Koopman Spectral Analysis of Lithium-Ion Battery Dynamics: State of Charge as a Marginally Stable Observable

Accurate state-of-charge (SOC) estimation in lithium-ion batteries remains a challenge due to its strongly nonlinear electrochemical dynamics, operating-condition-dependent parameters, and aging-induced model drift. Conventional equivalent circuit model (ECM) and Kalman-filter-based estimators require repeated parameter identification, while purely data-driven methods sacrifice physical interpretability. We propose a Koopman operator theoretic SOC estimation framework that leverages dynamic mode decomposition with control (DMDc) and Hankel time-delay embedding to identify a linear representation of battery dynamics from input-output measurements. Terminal voltage and current measurements from hybrid pulse power characterization (HPPC) tests are lifted into a high-dimensional observable space via Hankel embedding, enabling linear approximation of the underlying nonlinear dynamics. DMDc identifies the state-transition operator directly from data, and eigen-decomposition of this operator reveals the intrinsic Koopman spectral structure of the battery. The SOC dynamics emerge as the slowest marginally stable mode, with eigenvalue nearest to the unit circle, consistent with the integrator-type pole implied by charge conservation. The corresponding modal coordinate provides a physically grounded SOC-sensitive observable, extracted without any explicit circuit parameter identification. The proposed framework reconstructs terminal voltage with an RMSE of 0.0131 V and estimates SOC with an RMSE of 0.0043%, outperforming both Coulomb counting and the extended Kalman filter.