Information Flow Requirements for the Stability of Motion of Vehicles in a 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 chapter investigates this issue further for a homogeneous collection of vehicles, where in the motion of each vehicle is modeled as a point mass and is digitally controlled. The structure of the controller employed by the vehicles is as follows: $$U_i (z) = C(z)sum olimits_{j in S_i } {(X_i - X_j - frac{{L_{ij} z}}{{z - 1}})} $$, where Ui(z) is the (z- transformation of) control action for the ith vehicle, Xi is the position of the ith vehicle, Lij is the desired distance between the ith and the jth vehicles in the collection, C(z) is the discrete transfer function of the controller and Si is the set of vehicles that the ith vehicle can communicate with directly. This chapter further assumes that the information flow is undirected, i.e., i Sj j Si 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. We allow q(n) to vary with the size n of the collection. We first show that C(z) cannot have any zeroes at z = 1 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(z) has one or more poles located at z = 1, 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(1) 0 and C(1) must be well defined. We further show that if q(n)/n 0 as n then there is a low frequency sinusoidal disturbance of at most unit amplitude acting on each vehicle such that the maximum error in spacing response increase at least as $$Omega left( {sqrt { frac{{n^3 }}{{q^3 (n)}}} } ight)$$. A consequence of the results presented in this chapter 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 (n) other vehicles in the collection. We also show that there can be at most one vehicle that communicates with (n) vehicles and that any other vehicle in the collection can only communicate with at most p vehicles, where p depends only on the chosen controller and the its sampling time.