Identification of linear time-invariant systems with Dynamic Mode Decomposition

TL;DR

This paper analyzes the system identification capabilities of Dynamic Mode Decomposition (DMD), showing it can recover original linear dynamics under certain conditions and classifying errors when using Runge-Kutta discretizations.

Contribution

It provides theoretical insights into DMD's invariance properties and its ability to recover linear time-invariant system dynamics, including discretized and continuous cases.

Findings

DMD is invariant under linear transformations of data.

DMD can recover original dynamics for LTI systems under mild conditions.

Error analysis for DMD with Runge-Kutta discretizations, including continuous dynamics recovery.

Abstract

Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data is constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with a Runge-Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge-Kutta methods even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.