# Optimal and Cooperative Control of Vehicle Formations

## J. Alexander FaxPhD Dissertation

Control of vehicle formations has emerged as a topic of significant interest to the controls community. In applications such as microsatellites and underwater vehicles, formations have the potential for greater functionality and versatility than individual vehicles. In this thesis, we investigate two topics relevant to control of vehicle formations: optimal vehicle control and cooperative control.

The framework of optimal control is often employed to generate vehicle trajectories. We use tools from geometric mechanics to specialize the two classical approaches to optimal control, namely the calculus of variations and the Hamilton- Jacobi-Bellman (HJB) equation, to the case of vehicle dynamics. We employ the formalism of the covariant derivative, useful in geometric representations of vehicle dynamics, to relate variations of position to variations of velocity. When variations are computed in this setting, the evolution of the adjoint variables is shown to be governed by the covariant derivative, thus inheriting the geometric structure of the vehicle dynamics. To simplify the HJB equation, we develop the concept of time scalability enjoyed by many vehicle systems. We employ this property to eliminate time from the HJB equation, yielding a purely spatial PDE whose solution supplies both finite-time optimal trajectories and a time-invariant stabilizing control law.

Cooperation among vehicles in formation depends on intervehicle communication. However, vehicle communication is often subject to disruption, especially in an adversarial setting. We apply tools from graph theory to relate the topology of the communication network to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the eŽect of the graph on formation stability. We also propose a method for decentralized information exchange between vehicles. This approach realizes a dynamical sys- tem that supplies each vehicle with a common reference to be used for cooperative motion. We prove a separation principle that states that formation stability is achieved if the information flow is stable for the given graph and if the local controller stabilizes the vehicle. The information flow can be rendered highly robust to changes in the graph, thus enabling tight formation control despite limitations in intervehicle communication capability.

Technical Report (PDF, 135 pages, 825K)