Extended Dynamic Mode Decomposition with Learned Koopman Eigenfunctions for Prediction and Control

Abstract

This paper presents a novel learning framework to construct Koopman eigenfunctions for unknown, nonlinear dynamics using data gathered from experiments. The learning framework can extract spectral information from the full nonlinear dynamics by learning the eigenvalues and eigenfunctions of the associated Koopman operator. We then exploit the learned Koopman eigenfunctions to learn a lifted linear statespace model. To the best of our knowledge, our method is the first to utilize Koopman eigenfunctions as lifting functions for EDMD-based methods. We demonstrate the performance of the framework in state prediction and closed loop trajectory tracking of a simulated cart pole system. Our method is able to significantly improve the controller performance while relying on linear control methods to do nonlinear control.

Geometric considerations of a good dictionary for Koopman analysis of dynamical systems: Cardinality, “primary eigenfunction,” and efficient representation

“…Thus follows a new kind of empirical scheme to produce efficient functions, that we call a good dictionary. In the spirit of naming algorithms for empirical spectral analysis of the Koopman operator as DMD-like methods, here we call ours oKEEDMD, for "optimal KEEDMD" where EDMD means empirical dynamic mode decomposition for the idea of using basis functions and KE is the new prefix from [8] for Koopman eigenfunction, meaning those basis functions are chosen to be themselves eigenfunctions.…”

“…In [8], the authors have developed a variant of EDMD where the observation functions are themselves Koopman eigenfunctions, that they called Koopman Eigenfunction Extended Dynamic Mode Decomposition (KEEDMD). In broad sense this is what we do here, but with the extra technology so as to make for an optimally efficient representation.…”

Extracting Latent State Representations with Linear Dynamics from Rich Observations

“…Koopman Operator Theory Many recent works have approached the problem of linearizing the dynamics by finding a lifting based on the Koopman operator (an infinite-dimensional linear operator representation). Williams et al [2015] propose the extended dynamic mode decomposition algorithm to approximate the leading eigenfunctions of the Koopman operator, and variants of this idea include the use of kernels [Kawahara, 2016], dictionary learning [Li et al, 2017], and neural networks [Lusch et al, 2018, Yeung et al, 2019, as well as extensions to control problems [Folkestad et al, 2019]. The use of the Koopman operator is not appropriate Figure 1: A graphical illustration of our hidden subspace model.…”

Learning Stable Models for Prediction and Control

“…However, unless finite-dimensional Koopman invariant subspaces exist [28], [30], [31], the operator is infinitedimensional and renders practical use challenging. Studies seek finite-dimensional approximations using methods such as the Dynamic Mode Decomposition (DMD) [32] extended DMD (EDMD) [33], [34], Hankel-DMD [35], or closedform solutions [36], [37] which use state measurements to approximate Koopman operators. Data-driven Koopman operators have already been used in many applications, such as robotics [1], [2], [38], human locomotion [39], neuroscience [40], fluid mechanics [41], and climate forecast [42].…”