Extended Dynamic Mode Decomposition

Koopman operator theory provides a linear framework for analyzing nonlinear dynamical systems and has become a major tool for data-driven modeling. A central challenge, however, is that finite-dimensional approximations computed by methods such as extended dynamic mode decomposition (EDMD) require the dictionary to be specified a priori. Recent machine-learning approaches address this limitation by learning the dictionary from data, predominantly using artificial neural network (ANN) autoencoder architectures. Although kernel methods offer an alternative with greater interpretability and tractability for theoretical analysis, they have received little attention in this setting. We introduce extended dynamic mode decomposition with kernel-based dictionary learning (EDMD-kDL), a kernel-based method for learning finite-dimensional Koopman embeddings directly from data. The method combines ideas from collocation methods and bilevel optimization to simultaneously learn a kernel dictionary and the corresponding Koopman approximation. We evaluate EDMD-kDL against state-of-the-art ANN-based approaches on a range of numerical experiments, including global sea-surface-temperature forecasting and learning directly from video data. Across all tested settings, EDMD-kDL achieves performance comparable to or better than the ANN-based methods. Moreover, in contrast to standard kernel methods, the proposed approach is scalable to large datasets by design since the size of the required kernel matrices depends on the number of collocation points rather than the size of the training dataset.

Soft pneumatic actuators offer inherent compliance and safe interaction but remain difficult to model and control because of their highly nonlinear, distributed dynamics. We present a control-oriented data-driven modeling and control framework that decomposes actuator behavior into a nonlinear static equilibrium model and a linear residual dynamics model identified using Extended Dynamic Mode Decomposition with control (EDMDc). This representation enables feedforward compensation, task-space feedback control, and local closed-loop stability analysis through an augmented linear model. Experiments achieve approximately 1 mm root mean square error (RMSE) during low-speed (approximately 10 mm/s) trajectory tracking and below 10 mm RMSE at higher speeds (approximately 100 mm/s). The framework further achieves stable tracking of highly dynamic user-generated references with peak accelerations exceeding 25 m/s^2 while simultaneously performing real-time obstacle avoidance. Finally, the proposed stability analysis is experimentally validated by accurately predicting stable, marginal, and unstable operating regimes. These results demonstrate that structured, control-oriented learning provides an accurate and practical framework for soft actuator control.

Extended Dynamic Mode Decomposition (EDMD) approximates Koopman operators from data, but a single global operator is inefficient when different state-space regions exhibit distinct local dynamics. We introduce Cluster-Weighted EDMD (CW-EDMD), which jointly learns a soft phase-space partition and a per-cluster EDMD operator. Its Expectation-Maximization (EM) objective assigns each transition based on both geometric proximity and prediction residuals, so clusters specialize where local Koopman models are accurate rather than where the data are dense. On Lorenz, damped pendulum, and Duffing systems, across 36 configurations and 10 seeds, CW-EDMD improves matched-degree EDMD in one-step and 5s-rollout prediction. Across 288 paired comparisons, there are significant error reductions in 258 cases, increases in 4, and no differences in 26. Median one-step error reductions are 57x, 2.7x, and 12x on pendulum, Duffing, and Lorenz, respectively.

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. Data-driven techniques, like extended dynamic mode decomposition (EDMD), kernel EDMD, 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.

We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.

This work establishes a rigorous bridge between infinite-dimensional delay dynamics and finite-dimensional Koopman learning, with explicit and interpretable error guarantees. While Koopman analysis is well-developed for ordinary differential equations (ODEs) and partially for partial differential equations (PDEs), its extension to delay differential equations (DDEs) remains limited due to the infinite-dimensional phase space of DDEs. We propose a finite-dimensional Koopman approximation framework based on history discretization and a suitable reconstruction operator, enabling a tractable representation of the Koopman operator via kernel-based extended dynamic mode decomposition (kEDMD). Deterministic error bounds are derived for the learned predictor, decomposing the total error into contributions from history discretization, kernel interpolation, and data-driven regression. Additionally, we develop a kernel-based reconstruction method to recover discretized states from lifted Koopman coordinates, with provable guarantees. Numerical results demonstrate reliable prediction of nonlinear delay systems, with potential relevance to future control applications.

This paper investigates Koopman operator-based approaches for multivariable control of a two-spool turbofan engine. A physics-based component-level model is developed to generate training data and validate the controllers. A meta-heuristic extended dynamic mode decomposition is adapted, with a cost function designed to accurately capture both spool-speed dynamics and the engine pressure ratio (EPR), enabling the construction of a single Koopman model that can be reused across multiple control strategies. Using the identified time-varying Koopman model, an adaptive Koopman-based model predictive controller (AKMPC) with a disturbance observer is developed and compared with a Koopman-based feedback linearization controller (K-FBLC) and its integrator-augmented version (K-FBLC-I). The Koopman representation further enables nonlinear GTE output limiters, such as rotor-acceleration and turbine-inlet-temperature limits, to be expressed as linear constraints in the AKMPC. The controllers are evaluated for two control configurations of spool speeds and EPR, under both sea-level and varying flight conditions. The results demonstrate that the proposed identification approach enables accurate predictions of both spool speeds and EPR, allowing the Koopman model to be reused flexibly across different control formulations. While all strategies achieve comparable performance in sea-level conditions, the AKMPC demonstrates improved performance under varying flight conditions due to its ability to capture nonlinear dynamics, handle constraints, and compensate for model mismatch. Moreover, the EPR control strategy improves the thrust response. The study highlights the applicability of Koopman-based control and the advantages of the AKMPC framework for robust turbofan engine control.

Parametric models deployed in non-stationary environments degrade as the underlying data distribution evolves over time (a phenomenon known as temporal domain drift). In the current work, we present KOMET (Koopman Operator identification of Model parameter Evolution under Temporal drift), a model-agnostic, data-driven framework that treats the sequence of trained parameter vectors as the trajectory of a nonlinear dynamical system and identifies its governing linear operator via Extended Dynamic Mode Decomposition (EDMD). A warm-start sequential training protocol enforces parameter-trajectory smoothness, and a Fourier-augmented observable dictionary exploits the periodic structure inherent in many real-world distribution drifts. Once identified, KOMET's Koopman operator predicts future parameter trajectories autonomously, without access to future labeled data, enabling zero-retraining adaptation at deployment. Evaluated on six datasets spanning rotating, oscillating, and expanding distribution geometries, KOMET achieves mean autonomous-rollout accuracies between 0.981 and 1.000 over 100 held-out time steps. Spectral and coupling analyses further reveal interpretable dynamical structure consistent with the geometry of the drifting decision boundary.