Varying Speed Limit Control of Aw-Rascle-Zhang Traffic Model

This paper develops Varying Speed Limit(VSL) boundary control to reduce stop-and-go oscillations in congested traffic. The macroscopic traffic dynamics are governed by Aw-Rascle-Zhang(ARZ) model, consisting of second-order nonlinear partial differential equations(PDEs). The linear stability of uniform density and velocity of the ARZ model with relaxation term is discussed. Under certain densities in congested regime, small perturbations from the equilibrium transport upstream of traffic and grow into instabilities. To stabilize the inhomogeneous ARZ model around the uniform steady states, a boundary control input through VSL at outlet is considered. Using spatial transformation and backstepping method, we design a full-state feedback control law. By taking boundary measurement of density variation at the outlet, a collocated boundary observer is designed to estimate states on freeway segment. Therefore, we obtain the output feedback controller by combining the full state feedback control and collocated boundary observer. The exponential stability in $L_{2}$ sense and finite time convergence to equilibrium are achieved with boundary control law and boundary observer. In the end, we validate our result with simulation.