CN103399279B - Based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology - Google Patents

Abstract

Based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology, relate to a kind of cycle life of lithium ion battery Forecasting Methodology.Exist for different battery and the low problem of different operating state adaptive faculty to solve current these methods based on model.It comprises: one, the capacity data of on-line measurement lithium battery to be measured, preserves data and carries out pre-service to described data; Two, based on the parameter of EKF method determination lithium ion battery state-space model; Three, according to the lithium ion battery state-space model set up, state estimation is carried out to lithium ion battery to be measured, the output of described AR model is utilized to carry out the state updating of lithium ion battery to be measured, described lithium ion battery state-space model obtains the battery capacity data of each charge and discharge cycles, and described data are compared acquisition lithium ion battery residual life with the failure threshold of lithium ion battery to be measured.It is for predicting cycle life of lithium ion battery.

Description

Based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology

Technical field

The present invention relates to a kind of cycle life of lithium ion battery Forecasting Methodology, particularly one is based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology.

Background technology

At present for lithium ion battery residual life (RemainingUsefulLife, RUL) method predicted is roughly divided into physically based deformation model (Model-basedPrognostics) and based on data-driven (Data-Driven) method,, model complicated for failure mechanism is difficult to the electronics lithium battery to be measured set up, the method that major part research concentrates on based on data-driven.The statistics driving method of a class Corpus--based Method filtering is comprised as particle filter (ParticleFilter in data-driven method, PF), Kalman filtering (KalmanFilter, and EKF (ExtendedKalmanFilter KF), EKF), realize prediction by setting up lithium battery state transition equation to be measured and upgrade, take into full account lithium battery interior state transfer characteristics to be measured, but a certain degradation model lacks adaptability to dissimilar battery and different operating state; Another kind of be the method that drives based on clear data as autoregressive moving average (AutoregressiveMovingaverage, ARMA) model, have in mind and analyze the feature of data own and do not consider the characteristic of the lithium battery to be measured belonging to data.At present, the hybrid predicting framework that statistical filtering method and clear data driving method carry out merging constantly is suggested and improvement, the advantage of the two is carried out the defect occurred when combining to make up respective independent utility, but current these methods based on model exist for different battery and the low problem of different operating state adaptive faculty.

Summary of the invention

The object of the invention is to exist for different battery and the low problem of different operating state adaptive faculty to solve current these methods based on model, the invention provides a kind of based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology.

Of the present invention based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology, it comprises the steps:

Step one: the capacity data of on-line measurement lithium battery to be measured, preserves data and carries out pre-service to described data;

Step 2: the parameter based on EKF method determination lithium ion battery state-space model:

According to lithium ion battery experience degradation model and AR Construction of A Model lithium ion battery state-space model, utilize pretreated data and determine the parameter of described lithium ion battery state-space model according to EKF method; The sequence of observations after the prediction output valve of described AR model superposes with observation noise is the observed reading of the battery capacity of described lithium ion battery state-space model, and described AR model is the AR model utilizing pretreated data acquisition to merge autoregressive coefficient acquiring method to determine;

Step 3: state estimation is carried out to lithium ion battery to be measured according to the lithium ion battery state-space model that step 2 is set up, the output of described AR model is utilized to carry out the state updating of lithium ion battery to be measured, described lithium ion battery state-space model obtains the battery capacity data of each charge and discharge cycles, and described data are compared acquisition lithium ion battery residual life with the failure threshold of lithium ion battery to be measured.

The invention has the advantages that, the present invention utilizes EKF method and the prediction of AR model modeling, predicts, obtain long-term forecasting result and it can be used as the observed reading true value of Kalman filtering state updating link to input the long-term degradation trend of battery capacity.The feature that the lithium ion battery state-space model feature of this method and data itself embody combines, and improves the adaptability of this method for different battery, reduces the dependence of RUL prediction algorithm for experience degradation model.Method of the present invention possesses good status tracking ability, described method increases for the adaptability between different battery cell, NASAPCoE battery is tested, NASAPCoE Cell Experimentation An RUL prediction effect obviously improves, RUL Relative Error on average reduces about 14%, and capacity predict relative error on average reduces about 1.65%.

Accompanying drawing explanation

Fig. 1 is the schematic flow sheet based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology of the present invention.

The predicting residual useful life experiment of Fig. 2 for adopting the method described in this law to carry out No. 5 batteries in NASAPCoE center, for 60 modelings in early stage, the schematic diagram of 1 ~ 10 rank AR model AIC result of calculation.

The predicting residual useful life experiment of Fig. 3 for adopting the method described in this law to carry out No. 5 batteries in NASAPCoE center, for 80 modelings in mid-term, the schematic diagram of 1 ~ 10 rank AR model AIC result of calculation.

The predicting residual useful life experiment of Fig. 4 for adopting the method described in this law to carry out No. 5 batteries in NASAPCoE center, for later stage 100 modelings, the schematic diagram of 1 ~ 10 rank AR model AIC result of calculation.

Embodiment

Embodiment one: composition graphs 1 illustrates present embodiment, described in present embodiment based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology, it comprises the steps:

Step one: the capacity data of on-line measurement lithium battery to be measured, preserves data and carries out pre-service to described data;

Step 2: the parameter based on EKF method determination lithium ion battery state-space model:

According to lithium ion battery experience degradation model and AR Construction of A Model lithium ion battery state-space model, utilize pretreated data and determine the parameter of described lithium ion battery state-space model according to EKF method; The sequence of observations after the prediction output valve of described AR model superposes with observation noise is the observed reading of the battery capacity of described lithium ion battery state-space model, and described AR model is the AR model utilizing pretreated data acquisition to merge autoregressive coefficient acquiring method to determine;

Step 3: state estimation is carried out to lithium ion battery to be measured according to the lithium ion battery state-space model that step 2 is set up, the output of described AR model is utilized to carry out the state updating of lithium ion battery to be measured, described lithium ion battery state-space model obtains the battery capacity data of each charge and discharge cycles, and described data are compared acquisition lithium ion battery residual life with the failure threshold of lithium ion battery to be measured.

Embodiment two: present embodiment is to the further restriction based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology described in embodiment one,

Carrying out pretreated method to described data in described step one is:

Reject for point unusual in described data, the capacity orthogenesis excessive for amplitude carries out the level and smooth of trend.

Described unusual point comprises the data of larger measuring error and the data of mistake, the excessive capacity orthogenesis of described amplitude for show as several capacity rising parts in entire lowering trend in curve, the burr part namely in decline curve.

Embodiment three: present embodiment is to the further restriction based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology described in embodiment one,

In step 2, described according to lithium ion battery experience degradation model and AR Construction of A Model lithium ion battery state-space model, utilize pretreated data and determine that according to EKF method the method for the parameter of described lithium ion battery state-space model comprises the steps:

Steps A: according to lithium ion battery experience degradation model construct the state-space model of the parameter estimation of described degradation model:

$\left\{\begin{array}{cc}{a}_k=a_{k-1}+w_a& w_a~N(0,Q_a)\\ b_k=b_{k-1}+w_b& w_b~N(0,Q_b)\\ c_k=c_{k-1}+w_c& w_c~N(0,Q_c)\\ C_{k+1}=a_k{C}_k+b_k{e}^{(-c_k)}+v_k& v_k~N(0,R)\end{array}\right.$

Wherein $\left\{\begin{array}{cc}{a}_k=a_{k-1}+w_a& w_a~N(0,Q_a)\\ b_k=b_{k-1}+w_b& w_b~N(0,Q_b)\\ c_k=c_{k-1}+w_c& w_c~N(0,Q_c)\end{array}\right.$ For state transition equation, $C_{k+1}=a_k{C}_k+b_k{e}^{(-c_k)}+v_k$ For observation equation, a _k, b _kand c _kbe respectively the coulombic efficiency η in experience degradation model described in the current k moment _{c, k}, regenerated capacity parameter beta _{1, k}with regenerated capacity parameter beta _{2, k}estimated value;

C _kfor the discharge capacity in k moment in the degradation in capacity process of lithium battery to be measured, C _k+1for the discharge capacity in k+1 moment in the degradation in capacity process of lithium battery to be measured, η _{c, k}for the coulombic efficiency in charging and discharging lithium battery process to be measured; for lithium battery to be measured is at standing time of having a rest section △ t _kthe capacity of interior regeneration; w _a, w _band w _cbe respectively parameter a, white Gaussian noise that b and c comprises, Q _a, Q _band Q _cbe respectively w _a, w _band w _cvariance, noise w _a, w _band w _cmeet N (0, Q respectively _a), N (0, Q _b) and N (0, Q _c) Gaussian distribution; R is real number; v _kfor the observation noise of lithium battery to be measured, v _kobedience average is 0, v _kvariance be the Gaussian distribution of R; Step B: utilize pretreated data, adopts EKF method to carry out linearization, state estimation and state updating to the state-space model of the parameter estimation of described degradation model, determines the parameter a in the current K moment of described state-space model _k, b _kand c _k;

Step C: according to the parameter a in the current K moment of the state-space model of the parameter estimation of described degradation model _k, b _kand c _k, try to achieve the probability P that estimates of parameters under current k moment condition is parameter true value, namely the estimated value measured of current k moment is the probability of parameter true value is P; Be weighted on average according to described probability P, try to achieve a_s, b_s and the c_s in current k moment:

$m\_s=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}m\left(i\right)\·\frac{P\left(i\right)}{\underset{i=1}{\overset{N}{\mathrm{\Σ}}}P\left(i\right)}m=a,b,c$

Wherein, N is the length of the capacity data of on-line measurement lithium battery to be measured; The parameter a of m (i) corresponding to i-th discharge cycles _k, b _kor c _kor c, P (i) are the parameter a to i-th discharge cycles _k, b _kor c _kthe result carrying out estimating is the probability of the parameter actual value in the current k moment of state-space model;

Step D: using a_s, b_s and c_s of acquisition as the parameter of lithium ion battery state-space model, obtain described lithium ion battery state-space model:

$\left\{\begin{array}{cc}{C}_{k+1}=a\_s\·C_k+b\_s\·e^{(-c\_s)}+w_k& w_k~N(0,Q)\\ y_k=C_k+v_k& v_k~N(0,R)\end{array}\right.$

Wherein, w _kfor the process noise of lithium battery to be measured, obeying average is 0, and variance is the Gaussian distribution of Q, and Q is rational number, y _kfor systematic perspective measured value.

Construct the state-space model of the parameter estimation of described degradation model in the present embodiment

$\left\{\begin{array}{cc}{a}_k=a_{k-1}+w_a& w_a~N(0,Q_a)\\ b_k=b_{k-1}+w_b& w_b~N(0,Q_b)\\ c_k=c_{k-1}+w_c& w_c~N(0,Q_c)\\ C_{k+1}=a_k{C}_k+b_k{e}^{(-c_k)}+v_k& v_k~N(0,R)\end{array}\right.$ Time,

By lithium ion battery experience degradation model constant process is regarded in capacity regeneration as, and what obtain therefore is the final calculation result of part, therefore in order to simplify calculating, by △ t _kturn to constant 1, one can be reduced and propose data and pretreated process, to improve the counting yield of algorithm.

Be weighted on average based on this probability P in the present embodiment, P value is larger, and illustrate that corresponding parameter prediction result is more close to real parameter value, therefore should have higher weight, namely its confidence level is higher.

Embodiment four: present embodiment is to the further restriction based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology described in embodiment one,

In described step 2, the method that pretreated data acquisition merges the AR model that autoregressive coefficient acquiring method is determined is utilized to comprise the steps:

Step e:

Utilize pretreated data and ask for AR model according to AIC criterion model order p;

Step F: utilize pretreated data, asks for the autoregressive coefficient of described AR model respectively according to Yulle-Wallker method and Burg method, adopt the method that dynamic linear merges to export final autoregressive coefficient try to achieve two autoregressive coefficients

Step G: the final autoregressive coefficient obtained according to model order p and the step F of step e acquisition determine AR model.

The acquiring method of AR model order has a lot, as Final prediction error criterion, average information criterion etc., these methods are all the AR model of each order is carried out to the calculating of certain judgment criterion, and the model order corresponding to the minimum value in result of calculation is carried out follow-up modeling and forecasting as best order.In the present embodiment, average information criterion (AverageInformationCriterion, AIC) is adopted to judge model order.

Embodying such as formula shown in (3-3) of AIC criterion.

$\mathrm{AIC}\left(p\right)=N\ln {\mathrm{\σ}}_p^2+2p---(3-3)$

Wherein N is sequential element number, for p rank prediction error variance, p is the model order determined.Usually, the order of AR model more than 10 times, if more than 10 times, not only bad for raising precision of prediction, can not may make precision of prediction decline, and cause consuming time in algorithm implementation, that committed memory is larger phenomenon on the contrary.So in concrete order deterministic process, present embodiment calculates the traversal formula that 1 ~ 10 rank AR model carries out AIC result, and therefrom chooses the minimum order of AIC result of calculation as the final modeling order of present embodiment.

Utilize formula (3-3) to calculate AIC value corresponding to p=1 ~ 10 different order AR model, and judge size, model order p corresponding when AIC value is minimum is the optimum AR model order for current modeling data.Concrete implementation method is as follows:

A) the raw data input for training the capacity data F of modeling to judge as order is extracted;

B) standardization is carried out to F:

Zero-mean: the average Fmean asking for training modeling data F, can obtain the sequence f=F-Fmean of zero-mean;

Variance criterion: the standard deviation sigma asking for the modeling data f after zero-mean _f, obtain standardized data Y=f/ σ _f;

Whether the modeling data c) after criterion is applicable to setting up AR model, namely judges coefficient of autocorrelation and PARCOR coefficients truncation characteristic:

0 step autocovariance: $R_0=\underset{i=1}{\overset{L1}{\mathrm{\Σ}}}\frac{Y^2\left(i\right)}{L1}---(3-9)$

1 ~ 20 step autocovariance: $R\left(k\right)=\underset{i=k+1}{\overset{L1}{\mathrm{\Σ}}}\frac{Y\left(i\right)\·Y(i-k)}{L1}(k=\mathrm{1,2},...,20)---(3-10)$

Coefficient of autocorrelation: x=R/R ₀(3-11)

According to result of calculation, draw coefficient of autocorrelation curve, judge truncation characteristic, if truncation, be applicable to MA modeling.

Partial correlation coefficient: solve Yule-Wallker equation, draws partial correlation coefficient curve according to solving result, judges truncation characteristic, if truncation, is applicable to AR modeling.

Research shows, MA model can be similar to by the AR model of high-order, therefore, if data are applicable to MA modeling, also can carry out the prediction of AR model modeling to it with regard to illustrating.

D) AIC calculates:

Calculated by coefficient of autocorrelation: S=[R ₀, R (1), R (2), R (3)] and (3-12)

Calculate Toeplitz matrix: G=toeplitz (S) (3-13)

Calculating parameter: W=G ^-1[R (1), R (2), R (3), R (4)] ^t(3-14)

Model residual variance calculates: ${\mathrm{\σ}}_p^2=\frac{1}{L1-p}\underset{t=p+1}{\overset{L1}{\mathrm{\Σ}}}{[Y\left(t\right)-\underset{i=1}{\overset{p}{\mathrm{\Σ}}}W\left(i\right)\·Y(t-i)]}^2---(3-15)$

AIC calculates such as formula (3-3).

E) judge to be Optimal order by the model order p that AIC minimum value is corresponding.

F) asking for, for follow-up modeling of best model order under AIC criterion is carried out respectively to each modeling data sample of each battery data collection.

Special needs to be pointed out is, in algorithm principle introductory section present embodiment, the coefficient of AR model is carried out to the estimation of fusion way, each step merges to be needed to carry out the superposition of p to coefficient, and calculated amount is larger.So in actual applications, present embodiment adopts estimated parameter respectively and carries out the method predicted, obtains respective prediction capacity respectively, carries out dynamic superpose to capacity data, decrease the number of times of superposition, reduce calculated amount.But the two is identical in itself, the just different implementation methods of same thinking.Described Yule-Wallker method and Burg method:

(1) Yule-Wallker method

Yule-Wallker method, also known as correlation method, is solved by Yule-Wallker equation, and Yule-Wallker equation is as follows:

Matrix form is as follows:

In calculating process, utilize the parameter of AR (0) and AR (1) to ask for the parameter of AR (2) as starting condition, again according to the parameter of the parametric solution AR (3) of gained, gone out the parameter of AR (p) by Recursive Solution, recursion completes the parameter that can obtain AR model.

(2) Burg method

Burg method utilizes filtering error f forward _n,tfiltering error b backward _n,t, obtain and ensure that filtering error power is minimum calculate according to Levinson algorithm again forward filtering error, backward filtering error and filtering error power definition as follows:

$e_n^2=\frac{1}{2(N-n)}(\underset{t=n+1}{\overset{N}{\mathrm{\Σ}}}{f}_{n,t}^2+\underset{t=N-(n+1)}{\overset{1}{\mathrm{\Σ}}}{b}_{n,t}^2)---(3-8)$

Order can solve and ensure that filtering error power is minimum

The way of asking for of autoregressive coefficient has a variety of, and when enough large for sample, the coefficient that distinct methods obtains is close.But for the Small Sample Database collection that battery capacity data is such, the number of data is less, a kind of independent method ask for the real features that result is difficult to reflect exactly data.Therefore, select Yulle-Wallker method and Burg method independently to ask for model coefficient in present embodiment, then the method for carrying out simple dynamic linear combination export final coefficient results.Concrete parameter acquiring method is as follows:

Use Matlab to carry AR model modeling function arburg.m and aryule.m, utilize identical history modeling data computation model autoregressive coefficient respectively, obtain independently coefficient and ask for result with ;

Original fusion FACTOR P is set ₁and P ₂;

Along with the increase of prediction step, dynamic conditioning fusion coefficients: P ₁=P ₁-f (i), P ₂=P ₂+ f (i), wherein i is prediction step.It is pointed out that f (i) needs constantly to attempt adjustment in actual prediction process, but for same class battery, just no longer change once the concrete form determining f (i).That is, for a class battery characteristics, construct a kind of dynamic fusion coefficient, to a certain extent, this dynamic fusion coefficient also represent the degenerative character of a certain battery;

Fusion coefficients calculates: using the coefficient of this coefficient as the final AR model in order to capacity long-term degradation trend prediction.

Determining model order and model parameter, AR(p) model just set up.Then, utilize pretreated data, before input current time, the state value in p moment is as initial condition data (p is AR model order), substitutes into shown model, can obtain current status predication result.According to the continuous iterative computation of such step, just can obtain the battery capacity prediction result in current K moment, also just obtain degradation in capacity long-term forecasting output data set ARpredict.

Embodiment five: present embodiment is to the further restriction based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology described in embodiment three,

In described step B: utilize pretreated data, adopt EKF method to carry out linearization, state estimation and state updating to the state-space model of the parameter estimation of described degradation model, determine the parameter a in the current K moment of described state-space model _k, b _kand c _kmethod be:

Step B1: linearization is carried out to the state-space model of the parameter estimation of described degradation model:

To the state transition equation input state transition matrix F of the state-space model of the parameter estimation of described degradation model _kdescribed F _kfor:

$F_k=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right];$

Taylor expansion carried out to the observation equation of the state-space model of the parameter estimation of described degradation model and utilizes its first-order section to carry out the linearization approximate of nonlinear equation, obtaining the matrix H of the observation equation after linearization _k:

$H_k=[C_k,e^{-c_k},-b\·e^{-c_k}];$

Step B2: adopt the parameter of kalman filter method to the current K moment of the state-space model of the parameter estimation of the described degradation model after linearization to carry out state estimation and renewal:

Estimated by the parameter of the state transition equation after linearization to the current K moment of described model, obtain the estimated value of the parameter in current k moment:

[a _k ^-,b _k ^-,c _k ^-]=[a _k-1 ⁺,b _k-1 ⁺,c _k-1 ⁺]

$P_k^{-}=F_k{P}_{k-1}^{+}{F}_k^T+W_k{Q}_k{W}_k^T$

Wherein, a _k ^-, b _k ^-and c _k ^-represent k moment parameter a respectively _k, b _kand c _kestimated value, a _k-1 ⁺, b _k-1 ⁺and c _k-1 ⁺represent k-1 moment parameter a respectively _k, b _kand c _kupdated value, for the estimated value of the state covariance matrix of k moment lithium battery to be measured, 1 is the updated value of the state covariance matrix of k-1 moment lithium battery to be measured, W _kfor the process noise matrix of coefficients of lithium battery to be measured, Q _kfor the variance of described process noise;

The estimated value of the parameter in described current k moment is brought in the observation equation of the state-space model of the parameter estimation of degradation model, obtains the estimated value of battery capacity observed reading

The estimated value of described battery capacity observed reading and the observed reading true value of battery capacity are compared and obtains measuring remaining difference covariance and obtain optimum kalman gain that described estimated value is corrected $K_k=P_k^{-}{H}_k^T{S}_k^{-1},$ Utilize formula ${{[a}_k}^{+},{b_k}^{+},{c_k}^{+}]=[{a_k}^{-},{b_k}^{-},{c_k}^{-}]+K_k(C_k-C_k^{~})$ And formula carry out based on the state updating under minimum variance principle to the estimated value of the parameter in each moment of state-space model, obtain the parameter a in each moment of surely described state-space model _k, b _kand c _k,

Wherein, the discharge capacity in the kth cycle calculated based on estimated parameter, C _k-1for the discharge capacity in-1 cycle of kth of on-line measurement lithium battery to be measured, S _kthe covariance matrix measuring remaining difference, H _kfor the observing matrix of lithium battery to be measured, for the updated value of the state covariance matrix of k moment lithium battery to be measured, V _kfor the observation noise matrix of coefficients of lithium battery to be measured, R _kfor the variance of the observation noise of lithium battery to be measured, a _k ⁺, b _k ⁺and c _k ⁺represent k moment parameter a respectively _k, b _kand c _kupdated value, I is unit matrix.

In linearizing process, the process noise of lithium battery to be measured and observation noise are linear superposition noise, therefore linearization process noise and observation noise matrix of coefficients are such as formula shown in (2-18), (2-19).

$W_k=\left[\begin{array}{ccc}\frac{{\∂f}_a}{{\∂w}_a}& \frac{{\∂f}_a}{{\∂w}_b}& \frac{{\∂f}_a}{{\∂w}_c}\\ \frac{{\∂f}_b}{{\∂w}_a}& \frac{{\∂f}_b}{{\∂w}_b}& \frac{{\∂f}_b}{{\∂w}_c}\\ \frac{{\∂f}_c}{{\∂w}_a}& \frac{{\∂f}_c}{{\∂w}_b}& \frac{{\∂f}_c}{{\∂w}_c}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]---(2-18)$

$V_k=\frac{\∂h}{\∂v}=1---(2-19)$

Because each noise is independent mutually, then there is lithium battery process noise covariance matrix Q to be measured such as formula (2-20).

$Q=\left[\begin{array}{ccc}{Q}_{a,a}& Q_{a,b}& Q_{a,c}\\ Q_{b,a}& Q_{b,b}& Q_{b,c}\\ Q_{c,a}& Q_{c,b}& Q_{c,c}\end{array}\right]=\left[\begin{array}{ccc}{Q}_a& 0& 0\\ 0& Q_b& 0\\ 0& 0& Q_c\end{array}\right]---(2-20)$

Cycle life of lithium ion battery Forecasting Methodology of the present invention, be exactly the capacity long-term degradation prediction output valve ARpredict of the AR model that clear data is driven, input as the observed reading true value under Kalman filtering capacity predict framework after ARpredict superposes with white Gaussian noise, replace and upgrade link formula in like this, state updating link based on model prediction Output rusults just replaces to the state updating based on data-driven prediction Output rusults, utilize the characteristic information of data itself to upgrade the capacity estimation value based on experience degradation model, model information and data message are combined.

Before based on Kalman filtering algorithm prediction link being carried out to battery capacity based on the method for EKF method and AR Model Fusion in the present invention, need to carry out AR model modeling and ARpredict prediction output procedure, and obtain the true observation input upgrading link on this basis, replace and use based in the EKF algorithm of model merely carry out the process of state updating, the embodiment based on EKF method and AR Model Fusion is as follows:

1. the acquisition of state transition equation: utilize known modeling training dataset, based on expanded Kalman filtration algorithm through carrying out linearization, state estimation and state updating to state equation, realize carrying out dynamic estimation to experience degradation model parameter, and based on prediction probability, parameter weighting mean value computation is carried out to the model parameter that status tracking obtains, determine the model parameter used in later stage forecasting process.

2. observe the acquisition of true value:

(1) history modeling data collection is utilized to carry out modeling feasibility analysis: to carry out standardization to modeling data, calculate partial autocorrelation and coefficient of autocorrelation, judge its truncation characteristic, if PARCOR coefficients truncation, then be applicable to AR model modeling, if coefficient of autocorrelation truncation, be then applicable to MA model modeling, from preliminary analysis, MA model can be similar to by high-order AR model, therefore, if coefficient of autocorrelation truncation, also can judge that data sample is applicable to AR modeling.

(2) through type (3-12), (3-13), (3-14), (3-15), (3-3) carry out 1-10 rank AR model AIC and calculate, and based on AIC criterion, judge the best AR model order for current training sample.

(3) utilize Burg method and Yule-Wallker method to carry out the calculating of AR model coefficient and respective volume prediction output, adopt performance coeffcient to merge two kinds of methods, obtain fusion forecasting and export ARpredict.

3., through 1 and 2, lithium ion battery state-space model can be obtained

$\left\{\begin{array}{cc}{C}_{k+1}=a\_s\·C_k+b\_{s\·e}^{(-c\_s)}+w_k& w_k~N(0,Q)\\ y_k=C_k+v_k& v_k~N(0,R)\end{array}\right.,$ Pattern of fusion RUL forecasting process can be carried out:

(1) state estimation: utilized the lithium ion battery experience degradation model and the predicting the outcome of a upper moment that obtain in 1, and carried out $C_k^{-}=a\_s\·C_{k-1}^{+}+b\_s\·e^{(-c\_s)}$ With $P_k^{-}=F_k{P}_{k-1}^{+}{F}_k^T+Q_k$ The estimation of shown current time battery capacity.

(2) pattern of fusion state updating process: in this link, utilizes the capacity long-term degradation obtained based on data-driven algorithm in 2 to predict the outcome ARpredict structure observation true value sequence, and carries out state updating:

A () determines observation true value sequence:

C_real=ARpredict+a _t(3-16)

Wherein ARpredict is AR model prediction output sequence, and with modeling data unique characteristics, this feature and lithium battery model to be measured have nothing to do, and are only conceived to the variation characteristic of data self.A _tfor obedience variance is W, average is the white Gaussian noise of 0, i.e. observation noise a _t~ N (0, W).Namely two parts superposition constitutes and upgrades the sequence of observations of link with observation noise.

B () utilizes formula computation and measurement remaining difference covariance, and through type on this basis calculate optimum kalman gain, then utilize following formula to carry out the renewal process of state:

$C_k^{+}=C_k^{-}+K_k[C\_\mathrm{real}-C_k^{-}]---(3-17)$

In formula, C_real is the observed reading true value sequence based on data-driven algorithm determined in step (a), the definition of all the other parameters and formula identical.In this step, achieve data-driven algorithm and the fusion based on model method, the long-term forecasting Output rusults utilizing data-driven algorithm to obtain upgrades the state estimation result obtained based on model algorithm.That is, utilize the capacity predict result of the data characteristics of capacity data to the representative lithium battery interior state to be measured transfer characteristic obtained based on experience degradation model itself to carry out upgrading and correct.Like this, incorporated the characteristic information of data itself at the algorithm based on model, achieved the fusion of algorithm, the predicated error brought for experience degradation model bad adaptability plays certain correcting action.

(3) based on the capacity estimation in previous step and renewal, the battery capacity of each charge and discharge cycles after prediction starting point is made prediction, obtains the prediction output sequence of the battery capacity under fusion forecasting algorithm.

4. so far complete the prediction of battery capacity, by the comparison of failure threshold and capacity predict result, corresponding RUL predicted value can be provided, the error of capacity predict and RUL prediction can be analyzed simultaneously, compare with technical requirement.

In concrete confirmatory experiment, utilize the open battery data collection of NASAPCoE, certain battery testing data sample selected is as the capacity data of on-line measurement residual life lithium battery to be measured, carry out the length of described data sample being divided into data that length is L1 and length is the data of L2, length is the capacity data that in the digital simulation actual application of L1, on-line measurement obtains, for determining lithium ion battery state-space model training data, length is that the data of L2 belong to unknown data in the online life prediction process of reality, namely need the data utilizing prediction algorithm to predict it, in specific experiment, this part data are for verifying lithium ion battery residual life prediction effect, the separation of L1 and L2 is prediction starting point, the contrast experiment of different prediction starting point is carried out equally for same data sample, still select 30% of data total length, 50% and 70%(i.e. early stage, mid-term and later stage) as model training data set, checking prediction algorithm is for the adaptive faculty of the training data of same battery sample different length.Meanwhile, select 6 independently battery testing sample carry out prognostic experiment respectively, the Generalization Ability of verification algorithm between different cells.

One, NASAPCoE lithium ion battery RUL prognostic experiment

The present invention utilizes 5,6, No. 18 batteries of NASA normal temperature normal deterioration to carry out the checking of algorithm.In experimental verification process, according to NASAPCoE battery testing condition, delimiting 1.38Ah is battery capacity failure threshold, be roughly 70% of rated capacity, to each data sample, carry out predicting that starting point is in the prognostic experiment of the different reference positions in early stage (30%) of whole life cycle, mid-term (50%) and later stage (70%) for this data length, data before prediction starting point are in order to simulate the battery capacity data of actual online acquisition, experiment is launched to different sample, observes the prediction effect of algorithm.

For No. 5, NASAPCoE center battery, model and forecast process is described in detail below.

1. remove: remove all variablees in work space, Close All the Windows;

2. raw data imports: by being loaded into data and carrying out pre-service to data, import the capacity data being used for modeling:

(1) be loaded into B0005.mat file by load function, and extract the battery capacity data Capacity in discharge cycles;

(2) normal capacity threshold value U_MAX=1.85 is set, U_MIN=1.20, the singular point peeled off is rejected, and supplement corresponding capacity data according to the capability value of previous step;

(3) for orthogenesis too visibility point, adopt the average in proximity to replace the former data of this point, avoid excessive capacity regeneration effect algorithm convergence.

3. optimum configurations: dividing data, training modeling data length is still L ₁=60,80,100, U=1.38Ah is failure threshold;

4. based on the model parameter estimation of expanded Kalman filtration algorithm: by being loaded into data and carrying out pre-service to data, import the capacity data being used for modeling, lithium battery state to be measured is followed the tracks of, and carries out parameter estimation according to training data set experience degradation model:

(1) observation equation h is defined such as formula the d formula in (2-12) and the partial derivative matrix H being obtained observation equation by equation linearization process _kshown in (2-17);

(2) filtering initialization, defines the constant matrices occurred in algorithm.

Lithium battery state (parameter) initial value to be measured: [a ₀; b ₀; c ₀]=[1; 10; 10]

Status predication covariance matrix initial value: $P=\left[\begin{array}{ccc}10& 0& 0\\ 0& 10& 0\\ 0& 0& 10\end{array}\right]$

Constant matrices in lithium battery state model to be measured: $F=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right];$ V=1； $W=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Lithium battery noise parameter to be measured: $Q=\left[\begin{array}{ccc}0.0001& 0& 0\\ 0& 0.0001& 0\\ 0& 0& 0.0001\end{array}\right];$ R=0.0001

(3) input state tracking section concept of reality measured value and true capacity data Y_real=Capacity (1:L ₁);

(4) initialization space of matrices: the space MM of stored parameter estimated value, stores the space PP of covariance matrix value, stores the space Cappredict of the capability value that estimated parameter calculates;

(5) EKF parameter estimation procedure, carries out state estimation and state updating to model parameter value, namely utilizes expanded Kalman filtration algorithm to carry out battery status tracking.

State estimation: such as formula [a _k ^-, b _k ^-, c _k ^-]=[a _k-1 ⁺, b _k-1 ⁺, c _k-1 ⁺], based on die sinking in upper a period of time shape parameter predicted value and state transition equation, carry out the estimation of current time model parameter value;

State updating: carry out state updating process such as formula (2-23) ~ (2-27), obtains the parameter updated value after based on the correction of observation true value;

Capacity estimation value: $C_k^{~}=a_k^{-}{C}_{k-1}+b_k^{-}{e}^{(-c_k^{-})}---(2-23)$

Measure remaining difference covariance: $S_k=H_k{P}_k^{-}{H}_k^T+V_k{R}_k{V}_k^T---(2-24)$

Kalman gain: $K_k=P_{k|}^{-}{H}_k^T{S}_k^{-1}---(2-25)$

State updating: $[{a_k}^{+},{b_k}^{+},{c_k}^{+}]=[{a_k}^{-},{b_k}^{-},{c_k}^{-}]+K_k(C_k-C_k^{~})---(2-26)$

$P_k^{+}=(I-K_k{H}_k)P_k^{+}---(2-27)$

Result is preserved into default space of matrices: MM, PP and Cappredic.

(6) according to current estimates of parameters be the prediction probability P of parameter true value,

According to formula it is average that m=a, b, c carry out parameter weighting, obtains the model parameter a_s needed for capacity predict link, b_s and c_s.

5.AR model modeling and forecasting process:

(1) extract history modeling data according to prediction starting point, and carry out standardization to historical data, the average making the capacity data after processing is 0, and variance is 1, and detailed process is shown in the step that model order is asked for;

(2) calculate coefficient of autocorrelation and partial correlation coefficient, draw respective graphical, judge truncation characteristic, if partial correlation coefficient truncation, then meet AR modeling conditions, detailed process is shown in the step that model order is asked for;

(3) AIC criterion is utilized, the AIC value of carrying out 1-10 rank AR model calculates, step is such as formula shown in (3-12) ~ (3-15), the order selecting AIC value minimum is as the best order p of modeling, and pass through the applicability of white noise verification judgment models, even between white noise, related coefficient is namely approximate separate close to 0, then can think that model is suitable for;

(4) carry out AR model modeling and prediction respectively by Yule-Wallker and Burg method, obtain the capacity predict value under different parameters computing method respectively, Mobile state additive fusion of going forward side by side.By constantly testing and can obtaining, for NASAPCoE battery data, f (i)=0.0011 √ i (i=1 in fusion coefficients, 2, ...), determine this factor and be applied to the forecasting process of NASAPCoE center all batteries sample, finally obtain volume output sequence A Rpredict, the observed reading true value as subsequent step uses;

6. the battery capacity prediction process under pattern of fusion prediction framework:

(1) definition status equation of transfer: h is such as formula a formula in (2-29), and model parameter is obtained by the 4th step;

(2) filtering initialization: F=a_s; H=1; Q=0.0001; R=0.0001;

(3) actual value of input capacity predicted portions observed reading, to be superposed with white Gaussian noise by the capacity long-term degradation trend prediction value ARpredict of the AR model obtained in step 5 and forms, be i.e. C_real=ARpredict+a _t, wherein a _tfor single-point white Gaussian noise.

(4) original state is arranged, and adopt last capability value of history training modeling data as original state value, this is also true capacity value uniquely known in forecasting process, and the setting of covariance is identical with (2) in 2.3.2.1 step 4;

(5) space of matrices is arranged: the space MM of stored parameter estimated value, stores the space PP of covariance matrix value, stores the space Cappredict of the capability value that estimated parameter calculates;

(6) state estimation: such as formula (2-30), (2-31) based on upper moment status predication value and a state transition equation, carry out the estimation of current state value and battery capacity value;

$C_k^{-}=a\_s\·C_{k-1}^{+}+b\_s\·e^{(-c\_s)}---(2-30)$

$P_k^{-}=F_k{P}_{k-1}^{+}{F}_k^T+Q_k---(2-31)$

(7) state updating: carry out calculating based on the optimum kalman gain of the remaining difference of observation such as formula (2-32), (2-33), (2-35) and (3-17), realize data-driven algorithm and the fusion based on model algorithm;

$S_k=H_k{P}_k^{-}{H}_k^T+R_k---(2-32)$

$K_k=P_k^{-}{H}_k^T{S}_k^{-1}---(2-33)$

$C_k^{+}=C_k^{-}+K_k[C\_\mathrm{real}-C_k^{-}]---(2-34)$

$P_k^{+}=(I-K_k{H}_k)P_k^{-}---(2-35)$

(8) residual life exports: found the index position being less than failure threshold U in forecasting sequence by find function, its minimum value deducts 1 and is RUL value, can calculate true residual life RUL ₁with prediction residual life RUL ₂;

7. error calculation: the absolute error calculating the absolute average error of battery capacity prediction, root-mean-square error and RUL prediction according to formula (2-7), (2-8) and (2-9) respectively, quantitatively evaluating is carried out to algorithm, in addition the relative error of calculated capacity prediction and RUL prediction, compares with technical indicator;

Capacity predict mean absolute error: $\mathrm{MAE}=\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}|Y\_\mathrm{real}\left(i\right)-Y\_m\left(i\right)|---(2-7)$

Capacity predict root-mean-square error: $\mathrm{RMSE}=\sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{(Y\_\mathrm{real}\left(i\right)-Y\_m\left(i\right))}^2}---(2-8)$

Life prediction absolute error: errul=|RUL ₂-RUL ₁| (2-9)

8. Output rusults, judges whether to meet technical requirement.

Carried out the predicting residual useful life experiment of No. 5 batteries in NASAPCoE center by above-mentioned modeling procedure, for 60 modelings in early stage, 1 ~ 10 rank AR model AIC result of calculation as shown in Figure 2, AIC minimum value AICmin=-0.9875, the best order p=1 of corresponding modeling.Sequence of observations input fusion forecasting algorithm frame after AR model long-term forecasting output being superposed with noise, can obtain RUL and predict the outcome.

For 80 modelings in mid-term of No. 5 batteries, 1 ~ 10 rank AR model AIC result of calculation as shown in Figure 2, AIC minimum value AICmin=-117.1, the best order p=1 of corresponding modeling.Sequence of observations input fusion forecasting algorithm frame after AR model long-term forecasting output being superposed with noise, can obtain RUL and predict the outcome.

For 100 modelings of NASA5 battery later stage, 1 ~ 10 rank AR model AIC result of calculation as shown in Figure 4, AIC minimum value AICmin=-174.9, the best order p=1 of corresponding modeling.Sequence of observations input fusion forecasting algorithm frame after AR model long-term forecasting output being superposed with noise, can obtain RUL and predict the outcome.

This method adopts same method to carry out merging the modeling under framework and RUL prognostic experiment to No. NASAPCoE6 and No. 18 battery capacity data, and parameter adjustment is consistent with above-mentioned.And then obtain No. 6, No. 18 battery RUL and predict the outcome.The numerical result of AIC minimum value and corresponding best modeled order is as follows:

NASAPCoE6 battery AIC result of calculation:

In earlier stage 60 modeling: AICmin=-25.29; Best order p=1.

Mid-term 80 modeling: AICmin=-113.7; Best order p=1.

Later stage 100 modeling: AICmin=-201.1; Best order p=1.

NASAPCoE18 battery AIC result of calculation:

In earlier stage 30 modeling: AICmin=-7.936; Best order p=1.

Mid-term 50 modeling: AICmin=-2.847; Best order p=1.

Later stage 70 modeling: AICmin=-40.51; Best order p=1.

By 3 battery samples based on difference prediction starting point predicting the outcome and quantization error result gathers, concrete result of calculation is as shown in table 1.

Table 1NASABattery experimental result gathers

Can be found out by prediction curve intuitively, prediction degradation trend and true degradation trend obviously more identical, precision of prediction promotes to some extent.This illustrates the introducing of data characteristics, improves the adaptability of algorithm for different battery sample, can predict the different degenerative characters of different battery sample better.

Can be found out by the quantization error data in table 1, no matter be for different battery sample or the different modeling data collection for same battery sample, predicting the outcome of battery capacity can meet the error requirements of expection, and predicated error scope about 2% is to about 18%, and precision of prediction is higher.For the predicated error of battery RUL, the absolute value of error is less, is generally below 10 charge and discharge cycles.For the experiment sample of the overwhelming majority, the relative error of RUL can meet setting requirement, but has indivedual sample as the early stage of No. 6 battery later stages, No. 18 batteries, and Relative Error has exceeded 40%.Can find out, the prediction effect of RUL promotes to some extent, but can't meet the demands completely.For different individual cells, the otherness of prediction effect reduces, and adaptability improves.This illustrates, the problem of introducing to Model suitability difference of data-driven algorithm makes moderate progress, and serves certain improved action by data characteristics to aspect of model correction.

But for the prognostic experiment of the different modeling data sample of same battery, still show some differences, the Relative Error ratio of some battery different times even reaches about 10 times, and algorithm is still poor for the adaptability of different times.This is because no matter period of degenerating in difference is state transfer relationship or the feature of data own, all there will be larger difference, data-driven algorithm is still predict for modeling data in early stage, and the data characteristics of introducing is also the information from early stage, modeling data comprised.Therefore the change of the degradation in capacity feature doped in later stage degenerative process is difficult to.That is, the introducing of data-driven algorithm is difficult to the predictive ability that boosting algorithm itself changes different times degradation trend.

Claims (3)

1., based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology, it is characterized in that,

It comprises the steps:

Step one: the capacity data of on-line measurement lithium ion battery to be measured, preserve data and pre-service is carried out to described data:

Reject for point unusual in described data, the capacity orthogenesis excessive for amplitude carries out the level and smooth of trend; Step 2: the parameter based on EKF method determination lithium ion battery state-space model:

According to lithium ion battery experience degradation model and AR Construction of A Model lithium ion battery state-space model, utilize pretreated data and determine the parameter of described lithium ion battery state-space model according to EKF method; The sequence of observations after the prediction output valve of described AR model superposes with observation noise is the observed reading of the battery capacity of described lithium ion battery state-space model, and described AR model is the AR model utilizing pretreated data acquisition to merge autoregressive coefficient acquiring method to determine;

Step 3: state estimation is carried out to lithium ion battery to be measured according to the lithium ion battery state-space model that step 2 is set up, the output of described AR model is utilized to carry out the state updating of lithium ion battery to be measured, described lithium ion battery state-space model obtains the battery capacity data of each charge and discharge cycles, and described data are compared acquisition lithium ion battery residual life with the failure threshold of lithium ion battery to be measured.

2. according to claim 1 based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology, it is characterized in that, in step 2, described according to lithium ion battery experience degradation model and AR Construction of A Model lithium ion battery state-space model, utilize pretreated data and determine that according to EKF method the method for the parameter of described lithium ion battery state-space model comprises the steps:

Steps A: according to lithium ion battery experience degradation model construct the state-space model of the parameter estimation of described degradation model:

$\left\{\begin{array}{cc}{a}_k=a_{k-1}+w_a& w_a~N(0,Q_a)\\ b_k=b_{k-1}+w_b& w_b~N(0,Q_b)\\ c_k=c_{k-1}+w_c& w_c~N(0,Q_c)\\ C_{k+1}=a_k{C}_k+b_k{e}^{(-c_k)}+v_k& v_k~N(0,R)\end{array}\right.$

Wherein $\left\{\begin{array}{cc}{a}_k=a_{k-1}+w_a& w_a~N(0,Q_a)\\ b_k=b_{k-1}+w_b& w_b~N(0,Q_b)\\ c_k=c_{k-1}+w_c& w_c~N(0,Q_c)\end{array}\right.$ For state transition equation, $C_{k+1}=a_k{C}_k+b_k{e}^{(-c_k)}+v_k$ For observation equation, a _k, b _kand c _kbe respectively the coulombic efficiency η in experience degradation model described in the current k moment _{c, k}, regenerated capacity parameter beta _{1, k}with regenerated capacity parameter beta _{2, k}/ △ t _kestimated value;

C _kfor the discharge capacity in k moment in the degradation in capacity process of lithium ion battery to be measured, C _k+1for the discharge capacity in k+1 moment in the degradation in capacity process of lithium ion battery to be measured, η _{c, k}for the coulombic efficiency in lithium ion battery charge and discharge process to be measured; for lithium ion battery to be measured is at standing time of having a rest section △ t _kthe capacity of interior regeneration; w _a, w _band w _cbe respectively parameter a _k, b _kand c _kthe white Gaussian noise comprised, Q _a, Q _band Q _cbe respectively w _a, w _band w _cvariance, noise w _a, w _band w _cmeet N (0, Q respectively _a), N (0, Q _b) and N (0, Q _c) Gaussian distribution; R is real number; v _kfor the observation noise of lithium ion battery to be measured, v _kobedience average is 0, v _kvariance be the Gaussian distribution of R; Step B: utilize pretreated data, adopts EKF method to carry out linearization, state estimation and state updating to the state-space model of the parameter estimation of described degradation model, determines the parameter a in the current k moment of described state-space model _k, b _kand c _k;

Step C: according to the parameter a in the current k moment of described state-space model _k, b _kand c _k, try to achieve the probability P that estimates of parameters under current k moment condition is parameter true value, be weighted on average according to described probability P, try to achieve a in current k moment _k_ s, b _k_ s and c _k_ s:

$m\_s=\underset{i=1}{\overset{N}{\Σ}}m\left(i\right)\·\frac{P\left(i\right)}{\underset{i=1}{\overset{N}{\Σ}}P\left(i\right)},m=a_k,b_k,c_k$

Wherein, N is the length of the capacity data of on-line measurement lithium ion battery to be measured; The parameter a of m (i) corresponding to i-th discharge cycles _k, b _kor c _k, P (i) is the parameter a to i-th discharge cycles _k, b _kor c _kthe result carrying out estimating is the probability of the parameter actual value in the current k moment of state-space model;

Step D: by a obtained _k_ s, b _k_ s and c _k_ s, as the parameter of lithium ion battery state-space model, obtains described lithium ion battery state-space model:

$\left\{\begin{array}{cc}{C}_{k+1}=a_k\_s\·C_k+b_k\_s\·e^{(-c\_s)}+w_k& w_k~N(0,Q)\\ y_k=C_k+v_k& v_k~N(0,R)\end{array}\right.$

Wherein, w _kfor the process noise of lithium ion battery to be measured, obeying average is 0, and variance is the Gaussian distribution of Q, and Q is rational number, y _kfor systematic perspective measured value.

3. according to claim 2 based on EKF method and AR Model Fusion type cycle life of lithium ion battery Forecasting Methodology, it is characterized in that, in described step B: utilize pretreated data, adopt EKF method to carry out linearization, state estimation and state updating to the state-space model of the parameter estimation of described degradation model, determine the parameter a in the current k moment of described state-space model _k, b _kand c _kmethod be:

Step B1: linearization is carried out to the state-space model of the parameter estimation of described degradation model:

To the state transition equation input state transition matrix F of the state-space model of the parameter estimation of described degradation model _k, described F _kfor:

$F_k=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right];$

Taylor expansion carried out to the observation equation of the state-space model of the parameter estimation of described degradation model and utilizes its first-order section to carry out the linearization approximate of nonlinear equation, obtaining the matrix H of the observation equation after linearization _k:

$H_k=\[C_k,e^{-c_k},-b\·e^{-c_k}\];$

Step B2: adopt the parameter of kalman filter method to the current K moment of the state-space model of the parameter estimation of the described degradation model after linearization to carry out state estimation and renewal:

Estimated by the parameter of the state transition equation after linearization to the current K moment of described state-space model, obtain the estimated value of the parameter in current k moment:

[a _k ^-，b _k ^-，c _k ^-]＝[a _k-1 ⁺，b _k-1 ⁺，c _k-1 ⁺]

$P_k^{-}=F_k{P}_{k-1}^{+}{F}_k^T+W_k{Q}_k{W}_k^T$

Wherein, a _k ^-, b _k ^-and c _k ^-represent k moment parameter a respectively _k, b _kand c _kestimated value, a _k-1 ⁺, b _k-1 ⁺and c _k-1 ⁺represent k-1 moment parameter a respectively _k, b _kand c _kupdated value, for the estimated value of the state covariance matrix of k moment lithium ion battery to be measured, for the updated value of the state covariance matrix of k-1 moment lithium ion battery to be measured, W _kfor the process noise matrix of coefficients of lithium ion battery to be measured, Q _kfor the variance of described process noise;

The estimated value of the parameter in described current k moment is brought in the observation equation of the state-space model of the parameter estimation of degradation model, obtains the estimated value of battery capacity observed reading:

$C_k^{~}=a_k^{-}{C}_{k-1}+b_k^{-}{e}^{(-c_k^{-})};$

The estimated value of described battery capacity observed reading and the observed reading true value of battery capacity are compared and obtains measuring remaining difference covariance and obtain optimum kalman gain that described estimated value is corrected utilize formula $\[{a_k}^{+},{b_k}^{+},{c_k}^{+}\]=\[{a_k}^{-},{b_k}^{-},{c_k}^{-}\]+K_k(C_k-C_k^{~})$ And formula $P_k^{+}=(I-K_k{H}_k)P_k^{+}$ Carry out based on the state updating under minimum variance principle to the estimated value of the parameter in the current k moment of state-space model, obtain the parameter a in the current k moment of described state-space model _k, b _kand c _k,

Wherein, the discharge capacity in the kth cycle calculated based on estimated parameter, C _k-1for the discharge capacity in-1 cycle of kth of on-line measurement lithium ion battery to be measured, S _kthe covariance matrix measuring remaining difference, H _kfor the observing matrix of lithium ion battery to be measured, for the updated value of the state covariance matrix of k moment lithium ion battery to be measured, V _kfor the observation noise matrix of coefficients of lithium ion battery to be measured, R _kfor the variance of the observation noise of lithium ion battery to be measured, a _k ⁺, b _k ⁺and c _k ⁺represent k moment parameter a respectively _k, b _kand c _kupdated value, I is unit matrix.