Deep Robust Koopman Learning from Noisy Data

TL;DR

This work targets robustly learning Koopman representations for nonlinear control-affine systems in the presence of measurement noise. It introduces the Deep Robust Koopman Network (DRKN), an autoencoder-based architecture that learns Koopman observables while jointly training forward and backward dynamics to synthesize a reduced-bias operator $\boldsymbol{K_{proposed}}=\sqrt{\boldsymbol{K_{fm}}\boldsymbol{K_{bm}}^{-1}}$, with theoretical guarantees under mild noise assumptions. Theoretical analysis shows how forward/backward consistency mitigates noise-induced bias, and extensive simulations (Van der Pol, 4R, Franka FR3) plus real-world-like experiments validate substantial gains in prediction accuracy and closed-loop tracking under noisy data. The approach improves robustness for data-driven control in robotics, enabling reliable MPC-based planning under realistic sensing conditions, and sets the stage for online adaptation to further enhance generalizability.

Abstract

Koopman operator theory has emerged as a leading data-driven approach that relies on a judicious choice of observable functions to realize global linear representations of nonlinear systems in the lifted observable space. However, real-world data is often noisy, making it difficult to obtain an accurate and unbiased approximation of the Koopman operator. The Koopman operator generated from noisy datasets is typically corrupted by noise-induced bias that severely degrades prediction and downstream tracking performance. In order to address this drawback, this paper proposes a novel autoencoder-based neural architecture to jointly learn the appropriate lifting functions and the reduced-bias Koopman operator from noisy data. The architecture initially learns the Koopman basis functions that are consistent for both the forward and backward temporal dynamics of the system. Subsequently, by utilizing the learned forward and backward temporal dynamics, the Koopman operator is synthesized with a reduced bias making the method more robust to noise compared to existing techniques. Theoretical analysis is used to demonstrate significant bias reduction in the presence of training noise. Dynamics prediction and tracking control simulations are conducted for multiple serial manipulator arms, including performance comparisons with leading alternative designs, to demonstrate its robustness under various noise levels. Experimental studies with the Franka FR3 7-DoF manipulator arm are further used to demonstrate the effectiveness of the proposed approach in a practical setting.

Figures (7)

• Figure 1: Schematics of the proposed DRKN architecture
• Figure 2: Performance comparison for learning the dynamics of a Van der Pol oscillator a) Phase portrait generated using learned Koopman models from data with SNR 20 dB. b) Predictive performance comparison. c) Training time comparison with a dataset corrupted by 20dB. noise.
• Figure 3: Schematic of the proposed Koopman-MPC framework
• Figure 4: a) Prediction performance comparison for learning dynamics of the 4R manipulator arm. b) Closed-loop tracking error comparison with feedback corrupted by 30 dB noise.
• Figure 5: Closed-loop comparison of a) control effort, and b) per-step MPC computation time for the 4R manipulator using Koopman models trained at different noise levels with feedback measurements corrupted by 30dB noise.