Scaling in bidirectional platoons with dynamic controllers and proportional asymmetry

TL;DR

This paper analyzes how the stability and performance of bidirectional vehicle platoons with dynamic controllers scale with size, revealing conditions for exponential growth or string stability based on vehicle dynamics and network properties.

Contribution

It provides new insights into the scaling behavior of bidirectional platoons with dynamic, proportional asymmetric controllers, including conditions for stability and controller design strategies.

Findings

H-infinity norm scales exponentially if two or more integrators are in the open-loop transfer function.

String stability is achievable with a single integrator under certain conditions.

Controller design can always follow a predecessor strategy in the single integrator case.

Abstract

We consider platoons composed of identical vehicles with an asymmetric nearest-neighbor interaction. We restrict ourselves to intervehicular coupling realized with dynamic arbitrary-order onboard controllers such that the coupling to the immediately preceding vehicle is proportional to the coupling to the immediately following vehicle. Each vehicle is modeled using a transfer function and we impose no restriction on the order of the vehicle. The platoon is described by a transfer function in a convenient product form. We investigate how the H-infinity norm and the steady-state gain of the platoon scale with the number of vehicles. We conclude that if the open-loop transfer function of the vehicle contains two or more integrators and the Fiedler eigenvalue of the graph Laplacian is uniformly bounded from below, the norm scales exponentially with the growing distance in the graph. If there is just one integrator in the open loop, we give a condition under which the norm of the transfer function is bounded by its steady-state gain - the platoon is string-stable. Moreover, we argue that in this case it is always possible to design a controller the predecessor following strategy.