We propose a noise-robust learning framework for the Koopman operator of nonlinear dynamical systems, with guaranteed long-term stability and improved model performance for better model-based predictive control tasks. Unlike some existing approaches that rely on ad hoc observables or black-box neural networks in extended dynamic mode decomposition (EDMD), our framework leverages observables generated by the system dynamics, when the system dynamics is known, through a Hankel matrix, which shares similarities with discrete Polyflow. When system dynamics is unknown, we approximate them with a neural network while maintaining structural similarities to discrete Polyflow. To enhance noise robustness and ensure long-term stability, we developed a stable parameterization of the Koopman operator, along with a progressive learning strategy for rollout loss. To further improve the performance of the model in the phase space, a simple iterative data augmentation strategy was developed. Numerical experiments of prediction and control of classic nonlinear systems with ablation study showed the effectiveness of the proposed techniques over several state-of-the-art practices.

翻译：我们提出了一种用于非线性动力系统Koopman算子的噪声鲁棒学习框架，该框架保证了长期稳定性，并提升了模型性能，以更好地完成基于模型的预测控制任务。与现有某些在扩展动态模态分解（EDMD）中依赖启发式可观测量或黑盒神经网络的方法不同，当系统动力学已知时，我们的框架利用系统动力学通过Hankel矩阵生成可观测量，这与离散Polyflow有相似之处。当系统动力学未知时，我们使用神经网络对其进行近似，同时保持与离散Polyflow的结构相似性。为了增强噪声鲁棒性并确保长期稳定性，我们开发了Koopman算子的稳定参数化方法，以及用于滚动损失的渐进式学习策略。为了进一步提升模型在相空间中的性能，我们开发了一种简单的迭代数据增强策略。通过对经典非线性系统进行预测和控制的数值实验及消融研究，证明了所提技术相对于多种先进实践方法的有效性。