Online learning of Koopman operator using streaming data from different dynamical regimes

1. Problem Statement

The paper addresses the challenge of modeling and controlling complex nonlinear dynamical systems using streaming sensor data. Key challenges identified include:

• Computational Scalability: Traditional data-driven methods like Extended Dynamic Mode Decomposition (EDMD) rely on Singular Value Decomposition (SVD) and matrix inversion, which become computationally infeasible as the dataset grows indefinitely.
• Dynamic Regime Variability: Streaming data often contains intermittent samples from different dynamical regimes or rare events. Existing methods either retain all historical data (causing memory/compute overload) or use "forgetting factors" that may discard critical data from regimes that reappear later.
• Fixed Basis Functions: Many methods require pre-defined basis functions or fixed system orders, which limits adaptability to unknown or changing system dynamics.

The goal is to develop an online learning framework that updates the Koopman operator efficiently, adapts to new dynamical regimes, minimizes the data required for updates, and learns the system order and basis functions dynamically.

2. Methodology

The proposed framework combines Recursive Subspace System Identification (R-SSID) with Grassmannian distance-based data pruning and Gaussian Process (GP) regression.

A. Koopman Operator via Subspace Identification

The authors approximate the nonlinear system as a high-dimensional Linear Time-Invariant (LTI) system in a "lifted" space. Instead of pre-defining basis functions, they use subspace identification to:

1. Construct Hankel matrices from input-output data.
2. Project out input influence to isolate the extended observability matrix ($\Gamma$).
3. Use SVD to determine the system order ($r$) and the Koopman operator ($K$).

B. Recursive Update (R-SSID)

To handle streaming data without re-computing SVD on the entire dataset, the authors employ a recursive algorithm (based on Oku and Kimura, 1999).

• Instead of updating the full Hankel matrix, they update a symmetric data matrix $\Xi_N = Y_N \Pi^\perp_{U_N} Y_N^\top$, where $\Pi^\perp$ is the projection matrix orthogonal to the input.
• This allows the extended observability matrix and system matrices ($K, B, C, D$) to be updated incrementally as new data arrives.

C. Grassmannian Distance for Data Pruning

A core innovation is the use of the Grassmannian distance to filter incoming data segments:

• Concept: The Grassmannian distance measures the dissimilarity between the subspace of the current extended observability matrix ($\Gamma_i$) and the subspace of a new data segment ($\Gamma_{i+1}$).
• Logic:
  • If the distance $G < \epsilon$ (threshold): The new data is $\epsilon$-redundant (it belongs to the same dynamical regime). It is discarded to save computation.
  • If the distance $G > \epsilon$: The new data represents a novel regime. It is added to the archive, and the Koopman operator is recursively updated.
• Adaptivity: This mechanism allows the system to detect shifts in dynamics (e.g., from single-well to double-well oscillations) and update the model accordingly.

D. Adaptive Basis Learning

The paper utilizes Gaussian Process (GP) regression to map the original state space measurements to the lifted state space. This allows the basis functions to be learned adaptively from the data rather than being fixed a priori.

3. Key Contributions

1. Novel Online Framework: A method for online Koopman operator learning that avoids storing massive historical datasets by filtering redundant data via Grassmannian distance.
2. Dynamic System Order Identification: The algorithm can automatically adjust the system order ($r$) based on the singular values of the updated subspace, accommodating changes in system complexity.
3. Regime Detection: The ability to distinguish between different dynamical regimes and rare events without discarding them, ensuring the model remains valid for future visits to those regimes.
4. Basis Function Learning: Integration of GP regression to learn the lifting map (basis functions) directly from the identified subspace, removing the need for manual feature engineering.

4. Simulation Results

The authors validated the method on two examples:

• Simple Nonlinear System (Koopman Embedding):

  • A system with known analytical Koopman embedding was used.
  • Result: Out of 100 streaming datasets, only 10 were accepted for updates (90% redundancy filtered). The identified system order converged to the true order (3), and the Grassmannian distance between the identified and true subspaces converged to zero.

• Duffing Oscillator with Control:

  • A complex system exhibiting single-well and double-well oscillations depending on initial conditions.
  • Scenario: The first 600 datasets were single-well; datasets 601–900 introduced double-well dynamics.
  • Result:
    • The Grassmannian distance spiked at dataset 601, triggering an update and an increase in system order (from 9 to 11).
    • Prediction Accuracy: The updated model ($K_{900}$) significantly outperformed the model trained only on the first 600 datasets ($K_{600}$).
    • Efficiency: The proposed R-SSID algorithm (using 11 learned basis functions) achieved lower Root Mean Square Error (RMSE) than K-EDMD methods using 27 and 102 fixed basis functions.

5. Significance

This work is significant for the control and modeling of complex systems in real-world applications where:

• Data is continuous and voluminous: It solves the "big data" bottleneck in system identification.
• Dynamics are non-stationary: It handles systems that switch between different operating modes or encounter rare events.
• Computational resources are limited: By pruning redundant data and learning minimal basis functions, it enables the deployment of high-fidelity data-driven models on edge devices or in real-time control loops.

The paper demonstrates that quality of data (diversity of regimes) is more critical than quantity of data for accurate Koopman modeling, provided a mechanism exists to detect and prioritize that diversity.