Output-Feedback PDE Control of Traffic Flow on Cascaded Freeway Segments

Yu, Huan; Auriol, Jean; Krstić, Miroslav · 2020 · IFAC-PapersOnLine

DOI: 10.1016/j.ifacol.2020.12.1362

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Summary

This paper addresses the challenge of stabilizing traffic flow on an underactuated network consisting of two connected freeway segments: one incoming and one outgoing road joined at a junction. The research is motivated by the limitations of existing control strategies, which typically focus on single road segments or spatially discretized models, leaving the problem of simultaneously controlling upstream and downstream traffic with a single boundary actuation largely unexplored. The authors aim to develop a boundary output feedback control law that stabilizes the macroscopic traffic dynamics around a chosen reference steady state, specifically targeting the suppression of stop-and-go oscillations in congested regimes. The study utilizes the second-order Aw-Rascle-Zhang (ARZ) model, which describes traffic density and velocity through nonlinear hyperbolic partial differential equations (PDEs). The system is modeled as two interconnected PDE systems coupled through their boundaries at the junction. The control actuation is implemented via ramp metering at the outlet of the outgoing road, allowing only the traffic flux leaving the domain to be manipulated. The authors linearize the ARZ model around a steady state and employ a backstepping approach to design the controller. Specifically, they construct a delay-robust full state feedback control law and a boundary observer using measurements of traffic flux and velocity taken at the junction. The design involves invertible backstepping transformations that map the original coupled system into a decoupled target system, ensuring exponential stability. The main findings demonstrate that the proposed control law successfully stabilizes the traffic network on both road segments. The theoretical analysis proves that the closed-loop system achieves exponential convergence to the reference steady state in the sense of the $L^2$-norm. The control design is shown to be robust to delays in actuation, a critical feature for practical implementation. The paper provides explicit formulas for the control kernels and observer gains, derived from solving specific PDEs on triangular domains. The results confirm that the underactuated nature of the network—where only one boundary is controlled—can be effectively managed through the proposed output feedback strategy, provided certain assumptions regarding driver aggressiveness and traffic pressure ratios are met. The significance of this work lies in its contribution to the field of PDE control for traffic networks. It provides a rigorous theoretical framework for controlling interconnected traffic systems using boundary actuation, filling a gap in literature that previously focused on single segments. The development of a boundary observer for this underactuated network is a novel theoretical contribution, enabling practical implementation where full state information is not available. This approach offers flexibility for evaluating different performance objectives, such as fuel consumption and travel time, and provides a foundation for future research on more complex traffic network topologies.

Key finding

The proposed output feedback control law, combining a boundary observer and backstepping-based state feedback, achieves exponential stabilization of traffic flow on cascaded freeway segments using a single actuator.

Methodology

modeling

Provenance

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Topics

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• traffic density