  The current limitation in the synthesis of distributed $\mathscr{H}_2$
controllers for linear interconnected systems is scalability due to non-convex
or unstructured synthesis conditions. In this paper we develop convex and
structured conditions for the existence of a distributed $\mathscr{H}_2$
controller for discrete-time interconnected systems with an interconnection
structure that corresponds to an arbitrary graph. Neutral interconnections and
a storage function with a block-diagonal structure are utilized to attain
coupling conditions that are of a considerably lower computational complexity
compared to the corresponding centralized $\mathscr{H}_2$ controller synthesis
problem. Additionally, the developed conditions are adapted for the
corresponding decentralized $\mathscr{H}_2$ controller synthesis problem with
fixed supply functions for the interconnections. The effectiveness and
scalability of the developed distributed $\mathscr{H}_2$ controller synthesis
method is demonstrated for small- to large-scale oscillator networks on a cycle
graph.
