2. Jilin University, Changchun 130012, China;
3. Beijing Institute for Control and Electronic Technology, Beijing 100038, China
LITHIUMION batteries have high levels of energy and power density among electrochemical batteries. These attributes make them suitable to be the energy storage system in electric vehicle, hybrid electric vehicle, and plugin hybrid electric vehicle (EV/HEV/PHEV) [1]. One of the important requirements in automotive batteries is to monitor their real time state of charge (SOC). SOC of the battery is not readily available during charging and discharging cycles. SOC values have to be predicted based on the measured terminal voltage and current [2].
SOC reflects the remaining capacity that can be drawn from the battery pack and is used to ensure an optimum control of charging and discharging processes. However, several factors of SOC determination for batteries such as hysteresis phenomena, flat characteristic of open circuit voltage (OCV) over SOC, and limited voltage measurement accuracy can result in SOC estimation error [3].
In recent years, great effort has been exercised to improve the accuracy of SOC estimation. There are two main kinds of methods for SOC estimation at present. One is calculating SOC by chargedischarge current and OCV based on energy conservation and inner physical characteristics of the lithiumion cell. The classical approach of current integration (Coulomb counting) which samples the battery current and computes the accumulated charge and discharge to estimate SOC, is simple and inexpensive to implement, but it cannot solve the problems of accumulative error and inaccurate initial values. Open circuit voltage commonly needs relatively long rest periods in applications. Since the rest periods will only occur from time to time, the open circuit voltage measurement is usually combined with other techniques [4]. Other methods, such as discharge test or internal resistance characteristic, are usually considered as laboratory methods. The obvious disadvantages of nonmodel based estimation methods are the limited accuracy, long estimation time and only offline application in spite of the advantages as simple principle and easy implementation [5].
The other is indirectly estimating SOC based on the mathematical model of the lithiumion cell. State estimation with Luenberger observer has already been analyzed [6]. Kalman filter (KF) approach with its knowledge about statistical characteristics of process and measurement noise is intensively studied. Extended Kalman filter (EKF) and sigma point Kalman filter (SPKF) are two improvements to the classical KF which are investigated for nonlinear systems [7][9]. The support vector machine (SVM) and sliding mode observer (SMO) can simulate the complicated battery dynamics, but the performance of these techniques is sensitive to training data. Therefore, it is not appropriate for the battery online application [10], [11].
The wellknown inputtostate stability (ISS) property of deterministic systems originated in [12] and was investigated quite intensively in recent years [13], [14]. Especially, some concepts of ISS observer have appeared in [15]. Considering the parameter uncertainties and the unmodeled dynamics as disturbance inputs [16], analyzed the robust stability of the closed loop estimation error system based on the ISS theory, and gave estimation system parameter adjustment guidelines on this basis.
It is obvious that proper design, engineering and operation of these battery systems require an appropriate battery model [6]. In [17], we can see that the battery parameters, such as internal resistance, can be different in different charging and discharging rates or other conditions. So it is necessary for us to build a more accurate battery model such as parametervarying model with hysteresis effect.
The hysteresis constitutes a very significant internal variable for some battery technologies. Its value depends on the battery process, which can be divided into three possibilities: charging, discharging and charging/discharging transient. There is not yet a clear physical explanation for the hysteresis phenomenon, although the domain theory relates the hysteresis to separated regions inside the electrode that lead to different electrode equilibrium potentials during the charging and discharging [18], [19]. The hysteresis phenomenon also modifies the model parameters of the battery and therefore its dynamic response [20]. So we consider the influence of hysteresis on the OCV of the battery in charging/ discharging transient.
Therefore, the future study on SOC estimation will focus not only on highly accurate and strongly robust estimation methods, but also on modeling the cell accurately by means of studying the battery characteristics.
The remaining part of the paper is organized as follows. In Section Ⅱ an equivalent circuit model with hysteresis (ECMH) is presented to characterize the dynamics of lithiumion cell, and the detail procedures to identify the battery ECM parameters are also explained. In Section Ⅲ, the nonlinear observer based on the ISS theory is proposed to estimate SOC of lithiumion cell. The numerical algorithm for SOC estimation based on the nonlinear observer is verified by the cosimulation in Matlab/Simulink and AMESim respectively in Section Ⅳ. Finally, conclusions are given in Section Ⅴ.
Ⅱ. HYSTERESIS MODELING AND PARAMETER IDENTIFICATIONIn this section, the simplified equivalent circuit model with the hysteresis characteristic of lithiumion cell will be modeled. Timevarying elements in each RC network and the internal resistance are considered. The hysteresis element in the equivalent circuit model shows the nonlinear characteristics of the components inside the SOC estimating system [21].
A. Hysteresis Modeling of Lithiumion CellThe motion of the lithiumion in the cell is shown in Fig. 1. The equivalent circuit model with hysteresis characteristic of lithiumion cell is shown in Fig. 2.
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Fig. 1 Lithiumion battery schematics. 
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Fig. 2 Equivalent circuit model with hysteresis. 
This model consists of an ohmic internal resistance
According to Kirchhoff's law, the terminal voltage can be given by
$ \begin{equation} \label{eq:1} V=V_{\textrm{oc}}(SOC)+V_{1}+V_{2}+V_{h}+R_{\textrm{e}}(i)\cdot{i} \end{equation} $  (1) 
and the electrochemical polarization voltage
$ \begin{equation} \label{eq:2} \dot{V}_{1}={\frac{1}{R_{1}C_{1}}}V_{1}+{\frac{1}{C_{1}}}i \end{equation} $  (2) 
$ \begin{equation} \label{eq:3} \dot{V}_{2}={\frac{1}{R_{2}C_{2}}}V_{2}+{\frac{1}{C_{2}}}i \end{equation} $  (3) 
where
The OCV, as a function of SOC, is simplified as
$ \begin{equation} \label{eq:4} V_{\textrm{oc}}(SOC)={\lambda}\cdot{V_{\textrm{oc}}.{\textrm{C}}(SOC)}+(1{\lambda})\cdot{V_{\textrm{oc}}.{\textrm{D}}(SOC)} \end{equation} $  (4) 
where
The hysteresis voltage
$ \begin{equation} \label{eq:5} \dot{V}_{\textrm{h}}=(1e.{\left{\kappa}\cdot{i}\right})V_{\textrm{h}}\pm(1e.{\left{\kappa}\cdot{i}\right})H \end{equation} $  (5) 
where
$ \begin{equation} \label{eq:6} H={\frac{1}{2}}[V_{\textrm{oc}}.{\textrm{C}}(SOC)V_{\textrm{oc}}.{\textrm{D}}(SOC)]. \end{equation} $  (6) 
Choose state variables as
$ \begin{equation} \label{eq:7} \dot{x}=A(i)\cdot{x}+B(i)\cdot{u} \end{equation} $  (7) 
$ \begin{equation} \label{eq:8} y=g(x)+D(i)\cdot{u} \end{equation} $  (8) 
where
$ B(i)=\left[\begin{array}{cc} \frac{1}{{Q}_{N}}&0 \\ 0&1e.{\left{\kappa}\cdot{i}\right} \\ \frac{1}{{C}_{1}(i)}&0\\ \frac{1}{{C}_{2}(i)}&0 \end{array}\right]$, $D(i)=\begin{bmatrix}R_{e}(i)&0\end{bmatrix} $ 
The cell characterization and validation tests are carried out on single cell. The 2.5 Ah cells are used for experiment in this paper. The OCV boundary curve test and hysteresis test are done on a Neware battery test device BTS5V6A that comprises a voltage measurement accuracy of
The model parameterization test schedule includes cell capacity test and OCV test. The first charging circulation of the cells makes the cells' chemical characteristics fully activated.
During the first OCV test (boundary curve test), the cells are gradually discharged in 10
The charging and discharging idle curve of the battery test is shown in Fig. 4, in which there are two curves. One is the charging segment ① and charging idle segment ②. The other is the discharging segment ③ and idle segment ④.
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Fig. 3 Configuration of the battery test bench. 
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Fig. 4 Battery charge and discharge rest curve. 
Segment ① shows the charging process with SOC 0 % to 50 %; segment ② shows that the battery has been rested for 3 hours;
Segment ③ shows the discharging process with SOC 100 % to 50 %; segment ④ also shows that the battery has been rested for 3 hours;
Segment ④ is calibrated as shown in Fig. 5, where the initial point is set at zero for simply curve fitting. Here exponential function is taken to fit parameters of battery terminal voltages.
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Fig. 5 Fitting curve of Segment ④. 
According to the circuit relations in Fig. 1 and (1)(3), we can get the battery terminal voltage output equation as follows
$ \begin{equation} \label{eq:9} V=V_{\textrm{oc}}+R_{\textrm{e}}i+R_{1}i(1e.{t/{\tau_{1}}})+R_{2}i(1e.{t/{\tau_{2}}}). \end{equation} $  (9) 
Then we choose the exponential fitting function expression as
$ \begin{equation} \label{eq:10} V=k_{0}+k_{1}e.{{\lambda}_{1}t}+k_{2}e.{{\lambda}_{2}t} \end{equation} $  (10) 
where
In order to study the effect of charge and discharge current of battery on the model parameters, we had taken 16 sets of charging/discharging experiments, and 100 times per set. Discharging currents are1600 mA to200 mA, and charging currents are 200 mA to 1600 mA, with 200 mA interval current. Fitting the charging/discharging rest curve, the values of equivalent internal resistance, polarization resistances and polarization capacitances are shown in Table Ⅰ.
The second OCV test is for investigating the hysteresis effect.
Gradual partial cycles are applied to achieve the intermediate OCV curves between the OCV boundary curves as shown in Fig. 6. At first the cells (SOC =
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Fig. 6 Charging/discharging OCV curves. 
From Fig. 6 we can see that the OCV value changes through the hysteresis belt from the charging curve into discharging curve. We can get the maximum value of
To verify the accuracy of this parametervarying second order RC ECHM, build the battery simulation model in MATLAB/Simulink. Charge the battery from SOC
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Fig. 7 The parametervarying second order RC equivalent circuit model with hysteresis simulation verification results. 
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Fig. 8 Battery model error. 
To verify the accuracy and reliability of the model in complex process, design a currentvarying procedure as shown in Fig. 9, which contains current value switching and chargingdischarging switching. We contrast the voltage output of this model and a first order RC model with the measured values, and get the results as in Figs. 10 and 11.
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Fig. 9 Custom current conditions. 
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Fig. 10 Battery model verification results. 
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Fig. 11 Battery model error under custom current conditions. 
The voltage output values of the two models and the measured voltage are shown in Fig. 12. And the output errors of the two models are shown in Fig. 13. And those figures tell us obviously that the errors of parametervarying model with hysteresis characteristics mainly remain under 10 mV, which is a great improvement in accuracy than the first order one. In addition, the tracking performance progresses a lot as well especially when chargingdischarging switching.
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Fig. 12 ISS based observer simulation model block diagram. 
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Fig. 13 The SOC estimation effect validation of the nonlinear observer. 
The simulation results show that the accuracy and other model characters could be improved by adding into the model the consideration of the difference between charging and discharging process and the influence of the current on the coefficients. Besides, the errors of the two models both increase when chargingdischarging switching, but the addition of hysteresis influence would make the model better simulate the real response when switching and keep the error at about 10 mV.
Ⅲ. ISS BASED NONLINEAR OBSERVERIn this section, the parameter uncertainties and unmodeled dynamics are considered as disturbance inputs, and it can be proved that the SOC estimating closedloop error system has robust stability based on the ISS theory.
A. ISS Stability AnalysisTo analyze the ISS stability of the SOC estimator of battery, we rewrite the dynamic system (7) and (8) as follows
$ \begin{equation} \label{eq:11} \dot{x}=A(i)\cdot{x}+B(i)\cdot{u}+w \end{equation} $  (11) 
$ \begin{equation} \label{eq:12} y=g(x)+D(i)\cdot{u}+v \end{equation} $  (12) 
where
The nonlinear observer is designed in the form of
$ \begin{equation} \label{eq:13} \dot{\hat{x}}=A(i)\cdot{\hat{x}}+B(i)\cdot{u}+L(y\hat{y}) \end{equation} $  (13) 
$ \begin{equation} \label{eq:14} \hat{y}=g(\hat{x})+D(i)\cdot{u} \qquad\qquad \end{equation} $  (14) 
where
The observer error is defined as follows
$ \begin{equation} \label{eq:15} e=x\hat{x} \end{equation} $  (15) 
then the error dynamics are described by
$ \begin{equation} \label{eq:16} \begin{split} \dot{e}&=A(i){x}+B(i){u}+wA(i){\hat{x}}B(i){u}L(y\hat{y})\\ &=A(i){e}+wL[g(x)g(\hat{x})]Lv \end{split} \end{equation} $  (16) 
where
$ \begin{equation} \label{eq:17} g(x)g(\hat{x})=C(x)\cdot{x}C(\hat{x})\cdot{\hat{x}}+o(x). \end{equation} $  (17) 
Substituting (17) into (16) leads to
$ \begin{equation} \label{eq:18} \begin{split} \dot{e}&=[A(i)LC(\hat{x})]e+wL[\Delta{Cx}+o(x)+v]\\ &=[A(i)LC(\hat{x})]e+wLw_{1} \end{split} \end{equation} $  (18) 
where
Let the candidate Lyapunov function be in the form
$ \begin{equation} \label{eq:19} V={\frac{1}{2}}e.{ T}Pe \end{equation} $  (19) 
with positive definite symmetric matrix
The time derivative of (19) is
$ \begin{equation*} \label{eq:20} \begin{split} ~\dot{V} = &{\frac{1}{2}}[\dot{e}.T Pe+e.T P\dot{e}]\\ = &{\frac{1}{2}}[e.T(A(i).TPC(x).T L.T P+PA(i)PLC(x))e]\\ &+{\frac{1}{2}}(w.T Pew_{1}.T L.T Pe+e.T Pwe.T PLw_{1}).~~~~(20) \end{split} \end{equation*} $  (20) 
Let
$ \begin{align} \dot{V} = &{\frac{1}{2}}[e.T(A(i).TPC(x).TQ.T+PA(i)QC(x))e]\nonumber\\ &+(P_{1}e.T)v_{0}. \end{align} $  (21) 
Apply Young's Inequality [15] to achieve
$ \begin{equation} \label{eq:22} (P_{1}e.T)v_{0}\leq{\kappa_{1}e.TPe+{\frac{1}{4\kappa_{1}}}v_{0}.Tv_{0}} \end{equation} $  (22) 
where
Then (21) becomes
$ \begin{align} \label{eq:23} \dot{V}={\frac{1}{2}}\Big[&e.T(A(i).TPC(x).TQ.T+PA(i)\nonumber\\ &QC(x)+2{\kappa_{1}}P)e\Big]+{\frac{1}{4\kappa_{1}}}v_{0}.Tv_{0}. \end{align} $  (23) 
Choose
$ \begin{equation} \label{eq:24} A(i).TPC(x).{T}Q.T+PA(i)QC(x)+2{\kappa_{1}}P\leq{{\kappa_{2}}P} \end{equation} $  (24) 
with
Then (23) becomes
$ \begin{equation} \label{eq:25} \dot{V}\leq{{\frac{1}{2}}{\kappa_{2}}e.TPe+{\frac{1}{4\kappa_{1}}}v_{0}.Tv_{0}}. \end{equation} $  (25) 
Substitute (19) into (25), we have
$ \begin{equation} \label{eq:26} \dot{V}\leq{{\kappa_{2}}V+{\frac{1}{4\kappa_{1}}}v_{0}.Tv_{0}}. \end{equation} $  (26) 
Using the fact that [13]
$ \begin{equation} \label{eq:27} {{\frac{1}{2}}{\lambda_{\min}(P)}{\e\}.{2}}\leq{{\frac{1}{2}}{e.TPe}} \leq{{\frac{1}{2}}{\lambda_{\textrm{max}}(P)}{\e\}.{2}} \end{equation} $  (27) 
where
From (25) and (26), we have
$ \begin{equation} \label{eq:28} \dot{V}\leq{{\frac{1}{2}}{\kappa_{2}}{\e\}.{2}\sup{\lambda_{\textrm{max}}(P)} +{\frac{1}{4\kappa_{1}}}{\{v_{0}}\}.{2}}. \end{equation} $  (28) 
Upon multiplication of (26) by
$ \begin{equation} \label{eq:29} {\frac{d}{dt}}(Ve.{{\kappa_{2}}t})\leq{{\frac{1}{4\kappa_{1}}}v_{0}.Tv_{0}e.{{\kappa_{2}}t}}. \end{equation} $  (29) 
Integrating (29) over the interval
$ \begin{equation} \label{eq:30} V(t)=V(0)e.{{\kappa_{2}}t}+{\frac{1}{4\kappa_{1}}}\int_{0}.{t}{{e.{{\kappa_{2}(t\tau)}}}v_{0}.{T}v_{0}d\tau} \end{equation} $  (30) 
Hence, the properties of the error dynamics of the designed observer (13) and (14) are described as follows.
Theorem 1: Suppose the following:
1)
2) The observer gain
Then, the error dynamic property of the observer (13) and (14) are as follow:
1) Inputtostate stable, if
2) Exponentially stable with
Proof: It follows from (25) to (30), which shows that the error dynamics admit the inputtostate stability property if the model error
Taking
Remark 1: Now we give some discussions on the parameters
$ \begin{equation} \label{eq:31} {\{e(\infty)}\}.{2}\leq{({\frac{{{{\{v_{0}}\}_{\infty}.{2}}}}{4\kappa_{1}}})\lim_{t\rightarrow{\infty}}\int_{0}.{t}{e.{\kappa_{2}(t\tau)}d\tau}} \end{equation} $  (31) 
i.e.,
$ \begin{equation} \label{eq:32} {\{e(\infty)}\}.{2}\leq{{\frac{{{{\{v_{0}}\}_{\infty}.{2}}}}{4\kappa_{1}\kappa_{2}}}}. \end{equation} $  (32) 
Hence, one may choose larger
Remark 2: Equation (32) gives just an upper bound of the estimation error offset, if the bound of the model error is given. The real offset could be much smaller, due to the multiple use of inequalities in the above derivation.
B. Implementation Issues and SOC Observer DesignAccording to Theorem 1 and Remark 1, a systematic procedure is given to design the ECHM based observer in the form of (13), Equation (14) as follows, and the observer gain satisfies (24).
Step 1: Choose the parameter
Step 2: Choose the parameter
Step 3: Determine the observer gain
Step 4: Use (32) to compute the estimated upper bound of the offset for a given model error bound, and check if the offset bound is acceptable;
Step 5: If the offset bound is acceptable, end the design procedure. If not acceptable, go to Step 2.
It is well known that getting model error bounds is in general very difficult, if not impossible. As mentioned in Remark 2, for a given model error bound, (32) gives just an upper bound of the estimation error offset, but the calculation of static error after continuous inequality arithmetic might be much larger than the real offset. Hence, the stopping of iterations from {Steps 15} is somehow a ''rule of thumb", and we can obtain the most appropriate gain after repeated calculation [16].
We now give a solution of (24) for choosing
$ \begin{equation} \label{eq:33} {(A(i)~~C(x))}\in{Co\left\{(A_{1}~~C_{1}), (A_{2}~~C_{2}), {\ldots}, (A_{r}~~C_{r})\right\}} \end{equation} $  (33) 
where
$ \begin{equation} \label{eq:34} \beta_{1}, \beta_{2}, {\ldots}, \beta_{r}\geq{0} \end{equation} $  (34) 
that satisfy
$ \begin{equation} \label{eq:35} \sum_{n=1}.{r}{\beta_{n}}=1 \end{equation} $  (35) 
and make
$ \begin{equation} \label{eq:36} {(A(i)~~C(x))}=\sum_{n=1}.{r}{\beta_{n}(A_{n}~~C_{n})}. \end{equation} $  (36) 
Hence, conclusions are given as follows.
Theorem 2: Suppose that
$ \begin{equation} \label{eq:37} \begin{split} {A_{n}.{T}PC_{n}.{T}Q.{T}+}&{PA_{n}QC_{n}+2\kappa_{1}P+\kappa_{2}P}\leq{0}\\ &n=1, 2, {\ldots}, r \end{split} \end{equation} $  (37) 
meet the observer gain condition (24).
Proof : The substitution of (36) into (24) leads to
$ \begin{equation} \label{eq:38} \sum_{n=1}.{r}{\beta_{n}(A_{n}.{T}PC_{n}.{T}Q.{T}+PA_{n}QC_{n}+2\kappa_{1}P+\kappa_{2}P)}\leq{0}. \end{equation} $  (38) 
Due to (35), the satisfaction of (37) guarantees (24) and hence (38).
According to (35), if (37) is satisfied, (24) and (38) must be valid.
In (37),
By solving LMIs optimization problem (37), the solution gives then a constant observer gain with the lowest possible values satisfying the condition (24).
Under different circuit working conditions,
$ \begin{equation} \label{eq:39} P=\left[\begin{array}{cccc} ~~4.1921 ~~~&~~~ 0.0636 ~~~&~~~0.0048 ~~~&~~~ 0.0402 ~~\\[1mm] 0.0636&0.0011&1.0366e.{5} &6.1983e.{5} \\[1mm] 0.0048&1.0366e.{5}&1.6871&4.8214e.{5}\\[1mm] 0.0402&6.1983e.{5}&4.8214e.{5}&3.8635e.{5} \end{array}\right] \end{equation} $  (39) 
$ \begin{equation} \label{eq:40} Q=\begin{bmatrix} 0.0048&4.3605e.{5} &3.7697e.{5} & 4.3146e.{5}\end{bmatrix}.T \end{equation} $  (40) 
then the observer gain is
$ L=P.{1}Q=\begin{bmatrix}0.1553&0.8038&1.9750&17.3426\end{bmatrix}.T. $ 
The proposed ISS based SOC observer is verified by AMESim/Simulink cosimulation. First the observer is programmed in MATLAB/Simulink as in Fig. 12. Then a simulation model of EV is developed in AMESim environment. The model's parameters are configured based on driving cycles and the lithiumion batteries used in this paper, so that the battery module within will capture the transient dynamics.
Before simulation, set up the initial value of battery model SOC as
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Fig. 14 Detailed SOC estimation simulation before the 2000s. 
In Fig. 15, SOC estimation error and detailed simulation before the 2000 s are shown, and we can see that, the observer estimation error is within
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Fig. 15 SOC estimation error. 
Fig. 16 shows the comparison curve of the output voltage for the nonlinear observer and the terminal voltage of battery model. In Fig. 16 the red curve is the terminal voltage estimation value of the battery model, the blue curve is the terminal voltage of battery model with white noise, to simulate the actual sensor measurement noise. The resolution of the experimental equipment is 5 mV, the white Gaussian noise added into this simulation experimentation is with the average of 0, and the variance of 0.25. Hence, the estimation performance in Fig. 13 to Fig. 16 is much better. In the custom current conditions, the observer can accurately track the voltage of the battery, which shows that the observer can guarantee the precision of SOC estimation, at the same time, and can accurately estimate the other three state variables.
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Fig. 16 Nonlinear observer battery terminal voltage estimation effect verification. 
The proposed ISS based SOC observer is verified by AMESim/Simulink cosimulation. First the observer is programmed in Matlab/Simulink as in Fig. 10. Then a simulation model of EV is developed in AMESim environment. The model's parameters are configured based on driving cycles and the lithiumion batteries used in this paper, so that the builtin battery module will capture the transient dynamics.
Electric vehicle model and the battery module in AMESim are shown in Fig. 17.
We set up the cosimulation environment, and the AMESim/Simulink cosimulation flow chart is shown in Fig. 17. All the model ports allow parameter configuration. So before simulation, we need to load the relation between the battery's internal parameters (resistance
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Fig. 17 AMESim/Simulink cosimulation sketch. 
In practice, the cell obviously cannot provide the driven motor with sufficient voltage and current, so we need to make battery packs which can be series or parallel connected in several sets [9]. In the process of vehicle modeling, we changed the battery module parameters into the A12326650 battery which is studied and completely calibrated. We set that the battery module contains 40 series battery unit and every battery unit contains 100 parallel cells. At this point, the electric vehicle battery module is set completely.
In this paper we choose New European driving cycle (NEDC) which lasts for 1180 s. This driving cycle is a typical procedure for lightduty vehicle (less than 3500 kg) under European emission standards. It consists of 4 urban cycles and an expressway cycle (urban speed under 50 km/h, expressway speed under 120 km/h, frequent startstop state), and the whole process lasts 20 minutes. The details of the procedure are shown in Fig. 18.
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Fig. 18 NEDC speed working condition details. 
We need to verify if the precision of the battery model is satisfied in AMESim before SOC estimation. If the model accuracy meets the requirements, cosimulate to estimate battery SOC and to verify the accuracy of the estimation algorithm under the actual working condition. If it does not meet the precision requirements, then we need to adjust model parameters.
Under AMESim environment, set up the cycle condition as NEDC, and the initial value of battery SOC at
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Fig. 19 Working current of the cell under NEDC. 
The current value of the working condition and charge/ discharge switches frequently comparing the forestall custom condition, and it is in favor of validating battery model's accuracy in complex current conditions. The terminal voltage of cell in the battery pack through the simulation experiment is shown as the blue curve in Fig. 20. After finetuning the parameters of the model established in Section Ⅱ, we get the output voltage of the battery model shown as the red dotted line in Fig. 20. The battery model error is shown in Fig. 21.
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Fig. 20 Verification of the Liion battery under AMESim environment 
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Fig. 21 Modeling error of the Liion battery under AMESim environment. 
Within the simulation time of 0 s800 s, i.e., under the city working condition, the battery working current is relatively small, and the model error is within 15 mV; Later in the simulation, the car went into the high speed cycle, and the battery charge/discharge current increases. The error of the parameter identification resulted in the model error increased in this period. When the discharge current increases instantaneously, the model error value is up to 26 mV. Throughout the whole working condition cycle, by removing the influence of the individual ''spine" points, the model error can keep within 15 mV, which means the model accuracy meets the requirements.
After the model verification, we would test the SOC estimation accuracy of the proposed nonlinear observer under complex current procedure. Link AMESim and Simulink according to Fig. 17, set the chargingdischarging current as is shown in Fig. 18, and set the SOC initial value
The battery working current and SOC value constantly changes along with the vehicle's startstop, accelerating and braking. In urban cycle, the maximum discharging current is 40 A, while in expressway cycle it reaches 110 A. The battery's terminal voltage changes with current, but the range is limited within 10 V because of the flat voltage plateau of LiFePO4 batteries. With the observer's SOC initial value set at
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Fig. 22 SOC estimation of Liion battery pack under NEDC. 
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Fig. 23 SOC estimation error of Liion battery pack under NEDC. 
We set the observer with different SOC initial values in the AMESim battery model, because of the difficulty in determination of SOC initial value in application, and to verify the proposed observer's high accuracy, good robustness and the ability to overcome the incorrect initial value problem. In the beginning of simulation, the observer constantly corrects the error between estimation value and ''true value". And then with the influence of model error, it keeps the estimation value waving near ''true value" by correction. Finally, considering all the results of the whole cycle procedure, we get the SOC estimation error within
The simulation results show that the proposed ISS observer can not only guarantee the convergence of estimation error, but also effectively overcome the error of incorrect SOC initial value. Furthermore, the nonlinearity causing divergence can be effectively overcome in real vehicle running for the model nonlinearity is sufficiently taken into consideration during ISS observer design. Therefore, we believe that the proposed ISS observer can accurately estimate battery SOC in real vehicle running circumstances, in which the chargingdischarging current would change violently.
Ⅴ. CONCLUSIONThis paper proposes a nonlinear state observer based on the ISS theory to estimate SOC of lithiumion battery. We set up equivalent circuit model with two RC networks, considering the hysteresis characteristic when the lithiumion battery is in charging and discharging process. And the hysteresis model parameters are identified by the measurement data acquired from a commercial lithiumion battery on a test bench. Then the ISS based estimator has been set up to estimate SOC of the lithiumion battery. Simulation results denote the ISS based estimator for the SOC estimation has high accuracy, and improved robustness. The designed observer is also tested on AMESim and Simulink cosimulation. The simulation results show that the state observer based ISS theory in this paper could trace the theoretical SOC well with the modeling error, and the estimation error is restricted in the required bound even when the model error or the initial SOC value is large.
In future work, the underlying battery model will be extended taking battery aging (capacity fade and resistance rise) into account.
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