  We consider a decentralized bidirectional control of a platoon of N identical
vehicles moving in a straight line. The control objective is for each vehicle
to maintain a constant velocity and inter-vehicular separation using only the
local information from itself and its two nearest neighbors. Each vehicle is
modeled as a double integrator. To aid the analysis, we use continuous
approximation to derive a partial differential equation (PDE) approximation of
the discrete platoon dynamics. The PDE model is used to explain the progressive
loss of closed-loop stability with increasing number of vehicles, and to devise
ways to combat this loss of stability.
  If every vehicle uses the same controller, we show that the least stable
closed-loop eigenvalue approaches zero as O(1/N^2) in the limit of a large
number (N) of vehicles. We then show how to ameliorate this loss of stability
by small amounts of "mistuning", i.e., changing the controller gains from their
nominal values. We prove that with arbitrary small amounts of mistuning, the
asymptotic behavior of the least stable closed loop eigenvalue can be improved
to O(1/N) All the conclusions drawn from analysis of the PDE model are
corroborated via numerical calculations of the state-space platoon model.
