# Closed-Loop Koopman Operator Approximation

arXiv: https://arxiv.org/abs/2303.15318 · HTML: https://mlsanity.com/paper/2303.15318/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 50 references that are in the map.
- First posted on arXiv 27 Mar 2023.

## Builds on (most cited first, 10 of 10)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Active Learning of Dynamics for Data-Driven Control Using Koopman Operators](https://mlsanity.com/paper/1906.05194/) (2019)
- [Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control](https://mlsanity.com/paper/1902.02827/) (2019)
- [Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics](https://mlsanity.com/paper/2010.09961/) (2020)
- [Model-Based Control Using Koopman Operators](https://mlsanity.com/paper/1709.01568/) (2017)
- [LMI Properties and Applications in Systems, Stability, and Control Theory](https://mlsanity.com/paper/1903.08599/) (2019)
- [Koopman Linearization for Data-Driven Batch State Estimation of Control-Affine Systems](https://mlsanity.com/paper/2109.07000/) (2021)
- [Linear Matrix Inequality Approaches to Koopman Operator Approximation](https://mlsanity.com/paper/2102.03613/) (2021)
- [System Norm Regularization Methods for Koopman Operator Approximation](https://mlsanity.com/paper/2110.09658/) (2021)

## Built on by (most cited first, 1 of 1)

- [Uncertainty Modelling and Robust Observer Synthesis using the Koopman Operator](https://mlsanity.com/paper/2410.01057/) (2024)

## Abstract (arXiv)

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.