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.
A RMPC framework for unknown nonlinear systems with general nonlinear constraints based on data-driven bilinear Koopman realizations is proposed and robust satisfaction of the original nonlinear constraints is proved by the true closed-loop trajectory, recursive feasibility, and convergence to a neighborhood of the target state.
This paper derives sufficient conditions for input-dependent contraction and formally establish an input-to-state contraction property under bounded external excitations, and develops a novel deep learning framework that seamlessly incorporates time-varying control inputs while ensuring incremental exponential convergence via input-dependent contraction regularization.