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态系统的分析、（控制）设计和优化通常基于高维离散化模型，因为它们例如来自复杂几何形状上的偏微分方程的有限元离散化。结果常微分方程的数量通常非常大（10.000到100万），因此使数值计算效率低下。这促使使用模型降阶方法。当模型由子系统的耦合产生时，以端口-哈密顿（pH）形式对这样的系统进行建模是特别有利的，其中物理变量描述所存储的能量。因此，这个模型类将是这个项目的重点。一个目标是保留特征pH结构，以便在随后的控制设计或耦合中利用。传统的降阶方法采用了一个相关的可用自由度的一部分，这种结构的保护，提供更少的参数，以提高近似质量。这个项目的目标是引入新的自由度的结构保持模型的端口-哈密顿系统的降阶，从而扩展现有的结果一般状态空间模型到这个系统类。此外，严格的和全球的模型降阶误差的误差界被扩展到减少端口哈密顿系统。

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.