  We consider the optimal distributed controller design problem subject to two
structural requirements: locality, i.e. available measurements and
sub-controllers' interactions are governed by a graph structure, and relative
feedback, i.e. only differences of measurements are available to the
controller. We formalize controller locality in terms of the controller's
transfer function, state-space realization, or resulting closed-loop mapping.
We demonstrate that the relative feedback requirement can be written as a
convex constraint on the controller and (in special cases) on the resulting
closed-loop, and we characterize the allowable structures of relative feedback
controllers. We prove that sparse closed-loop design is a convex relaxation of
structured controller state-space design, even in the continuous time IIR
setting. This formalizes and extends results of the recently developed System
Level Synthesis framework. We take a first step toward quantifying the
performance gap associated with this convex relaxation by constructing a class
of examples (based on relative feedback requirements) for which the difference
in performance, measured by an H2 norm, is infinite. The results presented are
used to contrast several issues of structural constraints in distributed
control design that remain as open problems.
