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

Jul 16, 2026

Accurate lithium-ion battery state-of-charge (SOC) estimation is hindered by strongly nonlinear electrochemical dynamics, operating-condition-dependent parameters, and aging-driven model drift. The framework proposes a Koopman-operator theoretical SOC estimator using dynamic mode decomposition with control (DMDc) and Hankel time-delay embedding to build a linear observable representation from terminal voltage and current. Koopman eigen-structure identifies SOC as the slowest marginally stable mode, enabling physically grounded SOC sensitivity without circuit parameter identification. SOC and voltage are reconstructed with RMSE 0.0043% and 0.0131 V, outperforming Coulomb counting and the extended Kalman filter.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages14

Reading Time35 min

Introduction

Motivation for SOC Estimation

Limitations of Existing Methods

Model-Based and Physics-Based Approaches

arXiv :2607 .07594v2 [ ee ss . SY] 9 Jul 2026 Koopman Spectral Analysis of Lithium-Ion Battery Dynamics: State of Charge as a Marginally Stable Observable Bakhtiar Nafis 1 , Khalid Mahmud Labib 1 , Saad Waheed 1 , and Shabbir Ahmed 1,∗ 1 Dynamical Systems and Signals Lab (DSSL), Department of Mechanical Engineering, South Dakota State University, Brookings, SD ∗ Corresponding author: [shabbir.ahmed@sdstate.edu](shabbir.ahmed@sdstate.edu) Abstract 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-filterbased 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 timedelay 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 asthe 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. Keywords: Lithium-ion batteries, State of Charge estimation, Dynamic Mode Decomposition with control, Koopman operator, Hankel embedding, Battery Management System 1 Introduction Lithium-ion batteries have become the dominant energy storage technology across a broad range of modern applications owing to their high energy density, long cycle life, low self-discharge rate, and superior power capability [1–3] . They are extensively deployed in electric vehicles (EVs), portable electronics, grid-scale renewable energy storage systems, aerospace platforms, and emerging smartgrid infrastructures [4–6] . The safe and efficient operation of these systems depends critically on advanced battery management systems (BMS), which monitor and estimate key internal states including state of charge (SOC), state of health (SOH), and state of power (SOP) . [1–3] . Among the various internal battery states, the state of charge (SOC) is one of the most fundamental and operationally significant quantities. From an electrochemical perspective, SOC reflects the normalized concentration of lithium ions stored within the active electrode materials and is therefore directly related to the degree of lithium intercalation and deintercalation. Equivalently, it represents the fraction of electrochemically available charge remaining relative to the fully charged state. Accurate SOC estimation is essential for preventing overcharge and deep discharge, maximizing battery utilization, extending service life, and enabling reliable driving-range prediction in EVs [4,6–8] . Since SOC cannot be directly measured by any sensor, it must be inferred from externally accessible signals such as terminal voltage and applied current, making it an inherently challenging state estimation problem [7,8] . Numerous SOC estimation approaches have been proposed in the literature. Coulomb counting, which integrates the measured current over time, remains one

FAQs

Why is state-of-charge (SOC) estimation difficult for lithium-ion batteries?

SOC estimation is challenging because battery electrochemical dynamics are strongly nonlinear, key parameters depend on operating conditions, and aging can cause model drift. SOC also cannot be measured directly and must be inferred from external signals like terminal voltage and current.

What is the core idea of the proposed Koopman-based SOC estimation framework?

The method uses DMDc together with Hankel time-delay embedding to lift input-output measurements into a high-dimensional observable space. DMDc then identifies a linear state-transition operator whose eigen-decomposition reveals the intrinsic Koopman spectral structure.

How does the framework identify SOC within the Koopman spectrum?

The SOC dynamics emerge as the slowest marginally stable mode, with an eigenvalue closest to the unit circle. This matches the integrator-type pole implied by charge conservation and yields a physically grounded SOC-sensitive observable.

Jul 16, 2026

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