Koopman operator theory: fundamentals, control, and applications

arXiv:2607.01819 · eess.SY, cs.LG, cs.SY · Submitted 2026-07-02 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Koopman operator theory".

Dev: Detailed Research Summary: Koopman Operator Theory (Fundamentals, Control, and Applications) This research paper provides a comprehensive overview of the Koopman operator framework,

Rosa: First, who's behind it and why it matters.

Title and authors: Rosa: So we're looking at this paper called "Koopman operator theory: fundamentals, control, and applications," and it’s really about taking these super complicated nonlinear systems and turning them into something linear that we already understand.

Dev: Exactly. Think of it like finding a secret code that lets us describe how a complex system moves by using simple multiplication instead of those messy nonlinear equations.

Taro: It’s about getting the dynamics into a linear representation so we can actually use the tools from classical control theory, which is pretty powerful for figuring out what happens next.

Rosa: The paper talks about defining this Koopman operator, K, which maps observable functions to new functions based on the system's evolution function, F >

Dev: And it points out that if you can find these eigenfunctions and eigenvalues, you get a spectral analysis of the system's stability and how it settles down >

Taro: It also brings up something called Koopman Mode Decomposition, which is basically a way to break down the evolution of whatever observable you’re tracking into different modes based on those eigenvalues >

Rosa: And then there’s this idea of Koopman invariance, which lets us reduce the infinite-dimensional problem down to a finite-dimensional linear system representation using an approximation matrix K >

Dev: The paper gives a way to measure how good that approximation is with something called Invariance Proximity, IK(V), which tells us how close our learned model actually is to the true operator >

Taro: That’s important because it sets up the math for using data-driven methods, like Extended Dynamic Mode Decomposition, EDMD, to create these surrogate models from empirical data >

Rosa: Right. And they show that these data-driven methods can give us finite-dimensional approximations along with finite-data error bounds >

Dev: That’s a big deal because it means we aren't just guessing; we have a measurable way to know how much error we are actually making when training these models >

Taro: So, what this means for autonomous systems is that you can use these models to predict what the world does even when the rules are nonlinear, which helps with autonomy >

Rosa: And it’s not just about prediction; they show how you can use this theory to design controllers that actually keep the system stable using Koopman Control Family, KCF >

Dev: They suggest you can use these linear models for things like Linear Quadratic Regulator control or even Model Predictive Control, MPC, where you minimize a cost function subject to those dynamics constraints >

Taro: And they touch on how you can do state estimation too with Koopman Observers, KOF, which lets you recover the physical state by reading off the coordinates of specific eigenfunctions >

Rosa: So we’ve covered how they build the framework and how it connects to control design and estimation methods >

Dev: Now we need to look at how this theory actually intersects with modern machine learning because that's where things get really interesting for data scientists >

Title and authors: Taro: Because they explore applying Koopman ideas to world models, like those JEPA architectures, suggesting a shared underlying principle for learning dynamical processes >

Rosa: They also discuss how flow matching and diffusion models can be immediately applied through Koopman techniques because the iterative sampling process looks a lot like a dynamical system >

Dev: And they even look at things like pruning neural networks by recasting the training trajectory as a dynamical system under the Koopman operator >

Taro: Plus, for reinforcement learning, they suggest using spectral structure to handle out-of-distribution problems with something called Koopman Forward Conservative Q-learning, KFC >

Rosa: It really shows how this theory isn't just for pure mathematics; it’s a way to apply AI and machine learning tools to understand and model complex physical processes more rigorously >

Dev: The limitations they mention is that achieving the full invariance of the native space N is often required for the tightest error bounds in Kernel EDMD, which can be tricky to guarantee in practice >

Taro: That means if you aren't careful about the structure of your observable space, those error bounds might not hold as tightly as they predict >

Rosa: So we’ve talked about the theory, the data methods, and how it connects to control and AI applications in this paper called "Koopman operator theory: fundamentals, control, and applications" >

Dev: We've also covered how they improve things by suggesting ways to use input-output data alone to infer exponential stability of closed-loop systems >

Taro: I think the main implication is that we have a unified framework now that lets us tackle nonlinear dynamics in a way that bridges the gap between traditional physics and modern data learning tools >

Rosa: It really does. So, as we wrap up this discussion on this paper, what’s your final thought on where this research goes next?

Dev: I think the next step is refining those dictionaries or basis functions so they can prune away the noise while still capturing the most important dynamics >

Taro: And I think we need to look more into how to make these learning algorithms probabilistic so we can get better uncertainty quantification when training them >

Rosa: That sounds like a good path forward for making these models more reliable in real-world scenarios, especially as we move into robotics and planning >

Dev: Yeah, and connecting this theory more directly to contact-rich dynamics or soft dynamics in robotics is where I see the most immediate practical impact for me >

Taro: Because understanding how the system handles those unexpected interactions is critical for any system that needs to be robust autonomy.

Rosa: That’s a solid point about robustness, so we’ve explored the core ideas of this paper called "Koopman operator theory: fundamentals, control, and applications" and its potential across modeling and control >

The paper's summary: Rosa: So, this paper lays out this Koopman operator theory as basically a way to translate messy nonlinear system behavior into something linear that we can actually control or model easily.

Dev: It’s about finding this linear world for complex dynamics so we can use the same tools from traditional control engineering to figure things out.

Rosa: They’re showing how you define this operator, K, which is like a mathematical rule that takes whatever function describes the system and gives you a new function based on how the system is actually moving.

Dev: It’s not just a simple mapping; it highlights that if you find these eigenfunctions and their eigenvalues, you get this spectral analysis of how stable or oscillatory the system actually is.

Rosa: They introduce this thing called Koopman Mode Decomposition, which is essentially a formal way to break down the evolution of whatever observable you’re tracking into different patterns based on those eigenvalues.

Dev: And then there’s this idea of invariance, where they reduce that huge infinite problem down to a manageable finite system using some approximation matrix, K.

Rosa: They quantify how good that approximation is with this Invariance Proximity metric, IK(V), which tells you exactly how close your learned model is to the true system.

Dev: That’s a big deal because it sets up the math for using data-driven techniques, like Extended Dynamic Mode Decomposition, to build these surrogate models from real training data.

Rosa: They show that these methods can give us finite approximations along with error bounds that are tied directly to that IK(V) proximity we talked about earlier.

Dev: That means we aren't just guessing the model; we have a measurable way of knowing how much error we are actually making when training these models.

Rosa: So, what this really means is that you can use these linear models to predict what a nonlinear world does even when you don’t know the exact nonlinear equations governing it.

Dev: And it’s not just about prediction; they show how you can use this theory to design controllers that keep the system stable using something called the Koopman Control Family.

Rosa: They suggest you can use these linear models for things like standard control or even Model Predictive Control, MPC, where you minimize a cost function subject to those dynamics constraints.

Dev: They also touch on state estimation with Koopman Observers, KOF, which lets you recover the actual physical state by reading off coordinates of specific eigenfunctions.

Rosa: It really shows how this theory isn't just for pure math; it’s a way to apply AI and machine learning tools to understand complex physical processes more rigorously.

Dev: But they do flag that achieving the full invariance of the native space is often required for those tightest error bounds in Kernel EDMD, which can be tricky to guarantee when you're actually building something for real-time systems.

Rosa: That means if you aren't careful about how your observable space is structured, those error bounds might not hold as tightly as they predict.

Dev: So we’ve talked about the theory and the data methods, and now it’s time to look at how this connects directly to modern machine learning applications.

The paper's improvements: Tom: So, we’re looking at how the authors are trying to make this whole Koopman framework more practical and better suited for real-world use than just the math on paper.

Rosa: The biggest thing they push is this idea of "Deep Koopman" architectures, which means using neural networks that have a specific loss function.

Dev: They’re balancing three different kinds of losses: prediction error, reconstruction error, and a multi-step linearity loss.

Rosa: So the AI learns observables that cover a finite-dimensional subspace where the system actually behaves linearly enough for the network to work well.

Dev: That’s smart because it tackles one of the biggest problems in deep learning—the high dimensionality and noise—by forcing it to learn a simpler, more relevant space.

Rosa: Then they talk about improving controller design by using input-output data alone to infer if a closed-loop system will actually be stable.

Dev: That’s interesting because you usually need the actual physics model for stability analysis, not just data from running the system.

Rosa: They propose this way to bypass that, looking at the input and output patterns and seeing if they look like they lead to an exponentially stable state.

Dev: If that holds up, it could let us design controllers faster for systems where we don't have perfect physical equations.

Rosa: And they also discuss refining those dictionaries, the basis functions you use to build the linear model, so you can prune away the noise while still capturing what matters most.

Dev: Pruning is a huge topic in AI right now; if we can make that more theoretically sound, it means we can shrink these models down to be much faster for real-time applications.

Rosa: So, the implication here is that we move away from just trying to fit a model and start building models with built-in structural knowledge about how the dynamics work.

Dev: It shifts the focus from just getting a decent fit to ensuring that whatever we learn actually respects the underlying system structure.

Rosa: And this leads us right into how this relates to those cutting-edge applications in robotics, specifically contact-rich or soft dynamics, where things get really messy outside of clean lab settings.

Conclusion: Rosa: So we’ve seen how this paper on "Koopman operator theory: fundamentals, control, and applications" shows us how to take those complicated nonlinear systems and map them onto a linear representation that we can actually work with using standard tools.

Dev: It boils down to turning complexity into linearity so we can apply robust control techniques or even design better AI models for planning.

Rosa: The main thing is the connection between the data-driven modeling, like EDMD, and getting those rigorous error bounds tied back to the core theory of Koopman invariance.

Dev: So, for an engineer it means we can finally build a model that isn't just a curve fit but one that has some mathematical guarantee about how far off it might be when we use it in a real-time loop.

Rosa: And for someone who only listens to the show, this means that understanding how a system moves doesn't have to mean solving the original nonlinear equations directly.

Dev: It changes things because you can use established control methods like LQR or MPC on these linear approximations, which is much easier than trying to handle the original nonlinearity in every step.

Rosa: Taro, what’s your read on this for autonomy when things get weird?

Taro: I see this as a way to give autonomy systems a better internal language; if we can model the system linearly, we can predict how it will behave when the environment misbehaves in ways that are hard to calculate directly.

Dev: That makes sense. If the KCF works well, it could help us design controllers that handle those weird nonlinear interactions during operation without crashing or losing stability.

Rosa: It’s definitely about making the gap between simulation and real-world deployment smaller by giving us a more accurate, linear bridge across that gap.

Dev: Yeah, and looking ahead, they mentioned using input-output data alone to infer stability for closed-loop systems—that’s a pretty neat idea for reducing the amount of physical testing needed before we deploy something.

Rosa: Exactly. So "Koopman operator theory: fundamentals, control, and applications" gives us a unified way to think about nonlinear dynamics across modeling and control.

Dev: It sets up a solid foundation for future work on making these learning algorithms more probabilistic so we can get better uncertainty quantification when training them in complex scenarios.

Taro: I think the next big step is definitely connecting this theory directly to those contact-rich or soft dynamics problems in robotics because that’s where most of the interesting real-world complexity lives.

Department of Mechanical Engineering, University of California, Santa Barbara, USA · Department of Mechanical and Aerospace Engineering, University of California, San Diego, USA · Optimization-based Control Group, TU Ilmenau, Germany · Constrained Control of Complex Systems Lab, Control Systems Group, Department of Electrical Engineering, TU Eindhoven · Department of Electrical and Computer Engineering, National University of Singapore

eess.SY, cs.LG, cs.SY

Submitted: 2026-07-02

Updated: 2026-10-08

Code: https://github.com/KOT-tutorial/CDC26

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 87/100

The gist: Detailed Research Summary: Koopman Operator Theory (Fundamentals, Control, and Applications) This research paper provides a comprehensive overview of the Koopman operator framework, detailing its

Key concepts

Koopman Operator (K)
This is a mathematical tool that takes a complex nonlinear system's evolution function and maps it to an equivalent linear one. It allows researchers to analyze complicated dynamics using the well-understood tools of linear algebra, making nonlinear problems tractable through linearization.
Extended Dynamic Mode Decomposition (EDMD)
EDMD is a data-driven method used to estimate the Koopman operator from empirical data. It works by approximating the operator using a dictionary of basis functions and training examples, effectively creating a linear model that mimics the true nonlinear system's behavior.
Koopman Invariance Proximity (IK(V))
This metric measures how closely an approximate linear model matches the true dynamics within a specific subspace. A high proximity value indicates that the learned linear representation is a very good approximation of the original nonlinear system's behavior in that space.
Koopman Control Family (KCF)
KCF involves defining operators for every possible constant input to a system. By finding invariant subspaces under these operators, engineers can design controllers that are simpler and more effective than those designed for the original nonlinear system.

Terminology

Summary

Detailed Research Summary: Koopman Operator Theory (Fundamentals, Control, and Applications)

This research paper provides a comprehensive overview of the Koopman operator framework, detailing its theoretical foundations in dynamical systems theory, its application in control system design, and its burgeoning intersection with modern machine learning paradigms. The central thesis is that the Koopman operator offers a powerful method to translate complex nonlinear dynamics into an equivalent linear representation, thereby enabling the application of well-established classical control-theoretic techniques and providing a rigorous foundation for data-driven modeling.

I. Theoretical Foundations of the Koopman Operator

The paper begins by formally defining the Koopman operator (K), which serves as a global linear representation of highly complex dynamical systems. The operator maps observable functions g to new functions defined on the state space, satisfying Kg = g F, where F is the system's nonlinear evolution function. A critical property highlighted is that this operator is linear and remains bounded if the function space F upon which it acts is Koopman invariant.

Spectral Analysis and Invariance: The stability and dominant patterns of a dynamical system are characterized by its eigenfunctions (phi i) and corresponding eigenvalues (lambda i). For discrete time, this is defined by Kf = lambda f; for continuous time, Kt phi = e lambda t phi. A key structural feature is the closure of the eigenfunction set under pointwise multiplication: if phi 1 and phi 2 are eigenfunctions with eigenvalues beta 1 and beta 2, then their product, phi 1 phi 2, is also an eigenfunction with eigenvalue beta 1 beta 2.

Koopman Mode Decomposition (KMD): The paper introduces KMD as a formal decomposition of the observable's evolution. It decomposes the evolution of g into a projection onto the eigenspace associated with the eigenvalue 1 (P 0g(x)), a summation over other eigenvalues (sum lambda k i g, phi* i phi i(x)), and a residual term involving Koopman eigenfunctions and their eigenvalues. The point spectrum is explicitly linked to the almost periodic component of the system dynamics, which describes behavior converging toward equilibria.

Koopman Invariance and Approximation Error: A crucial concept for rigorous analysis is Koopman invariance, where Kg in V for all g in V, meaning a subspace V is invariant under the operator. This invariance allows for the reduction of the infinite-dimensional problem to a finite-dimensional linear system representation, KPV, which can be approximated by a matrix K. The quality of this approximation is quantified by the Invariance Proximity (IK(V)), defined as:

IK(V):= g in V|Kg| over|Kg - K approxg|

This proximity metric is essential for deriving tight error bounds on learned models.

II. Data-Driven Modeling and Surrogate Models

The paper bridges the theoretical framework with practical data-driven methods, focusing on approximating the Koopman generator using empirical data.

Extended Dynamic Mode Decomposition (EDMD): EDMD is presented as a primary data-driven approach to compute a surrogate model of the Koopman operator. It approximates the generator using an arbitrary dictionary of basis functions and training data, resulting in an estimate = K T-1, times. A specialized version, Kernel EDMD (kEDMD), utilizes kernels as the dictionary. The error analysis for kEDMD is detailed by splitting the approximation error into components related to projection and Koopman invariance proximity (IK(V)), emphasizing that the latter requires the invariance of the native space N.

Koopman Meets Machine Learning: The synergy between Koopman theory and modern AI is explored in several high-impact areas:

  1. World Models: Architectures like JEPA are noted as being structurally similar to Koopman-based models, suggesting a shared underlying principle for learning dynamical processes.

  2. Generative Models: Flow matching and diffusion models, which utilize iterative sampling interpreted as a dynamical system, are shown to be immediately applicable through Koopman techniques.

  3. Pruning Analysis: The training trajectory of neural networks can be recast as a dynamical system under the Koopman operator, providing a theoretical lens for understanding parameter pruning strategies (e.g., recasting gradient descent trajectories).

  4. Reinforcement Learning (RL): The Bellman equation governing optimal control is viewed as a dynamical object amenable to linearization via Koopman theory. Furthermore, the spectral structure of K can be exploited to address the out-of-distribution problem in offline RL through Koopman Forward Conservative Q-learning (KFC), which uses eigenfunctions to encode system symmetries for principled data augmentation.

III. Extension to Control Systems and Controller Design

The framework is extended from describing dynamics to designing controllers that act upon them.

Koopman Control Family (KCF): For systems with inputs (x+ = F(x, u)), the Koopman Control Family (KCF) is introduced, defined as the set of operators K u corresponding to every constant input u in U. Invariant subspaces under the KCF allow for an input-state separable form: (x+) = A(u) (x), where A: U to R n times n.

Controller Design Techniques: Controllers are designed by leveraging these linear models. This includes LQR control using linear Koopman models, robust controller design via bilinear representations, and Koopman Model Predictive Control (MPC). MPC utilizes the data-driven surrogate model to minimize a cost functional subject to dynamics constraints. Stability analysis for the resulting closed-loop system is achieved by finding Lyapunov functions V(x) that satisfy a Lyapunov decrease condition, with approximation errors compensated for by terms such as alpha 3(|x|).

Observer Design: State estimation is facilitated through linear infinite-dimensional observers (Koopman Observers (KOF)). By selecting a finite set of Koopman eigenfunctions whose span covers the state and output observables, a linear time-invariant system is formed in the lifted space (xi+ = xi, x = Mx xi, y = My xi), enabling standard Luenberger or Kalman observer design.

IV. Advanced Analysis and Future Directions

The paper concludes by detailing advanced analysis techniques and outlining future research avenues:

Error Analysis in EDMDc: Uniform pointwise error bounds for kEDMD are derived by partitioning the approximation error, linking it directly to the Koopman invariance proximity IK(V). The paper also discusses different operator norms (Linear, Bilinear, GeKo) and notes that while some metrics might yield lower median errors (e.g., GeKo), others offer better worst-case guarantees.

Input-Output Data: Extensions are proposed to leverage input-output data alone to infer the exponential stability of closed-loop systems.

Synthesis and Outlook: The overarching conclusion is that Koopman theory provides a unified, powerful framework for modeling, control, and learning from complex nonlinear systems. Future research priorities include:

  • Refining Dictionaries: Developing subspace pruning methods with theoretical guarantees to reduce modeling error while preserving relevant dynamics.

  • Probabilistic Learning: Exploring the probabilistic perspective on Koopman operator learning to enable scalable training methods with rigorous uncertainty quantification.

  • Theoretical Understanding of Invariance: Deepening the understanding of how approximate Koopman invariance impacts the complexity of learning problems.

  • Application Focus: Connecting theoretical progress more directly to high-impact applications in robotics (contact-rich, soft dynamics) and large-scale AI models for vision and planning.

In summary, this paper establishes the Koopman operator not merely as a mathematical tool but as a foundational paradigm that seamlessly integrates nonlinear system analysis with linear modeling, offering robust solutions across control engineering and cutting-edge machine learning research.

Improvements for AI systems

  1. Data-driven surrogate modeling for complex dynamics: The system can generate finite-dimensional approximations accompanied by finite-data error bounds using methods like EDMD and kernelized variants, allowing for tractable modeling of highly complex nonlinear systems.

  2. Controller design with closed-loop guarantees: Controllers can be designed using Koopman operator theory to ensure stability by leveraging the proportional error bound derived from the approximation, enabling rigorous verification of asymptotic stability for MPC closed-loop dynamics.

  3. Robust Model Predictive Control (MPC): The system can implement MPC that handles model-plant mismatch by utilizing input-lifted forms like KCF, which are shown to achieve tracking performance in cases where control-affineness fails, although this requires specialized solvers for real-time deployment.

  4. State estimation via Koopman observers: A Luenberger-type observer can be designed on the lifted space to recover the physical state by reading off principal eigenfunction coordinates, which allows for exponential convergence of the estimation error under specific observability conditions.

  5. Deep Learning-based Koopman models: The system can employ Deep Koopman architectures, utilizing a loss function that balances prediction loss, reconstruction loss, and multi-step linearity loss to learn observables spanning a finite-dimensional invariant subspace for deep learning applications.

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