Koopman Operator Lifting: Theory & Methods

• Koopman operator lifting is a framework that transforms nonlinear dynamics into a linear evolution of observables in infinite-dimensional spaces.
• It employs EDMD to generate finite-dimensional approximations, providing robust techniques for system identification, prediction, and control.
• The approach finds applications in robotics, networked systems, and uncertainty quantification while addressing challenges like basis selection and numerical conditioning.

Koopman operator lifting is a methodology that recasts the study of nonlinear dynamical systems as an equivalent, infinite-dimensional linear problem in a space of scalar-valued observables. This approach enables the use of linear identification, prediction, and control techniques for systems with fundamentally nonlinear state-space dynamics. Central to Koopman lifting is the selection or construction of a set of observables and the formulation of a finite-dimensional approximation—via extended dynamic mode decomposition (EDMD) or related methods—that serves as a practical surrogate for the infinite-dimensional Koopman operator. This article presents the theoretical principles, practical implementations, computational workflows, and limitations of Koopman operator lifting, with special emphasis on state-of-the-art developments in data-driven nonlinear system identification, control, and the rigorous technical subtleties associated with these methods.

1. Infinite-Dimensional Koopman Operator: Foundations

Given a discrete-time (or time-discretized) dynamical system $x_{k+1} = f(x_k)$, Koopman operator theory shifts the focus from the original state-space to the space of scalar observables, $g: \mathbb{R}^n \rightarrow \mathbb{R}$, typically elements of an infinite-dimensional Hilbert space $\mathcal{H}$. The Koopman operator $\mathcal{K}$ acts linearly on any observable $g$ via composition with the system map $f$:

$(\mathcal{K}g)(x) := g(f(x)).$

For two observables $g_1,g_2$ and scalars $\alpha,\beta$, linearity holds:

$$\mathcal{K}(\alpha g_1 + \beta g_2) = \alpha \mathcal{K}g_1 + \beta \mathcal{K}g_2.$$

Although the original map $g: \mathbb{R}^n \rightarrow \mathbb{R}$0 is typically nonlinear, the evolution of observables under $g: \mathbb{R}^n \rightarrow \mathbb{R}$1 is linear, but at the cost of infinite dimensionality (Snyder et al., 2021).

In the continuous-time setting, with $g: \mathbb{R}^n \rightarrow \mathbb{R}$2, an analogous result holds: the Koopman semigroup evolves observables according to $g: \mathbb{R}^n \rightarrow \mathbb{R}$3, with the action $g: \mathbb{R}^n \rightarrow \mathbb{R}$4 defining the infinitesimal generator (Mauroy et al., 2016).

2. Finite-Dimensional Lifting via Extended Dynamic Mode Decomposition (EDMD)

Practical computation requires a finite-dimensional approximation of the Koopman operator. The EDMD framework proceeds as follows (Snyder et al., 2021, Mauroy et al., 2016):

• Dictionary Selection: Choose $g: \mathbb{R}^n \rightarrow \mathbb{R}$5 scalar observables (basis functions) $g: \mathbb{R}^n \rightarrow \mathbb{R}$6, generating a lifted coordinate $g: \mathbb{R}^n \rightarrow \mathbb{R}$7.
• Data Collection: Gather $g: \mathbb{R}^n \rightarrow \mathbb{R}$8 snapshot pairs $g: \mathbb{R}^n \rightarrow \mathbb{R}$9, where $\mathcal{H}$0.
• Lifted Data Matrices:

$\mathcal{H}$1

• Operator Estimation: Seek best-fit $\mathcal{H}$2 minimizing $\mathcal{H}$3, with solution $\mathcal{H}$4 ($\mathcal{H}$5 is the Moore–Penrose pseudoinverse).

Equivalently, one may form Gram matrices $\mathcal{H}$6 and $\mathcal{H}$7, then $\mathcal{H}$8. This algorithm is robust to measurement noise, operates on snapshot (not derivative) data, and bypasses explicit nonlinear identification (Mauroy et al., 2016).

3. Koopman Eigenfunctions, Spectral Analysis, and Vector Field Recovery

Having identified $\mathcal{H}$9, one solves $\mathcal{K}$0 for eigenpairs. Each $\mathcal{K}$1 yields a Koopman eigenfunction $\mathcal{K}$2, with evolution $\mathcal{K}$3, discretizing continuous-time rates $\mathcal{K}$4 (Snyder et al., 2021).

For system identification, the EDMD approximation of the infinitesimal generator is $\mathcal{K}$5 (principal matrix logarithm). Given a monomial expansion of $\mathcal{K}$6 (the original vector field), one constructs generator matrices $\mathcal{K}$7 for each monomial and solves the overdetermined system

$\mathcal{K}$8

for the coefficients $\mathcal{K}$9 to reconstruct $g$0 (Mauroy et al., 2016).

Key advantages include no requirement for dense sampling and robust performance across unstable, chaotic, or open systems, provided the observables are expressive enough to capture the dynamics.

4. Koopman Lifting for Control and Input-Driven Systems

The Koopman lifting framework extends naturally to controlled systems $g$1. The dictionary can be augmented with input-dependent observables or by stacking input sequences, and the regression seeks best-fit matrices $g$2 in $g$3, with $g$4 (Snyder et al., 2021). This yields a data-driven linear state-space model where standard linear control design methods (including LQR and MPC) can be transferred to the nonlinear system without local linearization (Snyder et al., 2021).

Advanced techniques address causal lifting in physical systems, particularly when input-dependent observables induce anticausal dynamics: either integral observables are used (restoring causality), or the physical system is augmented with inertia or capacitance such that the lifted model becomes causal and linear (Selby et al., 2021). These methods have demonstrated high accuracy and robustness to measurement noise in practical simulations.

For complex underactuated or highly nonlinear systems, structured lifting—where nonlinear resistive, inertial, or coupling elements are made explicit—can achieve higher-fidelity, reduced-order lifted models suitable for convex MPC, as demonstrated in the modeling of bucket–soil interactions for autonomous excavation (Sotiropoulos et al., 2021).

5. Key Applications and Illustrative Results

Koopman operator lifting has been successfully applied to a broad range of systems:

• Oscillatory and underactuated systems: For simple systems (e.g., inverted pendulum), even a low-order polynomial dictionary suffices for accurate reconstruction. For systems such as cart–pole, additional nonlinear or Fourier basis functions may be necessary, yet finite dictionaries may not capture all coupling dynamics, highlighting intrinsic limitations (Snyder et al., 2021).
• Nonholonomic and nonlinear robotic systems: Systematic basis construction methods, such as Kronecker/Hermite polynomial dictionaries tailored to robot configuration space, provably guarantee completeness and rapid convergence in practice (Shi et al., 2021).
• Nonlinear networks and graphical systems: For large-scale nonlinear networks, block-sparse Koopman embeddings reflect underlying network topology, enabling convergence-certified joint state and topology estimation with polynomial complexity (Peng et al., 16 Jun 2026).
• Uncertainty quantification: Data-driven approaches leveraging robust positively invariant (RPI) sets in latent (lifted) coordinates provide certified uncertainty tubes for predicted state trajectories, even when using autoencoder-based, learned liftings (Kim et al., 2023).
• Industrial and control applications: Generalized bilinear Koopman realizations based on metaheuristically-optimized RBF-based lifting functions outperform linear Koopman models, particularly in strongly nonlinear and coupled MIMO industrial systems (Yahagi et al., 17 Feb 2026).

6. Limitations, Basis Choice, and Regularization

Several intrinsic and practical limitations of Koopman operator lifting are recognized:

• Dictionary expressiveness: Finite dictionaries do not, in general, span the true Koopman-invariant subspace except for very special systems, leading to truncation errors and incomplete capture of the nonlinear dynamics (Snyder et al., 2021, Mauroy et al., 2016).
• Diminishing returns and ill-conditioning: Increasing dictionary size beyond a certain point yields diminishing improvements and often induces ill-conditioning in the Gram matrix $g$5 (Snyder et al., 2021). Basis selection—polynomial, Fourier, RBFs, or learned functions—critically affects numerical stability and convergence rates.
• Noise and sampling: EDMD and similar techniques are robust to moderate noise and low-sampling regimes; but ill-chosen dictionaries or poor coverage can exacerbate residuals and destabilize identification (Mauroy et al., 2016).
• Sensitivity to Timestepping: The recovery of an infinitesimal generator via matrix logarithm is valid only if the Koopman matrix has eigenvalues in the correct domain; otherwise, multivaluedness or nonuniqueness can arise (Mauroy et al., 2016).

Mitigation strategies include basis pruning, regularization (e.g., Tikhonov, $g$6 sparsity), system-norm optimization for numerical conditioning, and design of physically interpretable dictionaries or automatic, data-driven basis construction (Shi et al., 2021, Dahdah et al., 2021, Kim et al., 2023).

7. Outlook and Theoretical Significance

Koopman operator lifting has provided a foundational shift in nonlinear systems analysis, identification, and control. By “lifting” system evolution to a (formally) linear process in the space of observables, it enables the transfer of decades of linear systems theory to settings previously dominated by nonlinear techniques. EDMD and its variants render this approach tractable and robust for diverse classes of applications.

However, the tension between finite practical implementability and infinite-dimensional theory remains central. The choice of lifting dictionary governs convergence, accuracy, and computational feasibility, with recent research focusing on structured, topology-aware, data-driven, or even learned dictionaries to balance expressiveness and stability (Shi et al., 2021, Kim et al., 2023).

Koopman operator lifting continues to drive new developments at the intersection of dynamical systems, control theory, numerical analysis, and data-driven modeling, with its theoretical and algorithmic subtleties under active investigation in nonlinear identification, uncertainty quantification, formal verification, and high-dimensional robotic systems (Snyder et al., 2021, Mauroy et al., 2016).