Koopman Spectral Analysis from Noisy Measurements based on Bayesian Learning and Kalman Smoothing

Koopman spectral analysis plays a crucial role in understanding and modeling nonlinear dynamical systems as it reveals key system behaviors and long-term dynamics. However, the presence of measurement noise poses a significant challenge to accurately extracting spectral properties. In this work, we propose a robust method for identifying the Koopman operator and extracting its spectral characteristics in noisy environments. To address the impact of noise, our approach tackles an identification problem that accounts for both systematic errors from finite-dimensional approximations and measurement noise in the data. By incorporating Bayesian learning and Kalman smoothing, the method simultaneously identifies the Koopman operator and estimates system states, effectively decoupling these two error sources. The method's efficiency and robustness are demonstrated through extensive experiments, showcasing its accuracy across varying noise levels.