This paper presents a systematic methodology for the discretization and reduction of a class of one-dimensional Partial Differential Equations (PDEs) with inputs and outputs collocated at the spatial boundaries. The class of system that we consider is known as Boundary-Controlled Port-Hamiltonian Systems (BC-PHSs) and covers a wide class of Hyperbolic PDEs with a large type of boundary inputs and outputs. This is, for instance, the case of waves and beams with Neumann, Dirichlet, or mixed boundary conditions. Based on a Partitioned Finite Element Method (PFEM), we develop a numerical scheme for the structure-preserving spatial discretization for the class of one-dimensional BC-PHSs. We show that if the initial PDE is passive (or impedance energy preserving), the discretized model also is. In addition and since the discretized model or Full Order Model (FOM) can be of large dimension, we recall the standard Loewner framework for the Model Order Reduction (MOR) using frequency domain interpolation. We recall the main steps to produce a Reduced Order Model (ROM) that approaches the FOM in a given range of frequencies. We summarize the steps to follow in order to obtain a ROM that preserves the passive structure as well. Finally, we provide a constructive way to build a projector that allows to recover the physical meaning of the state variables from the ROM to the FOM. We use the one-dimensional wave equation and the Timoshenko beam as examples to show the versatility of the proposed approach.

翻译：本文提出了一种系统化方法，用于对一类在空间边界配置输入与输出的一维偏微分方程进行离散化与降阶。所考虑的系统类别被称为边界控制端口哈密顿系统，其涵盖了具有多种边界输入与输出类型的广泛双曲型偏微分方程。例如，满足诺伊曼、狄利克雷或混合边界条件的波动方程与梁方程均属此类。基于分区有限元方法，我们针对一维边界控制端口哈密顿系统类开发了一种结构保持的空间离散化数值格式。我们证明，若原始偏微分方程具有无源性（或阻抗能量守恒性），则离散化模型同样保持该性质。此外，鉴于离散化模型（即全阶模型）可能具有较大维度，我们回顾了基于频域插值的标准Loewner框架模型降阶方法。我们概述了在给定频率范围内逼近全阶模型的降阶模型构建步骤，并总结了获得同样保持无源结构的降阶模型所需遵循的流程。最后，我们提供了一种构造性方法以建立投影算子，使得能够从降阶模型恢复状态变量到全阶模型的物理意义。以一维波动方程和铁木辛柯梁为例，我们展示了所提方法的普适性。