  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.
