  A seminal result in decentralized control is the development of fixed modes
by Wang and Davison in 1973 - that plant modes which cannot be moved with a
static decentralized controller cannot be moved by a dynamic one either, and
that the other modes which can be moved can be shifted to any chosen location
with arbitrary precision. These results were developed for perfectly
decentralized, or block diagonal, information structure, where each control
input may only depend on a single corresponding measurement. Furthermore, the
results were claimed after a preliminary step was demonstrated, omitting a
rigorous induction for each of these results, and the remaining task is
nontrivial.
  In this paper, we consider fixed modes for arbitrary information structures,
where certain control inputs may depend on some measurements but not others. We
provide a comprehensive proof that the modes which cannot be altered by a
static controller with the given structure cannot be moved by a dynamic one
either, and that the modes which can be altered by a static controller with the
given structure can be moved by a dynamic one to any chosen location with
arbitrary precision, thus generalizing and solidifying Wang and Davison's
results.
  This shows that a system can be stabilized by a linear time-invariant
controller with the given information structure as long as all of the modes
which are fixed with respect to that structure are in the left half-plane; an
algorithm for synthesizing such a stabilizing decentralized controller is then
distilled from the proof.
