Koopman operator theory: fundamentals, control, and applications

2026-07-02 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Robotics & Autonomous Systems · Depth: Intermediate, quick

Summary

The Koopman operator theory offers a method to represent complex nonlinear dynamical systems linearly by observing their behavior through real- or complex-valued functions. This tutorial introduces the theory and its application in systems and control, highlighting data-driven techniques such as Extended Dynamic Mode Decomposition (EDMD), its kernelized variant, and machine learning methods. These techniques facilitate finite-dimensional approximations with quantifiable error bounds. The paper specifically focuses on developing data-driven surrogate models, extending them to systems with inputs, and designing controllers using Koopman operator theory, exemplified by Koopman Model Predictive Control (MPC). Simulation studies, complete with source code on GitHub, are provided to allow readers to explore these concepts practically in systems and control.

Key takeaway

For Machine Learning Engineers or Control Systems Designers working with complex nonlinear systems, this tutorial provides a clear path to leveraging Koopman operator theory. You should explore data-driven techniques like EDMD and Koopman MPC to achieve linear representations for system analysis and controller design. Utilizing the provided GitHub source code can accelerate your understanding and practical implementation of these advanced control strategies.

Key insights

The Koopman operator linearly represents nonlinear dynamics through observable functions, enabling data-driven control and system analysis.

Principles

• Nonlinear dynamics can be linearized via observable functions.
• Data-driven methods approximate Koopman operators.
• Finite-data error bounds are achievable for approximations.

Method

The paper details using Extended Dynamic Mode Decomposition (EDMD) and its kernelized variant, alongside machine learning, to generate finite-dimensional Koopman operator approximations for control and surrogate modeling.

In practice

• Apply EDMD for data-driven system identification.
• Implement Koopman MPC for controller design.
• Utilize provided GitHub code for simulations.

Topics

• Koopman Operator Theory
• Nonlinear Dynamical Systems
• Extended Dynamic Mode Decomposition
• Data-Driven Control
• Model Predictive Control
• Systems and Control

Best for: AI Scientist, Machine Learning Engineer, Robotics Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.