# System Norm Regularization Methods for Koopman Operator Approximation

arXiv: https://arxiv.org/abs/2110.09658 · HTML: https://mlsanity.com/paper/2110.09658/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 47 references that are in the map.
- First posted on arXiv 18 Oct 2021.

## Builds on (most cited first, 11 of 11)

- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [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)
- [A note on approximating the nearest stable discrete-time descriptor system with fixed rank](https://mlsanity.com/paper/1807.04481/) (2018)
- [Learning Koopman Operator under Dissipativity Constraints](https://mlsanity.com/paper/1911.03884/) (2019)
- [A Convex Optimization Approach to Learning Koopman Operators](https://mlsanity.com/paper/2102.03934/) (2021)
- [Linear Matrix Inequality Approaches to Koopman Operator Approximation](https://mlsanity.com/paper/2102.03613/) (2021)

## Built on by (most cited first, 2 of 2)

- [Closed-Loop Koopman Operator Approximation](https://mlsanity.com/paper/2303.15318/) (2023)
- [Uncertainty Modelling and Robust Observer Synthesis using the Koopman Operator](https://mlsanity.com/paper/2410.01057/) (2024)

## Abstract (arXiv)

Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered. Even low-dimensional systems can yield unstable or ill-conditioned results in a high-dimensional lifted space. In this paper, Extended Dynamic Mode Decomposition (DMD) and DMD with control, two methods for approximating the Koopman operator, are reformulated as convex optimization problems with linear matrix inequality constraints. Asymptotic stability constraints and system norm regularizers are then incorporated as methods to improve the numerical conditioning of the Koopman operator. Specifically, the H-infinity norm is used to penalize the input-output gain of the Koopman system. Weighting functions are then applied to penalize the system gain at specific frequencies. These constraints and regularizers introduce bilinear matrix inequality constraints to the regression problem, which are handled by solving a sequence of convex optimization problems. Experimental results using data from an aircraft fatigue structural test rig and a soft robot arm highlight the advantages of the proposed regression methods.