This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. 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. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop system, to simultaneously identify the closed-loop and plant systems. The advantages of the proposed closed-loop Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper.

翻译：本文提出了一种方法，用于在已知控制器的情况下，辨识反馈控制系统的库普曼模型。库普曼算子通过引入无穷维的提升函数集合，将非线性系统重写为无穷维线性系统。通过选择有限维的提升函数子集，并在提升空间中求解回归问题，可以从数据中辨识出库普曼算子的有限维近似。现有方法仅适用于开环系统的辨识，然而对于某些系统（如不稳定系统）而言，开环实验不切实际或无法实现。所提方法利用库普曼算子的线性特性，结合控制器知识和闭环系统结构，同时辨识闭环系统与原始被控系统。通过杜芬振荡器的仿真实验和旋转倒立摆系统的实际实验，验证了所提闭环库普曼算子逼近方法的优势。本文还公开了所提方法的开源软件实现及实验数据集。