Koopman operator theory: fundamentals, control, and applications

2026-07-02

AI Summary

This study addresses the challenge of global linear modeling and control for highly nonlinear dynamical systems by leveraging Koopman operator theory. By introducing observable functions, the nonlinear dynamics are lifted into a higher-dimensional space where they admit an approximately linear representation. A data-driven surrogate model is constructed through a synergistic integration of Extended Dynamic Mode Decomposition (EDMD), kernelized EDMD, and machine learning techniques. The work innovatively extends the Koopman framework to input-affine systems, proposing a unified modeling approach and a corresponding Koopman-based Model Predictive Control (MPC) design methodology. Numerical simulations demonstrate that the proposed method achieves high-fidelity modeling accuracy and effective closed-loop control performance. Full reproducibility is supported by the accompanying open-source implementation.

Abstract

The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Recently proposed data-driven techniques, like extended dynamic mode decomposition (EDMD), its kernelized variant, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.

Problem

• Koopman operator
• nonlinear dynamical systems
• data-driven modeling
• control design
• linear representation

Innovation

• Koopman operator
• extended dynamic mode decomposition
• data-driven modeling
• Koopman MPC
• nonlinear control