TL;DR
A physics-informed Koopman representation based on generalized momenta is introduced, yielding a linear control-affine model in lifted coordinates with known input structure that avoids the bilinear state – input coupling inherent in standard Koopman approaches, enabling improved prediction accuracy and tractable controller synthesis.
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.
One of the main objectives in control theory is to obtain a linear representation of inherently nonlinear systems in order to leverage the analytical and theoretical tools developed for linear systems. In this context, the Koopman operator has attracted increasing interest in recent years.Koopman operator theory provides a framework in which nonlinear dynamical systems are represented by a linear operator acting on an infinite-dimensional Hilbert space. Since such an infinite-dimensional representation is not numerically tractable, numerous finite-dimensional approximation methods have been proposed. These approaches typically rely on time-series data and include extended dynamic mode decomposition as well as deep learning–based variants. In this paper, we propose an original machine-learning-based approach for the synthesis of a fixed-dimensional Koopman approximant (lifting) of continuous-time nonlinear systems. A differential state-space representation of the system (as opposed to a recurrent state model) is assumed to be available through its vector field (f). The proposed encoder departs from conventional approaches in that it does not directly output the current latent state, but instead generates samples of the latent trajectory evaluated at user-defined time instants (temporal discretization). This formulation enables the integration into the learning process of Physical & Latent Continuous Losses, enforcing consistency between the physical dynamics and the Koopman dynamics, as well as Physical & Latent Boundary Losses, ensuring consistency with the prescribed initial conditions. In parallel, we introduce a structural stability constraint on the Koopman operator. The effectiveness of the proposed methodology is demonstrated through the analysis and simulation of two polynomial dynamical systems.
This work proposes a data-driven predictive control framework for nonlinear systems that incorporates data column preferences according to their proximity to the current operating point through a weighted norm regularization, thereby localizing the predictor without discarding any data.
Data-driven model predictive control (MPC) using Koopman operator theory is a promising approach for constrained control of unknown nonlinear systems. While linear Koopman realizations are commonly used due to their simplicity, bilinear Koopman realizations can provide significantly higher approximation accuracy for nonlinear control systems. However, robust MPC (RMPC) formulations that account for modeling errors in bilinear Koopman realizations remain limited. This paper proposes a RMPC framework for unknown nonlinear systems with general nonlinear constraints based on data-driven bilinear Koopman realizations. A central difficulty is that finite-dimensional Koopman predictors need not preserve the manifold of valid lifted states, so multi-step prediction in lifted coordinates may leave the region where one-step error certificates apply. We address this issue by reprojecting each predicted lifted state back onto the manifold, thereby obtaining an error-aware discrete-time control-affine predictor in the original state space without impractical assumptions. For this predictor, we develop a discrete-time robust control contraction metric based homothetic tube construction, and then formulate a tube-based RMPC problem with terminal ingredients. Under the proposed formulation, we prove robust satisfaction of the original nonlinear constraints by the true closed-loop trajectory, recursive feasibility, and convergence to a neighborhood of the target state. Numerical experiments demonstrate robust stabilization of nonlinear systems and the advantages of the proposed method over existing Koopman-based RMPC approaches in terms of performance.
Learning continuous-time representations of dynamical systems from observation data has emerged as a cornerstone of data-driven control and scientific machine learning. However, existing neural differential equations either treat external control inputs heuristically without providing strict structural guarantees, or enforce stability properties under the restrictive assumption of constant or vanishing inputs. This paper proposes the Input-Contraction Neural Differential Model (ICNDM), a novel deep learning framework that seamlessly incorporates time-varying control inputs while ensuring incremental exponential convergence via input-dependent contraction regularization. By leveraging an embedded input encoder and a parameterized metric network, the proposed architecture learns both the non-autonomous neural vector fields and a generalized Riemannian contraction metric simultaneously. We derive sufficient conditions for input-dependent contraction and formally establish an input-to-state contraction property under bounded external excitations. Extensive numerical evaluations on highly nonlinear chaotic oscillators and experimental data from a Permanent Magnet Synchronous Motor (PMSM) drive system demonstrate that ICNDM yields substantial reductions in long-horizon rollout errors and exhibits superior structural robustness against input perturbations compared with state-of-the-art neural differential benchmarks.