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.

中文翻译：

---

一类网络线性控制系统性能的显式表征

我们量化具有任意节点动力学的一类互连线性动力学网络的性能。结果表明，输出的稳态方差等于平方的平方$ \ mathcal {H} _2 $网络的范数作为一种性能度量，可以明确表示为网络基础图的拉普拉斯特征值上有理函数的总和。我们的结果概括了关于$ \ mathcal {H} _2 $ 一阶和二阶共识网络到具有任意节点动力学的一般共识类型线性动力学网络的-范数刻画。我们描述了此类网络性能的几个基本限制和缩放定律。为了展示我们的结果的实用性，我们讨论了如何设计基于分散的基于观察者的输出反馈机制，以及如何分析复合网络的性能。许多例子支持我们的理论贡献。