AI 中文总结
本文提出pqSEDMD算法，结合pqEDMD与子空间识别方法，在线性函数空间中稳健近似非线性系统，以Duffing振荡器为基准验证其有效性。
AI 中文摘要
动态模态分解（DMD）及其用于非线性系统的变体——扩展动态模态分解（EDMD），是从测量数据中提取（非）线性动力系统有意义的时空特征的有力工具。尽管在处理真实世界数据时已有一些努力来应对识别任务，但基于分解的方法面临一个关键挑战：真实世界数据具有两个固有的不确定性来源，即过程噪声和测量噪声。子空间识别方法是稳健的工具，能够直接从输入输出数据为多变量线性系统提供精确的状态空间模型。结合这两种方法，我们引入了p-q拟范数子空间扩展动态模态分解（pqSEDMD）。该算法利用我们先前对EDMD的改进，即在正交多项式基上使用p-q拟范数约简的pqEDMD算法，并结合子空间识别方法。其结果是在线性函数空间中对非线性系统的稳健近似，结合了两种方法的优势。在整篇论文中，我们将使用Duffing振荡器作为基准问题来展示该算法的有效性，并阐述与开发相关的许多重要方面。
英文摘要
The dynamic mode decomposition (DMD), along with its variant for nonlinear systems, the extended DMD (EDMD), are powerful tools for the extraction of meaningful spatio-temporal characteristics of (non)linear dynamical systems from measurement data. Despite some efforts to handle the identification task when dealing with real-world data, the decomposition based methods face a critical challenge: real-world data has two inherent sources of uncertainty, the process and measurement noise. Subspace identification methods, are robust tools able to provide accurate state-space models for multi-variable linear systems directly from input-output data. Combining these two methods, we introduce the p-q quasi-norm Subspace Extended Dynamic Mode Decomposition (pqSEDMD). An algorithm that uses our previous improvements to the EDMD by the use of a p-q-quasi-norm reduction on an orthogonal polynomial basis, the pqEDMD algorithm, along with subspace identification methods. The result is a robust approximation of nonlinear systems in a linear function space, combining the strengths of the two methodologies. Throughout the paper we will use the Duffing oscillator as a benchmark problem to show the effectiveness of the algorithm and illustrate many important aspects related to the development.
Comments14 pages, 4 figures
