Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories

• URL: http://arxiv.org/abs/2209.09977v3
• Date: Tue, 24 Oct 2023 21:36:38 GMT
• Title: Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories
• Authors: Samuel E. Otto, Sebastian Peitz, Clarence W. Rowley
• Abstract summary: We write the dynamics of observables governed by the Koopman generator as a bilinear hidden Markov model. We demonstrate the performance of this method on three examples.
• Score: 1.534667887016089
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: 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, then it becomes necessary to account for the dependence of the output time series on the sequence of supplied inputs when constructing observables to approximate Koopman operators. To address 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.

Related papers

• Uncertainty Modelling and Robust Observer Synthesis using the Koopman Operator [5.317624228510749]
  The Koopman operator allows nonlinear systems to be rewritten as infinite-dimensional linear systems. A finite-dimensional approximation of the Koopman operator can be identified directly from data. A population of several dozen motor drives is used to experimentally demonstrate the proposed method.
  arXiv  Detail & Related papers  (2024-10-01T20:31:18Z)
• Latent Space Energy-based Neural ODEs [73.01344439786524]
  This paper introduces a novel family of deep dynamical models designed to represent continuous-time sequence data. We train the model using maximum likelihood estimation with Markov chain Monte Carlo. Experiments on oscillating systems, videos and real-world state sequences (MuJoCo) illustrate that ODEs with the learnable energy-based prior outperform existing counterparts.
  arXiv  Detail & Related papers  (2024-09-05T18:14:22Z)
• Koopman-Based Surrogate Modelling of Turbulent Rayleigh-Bénard Convection [4.248022697109535]
  We use a Koopman-inspired architecture called the Linear Recurrent Autoencoder Network (LRAN) for learning reduced-order dynamics in convection flows. A traditional fluid dynamics method, the Kernel Dynamic Mode Decomposition (KDMD) is used to compare the LRAN. We obtained more accurate predictions with the LRAN than with KDMD in the most turbulent setting.
  arXiv  Detail & Related papers  (2024-05-10T12:15:02Z)
• Multistep Inverse Is Not All You Need [87.62730694973696]
  In real-world control settings, the observation space is often unnecessarily high-dimensional and subject to time-correlated noise. It is therefore desirable to learn an encoder to map the observation space to a simpler space of control-relevant variables. We propose a new algorithm, ACDF, which combines multistep-inverse prediction with a latent forward model.
  arXiv  Detail & Related papers  (2024-03-18T16:36:01Z)
• Koopman-Assisted Reinforcement Learning [8.812992091278668]
  The Bellman equation and its continuous form, the Hamilton-Jacobi-Bellman (HJB) equation, are ubiquitous in reinforcement learning (RL) and control theory. This paper explores the connection between the data-driven Koopman operator and Decision Processes (MDPs) We develop two new RL algorithms to address these limitations.
  arXiv  Detail & Related papers  (2024-03-04T18:19:48Z)
• Data-driven Nonlinear Model Reduction using Koopman Theory: Integrated Control Form and NMPC Case Study [56.283944756315066]
  We propose generic model structures combining delay-coordinate encoding of measurements and full-state decoding to integrate reduced Koopman modeling and state estimation. A case study demonstrates that our approach provides accurate control models and enables real-time capable nonlinear model predictive control of a high-purity cryogenic distillation column.
  arXiv  Detail & Related papers  (2024-01-09T11:54:54Z)
• Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems [8.259767785187805]
  Dynamic Mode Decomposition (DMD) is the poster child of projection-based methods. We introduce the concept of variance-pseudospectra to gauge statistical coherency. Our study concludes with practical applications using both simulated and experimental data.
  arXiv  Detail & Related papers  (2023-08-21T13:05:12Z)
• Stochastic Trajectory Prediction via Motion Indeterminacy Diffusion [88.45326906116165]
  We present a new framework to formulate the trajectory prediction task as a reverse process of motion indeterminacy diffusion (MID) We encode the history behavior information and the social interactions as a state embedding and devise a Transformer-based diffusion model to capture the temporal dependencies of trajectories. Experiments on the human trajectory prediction benchmarks including the Stanford Drone and ETH/UCY datasets demonstrate the superiority of our method.
  arXiv  Detail & Related papers  (2022-03-25T16:59:08Z)
• Estimating Koopman operators for nonlinear dynamical systems: a nonparametric approach [77.77696851397539]
  The Koopman operator is a mathematical tool that allows for a linear description of non-linear systems. In this paper we capture their core essence as a dual version of the same framework, incorporating them into the Kernel framework. We establish a strong link between kernel methods and Koopman operators, leading to the estimation of the latter through Kernel functions.
  arXiv  Detail & Related papers  (2021-03-25T11:08:26Z)
• Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems [14.211417879279075]
  This paper presents a generalizable methodology for data-driven identification of nonlinear dynamics that bounds the model error. We construct a Koopman operator-based linear representation and utilize Taylor series accuracy analysis to derive an error bound. When combined with control, the Koopman representation of the nonlinear system has marginally better performance than competing nonlinear modeling methods.
  arXiv  Detail & Related papers  (2020-10-12T15:15:13Z)
• Stochastically forced ensemble dynamic mode decomposition for forecasting and analysis of near-periodic systems [65.44033635330604]
  We introduce a novel load forecasting method in which observed dynamics are modeled as a forced linear system. We show that its use of intrinsic linear dynamics offers a number of desirable properties in terms of interpretability and parsimony. Results are presented for a test case using load data from an electrical grid.
  arXiv  Detail & Related papers  (2020-10-08T20:25:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.