$\mathcal{L}_2$ and $\mathcal{L}_{\infty}$ Stability Analysis of Heterogeneous Traffic With Application to Parameter Optimization for the Control of Automated Vehicles

The presence of (partially) automated vehicles on the roads presents an opportunity to compensate the unstable behavior of conventional vehicles. Vehicles subject to perturbations should: 1) recover their equilibrium speed and 2) react not to propagate but absorb perturbations. In this paper, we start with considering vehicle systems consisting of heterogeneous vehicles updating their dynamics according to realistic behavioral car-following models. Definitions of all types of stability that are of interest in the vehicle system, namely, input-output stability, scalability, weak and strict string stability, are introduced based on recent studies. Then, frequency domain linear stability analyses are conducted after linearization of the modeled system of vehicles, leading to conditions for input-output stability, strict and weak string stability over the behavioral parameters of the system, for finite and infinite systems of homogeneous and heterogeneous vehicles. This provides a solid basis that was missing for car-following model-based control design in mixed traffic systems where only a proportion of vehicles can be controlled. After visualization of the theoretical results in simulation, we formulate an optimization strategy with linear matrix inequality constraints to tune the behavioral parameters of the automated vehicles in order to maximize the L
_∞
string stability of the mixed traffic flow while considering the comfort of automated driving. The optimization strategy systematically leads to increased traffic flow stability. We show that very few automated vehicles are required to prevent the propagation of realistic disturbances.