. Simulation results on the linear ARZ model demonstrate substantial reductions in driver discomfort and notable safety improvements, with a tunable trade-off controlled by the resource-aware parameter . The work lays groundwork for practical VSL implementations under event-based schemes and points to future extensions to nonlinear ARZ dynamics and quantization effects.

Abstract

For stabilizing stop-and-go oscillations in traffic flow by actuating a variable speed limit (VSL) at a downstream boundary of a freeway segment, we introduce event-triggered PDE backstepping designs employing the recent concept of performance-barrier event-triggered control (P-ETC). Our design is for linearized hyperbolic Aw-Rascle-Zhang (ARZ) PDEs governing traffic velocity and density. Compared to continuous feedback, ETC provides a piecewise-constant VSL commands-more likely to be obeyed by human drivers. Unlike the existing regular ETC (R-ETC), which enforces conservatively a strict decrease of a Lyapunov function, our performance-barrier (P-ETC) approach permits an increase, as long as the Lyapunov function remains below a performance barrier, resulting in fewer control updates than R-ETC. To relieve VSL from continuously monitoring the triggering function, we also develop periodic event-triggered (PETC) and self-triggered (STC) versions of both R-ETC and P-ETC. These are referred to as R/P-PETC and R/P-STC, respectively, and we show that they both guarantee Zeno-free behavior and exponential convergence in the spatial norm. With comparative simulations, we illustrate the benefits of the performance-barrier designs through traffic metrics (driver comfort, safety, travel time, fuel consumption). The proposed algorithms reduce discomfort nearly in half relative to driver behavior without VSL, while tripling the driver safety, measured by the average dwell time, relative to the R-ETC frequent-switching VSL schedule.
Paper Structure (24 sections, 14 theorems, 78 equations, 3 figures, 1 table)

This paper contains 24 sections, 14 theorems, 78 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Boundary control of ARZ traffic model: an embodiment of coupled hyperbolic PDE systems
  3. Sampled-data and event-based control of PDE systems
  4. Results
  5. Contributions
  6. Notation
  7. Organization
  8. Continuous-time Control and Emulation
  9. Aw–Rascle–Zhang (ARZ) model
  10. Continuous-time PDE Backstepping Control
  11. Emulation of the PDE Backstepping Control
  12. Regular Event-Triggered Control (R-ETC)
  13. Regular Continuous-time Event-triggered Control (R-CETC)
  14. Regular Periodic Event-triggered Control (R-PETC)
  15. Regular Self-triggered Control (R-STC)
  16. ...and 9 more sections

Key Result

Proposition 1

For given $\left(\bar{w} (\cdot, t_k), \bar{v} (\cdot, t_k)\right)^T \in L^2\left((0,\ell); \mathbb{R}^2\right)$, there exists a unique solution $(\bar{w}, \bar{v})^T \in \mathcal{C}^0\left(\left[t_k, t_{k+1}\right] ; L^2\left((0,\ell) ; \mathbb{R}^2\right)\right)$ to the system eq:sys-wv-wbar-eq:sy

Figures (3)

  • Figure 1: Interrelation among various technical and principal results in the paper.
  • Figure 2: Comparison of the Lyapunov function, control update and dwell-times under the R/P-CETC, R/P-PETC and R/P-STC.
  • Figure 3: Comparison of $\rho(x,t)$ and $v(x,t)$.

Theorems & Definitions (24)

  • Proposition 1: Well-Posedness between control updates
  • Remark 1
  • Theorem 1: Results under R-CETC
  • Remark 2
  • Theorem 2: Results under R-STC (resp. R-PETC)
  • Lemma 1
  • Lemma 2
  • Theorem 3: Results under P-CETC
  • proof : Proof
  • Remark 3
  • ...and 14 more