Fractional-Order Sliding-Mode Observers for the Estimation of State-of-Charge and State-of-Health of Lithium Batteries

: Lithium batteries are widely used in power storage and new energy vehicles due to their high energy density and long cycle life. The accurate and real-time estimation for the state-of-charge (SoC) and the state-of-health (SoH) of lithium batteries is of great signiﬁcance to improve battery life, reliability, and utilization efﬁciency. In this paper, three cascaded fractional-order sliding-mode observers (FOSMOs) are designed for the estimation of SoC by observing the terminal voltage, the polarization voltage, and the open-circuit voltage of a lithium cell, respectively. Furthermore, to calculate the value of the SoH, two FOSMOs are developed to estimate the capacity and internal resistance of the lithium cell. The control signals of the observers are continuous by utilizing fractional-order sliding manifolds without low-pass ﬁlters. Compared with the existing sliding-mode observers for SoC and SoH, weaker chattering, faster response, and higher estimation accuracy are obtained in the proposed method. Finally, the experiment tests demonstrate the validity and feasibility of the proposed observer design method.


Introduction
With the depletion of global energy, new energy vehicles are gradually replacing fuel vehicles.Batteries, represented by lithium batteries, are the core components of newenergy vehicles.The real-time estimation for SoC and SoH is vital and indispensable for the safety, reliability, and efficiency of lithium batteries [1,2].The model of the battery is the mathematical expression of its characteristics, and accurate battery models can not only reflect the relationship between battery characteristics and influencing factors, but also provide an important basis for accurate state estimation.The mathematical models of lithium batteries are generally divided into two types: the electrochemical model [3] and the equivalent circuit model (ECM) [4].Electrochemical models take into account the impact of the distributed behaviors inside the cells, including the potentials in the electrolyte and the concentration of Li-ions [5].The equivalent circuit model has been widely used in the modeling and state estimation of various power lithium batteries because of its simple structure, easy integration, easy real-time calculation, and low complexity.In recent years, scholars proposed a variety of equivalent circuit models such as the Rint model, Thevenin model, Partnership for a New Generation of Vehicles (PNGV) model, and Dual Polarization (DP) model.The Thevenin model, also called the RC equivalent circuit model, which consists of an internal resistance and some resistance-capacitance loops, can simulate the static and dynamic performance precisely.However, because of their stochastic internal parameter variations, the uncertainties existing in the lithium batteries would be hard to estimate and compensate practically [6,7].The main estimation methods for lithium batteries based on ECMs include the impedance measurement method [8], machine learning algorithm [9], extended Kalman filter (EKF) algorithm [10], and sliding-mode observerbased method [11].
The impedance measurement method can obtain the functional relationship between the SoC and the lithium battery model by stimulating the currents with different frequencies [12].However, the functional relationship cannot be obtained accurately because the internal resistance of the battery is affected by its temperature and SoH [13], and the measurement can only be carried out when the batteries are off-line [14].The machine learning algorithm regards the lithium batteries as a black box and performs tests and training only by using a large amount of experimental data, regardless of the battery's internal relations [15].In [16], the machine learning algorithm is used to estimate the SoC of Li-ion batteries, which achieves high accuracy and fast estimation.However, the process of estimation is extremely reliant on the training number and the count of neurons in the hidden layer.The extended Kalman filter [17] uses the estimation state values of the present moment and the previous moment to observe the currents, while EKF needs precise modeling and other factors such as aging and environmental changes.
Generally, the aforementioned estimation methods of SoC and SoH depend on accurate modeling, however the existing ECMs have their own limitations [18].The sliding-mode observer (SMO) is insensitive to the internal parameter variations and robust to external disturbances, so it is suitable for the state observation of nonlinear and time-varying systems [19].In [20], a multi-time scale dual-SMO is designed to estimate the SoC and SoH of lithium batteries simultaneously.Based on Thevenin model, [21] designed a dual sliding-mode observer to observe the SOC and SoH of the battery, but it is difficult to obtain the equivalent control signal by relying on the filter with complex structure.In [22], the estimation of the SoC and SoH state is carried out by comparing the extended Kalman filter with Particle Filtering.It is found that Particle Filtering has higher estimation accuracy.However, it also has high dependence on the model and poor robustness.In [23], the variable parameters discrete sliding-mode observer is proposed in the time-varying circumstance to estimate the SoC of a lithium battery.However, the low availability of data and the phase lag of the low-pass filter (LPF) make the existing SMO have limitations in the practical application.In [24], Alexander used the least square method to estimate the model parameters in the electrochemical model of lithium batteries, considering the aging of lithium batteries.Ref. [25] proposed a sliding-mode observer based on an AEM model to estimate the SoH of a battery, but its control signal is discontinuous, which is not conducive to practical application.
For accurate and real-time estimation, the fractional-order sliding-mode observers are proposed for the SoC and SoH using the RC equivalent circuit model, which not only avoids the phase lag caused by LPF, but also smooths the observation signals by fractional order integrator.Furthermore, the observers are designed in the cascade type for the high accuracy of the global estimation.The contributions of the paper can be summarized as follows: (1) The accuracy of estimation for SoC and SoH is enhanced without utilizing the LPF.
Owing to the fractional-order sliding manifolds, the control signals of the observers are smoothed by the α + 1 order integrator, which can be directly used to estimate SoC and SoH.(2) The accuracy of the data is improved.The proposed FOSMOs adopt the data of measured voltage and current to estimate the significant data, which will be used in the next development of FOSMOs.
This paper is organized as follows: Section 2 introduces the RC equivalent circuit model of lithium batteries.In Section 3, three FOSMOs are designed in the estimation process of SoC.In Section 4, two novel observers are presented in the process of estimation of SoH.Section 5 gives the experimental results.Finally, conclusions are shown in Section 6.

The Mathematical Model of the Battery
The establishment of a high precision mathematical model is the basis of SoC and SoH estimation of lithium batteries.The lithium battery is a typical nonlinear system.The mathematical model of lithium batteries mainly includes the electrochemical model and ECM [26].The ECM has a simpler structure and lower computational complexity than the electrochemical model [27].Therefore, the RC equivalent circuit model is widely used in the observation and control of lithium batteries.The RC equivalent circuit model presents different modes according to the SoC, the current, and temperature, and it can simulate the static and dynamic performance of the lithium batteries properly [28].The RC equivalent circuit model of lithium batteries is shown in Figure 1.
model of lithium batteries.In Section 3, three FOSMOs are designed in the estimation process of SoC.In Section 4, two novel observers are presented in the process of estimation of SoH.Section 5 gives the experimental results.Finally, conclusions are shown in Section 6.

The Mathematical Model of the Battery
The establishment of a high precision mathematical model is the basis of SoC and SoH estimation of lithium batteries.The lithium battery is a typical nonlinear system.The mathematical model of lithium batteries mainly includes the electrochemical model and ECM [26].The ECM has a simpler structure and lower computational complexity than the electrochemical model [27].Therefore, the RC equivalent circuit model is widely used in the observation and control of lithium batteries.The RC equivalent circuit model presents different modes according to the SoC, the current, and temperature, and it can simulate the static and dynamic performance of the lithium batteries properly [28].The RC equivalent circuit model of lithium batteries is shown in Figure 1.
RC equivalent circuit model of the lithium battery.
In Figure 1, the controlled voltage source Voc (Z) is used to describe the open-circuit voltage of the lithium battery.The Voc (Z) and SoC of the battery meet a certain function relationship, and the Voc is affected by the ambient temperature.Z stands for SoC [29].The internal resistance of the battery is generally uncertain, which contains ohmic resistance Rt and polarization resistance Rp.Cp is the polarization capacitance.The polarization voltage Vp is used to describe the polarization effect of the lithium battery in the charge and discharge processes.The three polarization parameters are jointly affected by the charge and discharge currents, SoC, and ambient temperature.Vt is the terminal output voltage, and Δf represents the uncertainties.The dynamics of the SoC can be expressed as: where Cn is the rated capacity.The SoC and the Voc satisfy the following: where the variable ҡ, the slope at point Z, can be obtained from: In Figure 1, the controlled voltage source V oc (Z) is used to describe the open-circuit voltage of the lithium battery.The V oc (Z) and SoC of the battery meet a certain function relationship, and the V oc is affected by the ambient temperature.Z stands for SoC [29].The internal resistance of the battery is generally uncertain, which contains ohmic resistance R t and polarization resistance R p .C p is the polarization capacitance.The polarization voltage V p is used to describe the polarization effect of the lithium battery in the charge and discharge processes.The three polarization parameters are jointly affected by the charge and discharge currents, SoC, and ambient temperature.V t is the terminal output voltage, and ∆f represents the uncertainties.The dynamics of the SoC can be expressed as: where C n is the rated capacity.The SoC and the V oc satisfy the following: .
where the variable κ, the slope at point Z, can be obtained from: where the nonlinear function f voc (Z) is the mapping of V oc versus Z.
The mathematical relationship of V t , V oc , and V p in Figure 1 is expressed as: . . . where The disturbance ∆ f is unknown, which is Lipschitz and bounded, then: where F f1 and F f2 are known positive constants.The fractional-order derivate of the estimation error of the open-circuit voltage and the estimation error of the polarization voltage are assumed to be bounded as: where F oc > 0 and F p > 0 are certain constants, and Voc and Vp are the estimations of V oc and V p .

High Order Sliding-Mode Observer
The high order sliding-mode control (HOSMC) proposed by Levant in 1996 can eliminate chattering and improve the control accuracy effectively [30].As a special type of the HOSMC, the super twisting algorithm (STA) can realize convergency of the system state to the ideal sliding manifold s = .s = 0 in finite time, but the rate of the convergence is restricted if the initial state is far away from the equilibrium point [31].
Consider a first-order system as follows: .
where x is the state variable, f (x) a known function, u the control input of the system, and d the disturbance.Then, a high order sliding-mode observer (HOSMO) [32] can be designed as: where x is the observed value of x, v the observer control signal, and the estimating error is defined as e = x − x.Then, we can obtain the follow form: .
The finite time convergence of the system is guaranteed if the observer control law is designed as follows: The system state could converge to the equilibrium point in finite time and the sliding manifold could achieve s = .s = 0.The lemmas are given below to prove the theorems presented in Sections 3 and 4. To prove the stability of the fractional differential systems, Lemma 1 is given as the theoretical basis to design the fractional-order sliding manifold.Lemma 1. (Fractional-order sliding-mode [33]): Consider a fractional-order system expressed by where v is the fractional-order parameter, 0 < v < 2, x ∈ R n , A ∈ R n×n , if |arg(eig(A)|> vπ/2, and the solution of the system is asymptotically stable.
To prove the finite-time convergence of the sliding manifold, Lemma 2 is given below as the theoretical basis.
Lemma 2. (Finite-time convergence theorem [34]): if there exists a positive definite continuous function V(x): U→R, real numbers c > 0 and 0 < α < 1, and an open neighborhood U 0 ⊂U of the origin such that

The Estimation Method for SoC
The SoC of the lithium battery is generally defined as the ratio of the remaining capacity to the rated capacity [35].The SoC is related to the nature of the battery itself, but also affected by the rate of discharge, environment temperature, and ambient noise [36].The SoC of the lithium battery is mainly defined in two ways: (1) The electric quantity perspective-based definition is given as [35]: where C s and C N are the discharge quantity and rated capacity of the lithium battery, i s (t) is the discharge current, and η s is the discharge efficiency factor.(2) The energy perspective-based definition is shown by [37]: where W s and W N are the released energy and total energy of the lithium battery, and V(t) is the terminal voltage.
In the paper, the SoC definition based on the electric quantity perspective is selected.To estimate SoC, three observers estimating the terminal voltage V t , the open-circuit voltage V oc , and the polarization voltage V p are designed, respectively.

Terminal Voltage Observer
According to (4), an observer for V t is proposed as follows: .
where v 1 is the observer control signal.The estimating error of the V t is defined as e 1 = V t − Vt , which can be obtained from ( 4) and (16).Then, the dynamics of the estimation error can be expressed as: .
Theorem 1.The finite-time convergence of the error dynamics ( 17) is guaranteed if the fractionalorder sliding manifold ( 18) is utilized, and the following observer control law is designed as: where k 1 = a 1 F oc + η 1 , F oc is defined in (8), and η 1 is a positive constant.On the fractional-order sliding manifold, the estimation error of the open-circuit voltage e 2 = V oc − Voc satisfies: The proof is set out in Appendix A. In conclusion, the terminal voltage V t can be estimated directly, and the open circuit voltage V oc is unknown from (4).The observer control law ( 19) is applied to guarantee the convergence of e 1 .Although the V oc is unknown, the error e 2 = V oc − Voc can be obtained by (22), which is further adopted for the observer of V oc .

Open-Circuit Voltage Observer
According to (5), an observer for the open-circuit voltage is designed as: where Vp is the estimation of V p , v 2 the observer control law.Defining the voltage error between V p and its estimation as e 3 = V p − Vp , the error dynamics of the V oc can be expressed based on (5) and (23) as: .
Theorem 2. The error dynamics (24) can approach s 2 = 0 in finite time, then converge to zero along s 2 = 0 if the fractional-order sliding manifold s 2 is chosen as (25) and the observer control law is designed as: where k 2 = −κa 2 F P + η 2 , F p is a certain constant defined in (8), and η 2 is a positive constant.When the sliding manifold satisfies s 2 = 0, then the voltage error e 3 can be obtained as: After the first observer ( 16) tracks the variable of the system (4), e 2 is given by ( 22) and utilized in the observer control law (26).Then, e 3 can be observed by (29) and will be transferred to the observer for V p after e 2 converges to zero.

Polarization Voltage Observer
An observer based on the dynamics of the polarization voltage ( 6) can be designed by: .
where v 3 is the observer control signal.Combining ( 6) and ( 30), the polarization voltage error dynamics are governed by: .
Theorem 3. When the error dynamics (24) stays on the fractional-order sliding manifold s 2 = 0, the error dynamics (31) can reach s 3 = 0 in finite time and converge to zero along s 3 = 0, if the fractional-order sliding-mode s 3 is selected as (32) and the observer control law is presented below: where k 3 = F f2 + η 3 , F f2 is from (7), η 3 > 0. When the trajectory of the error dynamics (31) reaches the sliding manifold s 3 = 0, we have D α+1 e 3 + β 3 e 3 = 0, then e 3 reaches zero from Lemma 1.
From (6), we can know that only the current can be measured.When the V oc in ( 5) is estimated, e 3 can be obtained by (29) and used in the observer control law (33).When the polarization voltage error e 3 converges to zero, all the estimation errors, i.e., e 1 , e 2 and e 3 , converge to zeros.

Calculation of SoC
In Theorems 1-3, three observers based on fractional-order sliding-mode theory are designed for the terminal voltage V t , the open circuit voltage V oc , and the polarization voltage V p in the model of lithium batteries.Then, Voc , the estimation of V oc (Z) in the second observer, is determined and can be used to calculate the SoC as follows: where f voc is the known function defined in (3).

The Estimation Method for SoH
The estimation of SoH needs the capacity and the resistance.During the process of estimation of SoC, C n and R t are assumed to be constant, while in the estimation of SoH, they should be seen as variables.The definition of the SoH can be expressed in two ways.
(1) The capacity-based definition is shown by [36]: where C n and C nom are the actual capacity and nominal capacity of the lithium battery.(2) The internal resistance-based definition is given as [38]: where R t and R nom are the actual and nominal resistance, and R EoL is the resistance at the end of life [39].In the SoH estimation, the capacity and resistance are monotonically changing with aging, environmental temperature, magnitude of current, and depth of discharge.The observer control signals can be viewed as continuous.
Therefore, the following assumptions can be given: where Ĉn and Rt are the estimation value, and F c , F cn , F t , and F rt are all known positive constants.

Battery Capacity Observer
According to the dynamics of the SoC (1), the observer can be designed as follows: .
where Ẑsoh is the estimated variable of Z under the time-varying C n in the design the SoH observer, and v 4 is the observer control.When e 2 converges to zero, Ẑ is available and satisfies .Ẑ = (1/C n )I(t).In order to obtain C n , e 4 = Ẑ − Ẑsoh is defined, then: .
Theorem 4. The sliding-mode manifold is selected as (43) and the observer control law is designed as follows, then the state trajectory of the error dynamics (42) can reach s 4 = 0 from any s 4 (0) = 0 in a finite-time t 4r ≤ |s 4 (0)|/η 4 : where k 4 = F c + η 4 , η 4 > 0. The proof is set out in Appendix A.

Battery Inner Resistance Observer
In addition to C n , in the SoH estimation, R t is also seen as a time varying variable.Therefore, b 1 in (4) can be rewritten as a function of C n and R t : Different from the observer in the SoC estimation (16), another observer can be designed in the following form: where Vtsoh is the estimate of the terminal voltage in SoH observer, and v 5 is the observer control law.Assuming that the current of the battery cell is constant, we have .I = 0. Define e 5 = V t − Vtsoh , then: .
Theorem 5.If s 5 is chosen as (50) and the observer control law is designed as follows, the error dynamics (49) can reach s 5 = 0 in finite-time after the error dynamics (42) converges to zero: A fractional-order sliding manifold is chosen for the error dynamics (49): Where β5 > 0 is a constant, and α satisfies 0 < α < 1.
Theorem 5.If s5 is chosen as (50) and the observer control law is designed as follows, the error dynamics (49) can reach s5=0 in finite-time after the error dynamics (42) converges to zero:

SoC-Estimation
As shown in Figure 3, the battery testing equipment consists of the Arbin BT2000 Battery Test System and the ESPEC PRA-3AP Temperature and Humidity Chamber.The Arbin BT2000 is used to record the cumulative time, voltage, current, capacity, and other data of the lithium battery.The ESPEC PRA-3AP chamber is utilized to control the tem-   Test Manual, which is simplified from FUDST operating conditions.It is easy to operate and effectively reflect the dynamic charging and discharging characteristics of lithium batteries.The FUDST is commonly used in industry validation tests such as to test the impact of the variable power requirements on the battery while the vehicle is in motion.The FUDST tests the car at its usual peak power condition and the high frequency of the charging and discharging process caused by acceleration and deceleration.The SoC and SoH tests of the battery will be carried out using the above equipment under a constant temperature condition.A comparison experiment is carried out between the HOSMO and the proposed FOSMO.The sliding manifolds and the observer control laws under the HOSMO method are designed as follows:  The battery tests, Dynamic Stress Test (DST) and Federal Urban Dynamic Stress Test (FUDST), are conducted to estimate the SoC of the lithium battery.The DST operating condition test is carried out according to the US Advanced Battery Consortium Battery Test Manual, which is simplified from FUDST operating conditions.It is easy to operate and effectively reflect the dynamic charging and discharging characteristics of lithium batteries.The FUDST is commonly used in industry validation tests such as to test the impact of the variable power requirements on the battery while the vehicle is in motion.The FUDST tests the car at its usual peak power condition and the high frequency of the charging and discharging process caused by acceleration and deceleration.The SoC and SoH tests of the battery will be carried out using the above equipment under a constant temperature condition.A comparison experiment is carried out between the HOSMO and the proposed FOSMO.The sliding manifolds and the observer control laws under the HOSMO method are designed as follows: where i = 1, 2, • • • , 5. When estimating the SoC, the charge and discharge current I and the terminal voltage V t are usually estimated, which are the most basic measurement data that can be obtained.Figure 4 shows the terminal voltage V t and the measured current I of the lithium battery from DST.In Figure 4, it can be intuitively seen that the current passing through the lithium battery basically fluctuates in the range of −4 A to 2 A, and the terminal voltage shows a trend of gradual decline.In Figure 5, the SoC state estimation of the battery was performed by DST testing.Figure 5 shows the true SoC values compared with the SoC estimation value under the FOSMO method and under the HOSMO method.
Both FOSMO and HOSMO can converge to the actual values of SOC, and there is a certain drift error between the observation results and the actual value of the SOC.The FOSMO have better accuracy and faster dynamical response compared with the HOSMO method.
Batteries 2023, 9,213 where  = 1, 2, ⋯ ,5.When estimating the SoC, the charge and discharge current I a terminal voltage Vt are usually estimated, which are the most basic measurement da can be obtained.Figure 4 shows the terminal voltage Vt and the measured current lithium battery from DST.In Figure 4, it can be intuitively seen that the current p through the lithium battery basically fluctuates in the range of −4A to 2A, and the te voltage shows a trend of gradual decline.In Figure 5, the SoC state estimation of t tery was performed by DST testing.Figure 5 shows the true SoC values compare the SoC estimation value under the FOSMO method and under the HOSMO metho FOSMO and HOSMO can converge to the actual values of SOC, and there is a certa error between the observation results and the actual value of the SOC.The FOSMO better accuracy and faster dynamical response compared with the HOSMO metho

Time(s)
Current(A)  The terminal voltage V t and the measured current I of the lithium battery under FUDST are shown in Figure 6.We can see that the rate of change of current I and terminal voltage V t is faster under the FUDST.Figure 7 shows the actual value of the SoC compared with the estimated SoC under FOSMO and HOSMO under FUDST.In Figure 7, the estimated results under the HOSMO and FOSMO can track the actual SOC with high precision, and there is some drift error between the observed results and the actual values.The proposed method under FOSMO has a better performance in accuracy and dynamical response, and shows much less estimation error.It is smoother than that under HOSMO, owing to the fractional-order integral observer control law.The terminal voltage Vt and the measured current I of the lithium battery unde FUDST are shown in Figure 6.We can see that the rate of change of current I and termina voltage Vt is faster under the FUDST.Figure 7 shows the actual value of the SoC compared with the estimated SoC under FOSMO and HOSMO under FUDST.In Figure 7, the esti mated results under the HOSMO and FOSMO can track the actual SOC with high preci sion, and there is some drift error between the observed results and the actual values.The proposed method under FOSMO has a better performance in accuracy and dynamica response, and shows much less estimation error.It is smoother than that under HOSMO owing to the fractional-order integral observer control law.By analyzing the data in Figures 5 and 7, the estimation value of the SoC under DST and FUDST conditions are conducted with the mean absolute error (MAE), root mean squared err (RMSE), and mean relative error (MRE), as shown in Table 2.We can conclude that the real-time estimation error fluctuation in FOSMO is smaller than the HOSMO-based method, especially the RMSE and MRE, with better performance.

SoH-Estimation
The SoC describes the short-term changes in current parameters.While the SoH does not need to be carried out continuously, it can be obtained by periodic measurement.The data from the INR 18650-20R cell under DST in Figure 4 is used for the SoH estimation.In this paper, the SoH algorithm is described by the battery capacity C n and the internal resistance R t .
In Figure 8, the real battery capacity C n , the estimation results of capacity under FOSMO and HOSMO are depicted.It can be seen that in FUDST condition the estimates of C n under FOSMO can converge to the real value with faster dynamical performance than that under HOSMO and fluctuate around the reference value.Furthermore, the observed results and the actual value maintain a drift error owing to the inherent error between the established mathematical model and the actual system.The estimation results of the battery's inner resistance R t is shown in Figure 9.It shows that the estimation value of R t under the proposed FOSMO method has better dynamic response speed and higher tracking accuracy.
served results and the actual value maintain a drift error owing to the inherent error between the established mathematical model and the actual system.The estimation results of the battery's inner resistance Rt is shown in Figure 9.It shows that the estimation value of Rt under the proposed FOSMO method has better dynamic response speed and higher tracking accuracy.According to the Figures 8 and 9, Table 3 shows the mean absolute error (MAE), root mean squared error (RMSE), and mean relative error (MRE) for the estimation value of Rt and Cn.We can conclude that, in the FOSMO method, the estimation error fluctuation is smaller than the HOSMO method.The MRE value of capacitance under FOSMO is better, which noticeably improves the performance of tracking.According to the Figures 8 and 9, Table 3 shows the mean absolute error (MAE), root mean squared error (RMSE), and mean relative error (MRE) for the estimation value of R t and C n.We can conclude that, in the FOSMO method, the estimation error fluctuation is smaller than the HOSMO method.The MRE value of capacitance under FOSMO is better, which noticeably improves the performance of tracking.

Discussion
In this paper, the fractional-order sliding-mode observers are designed for the estimation of the SoC and SoH of lithium batteries, and the corresponding experimental verifications are carried out.The SoC and SoH are the basis of the battery management system, and accurate estimation values are directly related to efficiency and safety, especially in vehicle operation.Owing to the nonlinear characteristics of battery reaction and the interference of many environmental factors, accurate estimation for the battery system is still a challenge.From the experiment result, we can see that the proposed observers have better performance in accuracy and rapidity than the HOSMO method.However, there still exist some deviations between the observed results and the actual values because of the inherent error between the established model and the actual SoC and SoH.Therefore, to design observers of higher precision, stronger robustness, and more simplification in battery management systems is our future work.

Conclusions
In the paper, fractional-order sliding-mode observers are proposed to estimate the SoC and SoH of the lithium cell with high accuracy and rapidity.The uncertainties, including parameter perturbations and external disturbances, are considered in the system.The main contributions of the paper can be summarized as: (1) The designed sequential connection of FOSMOs improve the global accuracy and rapidity of estimation; (2) The fractional-order sliding manifolds attenuate the chattering in the injection output signals and guarantee the smooth response of SoC; (3) The experimental tests are carried out under the DST and FUDST conditions, which verified the superiority of the proposed method compared with the existing HOSMO method.
Author Contributions: M.Z.: Conceptualization, algorithm innovation, methodology, writing and original draft; K.W.: data and formal analysis, investigation, software, simulation, writing and original draft.X.W.: conceptualization, simulation, investigation, methodology, original draft; L.W.: investigation, review, and editing; H.S.: formal analysis and editing; D.W.: project administration and writing; Y.Z. and J.L.: data and formal analysis, software, simulation.All authors have read and agreed to the published version of the manuscript.where k cn is a positive constant, F cn is defined in (39), and the derivative of e cn is expressed as: . e cn = .
C n − (F cn + k cn )sgn(e cn ) A Lyapunov function V cn = 0.5e 2 cn is considered, and its derivative is shown below: . The estimation of R t can be obtained by: .Rt = (F rt + k r )sgn R p C p I D −α−1 (k 5 sgn(s 5 )) = (F rt + k r )sgn(e rt ) (A9) where k r is a positive constant, and F rt is defined in (40).If a Lyapunov function is chosen as V tr = 0.5e 2 rt , then it can be deduced as: .

Figure 1 .
Figure 1.RC equivalent circuit model of the lithium battery.

Figure 2 .
Figure 2. Block diagram of SoC and SoH algorithms.

Figure 2 .
Figure 2. Block diagram of SoC and SoH algorithms.

Figure 4 .
Figure 4.The current and the terminal voltage of the lithium battery from DST.

Figure 4 .
Figure 4.The current and the terminal voltage of the lithium battery from DST.

Figure 4 .Figure 5 .Figure 5 .
Figure 4.The current and the terminal voltage of the lithium battery from DST.

Figure 7 .
Figure 7. True value of the SoC and the estimated SoC under FOSMO and HOSMO from FUDST.

Figure 8 .
Figure 8.Comparison of Cn estimated by FOSMO and HOSMO with the true value of Cn.

Figure 8 .Figure 9 .
Figure 8.Comparison of C n estimated by FOSMO and HOSMO with the true value of C n .

Figure 9 .
Figure 9.Comparison of R t estimated by FOSMO and HOSMO with the true value of R t .

Funding:AV 4 = s 4 .s 4 = s 4 n− s 4 k 4 sgn(s 4 )I C − 1 n − Ĉ− 1 n+ |s 4 |
This research was funded by the National Natural Science Foundation of China, grant number U21A20145; by the Heilongjiang Industrial Revitalization Major Project on Engineering and Science, grant number 2019ZX02A01.Institutional Review Board Statement: Not applicable.Informed Consent Statement: Not applicable.Data Availability Statement: Not applicable.differentiatings 4 above with respect to time t and substituting the control law (46) into the above yields:.Lyapunov function for the error dynamics V 4 = 0.5s 4 2 is considered, then the derivative of V 4 can be obtained as:.≤ |s 4 | D α .D α I −.means that the error dynamics (42) can approach s 4 = 0 in a finite time t 4r ≤ |s 4 (0)| /η 4 according to Lemma 2. When s 4 = 0, from (A4) we have: Define the error of the capacity e cn = C n − Ĉn , then the signum function of the e cn is:sgn(e cn ) = sgn(C n − Ĉn ) = −sgn( 1 C n − 1 Ĉn ) = −sgn( v 4n I ) = −sgn( k 4 I D −α−1 sgns 4 )The estimation of C n is designed by: .Ĉn = −(F cn + k cn )sgn( k 4 I D −α−1 sgns 4 ) = (F cn + k cn )sgn(e cn ) (A6)

Table 2 .
The SoC estimation under the DST and FUDST testing.

Table 3 .
The measurement error in SoH estimation.

Table 3 .
The measurement error in SoH estimation.
rt = e rt .−(F rt + k r )sgn(e rt ) ≤ −k r |e rt | < 0which can be concluded that e rt can converge to zero from e rt (0) = 0 and Rt in (A9) can track R t in finite time, i.e., Rt = R t .This completes the proof.