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

TL;DR

This paper introduces a noise-robust, input-capable Koopman operator approximation method that guarantees asymptotic stability, improving data-driven nonlinear system modeling.

Contribution

It proposes a total least-squares approach with stability constraints for Koopman models using noisy data and inputs, advancing existing methods.

Findings

Reduces bias in Koopman models with noisy data

Ensures asymptotic stability through LMI constraints

Outperforms standard EDMD and FBEDMD methods in experiments

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.