  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 boundary observer design so that estimates of aggregated traffic
states in a freeway segment are obtained simply from boundary measurement of
flow and velocity. 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 with a copy of the nonlinear plant and output injection of
boundary measurement errors. The output injection gains are designed for the
error system of the linearized ARZ model so that the exponential stability of
error system in the $L^2$ norm and finite-time convergence to zero are
guaranteed. Simulations are conducted to validate the boundary observer design
for nonlinear ARZ model without knowledge of initial conditions.
