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

TL;DR

This work tackles the nonlinear dynamics of lithium-ion batteries by applying Dynamic Mode Decomposition (DMD) and Dynamic Mode Decomposition with Control (DMDc) to voltage time-series data from HPPC tests. By using Hankel (time-delay) embeddings, the authors construct high-dimensional linear representations that capture nonlinear behavior while remaining interpretable, with DMDc explicitly incorporating current as a control input. Key findings show that DMDc with delay-embedded inputs achieves substantially lower residual errors than plain DMD (RSS as low as $1.74$) and remains robust when predicting degraded battery states, even when the system is aged. The results suggest that DMD-based, voltage-only models can form the foundation of adaptive, transparent battery diagnostics and prognostics within BMS frameworks, capable of tracking evolving nonlinear dynamics as SoC and health change.

Abstract

The complex electrochemical behavior of lithium-ion batteries results in non-linear dynamics and appropriate modeling of this non-linear dynamical system is of interest for better management and control. In this work, we proposed a family of dynamic mode decomposition (DMD)-based data-driven models that do not require detailed knowledge of the composition of the battery materials but can essentially capture the non-linear dynamics with higher computational efficiency. Only voltage and current data obtained from hybrid pulse power characterization (HPPC) tests were utilized to form the state space matrices and subsequently used for predicting the future terminal voltage at different state of charge (SoC) and aging levels. To construct the system model, 60\% of the data from a single HPPC test was utilized to generate time-delay embedded snapshots, with embedding dimension ranging from 40 to 2000. Among these, an embedding dimension of 1810 resulted in the least residual sum of squares (RSS) error of 3.86 for the dynamic mode decomposition with control (DMDc) model and 30 for the standard DMD model. For DMDc model, delay embeddings (ranging from 1 to 12) were also incorporated into the input current signals. For the input matrix, an embedding dimension of 6 resulted in a minimum RSS error of 1.74. Furthermore, the system matrices A and B, identified from the HPPC test when the cell is in its healthy state, were held fixed and used to simulate the system dynamics for aged batteries by updating only the control input. Despite the presence of nonlinear degradation effects in later cycles, the DMDc model effectively captured key inner dynamics such as voltage dips and transient responses for subsequent charge and discharge cycles.

Figures (5)

• Figure 1: Schematic representation of the experimental setup and modeling framework
• Figure 2: Comparison of RSS for different delay embeddings (a) Voltage(output) (b) current(input) in DMD and DMDc models.
• Figure 3: Comparison of discharge profiles with actual measurements, DMD and DMDc models: (a) shows the complete voltage response curve with training and forecast part, (b) prediction from 1 hour to 2.5 hours; (c) prediction from 5 hours to 6.5 hours; (d) prediction from 10 hours to 12.5 hours.
• Figure 4: Comparison of experimental measurements with DMDc model predictions, where the model was trained using data from the 20th cycle: (a) prediction for the 80th cycle, and (b) prediction for the 340th cycle.
• Figure 5: Residual sum of squares (RSS) values at different degradation levels of the battery obtained from the identified DMD and DMDc model.