Stability Margin Scaling Laws for Distributed Formation Control as a Function of Network Structure

We describe a methodology for modeling, analysis and distributed control design of a large vehicular formation whose information graph is a D-dimensional lattice. We derive asymptotic formulae for the closed-loop stability margin based on a partial differential equation (PDE) approximation of the formation. We show that the exponent in the scaling law for the stability margin is influenced by the structure of the information graph and by the control architecture (symmetric or asymmetric). For a given fixed number of vehicles, we show that the scaling law can be improved significantly by employing a higher dimensional information graph and/or by introducing small asymmetry (mistuning) in the nominally symmetric proportional control gains. We also provide a characterization of the error introduced by the PDE approximation.