Nonzero bound on Fiedler eigenvalue causes exponential growth of H-infinity norm of vehicular platoon

TL;DR

This paper demonstrates that in asymmetric distributed vehicle platoons, a nonzero Fiedler eigenvalue leads to exponential growth in the H-infinity norm, indicating poor scalability regardless of control design.

Contribution

It proves that asymmetric control schemes ensure a nonzero Fiedler eigenvalue but cause exponential growth in the H-infinity norm, revealing fundamental scalability limitations.

Findings

Fiedler eigenvalue is bounded away from zero in such platoons.

H-infinity norm grows exponentially with the number of vehicles.

Asymmetric control schemes lead to poor scaling properties.

Abstract

We consider platoons composed of identical vehicles and controlled in a distributed way, that is, each vehicle has its own onboard controller. The regulation errors in spacing to the immediately preceeding and following vehicles are weighted differently by the onboard controller, which thus implements an asymmetric bidirectional control scheme. The weights can vary along the platoon. We prove that such platoons have a nonzero uniform bound on the second smallest eigenvalue of the graph Laplacian matrix - the Fiedler eigenvalue. Furthermore, it is shown that existence of this bound always signals undesirable scaling properties of the platoon. Namely, the H-infinity norm of the transfer function of the platoon grows exponentially with the number of vehicles regardless of the controllers used. Hence the benefits of a uniform gap in the spectrum of a Laplacian with an asymetric distributed controller are paid for by poor scaling as the number of vehicles grows.