Information flow and its relation to stability of the motion of vehicles in a rigid formation

In this note, we consider the effect of information flow on the propagation of errors in spacing in a collection of vehicles trying to maintain a rigid formation during translational maneuvers. The motion of each vehicle is described using a linear time-invariant (LTI) system. We consider undirected and connected information flow graphs, and assume that each vehicle can communicate with a maximum of q vehicles, where q may vary with the size n of the collection. We consider translational maneuver of a reference vehicle, where its steady state velocity is different from its initial velocity. In the absence of any disturbing forces acting on the vehicles during the maneuver, it is desired that the collection be controlled in such a way that its motion asymptotically resembles that of a rigid body. In the presence of bounded disturbing forces acting on the vehicles, it is desired that the maximum deviation of the motion of the collection from that of a rigid body be bounded and be independent of the size of the collection. We consider a decentralized feedback control scheme, where the controller of each vehicle takes into account the aggregate errors in position and velocity from the vehicles with which it is in direct communication. We assume that all vehicles start at their respective desired positions and velocities. Since the displacement of every vehicle at the end of the maneuver of the reference vehicle must be the same, we show 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, then the motion of the collection is unstable, that is, its deviation from the rigid body motion is arbitrarily large, if the size of the formation is sufficiently large. If l is the number of poles of the transfer function relating the position of a vehicle with its control input, we show that if (q(n)/n)rarr0 as nrarrinfin, then there is a low frequency sinusoidal disturbance- - of at most unit amplitude acting on each vehicle such that the maximum errors in spacing response increase at least as Omega(((radicn/q(n))^l+1)).A function p(n) is Omega(q(n)) if there exists a nonzero constant c>0 and a N^* such that |p(n)|gesc|q(n)| for all n>N^* . A consequence of the results presented in this note is that the maximum errors 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 Omega(n) other vehicles in the collection