Eigenfunction construction for the Koopman operator

Application of subspace methods and the construction of derived bilinear models

Master thesis (2023)

Authors

M.T.P. van Laarhoven Mechanical Engineering

Contributors

M.H.G. Verhaegen Team Michel Verhaegen - Mechanical, Maritime and Materials Engineering (supervisor 1)
Faculty
Mechanical Engineering
Copyright: © 2023 Menno van Laarhoven
System Identification DMD ESPRIT Koopman Operator Eigenfunctions Data-driven Modelling Nonlinear identification
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Published Date

31-08-2023

Language

English

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Faculty

Mechanical Engineering

Abstract


We expand on the framework by Korda and Mezic (2020) to construct eigenfunctions directly from data by exploiting the eigenfunction PDE, guaranteeing closure and eliminating the need for a prior data dictionary.
The constructed models are extended to forced systems through the multi-step prediction error of a linear state-space model.

By identifying a relationship between ESPRIT and DMD applied to Hankel matrices, we simplify the original optimisation problem and significantly reducing the required model order.
A detailed numerical investigation of both autonomous dynamics and forced dynamics follows. For autonomous dynamics we report a VAF up to 90 % for a toy model and the Van der Pol oscillator, whilst the original work is unable to reconstruct the underlying dynamics on longer time scales. We were unable to reproduce accurate multi-step predictions under the influence of forcing.

We extend the constructed models by inclusion of monomial terms into the dynamics. This can be interpreted as a linear model with nonlinear output, approximated by a polynomial. Results on the Koopman generator and the inclusion of monomial terms suggest the construction of a bilinear model. A multi-step prediction is formulated, simplified and solved, expanding the predictive capabilities of the model. Whilst the inclusion of monomial terms improved the prediction of autonomous dynamics, the bilinear models failed to converge for the Duffing oscillator and Van der Pol oscillator.

We perform a further study on the constructed eigenfunctions by designing a new neural network architecture, aimed at learning Koopman eigenfunctions. The network architecture accurately recovers the autonomous dynamics of the system. The learned eigenfunctions suggest that the constructed eigenfunctions can be severely limited by the choice of the initial condition set Γ, opening the door for future research.