Simultaneous Stabilization of Traffic Flow on Two Connected Roads

Yu, Huan; Auriol, Jean; Krstić, Miroslav · 2020 · Unknown

DOI: 10.23919/acc45564.2020.9147929

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Summary

This paper addresses the challenge of simultaneously stabilizing traffic flow on two connected road segments—an incoming road and an outgoing road linked by a junction—to suppress stop-and-go congestion. While previous control strategies could stabilize traffic either upstream or downstream of a ramp, they failed to address the coupled dynamics of both segments simultaneously. The authors aim to design a boundary state feedback control law that stabilizes both the upstream and downstream traffic to specific spatially uniform steady states, despite the segments having different equilibrium conditions (e.g., lane counts and maximum densities). The study utilizes the second-order Aw-Rascle-Zhang (ARZ) macroscopic traffic model, which describes traffic density and velocity via nonlinear hyperb partial differential equations (PDEs). The system is modeled as a network of two interconnected PDE systems coupled at the junction boundary ($x=0$). The control actuator is a ramp metering mechanism at the junction that regulates the traffic flow rate entering from an on-ramp. The authors linearize the ARZ model around chosen steady states and apply a backstepping control methodology. This involves using Volterra integral transformations to map the original underactuated system, which has in-domain coupling terms, into a target system where these couplings are moved to the actuated boundary. A full-state feedback control law is then derived to eliminate these boundary terms, ensuring the target system is exponentially stable. The theoretical results are validated through numerical simulations. The simulation setup consists of two 2 km road segments with a total length of 4 km. The downstream segment has six lanes with a maximum density of 800 vehicles/km, while the upstream segment has fewer lanes and a maximum density of 700 vehicles/km. Both segments are initialized in a congested regime with sinusoidal disturbances. The control law successfully stabilizes the traffic flow to the desired steady states (600 vehicles/km for downstream, 488.6 vehicles/km for upstream) in an $L^2$ sense. The results demonstrate exponential convergence of the traffic density and velocity deviations to zero, confirming that the ramp metering control effectively suppresses oscillations across the entire network. The significance of this work lies in providing an explicit control design for the simultaneous stabilization of cascaded traffic networks, a problem previously left unanswered in single-segment studies. By treating the junction as a coupling boundary and using backstepping to handle the underactuated nature of the network, the authors establish a foundational approach for controlling macroscopic traffic models on road networks. This method preserves robustness margins and offers a scalable framework for managing traffic congestion in more complex network structures.

Key finding

A backstepping-based boundary state feedback control law using ramp metering at a junction can simultaneously stabilize traffic flow on two connected road segments with different equilibrium conditions, achieving exponential convergence to steady states.

Methodology

simulation_modeling

Provenance

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Topics

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• traffic density