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Koopman Operator Dynamical Models: Learning, Analysis and Control
Petar Bevanda, Stefan Sosnowski, Sandra Hirche
TL;DR
Nonlinear systems need representations that support efficient prediction, analysis, and control, but Koopman operators are generally infinite-dimensional and lack a general finite-realization framework. The paper systematically reviews data-driven Koopman models, relates them to system theory and control, and concludes that invariant-coordinate learning, model selection, and error quantification remain central challenges.
Problem
Koopman operators offer globally linear representations of nonlinear systems, but finite approximations lack an overarching framework and suitable coordinates may be difficult to learn and reconstruct from.
Method
The paper systematically reviews structured and unstructured data-driven Koopman representations, system-theoretic connections, and control applications.
Results
The review finds that current approaches represent transient and quasi-periodic behavior well, while non-dissipative dynamics and continuous spectra remain open research questions.
Takeaways & Limitations
Efficiently learned, well-posed Koopman representations can support prediction, analysis, and control through linear techniques applied to nonlinear systems.
Takeaways & Limitations
Safety-critical applications remain constrained by the lack of tractable a priori error quantification, while genuinely invariant coordinates and principled model-order selection remain unresolved.
Abstract
from arXiv · showhide
The Koopman operator allows for handling nonlinear systems through a (globally) linear representation. In general, the operator is infinite-dimensional - necessitating finite approximations - for which there is no overarching framework. Although there are principled ways of learning such finite approximations, they are in many instances overlooked in favor of, often ill-posed and unstructured methods. Also, Koopman operator theory has long-standing connections to known system-theoretic and dynamical system notions that are not universally recognized. Given the former and latter realities, this work aims to bridge the gap between various concepts regarding both theory and tractable realizations. Firstly, we review data-driven representations (both unstructured and structured) for Koopman operator dynamical models, categorizing various existing methodologies and highlighting their differences. Furthermore, we provide concise insight into the paradigm's relation to system-theoretic notions and analyze the prospect of using the paradigm for modeling control systems. Additionally, we outline the current challenges and comment on future perspectives.
1. INTRODUCTION
The paper motivates Koopman-based representations as globally linear alternatives for nonlinear systems and reviews their data-driven, system-theoretic, and control perspectives. It addresses the lack of a general finite-realization framework by organizing methods and connecting theory with tractable modeling approaches.
- Complex systems often lack accurate first-principles models, motivating flexible machine-learning techniques for prediction, control, and analysis.
- Koopman realizations trade infinite-dimensionality for linearity by discovering coordinates that produce globally linear models of nonlinear systems.
- Koopman representations support nonlinear mode analysis, dynamic-regime decomposition, and analysis using linearly evolving coordinates.
- Finite Koopman realizations lack an overarching framework, creating a need to organize diverse data-driven approaches and clarify their theoretical properties.
- The article systematically reviews Koopman representations, system-theoretic analysis, control integration, applications, and future perspectives.
Notation
The notation distinguishes vectors, matrices, continuous- and discrete-time variables, function spaces, norms, and Koopman operator families associated with flows and maps.
- Bold lowercase and uppercase symbols denote vectors and matrices, while N, R, and C denote natural, real, and complex numbers.
- Continuous- and discrete-time dependence is written as y_t(·) and y_k, with t and k ranging over nonnegative reals and integers.
- A flow F^t(x) and its composition-operator family represent continuous-time evolution, while map iterates use a corresponding discrete-time notation.
2. Koopman operator theory
Koopman theory represents nonlinear state evolution through linear composition operators acting on observables, with eigenfunctions providing evolving coordinates and spectra separating dynamical components. The theory includes continuous- and discrete-time operators, generators, modes, and adjoint formulations.
- The Koopman framework replaces point-orbit evolution with propagation of observables over the state space, while modes combine evolving eigenfunctions to describe nonlinear evolution.
- The Koopman operator acts on observable functions by composing them with the system flow or discrete-time map.
- Koopman operators are linear in observables, although the underlying state dynamics may remain nonlinear.
- For bounded generators, the Koopman semigroup is obtained as K_t = exp(G_K t), providing a continuous-time linear evolution of observables.
- Koopman eigenfunctions provide linearly evolving coordinates that enable linear representations of nonlinear systems.
- Spectral decomposition separates transient and quasi-periodic behavior from continuous-spectrum components associated with chaotic on-attractor dynamics.
- Products of Koopman eigenfunctions are eigenfunctions, allowing principle eigenpairs to generate additional eigenpairs.
- A nonlinear discrete-time example uses closed eigenfunction products to construct finite linear-in-time coordinates, illustrating global representation of nonlinear trajectories.
3. Data-driven Koopman operator-based dynamical models
Finite-dimensional Koopman models are needed because the operator is generally infinite-dimensional, but choosing suitable invariant coordinates remains difficult. The section organizes data-driven approaches by their lifting coordinates and contrasts unstructured observable lifting with structurally constrained, spectrally direct methods.
- Problem: Finite-dimensional representations are required because nonlinear systems may need infinitely many dimensions for an exactly linear Koopman representation.Predetermined arbitrary bases may require infinitely many functions, whereas Koopman-invariant bases can avoid this requirement.
- Observable-based approaches: DMD approximates Koopman evolution by fitting a transition matrix between snapshot matrices, with SVD supporting reduced-order modeling of high-dimensional data.The closed-form matrix uses the Moore–Penrose pseudoinverse.
- Classification: Koopman discovery methods are tentatively classified as spectrally indirect observable lifting, spectrally direct eigenfunction coordinates, or time-delay coordinates for on-attractor behavior.The classification concerns the coordinates used to predict outputs of interest and includes linear or nonlinear state reconstruction choices.
- Observable-based approaches: EDMD lifts data through often heuristically chosen nonlinear features before solving a compatible regression problem, with kernel, generator, and structured extensions.Vanilla DMD is recovered when the lifting map is the identity, ψ(x) = x.
- Limitations and structural approaches: Unstructured observable lifting can yield only locally accurate predictions and spectra that misrepresent the underlying system, motivating direct learning of genuine eigenfunctions.The section presents spectrally direct learning as better posed when long-term prediction and analysis require linearly evolving coordinates.
- Structurally aware approaches: Structurally aware approaches exploit algebraic, geometric, and topological properties to provide supervision for learning Koopman eigenfunctions.Topological conjugacy constrains eigenvalues and coordinate bases to genuine Koopman eigenfunctions, producing a more restricted solution space.
4. System analysis and Koopman operator paradigm
The Koopman paradigm connects nonlinear-system analysis with spectral and Lyapunov concepts, providing system information through observables and eigenfunctions. Its stability conclusions depend on spectral properties, observable spaces, and assumptions such as invariance and regularity.
- Koopman representations expose invariant manifolds, fixed points, attractors, and periodic orbits as information useful for system analysis.
- 4.1. Classical stability analysis and Koopman paradigm: Lyapunov functions are observables whose generator derivative is negative semidefinite, linking classical Lyapunov analysis to Koopman operator theory.
- 4.1. Classical stability analysis and Koopman paradigm: A proper Lyapunov function with negative Lie-derivative behavior outside the equilibrium implies asymptotic stability.
- 4.1. Classical stability analysis and Koopman paradigm: Spectral analysis of nonlinear maps can mirror linear stability analysis by using linear techniques to construct a Lyapunov function representing nonlinear flow, although discrete-time spectral stability connections remain insufficiently rigorous.
- 4.2. Analysis via the Koopman paradigm: Koopman spectral analysis can extend stability reasoning globally: C1 eigenfunctions imply global asymptotic stability under the stated full-rank and stable-linearization assumptions.
- 4.2. Analysis via the Koopman paradigm: For a positive invariant compact set, a continuous eigenfunction with eigenvalue satisfying Re{s} < 0 has a zero level set that is forward invariant and globally asymptotically stable.
5. Koopman-based control approaches and applications
This section turns to non-autonomous systems and approaches for efficient control-system representations using the Koopman operator paradigm.
- The focus shifts to non-autonomous systems and efficient Koopman-based representations of control systems.
5.1. Extensions to control systems
The paper modifies Koopman models for controlled systems, deriving structured representations for control-affine dynamics and discussing broader nonlinear cases with important scope limitations.
- Extensions to control systems: Koopman models formed for autonomous systems require modifications to represent non-autonomous systems.The paper uses an atomic approach based on the Koopman eigenfunction evolution PDE.
- Control-affine nonlinear systems: A novel derivation obtains Koopman eigenfunctions for single-input, single-output control-affine systems.
- Control-affine nonlinear systems: Control-affine dynamics can have bilinear eigenfunction evolution when control-vector-field eigenfunctions lie in the span of drift eigenfunctions.
- Control-affine nonlinear systems: The resulting continuous-time formulation yields a control-parameterized family of Koopman semigroups and a corresponding discrete-time representation.
- Control-affine nonlinear systems: The common lifted model zdot = Az + Bu represents control effects only locally, explaining its suitability mainly for short-term prediction.
- General nonlinear control systems: General nonlinear systems f(x,u) lack a closed-form bilinear prediction representation, although joint state-and-input eigenfunctions and state inflation provide alternatives.
- General nonlinear control systems: Restricting input dependence to finitely many inputs can produce switched Koopman operators, which have been used for efficient model predictive control.
5.2. Control-oriented frameworks
The paper surveys control-oriented Koopman frameworks spanning energy-based, reduced-order, optimization-based, stabilization, observer, and filtering applications.
- Control-oriented frameworks: Energy-based and reduced-order approaches use intrinsic Koopman dynamics for control and data-driven switched control of PDE-governed systems.
- Control-oriented frameworks: Koopman-based MPC can improve accuracy and validity over locally linearized models while reducing computational burden relative to nonlinear optimization.
- Control-oriented frameworks: Lifted-space control Lyapunov functions have shown promise for nonlinear optimal stabilization.
- Control-oriented frameworks: Koopman dynamical models have also been applied to observer synthesis and filtering, with reported improved efficacy.
5.3. Why not linearize via feedback?
The paper motivates Koopman-based global linearizations as alternatives to feedback linearization, particularly when useful nonlinearities and modeling uncertainty make cancellation undesirable.
- Why not linearize via feedback?: Non-local linear representations can support efficient controller design in computation and control effort.
- Why not linearize via feedback?: Koopman-based linearizations avoid compensating input injections required by exact input-output and feedback linearization.
- Why not linearize via feedback?: For underwater vehicles, hydrodynamic damping creates useful nonlinearities that feedback linearization may cancel.
- Why not linearize via feedback?: Feedback linearization can destabilize an underwater vehicle under small modeling errors and require excessive control effort.
6. CONCLUSION
The review finds that Koopman-based modeling can simplify prediction, analysis, and control when suitable representations are learned, but key theoretical and practical gaps remain. These include invariant-coordinate learning, error quantification, principled comparisons, robustness assessment, and adaptive or online methods.
- Koopman representations can greatly simplify prediction, analysis, and control if they are efficiently learned in a well-posed manner.
- The main challenge is learning genuinely invariant coordinates while rigorously selecting their relevance and the model order.
- Safety-critical control still lacks tractable a priori error quantification, hindering wider applicability of Koopman-based frameworks.
- The field lacks an overarching data-driven framework connecting system-theoretic considerations with realizations and enabling principled comparisons to conventional identification and control.
- Koopman-based controllers require comparisons with exact feedback linearization and robustness assessment under external disturbances.
- Adaptive control and online learning under changing conditions remain largely unexplored.