Modeling of Non-linear Dynamics of Lithium-ion Batteries via Delay-Embedded Dynamic Mode Decomposition

1. Problem Statement

Lithium-ion batteries exhibit complex, non-linear electrochemical dynamics that are difficult to model accurately for effective Battery Management Systems (BMS). Existing modeling approaches face specific limitations:

• Physics-based models: Require detailed knowledge of internal material composition and complex partial differential equations, making them computationally expensive.
• Equivalent Circuit Models (ECMs): While widely used, they rely on simplified circuit topologies. Their accuracy degrades with battery aging and requires careful parameterization, which is sensitive to noise and degradation.
• Data-driven models (e.g., Machine Learning): Often lack interpretability (acting as "black boxes") and may require derived features like State of Charge (SoC) or capacity, which are not directly measurable and must be estimated.
• Sparse Identification (SINDy/NARX): These methods may struggle with high-dimensional data, require time derivatives (sensitive to noise), and do not readily provide the state-transition ($A$) and input ($B$) matrices required for standard state-space control and filtering (e.g., Kalman filters).

The core challenge is to develop a data-driven, computationally efficient, and interpretable model that captures non-linear battery dynamics using only raw voltage and current data, without relying on internal state estimates or complex physical parameters.

2. Methodology

The authors propose a framework using Dynamic Mode Decomposition (DMD) and Dynamic Mode Decomposition with Control (DMDc), enhanced by time-delay embedding via Hankel matrices.

A. Data Acquisition

• Subject: A 30 Ah Lithium-ion battery.
• Test Protocol: Hybrid Pulse Power Characterization (HPPC) tests were conducted.
  • The protocol involved constant current (CC) and constant voltage (CV) charging, followed by rest periods and pulse discharge/charge cycles.
  • Tests were performed on a fresh (healthy) battery and repeated after every 20 cycles up to 360 cycles to induce and characterize degradation.
• Inputs/Outputs: Only raw terminal voltage (output) and current (input) data were utilized. No internal states (SoC, SoH) were used as model inputs.

B. Mathematical Formulation

1. Hankel Matrix Construction (Time-Delay Embedding):

   • Since the true internal state is unobserved, the authors constructed a block Hankel matrix from the voltage time series.
   • This transforms a 1D signal into a higher-dimensional space ($m$-dimensional), where columns stack $m$ consecutive delayed samples. This "lifts" the temporal structure, allowing linear operators to approximate non-linear dynamics.
   • Embedding Dimension ($m$): Ranged from 40 to 2000. The optimal dimension was found to be 1810.

2. Dynamic Mode Decomposition (DMD):

   • Assumes a linear relationship: $x_{k+1} = A x_k$.
   • The operator $A$ is estimated via Least Squares (using SVD) to advance the snapshot matrix $X$ to $X'$.
   • Limitation: Does not account for external control inputs (current).

3. Dynamic Mode Decomposition with Control (DMDc):

   • Extends DMD to include control inputs: $x_{k+1} = A x_k + B u_k$.
   • Here, $u_k$ represents the current.
   • Input Embedding: The current input was also delay-embedded (Hankel matrix of input) to capture temporal dependencies in the excitation signal. The optimal input embedding dimension was found to be 6.
   • The algorithm solves for matrices $A$ (state transition) and $B$ (input map) simultaneously.

C. Training and Validation Strategy

• Training: 60% of the data from a single HPPC test (healthy state) was used to identify matrices $A$ and $B$.
• Validation: The remaining 40% of the data was used for prediction.
• Degradation Testing: The matrices identified from the healthy battery were held fixed. The model was then tested on data from aged batteries (up to cycle 360) by only updating the control input sequence ($u_k$), without re-identifying the system matrices.

3. Key Contributions

1. Delay-Embedded Linearization: Demonstrated that non-linear battery dynamics can be effectively captured within a linear state-space framework by using time-delay embedding (Hankel matrices) on raw voltage data.
2. Input-Aware Modeling: Showed that incorporating current as a control input (DMDc) significantly outperforms standard DMD, especially for capturing transient responses and voltage dips.
3. Robustness to Aging: Proved that system matrices ($A$ and $B$) identified from a healthy battery can predict the behavior of aged batteries with reasonable accuracy, provided the input (current) is updated. This suggests the underlying linear operator structure remains relatively stable despite non-linear degradation.
4. Interpretability without Internal States: The method provides interpretable $A$ and $B$ matrices (suitable for control theory) without requiring knowledge of SoC, capacity, or internal electrochemical parameters.

4. Results

• Optimal Parameters:
  • Voltage Embedding: Dimension 1810 yielded the lowest error.
  • Current Embedding: Dimension 6 yielded the lowest error.
• Prediction Accuracy (Healthy State):
  • Standard DMD: Residual Sum of Squares (RSS) = 30.
  • DMDc: RSS = 3.86 (with input embedding) and as low as 1.74 with optimized input embedding.
  • DMDc significantly reduced errors in predicting voltage dips and transient responses compared to standard DMD.
• Long-Horizon Prediction:
  • Standard DMD errors grew significantly after ~6.5 hours.
  • DMDc maintained high accuracy even for far-future predictions (up to 12.5 hours) with negligible deviation.
• Degradation Adaptability:
  • When applied to aged batteries (Cycle 360) using matrices trained on healthy data:
    • DMD RSS: Increased to 143.
    • DMDc RSS: Increased to 76.
  • While errors increased with degradation, DMDc remained robust, successfully capturing the altered dynamics and voltage profiles of the degraded cell.

5. Significance and Implications

• Alternative to ECMs: This approach offers a middle ground between the complexity of physics-based models and the "black box" nature of deep learning. It provides a linear, interpretable state-space model that is computationally efficient.
• BMS Integration: The ability to derive $A$ and $B$ matrices directly from data makes this framework highly suitable for integration into Model Predictive Control (MPC) and Kalman Filtering for real-time battery management.
• Scalability: The method does not require re-parameterization for different battery chemistries or aging states, as it learns the dynamics directly from the data.
• Future Applications: The authors suggest this framework could serve as a foundation for adaptive battery diagnostics and prognostics, potentially extending to other energy storage technologies beyond lithium-ion.

In conclusion, the paper establishes that Delay-Embedded DMDc is a powerful tool for modeling non-linear battery dynamics, offering superior accuracy and robustness to aging compared to standard DMD, while maintaining the interpretability required for advanced control systems.