${\mathcal {H}_2}$ Performance of Series-Parallel Networks: A Compositional Perspective

We examine the $\mathcal {H}_2$ norm of matrix-weighted leader-follower consensus on series-parallel networks. By using an extension of electrical network theory on matrix-valued resistances, voltages, and currents, we show that the computation of the ${\mathcal {H}_2}$ norm can be performed efficiently by decomposing the network into atomic elements and composition rules. Finally, we examine the problem of efficiently adapting the matrix-valued edge weights to optimize the ${\mathcal {H}_2}$ norm of the network.