On Structured-Closed-Loop versus Structured-Controller Design: the Case of Relative Measurement Feedback

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.