Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition
Abstract
:1. Introduction
- What is the impact of transformations of the data on the resulting DMD approximation?
- Assume that the data used to generate the DMD approximation are obtained from a linear differential equation. Can we estimate the error between the continuous dynamics and the DMD approximation?
- Are there situations in which we are even able to recover the original dynamical system from its DMD approximation?
- We show in Theorem 1 that DMD is invariant in the image of the data under linear transformations of the data.
- Theorem 2 details that DMD is able to identify discrete-time dynamics, i.e., for every initial value in the image of the data, the DMD approximation exactly recovers the discrete-time dynamics.
- In Theorem 3, we show that if the DMD approximation is constructed from data that are obtained via a RKM, then the approximation error of DMD with respect to the ordinary differential equation is in the order of the error of the RKM. If a one-stage RKM is used and the data are sufficiently rich, then the continuous-time dynamics, i.e., the matrix F in Figure 1, can be recovered cf. Lemma 1.
Notation
2. Preliminaries
2.1. Runge–Kutta Methods
2.2. Dynamic Mode Decomposition
3. System Identification and Error Analysis
3.1. Data Scaling and Invariance of the DMD Approximation
3.2. Discrete-Time Dynamics
3.3. Continuous-Time Dynamics and RK Approximation
4. Numerical Examples
- If we first transform the data with the matrix
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
DMD | dynamic mode decomposition |
IVP | initial value problem |
ODE | ordinary differential equation |
RKM | Runge–Kutta method |
SVD | singular value decomposition |
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Method | Lemma 1 | |
---|---|---|
explicit Euler | ||
implicit Euler | ||
implicit midpoint rule |
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Heiland, J.; Unger, B. Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition. Mathematics 2022, 10, 418. https://doi.org/10.3390/math10030418
Heiland J, Unger B. Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition. Mathematics. 2022; 10(3):418. https://doi.org/10.3390/math10030418
Chicago/Turabian StyleHeiland, Jan, and Benjamin Unger. 2022. "Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition" Mathematics 10, no. 3: 418. https://doi.org/10.3390/math10030418
APA StyleHeiland, J., & Unger, B. (2022). Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition. Mathematics, 10(3), 418. https://doi.org/10.3390/math10030418