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

T0 review · 4 major / 6 minor · reviewed 2026-07-10 · grok-4.5

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Pith's one-line read State of charge appears as the battery's slowest, marginally stable Koopman mode, recovered from voltage and current alone without circuit parameters.

desk verdict Solid application of Hankel-DMDc that recovers the charge-conservation integrator as a near-unit eigenvalue; useful for BMS people, but the 0.0043% SOC RMSE and single-trajectory min-max scaling keep it provisional. read the letter →

arxiv 2607.07594 v2 pith:FEJHOGPN submitted 2026-07-08 eess.SY cs.SY

Bakhtiar Nafis , Khalid Mahmud Labib , Saad Waheed , Shabbir Ahmed This is my paper · ORCID
classification eess.SYcs.SY
keywords stateofchargeestimationDynamicModeDecompositionwithcontrolKoopmanoperatorHankelembeddinglithium-ionbatteriesbatterymanagementsystemmarginallystable
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Lithium-ion batteries are nonlinear and change with temperature, load, and age, so conventional SOC estimators either keep re-fitting circuit parameters or abandon physical meaning. This paper shows that lifting terminal voltage into a high-dimensional Hankel space and applying Dynamic Mode Decomposition with control recovers a linear operator whose spectrum already contains the battery's intrinsic timescales. The mode whose discrete eigenvalue sits nearest the unit circle is precisely the integrator that charge conservation demands; its modal coordinate, after a simple min-max map, is an SOC estimate. On HPPC data the method reconstructs voltage to 0.0131 V RMSE and SOC to 0.0043 % RMSE, beating both Coulomb counting and a 2RC extended Kalman filter, all without ever writing down resistances or capacitances. A sympathetic reader cares because the same spectral picture that yields the estimate also explains why SOC is the slowest, non-decaying mode.

What carries the argument

Hankel-DMDc operator: voltage is stacked into a delay-embedded Hankel matrix so that the nonlinear battery map becomes approximately linear; DMDc then yields the pair (A,B) whose eigen-decomposition of A produces the Koopman eigenvalues and modes that separate SOC from faster polarization dynamics.

What would settle it

Re-run the identical Hankel-DMDc pipeline on a second cell of the same chemistry under a different temperature or after measurable aging; if the eigenvalue nearest unity no longer tracks reference SOC (or another mode becomes closer to the unit circle), the spectral identification claim fails.

Watch

Extended reading notes

Core claim

When battery voltage and current from HPPC testing are lifted by Hankel delay embedding and the resulting input-output map is identified by DMDc, eigen-decomposition of the learned state-transition matrix isolates SOC dynamics as the unique slowest, marginally stable mode (eigenvalue closest to 1). The associated modal coordinate, after min-max normalization over the operating range, supplies a quantitatively usable SOC estimate without any explicit equivalent-circuit parameter identification.

Load-bearing premise

A single linear operator learned from one HPPC trajectory in a 2000-dimensional delay space remains faithful enough that the eigenvalue nearest unity can be unambiguously labeled the SOC mode and simply rescaled by min-max to give quantitative state of charge.

Editorial extensions

If this is right

  • SOC can be read directly from the spectrum of a data-driven operator rather than from an equivalent-circuit model that must be re-identified whenever temperature or age changes.
  • Voltage reconstruction and SOC estimation become a single eigen-decomposition step once the Hankel-DMDc operator is known.
  • Long-term drift of the same eigenvalue or its associated mode may later serve as a spectral signature of capacity fade.
  • The same workflow can be re-applied to other chemistries or packs without rewriting physics-based state equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If mode mixing is the main obstacle, sparse or orthogonalized Koopman dictionaries could further separate SOC from hysteresis and diffusion without enlarging the Hankel dimension.
  • Adaptive re-identification of the DMDc operator on a sliding window would turn the present offline spectral snapshot into an online BMS estimator that tracks aging.
  • The same marginally stable mode idea may transfer to other electrochemical storage devices (fuel cells, supercapacitors) whose charge conservation likewise produces an integrator pole.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes a data-driven SOC estimation framework based on Dynamic Mode Decomposition with control (DMDc) applied to Hankel time-delay embeddings of terminal voltage, with applied current as the control input, using HPPC measurements. After identifying a linear lifted operator, eigen-decomposition is used to argue that SOC appears as the slowest, marginally stable Koopman mode (eigenvalue nearest the unit circle), consistent with the continuous-time integrator pole of charge conservation. The corresponding modal coordinate, after min–max normalization over the operating SOC range, is reported as an SOC estimate (RMSE 0.0043%), while the same model reconstructs terminal voltage with RMSE 0.0131 V, outperforming a 2RC-ECM EKF and Coulomb counting on the same HPPC cycle.

Significance. If the spectral identification of SOC as a robust, physically grounded Koopman mode holds under broader conditions, the work would offer a useful middle path between ECM/Kalman estimators (which require repeated parameter identification) and black-box learning methods (which often lack interpretability). The theoretical link from charge conservation (pole at s=0) to a discrete eigenvalue at unity is standard and correctly stated, and framing SOC as a marginally stable observable is a clear, falsifiable spectral claim. Voltage reconstruction performance is credible and supports the utility of Hankel–DMDc for input–output battery modeling. The contribution is primarily methodological and interpretive rather than a fully validated BMS-ready estimator; its lasting value depends on whether mode selection and quantitative SOC mapping remain reliable beyond a single calibrated HPPC trajectory.

major comments (4)
  1. Quantitative SOC is obtained only after min–max normalization of the modal coordinate over the experimental operating SOC range (Section 2.5, Eq. 9; Section 3.5). This is an affine calibration that requires knowledge of the SOC bounds (or equivalent end-point labels). The abstract and Table 2 present RMSE 0.0043% as if the spectrum alone supplies a quantitative SOC observable; the manuscript should clearly separate (i) spectral mode identification from (ii) the subsequent data-dependent scaling, and report accuracy of the raw modal coordinate (or a fixed calibration) rather than only the post-normalized signal. The conclusion already flags this limitation; it should be reflected in the claims and tables.
  2. Table 2 reports SOC RMSE of 0.0043% (DMDc), 0.0047% (Coulomb counting), and 0.0078% (EKF). These values are orders of magnitude below typical laboratory SOC estimation errors and below what sensor noise and Coulombic-efficiency uncertainty usually allow on multi-hour HPPC profiles. The reference is described as “derived from experimental discharge capacity data,” which is often itself an ampere-hour integral of the same current. The paper must specify exactly how the reference SOC is constructed, whether it is independent of the Coulomb-counting baseline, and why all three methods achieve sub-0.01% RMSE. Without that, the performance comparison does not support the claim of a superior SOC-sensitive observable.
  3. Section 3.4 identifies mode 457 with λ_SOC = 1.000001 among 2000 modes as the unique SOC mode because it is “closest to the unit circle.” The reported eigenvalue lies slightly outside the unit circle (mildly unstable), with no discussion of numerical tolerance, isolation gap to the next-nearest eigenvalues, or sensitivity to Hankel dimension, SVD truncation rank, or train/test split. A load-bearing claim that SOC “naturally emerges” requires evidence that this selection is unique and stable (e.g., eigenvalue gap plot, ablation over embedding dimension/rank, and at least one independent trajectory or operating condition).
  4. All identification and evaluation use a single HPPC trajectory (≈75% train / held-out remainder of the same test; Sections 3.1–3.3). There is no second cell, temperature, aging state, or drive-cycle hold-out that freezes the learned operator and tests only mode selection and the SOC coordinate. Given the free parameters (embedding dimension d=2000, SVD truncation, min–max bounds), the central claim that the Koopman spectrum itself supplies a robust SOC observable remains under-supported for generalization beyond this dataset.
minor comments (6)
  1. Table 1: EKF voltage MAE is listed as 0.0007 V while RMSE is 0.0439 V. Such a large RMSE–MAE gap is possible only with rare extreme outliers; please verify the numbers and, if correct, briefly explain the residual structure (e.g., pulse spikes).
  2. Figure 3A caption refers to “long-timescale capacity fade behavior” for the SOC mode; the experiment is a single HPPC cycle and does not measure capacity fade. Align the caption with the charge-conservation / OCV interpretation used in the text.
  3. Notation: continuous-time eigenvalue is written λ_c = 0 and discrete λ_d = 1 (Eqs. 7–8), while the identified value is 1.000001; state consistently whether “closest to unity” or “on the unit circle” is the selection criterion.
  4. Section 2.6 and Eq. (14): EKF noise covariances are given without tuning rationale or sensitivity; a short note would help readers reproduce the baseline.
  5. References [2] and [3] share nearly identical generic titles (“Review of battery management systems”); expand bibliographic detail for traceability.
  6. Abstract and keywords: “0.0043%” should be reconciled with whatever unit convention is adopted after addressing the reference-SOC definition (percentage points vs. absolute SOC fraction).

Circularity Check

1 steps flagged · score 4.0 of 10

Theoretical SOC-mode selection (nearest unit-circle eigenvalue) is independent, but quantitative SOC values are obtained by min-max normalization of the modal coordinate over the same experiment’s operating range, forcing absolute scale by construction.

  1. fitted input called prediction [Section 2.5 (SOC Mode Identification), Eq. 9 and following paragraph]
    "The row of Y corresponding to the SOC eigenvalue (λ_d ≈1) yields the raw SOC-sensitive modal coordinate. After Min–Max Normalization of this raw coordinate over the operating SOC range, the resulting signal provides the DMDc-based SOC estimate."

    Min-max normalization is an affine map whose two free parameters are fixed by the min and max of the modal coordinate and the known SOC endpoints of the same HPPC experiment. Absolute SOC values are therefore forced by construction; only the intermediate shape remains free. The reported RMSE of 0.0043 % consequently evaluates a data-calibrated signal rather than an independent prediction of SOC.

full rationale

The derivation that charge conservation implies a continuous-time pole at s=0 and therefore a discrete eigenvalue of exactly 1 (Eqs. 6–8) is self-contained and non-circular; selecting the empirical eigenvalue nearest the unit circle (reported λ_SOC=1.000001) therefore has independent theoretical grounding. Voltage reconstruction via Hankel-DMDc is a standard least-squares fit evaluated on a held-out portion of the HPPC trajectory and is likewise non-circular. The only load-bearing circular step is the conversion of the raw modal coordinate into a numerical SOC estimate: min-max normalization maps the extrema of that coordinate exactly onto the known SOC operating range of the identical experiment, so absolute SOC values are forced by an affine map fitted to the same data whose shape is then scored. The resulting RMSE (0.0043 %) therefore measures fidelity of a calibrated signal rather than an uncalibrated, parameter-free prediction. No self-citation is load-bearing for the central claim, and no uniqueness theorem or ansatz is smuggled in. The circularity is partial and confined to the final quantitative mapping; the spectral identification itself remains independent.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The central claim rests on (i) the standard discrete-time mapping of a continuous integrator to eigenvalue 1, (ii) the empirical adequacy of a high-dimensional linear DMDc model in Hankel coordinates, and (iii) a post-hoc min-max map that converts the modal coordinate into percent SOC. Free parameters (embedding dimension, truncation, normalization bounds, EKF covariances) are chosen to make the numbers work; no new physical entities are postulated.

free parameters (5)
  • Hankel embedding dimension d = 2000
    Set to 2000 without cross-validation or sensitivity analysis; directly determines the size of the lifted space and the number of modes.
  • SVD truncation rank for DMDc pseudoinverse
    Implicit free parameter controlling numerical rank of the operator; value never stated.
  • min-max normalization bounds for SOC modal coordinate = operating SOC range of the HPPC test
    Affine map that converts the raw modal trajectory into the reported percent SOC; bounds taken from the operating range of the same experiment.
  • EKF process and measurement noise covariances = Q=diag(1e-6,1e-3,1e-3), R=1e-4
    Hand-tuned diagonal matrices used for the baseline comparison; affect the reported EKF RMSE.
  • train/test split fraction = ~75 %
    Approximately 75 % of the HPPC trajectory used for operator identification; exact indices not given.
assumptions (4)
  • domain assumption Charge conservation implies a continuous-time pole at s=0, which maps under sampling to a discrete eigenvalue exactly equal to 1.
    Derived in Section 2.5 from the Coulomb-counting integral; standard and uncontroversial.
  • domain assumption A finite Hankel delay embedding of terminal voltage yields a Koopman-invariant subspace in which the battery dynamics are approximately linear.
    Invoked via Takens and recent DMD-Hankel theory (citations 36-41); assumed to hold for the chosen d=2000.
  • ad hoc to paper The eigenvalue of the identified DMDc operator that lies closest to the unit circle can be uniquely identified with the SOC mode.
    Stated in Section 3.4; relies on the absence of other near-unit modes and on numerical accuracy of the eigen-decomposition.
  • ad hoc to paper Min-max normalization of the corresponding modal coordinate over the experimental SOC range produces a quantitatively accurate SOC estimate.
    Section 2.5; converts an unscaled observable into the numbers reported in Table 2.
invented entities (1)
  • SOC-sensitive modal coordinate (row of Y = Φ^{-1} H_v corresponding to λ≈1)
    purpose: Serves as the data-driven observable that is later normalized to percent SOC.
    Defined by the spectral decomposition of the learned operator; no independent physical measurement of this coordinate exists outside the paper’s pipeline.

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Pith. "Pith review of Koopman Spectral Analysis of Lithium-Ion Battery Dynamics: State of Charge as a Marginally Stable Observable." pith.science (2026). https://pith.science/paper/FEJHOGPN

@misc{pith2026260707594,
  author       = {Pith},
  title        = {Pith review of: Koopman Spectral Analysis of Lithium-Ion Battery Dynamics: State of Charge as a Marginally Stable Observable},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FEJHOGPN}},
  note         = {Machine review of arXiv:2607.07594}
}
read the original abstract

Accurate state-of-charge (SOC) estimation remains a fundamental challenge in lithium-ion battery management systems because battery dynamics are highly nonlinear, operating-condition dependent, and sensitive to parameter variations caused by aging and temperature. Conventional model-based estimators, such as equivalent circuit model (ECM) and Kalman-filter-based approaches, rely heavily on repeated parameter identification and accurate electrochemical modeling, whereas purely data-driven methods often sacrifice physical interpretability. This work proposes a Koopman-theoretic, data-driven framework for SOC estimation using Dynamic Mode Decomposition with control (DMDc) combined with Hankel time-delay embedding. Instead of explicitly identifying ECM parameters, the proposed approach reconstructs a lifted dynamical state space directly from measured terminal voltage and current obtained through Hybrid Pulse Power Characterization (HPPC) testing. Spectral decomposition of the identified DMDc operator reveals intrinsic battery dynamics in terms of Koopman modes and eigenvalues. The SOC dynamics naturally emerge as the slowest marginally stable mode whose eigenvalue lies closest to the unit circle, consistent with the integrator-type behavior of charge conservation. The corresponding modal coordinate is subsequently utilized as an SOC-sensitive observable.

Figures

Figures reproduced from arXiv: 2607.07594 by the authors.

Figure 1
Figure 1. Workflow of the proposed DMDc-based SOC estimation framework. HPPC current [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 1
Figure 1. Workflow of the proposed DMDc-based SOC estimation framework. HPPC current [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. DMDc-based terminal voltage prediction evaluated on the full HPPC test cycle. ( [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figures from the paper (3 more)
Figure 2
Figure 2. Figure 2: DMDc-based terminal voltage prediction evaluated on the full HPPC test cycle. ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png]
Figure 3
Figure 3. Figure 3: Koopman-theoretic analysis of Li-ion battery dynamics via DMDc. ( [PITH_FULL_IMAGE:figures/full_fig_p009_3.png]
Figure 3
Figure 3. Figure 3: Koopman-theoretic analysis of Li-ion battery dynamics via DMDc. ( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png]

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Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Towards a smarter battery management system: A critical review on battery state of health monitoring methods

    Xiong, Rui, Li, Linlin and Tian, Jinpeng. “Towards a smarter battery management system: A critical review on battery state of health monitoring methods.”Journal of Power Sources Vol. 405 (2018): pp. 18–29. DOI 10.1016/j.jpowsour.2018.10.019

    work page doi:10.1016/j.jpowsour.2018.10.019 2018

  2. [2]

    Review of battery management systems

    Wang, Y et al. “Review of battery management systems.”Applied EnergyVol. 228 (2017): pp. 1512–1525

    work page 2017

  3. [3]

    Review of battery management systems

    Feng, X et al. “Review of battery management systems.”Renewable and Sustainable Energy ReviewsVol. 103 (2019): pp. 60–75

    work page 2019

  4. [4]

    State-of-charge estimation of lithium-ion batteries

    He, Hongwen et al. “State-of-charge estimation of lithium-ion batteries.”EnergiesVol. 4 No. 4 (2011): pp. 582–598

    work page 2011

  5. [5]

    Critical review of state of health estimation methods

    Berecibar, M et al. “Critical review of state of health estimation methods.”Renewable and Sustainable Energy ReviewsVol. 56 (2016): pp. 572–587. 11

    work page 2016

  6. [6]

    State estimation techniques for lithium-ion batteries

    Zheng, Y et al. “State estimation techniques for lithium-ion batteries.”EnergiesVol. 11 (2018): p. 1822

    work page 2018

  7. [7]

    A Critical Look at Coulomb Counting Approach for State of Charge Estimation in Batteries

    Movassagh, Koosha, Raihan, Ahmed, Bhattacharya, Bhaskar and Pattipati, Krishna. “A Critical Look at Coulomb Counting Approach for State of Charge Estimation in Batteries.” EnergiesVol. 14 No. 14 (2021): p. 4074. DOI 10.3390/en14144074

    work page doi:10.3390/en14144074 2021

  8. [8]

    Methods for state-of-charge determination

    Piller, S et al. “Methods for state-of-charge determination.”Journal of Power SourcesVol. 96 (2001): pp. 113–120

    work page 2001

Show all 45 references
  1. [9]

    Enhanced coulomb counting method

    Ng, K et al. “Enhanced coulomb counting method.”Applied EnergyVol. 86 (2011): pp. 1506–1511

    2011

  2. [10]

    A comparative study of equivalent circuit models for Li-ion batteries

    Hu, Xiaosong, Li, Shengbo and Peng, Huei. “A comparative study of equivalent circuit models for Li-ion batteries.”Journal of Power SourcesVol. 198 (2012): pp. 359–367. DOI 10.1016/j. jpowsour.2011.10.013

    2012 doi

  3. [11]

    A comparative study of different equivalent circuit models for estimating state-of-charge of lithium-ion batteries

    Lai, Xin, Zheng, Yuejiu and Sun, Tao. “A comparative study of different equivalent circuit models for estimating state-of-charge of lithium-ion batteries.”Electrochimica ActaVol. 259 (2018): pp. 566–577. DOI 10.1016/j.electacta.2017.10.153

    2018 doi

  4. [12]

    Review on battery modeling methods

    Zhang, S. “Review on battery modeling methods.”Renewable and Sustainable Energy Reviews Vol. 15 (2011): pp. 3126–3131

    2011

  5. [13]

    Identification of battery model parameters

    Barai, A et al. “Identification of battery model parameters.”Journal of Power SourcesVol. 280 (2015): pp. 74–80

    2015

  6. [14]

    Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 1. Background

    Plett, Gregory L. “Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 1. Background.”Journal of Power SourcesVol. 134 No. 2 (2004): pp. 252–261. DOI 10.1016/j.jpowsour.2004.02.031

    2004 doi

  7. [15]

    Unscented Kalman filtering for battery management systems

    Plett, Gregory L. “Unscented Kalman filtering for battery management systems.”Journal of Power SourcesVol. 134 (2004): pp. 277–292

    2004

  8. [16]

    Sigma-point Kalman filtering for battery management systems

    Plett, Gregory L. “Sigma-point Kalman filtering for battery management systems.”Journal of Power SourcesVol. 161 (2004): pp. 1354–1368

    2004

  9. [17]

    Adaptive EKF for lithium-ion battery SOC estimation

    Sun, Fengchun et al. “Adaptive EKF for lithium-ion battery SOC estimation.”EnergyVol. 36 (2011): pp. 3531–3540

    2011

  10. [18]

    A Comparative Study of SOC Estimation Based on Equivalent Circuit Models

    Wei, Meng, Ye, Min, Li, Jian-Bo, Wang, Qiao and Xu, Xueqiang. “A Comparative Study of SOC Estimation Based on Equivalent Circuit Models.”Frontiers in Energy ResearchVol. 10 (2022): p. 914291. DOI 10.3389/fenrg.2022.914291

    2022 doi

  11. [19]

    Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell

    Doyle, Marc, Fuller, Thomas F. and Newman, John. “Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell.”Journal of the Electrochemical SocietyVol. 140 No. 6 (1993): pp. 1526–1533. DOI 10.1149/1.2221597

    1993 doi

  12. [20]

    Modeling of lithium-ion batteries

    Newman, John and Thomas-Alyea, Karen. “Modeling of lithium-ion batteries.”Electrochem- ical Systems(2004)

    2004

  13. [21]

    Review of models for predicting battery performance

    Santhanagopalan, S and White, R. “Review of models for predicting battery performance.” Journal of Power SourcesVol. 156 (2006): pp. 620–628. 12

    2006

  14. [22]

    Control oriented battery models

    Smith, K and Wang, C. “Control oriented battery models.”Journal of Power SourcesVol. 160 (2007): pp. 662–673

    2007

  15. [23]

    Long short-term memory networks for accurate state-of-charge estimation of Li-ion batteries

    Chemali, Ephrem, Kollmeyer, Phillip J., Preindl, Matthias, Ahmed, Ryan and Emadi, Ali. “Long short-term memory networks for accurate state-of-charge estimation of Li-ion batteries.” IEEE Transactions on Industrial ElectronicsVol. 65 No. 8 (2018): pp. 6730–6739. DOI 10.1109/TIE...

    2018 doi

  16. [24]

    Deep learning based battery state estimation

    Li, Y et al. “Deep learning based battery state estimation.”Applied EnergyVol. 250 (2019): pp. 983–993

    2019

  17. [25]

    State estimation using deep neural networks

    Yang, F et al. “State estimation using deep neural networks.”EnergyVol. 201 (2020): p. 117664

    2020

  18. [26]

    Battery health prediction using machine learning

    Sehgal, A et al. “Battery health prediction using machine learning.”Applied EnergyVol. 250 (2019): pp. 1110–1121

    2019

  19. [27]

    Dynamic mode decomposition of numerical and experimental data

    Schmid, Peter J. “Dynamic mode decomposition of numerical and experimental data.”Journal of Fluid MechanicsVol. 656 (2010): pp. 5–28. DOI 10.1017/S0022112010001217

    2010 doi

  20. [28]

    Spectral analysis of nonlinear flows

    Rowley, Clarence et al. “Spectral analysis of nonlinear flows.”Journal of Fluid Mechanics Vol. 641 (2009): pp. 115–127

    2009

  21. [29]

    Spectral properties of dynamical systems

    Mezic, Igor. “Spectral properties of dynamical systems.”Nonlinear DynamicsVol. 41 (2005): pp. 309–325

    2005

  22. [30]

    Analysis of fluid flows via Koopman operator

    Mezic, Igor. “Analysis of fluid flows via Koopman operator.”Annual Review of Fluid Me- chanicsVol. 45 (2013): pp. 357–378

    2013

  23. [31]

    On dynamic mode decomposition

    Tu, Jonathan H et al. “On dynamic mode decomposition.”Journal of Computational Dynamics Vol. 1 (2014): pp. 391–421

    2014

  24. [32]

    Dynamic Mode Decomposition with Control

    Proctor, Joshua L., Brunton, Steven L. and Kutz, J. Nathan. “Dynamic Mode Decomposition with Control.”SIAM Journal on Applied Dynamical SystemsVol. 15 No. 1 (2016): pp. 142–

    2016

  25. [33]

    DOI 10.1137/15M1013857

    doi

  26. [34]

    Springer (2020)

    Mauroy, Alexandre et al.The Koopman Operator in Systems and Control. Springer (2020)

    2020

  27. [35]

    Koopman operator based model reduction

    Peitz, Sebastian and Klus, Stefan. “Koopman operator based model reduction.”Automatica Vol. 106 (2019): pp. 184–192

    2019

  28. [36]

    Linear observer synthesis via Koopman operator

    Surana, Amit and Banaszuk, Andrzej. “Linear observer synthesis via Koopman operator.” IF ACVol. 49 (2016): pp. 716–723

    2016

  29. [37]

    Detecting strange attractors in turbulence

    Takens, Floris. “Detecting strange attractors in turbulence.”Lecture Notes in Mathematics Vol. 898 (1981): pp. 366–381

    1981

  30. [38]

    Ergodic theory, Koopman analysis and computation

    Arbabi, Hassan and Mezic, Igor. “Ergodic theory, Koopman analysis and computation.”SIAM Journal on Applied Dynamical SystemsVol. 16 (2017): pp. 2096–2126

    2017

  31. [39]

    Chaos as an inter- mittently forced linear system

    Brunton, Bingni W., Proctor, Joshua L. and Kutz, J. Nathan. “Chaos as an inter- mittently forced linear system.”Nature CommunicationsVol. 8 (2017): p. 19. DOI 10.1038/s41467-017-00030-8. 13

    2017 doi

  32. [40]

    Time-Delay Observ- ables for Koopman: Theory and Applications

    Kamb, Mason, Kaiser, Eurika, Brunton, Steven L. and Kutz, J. Nathan. “Time-Delay Observ- ables for Koopman: Theory and Applications.”SIAM Journal on Applied Dynamical Systems Vol. 19 No. 2 (2020): pp. 886–917. DOI 10.1137/18M1216572

    2020 doi

  33. [41]

    SIAM (2016)

    Kutz, Nathan et al.Dynamic Mode Decomposition. SIAM (2016)

    2016

  34. [42]

    Cambridge University Press (2022)

    Brunton, Steven and Kutz, Nathan.Data-Driven Science and Engineering, 2nd ed. Cambridge University Press (2022)

    2022

  35. [43]

    Pulse power characterization of lithium-ion batteries

    Smith, K et al. “Pulse power characterization of lithium-ion batteries.”Journal of Power SourcesVol. 196 (2010): pp. 8723–8731

    2010

  36. [44]

    Evaluation of commercial lithium-ion cells

    Dubarry, M and Liaw, B. “Evaluation of commercial lithium-ion cells.”Journal of Power SourcesVol. 195 (2010): pp. 5415–5425

    2010

  37. [45]

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

    Labib, Khalid Mahmud and Ahmed, Shabbir. “Modeling of Non-linear Dynamics of Lithium-ion Batteries via Delay-Embedded Dynamic Mode Decomposition.”arXiv preprint arXiv:2601.22403(2026)URLhttps://arxiv.org/abs/2601.22403. 14

    2026

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