Asymptotically Stable Data-Driven Koopman Operator Approximation with Inputs using Total Extended DMD

TL;DR

Bias is reduced by using a total least-squares, modified to accommodate inputs in addition to lifted inputs to enforce asymptotic stability of the approximate Koopman model, and linear matrix inequality constraints are augmented to the identification problem.

Abstract

The Koopman operator framework can be used to identify a data-driven model of a nonlinear system. Unfortunately, when the data is corrupted by noise, the identified model can be biased. Additionally, depending on the choice of lifting functions, the identified model can be unstable, even when the underlying system is asymptotically stable. This paper presents an approach to reduce the bias in an approximate Koopman model, and simultaneously ensure asymptotic stability, when using noisy data. Additionally, the proposed data-driven modeling approach is applicable to systems with inputs, such as a known forcing function or a control input. Specifically, bias is reduced by using a total least-squares, modified to accommodate inputs in addition to lifted inputs. To enforce asymptotic stability of the approximate Koopman model, linear matrix inequality constraints are augmented to the identification problem. The performance of the proposed method is then compared to the well-known extended dynamic mode decomposition method and to the newly introduced forward-backward extended dynamic mode decomposition method using a simulated Duffing oscillator dataset and experimental soft robot arm dataset.

Figures (6)

• Figure 1: Eigenvalues of the identified Koopman matrices using the simulated Duffing oscillator dataset at an SNR of 28 dB. All the methods with asymptotic stability constraints, EDMD-AS, fbEDMD-AS, and TEDMD-AS, have their largest eigenvalue bounded strictly within the unit circle, which means that the identified Koopman systems are asymptotically stable. Forward-backward EDMD and TEDMD produce unstable systems with some eigenvalues of the Koopman matrices outside the unit circle. Although EDMD does not enforce asymptotic stability, it still identified a stable Koopman system.
• Figure 2: Prediction error plots and multi-step trajectories of the first test episode for all three asymptotically stable Koopman systems identified with the simulated Duffing oscillator dataset at an SNR of (i) 28 dB and (ii) 18 dB. At each step of the prediction, the states are recovered and re-lifted. The prediction starts at $t_\mathrm{i} = 0 \;\mathrm{s}$ and finishes at $t_\mathrm{f} = 21 \; \mathrm{s}$.
• Figure 3: Eigenvalues of the identified Koopman matrices using the experimental soft robot arm dataset at an SNR of 28 dB. All the methods with asymptotic stability constraints, EDMD-AS, fbEDMD-AS, and TEDMD-AS, have their largest eigenvalue bounded strictly within the unit circle, which means that the identified Koopman systems are asymptotically stable. Extended DMD, fbEDMD, and TEDMD produce unstable systems with some eigenvalues of the Koopman matrices outside the unit circle.
• Figure 4: Prediction error plots and multi-step trajectories of the second test episode for all three asymptotically stable Koopman systems identified with the experimental soft robot arm dataset at an SNR of (i) 28 dB and (ii) 18 dB. At each step of the prediction, the states are recovered and re-lifted. The prediction starts at $t_\mathrm{i} = 0 \;\mathrm{s}$ and finishes at $t_\mathrm{f} = 24.5 \; \mathrm{s}$.
• Figure 5: The root-mean-square (RMS) and mean multi-step prediction errors for Koopman matrices identified using the four test episodes given in the experimental soft robot arm dataset at an SNR of 28 dB (i) and 18 dB (ii).