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We study the stability and robustness of large-scale vehicular formations, inwhich each vehicle is modeled as a double-integrator. Two types of informationgraphs are considered: directed trees and undirected graphs. We prove stabilityof the formation with arbitrary number of vehicles for linear as well as aclass of nonlinear controllers. In the case of linear control, we providequantitative scaling laws of the stability margin and sensitivity to externaldisturbances (H-infinity norm) with respect to the number of vehicles $N$ inthe formation. It is shown that the formation with directed tree graph achievessize-independent stability margin but suffers from high algebraic growth ofinitial errors. The stability margin in case of the undirected graph decays to0 as at least $O(1/N)$. In addition, we show that the sensitivity to externaldisturbances in directed tree graphs is geometric in $k$ where $k <= N$ is thenumber of generations of the directed tree, while that of the undirected graphis only quadratic in $N$. In particular, for 1-D vehicular platoons, we obtainprecise formulae for the H-infinity norm of the transfer function from thedisturbances to the position errors. It is shown that the H-infinity normscales as $O(\alpha^N) (\alpha>1)$ for predecessor-following architecture, butonly as $O(N^3)$ for symmetric bidirectional architectures. For a class ofnonlinear controllers, numerical simulations show that the transient responsedue to initial errors and sensitivity to external disturbances are improvedconsiderably for the formation with directed tree graphs. However, by using thenonlinear controller considered, little improvement can be made for that withundirected graphs.

Decentralized control of large vehicular formations: stability margin and sensitivity to external disturbances