Stability of Ring Roads and String Stability of Car Following Models

The ring road, a closed circular track, provides a controlled environment for studying car-following behavior in dynamic traffic flow. This work examines the ring road from a stability analysis perspective to shed light on its underlying stability mechanisms. We model the ring road as an interconnected system composed of subsystems of ODEs which can represent car-following dynamics. We analyze its stability based on the string stability of these subsystems and then apply the findings to ring road models. Our study addresses both interconnected systems consisting of homogeneous (identical) and heterogeneous subsystems. For systems with heterogeneous subsystems, we demonstrate that either string stability or collective string stability of the subsystems provides a sufficient condition for local asymptotic stability. For systems with identical subsystems, we show that string stability of each subsystem is both a necessary and sufficient condition for asymptotic stability. Numerical simulations validate our findings, illustrating that string-unstable car-following models lead to an unstable ring road, while string-stable models ensure stability. Additionally, using the string stability condition, we design a controller for an autonomous vehicle to stabilize a previously unstable ring road. Simulations using both linear and nonlinear car-following models further confirm our results.