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

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 paper investigates this issue further for a homogeneous collection of vehicles, where in the motion of each vehicle is modeled as a point mass. The structure of the controller employed by the vehicles is as follows: U_i(s)=C(s)Σ _j∈si(X_i - X_j - L_ij/s) where U_i(s) is the (Laplace transformation of) control action for the i^th vehicle, L_ijis the position of the i^th vehicle, L_ij is the desired distance between the i^th and the j^th vehicles in the collection, C(s) is the controller transfer function and S_i is the set of vehicles that the i^th vehicle can communicate with directly. This paper further assumes that the information flow is undirected, i.e., i∈S_j/j∈S_i, and the information flow graph is connected. We consider information flow in the collection, where each vehicle can communicate with a maximum of q(n) vehicles, such that q(n) may vary with the size n of the collection. We first show that C(s) cannot have any zeroes at the origin to ensure that relative spacing is maintained in response to a reference vehicle making a maneuver where its velocity experiences a steady state offset. We then show that if the control transfer function C(s) has one or more poles located at the origin of the complex plane, then the motion of the collection of vehicles will become unstable if the size of the collection is sufficiently large. These two results imply that C(0)≠0 and C(0) is well defined. We further show that if q(n)³/n²→0 as n →∞ 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 O (√(n²)/q(n)³). A 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(n²- 3/) other vehicles in the collection.