Forward-Backward Extended DMD with an Asymptotic Stability Constraint

Abstract

This paper presents a data-driven method to identify an asymptotically stable Koopman system from noisy data. In particular, the proposed approach combines approximations of the system's forward- and backward-in-time dynamics to reduce bias caused by noisy data while enforcing asymptotic stability. A Koopman model of an inherently asymptotically stable system can be unstable due to noisy data and a poor choice of lifting functions. To prevent identifying an unstable model, the proposed approach imposes an asymptotic stability constraint on the Koopman model. The proposed method is formulated as a semidefinite program and its performance is compared to state-of-the-art methods with a simulated Duffing oscillator dataset and experimental soft robot dataset.

Figures (7)

• Figure 1: Identification process overview of the Koopman system with proposed bias reduction method and constraints, including the resulting effects of the proposed method. $(a)$ First, the forward- and backward-in-time snapshot matrices are collected using noisy data from the system. $(b)$ Then, the forward- and backward-in-time Koopman matrices are approximated using asymptotic stability constraints. $(c)$ Finally, the Koopman matrix with reduced bias is computed. The proposed method identifies an asymptotically stable Koopman system representation with a reduced bias when noisy data is used.
• Figure 2: Eigenvalues of the Koopman dynamics matrix identified with the Duffing oscillator dataset. The dynamics matrix identified with fbEDMD has eigenvalues outside the unit circle and therefore produces an unstable model, while the Koopman systems identified with EDMD, EDMD-AS, and fbEDMD-AS are asymptotically stable, since their respective dynamics matrix has its largest eigenvalue within the unit circle.
• Figure 3: Multi-step trajectory $(a)$ and prediction error plot $(b)$ of Koopman models approximated with the simulated Duffing oscillator dataset. States are recovered and re-lifted between prediction timesteps. Bias in the eigenvalues of dynamics matrices identified with EDMD and EDMD-AS leads to overestimated decay rates. The trajectories obtained with the Koopman systems identified with fbEDMD and fbEDMD-AS track better the true trajectory, since most of the bias in the models is removed.
• Figure 4: The relative Frobenius norm error of the approximated Koopman, dynamics, and input matrices at different signal-to-noise ratios (SNRs). A Koopman model approximated with FBEDMD methods is more accurate than with EDMD methods under 35 dB. Although FBEDMD-AS uses additional constraints compared to FBEDMD, it approximates a Koopman model with a similar accuracy to FBEDMD.
• Figure 5: Eigenvalues of the Koopman dynamics matrix identified with the soft robot dataset at a SNR of 28 dB. Both dynamics matrices identified with EDMD and fbEDMD have eigenvalues outside the unit circle and therefore produce unstable models. Koopman systems identified with EDMD-AS and fbEDMD-AS are asymptotically stable, since their respective dynamics matrix has its largest eigenvalues bounded within the unit circle.