Collision Potential Analysis in First and Second Order Integrator Networks Over Strongly Connected Digraphs

This paper investigates the collision potential of any two nodes in first and second order integrator networks subject to distributed disturbances. Our results extend previous analysis quantifying this notion of robustness using an induced L
₂
to L
_∞
norm for networks with symmetric feedback interconnection to systems with directed feedback. We focus on the special case where the underlying feedback interconnection is represented by a strongly connected digraph described by a diagonalizable weighted graph Laplacian matrix. We derive analytical expressions for system robustness for networks of first and second order systems with two different combinations of absolute and relative state feedback. A numerical example simulating a second order system connected over a line graph (a vehicle platoon) is employed to investigate the effect of asymmetric feedback control laws on collision potential. Our numerical results show that in contrast to previous results showing improved stability margin through asymmetric feedback control laws, this type of control law can actually increase the collision potential (decrease system robustness) for certain vehicle pairs in platoons with and without leaders.