# Data-driven discovery of Koopman eigenfunctions for control

**Authors:** Eurika Kaiser, J. Nathan Kutz, Steven L. Brunton  
**arXiv:** [1707.01146](https://arxiv.org/abs/1707.01146)  
**Date:** 2017-07-04  
**Summary page:** https://www.summarizepaper.com/en/paper/1707.01146v1/

## Key points

- The paper explores the use of data-driven transformations to reformulate nonlinear systems in a linear framework
- The authors focus on the Koopman operator, which is a principled linear embedding of nonlinear dynamics
- Koopman eigenfunctions establish intrinsic coordinates along which the dynamics behave linearly
- The authors propose a data-driven control architecture called KRONIC, which utilizes Koopman eigenfunctions for control over nonlinear systems
- Approximation of these eigenfunctions is done through data-driven regression and power series expansions based on the partial differential equation governing the infinitesimal generator of the Koopman operator
- Lightly damped eigenfunctions can be accurately extracted using sparse regression, and they are relevant for control as they correspond to nearly conserved quantities associated with persistent dynamics such as Hamiltonian energy
- Nonlinear controllers are designed using standard linear optimal control theory in the intrinsic eigenfunction coordinates
- KRONIC is demonstrated on various examples including a system with a known linear embedding, different types of Hamiltonian systems, and a high-dimensional double-gyre model for ocean mixing
- This work presents an innovative approach to utilizing data-driven techniques for controlling strongly nonlinear dynamics by leveraging principles from linear systems theory through the use of Koopman eigenfunctions

## Summary

The paper titled "Data-driven discovery of Koopman eigenfunctions for control" explores the use of data-driven transformations to reformulate nonlinear systems in a linear framework. This approach has the potential to enable the prediction, estimation, and control of strongly nonlinear dynamics using standard linear systems theory. The authors focus on the Koopman operator, which is a principled linear embedding of nonlinear dynamics. The eigenfunctions of the Koopman operator establish intrinsic coordinates along which the dynamics behave linearly. To manipulate nonlinear systems using linear systems theory, the authors propose a data-driven control architecture called Koopman Reduced Order Nonlinear Identification and Control (KRONIC). This architecture utilizes Koopman eigenfunctions to achieve control over nonlinear systems. The approximation of these eigenfunctions is done through data-driven regression and power series expansions based on the partial differential equation that governs the infinitesimal generator of the Koopman operator. Previous regression-based methods have been hindered by spurious dynamics; however, this work demonstrates that lightly damped eigenfunctions can be accurately extracted using sparse regression. These lightly damped eigenfunctions are particularly relevant for control as they correspond to nearly conserved quantities associated with persistent dynamics such as Hamiltonian energy. The paper further derives the form of control in these intrinsic eigenfunction coordinates and designs nonlinear controllers using standard linear optimal control theory. To validate their approach, KRONIC is demonstrated on various examples including a nonlinear system with a known linear embedding, different types of Hamiltonian systems, and a high-dimensional double-gyre model for ocean mixing. Overall, this work presents an innovative approach to utilizing data-driven techniques for controlling strongly nonlinear dynamics by leveraging principles from linear systems theory through the use of Koopman eigenfunctions. The results demonstrate promising applications in various domains and highlight how this methodology can enable effective prediction, estimation, and control in complex dynamical systems.