  This paper considers the robustness analysis and distributed
$\mathcal{H}_{\infty}$ (H-infinity) controller synthesis for a platoon of
connected vehicles with undirected topologies. We first formulate a unified
model to describe the collective behavior of homogeneous platoons with external
disturbances using graph theory. By exploiting the spectral decomposition of a
symmetric matrix, the collective dynamics of a platoon is equivalently
decomposed into a set of subsystems sharing the same size with one single
vehicle. Then, we provide an explicit scaling trend of robustness measure
$\gamma$-gain, and introduce a scalable multi-step procedure to synthesize a
distributed $\mathcal{H}_{\infty}$ controller for large-scale platoons. It is
shown that communication topology, especially the leader's information, exerts
great influence on both robustness performance and controller synthesis.
Further, an intuitive optimization problem is formulated to optimize an
undirected topology for a platoon system, and the upper and lower bounds of the
objective are explicitly analyzed, which hints us that coordination of multiple
mini-platoons is one reasonable architecture to control large-scale platoons.
Numerical simulations are conducted to illustrate our findings.
