On string stability of a mixed and heterogeneous traffic flow: A unifying modelling framework

Urged by a close future perspective of a traffic flow made of a mix of human-driven vehicles and connected, automated vehicles (CAVs), research has recently focused at making the most of CAVs capabilities to mitigate the instability of the whole, i.e. mixed, traffic flow. In all works, however, either the two sub-flows are studied under a simplifying but unrealistic assumption of flow homogeneity, or drivers’ and vehicles heterogeneity is not correctly taken into account within each sub-flow. We show here that the only condition developed so far to study a car-following model string stability for a heterogeneous flow, is inaccurate. Therefore, we propose a methodology to model string stability that considers drivers’ and vehicles heterogeneity, which is the essence of a real traffic. Uncertain transfer functions are introduced to map the probability distributions of car-following model parameters into a L2 stability measure of a mixed and heterogeneous traffic. Specifically, they allow us to move from the stability analysis of a car-following model, or of a controller, to the stability analysis of a traffic flow, as interpreted by that model, or controller. Eventually, several other theoretical contributions on stability analysis are given in the paper, aiming at reconciling approaches from different fields. Among these, a mathematical justification of the equivalence between the asymptotic stability of a closed-loop platoon system – which has been studied through the famous “traffic wave ansatz” on a ring-road – and the L2 stability of an open-loop platoon system.