Learning Noise-Robust Stable Koopman Operator for Control with Hankel DMD

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.