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 that leads 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 because the upstream traffic demand exceeds its capacity. Therefore, it is important to regulate the incoming traffic flow of the segment to avoid overloading the bottleneck area. Traffic densities on the freeway segment are described with the Lighthill-Whitham-Richards (LWR) macroscopic Partial Differential Equation (PDE) model. To mitigate the traffic congestion upstream of the bottleneck, incoming flow at the inlet of the freeway segment is controlled so that the optimal density that maximizes the outgoing flow is reached. The density and traffic flow relation at the bottleneck area, described with fundamental diagram, is considered to be un-known. 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.

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