Delay-Compensated Distributed PDE Control of Traffic With Connected/Automated Vehicles

We develop an input delay-compensating design for stabilization of an Aw–Rascle–Zhang (ARZ) traffic model in congested regime, which is governed by a
$2\times 2$
first-order hyperbolic nonlinear partial differential equation (PDE). The traffic flow consists of both adaptive cruise control-equipped (ACC-equipped) and manually driven vehicles. The control input is the time gap of ACC-equipped and connected vehicles, which is subject to delays resulting from communication lag. For the linearized system, a novel three-branch bakcstepping transformation with explicit kernel functions is introduced to compensate the input delay. The transformation is proved to be bounded, continuous, and invertible, with explicit inverse transformation derived. Based on the transformation, we obtain the explicit predictor-feedback controller. We prove exponential stability of the closed-loop system with the delay compensator in
$L_{2}$
norm. The performance improvement of the closed-loop system under the proposed controller is illustrated in simulation.