Koopman operator theory: fundamentals, control, and applications

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.

Characterization of Thermal Systems from Noisy and Low-resolution Measurements Using Dynamic Mode Decomposition

Thermal monitoring in practical applications is often constrained by sparse sensing, measurement noise, and limited spatial resolution, which hinder the identification of heat transfer dynamics. In such settings, calibrating high-fidelity physical models is computationally demanding, motivating data-driven approaches. Dynamic Mode Decomposition (DMD) provides a framework for extracting spatiotemporal structures from measurement data, but its standard formulation is sensitive to noise and degraded observations. This chapter examines the use of DMD under these constraints, focusing on preprocessing and truncation strategies that affect stability and interpretability. Two cases are considered: forced convection with thermocouple data and transient heat conduction from degraded thermal images. The number of retained modes is treated as a modeling parameter that governs the trade-off between reconstruction fidelity and noise sensitivity. The results indicate that DMD recovers dominant thermal behavior from both sparse and degraded datasets when the truncation level is appropriately selected. Low-rank models provide stable but simplified descriptions, while higher-rank models improve spatial detail at the cost of increased noise sensitivity.