# On the information flow required for the scalability of the stability of motion of approximately rigid formation

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It is known in the literature on Automated Highway Systems that information flow can significantly affect the propagation of errors in spacing in a collection of vehicles. This thesis investigates this issue further for a homogeneous collection of vehicles. Specifically, we consider the effect of information flow on the propagation of errors in spacing and velocity in a collection of vehicles trying to maintain a rigid formation. The motion of each vehicle is modeled using a Linear Time Invariant (LTI) system. We consider undirected and connected information flow graphs, and assume that that each vehicle can communicate with a maximum of q(n) vehicles, where q(n) may vary with the size n of the collection. The feedback controller of each vehicle takes into account the aggregate errors in position and velocity of the vehicles, with which it is in direct communication. The controller is chosen in such a way that the resulting closed loop system is a Type-2 system. This implies that the loop transfer function must have at least two poles at the origin. We then show that if the loop transfer function has three or more poles at the origin, and if the size of the formation is sufficiently large, then the motion of the collection is unstable. Suppose l is the number of poles of the transfer function relating the position of a vehicle with the control input at the origin of the complex plane, and if the number (q(n)l+1)/(nl) -> 0 as n -> (Infinity), then we show that there is a low frequency sinusoidal disturbance with unity maximum amplitude acting on each vehicle such that the maximum errors in spacing response increase at least as much as O (square_root(n^l/(q(n)^(l+1)) ) consequence of the results presented in this paper is that the maximum of the error in spacing and velocity of any vehicle can be made insensitive to the size of the collection only if there is at least one vehicle in the collection that communicates with at least O(square_root(n)) other vehicles in the collection.