Neural Operators for Boundary Stabilization of Stop-and-go Traffic

Zhang, Yihuai; Zhong, Ruiguo; Yu, Huan · 2023 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2312.10374

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Summary

This paper addresses the challenge of stabilizing stop-and-go traffic oscillations in congested flow by designing efficient boundary control strategies. Traffic dynamics are modeled using the Aw-Rascle-Zhang (ARZ) system, a set of coupled hyperbolic partial differential equations (PDEs). While the backstepping method is a standard approach for stabilizing such PDEs, it requires solving complex kernel equations, a process that is computationally intensive and demands significant expertise. To overcome these limitations, the authors propose using neural operators (NOs) to approximate the control design, thereby accelerating computation and simplifying implementation. The study introduces two distinct neural operator learning schemes based on the DeepONet architecture. The first scheme approximates the backstepping gain kernels, embedding these learned kernels into a predefined backstepping controller structure. The second scheme directly learns the mapping from system parameters to the boundary control law, bypassing the kernel computation entirely. The authors provide rigorous theoretical analysis using Lyapunov methods to prove the stability of both approaches. They demonstrate that the closed-loop system with NO-approximated kernels is exponentially stable, while the system with the directly approximated control law is practically exponentially stable, provided the neural operator approximation error remains within specific bounds. Experimental validation was conducted using numerical simulations on a 500-meter road section with sinusoidal initial disturbances to mimic traffic waves. The neural operators were trained on 900 instances of varying equilibrium densities. The results were compared against a traditional PDE-based backstepping controller and a Proportional Integral (PI) controller. The NO-approximated kernel method successfully stabilized the traffic system, with density and velocity converging to equilibrium points similar to the backstepping baseline. The direct NO control law also stabilized the system but exhibited persistent small errors due to the approximation nature of the mapping. Crucially, the neural operator methods demonstrated significantly faster computation times than both the backstepping and PI controllers. Specifically, the NO-approximated kernel method took 0.0021 seconds and the direct NO control law took 0.0012 seconds, compared to 0.0847 seconds for the backstepping controller. Furthermore, the NO methods achieved lower average L2 errors than the PI controller, which required more control effort and converged more slowly. The significance of this work lies in demonstrating that neural operators can effectively replace computationally expensive PDE-based control designs in intelligent transportation systems. By drastically reducing computation time while maintaining stability and outperforming simpler PI controllers, this approach facilitates the potential for real-time, online application of advanced boundary control strategies in traffic management. The study establishes a framework for using operator learning to simplify and accelerate the deployment of robust control laws for hyperbolic PDE systems.

Key finding

Neural operator-based controllers for traffic boundary stabilization achieve faster computation speeds and lower errors than proportional-integral controllers, offering a viable alternative to computationally intensive backstepping methods.

Methodology

simulation_modeling

Provenance

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Topics

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• teleoperation remote driving