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Observability analysis and state estimation of lithium-ion batteries in the presence of sensor biases

Shi Zhao, Stephen R. Duncan, David A. Howey

arXiv:1510.06553v3eess.SYcs.SY

TL;DR

Battery-model observability is often overlooked, even though state estimation requires an observable model, and sensor biases can impair SOC estimation. The paper derives observability conditions for a second-order RC ECM and incorporates sensor biases into an augmented state model for nonlinear Kalman-filter estimation. With only a −100 mA current-measurement bias, UKF SOC RMSE was 0.43 % between t = 1000 s and t = 2000 s, versus 2.80 % for the first-order EKF.

  • Problem

    Battery-model observability is often overlooked, even though state estimation requires an observable model, and sensor biases can impair SOC estimation.

    Source: §I · p. 2, §IV · p. 4

  • Method

    The paper derives observability conditions for a second-order RC ECM and incorporates sensor biases into an augmented state model for nonlinear Kalman-filter estimation.

    Source: §I · p. 2, A. Battery ECM · p. 3, §IV · p. 4

  • Results

    With only a −100 mA current-measurement bias, UKF SOC RMSE was 0.43 % between t = 1000 s and t = 2000 s, versus 2.80 % for the first-order EKF.

    Source: B. State estimation using various… · p. 7

  • Takeaways & Limitations

    In the tested sensor-bias cases, the UKF estimates SOC more accurately than the first-order EKF, and the paper explains the filter difference through observability.

    Source: §I · p. 2, B. State estimation using various… · p. 7

    With biases in both voltage and current sensors, accurate simultaneous estimation of SOC and biases may not be possible, even when the augmented model appears locally observable.

    Source: B. State estimation using various… · p. 8

Abstract

from arXiv · show

This paper investigates the observability of one of the most commonly used equivalent circuit models (ECMs) for lithium-ion batteries and presents a method to estimate the state of charge (SOC) in the presence of sensor biases, highlighting the importance of observability analysis for choosing appropriate state estimation algorithms. Using a differential geometric approach, necessary and sufficient conditions for the nonlinear ECM to be observable are derived and are shown to be different from the conditions for the observability of the linearised model. It is then demonstrated that biases in the measurements, due to sensor ageing or calibration errors, can be estimated by applying a nonlinear Kalman filter to an augmented model where the biases are incorporated into the state vector. Experiments are carried out on a lithium-ion pouch cell and three types of nonlinear filters, the first-order extended Kalman filter (EKF), the second-order EKF and the unscented Kalman filter (UKF) are applied using experimental data. The different performances of the filters are explained from the point of view of observability.

I. INTRODUCTION

Battery SOC must be inferred from measurements, and equivalent circuit models are commonly used despite limits in interpretability. This brief argues that observability—not output fit alone—matters for choosing state-estimation methods, and analyzes a nonlinear ECM and sensor-bias estimation.

I. INTRODUCTION

  • ECMs are commonly used in BMSs, while electrochemical models can describe battery dynamics across a wider operating range but are harder to implement because of computational complexity.The single-particle model is valid only at low C-rates, and parameter estimation for a reported full P2D-model approach remains challenging.Source: §I · p. 1
  • Observability is often overlooked in battery estimation, although an unobservable model cannot support state estimation regardless of its input-output fit.Model choice should consider observability as well as output error, and a first-order EKF may struggle when the nonlinear model is observable but its linearisation is not.Source: §I · pp. 1–2
  • The brief derives necessary and sufficient nonlinear observability conditions for a second-order RC battery model and shows that unknown sensor biases can be estimated in a locally observable augmented model.It evaluates first-order and second-order EKFs and a UKF on experimental data from an NMC pouch cell; UKF estimation is reported as more accurate than first-order EKF with sensor bias.Source: §I · p. 2

II. NONLINEAR OBSERVABILITY

  • Nonlinear and linear observability differ, so linearising a system is not by itself a reliable test of nonlinear observability.The paper introduces nonlinear observability criteria before analyzing the battery ECM.Source: §II · p. 2, §I · p. 2

A. Distinguishability and observability

The section defines local observability through distinguishability of nearby states and presents a Lie-derivative rank test for nonlinear systems. It also notes that the general test has practical limits.

A. Distinguishability and observability

  • A system is locally observable at a state when no other state in a neighborhood is indistinguishable from it under some input.Two states are distinguishable if there exists an input that produces different outputs for a finite time.Source: A. Distinguishability and observability · p. 2

B. Observability rank test

  • The nonlinear observability rank test checks whether Lie-derivative rows provide enough linearly independent directions to reach the state dimension.The test is stated using output-function gradients and repeated Lie derivatives with respect to the system vector fields.Source: B. Observability rank test · pp. 2–3
  • For a general nonlinear system, the rank test may require an unbounded number of Lie derivatives and is sufficient but not necessary for local observability.The linear-system observability rank test follows as a special case.Source: B. Observability rank test · p. 3

III. OBSERVABILITY OF AN ECM

The paper analyzes a widely used second-order RC equivalent circuit model for batteries. Its states represent the two RC-pair voltages and normalized SOC, while its output includes nonlinear open-circuit voltage.

A. Battery ECM

  • The analyzed ECM is a second-order RC model, one of the widely used equivalent circuit models in the literature.Its parameters are identified to match experimental measurements; allowing them to vary with SOC and temperature may improve model accuracy at greater computational cost.Source: A. Battery ECM · p. 3
  • The continuous model uses V1 and V2 for the RC-pair voltages, Z for normalized SOC, Q for capacity, I for applied current, and V for terminal voltage.Current is positive during discharge and negative during charge; τ1 and τ2 denote R1C1 and R2C2, respectively.Source: A. Battery ECM · p. 3
  • The ECM's state equation is linear, while its output equation contains the nonlinear open-circuit voltage function VOC(Z).The model is rewritten in nonlinear state-space form for observability analysis; the output feedthrough term does not affect application of the theorem.Source: A. Battery ECM · p. 3

B. Observability analysis

For the second-order RC model, nonlinear observability depends on the derivatives of the open-circuit-voltage function, and can hold even when the linearised model is unobservable. The analysis assumes distinct RC time constants.

B. Observability analysis

  • The ECM analysis obtains its observability condition from output-function gradients and Lie derivatives, with mixed Lie derivatives involving both f and g contributing only constants.The constant terms can be ignored in the rank test.Source: B. Observability analysis · p. 3
  • With distinct RC time constants, the ECM is observable if the derivatives of VOC are not all zero simultaneously.If the condition fails, the system is not locally observable; equal time constants are excluded because the two RC pairs then combine and their individual voltages cannot be uniquely determined.Source: B. Observability analysis · p. 4
  • The nonlinear ECM may be locally observable even when its linearisation is not, so the linearised-model rank condition is not necessary for nonlinear observability.The linearised model has full rank only under its stated condition, which the paper says is not necessary for local observability of the nonlinear model.Source: B. Observability analysis · p. 4

IV. ESTIMATION WITH SENSOR BIASES

Sensor measurements are modeled with zero-mean Gaussian noise, but ageing or calibration errors can also introduce unknown biases. Augmenting the model offers a way to estimate those biases and battery SOC together without adding sensors.

IV. ESTIMATION WITH SENSOR BIASES

  • Current and voltage sensors are commonly modeled with zero-mean Gaussian noise, whose variances describe sensor precision.Source: §IV · p. 4
  • Sensor ageing or calibration errors can create unknown biases, treated as sensor bias faults.Source: §IV · p. 4
  • Model augmentation can estimate sensor biases and battery SOC simultaneously without additional sensors, with biases modeled as constants.Source: §IV · p. 4

A. ECM with sensor biases

For a voltage sensor bias, the model incorporates the unknown offset into the state vector and uses the measured voltage to reformulate the battery model. Current-sensor bias can be handled similarly; the analysis focuses mainly on the voltage-bias case.

A. ECM with sensor biases

  • For a voltage sensor bias Ve, subtracting the unknown offset from measured voltage Vm gives the actual terminal voltage V and supports reformulating the model.Source: A. ECM with sensor biases · p. 4
  • The reformulated model is called the augmented model; its state includes V1, V2, SOC Z, and voltage bias Ve, and it is not controllable.Source: A. ECM with sensor biases · p. 4
  • For current-sensor bias Ie, the actual applied current is I = Im − Ie, and the battery model is reformulated accordingly.Source: A. ECM with sensor biases · pp. 4–5
  • When both sensors are biased, both offsets can be included in the augmented state, but the analysis focuses mainly on voltage bias and says it extends to other cases.Source: A. ECM with sensor biases · p. 5

B. Observability analysis

The observability analysis uses a rank test for the nonlinear battery model and distinguishes full rank of the test matrix from local observability. The nonlinear OCV relationship matters, and the same stated condition applies to the related model.

B. Observability analysis

  • For model (6), local observability is tested by whether its observability set has four linearly independent rows at the state of interest.The analysis notes that Lie derivatives involving both f and g can be ignored in the rank test because they are constants.Source: B. Observability analysis · p. 5
  • If condition (8) fails, the observability matrix has rank at most 3; when it holds, the matrix has full rank, making the condition necessary and sufficient for full rank.Source: B. Observability analysis · p. 5
  • Because system (6) is not controllable, condition (8) is sufficient, but not necessary, for local observability.Source: B. Observability analysis · p. 5
  • For a linear OCV–SOC relationship, the rank condition fails at every state and system (6) is unobservable; model (7) has the same stated local-observability condition as model (6).Source: B. Observability analysis · p. 5

V. EXPERIMENTAL VALIDATION

The experimental section applies several nonlinear Kalman filters to experimental data to estimate battery states under an unknown voltage-sensor bias.

V. EXPERIMENTAL VALIDATION

  • Using experimental data, the study implements several nonlinear Kalman filters based on model (6) to estimate battery states with an unknown voltage-sensor bias.Source: §V · p. 5

A. Battery test

The battery tests characterize a Kokam lithium-ion pouch cell and identify model parameters using a FUDS load. The model is then validated against experimental voltage measurements under a UDC load.

A. Battery test

  • The 740 mAh Kokam NMC pouch cell was tested at 20 °C; GITT measurements at 0.1C sampled charge and discharge OCV–SOC curves, whose discrepancy was attributed to hysteresis.The averaged curves were used as the OCV–SOC function.Source: A. Battery test · p. 5
  • The OCV polynomial’s maximum approximation error was 17.97 mV and its RMSE was 3.85 mV over the 10%–100% SOC range.Errors are relative to the averaged OCV–SOC curve.Source: A. Battery test · p. 5
  • Under UDC validation, the model’s maximum error was 21.1 mV and RMSE was 4.78 mV, or 0.13% of mean terminal voltage.Fig. 3(a–b) shows the FUDS training and UDC validation current profiles; panels (c–d) compare experimental and model outputs under those loads.Source: A. Battery test · p. 6, Figure 3 · p. 6
  • Model parameters were identified from current and voltage recorded under a FUDS profile with the cell initially at 100% SOC, then reused for UDC validation.The parameter values are reported in Table II.Source: A. Battery test · p. 6, Table II · p. 6

B. State estimation using various Kalman filters

Experiments compare three nonlinear Kalman filters for SOC and sensor-bias estimation using battery measurements. With a voltage bias, the UKF gives the most accurate estimates, while first-order EKF performance reflects the model’s observability loss under linearisation.

B. State estimation using various Kalman filters

  • The experiments use UDC-load cell measurements, add a constant to recorded voltage to simulate bias, and treat Coulomb-counting SOC as true SOC.The three filters are first-order EKF, second-order EKF, and UKF.Source: B. State estimation using various… · p. 6
  • With a 100 mV voltage bias, first-order EKF estimation using model (2) has 15.6% SOC RMSE, and second-order EKF or UKF use does not improve accuracy.The example starts with true SOC about 85% and an estimated SOC of 95%.Source: B. State estimation using various… · p. 6, Figure 4 · p. 7
  • Model (6) is locally observable because of its nonlinearity, but linearisation removes observability, helping explain the first-order EKF’s poor estimates.The analysis gives a sufficient condition for local observability, satisfied everywhere for the polynomial VOC function considered.Source: B. State estimation using various… · p. 7
  • The UKF’s voltage-bias estimate converges toward the true value, with 3.41 mV bias RMSE and 0.33% SOC RMSE; SOC error quickly falls below 1%.The second-order EKF has some advantages over the first-order EKF, but the UKF provides the most accurate estimation.Source: Figure 5 · p. 7, B. State estimation using various… · p. 7
  • For a −100 mA current-sensor bias, first-order EKF estimation using model (2) has 2.80% SOC RMSE, while UKF estimation using model (7) reduces it to 0.43%.The UKF tracks the current-sensor bias accurately; the second-order EKF has similar accuracy, and the first-order EKF is least accurate for SOC and bias estimation.Source: B. State estimation using various… · p. 7
  • The comparison is not evidence that UKF is always superior: without sensor bias, first-order EKF using model (2) is as accurate as UKF.The paper also presents bias estimation without adding sensors.Source: B. State estimation using various… · p. 8
  • When both voltage and current sensors are biased, simultaneous accurate estimation of SOC and both biases may not be possible, even if the augmented model appears locally observable.The paper attributes this possibility to an ill-conditioned observability matrix and ambiguity about which sensor caused an output error.Source: B. State estimation using various… · p. 8

VI. CONCLUSION

The conclusion distinguishes nonlinear observability from observability of a linearised model and reports bias-aware SOC estimation using an augmented state model. Experiments underscore the role of nonlinearity and identify voltage accuracy as critical, while noting that changing model parameters and the OCV curve make estimation more challenging.

VI. CONCLUSION

  • The nonlinear ECM’s local observability at a point differs from the observability of the system linearised around that point.The conclusion says nonlinear observability must be considered in state estimation.Source: §VI · p. 8
  • An augmented model incorporating sensor biases into the state vector supports SOC estimation, and its observability depends on model nonlinearity.Source: §VI · p. 8
  • Experimental validation shows that estimation algorithms must account for model nonlinearity; accurate voltage measurement is critical, while the filters are relatively robust to current-sensor bias.The stated robustness rests on knowledge of the OCV-SOC ground truth and ECM parameters.Source: §VI · p. 8
  • Changes over time in model parameters and the OCV curve make the estimation problem significantly more challenging and are left for future investigation.Source: §VI · p. 8
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