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Optimal and Cooperative Control of Vehicle Formations

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J. Alexander Fax

PhD 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)

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Richard Murray
(murray@cds. caltech.edu)