Abstract
We quantify performance of a class of interconnected linear dynamical networks with arbitrary nodal dynamics. It is shown that the steady-state variance of the output, which is equivalent to the squared $\mathcal {H}_2$ norm of the network, as a performance measure, can be expressed explicitly as a summation of a rational function over the Laplacian eigenvalues of the underlying graph of the network. Our results generalize the existing works on $\mathcal {H}_2$ -norm characterization for the first- and second-order consensus networks to general consensus-type linear dynamical networks with arbitrary nodal dynamics. We characterize several fundamental limits and scaling laws for the performance of this class of networks. To show the practical usefulness of our results, we discuss how to design decentralized observer-based output feedback mechanisms and how to analyze the performance of composite networks. Numerous examples support our theoretical contributions.
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