The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions instead of a state vector. The main feature of this representation is its linearity, making it compatible with existing linear systems theory. A finite-dimensional approximation of the Koopman operator can be identified from experimental data by choosing a finite subset of lifting functions, applying it to the data, and solving a least squares problem in the lifted space. Existing Koopman operator approximation methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems without a feedback controller. Unfortunately, the introduction of feedback control results in correlations between the system's input and output, making some plant dynamics difficult to identify if the controller is neglected. This paper addresses this limitation by introducing a method to identify a Koopman model of the closed-loop system, and then extract a Koopman model of the plant given knowledge of the controller. This is accomplished by leveraging the linearity of the Koopman representation of the system. The proposed approach widens the applicability of Koopman operator identification methods to a broader class of systems. The effectiveness of the proposed closed-loop Koopman operator approximation method is demonstrated experimentally using a Harmonic Drive gearbox exhibiting nonlinear vibrations.

翻译：库普曼算子通过使用无限维的升维函数集合而非状态向量来重构非线性系统，将其转化为无限维线性系统。该表示的主要特征是其线性性质，使其能与现有线性系统理论兼容。通过选择有限维升维函数子集、将其应用于实验数据并在升维空间中求解最小二乘问题，可从实验数据中识别库普曼算子的有限维近似。现有库普曼算子近似方法旨在识别开环系统，但对于某些系统而言，若无反馈控制器则无法或难以进行实验。然而，引入反馈控制会导致系统输入与输出之间存在相关性，若忽略控制器，部分受控对象动力学特性将难以识别。本文针对这一局限，提出一种方法：先识别闭环系统的库普曼模型，再利用已知的控制器信息提取受控对象的库普曼模型。该方法通过利用系统库普曼表示的线性性质实现。所提出的方法拓宽了库普曼算子识别方法对更广泛系统类别的适用性。通过使用表现出非线性振动的谐波传动减速器进行实验，验证了所提出的闭环库普曼算子近似方法的有效性。