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Learning Koopman Models From Data Under General Noise Conditions

Lucian Cristian Iacob, Máté Szécsi, Gerben Izaak Beintema, Maarten Schoukens, Roland Tóth

TL;DR

The paper tackles identifying Koopman models for nonlinear systems with inputs under general noise by combining a deep state-space encoder with a multiple-shooting loss. It derives an exact finite-dimensional Koopman representation that includes an innovation noise term and demonstrates how to estimate lifted states from input-output data using reconstructability concepts. The method offers convergence and consistency guarantees and shows strong performance across Wiener-Hammerstein, Bouc-Wen hysteresis, and a Crazyflie quadrotor, including real-world experiments. Overall, it provides a scalable, theoretically grounded framework for data-driven, noise-robust Koopman identification with partial observations and input channels.

Abstract

This paper presents a novel identification approach of Koopman models of nonlinear systems with inputs under rather general noise conditions. The method uses deep state-space encoders based on the concept of state reconstructability and an efficient multiple-shooting formulation of the squared loss of the prediction error to estimate the dynamics and the lifted state from input-output data. Furthermore, the Koopman model structure includes an innovation noise term that is used to handle process and measurement noise. It is shown that the proposed approach is statistically consistent and computationally efficient due to the multiple-shooting formulation where, on subsections of the data, multi-step prediction errors can be calculated in parallel. The latter allows for efficient batch optimization of the network parameters and, at the same time, excellent long-term prediction capabilities of the obtained models. The performance of the approach is illustrated by nonlinear benchmark examples and an experimental quadcopter setup.

Learning Koopman Models From Data Under General Noise Conditions

TL;DR

The paper tackles identifying Koopman models for nonlinear systems with inputs under general noise by combining a deep state-space encoder with a multiple-shooting loss. It derives an exact finite-dimensional Koopman representation that includes an innovation noise term and demonstrates how to estimate lifted states from input-output data using reconstructability concepts. The method offers convergence and consistency guarantees and shows strong performance across Wiener-Hammerstein, Bouc-Wen hysteresis, and a Crazyflie quadrotor, including real-world experiments. Overall, it provides a scalable, theoretically grounded framework for data-driven, noise-robust Koopman identification with partial observations and input channels.

Abstract

This paper presents a novel identification approach of Koopman models of nonlinear systems with inputs under rather general noise conditions. The method uses deep state-space encoders based on the concept of state reconstructability and an efficient multiple-shooting formulation of the squared loss of the prediction error to estimate the dynamics and the lifted state from input-output data. Furthermore, the Koopman model structure includes an innovation noise term that is used to handle process and measurement noise. It is shown that the proposed approach is statistically consistent and computationally efficient due to the multiple-shooting formulation where, on subsections of the data, multi-step prediction errors can be calculated in parallel. The latter allows for efficient batch optimization of the network parameters and, at the same time, excellent long-term prediction capabilities of the obtained models. The performance of the approach is illustrated by nonlinear benchmark examples and an experimental quadcopter setup.

Paper Structure

This paper contains 22 sections, 3 theorems, 63 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koopman embedding of nonlinear systems
  4. Linear representations subject to nonlinear constraints
  5. Observability and reconstructability
  6. Identification approach
  7. Data generating system
  8. Koopman model structure
  9. Identification problem
  10. Subspace encoder
  11. Model estimation via multiple shooting and subspace encoding
  12. Consistency analysis
  13. Convergence
  14. Consistency
  15. Experiments and results
  16. ...and 7 more sections

Key Result

Theorem 3.1

Under Assumptions assumption:exact_embedding_aut and assumption:exact_embedding_output_aut, the nonlinear system eq:data_gen can be written into the form: which is an exact finite dimensional Koopman form with innovation noise structure and $z_k=\Phi(x_k)$, with $z_k\in\mathbb{R}^{n_\mathrm{z}}$ being the lifted state and $n_\mathrm{z}=n_\mathrm{f}$.

Figures (11)

  • Figure 1: Compliant surface corresponding to $\Psi$ (in blue), compliant trajectories of the lifted system (in green), non-compliant trajectories (in red) of the lifted system and trajectories of the original nonlinear system (in black).
  • Figure 1: Network architecture. The lifted state at moment $k$, i.e., $\hat{z}_{k|k}$, is estimated using the encoder function $\bar{\Pi}^\eta_{\mathrm{z},n}$ based on previously measured input and output data.
  • Figure 1: Wiener-Hammerstein system
  • Figure 2: NRMS of the simulation responses of the process part of the Koopman models w.r.t. a noiseless test data set, when the Koopman models are estimated with noisy data under various SNR levels (Wiener-Hammerstein system).
  • Figure 3: NRMS of the one-step-ahead prediction by the Koopman models w.r.t. noisy test data sets with different SNRs, when the Koopman models are also estimated with noisy data under these SNR levels (Wiener-Hammerstein system).
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • Proof 1
  • Theorem 4.1
  • Proof 2
  • Theorem 4.2
  • Proof 3