New degrees of freedom and rigorous error bounds for the structure-preserving model order reduction of port-Hamiltonian systems

具有空间分布参数的动力系统的分析、(控制)设计和优化通常基于高维离散化模型，因为它们是通过对复杂几何形状上的偏微分方程组进行有限元离散而得到的。由此产生的常微分方程组的数量通常非常大(10.000到100万)，因此使得数值计算效率低下。这促使了模型降阶方法的使用。以端口哈密顿(PH)形式对这类系统进行建模尤其有利，因为模型是由子系统的耦合产生的，其中物理变量描述了存储的能量。因此，这个模型类将是本项目的重点。一个目标是保留特有的pH结构，以便在后续的控制设计或耦合中利用。传统的约化方法利用了可用自由度中的相关部分来保持这种结构，提供了较少的参数来提高逼近质量。本项目的目的是在港口-哈密顿系统的保结构模型降阶中引入新的自由度，从而将已有的一般状态空间模型的结果推广到这类系统。进一步，将模型降阶误差的严格的全局误差界推广到端口-哈密顿系统的降阶。

The analysis, (control) design and optimization of dynamical systems with spatially distributed parameters is often based on high-dimensional discretized models, as they result e.g. from a finite element discretization of partial differential equations over a complex geometry. The amount of resulting ordinary differential equations is often very large (10.000 up to 1 million), hence making numerical computations unefficient. This motivates the use of model order reduction methods. Modelling of such systems in port-Hamiltonian (pH) form is particularly advantageous when the models result by thecoupling of subsystems, in which the physical variables describe the energy stored. Therefore, this model class will be the focus of this project. One goal is to preserve the characteristic pH-structure, to be taken advantage of in a subsequent control design or coupling. Convential reduction methods employ a relevant part of the availabledegrees of freedom for this structure preservation, providing less parameters to increase the approximation quality. The goal of this project is to introduce new degrees of freedom in the structure-preserving model order reduction of port-Hamiltonian systems andhence to extend existing results for general state-space models to this system class. Further, rigorous and global error bounds for the model reduction error are to be extended to the reduction of port-Hamiltonian systems.