Koopman operator theory: fundamentals, control, and applications
It serves as an educational resource for researchers and practitioners in systems and control seeking to apply Koopman operator methods.
This tutorial provides a concise introduction to Koopman operator theory for global linear representation of nonlinear dynamics, focusing on data-driven surrogate models, systems with inputs, and controller design via EDMD and Koopman MPC, with simulation studies and source code provided.
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.