Effect of network structure on the stability margin of large vehicle formation with distributed control

We study the problem of distributed control of a large network of double-integrator agents to maintain a rigid formation. A few lead vehicles are given information on the desired trajectory of the formation; while every other vehicle uses linear controller which only depends on relative position and velocity from a few other vehicles, which are called its neighbors. A predetermined information graph defines the neighbor relationships. We limit our attention to information graphs that are D-dimensional lattices, and examine the stability margin of the closed loop, which is measured by the real part of the least stable eigenvalue of the state matrix. The stability margin is shown to decay to 0 as O(1/N2/D) when the graph is “square“, where N is the number of agents. Therefore, increasing the dimension of the information graph can improve the stability margin by a significant amount. For a non-square information graph, the stability margin can be made independent of N by choosing the “aspect ratio“ appropriately. An information graph with large D may require nodes that are physically apart to exchange information. Similarly, choosing an aspect ratio to improve stability margin may entail an increase in the number of lead vehicles. These results are useful to the designer in making trade-offs between performance and cost in designing information exchange architectures for decentralized control.