Operator-Theoretic Joint Estimation of Aging-Aware State of Charge and Control-Informed State of Health

TL;DR

The paper tackles the critical problem of jointly estimating aging-aware state of charge (SoC) and state of health (SoH) in lithium-ion batteries. It introduces a two-path operator-theoretic framework that couples a Koopman-based latent model for cycle-level capacity evolution with a Fourier Neural Operator for intra-cycle SoC dynamics, trained end-to-end with aging-informed corrections. Stability is guaranteed via spectral-radius clipping of the Koopman operator, and the framework supports zero-shot and few-shot out-of-distribution generalization across batteries, temperatures, C-rates, and chemistries. Empirical results on real-world datasets show real-time capability, superior accuracy and stability compared to baselines, and robust capacity fade tracking as a surrogate for SoH across diverse operating conditions.

Abstract

Accurate estimation of a battery's state of charge and state of health is essential for safe and reliable battery management. Existing approaches often decouple these two states, lack stability guarantees, and exhibit limited generalization across operating conditions. This study introduces a unified operator-theoretic framework for aging-aware state of charge and control-informed state of health estimation. The architecture couples a Koopman-based latent dynamics model, which enables linear forecasting of nonlinear discharge-capacity evolution under varying operational conditions, with a neural operator that maps measurable intra-cycle signals to state of charge. The predicted discharge capacity is incorporated as a static correction within the neural operator pathway, yielding an age-aware state of charge estimate. Stability is ensured through spectral-radius clipping of the Koopman operator. The overall framework is trained end-to-end and evaluated on real-world lithium-ion battery datasets, demonstrating real-time capability while maintaining stable dynamics. To handle condition shifts and unseen regimes, the method integrates both zero-shot and few-shot out-of-distribution adaptation using only a limited number of cycles. Results show accurate and stable capacity forecasts, competitive state of charge trajectories on held-out cycles, and a direct, model-consistent mechanism for tracking capacity fade as a surrogate for state of health across diverse operating conditions.

Figures (8)

• Figure 1: The proposed operator-theoretic joint estimation framework, composed of two coupled pathways: a Koopman-based latent dynamics model for aging-aware capacity forecasting (top) and an FNO-based SoC estimator (bottom)
• Figure 2: Flow chart of the training process of the methodology
• Figure 3: Eigenvalue spectra of the learned Koopman operators for various cycle sampling times ($N_c$) and test set sizes ($ts$). Blue dots denote the eigenvalues in the complex plane, and the red dashed circle is the unit circle ($|\Lambda|=1$).
• Figure 4: Plot of one cycle of SoC models (True trajectory (black solid line) versus predictions from the FFN, RNN, BiLSTM, CNN, coupled FNO, and decoupled FNO) across test sizes and cycle counts ($N_c$). A grey-highlighted segment in each plot denotes a zoomed-in region for detailed comparison of the predictions.
• Figure 5: Plot of $Q^{\max}$ (True trajectory (black solid line) versus predictions from the RNN, BiLSTM, TCN, CNN, coupled Koopman, and decoupled Koopman) across test sizes and cycle counts ($N_c$). A grey-highlighted segment in each plot denotes a zoomed-in region for detailed comparison of the predictions.