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Temporally-Consistent Bilinearly Recurrent Autoencoders for Control Systems

Ananda Chakrabarti, Indranil Nayak, Debdipta Goswami

TL;DR

tcBLRAN advances data-driven modeling of control-affine nonlinear systems by enforcing temporal consistency in a bilinearly recurrent Koopman latent space. By requiring the latent subspace to be invariant under drift and control operators, and by regularizing predictions across multiple time horizons, tcBLRAN improves long-horizon forecasting under limited and noisy data beyond traditional KAE/BLRAN approaches. The method is validated on pendulum, Van der Pol, and Duffing oscillators, showing superior accuracy in both clean and noisy settings and demonstrating robustness to measurement noise. This approach enhances the practical applicability of Koopman-based models for control tasks, with potential impact on data-driven MPC and autonomous system design.

Abstract

This paper introduces the temporally-consistent bilinearly recurrent autoencoder (tcBLRAN), a Koopman operator based neural network architecture for modeling a control-affine nonlinear control system. The proposed method extends traditional Koopman autoencoders (KAE) by incorporating bilinear recurrent dynamics that are consistent across predictions, enabling accurate long-term forecasting for control-affine systems. This overcomes the roadblock that KAEs face when encountered with limited and noisy training datasets, resulting in a lack of generalizability due to inconsistency in training data. Through a blend of deep learning and dynamical systems theory, tcBLRAN demonstrates superior performance in capturing complex behaviors and control systems dynamics, providing a superior data-driven modeling technique for control systems and outperforming the state-of-the-art Koopman bilinear form (KBF) learned by autoencoder networks.

Temporally-Consistent Bilinearly Recurrent Autoencoders for Control Systems

TL;DR

tcBLRAN advances data-driven modeling of control-affine nonlinear systems by enforcing temporal consistency in a bilinearly recurrent Koopman latent space. By requiring the latent subspace to be invariant under drift and control operators, and by regularizing predictions across multiple time horizons, tcBLRAN improves long-horizon forecasting under limited and noisy data beyond traditional KAE/BLRAN approaches. The method is validated on pendulum, Van der Pol, and Duffing oscillators, showing superior accuracy in both clean and noisy settings and demonstrating robustness to measurement noise. This approach enhances the practical applicability of Koopman-based models for control tasks, with potential impact on data-driven MPC and autonomous system design.

Abstract

This paper introduces the temporally-consistent bilinearly recurrent autoencoder (tcBLRAN), a Koopman operator based neural network architecture for modeling a control-affine nonlinear control system. The proposed method extends traditional Koopman autoencoders (KAE) by incorporating bilinear recurrent dynamics that are consistent across predictions, enabling accurate long-term forecasting for control-affine systems. This overcomes the roadblock that KAEs face when encountered with limited and noisy training datasets, resulting in a lack of generalizability due to inconsistency in training data. Through a blend of deep learning and dynamical systems theory, tcBLRAN demonstrates superior performance in capturing complex behaviors and control systems dynamics, providing a superior data-driven modeling technique for control systems and outperforming the state-of-the-art Koopman bilinear form (KBF) learned by autoencoder networks.

Paper Structure

This paper contains 11 sections, 2 theorems, 24 equations, 6 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Mathematical Preliminaries
  3. Koopman theory: an overview
  4. Koopman bilinear form: Koopman approximation of control affine system
  5. Enforcing invariance via neural-networks: Koopman autoencoders
  6. Temporally-Consistent Bilinearly Recurrent Autoencoders
  7. Numerical Examples
  8. Simple pendulum
  9. Van der Pol oscillator
  10. Forced Duffing oscillator
  11. Conclusion

Key Result

Theorem 1

Goswami2017Goswami2021 The system Eq: System is bilinearizable in a countable (possibly infinite) basis if the eigenspace of $L_{\mathbf{f}}$, i.e., the Koopman generator corresponding to the drift vector field is an invariant subspace of $L_{\mathbf{g}_i}, i=1,\ldots,m$, i.e., the Koopman generator

Figures (6)

  • Figure 1: Schematic of BLRAN (subscript $k$ denotes the time-sample at $t_0 + k\Delta t$)
  • Figure 2: Schematic of temporal consistency regularization (subscript $k$ denotes the time-sample at $t_0 + k\Delta t$)
  • Figure 3: Time-average relative prediction error for simple pendulum \ref{['Eq: Pendulum']} over 30 initial conditions and 10 seeds: (a) Trained on clean data, (b) Trained on noisy data with $20$ dB SNR.
  • Figure 4: Relative prediction error with time for simple pendulum \ref{['Eq: Pendulum']}
  • Figure 5: Time-average relative prediction error for Van der Pol oscillator \ref{['Eq: VDP']} over 30 initial conditions and 10 seeds: (a) Trained on clean data, (b) Trained on noisy data with $20$ dB SNR.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4