Bilateral Boundary Control of Moving Shockwave in LWR Model of Congested Traffic

Yu, Huan; Diagne, Mamadou; Zhang, Liguo; Krstić, Miroslav · 2020 · IEEE Transactions on Automatic Control

DOI: 10.1109/tac.2020.2994031

archive: archived pipeline: cataloged verified

Get this paper ↗ (DOI — opens at the source; we link to it, we don't host it)

Summary

This paper addresses the problem of stabilizing moving shockwaves in congested freeway traffic using bilateral boundary control. Moving shockwaves, characterized by a discontinuity in traffic density where light traffic upstream transitions to heavy traffic downstream, propagate upstream due to flux differences, causing congestion and unsafe driving conditions. The authors aim to halt this upstream propagation and drive the shockwave front to a static setpoint position using boundary actuations, such as ramp metering and varying speed limits. The study is motivated by the need for macroscopic traffic control strategies that can dissipate congestion by regulating traffic density and velocity at the boundaries of a freeway segment. The traffic dynamics are modeled using the Lighthill-Whitham-Richards (LWR) model, a first-order hyperbolic partial differential equation (PDE) describing traffic density. The moving shockwave creates a coupled PDE-ODE system: two PDEs govern the traffic density in the upstream (free) and downstream (congested) regimes, while an ordinary differential equation (ODE), derived from the Rankine-Hugoniot condition, governs the position of the moving interface. The authors linearize this nonlinear coupled system around steady-state values. To design the controller, they transform the PDE-ODE system into an equivalent representation of an ODE with two state-dependent input delays. Using a predictor-based backstepping transformation, they design state feedback controllers that cooperatively compensate for these delays. The control inputs are applied at both the inlet and outlet of the freeway segment to regulate the incoming and outgoing traffic flux. The main finding is the development of a predictor feedback control law that ensures the local exponential stability of the closed-loop system in the $H^1$ norm. Through Lyapunov stability analysis, the authors prove that the proposed controllers can stabilize both the PDE states (traffic densities) and the ODE state (shockwave position) to their respective setpoints. This effectively halts the upstream propagation of the shockwave, keeping it stationary within the freeway segment. The theoretical results are validated through numerical simulations, which demonstrate the effectiveness of the control strategy in stabilizing the moving shockwave and maintaining desired traffic states. The significance of this work lies in providing the first theoretical result on the control of two PDE state-dependent input delays to an ODE, extending previous backstepping methods that addressed single or constant delays. Furthermore, it addresses a novel application of boundary control to traffic moving shockwaves, a problem previously unaddressed in the literature. The proposed method offers a rigorous framework for using boundary actuations to manage traffic congestion caused by road geometry changes, such as hills and curves, thereby improving traffic flow safety and efficiency.

Key finding

The proposed predictor-based backstepping controller successfully stabilizes the moving shockwave position and traffic densities, ensuring local exponential stability of the closed-loop PDE-ODE system.

Methodology

simulation_modeling

Provenance

The full processing record for this entry. Every stage of this paper's journey through the pipeline is logged — what ran, with which tool and model, how many attempts it took, and when it last completed. Discovered via author_sweep_intake on 2026-05-28.

Summary generated by qwen3.6-27b-prismaquant on 2026-06-10; verification: verified.

Topics

Ranked by relevance to this paper. Hover a topic for its definition.

• traffic density