We consider distributed control of a large two-dimensional (planar) vehicular formation. An individual vehicle in the formation is assumed to be a fully actuated point mass. The control objective is to move the formation with a constant pre-specified velocity while maintaining constant inter-vehicle separation between any pair of nearby vehicles. The control law is distributed in the sense that the control action at each vehicle depends on the relative position measurements with nearby vehicles and its own velocity measurement. For this problem, a partial differential equation (PDE) model is derived to describe the spatio-temporal evolution of velocity perturbations for large number of vehicles, Nveh. The PDE model is used to deduce asymptotic formulae for the stability margin (absolute value of the real part of the least stable eigenvalue). We show that the stability margin of the closed loop decays to 0 as the number of vehicles increases, but the decay rate in 2D formation is much slower than in 1D platoons. In addition, the PDE is used to optimize the stability margin using a mistuning-based approach, in which the control gains of the vehicles are changed slightly from their nominal values. We show that the mistuning design reduces the loss of stability margin significantly even with arbitrarily small amount of mistuning. The results of the analysis with the PDE model are corroborated with numerical computation of eigenvalues with the state-space model of the vehicular formation.

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