The model reduction problem in the Loewner framework for port-Hamiltonian and network systems on graphs is studied. In particular, given a set of right-tangential interpolation data, the (subset of) left-tangential interpolation data that allow constructing an interpolant possessing a port-Hamiltonian structure is characterized. In addition, conditions under which an interpolant retains the underlying port-Hamiltonian structure of the system generating the data are given by requiring a particular structure of the generalized observability matrix. a characterization of the reduced order model in terms of Dirac structure with the aim of relating the Dirac structure of the underlying port-Hamiltonian system with the Dirac structure of the constructed interpolant is given. This result, in turn, is used to solve the model reduction problem in the Loewner framework for network systems described by a weighted graph. The problem is first solved, for a given clustering, by giving conditions on the right- and left-tangential interpolation data that yield an interpolant possessing a network structure. Thereafter, for given tangential data obtained by sampling an underlying network system, we give conditions under which we can select a clustering and construct a reduced model preserving the network structure. Finally, the results are illustrated by means of a second order diffusively coupled system and a first order network system.

中文翻译：

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Loewner 框架中的端口哈密尔顿和网络系统的数据驱动模型简化


研究了Loewner 框架中图上的端口哈密尔顿系统和网络系统的模型简化问题。具体地，给定一组右切向插值数据，表征允许构造具有端口哈密尔顿结构的插值的左切向插值数据（的子集）。此外，插值保留生成数据的系统的底层端口哈密尔顿结构的条件是通过需要广义可观测性矩阵的特定结构来给出的。给出了根据狄拉克结构的降阶模型的表征，其目的是将底层端口哈密尔顿系统的狄拉克结构与构造的插值的狄拉克结构联系起来。该结果又用于解决 Loewner 框架中加权图描述的网络系统的模型简化问题。对于给定的聚类，首先通过给出右切向和左切向插值数据的条件来解决该问题，这些条件产生具有网络结构的插值。此后，对于通过对底层网络系统进行采样获得的给定切向数据，我们给出了可以选择聚类并构建保留网络结构的简化模型的条件。最后，通过二阶扩散耦合系统和一阶网络系统说明了结果。