Modeling and Control Problems in Sustainable Transportation and Power Systems
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Xue Li, Jiuchun Jiang, Caiping Zhang, Le Yi Wang, Linfeng Zheng, "Robustness of SOC Estimation Algorithms for EV LithiumIon Batteries against Modeling Errors and Measurement Noise", Mathematical Problems in Engineering, vol. 2015, Article ID 719490, 14 pages, 2015. https://doi.org/10.1155/2015/719490
Robustness of SOC Estimation Algorithms for EV LithiumIon Batteries against Modeling Errors and Measurement Noise
Abstract
State of charge (SOC) is one of the most important parameters in battery management system (BMS). There are numerous algorithms for SOC estimation, mostly of modelbased observer/filter types such as Kalman filters, closedloop observers, and robust observers. Modeling errors and measurement noises have critical impact on accuracy of SOC estimation in these algorithms. This paper is a comparative study of robustness of SOC estimation algorithms against modeling errors and measurement noises. By using a typical battery platform for vehicle applications with sensor noise and battery aging characterization, three popular and representative SOC estimation methods (extended Kalman filter, PIcontrolled observer, and observer) are compared on such robustness. The simulation and experimental results demonstrate that deterioration of SOC estimation accuracy under modeling errors resulted from aging and larger measurement noise, which is quantitatively characterized. The findings of this paper provide useful information on the following aspects: (1) how SOC estimation accuracy depends on modeling reliability and voltage measurement accuracy; (2) pros and cons of typical SOC estimators in their robustness and reliability; (3) guidelines for requirements on battery system identification and sensor selections.
1. Introduction
Electric vehicles (EVs), including hybrid electric vehicles (HEVs), battery electric vehicles (BEVs), and plugin hybrid electric vehicles (PHEVs), have become a critical driving force for green economy and attracted great research effort recently. An appropriate battery management system (BMS) is indispensable for safe, reliable, and efficient operations of EV battery systems [1]. The state of charge (SOC) is one of the most important state variables in BMS. Failure to estimate SOC accurately may cause overdischarging or overcharging, resulting in decreased battery longevity and even causing dangerous accidents [2].
There are many methods to estimate the SOC, with their own pros and cons. The Coulomb integral method [3] is easy to implement, but it needs the prior knowledge of the initial SOC and suffers from accumulated errors from measurement noise and bias. The open circuit voltage (OCV) method is a more reliable approach for SOC estimation. There is a monotonic relationship between the OCV and SOC. However, this relationship is accurate only at a steadystate after several hours of open circuit condition. As a result, the OCV method cannot be used reliably for online SOC estimation [4]. The Kalman filter [5, 6] and extended Kalman filter (EKF) [7–12] have the appealing property of minimizing the meansquare estimation errors when the state and output measurement noises are additive, independent, zero mean, and Gaussian. On the other hand, they are susceptible to modeling errors and noise feature variations. The nonlinear observer method [13] employs a feedback mechanism to correct SOC estimation errors. Although this method works well under noisefree environment, its feedback gain must be carefully designed to achieve noise attenuation and robustness, which are highly challenging.
The accuracy of the model parameters is one of the main reasons affecting the SOC estimation accuracy. In [14], the researchers analyze the effects of the model parameters on the SOC estimation accuracy, when the model parameters drift due to battery aging. Various degrees of impact of different model parameters are established, leading to some parameter updating guidelines to focus on highimpact parameters so that computational complexity can be reduced.
The paper is a comparative study of several typical SOC estimation algorithms on their robustness against modeling errors and measurement noises. We focus on variations of model parameters caused by battery aging. Our simulation results demonstrate that the Ohmic resistance , polarization resistance , and the open circuit voltage OCV are the key parameters affecting SOC estimation accuracy. However, the polarization capacitor which is an important parameter only influences the dynamic response characteristics of SOC estimation and does not have noticeable effects on the steadystate accuracy of SOC estimation. Within the extended Kalman filter algorithm, observer, and PI observer studied in this paper, our results indicate that the robustness and the estimation accuracy of the three methods against modeling errors and measurement noises are similar. However, in the view of application and SOC accuracy, the PI observer has advantages over the observer and the EKF algorithm to be applied in BMS.
This paper reveals that SOC estimation accuracy depends critically on voltage measurement errors. While random noises in voltage measurements can be effectively filtered out, any bias or persistent errors will cause substantial deterioration on SOC estimation accuracy, which is a major reason for many algorithms to fail. Since voltage measurement accuracy varies substantially among BMS manufacturers, careful examination and enhancement of robustness of algorithms by design improvement and online parameter estimation are of essential importance. This paper utilizes some common scenarios of battery aging and parameter variations to study this issue and provide some related guidelines on how to select a robust method which has a strong tolerance towards voltage measurement errors.
The main contributions of this paper are in the following aspects which are essential for BMS design: () a clear analysis of the influence of each model parameter on the SOC estimation precision; () comparison of the robustness of various SOC estimation algorithms against model errors; () establishment of the quantitative relationship between measurement noise and SOC estimation accuracy.
The remainder of this paper is organized as follows. The battery model is introduced in Section 2. The three observerbased algorithms under study are described. Estimation accuracy of these algorithms is evaluated in Section 3 and their robustness is quantitatively compared. Section 4 investigates adaptability of the three algorithms against system uncertainties. Finally, some conclusions are drawn in Section 5.
2. Battery Model
Lithiumion battery is a complex, nonlinear electrochemical system. It is difficult to find a very accurate model to describe the complex changes in its charging and discharging processes. Extensive research on batteries has generated many battery models [15–19]. In these battery models, the equivalent circuit model is used commonly, including the Rint model, the firstorder RC model, and the secondorder RC model [9, 20]. In general, an accurate battery model is essential for precise battery state estimation. However, highfidelity battery models need more complex structures, more parameters, and carry high computational complexity. Therefore, it is necessary to find a compromise between accuracy of SOC estimation and simplicity of the model. In this paper, a firstorder RC model is employed, shown in Figure 1, in which the battery terminal voltage error is within ±20 mV, meeting the requirements of the estimation accuracy.
One of the most important state variables in BMS is state of charge (SOC), which is defined aswhere is the SOC at time , the initial value, the battery nominal capacity, the current at time , and the coulomb efficiency. The coulomb efficiency can be considered to be 1 [21, 22]. The influence of the selfdischarge on battery SOC estimation can be neglected.
According to Kirchhoff’s current and voltage laws, it is easy to obtain the following mathematical relationships:where is the Ohmic resistance, the polarization resistance, the polarization capacitor, the open circuit voltage, the terminal voltage, the voltage across , and the polarization voltage.
According to (1), it can be converted into the derivative equation as follows:
The relationship between the SOC and OCV is nonlinear. In this paper, this function is represented by piecewise linear segments,where is the slope of the th line segment and is the intercept. Their values are listed as in Table 1.

If the state of the battery is defined as , then the state equations of the battery in each segment are linearwhere , , , , , and .
3. SOC Estimation Algorithms and Their Robustness
In this paper, three closedloop observers are evaluated on their accuracy and robustness in SOC estimation. The key control principle of the closedloop observers is to use the difference between the measured terminal voltage and the estimated value as the input to the feedback module with a gain matrix to update the polarization voltage and SOC. The general structure of the closedloop observers is shown in Figure 2.
3.1. The Observer
The control theory was initiated by Zames in his seminal paper [23]. Since then extensive theoretical development, efficient solutions using frequency domain methods, state space models, numerical algorithms, and software packages have resulted in a rich treatise in this field [24–26]. Numerous successful applications have also been documented. In particular, numerical solutions to standard observers can be found by using the Robust Control Tool Box and LMI model in MATLAB.
The main advantages of the observer are as follows: () it is designed to attenuate disturbances of broader types than Kalman filters and Wiener filters which target Gaussian white noises; () it is robust against unstructured model uncertainty. However, as a worstcase robust design approach, it may be conservative, namely, nonoptimal, if the noise spectrum is actually known.
Consider a generic nonlinear battery system described bywhere , , , , , , and are coefficient matrices, which depend on the actual battery system, is the state, and is the output. Consider , , , , and .
The structure of the observer is where and are the estimates for and and is the gain vector of the observer.
Define the state estimation error . Then the error dynamics is
The goal of the observer design is disturbance attenuation: for a given (acceptable) sensitivity coefficient , design the observer gain such that the error system (10) is stable and that the following inequality is met under the zero initial condition:
The gain may be numerically solved by using the LMI approach [27]where , , for which the LMI Toolbox in MATLAB can be used. The derivation of (12) is similar to the derivation in [13].
Reference [13] has verified that the battery state space model (8) is observable. As a result, the LMI approach is applicable to design the observer to estimate the SOC of the battery system.
3.2. The PI Observer
The proportional control law is the simplest most common control law. However, it carries steadystate error which limits its applications alone. The integral controller is not only related to the size of the input bias but also related to the existence of time deviation. As long as the bias exists, the output will continue to accumulate until the bias is zero; it will stop accumulating. Therefore, the integral control can eliminate steadystate error. Although the integral control can eliminate residual error, it slows down the control action, and as such it has detrimental effect on stability and transient performance. By combining these two control actions, the proportional and integral (PI) control inherits the advantages of both.
The observability and stability of the PI observer estimating the battery SOC are proved in [28]. Therefore, the PI observer can be utilized to estimate the battery SOC.
The relationship of the input and the output in the PI observer iswhere is the proportional gain of the observer, the time constant of integration, the integral gain of the observer, the output of the feedback system in the observer, and the error between the estimated voltage and measured value, which is the input to the feedback loop in the observer.
The ZieglerNichols tuning method [29] is a heuristic method of tuning a PID controller. It was developed by Nichols and Ziegler. It is performed by setting the integral gain to zero. The proportional gain is then increased (from zero) until it reaches the ultimate gain , at which the output of the control loop oscillates with a constant amplitude. The ultimate gain and the oscillation period are used to set the proportional gain and the integral gain . According to [29], the proportional gain and the integral gain in the simulation are set as follows:
The PI observer takes the advantage of the proportional control to generate control action immediately and that of the integral control to eliminate residual error. The control parameters must be properly designed to achieve a desirable balance between dynamic quality and steadystate performance of the observer.
3.3. The Extended Kalman Filter
In 1960, Kalman published his famous paper describing a recursive solution to optimal discretetime linear filtering problems under additive and independent Gaussian noise [30]. A Kalman filter estimates the state of a dynamic system with a linear process model and measurement model [31]. Its extension to nonlinear systems employs local linearization, leading to the extended Kalman filter (EKF) [32].
In its application to the battery systems considered in this paper, the discretetime nonlinear system with additive noise is given by
By substituting (5) into (2), the terminal voltage can be expressed as
Therefore, coefficient matrices can be derived as
The EKF algorithm involves the following steps:(i)Prediction update:(ii)Measurement update: where and are the variances of the noises and , respectively, and is the updated state estimate from the previous estimate .
Implementation of the EKF is depicted by the flowchart in Figure 3.
3.4. Experiments
In this paper, one battery with a nominal capacity of 92 Ah, whose anode is lithium manganese oxide and whose cathode is graphite, is used in our experiments to verify parameter identification and SOC estimation accuracy. All experiments are accomplished on the battery testing platform, shown in Figure 4, which includes the Arbin Testing System, thermostat, PC, BMS, and a high precision multimeter. The charge and discharge tests are finished by the Arbin Instrument BT2000 battery testing system, whose maximum voltage and charge/discharge current are 5 V and 400 A, respectively, in which the current can be set to the low range (−1 A~1 A), the middle range (−50 A~50 A), or the high range (−400 A~400 A), according to the required maximum testing current. The controllable temperature range of the thermostat is −373.15 K~233.15 K. The BMS is manufactured by Huizhou Epower Electronic Co., Ltd. The digital multimeter with a 6.5digit resolution has a precision of 0.1 mV, so the voltage measured by this device is considered as true values. The noise is acquired by subtracting the BMS measured value from the digital multimeter measured value, whose statistical distribution is shown in Figure 5. The statistical characteristics of the terminal voltage measurement noise of the BMS are specified in Table 2.

The tests are composed of two parts. One part is parameter identification experiment and the other part is SOC estimation accuracy verification experiment. A 1/3 C constant current is used to charge the battery to 4.2 V, and the capacity in this paper refers to the maximum available capacity of the battery in current state of health if not figured out specifically. These experiments were first performed when the battery was brandnew. When the capacity of the battery was reduced to 74.5 Ah, the experiments were repeated to analyze the impact of the model parameter variations caused by battery aging. Data points are acquired at 1 Hz during the tests.
The battery model is built on the MATLAB/Simulink platform. The measured data including the voltage and current from the test bench are used as the input information to the three algorithms to estimate the SOC value. The SOC estimation accuracies are compared with the current integration values which serve as the SOC reference values.
The current integration method has two disadvantages: (a) it needs to know the initial SOC value in advance; (b) there is an accumulated error caused by the current measurement that is not accurate.
First, in this paper, the battery is discharged entirely in advance, so the initial SOC value is 0. Therefore, the initial SOC value is known. Second, the data used in the paper is obtained by the Arbin Instrument BT2000 battery testing system, and the precision of the current is 0.1% of the measuring range. The current range in experiments is set to the middle range (−50 A~50 A), so the precision of the current sampling is 0.1 A. When the battery is charged in 1/3 C current rate, the maximum SOC accumulated error in one full charging is , which can be neglected. Therefore, the SOC value calculated by the current integration method in instrument can serve as the SOC reference values in this paper, which overcomes the shortcomings of the current integration method effectively.
3.5. Verification
The SOC is estimated from experimental data by using the three estimation methods, respectively. Figure 6(a) shows the results of SOC estimation from the observer; Figure 6(b) shows the results of SOC estimation from the PI observer; and Figure 6(c) shows the results of SOC estimation from the EKF algorithm. The left axis is the SOC value, including the estimated value and experimental data, and the right axis is the SOC estimation error, and the axis is the time. All of three observers demonstrate good SOC estimation accuracy and convergence to the true SOC value, with a very short response time.
(a) observer
(b) PI observer
(c) EKF algorithm
(d) Three estimation methods
While the SOC estimation errors of the three estimation methods have the same trends, there are important differences among them, as indicated by Figure 6(d). In Figure 6(d), it is apparent that the EKF algorithm has the largest transient volatility, which is partially due to the initial value of , , and . After SOC estimation reaches steadystate, the observer has the largest SOC estimation errors, with the upper limit 1.67% and lower limit −0.79% implying that the observer is least accurate. This may be attributed to the fact that the observer is a conservative estimation method, which does not attenuate noise optimally.
The steadystate SOC estimation errors of the three estimation methods can maintain between the 2% band and −2% band when the SOC initial error is 20%, see Figure 6(d).
4. Robust Analysis of the Algorithms against System Uncertainties
In this section, the adaptability of the noise characteristics is discussed in detail. The noise is divided into two parts. One is the modelling error due to parameter changes caused by battery aging and the other is the terminal voltage measurement noise caused by BMS sampling accuracy. The adaptability of the model noise is discussed in Section 4.1, and the adaptability of the measurement noise is analyzed in Section 4.2.
4.1. The Model Parameter Perturbation
When the capacity of the battery with nominal capacity 92 Ah declines to 74.5 Ah, other parameters of the battery will change too. If the parameters for the new battery of capacity 92 Ah are used to estimate the SOC of the old battery of capacity 74.5 Ah, the SOC estimation accuracy of the three estimation methods will be affected significantly. Therefore, the impacts of the inaccurate battery parameters caused by battery aging to the SOC estimation accuracy are of essential importance.
In order to analyze the effects of parameter variations on SOC estimation accuracy, four cases are considered in our simulation.
Case 1. It is the process of estimating the SOC of the old battery using the parameters of the new battery.
Figure 7 shows the results of the SOC estimation in Case 1 using three algorithms. Using the observer, the maximum value of the SOC estimation error is 6.39%, and the minimum value of the SOC estimation error is 0.4%. Using the PI observer, the maximum value of the SOC estimation error is 6.33%, and the minimum value of the SOC estimation error is 0.42%. Using the EKF algorithm, the maximum value of the SOC estimation error is 6.33%, and the minimum value of the SOC estimation error is 1.54%.
(a) SOC curves with three algorithms and experiment
(b) SOC estimation error curves
Case 2. It is the process of estimating the SOC of the old battery by updating the battery’s parameters used in Case 1 to the old battery’s parameters except the SOCOCV curve.
Figure 8 shows the results of the SOC estimation in Case 2 using three algorithms. Using the observer, the maximum value of the SOC estimation error is 1.69%, and the minimum value of the SOC estimation error is −4.96%. Using the PI observer, the maximum value of the SOC estimation error is 1.59%, and the minimum value of the SOC estimation error is −4.42%. Using the EKF algorithm, the maximum value of the SOC estimation error is 1.59%, and the minimum value of the SOC estimation error is −4.41%.
(a) SOC curves with three algorithms and experiment
(b) SOC estimation error curves
Case 3. It is the process of estimating the SOC of the old battery by updating only the SOCOCV curve to the old battery’s parameters, while the other parameters are the same as in Case 1.
Figure 9 shows the results of the SOC estimation in Case 3 using three algorithms. Using the observer, the maximum value of the SOC estimation error is 8.54%, and the minimum value of the SOC estimation error is 0.38%. Using the PI observer, the maximum value of the SOC estimation error is 8.38%, and the minimum value of the SOC estimation error is 0.40%. Using the EKF algorithm, the maximum value of the SOC estimation error is 8.38%, and the minimum value of the SOC estimation error is 1.40%.
(a) SOC curves with three algorithms and experiment
(b) SOC estimation error curves
Case 4. It is the process of estimating the SOC of the old battery by updating all the parameters to the old battery’s parameters.
Figure 10 shows the results of the SOC estimation in Case 4 using three algorithms. Using the observer, the maximum value of the SOC estimation error is 1.72%, and the minimum value of the SOC estimation error is −1.69%. Using the PI observer, the maximum value of the SOC estimation error is 1.64%, and the minimum value of the SOC estimation error is −1.63%. Using the EKF algorithm, the maximum value of the SOC estimation error is 1.63%, and the minimum value of the SOC estimation error is −1.62%.
(a) SOC curves with three algorithms and experiment
(b) SOC estimation error curves
Discussions. Note thatwhere is the measured terminal voltage and is the estimated terminal voltage.
From (21), since the measured terminal voltage is known, the estimated terminal voltage determining the terminal voltage error affects the accuracy and convergence time of the three SOC estimation methods. The estimated terminal voltage includes open circuit voltage , polarization voltage , and , as (20) shows. Therefore, they have obvious effects on estimation accuracy and robustness of the three SOC estimation methods, as (22) shows, demonstrated by Figures 7–10.
The parameters changes from the new battery to the old battery’s parameters are shown as in Table 3.

When the battery is aging from 92 Ah to 74.5 Ah, during a complete charging process in 1/3 C rate, the average polarization voltage increase is 16 mV, and the average Ohmic voltage increase is 7 mV. However, the average OCV decrease is 2.5 mV.
When we use the parameters of the old battery to estimate the SOC of the old battery, the SOC estimation error is only caused by the battery model error. The terminal voltage error caused by the battery model is considered to be , and the SOC estimation result is shown in Figure 10.
However, using the parameters of the new battery to estimate the SOC of the old battery, which means that the parameters are not updated, the estimated value of the terminal voltage is mV smaller than the measured terminal voltage value. This causes a great SOC estimation error, whose maximum is between 2% and 8%, as shown in Figures 7 and 10. By updating the Ohmic resistance and the polarization resistance to the true values of the old battery, the estimated value of the terminal voltage is mV smaller than the measured terminal voltage value. This causes only a little SOC estimation error, whose minimum is less than −2%, as shown in Figures 8 and 10. If the OCV is updated, but not other parameters, to the value of the old battery, the estimated value of the terminal voltage is mV smaller than the measured terminal voltage value. Because mV is larger than mV, estimation errors by using the updated OCV of old battery are worst among all cases, whose maximum is more than 8%, larger than , as shown in Figures 9 and 10. The clear comparison is shown in Table 4.
In our recent studies [33], it is shown that SOC estimation accuracy is dependent on the curve of SOCOCV significantly, if the SOCOCV curve varies substantially. In this study during the progress of the battery aging, the SOCOCV curve does not change much. The experimental results are included in Figure 11. As a result, the impact of aging on the OCV has negligible effect on SOC estimation accuracy, as analyzed in Section 4.1. In contrast, effect of aging on other model parameters plays much more prominent roles in SOC estimation accuracy.
In addition, we may conclude from comparing Figures 7, 8, 9, and 10 that the SOC estimation error when using the parameters of the new battery to estimate the SOC of the old battery is not a linear superposition of the SOC estimation error caused by the changes of the resistance and capacitance and the changes of the curve of SOCOCV. This is clearly indicated in Figure 12, taking the SOC estimation results of the PI observer as an example.
4.2. The Terminal Voltage Measurement Errors
In order to find the relationship between the SOC estimation accuracy and the statistical characteristics of the terminal voltage measurement noise, we further add measurement noises with different means and variances to the experimental data of the terminal voltage in Section 3.5. There are different SOC estimation results using different estimation methods.
(i) The Observer. When the mean of the measurement noise changes from −6 to 4 and standard deviation of the measurement noise increases from 0 to 10 mV, the SOC accuracy is depicted in Figure 13.
If the SOC estimation error is required to be less than 2%, the mean and standard deviation of the measurement noise must be confined from −3.172 mV to 1.172 mV and 0 to 3.43 mV, respectively.
(ii) The PI Observer. When the mean of the measurement noise changes from −6 to 4 and standard deviation of the measurement noise increases from 0 to 10 , the SOC accuracy is shown in Figure 14.
If the SOC estimation error is required to be less than 2%, the mean and standard deviation of the measurement noise must be confined from −3.36 mV to 1.172 mV and 0 to 5.15 mV, respectively.
(iii) The EKF Algorithm. When the mean of the measurement noise changes from −6 to 4 and standard deviation of the measurement noise increases from 0 to 10 mV, the SOC accuracy is illustrated in Figure 15.
If the SOC estimation error is required to be less than 2%, the mean and standard deviation of the measurement noise must be confined from −3.36 mV to 1.3 mV, 0 to 3.535 mV, respectively.
(iv) Discussions. The principle of the observer is similar to the Luenberger observer, whose feedback gain is a proportional gain. It controls quite aggressively with an expected fast response. However, it shows certain levels of overreaction, causing nonsmooth contours of SOC estimation errors. In contrast the PI observer with its integral part acts as a signal smoother (a lowpass filter type), so the contours of its SOC estimation errors are smoother than that of the observer. Note that the gain of the EKF algorithm is updated timely, so the contours of its SOC estimation errors are the smoothest.
To capture transient errors more concretely, in Figures 13–15, we define the area enclosed by the axis and the contour line as the total absolute error. The total absolute error ratio is then calculated as the total absolute error divided by the area which is equal to the product of the maximum value of axis and maximum value of axis in the figure. The ratios of the three observers are shown in Figure 16.
Under the same SOC estimation accuracy, the adaptation range against the measurement noise of the PI observer is the biggest and that of the observer is the smallest, as Figure 16 shows. It is worth noting that Figures 13, 14, and 15 are not symmetrical about the mean, result from the total effect of the terminal voltage noise mean and the inaccurate parameters on the SOC estimation accuracy.
5. Conclusions
In this paper, several typical SOC estimation algorithms including the observer, PI observer, and extended Kalman filter are applied to estimate SOC. By considering four categories of variations of model parameters caused by battery aging and studying the influence of each category on the SOC estimation precision, we compare the algorithms in terms of their robustness against modeling errors. In addition, their tolerance to voltage measurement errors is quantitatively evaluated.
The robustness and the estimation accuracy of the three methods against modeling errors and measurement noises are similar. However, the observer needs to know the gain , which needs to calculate the Linear Matrix Inequalities in advance; the EKF algorithm needs to know the distribution of the measurement noise, which is difficult to be obtained in fact. Compared to these two methods, the PI observer can acquire the proportional gain and the integral gain depending on the experiences. Therefore, in the view of application and SOC accuracy, the PI observer has advantages over the observer and the EKF algorithm to be applied in BMS. Through simulation results we reach the following conclusions:(1)The Ohmic resistance , polarization resistance , and the open circuit voltage OCV are the key parameters affecting SOC estimation accuracy. However, the polarization capacitor , which is an important parameter, only influences the dynamic response characteristics of SOC estimation but does not have noticeable effects on the steadystate accuracy of SOC estimation.(2)Under the same SOC estimation accuracy and the robustness against modeling errors and measurement noises, the PI observer has advantages over the observer and the EKF algorithm to be applied in BMS.(3)The relationship between SOC estimation accuracy and voltage measurement errors has been resolved, and some related guidelines on how to select a robust method which has a strong tolerance against voltage measurement errors are provided.
There are several important related topics that are not covered in this paper. First, optimal design of PI observers requires essential statistical information on measurement noises and individualized models. Learning algorithms for noise characterizations and parameter estimation can lead to adaptive PI observers with improved SOC estimation accuracy. Furthermore, implementation of SOC estimators in battery management systems on electric vehicles encounters hardware and computational complexity constraints. For example, sensor precision levels and sampling rates will limit data flow rates and reliability. Analysis of the influence of synchronous sampling, sampling rates, asynchronous sampling, and quantization on SOC estimation accuracy will be pursued in our future studies.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work was supported by National Natural Science Foundation of China under Grant no. 51477009.
References
 K. W. E. Cheng, B. P. Divakar, H. Wu, K. Ding, and H. F. Ho, “Batterymanagement system (BMS) and SOC development for electrical vehicles,” IEEE Transactions on Vehicular Technology, vol. 60, no. 1, pp. 76–88, 2011. View at: Publisher Site  Google Scholar
 V. Pop, H. J. Bergveld, D. Danilov, P. P. L. Regtien, and P. H. L. Notten, Battery Management Systems: Accurate StateofCharge Indication for BatteryPowered Applications, Springer, New York, NY, USA, 2008.
 S. Piller, M. Perrin, and A. Jossen, “Methods for stateofcharge determination and their applications,” Journal of Power Sources, vol. 96, no. 1, pp. 113–120, 2001. View at: Publisher Site  Google Scholar
 S. Lee, J. Kim, J. Lee, and B. H. Cho, “Stateofcharge and capacity estimation of lithiumion battery using a new opencircuit voltage versus stateofcharge,” Journal of Power Sources, vol. 185, no. 2, pp. 1367–1373, 2008. View at: Publisher Site  Google Scholar
 J. Xu, C. C. Mi, B. Cao, and J. Cao, “A new method to estimate the state of charge of lithiumion batteries based on the battery impedance model,” Journal of Power Sources, vol. 233, pp. 277–284, 2013. View at: Publisher Site  Google Scholar
 M. Mastali, J. VazquezArenas, R. Fraser, M. Fowler, S. Afshar, and M. Stevens, “Battery state of the charge estimation using Kalman filtering,” Journal of Power Sources, vol. 239, pp. 294–307, 2013. View at: Publisher Site  Google Scholar
 C. P. Zhang, J. C. Jiang, W. G. Zhang, and S. M. Sharkh, “Estimation of state of charge of lithiumion batteries used in HEV using robust extended Kalman filtering,” Energies, vol. 5, no. 4, pp. 1098–1115, 2012. View at: Publisher Site  Google Scholar
 Z. Chen, Y. Fu, and C. C. Mi, “State of charge estimation of lithiumion batteries in electric drive vehicles using extended Kalman filtering,” IEEE Transactions on Vehicular Technology, vol. 62, no. 3, pp. 1020–1030, 2013. View at: Publisher Site  Google Scholar
 R. Xiong, H. He, F. Sun, and K. Zhao, “Evaluation on state of charge estimation of batteries with adaptive extended kalman filter by experiment approach,” IEEE Transactions on Vehicular Technology, vol. 62, no. 1, pp. 108–117, 2013. View at: Publisher Site  Google Scholar
 G. L. Plett, “Extended Kalman filtering for battery management systems of LiPBbased HEV battery packs: Part 3. State and parameter estimation,” Journal of Power Sources, vol. 134, no. 2, pp. 277–292, 2004. View at: Publisher Site  Google Scholar
 G. L. Plett, “Extended Kalman filtering for battery management systems of LiPBbased HEV battery packs: part 2. Modeling and identification,” Journal of Power Sources, vol. 134, no. 2, pp. 262–276, 2004. View at: Publisher Site  Google Scholar
 G. L. Plett, “Extended Kalman filtering for battery management systems of LiPBbased HEV battery packs. Part 1. Background,” Journal of Power Sources, vol. 134, no. 2, pp. 252–261, 2004. View at: Publisher Site  Google Scholar
 F. Zhang, G. Liu, L. Fang, and H. Wang, “Estimation of battery state of charge with ${H}_{\infty}$ observer: applied to a robot for inspecting power transmission lines,” IEEE Transactions on Industrial Electronics, vol. 59, no. 2, pp. 1086–1095, 2012. View at: Publisher Site  Google Scholar
 X. Hu, S. Li, H. Peng, and F. Sun, “Robustness analysis of StateofCharge estimation methods for two types of Liion batteries,” Journal of Power Sources, vol. 217, pp. 209–219, 2012. View at: Publisher Site  Google Scholar
 M. Chen and G. A. RincónMora, “Accurate electrical battery model capable of predicting runtime and IV performance,” IEEE Transactions on Energy Conversion, vol. 21, no. 2, pp. 504–511, 2006. View at: Publisher Site  Google Scholar
 L. Gao, S. Liu, and R. A. Dougal, “Dynamic lithiumion battery model for system simulation,” IEEE Transactions on Components and Packaging Technologies, vol. 25, no. 3, pp. 495–505, 2002. View at: Publisher Site  Google Scholar
 V. H. Johnson, “Battery performance models in ADVISOR,” Journal of Power Sources, vol. 110, no. 2, pp. 321–329, 2002. View at: Publisher Site  Google Scholar
 T.S. Dao, C. P. Vyasarayani, and J. McPhee, “Simplification and order reduction of lithiumion battery model based on porouselectrode theory,” Journal of Power Sources, vol. 198, pp. 329–337, 2012. View at: Publisher Site  Google Scholar
 A. Szumanowski and Y. Chang, “Battery management system based on battery nonlinear dynamics modeling,” IEEE Transactions on Vehicular Technology, vol. 57, no. 3, pp. 1425–1432, 2008. View at: Publisher Site  Google Scholar
 H. He, R. Xiong, X. Zhang, F. Sun, and J. Fan, “Stateofcharge estimation of the lithiumion battery using an adaptive extended kalman filter based on an improved Thevenin model,” IEEE Transactions on Vehicular Technology, vol. 60, no. 4, pp. 1461–1469, 2011. View at: Publisher Site  Google Scholar
 G. L. Wu, R. G. Lu, C. Zhu, and C. C. Chan, “An improved Amperehour method for battery state of charge estimation based on temperature, coulomb efficiency model and capacity loss model,” in Proceedings of the IEEE Vehicle Power and Propulsion Conference (VPPC '10), pp. 1–4, Lille, France, September 2010. View at: Publisher Site  Google Scholar
 M.H. Chang, H.P. Huang, and S.W. Chang, “A new state of charge estimation method for LiFePO_{4} battery packs used in robots,” Energies, vol. 6, no. 4, pp. 2007–2030, 2013. View at: Publisher Site  Google Scholar
 G. Zames, “Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses,” IEEE Transactions on Automatic Control, vol. 26, no. 2, pp. 301–320, 1981. View at: Publisher Site  Google Scholar  MathSciNet
 J. Huang and C.F. Lin, “Numerical approach to computing nonlinear ${H}_{\infty}$ control laws,” Journal of Guidance, Control, and Dynamics, vol. 18, no. 5, pp. 989–996, 1995. View at: Publisher Site  Google Scholar
 J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “Statespace solutions to standard ${\text{H}}_{\text{2}}$ and ${H}_{\infty}$ control problems,” IEEE Transactions on Automatic Control, vol. 34, no. 8, pp. 831–847, 1989. View at: Publisher Site  Google Scholar
 J. W. Helton and A. Sideris, “Frequency response algorithms for ${H}_{\infty}$ optimization with time domain constraints,” IEEE Transactions on Automatic Control, vol. 34, no. 4, pp. 427–434, 1989. View at: Publisher Site  Google Scholar
 K. Zhou and J. Doyle, Essentials of Robust Control, Prentice Hall, Hoboken, NJ, USA, 1998.
 J. Xu, C. C. Mi, B. Cao, J. Deng, Z. Chen, and S. Li, “The state of charge estimation of lithiumion batteries based on a proportionalintegral observer,” IEEE Transactions on Vehicular Technology, vol. 63, no. 4, pp. 1614–1621, 2014. View at: Publisher Site  Google Scholar
 N. B. Nichols and J. G. Ziegler, “Optimum settings for automatic controllers,” Journal of Dynamic Systems, Measurement, and Control, vol. 115, no. 2, pp. 220–222, 1993. View at: Publisher Site  Google Scholar
 R. E. Kalman, “A new approach to linear filtering and prediction problems,” Journal of Fluids Engineering, vol. 82, no. 1, pp. 35–45, 1960. View at: Google Scholar
 B. Ristic, S. Arulampalam, and N. Gordon, Beyond the Kalman Filter, Artech House, Boston, Mass, USA, 2004.
 M. Hoshiya and E. Saito, “Structural identification by extended Kalman filter,” Journal of Engineering Mechanics, vol. 110, no. 12, pp. 1757–1770, 1984. View at: Publisher Site  Google Scholar
 C. P. Zhang, L. Y. Wang, X. Li et al., “Robust and adaptive estimation of state of charge for lithiumion batteries,” IEEE Transactions on Industrial Electronics, vol. 62, no. 8, pp. 4948–4957, 2015. View at: Publisher Site  Google Scholar
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