  This paper develops boundary control for freeway traffic with a downstream
bottleneck. Traffic on a freeway segment with capacity drop at outlet of the
segment is a common phenomenon leading to traffic bottleneck problem. The
capacity drop can be caused by lane-drop, hills, tunnel, bridge or curvature on
the road. If incoming traffic flow remains unchanged, traffic congestion forms
upstream of the bottleneck due to outgoing traffic overflowing its capacity.
Therefore, it is important for us to regulate the incoming traffic flow of the
segment so that the outgoing traffic at the bottleneck can be discharged with
the maximum flow rate. Traffic densities on the freeway segment are described
with Lighthill-Whitham-Richards (LWR) macroscopic Partial Differential Equation
(PDE) model. To prevent the traffic congestion forming upstream of the
bottleneck, incoming flow at the inlet of the freeway segment is controlled so
that the optimal density could be achieved to maximize the outgoing flow and
not to surpass the capacity at outlet. The density and traffic flow relation,
described with fundamental diagram, is assumed to be unknown at the bottleneck
area. We tackle this problem using Extremum Seeking (ES) Control with delay
compensation for LWR PDE. ES control, a non-model based approach for real-time
optimization, is adopted to find the optimal density for the unknown
fundamental diagram. A predictor feedback control design is proposed to
compensate the delay effect of traffic dynamics in the freeway segment. In the
end, simulation results validate a desired performance of the controller on the
nonlinear LWR model with an unknown fundamental diagram.
