Bilateral Boundary Control of Moving Shockwave in LWR Model of Congested Traffic

We develop backstepping state feedback control to stabilize a moving shockwave in a freeway segment under bilateral boundary actuations of traffic flow. A moving shockwave, consisting of light traffic upstream of the shockwave and heavy traffic downstream, is usually caused by changes of local road situations. The density discontinuity travels upstream and drivers caught in the shockwave experience transitions from free to congested traffic. Boundary control design in this paper brings the moving shockwave front to a static setpoint position, hindering the upstream propagation of traffic congestion. The traffic dynamics are described with Lighthill-Whitham-Richard (LWR) model, leading to a system of two first-order hyperbolic partial differential equations (PDEs). Each represents the traffic density of a spatial domain segregated by the moving interface. By Rankine-Hugoniot condition, the interface position is driven by flux discontinuity and thus governed by a PDE state dependent ordinary differential equation (ODE). For the PDE-ODE coupled system. the control objective is to stabilize both the PDE states of traffic density and the ODE state of moving shock position to setpoint values. Using delay representation and backstepping method, we design predictor feedback controllers to cooperatively compensate state-dependent input delays to the ODE. From Lyapunov stability analysis, we show local stability of the closed-loop system in $H^1$ norm. The performance of controllers is demonstrated by numerical simulation.