Mean-square exponential stabilization of mixed-autonomy traffic PDE system

Zhang, Yihuai; Yu, Huan; Auriol, Jean; Pereira, Mike · 2024 · Automatica

DOI: 10.1016/j.automatica.2024.111859

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Summary

This paper addresses the boundary stabilization of mixed-autonomy traffic systems on freeways, where Human-driven Vehicles (HVs) and Autonomous Vehicles (AVs) coexist. The research is motivated by the prevalence of stop-and-go traffic oscillations, which increase travel time, fuel consumption, and accident risks. While AVs were initially expected to improve road capacity, current commercial AVs often maintain larger spacing than HVs due to safety concerns, leading to complex interactions such as overtaking. The authors aim to stabilize these oscillations by modeling the traffic dynamics as uncertain coupled hyperbolic partial differential equations (PDEs) with Markov jumping parameters, specifically accounting for the stochastic nature of AV spacing policies caused by communication losses and control delays. The study develops an extended Aw-Rascle-Zhang (ARZ) model to describe the macroscopic spatial-temporal dynamics of HVs and AVs. The model incorporates area occupancy to capture interactions between vehicle classes and uses a continuous-time Markov chain to govern the stochastic impact area of AVs. The system is linearized around its equilibrium state, resulting in a stochastic linear hyperbolic PDE system. To stabilize the system, the authors employ a backstepping design method to derive a full-state feedback boundary control law. This control input is implemented via ramp metering. The theoretical analysis utilizes Lyapunov methods to prove the stability of the closed-loop system, introducing a novel stochastic Lyapunov candidate to handle the Markov-jumping parameters. The main findings demonstrate that the nominal backstepping control law, designed for the deterministic system, successfully stabilizes the stochastic mixed-autonomy traffic system provided the nominal parameters are sufficiently close to the stochastic ones on average. The authors derive specific conditions for mean-square exponential stability, ensuring that the traffic oscillations decay exponentially in the mean-square sense. These theoretical results are validated through numerical simulations, which confirm the effectiveness of the proposed control strategy in stabilizing the traffic flow despite the stochastic variations in AV behavior. The significance of this work lies in its novel modeling approach and control strategy for stochastic mixed-autonomy traffic. It is the first model to describe the macroscopic dynamics of HVs and AVs using an extended ARZ model with Markov-jumping parameters. The development of a robust stabilization method using a nominal controller for a stochastic system provides a theoretical foundation for managing mixed traffic scenarios. This research paves the way for practical applications in traffic control systems, offering a method to mitigate instabilities arising from the heterogeneous and stochastic nature of autonomous and human-driven vehicles sharing the road.

Key finding

A backstepping-based boundary control law achieves mean-square exponential stabilization of mixed-autonomy traffic systems with Markov jumping parameters, provided the nominal parameters are sufficiently close to the stochastic parameters on average.

Methodology

simulation_modeling

Provenance

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Topics

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• traffic density