PDE Traffic Observer Validated on Freeway Data

TL;DR

This paper introduces a boundary observer for freeway traffic state estimation based on the ARZ PDE model, enabling accurate real-time traffic monitoring from limited boundary data.

Contribution

It develops a PDE backstepping boundary observer for the nonlinear ARZ traffic model, ensuring exponential stability and finite-time convergence.

Findings

Observer accurately estimates traffic states from boundary measurements.

Numerical simulations validate the observer's effectiveness.

Model calibration with vehicle data confirms practical applicability.

Abstract

This paper develops boundary observer for estimation of congested freeway traffic states based on Aw-Rascle-Zhang (ARZ) partial differential equations (PDE) model. Traffic state estimation refers to acquisition of traffic state information from partially observed traffic data. This problem is relevant for freeway due to its limited accessibility to real-time traffic information. We propose a model-driven approach in which estimation of aggregated traffic states in a freeway segment are obtained simply from boundary measurement of flow and velocity without knowledge of the initial states. The macroscopic traffic dynamics is represented by the ARZ model, consisting of $2 \times 2$ coupled nonlinear hyperbolic PDEs for traffic density and velocity. Analysis of the linearized ARZ model leads to the study of a hetero-directional hyperbolic PDE model for congested traffic regime. Using spatial transformation and PDE backstepping method, we construct a boundary observer consisting of a copy of the nonlinear plant with output injections from boundary measurement errors. The output injection gains are designed for the estimation error system so that the exponential stability of the error system in the $L^2$ norm and finite-time convergence to zero are guaranteed. Numerical simulations are conducted to validate the boundary observer design for estimation of the nonlinear ARZ model. In data validation, we calibrate model parameters of the ARZ model and then use vehicle trajectory data to test the performance of the observer design.