论文标题
具有不均匀初始条件的二阶系统的模型降低
Model reduction for second-order systems with inhomogeneous initial conditions
论文作者
论文摘要
在本文中，我们考虑了找到具有不均匀初始条件的大规模二阶线性时间不变系统的替代模型的问题。对于这类系统，叠加原理使我们能够将系统行为分解为三个独立的组件。第一个行为对应于输入和输出零初始条件之间的转移。相比之下，其他两个对应于初始位置和具有零输入的初始速度条件之间的转移。基于系统的叠加，我们的目标是提出模型还原方案，允许在替代模型中保留二阶结构。为此，我们为每个系统组件介绍了量身定制的二阶Gramians，并以数值计算它们，以求解Lyapunov方程。结果，提出了两种方法。第一个是使用合适的平衡截断过程独立降低每个组件的组成。这些还原系统的总和提供了原始系统的近似值。此方法可以灵活地在还原阶模型的顺序上进行灵活性。第二种提出的方法包括从Gramians的总和中提取主要子空间，以构建导致替代模型的投影矩阵。此外，我们讨论了整体输出近似值的误差界限。最后，通过基准问题来说明所提出的方法。
In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to decompose the system behavior into three independent components. The first behavior corresponds to the transfer between the input and output having zero initial conditions. In contrast, the other two correspond to the transfer between the initial position and the initial velocity conditions having zero input, respectively. Based on this superposition of systems, our goal is to propose model reduction schemes allowing to preserve the second-order structure in the surrogate models. To this aim, we introduce tailored second-order Gramians for each system component and compute them numerically, solving Lyapunov equations. As a consequence, two methodologies are proposed. The first one consists in reducing each of the components independently using a suitable balanced truncation procedure. The sum of these reduced systems provides an approximation of the original system. This methodology allows flexibility on the order of the reduced-order model. The second proposed methodology consists in extracting the dominant subspaces from the sum of Gramians to build the projection matrices leading to a surrogate model. Additionally, we discuss error bounds for the overall output approximation. Finally, the proposed methods are illustrated by means of benchmark problems.
