3rd International Workshop on Linear Systems (LinSys2007)

Solution of a distributed linear system stabilisation problem
Brian D O Anderson

This talk deals with the following problem. Given a real square matrix A, when is there a real diagonal matrix D such that DA has all its eigenvalues with negative real part, and how may such a matrix be constructed? This problem arose in studying how to construct decentralized feedback control laws used by agents in a two-dimensional formation, where the shape of the formation is to be preserved through a sufficient number of agent pairs each maintaining a prescribed separation, and the open-loop system may be unstable.

A sufficient condition (which in a sense is not "far" from a necessary condition) is obtained, involving the principal minors of A, and it is fulfilled in the application problem. Some associated open problems are also exposed.