Final project for Sistem Kendali Prediktif & Adaptif: online identification of battery model parameters and State of Charge (SOC) estimation using FFRLS, Levenberg–Marquardt, and an Extended Kalman Filter in MATLAB.
The battery is modelled as a 2nd-order Thevenin equivalent circuit (2-RC):
The pipeline works in four stages:
- OCV–SOC curve is derived from OCV test data by polynomial fitting (orders 2–12, lowest RMSE wins, fitted separately for charge and discharge).
- FFRLS (Forgetting Factor Recursive Least Squares, λ = 0.98) identifies the discrete transfer-function parameters θ₁…θ₆ online from current/voltage data, which are then inverted into R0, R1, R2, C1, C2, Uoc.
- Levenberg–Marquardt refines the FFRLS estimate at each time step through nonlinear optimisation against θ.
- EKF uses the LM parameters to estimate the state
[U1; U2; SOC], benchmarked against Coulomb counting.
Battery test data for an A123 18650 LiFePO4 cell, 1100 mAh, 25 °C (CALCE format):
| File | Contents | Columns used |
|---|---|---|
A1-007-OCV-25-20120905.xlsx |
OCV test, cell #007 | 1 = time, 2 = current, 3 = voltage |
A1-008-OCV-25-20120905.xlsx |
OCV test, cell #008 | same |
A1-007-DST-US06-FUDS-25-20120827.xlsx |
Dynamic DST/US06/FUDS profile, cell #007 | 1 = time, 4 = current, 5 = voltage |
A1-008-DST-US06-FUDS-25-20120827.xlsx |
Dynamic profile, cell #008 | same |
Nominal capacity used throughout: Cn = 1.1 Ah.
The scripts are hardcoded to cell A1-007. To run cell 008, change the filename in the Read Data section of each notebook.
Live scripts, in execution order:
| File | Role | Reads | Writes |
|---|---|---|---|
| SoCApprox.mlx | OCV–SOC polynomial fitting | A1-007-OCV-*.xlsx |
_polynomialEstimate.txt |
| FFRLS.mlx | Online parameter identification | A1-007-DST-*.xlsx |
_FFRLSParameters.txt, _discreteParameters.txt |
| LM.mlx | Levenberg–Marquardt refinement | _discreteParameters.txt |
_LMParameters.txt |
| EKF.mlx | SOC estimation | _LMParameters.txt, _polynomialEstimate.txt |
SOC / Ut / Uoc plots + error |
Helper functions:
- UocCurve.m — evaluates OCV from SOC, selecting the charge or discharge coefficient set based on current sign, and clamping below the minimum fitted SOC.
- dUocCurve.m — dOCV/dSOC derivative, used as the SOC entry of the EKF output Jacobian.
The .txt files above are intermediate artefacts and are not tracked in the repository, so run the stages in order on a fresh clone — each stage consumes what the previous one produced.
Requires MATLAB (tested on R2024b). No extra toolboxes — only polyfit, readmatrix, and readtable.
% from the project folder
run SoCApprox.mlx % 1. OCV-SOC curve
run FFRLS.mlx % 2. parameter identification
run LM.mlx % 3. refinement
run EKF.mlx % 4. SOC estimationOrder-12 polynomials gave the lowest RMSE for both directions: 0.04234 V for charge and 0.01888 V for discharge. Below the minimum SOC present in the fitting data, OCV is held flat at the minimum measured value — the horizontal segment on the left of each curve.
Approximation error across polynomial orders — fainter lines are lower orders, the most visible line is order 12:
Discrete parameter estimates θ₁…θ₆ with λ = 0.98:
Inverted into RC-model parameters:
To keep the estimate from diverging when the covariance matrix P becomes ill-conditioned, an improved FFRLS step checks each parameter against a ±3σ band over the last 100 samples. On a violation it backtracks up to 40 previous θ estimates and keeps the one with the smallest prediction error, exiting early once the error drops below 0.05 V.
LM is initialised from the FFRLS estimate and refined per time step (μ₀ = 1e3, c₁ = 0.1, c₂ = 10, maxIter = 20, tol = 1e-2), with the Jacobian computed by finite differences (δ = 1e-6).
Compared side by side against FFRLS:
Three states are estimated: [U1, U2, SOC], with Q = diag([1e-6, 1e-6, 1e-6]), R = 1, and P₀ = I.
SOC estimate against the Coulomb-counting reference, with percentage error:
Reconstructed open-circuit voltage:
Reconstructed terminal voltage:
