  We study the stability and robustness of large-scale vehicular formations, in
which each vehicle is modeled as a double-integrator. Two types of information
graphs are considered: directed trees and undirected graphs. We prove stability
of the formation with arbitrary number of vehicles for linear as well as a
class of nonlinear controllers. In the case of linear control, we provide
quantitative scaling laws of the stability margin and sensitivity to external
disturbances (H-infinity norm) with respect to the number of vehicles $N$ in
the formation. It is shown that the formation with directed tree graph achieves
size-independent stability margin but suffers from high algebraic growth of
initial errors. The stability margin in case of the undirected graph decays to
0 as at least $O(1/N)$. In addition, we show that the sensitivity to external
disturbances in directed tree graphs is geometric in $k$ where $k <= N$ is the
number of generations of the directed tree, while that of the undirected graph
is only quadratic in $N$. In particular, for 1-D vehicular platoons, we obtain
precise formulae for the H-infinity norm of the transfer function from the
disturbances to the position errors. It is shown that the H-infinity norm
scales as $O(\alpha^N) (\alpha>1)$ for predecessor-following architecture, but
only as $O(N^3)$ for symmetric bidirectional architectures. For a class of
nonlinear controllers, numerical simulations show that the transient response
due to initial errors and sensitivity to external disturbances are improved
considerably for the formation with directed tree graphs. However, by using the
nonlinear controller considered, little improvement can be made for that with
undirected graphs.
