Analysis and design of a variable speed limit control system at a freeway lane-drop bottleneck: A switched systems approach

As traffic systems are genuinely nonlinear with uncongested and congested patterns, advanced control and management strategies are needed to improve their safety, mobility, and environmental impacts. In this study, we consider the control design problem for a traffic system upstream to a lane-drop bottleneck regulated by variable speed limits. The design goal is to mitigate the impacts of capacity drop by maximizing the out-flux and minimizing the delay as well as stabilizing the system at an ideal equilibrium state. Through analyzing the equilibrium states and their stability properties of an open-loop switched system, we first design the set point for density and demonstrate the hysteretic relation between the speed limit and the equilibrium state. For the closed-loop switched system with a PI-controller, we derive the region of the controller parameters for the system to be asymptotically stable and derive a Poincaré map to prove the existence of stable limit cycles when the system is unstable. Through this study, we have the following findings for a system with excessive demands: without control, the system converges to the congested equilibrium state with reduced out-fluxes; with open-loop control, the ideal uncongested equilibrium state is introduced, but the congested equilibrium state still exists and is asymptotically stable; with a well-designed PI feedback controller, the congested equilibrium state is removed, and the system can be stablized at the ideal uncongested equilibrium state with maximum out-fluxes. With numerical simulations, we verify the analytical results and findings.