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

Julien Monteil, Melanie Bouroche, Douglas J. Leith

Published 2016-09-22Version 1

The presence of autonomous vehicles on the roads presents an opportunity to compensate the unstable behaviour of conventional vehicles. Ideally vehicles should (i) be able to recover their equilibrium speed, and (ii) react so as not to propagate but absorb perturbations. However, stability analysis in the literature is found to mainly deal with infinite and homogeneous platoons of vehicles, assuming every vehicle behaves in the same way. In this paper, we address the stability analysis of platoon systems consisting of heterogeneous vehicles updating their dynamics according to realistic behavioural car-following models. In the vehicle dynamics the actuator is the throttle pedal and the outputs are the observed speeds and positions. First, definitions of all types of stability that are of interest in the platoon system -asymptotic, input-output, weak and strict string stability- are carefully presented based on recent studies. Then, frequency domain linear stability analyses are conducted after linearisation of the modelled system of vehicles. They lead to conditions for input-output stability, strict and weak string stability over the behavioural parameters of the system, for finite and infinite platoons of homogeneous and heterogeneous vehicles. This provides a solid basis that was missing for car-following model-based control design in mixed traffic environments. After visualisation of the theoretical results in simulation, we formulate an optimisation strategy with LMI constraints to tune the behavioural parameters of the (partially)-autonomous vehicles in mixed and heterogeneous traffic in order to maximise the stability of the traffic flow while considering the comfort of autonomous driving.