Wave Equation of Suppressed Traffic Flow Instabilities

Abstract

Traffic congestion wastes fuel and commuters’ time, and adds to CO2 emissions. Stop-and-go traffic instabilities can be suppresses using bilateral control—which differs from “car following” and adaptive cruise control in that, counter-intuitively, it uses information about the following vehicle (as well as about the leading vehicle). Stability can be proven mathematically, and can be demonstrated in simulation. A physical analog of a sequence of vehicles using bilateral control is a chain of masses connected by springs and dampers—a system which is inherently stable, since it lacks an external energy source. Here, in order to further understand bilateral control and its capacity to suppress instabilities, we move from a microscopic view (interaction of individual vehicles) to a macroscopic view (densities and flow rates). This leads us to the damped wave equation governing traffic under bilateral control. That equation allows us to determine the speed of propagation of disturbances, as well as their rate of decay. The equation is also useful in fine tuning parameters of bilateral control systems.