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.

翻译：库普曼算子理论已成为一种领先的数据驱动方法，其依赖于精心选择的观测函数，在提升的观测空间中实现非线性系统的全局线性表示。然而，现实世界的数据通常包含噪声，这使得难以获得准确且无偏的库普曼算子近似。从噪声数据集生成的库普曼算子通常会受到噪声诱导偏差的污染，严重降低预测及下游跟踪性能。为解决这一缺陷，本文提出了一种新颖的基于自编码器的神经架构，以从噪声数据中联合学习合适的提升函数和低偏差库普曼算子。该架构首先学习对于系统前向与后向时间动力学均一致的库普曼基函数。随后，通过利用已学习的前向与后向时间动力学，综合生成具有降低偏差的库普曼算子，使得该方法相较于现有技术对噪声更具鲁棒性。理论分析被用来证明在存在训练噪声的情况下显著的偏差降低。针对多个串联机械臂进行了动力学预测与跟踪控制仿真，包括与领先替代设计的性能比较，以展示其在各种噪声水平下的鲁棒性。进一步使用Franka FR3 7自由度机械臂进行实验研究，以证明所提方法在实际场景中的有效性。