This paper presents a 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 or Dirichlet boundary conditions at both sides and mixed boundary conditions. In addition, we recall the Loewner framework to reduce the discretized model. We show that if the initial PDE is {\it passive}, the discretized model is also. Moreover, if the initial PDE is {\it impedance energy preserving}, the discretized model is also. The {\it passive} structure is also preserved in the reduced-order if the selected frequency data has positive real part. We use the one-dimensional wave equation and the Timoshenko beam as examples to show the versatility of the proposed approach.

翻译：本文提出了一种对具有空间边界共位输入输出的一类一维偏微分方程进行离散化和降阶的方法。我们考虑的系统类型称为边界控制端口哈密顿系统，涵盖了大量具有多种边界输入输出类型的双曲型偏微分方程，例如在两侧采用诺伊曼或狄利克雷边界条件及混合边界条件的波和梁问题。此外，我们回顾了用于对离散化模型进行降阶的洛纳框架。研究表明，若原始偏微分方程是被动的，则离散化模型也保持被动性；若原始偏微分方程是阻抗能量守恒的，则离散化模型同样保持该特性。当所选频率数据具有正实部时，降阶模型也能保持被动结构。我们以一维波动方程和铁木辛柯梁为例，展示了所提方法的广泛适用性。