Spatially Aware Dictionary-Free Eigenfunction Identification for Modeling and Control of Nonlinear Dynamical Systems

TL;DR

The paper introduces a dictionary-free, spatially aware method (SADFED) to identify Koopman eigenfunctions from data by leveraging a reference trajectory, transforming the temporal basis, and projecting multiple trajectories to obtain eigenfunction initial values Φ0 via ridge regression. Eigenvalues are globally optimized (PSO/Nelder-Mead) while enforcing the Koopman PDE through a gradient-based KPDE cost, yielding robust, low-dimensional predictors suitable for control design. The approach demonstrates improved accuracy over traditional dictionary-based methods, reveals state-space structure such as isostables and isochrons, and provides a mechanism to estimate input dynamics gradients for LPV control, as illustrated on systems including a 2-spool turbojet engine. Limitations include the need for relatively dense sampling and substantial computation, particularly for higher-dimensional systems, motivating future work on scaling and automatic reference trajectory selection.

Abstract

A new approach to data-driven discovery of Koopman eigenfunctions without a pre-defined set of basis functions is proposed. The approach is based on a reference trajectory, for which the Koopman mode amplitudes are first identified, and the Koopman mode decomposition is transformed to a new basis, which contains fundamental functions of eigenvalues and time. The initial values of the eigenfunctions are obtained by projecting trajectories onto this basis via a regularized least-squares fit. A global optimizer was employed to optimize the eigenvalues. Mapping initial-state values to eigenfunction values reveals their spatial structure, enabling the numerical computation of their gradients. Thus, deviations from the Koopman partial differential equation are penalized, leading to more robust solutions. The approach was successfully tested on several benchmark nonlinear dynamical systems, including the FitzHugh-Nagumo system with inputs, van der Pol and Duffing oscillators, and a 2-spool turbojet engine with control. The study demonstrates that incorporating principal eigenvalues and spatial structure integrity promotion significantly improves the accuracy of Koopman predictors. The approach effectively discovers Koopman spectral components even with sparse state-space sampling and reveals geometric features of the state space, such as invariant partitions. Finally, the numerical approximation of the eigenfunction gradient can be used for input dynamics modeling and control design. The results support the practicality of the approach for use with various dynamical systems.

Figures (21)

• Figure 1: Schematic diagram of Koopman eigenfunction lifting and evolution.
• Figure 2: Flowchart of the eigenfunction identification process with the objective function.
• Figure 3: Flowchart of the entire framework, including the input dynamics fitting and possible control design. The example systems considered in this paper are shown as well for each case of the a priori dynamics knowledge.
• Figure 4: Sparse selection of trajectories from the full $21 \times 21$ (blue) grid to $6 \times 6$ (black) grid using every second odd rows and columns.
• Figure 5: Fitted trajectories in the state space - ground truth (green solid) vs. fitted (black dashed). The red line highlights the reference trajectory. Only samples corresponding to odd rows and columns are plotted for clarity.