$\mathcal{L}_2$ and $\mathcal{L}_{\infty}$ stability analysis of heterogeneous traffic with application to parameter optimisation for the control of automated vehicles

The presence of (partially) automated vehicles on the roads presents an opportunity to compensate the unstable behaviour of conventional vehicles. Vehicles subject to perturbations should (i) recover their equilibrium speed, (ii) react not to propagate but absorb perturbations. In this work, we start with considering vehicle systems consisting of heterogeneous vehicles updating their dynamics according to realistic behavioural 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 linearisation of the modelled system of vehicles, leading to conditions for input-output stability, strict and weak string stability over the behavioural 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 visualisation of the theoretical results in simulation, we formulate an optimisation strategy with LMI constraints to tune the behavioural parameters of the automated vehicles in order to maximise the L1 string stability of the mixed traffic flow while considering the comfort of automated driving. The optimisation strategy systematically leads to increased traffic flow stability. We show that very few automated vehicles are required to prevent the