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On the Predictive Capability of Dynamic Mode Decomposition for Nonlinear Periodic Systems with Focus on Orbital Mechanics

Sriram Narayanan, Mohamed Naveed Gul Mohamed, Indranil Nayak, Suman Chakravorty, Mrinal Kumar

TL;DR

The paper investigates the predictive capability of Dynamic Mode Decomposition (DMD) for nonlinear periodic orbital dynamics, establishing a theoretical lower bound on time delays $l^* = 2M$ and leveraging Hankel-DMD time-delay embedding to recover this structure from data. It connects DMD to the autoregressive representation and the Koopman operator, providing a data-driven approach to learn satellite trajectories without explicit analytical models. Through case studies on the ISS and Molniya-3-50 under Keplerian, J2, drag, and SGP4 perturbations, the work offers practical guidance on selecting the training window $W_{TRN}$, prediction window $W_{PRED}$, and the minimal delay $l^*$, while highlighting the impact of sampling rate and spectral richness on accuracy. The findings demonstrate the potential and limitations of DMD-based orbit propagation, emphasizing the need for case-specific parameter tuning and signaling opportunities for future extensions to extended-DMD with chosen basis functions.

Abstract

This paper discusses the predictive capability of Dynamic Mode Decomposition (DMD) in the context of orbital mechanics. The focus is specifically on the Hankel variant of DMD which uses a stacked set of time-delayed observations for system identification and subsequent prediction. A theory on the minimum number of time delays required for accurate reconstruction of periodic trajectories of nonlinear systems is presented and corroborated using experimental analysis. In addition, the window size for training and prediction regions, respectively, is presented. The need for a meticulous approach while using DMD is emphasized by drawing comparisons between its performance on two candidate satellites, the ISS and MOLNIYA-3-50.

On the Predictive Capability of Dynamic Mode Decomposition for Nonlinear Periodic Systems with Focus on Orbital Mechanics

TL;DR

The paper investigates the predictive capability of Dynamic Mode Decomposition (DMD) for nonlinear periodic orbital dynamics, establishing a theoretical lower bound on time delays and leveraging Hankel-DMD time-delay embedding to recover this structure from data. It connects DMD to the autoregressive representation and the Koopman operator, providing a data-driven approach to learn satellite trajectories without explicit analytical models. Through case studies on the ISS and Molniya-3-50 under Keplerian, J2, drag, and SGP4 perturbations, the work offers practical guidance on selecting the training window , prediction window , and the minimal delay , while highlighting the impact of sampling rate and spectral richness on accuracy. The findings demonstrate the potential and limitations of DMD-based orbit propagation, emphasizing the need for case-specific parameter tuning and signaling opportunities for future extensions to extended-DMD with chosen basis functions.

Abstract

This paper discusses the predictive capability of Dynamic Mode Decomposition (DMD) in the context of orbital mechanics. The focus is specifically on the Hankel variant of DMD which uses a stacked set of time-delayed observations for system identification and subsequent prediction. A theory on the minimum number of time delays required for accurate reconstruction of periodic trajectories of nonlinear systems is presented and corroborated using experimental analysis. In addition, the window size for training and prediction regions, respectively, is presented. The need for a meticulous approach while using DMD is emphasized by drawing comparisons between its performance on two candidate satellites, the ISS and MOLNIYA-3-50.
Paper Structure (18 sections, 2 theorems, 43 equations, 10 figures, 7 tables)

This paper contains 18 sections, 2 theorems, 43 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Theoretical Background
  4. Methodology: Time - Delay Dynamic Mode Decomposition or Hankel-DMD
  5. Hankel-DMD
  6. Results
  7. Nonlinear Oscillator
  8. ISS Case Study
  9. Keplerian dynamics
  10. Non-Keplerian dynamics with J2
  11. Non-Keplerian Dynamics with Drag
  12. Simplified General Pertubations model (SGP4)
  13. Note on FFT comparison
  14. Molniya-3-50: The Case of a Highly Eccentric Orbit
  15. Effect of Sampling Rate
  16. ...and 3 more sections

Key Result

Proposition 1

Given a nonlinear dynamical system as in Eq. Eq:state_equation, a periodic trajectory from the system can always be modeled using a linear autoregressive model as shown in Eq. Eq:AR_model. The minimal number of time delays required for the AR model is twice the number of frequencies $M$ present in t

Figures (10)

  • Figure 1: Nonlinear Oscillator model.
  • Figure 2: ISS (Keplerian dynamics)
  • Figure 3: ISS (Non-Keplerian Dynamics with J2)
  • Figure 4: ISS (Non-Keplerian dynamics with Drag)
  • Figure 5: ISS (SGP4)
  • ...and 5 more figures

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Remark 1