preprint
Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories
Samuel E.
Otto
author
Sebastian
Peitz
author 474270000-0002-3389-793X
Clarence W.
Rowley
author
655
department
PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing
project
Data-driven models for nonlinear dynamical systems based on approximating the
underlying Koopman operator or generator have proven to be successful tools for
forecasting, feature learning, state estimation, and control. It has become
well known that the Koopman generators for control-affine systems also have
affine dependence on the input, leading to convenient finite-dimensional
bilinear approximations of the dynamics. Yet there are still two main obstacles
that limit the scope of current approaches for approximating the Koopman
generators of systems with actuation. First, the performance of existing
methods depends heavily on the choice of basis functions over which the Koopman
generator is to be approximated; and there is currently no universal way to
choose them for systems that are not measure preserving. Secondly, if we do not
observe the full state, we may not gain access to a sufficiently rich
collection of such functions to describe the dynamics. This is because the
commonly used method of forming time-delayed observables fails when there is
actuation. To remedy these issues, we write the dynamics of observables
governed by the Koopman generator as a bilinear hidden Markov model, and
determine the model parameters using the expectation-maximization (EM)
algorithm. The E-step involves a standard Kalman filter and smoother, while the
M-step resembles control-affine dynamic mode decomposition for the generator.
We demonstrate the performance of this method on three examples, including
recovery of a finite-dimensional Koopman-invariant subspace for an actuated
system with a slow manifold; estimation of Koopman eigenfunctions for the
unforced Duffing equation; and model-predictive control of a fluidic pinball
system based only on noisy observations of lift and drag.
2022
eng
arXiv:2209.09977
2209.09977
S.E. Otto, S. Peitz, C.W. Rowley, ArXiv:2209.09977 (2022).
S. E. Otto, S. Peitz, and C. W. Rowley, “Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories,” <i>arXiv:2209.09977</i>. 2022.
Otto, Samuel E., et al. “Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories.” <i>ArXiv:2209.09977</i>, 2022.
Otto, S. E., Peitz, S., & Rowley, C. W. (2022). Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories. In <i>arXiv:2209.09977</i>.
@article{Otto_Peitz_Rowley_2022, title={Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories}, journal={arXiv:2209.09977}, author={Otto, Samuel E. and Peitz, Sebastian and Rowley, Clarence W.}, year={2022} }
Otto SE, Peitz S, Rowley CW. Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories. <i>arXiv:220909977</i>. Published online 2022.
Otto, Samuel E., Sebastian Peitz, and Clarence W. Rowley. “Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories.” <i>ArXiv:2209.09977</i>, 2022.
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