Simultaneous downstream and upstream output-feedback stabilization of cascaded freeway traffic

Abstract

In this paper, we develop boundary output feedback control laws for traffic flow on two cascaded freeway segments connected by a junction. The macroscopic traffic dynamics are governed by the Aw–Rascle–Zhang (ARZ) model in which two subsystems of second-order nonlinear partial differential equations (PDEs) describe the evolution of traffic density and velocity on each segment. Due to the change of road access at the junction, different equilibria are considered for the two connected segments. To suppress stop-and-go traffic oscillations on the cascaded roads, we consider a ramp metering that regulates the traffic flow rate entering from the on-ramp to the mainline freeway. Different control designs are proposed such that the output feedback stabilization is realized with either the ramp metering located at the middle junction or the outlet with only boundary measurements of flow rate and velocity. The control objective is to simultaneously stabilize the upstream and downstream traffic to a given spatially-uniform constant steady-state. The distinct actuation locations motivate our design of two different delay-robust full state feedback control laws. The proposed designs are based on the PDE backstepping methodology and guarantee the exponential stability of the under-actuated network of two systems of two hyperbolic PDEs. Collocated boundary observers are proposed to construct output feedback controllers. Numerical simulations are performed to validate the control designs. Stabilization performance of the two output feedback is compared and evaluated with Proportional Integral (PI) boundary feedback controllers. Robustness to delays is also investigated.