An overview of Koopman-based control: From error bounds to closed-loop guarantees

arXiv (Cornell University) · 2025-09-02

Controlling nonlinear dynamical systems remains a central challenge in a wide range of applications, particularly when accurate first-principle models are unavailable. Data-driven approaches offer a promising alternative by designing controllers directly from observed trajectories. A wide range of data-driven methods relies on the Koopman-operator framework that enables linear representations of nonlinear dynamics via lifting into higher-dimensional observable spaces. Finite-dimensional approximations, such as extended dynamic mode decomposition (EDMD) and its controlled variants, make prediction and feedback control tractable but introduce approximation errors that must be accounted for to provide rigorous closed-loop guarantees. This survey provides a systematic overview of Koopman-based control, emphasizing the connection between data-driven surrogate models, approximation errors, controller design, and closed-loop guarantees. We review theoretical foundations, error bounds, and both linear and bilinear EDMD-based control schemes, highlighting robust strategies that ensure stability and performance. Finally, we discuss open challenges and future directions at the interface of operator theory, approximation theory, and nonlinear control.

Koopman-Driven Grip Force Prediction Through EMG Sensing

IEEE Transactions on Neural Systems and Rehabilitation Engineering · 2025-01-01 · 2 citations

Loss of hand function due to conditions like stroke or multiple sclerosis impacts daily activities. Robotic rehabilitation provides tools to restore hand function, while surface electromyography (sEMG) enables the adaptation of the device's force output to the user's condition, thus enhancing rehabilitation outcomes. This study focuses on accurately predicting grip force during medium wrap grasps using a single sEMG sensor pair, addressing the challenge of escalating sensor requirements. We conducted sEMG measurements on 13 subjects at two forearm positions, validating results with a hand dynamometer. Established flexible signal processing steps achieved high peak cross-correlations between the processed sEMG signal and grip force. Influential parameters were subsequently identified through sensitivity analysis. Leveraging a novel data-driven Koopman-based approach and problem-specific data lifting, we devised a method for the estimation and short-term prediction of grip force from processed sEMG signals. The method achieved a weighted mean absolute percentage error (wMAPE) of ~5.5% for grip force estimation and ~17.9% for 0.5-second predictions. The methodology proved robust regarding precise electrode positioning, as the effect of sensing position on error metrics was non-significant. The algorithm executes exceptionally fast, processing, estimating, and predicting a 0.5-second sEMG signal batch in just ~30 ms, facilitating real-time implementation.

Two Roads to Koopman Operator Theory for Control: Infinite Input Sequences and Operator Families

ArXiv.org · 2025-10-16

The Koopman operator, originally defined for dynamical systems without input, has inspired many applications in control. Yet, the theoretical foundations underpinning this progress in control remain underdeveloped. This paper investigates the theoretical structure and connections between two extensions of Koopman theory to control: (i) Koopman operator via infinite input sequences and (ii) the Koopman control family. Although these frameworks encode system information in fundamentally different ways, we show that under certain conditions on the function spaces they operate on, they are equivalent. The equivalence is both in terms of the actions of the Koopman-based formulations in each framework as well as the function values on the system trajectories. Our analysis provides constructive tools to translate between the frameworks, offering a unified perspective for Koopman methods in control.

Koopman learning with episodic memory

Chaos An Interdisciplinary Journal of Nonlinear Science · 2025-01-01 · 2 citations

Koopman operator theory has found significant success in learning models of complex, real-world dynamical systems, enabling prediction and control. The greater interpretability and lower computational costs of these models, compared to traditional machine learning methodologies, make Koopman learning an especially appealing approach. Despite this, little work has been performed on endowing Koopman learning with the ability to leverage its own failures. To address this, we equip Koopman methods-developed for predicting non-autonomous time series-with an episodic memory mechanism, enabling global recall of (or attention to) periods in time where similar dynamics previously occurred. We find that a basic implementation of Koopman learning with episodic memory leads to significant improvements in prediction on synthetic and real-world data. Our framework has considerable potential for expansion, allowing for future advances, and opens exciting new directions for Koopman learning.

Clarifying the Effect of Mean Subtraction on Dynamic Mode Decomposition

SIAM Journal on Applied Dynamical Systems · 2025-09-02

An overview of Koopman-based control: From error bounds to closed-loop guarantees

Annual Reviews in Control · 2025-11-12 · 7 citations

Controlling nonlinear dynamical systems remains a central challenge in a wide range of applications, particularly when accurate first-principle models are unavailable. Data-driven approaches offer a promising alternative by designing controllers directly from observed trajectories. A wide range of data-driven methods relies on the Koopman-operator framework that enables linear representations of nonlinear dynamics via lifting into higher-dimensional observable spaces. Finite-dimensional approximations, such as extended dynamic mode decomposition (EDMD) and its controlled variants, make prediction and feedback control tractable but introduce approximation errors that must be accounted for to provide rigorous closed-loop guarantees. This survey provides a systematic overview of Koopman-based control, emphasizing the connection between data-driven surrogate models, approximation errors, controller design, and closed-loop guarantees. We review theoretical foundations, error bounds, and both linear and bilinear EDMD-based control schemes, highlighting robust strategies that ensure stability and performance. Finally, we discuss open challenges and future directions at the interface of operator theory, approximation theory, and nonlinear control.

Koopman Reduced-Order Modeling with Confidence Bounds

SIAM Journal on Applied Dynamical Systems · 2025-07-29

On Higher Order Drift and Diffusion Estimates for Stochastic SINDy

SIAM Journal on Applied Dynamical Systems · 2024-06-14 · 8 citations

Identifying Equivalent Training Dynamics

2024-01-01

A Koopman operator-based prediction algorithm and its application to COVID-19 pandemic and influenza cases

Scientific Reports · 2024-03-09 · 11 citations

Future state prediction for nonlinear dynamical systems is a challenging task. Classical prediction theory is based on a, typically long, sequence of prior observations and is rooted in assumptions on statistical stationarity of the underlying stochastic process. These algorithms have trouble predicting chaotic dynamics, "Black Swans" (events which have never previously been seen in the observed data), or systems where the underlying driving process fundamentally changes. In this paper we develop (1) a global and local prediction algorithm that can handle these types of systems, (2) a method of switching between local and global prediction, and (3) a retouching method that tracks what predictions would have been if the underlying dynamics had not changed and uses these predictions when the underlying process reverts back to the original dynamics. The methodology is rooted in Koopman operator theory from dynamical systems. An advantage is that it is model-free, purely data-driven and adapts organically to changes in the system. While we showcase the algorithms on predicting the number of infected cases for COVID-19 and influenza cases, we emphasize that this is a general prediction methodology that has applications far outside of epidemiology.