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arxiv: 2607.01819 · v1 · pith:RRVDV67Znew · submitted 2026-07-02 · 📡 eess.SY · cs.LG· cs.SY

Koopman operator theory: fundamentals, control, and applications

Igor Mezi\'c , Jorge Cort\'es , Karl Worthmann , Mircea Lazar , Armin Lederer This is my paper

Pith reviewed 2026-07-03 08:03 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SY
keywords Koopman operatordynamical systemsextended dynamic mode decompositiondata-driven modelingmodel predictive controlnonlinear systemssystems and control
0
0 comments X

The pith

Koopman operator turns nonlinear dynamics into linear representations using observable functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents Koopman operator theory as a framework that represents complex nonlinear dynamical systems through linear evolution of observable functions. It explains how data-driven methods such as extended dynamic mode decomposition generate finite-dimensional approximations that include explicit error bounds for finite data. The tutorial emphasizes extensions to input-driven systems and their use in controller synthesis, including Koopman-based model predictive control. A sympathetic reader would care because the approach lets standard linear control tools apply directly to systems that are otherwise nonlinear.

Core claim

The Koopman operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions, and recently proposed data-driven techniques like EDMD can generate finite-dimensional approximations accompanied by finite-data error bounds.

What carries the argument

The Koopman operator, which evolves observable functions linearly to capture the full nonlinear state evolution.

If this is right

  • Finite-dimensional EDMD approximations become practical surrogate models for simulation and prediction.
  • Systems with inputs admit direct extensions that preserve the linear structure for control design.
  • Koopman MPC applies linear predictive control to originally nonlinear plants while retaining stability guarantees from the linear theory.
  • Kernelized and machine-learning variants of EDMD improve scalability when the observable space must be learned from data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could be tested on high-dimensional fluid or power-system models where traditional linearization fails at large deviations.
  • Error-bound results open a route to certified learning-based controllers whose guarantees do not rely on local linearization.
  • Choice of observables may be automated by combining the theory with dictionary-learning algorithms that minimize the reported error bounds.

Load-bearing premise

Suitable observable functions exist that yield a useful global linear representation for the systems of interest.

What would settle it

A concrete dynamical system for which every choice of observable functions produces approximations whose error bounds grow without bound or fail to capture the dynamics on a positive-measure set.

Figures

Figures reproduced from arXiv: 2607.01819 by Armin Lederer, Igor Mezi\'c, Jorge Cort\'es, Karl Worthmann, Mircea Lazar.

Figure 1
Figure 1. Figure 1: Illustration of the lack of Koopman invariance of the subspace [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the state space reprojections commonly used to [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: One-step prediction errors for different dictionaries. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Average one-step prediction and training errors depending on [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example trajectories (black) for long-term prediction using different [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: Additionally, the RKHS of these kernels is well un [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of different kernel functions [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Phase trajectory plot, Duffing. training trajectory we extract sliding windows of horizon Ht, select a diverse subset of M windows by k-means on the stacked window descriptor [ z (0); z (H/2); z (H) ] (start, midpoint, and end of the lifted window). Then we keep the window nearest to each cluster centre, and unfold each win￾dow into H one-step pairs {(z (k) , u(k) , z(k+1))} Ht−1 k=0 . The resulting N = HM… view at source ↗
Figure 10
Figure 10. Figure 10: Phase trajectory plot, DC motor. Training and evaluation setting. We fit each form with one￾step and multi-step regression: one-step pairs are subsampled uniformly from all training transitions, and multi-step pairs are extracted from the selected sliding windows ( Ht = 20, γ = 10−4 ), matched in number so the two regimes differ only in the regression objective. Over Ntest = 10 test initial conditions we … view at source ↗
Figure 11
Figure 11. Figure 11: State error rollout, DC motor. TABLE III FORCED DUFFING OSCILLATOR. KCF COUPLES THE INPUT TO 6 OF THE 15 LIFT FEATURES (s = 33). MULTI-STEP OPEN-LOOP STATE ERROR OVER Ntest = 10 ICS, Ttest = 150. MAE ∥xˆ − x ∗∥ (multi-step) Op. norm Form median mean worst ∥ · ∥F Linear 1.06 1.16 2.11 4.09 Bilinear 0.52 0.66 1.62 4.26 GeKo 0.55 0.83 2.98 4.78 KCF 0.74 0.74 1.05 4.31 input-coupling overhead: Bilinear and Ge… view at source ↗
Figure 12
Figure 12. Figure 12: Closed-loop Koopman MPC results, Duffing oscillator. [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Closed-loop Koopman MPC results, DC motor. [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript is a tutorial paper introducing Koopman operator theory for global linear representations of nonlinear dynamics via observables, data-driven finite-dimensional approximations such as EDMD (including kernelized and ML variants) with finite-data error bounds, extensions to systems with inputs, and control applications including Koopman MPC. It includes simulation studies and GitHub source code for step-by-step demonstration.

Significance. As a tutorial, the work has value in consolidating established Koopman methods for the systems and control audience and in providing reproducible simulation examples with code; this supports accessibility and adoption without advancing new theorems or empirical claims.

minor comments (2)
  1. [Abstract] Abstract: the phrasing 'recently proposed data-driven techniques... can be used to generate... accompanied by finite-data error bounds' could more clearly attribute the error-bound results to the cited literature rather than the tutorial itself.
  2. The manuscript would benefit from an explicit statement early in the introduction that it is an expository tutorial rather than a research contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript as a tutorial consolidating Koopman methods with reproducible examples and code, and for the recommendation to accept. The referee's summary accurately reflects the paper's scope and contributions. There are no major comments to address.

Circularity Check

0 steps flagged

Tutorial paper with no novel derivations or fitted results

full rationale

The paper is explicitly a tutorial providing a concise introduction to established Koopman operator theory, EDMD, and related control methods, along with demonstrations and open-source simulations. No new derivation chain, parameter fitting, or predictive claim is advanced that could reduce to its own inputs by construction. All referenced results (operator properties, finite-data error bounds) are attributed to prior literature without self-citation load-bearing on novel content. The central purpose is exposition rather than advancement of a theorem or empirical result, making the derivation chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a tutorial on established Koopman operator theory with no new mathematical contributions, free parameters, axioms, or invented entities introduced.

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discussion (0)

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Reference graph

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