MJ1 Adaptive 2RC–EKF SOC Estimation

Experimentally parameterised battery modelling, EKF state estimation, autonomous excitation detection, and gated online Bayesian R0 adaptation for the LG INR18650 MJ1 in MATLAB and Simulink.

This repository extends a frozen SOC-dependent 2RC Thevenin EKF baseline with two additional layers:

• an autonomous current-based excitation gate; and
• a causal online Bayesian update of the ohmic-resistance multiplier.

The frozen state-transition model and RC dynamics remain unchanged.

Adaptive v1 validation · K=4 extension closeout · Frozen-EKF validation · Model-selection analysis · Reproduction guide

Architecture

flowchart LR
    I[Measured current] --> G[Autonomous excitation gate]
    I --> B[Online Bayesian R0 updater]
    V[Measured voltage] --> B
    G -->|gate + detected period| B
    B -->|alpha_R0| E[2RC EKF observer]
    I --> E
    V --> E
    E --> S[SOC estimate]
    E --> VH[Voltage estimate]

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The adaptive layer changes only the ohmic-resistance contribution used by the EKF measurement model:

R0*(SOC) = alpha_R0 × R0(SOC)

When the gate is closed, alpha_R0 = 1, so the adaptive estimator falls back numerically to the frozen EKF.

Key v1.0 result

The released v1.0 development condition is a fast periodic DC–AC excitation with measured frequency approximately 0.144 Hz and period approximately 6.96 s.

Metric	Frozen EKF	Autonomous adaptive EKF
Posterior-voltage RMSE, 20–80% SOC	4.021863 mV	3.805565 mV
Relative RMSE change	—	−5.378%
Approx. squared-error reduction	—	10.5%
Final alpha_R0	1.000000	1.018477
Accepted online updates	0	50

The adaptive layer reduced the fast-condition posterior-voltage RMSE by 5.378% while leaving the physical plant/state-transition path unchanged.

How v1.0 adaptation works

1. The measured current is analysed online by the excitation gate.
2. Parameter learning is allowed only when the excitation satisfies the validated fast-excitation conditions.
3. The Bayesian updater estimates a scalar multiplier alpha_R0.
4. The EKF measurement model uses R0*(SOC) = alpha_R0 × R0(SOC).
5. If the excitation is not sufficiently informative, the gate stays closed and the estimator remains identical to the frozen baseline.

The gate uses a rolling current window, linear detrending, spectral concentration tests, a fast-branch frequency criterion, and open/close hysteresis. Full implementation details and exact validation plots are in the Adaptive v1 validation report.

v1.0 fail-safe behaviour

The final v1.0 A/B matrix contains nine real conditions.

Condition family	Measured frequency / period	Gate	Online R0 update
Fast periodic development case	≈0.144 Hz / ≈6.96 s	Open	50 updates
Slow periodic, three amplitudes	≈0.0143 Hz / ≈70 s	Closed	0
Slow periodic, three amplitudes	≈0.00143 Hz / ≈700 s	Closed	0
Ultra-low-frequency	≈0.000412 Hz / ≈2427 s	Closed	0
Constant-current holdout	No finite excitation period	Closed	0

Across all eight non-fast conditions:

• gate-open fraction remained zero;
• update count remained zero;
• alpha_R0 remained exactly 1 within numerical precision;
• plant outputs were unchanged; and
• the adaptive estimator numerically collapsed to the frozen EKF.

The reviewed final real-condition matrix is 9/9 PASS.

v1.0 sensor-bias robustness

The released v1.0 architecture was stress-tested on its development dataset with:

• current offsets: ±10 mA, ±20 mA;
• voltage offsets: ±5 mV, ±10 mV.

All nine cases, including the unbiased baseline, passed the stated v1.0 robustness checks.

The maximum absolute shift in final alpha_R0 was approximately 1.96 × 10⁻⁴, and the maximum update-count shift was 1.

Current and voltage offsets are robustness stressors in v1.0; they are not augmented EKF states.

Extension Track 1 — finite posterior memory

Extension Track 1 investigated a failure of the frozen v1.0 cumulative Bayesian memory under an additional high-amplitude fast condition. The released v1.0 implementation was not modified.

A rolling-posterior family with memory lengths K = {1, 2, 4, 8, 16, 32} accepted windows was evaluated. After inspecting the sweep, the extension selection rule was defined as:

choose the largest finite K that preserves plant invariance, does not degrade any current A/B/C case relative to the Frozen EKF, and does not degrade the formal A/B cases relative to the original v1.0 adaptive estimator.

Under that post-sweep, non-preregistered rule, K=4 was selected.

Aggregate K=4 evidence

Case	Role	K4 vs Frozen	K4 vs frozen v1.0
A: 0.2C + 0.3C, fast band	Formal native case	−1.917%	−0.518%
B: 0.3C + 0.4C, fast band	Formal native case	−1.725%	−1.522%
C: 0.2C + 0.8C, fast band	Sensitivity-only case	−0.403%	−2.094%

The K=4 standalone core, integrated Simulink implementation, plant invariance, and gate-closed fail-safe behaviour all reached numerical parity with their corresponding reference calculations.

Evidence that prevents promotion to v1.1

K=4 remains a frozen extension candidate, not an independently validated release.

• The predeclared K=4 sensor-bias qualification remains FAIL because Case A at ±20 mA produced an update-count shift of −2 against the predefined |ΔN| ≤ 1 criterion. Later diagnostics do not rewrite that verdict.
• An archived raw 0.3C + 0.7C candidate (EXP_0037) was shown to contain the complete development electrical sequence sample-for-sample and is therefore not an independent physical holdout.
• No untouched independent fast-band holdout remains available in the current archive.
• Leave-one-condition-out selection stability is 2/3 PASS. When Case C is hidden, A+B select K=8; that K degrades the held-out C condition by +0.381% versus Frozen.
• Case C is sensitivity-only because its replay-ready record ends at approximately 78.07% SOC, not the complete 20–80% band.

SOC-local limitation

The aggregate Case-C improvement is not uniform across SOC. Five-percentage-point analysis shows sustained local degradation beginning around 55% SOC:

• 55–60%: +29.39% vs Frozen;
• 60–65%: +45.59% vs Frozen;
• 65–70%: +2.74% vs Frozen;
• 70–75%: +7.47% vs Frozen;
• 75–78.07%: +40.74% vs Frozen.

The strongest local regression occurs at 60–65% SOC.

Mechanistic closeout

A fixed-state conditional-alpha oracle shows that the high-SOC Case-C K=4 multiplier is systematically above the value that minimizes posterior-voltage error on the realized EKF state path. Over accepted windows at ≥55% SOC:

• mean shadow-window alpha ≈ 1.01453;
• mean EKF conditional-oracle alpha ≈ 1.00031;
• mean shadow-minus-oracle divergence ≈ +0.01422;
• 96.55% of windows have shadow alpha above the EKF conditional oracle.

This shadow-to-EKF target mismatch is not unique to Case C; related divergence is also present in A/B. Case C becomes problematic because the realized correction energy can exceed the residual-cancellation benefit.

Exact per-bin SSE decomposition gives:

SSE_K4 − SSE_Frozen = 2 e_Frozenᵀ ΔV + ||ΔV||²

The cross term measures whether the correction cancels or reinforces the pre-existing residual; the quadratic term is the correction-energy cost. In Case C:

• 55–60% SOC includes material state-path degradation;
• approximately 60–75% SOC is dominated by alpha-layer over-correction that reverses otherwise beneficial state-path effects;
• near 75–78% SOC both state-path and adaptive-layer contributions become unfavourable.

These oracle and decomposition results are diagnostic counterfactuals, not independently validated controller redesigns.

Full scope and evidence boundaries are documented in EXT1 K=4 candidate closeout.

Frozen 2RC–EKF baseline

The adaptive architecture preserves the previously validated frozen estimator.

The reference baseline benchmark uses a measured 1C discharge at −3.4 A, sampled every 1 s, covering approximately 50% to 17% SOC.

Initial SOC condition	SOC RMSE [pp]	Posterior-voltage RMSE [mV]
Correct initialization	2.445	10.182
−20 pp offset	2.509	10.925
+15 pp offset	2.455	11.187

SOC errors are evaluated against a Coulomb-counting consistency reference, not an independent SOC ground truth.

MATLAB and Simulink implementations were also checked for numerical agreement under constant-current and synthetic variable-current step tests.

Model

Configuration	Value
Cell	LG INR18650 MJ1
Nominal capacity	3.5 Ah
Model reference capacity	3.335 Ah
Frozen lookup-table SOC interval	17–85%
Current convention	Positive for charge; negative for discharge
State order	[v1, v2, SOC]
Reference sampling interval	1 s

The frozen EKF uses numerical Jacobians, Joseph-form covariance updates, and SOC-bound enforcement.

EKF setting	Reference value
Process covariance	diag([1e-7, 1e-7, 1e-9])
Voltage measurement covariance	(0.020 V)^2
Initial state covariance	diag([0.02^2, 0.02^2, 0.20^2])

Frequency descriptions

Historical tau-labels are retained only as dataset aliases. Public technical descriptions use measured frequency and period.

Historical alias	Public description
0.1τ	fast-band, ≈0.144 Hz, ≈6.96 s
1τ	slow-band, ≈0.0143 Hz, ≈70 s
10τ	slow-band, ≈0.00143 Hz, ≈700 s
ULF	ultra-low-frequency, ≈0.000412 Hz, ≈2427 s

Repository contents

Directory	Contents
data/	Frozen model lookup tables and benchmark data
matlab/	EKF equations, Jacobians, v1.0 Bayesian updater/gate, and the K=4 extension core
model/	Frozen plant/EKF models, released v1.0 adaptive model, and K=4 extension-candidate model
figures/validation/	Detailed validation plots
docs/validation/	Validation methods, results, evidence boundaries, and extension closeout notes
results/adaptive_v1/	Reviewed frozen-v1.0 A/B and sensor-bias evidence
results/extension_track1/	K=4 model-selection, robustness, limitation, and mechanism evidence
results/validation/	Archived frozen-EKF validation evidence
tools/	Offline evidence-verification utilities

Current limitations

The evidence supports same-frequency cross-amplitude transfer of frozen v1.0 on two independent native fast cases, but independent cross-frequency fast-band generalization has not been demonstrated.

For the K=4 extension candidate, A/B/C are consumed development/model-selection evidence. No untouched independent fast-band holdout remains available in the current archive.

Other scope boundaries:

• the Coulomb-counting SOC reference is not independent SOC ground truth;
• the autonomous gate is validated for the project's periodic/sinusoidal excitation family, not arbitrary automotive drive cycles;
• K=4 does not provide uniform SOC-local improvement under the high-amplitude sensitivity case;
• the K=4 predeclared sensor-bias qualification remains failed even though later output-impact attribution limits the engineering significance of that failure;
• unconditional all-timescale scalar-R0 adaptation is not supported;
• cross-cell and temperature generalization have not been demonstrated;
• current and voltage biases are stressors, not online estimated states.

Future work

Extension Track 1 is closed without promotion to v1.1.

A future candidate should not be tuned further on the consumed v1.0/A/B/C evidence and then evaluated on the same cases as if they were independent validation. Any architectural change should define a new candidate and a new prospective validation protocol.

The strongest next evidence would be an untouched fast-band experiment or external dataset that satisfies the frozen holdout protocol. Cross-frequency, temperature, and cross-cell evidence remain open.

Author

Jiaxing Lu — battery testing, modelling, diagnostics, and BMS-oriented state estimation.

For data terms, see DATA_LICENSE.md.