CN101604005B - Estimation method of battery dump energy based on combined sampling point Kalman filtering - Google Patents

Abstract

The invention relates to an estimation method of battery dump energy based on combined sampling point Kalman filtering, and aims to solve the problem that the prior method cannot meet the requirement of online detection and has poor precision. The estimation method comprises the following steps: firstly, measuring battery terminal voltage yk and battery supply current ik at k moment through a measuring circuit; secondly, showing battery state of charge at each moment by a state equation and an observation equation; and finally, estimating the battery dump energy by adopting standard sampling point Kalman filtering. The estimation method has the advantages that the method can conveniently carry out quick estimation of battery SOC and has fast convergence speed and high estimation precision; moreover, the method is suitable for quick estimation of SOC of various batteries.

Description

A kind of estimation method of battery dump energy based on combined sampling point Kalman filtering

Technical field

The invention belongs to technical field of lithium batteries, be specifically related to a kind of estimation method of battery dump energy based on combined sampling point Kalman filtering.

Background technology

Battery has obtained in fields such as communication, electric system, military equipments using widely as standby power supply.Compare with traditional fuel-engined vehicle, electric automobile can be realized zero-emission, is the main developing direction of following automobile therefore.Battery is directly as the active energy supply part in electric automobile, so the quality of its duty is directly connected to the driving safety and the operational reliability of whole automobile.Good for guaranteeing the battery performance in the electric automobile, prolong electric battery serviceable life, must be in time, exactly the running status of electrolytic cell, battery is carried out rational and effective management and control.

The accurate estimation of battery charge state (State of Charge is hereinafter to be referred as SOC) is a technology most crucial in the battery management system (bms).The SOC of battery can't directly record with a kind of sensor, and it must pass through the measurement to some other physical quantitys, and adopts certain mathematical model and algorithm to estimate to obtain.Battery SOC method of estimation commonly used at present has open-circuit voltage method, ampere-hour method etc.Adopt the open-circuit voltage method, battery must leave standstill the long period reaching steady state (SS), and only is applicable to that the SOC of electric automobile under dead ship condition estimates, can not satisfy online detection requirements; Adopt the ampere-hour method, be subjected to the influence of current measurement precision easily, under high temperature or the violent situation of current fluctuation, precision is very poor.

Summary of the invention

Purpose of the present invention overcomes the deficiencies in the prior art exactly, proposes a kind of battery dump energy method for quick estimating based on combined sampling point Kalman filtering, go for all batteries, and estimated accuracy is higher.

The concrete steps of the estimation method of battery dump energy based on combined sampling point Kalman filtering of the present invention are:

Step (1) records the battery terminal voltage y constantly at k by metering circuit _kWith the powered battery current i _k, k=1,2,3 ...

Each constantly state-of-charge that step (2) is represented battery with state equation and observation equation (State of Charge, SOC)

State equation: $z_{k+1}=f(z_k,i_k)+u_k=z_k-\left(\frac{{\mathrm{\η}}_i\mathrm{\Δt}}{Q_n}\right)i_k+u_k$

Observation equation: $y_k=g(p,z_k,i_k)+v_k=K_0-{\mathrm{Ri}}_k-\frac{K_1}{z_k}-K_2{z}_k+K_3\ln z_k+K_4\ln (1-z_k)+v_k$

Z is the state-of-charge of battery, i.e. dump energy, and z=100% represents that battery electric quantity is in full state, z=0% represents that battery electric quantity is in spent condition; η _iBe the discharge scale-up factor of battery, reflection be discharge rate, temperature, self discharge, factor such as aging influence degree to battery SOC; Q _nBe battery getable specified total electric weight under 25 ℃ of conditions of room temperature, when discharging with the discharge rate of 1/30 times of rated current; R is the internal resistance of battery; K ₀, K ₁, K ₂, K ₃, K ₄Be constant, p=[K ₀R K ₁K ₂K ₃K ₄] ^T, p is the parameter of battery observation model, is a column vector, they are constant to battery of the same type; Δ t is a measuring intervals of TIME, and u is for handling noise, and v is an observation noise, and subscript k is for measuring constantly.

Wherein, discharge scale-up factor η _iDefinite step be:

(a) will be full of the battery of electricity fully with different discharge rate C _i(0＜C _i≤ C, C are the nominal discharge current of battery) constant-current discharge N (N＞10) is inferior, calculates the total electric weight Q of battery under the corresponding discharge rate _i, 1≤i≤N.

(b) simulate Q according to least square method _iWith C _iBetween quafric curve relation, promptly under minimum mean square error criterion, obtain simultaneously and satisfy $Q_i=a{C}_i^2+b{C}_i+c$ (the optimal coefficient a of 1≤i≤N), b, c.

(c) be i at discharge current _kThe time, corresponding discharge scale-up factor η _iFor:

${\mathrm{\η}}_i=\frac{Q_n}{{\mathrm{ai}}_k^2+{\mathrm{bi}}_k+c}$

Herein, optimal coefficient a, b, c only need determine once for the battery of same type, can be used as the remaining capacity estimation that known constant is directly used in all batteries of the same type after determining.

The battery model parameter p adopts equation of the ecentre sampling point Kalman filtering algorithm to determine that its concrete steps are:

(d) under 25 ℃ of conditions of room temperature, with 1/30 times of rated current the battery that is full of electricity is carried out steady current discharge and exhausts until electric weight;

(e) in discharge process, measure battery at s terminal voltage y constantly with time interval Δ t by metering circuit _s, s=0,1,2 ... M, the initial discharging time after wherein the corresponding battery of s=0 is full of, the termination that the corresponding battery electric quantity of s=M exhausts is constantly.

(f) calculate s dump energy z constantly _s, z _s=1-s/M.

(g) an optional initial parameter $p={\hat{p}}_0,$ Setting its square root mean square deviation matrix is

$S_{p_0}=I_6;$ I wherein ₆It is 6 * 6 unit matrix; Choose proportionality constant h, h＞1; Set variable $R_r=\sqrt{{10}^{-3}}{I}_6;$ Set weighting coefficient $W_0^{\left(m\right)}=\frac{h^2-7}{h^2},$ $W_i^{\left(m\right)}=\frac{1}{2h^2},$ $W_i^{\left(c1\right)}=\frac{1}{2h},$ $W_i^{\left(c2\right)}=\frac{\sqrt{h^2-1}}{2h^2},$ i＝1，2，…，12。

To s=1,2 ..., M, (h)～(j) carries out successive iteration as follows:

(h) computing time area update:

The estimated value of computation model parameter

${\hat{p}}_s^{-}={\hat{p}}_{s-1}$

The estimated value of the square root mean square deviation matrix of computation model parameter

$S_{p_s}^{-}=S_{p_{s-1}}+D_{r_{s-1}},$ Wherein, $D_{r_{s-1}}=-\mathrm{diag}\left\{S_{p_{s-1}}\right\}+\sqrt{\mathrm{diag}{\left\{S_{p_{s-1}}\right\}}^2+\mathrm{diag}\left\{R_r\right\}},$ Diag{} is the column vector that the diagonal entry of corresponding matrix constitutes.

(i) calculate

The sampled point sequence

Be 6 * 1 column vectors,

Be 6 * 6 matrixes, so

Be 6 * 13 matrixes.

(j) measure renewal by following various calculating:

The observation sequence of calculating sampling point

Be 6 * 13 matrixes;

The calculating observation sequence

Estimated value

For I row;

The calculating observation sequence Square root mean square deviation matrix

Calculate covariance matrix

Calculate kalman gain K _s: $K_s=(P_{p_s{d}_s}/S_{{\tilde{d}}_s}^T)/S_{{\tilde{d}}_s};$

Calculating parameter upgrades ${\hat{p}}_s={\hat{p}}_s^{-}+K_s(y_s-{\hat{d}}_s^{-});$

Calculate temporary variable U: $U=K_s{S}_{{\tilde{d}}_s};$

The renewal of the square root mean square deviation matrix of computation model parameter $S_{p_s}=\mathrm{cholupdate}\{S_{p_s}^{-},U,-1\};$

Wherein qr{} represents to ask the ORTHOGONAL TRIANGULAR DECOMPOSITION of matrix, and returns the upper triangular matrix that obtains; () ^TBe the transpose of a matrix operation;

Matrix is asked in expression

Cholesky decompose.

By above-mentioned steps, final iteration obtains

Be the estimated battery model parameter that obtains.

To the battery of same type, these parameters only need be determined once, can be used as the remaining capacity estimation that known constant is directly used in all batteries of the same type after determining.

Step (3) adopts standard sample point Kalman filtering to carry out the estimation of battery dump energy, specifically:

1. carry out following initialization procedure:

Initial state

And variance P ₀Be respectively:

${\hat{z}}_0={\mathrm{SOC}}_0=100\%,$ P ₀＝var(z ₀)＝10 ^-2，

Handle noise variance R _wWith observation noise variance R _vBe respectively:

R _w＝10 ^-5，R _v＝10 ^-2

State vector after the expansion

And covariance P ₀ ^aFor:

${\hat{z}}_0^a={\left[\begin{array}{ccc}{z}_0& 0& 0\end{array}\right]}^T,$ $p_0^a=\left[\begin{array}{ccc}{P}_0& 0& 0\\ 0& R_w& 0\\ 0& 0& R_v\end{array}\right]$

Scale parameter γ is:

$\mathrm{\γ}=\sqrt{3}$

Average weighting coefficient w _i ^(m), i=0,1,2 ..., 6 and variance weighted coefficient w _i ^(c), i=0,1,2 ..., 6 are respectively:

$w_0^{\left(m\right)}=0,$ $w_0^{\left(c\right)}=2,$ $w_i^{\left(m\right)}=w_i^{\left(c\right)}=1/6,$ 1≤i≤6

2. adopt the standard sample point Kalman filtering algorithm recursion that circulates:

Measuring k=1 constantly, 2,3 ..., the battery terminal voltage y in the real work that records according to metering circuit _kAnd the supply current i of battery _k, calculate by the following various recursion of carrying out:

(l) according to k-1 extended mode vector constantly

And covariance P _K-1 ^a, calculate all sampled point sequences in this moment

(m) carrying out time domain according to state equation upgrades:

By the sampled point sequence

Upgrade according to state equation calculating sampling point

Sampled point is upgraded

Be weighted, computing mode is estimated

Computing mode is estimated

Variance

(n) finishing measurement according to observation equation according to following formula upgrades:

There is sampled point to upgrade

Calculating measurement according to observation equation upgrades

Upgrade measuring Be weighted, calculate to measure and estimate

Calculate to measure and estimate

Variance

Calculate

With

Cross covariance

Calculate kalman gain K _k: $K_k=P_{z_k{y}_k}{P}_{{\tilde{y}}_k}^{-1}$

Computing mode is upgraded

${\hat{z}}_k=z_k^{-}+K_k(y_k-{\hat{y}}_k^{-})$

Computing mode is upgraded

Variance

$P_{x_k}=P_{x_k}^{-}-K_k{P}_{{\tilde{y}}_k}{K}_k^T$

The resulting state updating value of recursion

Be the estimated battery dump energy that obtains of current time k.The whole circulation recursive process is online finishing, and promptly finishes each estimation of battery dump energy constantly in the battery practical work process synchronously.

The present invention can carry out the Fast estimation of battery SOC easily, this method fast convergence rate, and the estimated accuracy height, and be applicable to the Fast estimation of various battery SOCs.

According to a first aspect of the invention, disclose the measuring amount that a kind of sampling point Kalman filtering method that is used for the estimating battery dump energy is relied on, be respectively the terminal voltage of battery and the supply current of battery.

According to a second aspect of the invention, a kind of state equation and observation equation that is used for the sampling point Kalman filtering of estimating battery dump energy disclosed.Wherein the battery model parameter adopts equation of the ecentre sampling point Kalman filtering algorithm to determine in the observation equation.

According to a third aspect of the invention we, the initial value that a kind of standard sample point Kalman filtering that is used for estimating battery SOC is relied on is disclosed.Comprise initial SOC, the variance of initial SOC, the variance of handling noise and observing noise, and the weights of sampled point correspondence.The value of wherein initial SOC and initial SOC variance needn't be very accurate, in the successive iterations process of sampling point Kalman filtering their very rapid convergence near actual value.

According to a forth aspect of the invention, disclose a kind of application standard sampling point Kalman filtering and carried out the idiographic flow that battery SOC is estimated.Mainly comprise: calculating sampling point is by the weighted value after the state equation conversion, as the estimated value of state, and then the variance by the weighted calculation state estimation; Calculating sampling point is by the weighted value after the observation equation conversion, as the estimated value of observation; Calculate kalman gain; The renewal of computing mode and variance thereof etc.

Embodiment

Concrete grammar based on the estimation method of battery dump energy of combined sampling point Kalman filtering is:

Step (1) records the battery terminal voltage y constantly at k by metering circuit _kWith the powered battery current i _k, k=1,2,3 ...

Step (2) is represented each state-of-charge constantly, wherein state equation of battery with state equation and observation equation: $z_{k+1}=f(z_k,i_k)+u_k=z_k-\left(\frac{{\mathrm{\η}}_i\mathrm{\Δt}}{Q_n}\right)i_k+u_k$

Observation equation: $y_k=g(p,z_k,i_k)+v_k=K_0-{\mathrm{Ri}}_k-\frac{K_1}{z_k}-K_2{z}_k+K_3\ln z_k+K_4\ln (1-z_k)+v_k$

Z is the state-of-charge of battery, i.e. dump energy, and z=100% represents that battery electric quantity is in full state, z=0% represents that battery electric quantity is in spent condition; η _iBe the discharge scale-up factor of battery, reflection be discharge rate, temperature, self discharge, factor such as aging influence degree to battery SOC; Q _nBe battery getable specified total electric weight under 25 ℃ of conditions of room temperature, when discharging with the discharge rate of 1/30 times of rated current; R is the internal resistance of battery; K ₀, K ₁, K ₂, K ₃, K ₄Be constant, p=[K ₀R K ₁K ₂K ₃K ₄] ^T, p is the parameter of battery observation model, is a column vector, they are constant to battery of the same type; Δ t is a measuring intervals of TIME, and u is for handling noise, and v is an observation noise, and subscript k is for measuring constantly.

Wherein, discharge scale-up factor η _iDefinite step be:

(a) will be full of the battery of electricity fully with different discharge rate C _i(0＜C _i≤ C, C are the nominal discharge current of battery) constant-current discharge N (N＞10) is inferior, calculates the total electric weight Q of battery under the corresponding discharge rate _i, 1≤i≤N.

(b) simulate Q according to least square method _iWith C _iBetween quafric curve relation, promptly under minimum mean square error criterion, obtain simultaneously and satisfy $Q_i=a{C}_i^2+b{C}_i+c$ (the optimal coefficient a of 1≤i≤N), b, c.

(c) be i at discharge current _kThe time, corresponding discharge scale-up factor η _iFor:

${\mathrm{\η}}_i=\frac{Q_n}{{\mathrm{ai}}_k^2+{\mathrm{bi}}_k+c}$

Herein, optimal coefficient a, b, c only need determine once for the battery of same type, can be used as the remaining capacity estimation that known constant is directly used in all batteries of the same type after determining.

The battery model parameter p adopts equation of the ecentre sampling point Kalman filtering algorithm to determine that its concrete steps are:

(d) under 25 ℃ of conditions of room temperature, with 1/30 times of rated current the battery that is full of electricity is carried out steady current discharge and exhausts until electric weight;

(e) in discharge process, measure battery at s terminal voltage y constantly with time interval Δ t by metering circuit _s, s=0,1,2 ... M, the initial discharging time after wherein the corresponding battery of s=0 is full of, the termination that the corresponding battery electric quantity of s=M exhausts is constantly.

(f) calculate s dump energy z constantly _s, z _s=1-s/M.

(g) an optional initial parameter $p={\hat{p}}_0,$ Setting its square root mean square deviation matrix is

$S_{p_0}=I_6;$ I wherein ₆It is 6 * 6 unit matrix; Choose proportionality constant h, h＞1; Set variable $R_r=\sqrt{{10}^{-3}}{I}_6;$ Set weighting coefficient $W_0^{\left(m\right)}=\frac{h^2-7}{h^2},$ $W_i^{\left(m\right)}=\frac{1}{2h^2},$ $W_i^{\left(c1\right)}=\frac{1}{2h},$ $W_i^{\left(c2\right)}=\frac{\sqrt{h^2-1}}{2h^2},$ i＝1，2，…，12。

To s=1,2 ..., M, (h)～(j) carries out successive iteration as follows:

(h) computing time area update:

The estimated value of computation model parameter

${\hat{p}}_s^{-}={\hat{p}}_{s-1}$

The estimated value of the square root mean square deviation matrix of computation model parameter

$S_{p_s}^{-}=S_{p_{s-1}}+D_{r_{s-1}},$ Wherein, $D_{r_{s-1}}=-\mathrm{diag}\left\{S_{p_{s-1}}\right\}+\sqrt{\mathrm{diag}{\left\{S_{p_{s-1}}\right\}}^2+\mathrm{diag}\left\{R_r\right\}},$ Diag{} is the column vector that the diagonal entry of corresponding matrix constitutes.

(i) calculate The sampled point sequence

Be 6 * 1 column vectors,

Be 6 * 6 matrixes, so

Be 6 * 13 matrixes.

(j) measure renewal by following various calculating:

The observation sequence of calculating sampling point

Be 6 * 13 matrixes;

The calculating observation sequence

Estimated value

For

I row;

The calculating observation sequence

Square root mean square deviation matrix

Calculate covariance matrix

Calculate kalman gain K _s: $K_s=(P_{p_s{d}_s}/S_{{\tilde{d}}_s}^T)/S_{{\stackrel{\‾}{d}}_s};$

Calculating parameter upgrades

${\hat{p}}_s={\hat{p}}_s^{-}+K_s(y_s-{\hat{d}}_s^{-});$

Calculate temporary variable U: $U=K_s{S}_{{\tilde{d}}_s};$

The renewal of the square root mean square deviation matrix of computation model parameter

$S_{p_s}=\mathrm{cholupdate}\{S_{p_s}^{-},U,-1\};$

Wherein qr{} represents to ask the ORTHOGONAL TRIANGULAR DECOMPOSITION of matrix, and returns the upper triangular matrix that obtains; () ^TBe the transpose of a matrix operation;

Matrix is asked in expression

Cholesky decompose.

By above-mentioned steps, final iteration obtains

Be the estimated battery model parameter that obtains.

To the battery of same type, these parameters only need be determined once, can be used as the remaining capacity estimation that known constant is directly used in all batteries of the same type after determining.

Step (3) adopts standard sample point Kalman filtering to carry out the estimation of battery dump energy, specifically:

1. carry out following initialization procedure:

Initial state And variance P ₀Be respectively:

${\hat{z}}_0={\mathrm{SOC}}_0=100\%,$ P ₀＝var(z ₀)＝10 ^-2，

Handle noise variance R _wWith observation noise variance R _vBe respectively:

R _w＝10 ^-5，R _v＝10 ^-2

State vector after the expansion

And covariance P ₀ ^aFor:

${\hat{z}}_0^a={\left[\begin{array}{ccc}{z}_0& 0& 0\end{array}\right]}^T,$ $P_0^a=\left[\begin{array}{ccc}{P}_0& 0& 0\\ 0& R_w& 0\\ 0& 0& R_v\end{array}\right]$

Scale parameter γ is:

$\mathrm{\γ}=\sqrt{3}$

Average weighting coefficient w _i ^(m), i=0,1,2 ..., 6 and variance weighted coefficient w _i ^(c), i=0,1,2 ..., 6 are respectively:

$w_0^{\left(m\right)}=0,$ $w_0^{\left(c\right)}=2,$ $w_i^{\left(m\right)}=w_i^{\left(c\right)}=1/6,$ 1≤i≤6

2. adopt the standard sample point Kalman filtering algorithm recursion that circulates:

Measuring k=1 constantly, 2,3 ..., the battery terminal voltage y in the real work that records according to metering circuit _kAnd the supply current i of battery _k, calculate by the following various recursion of carrying out:

(l) according to k-1 extended mode vector constantly

And covariance P _K-1 ^a, calculate all sampled point sequences in this moment

(m) carrying out time domain according to state equation upgrades:

By the sampled point sequence

Upgrade according to state equation calculating sampling point

Sampled point is upgraded

Be weighted, computing mode is estimated

Computing mode is estimated

Variance

(n) finishing measurement according to observation equation according to following formula upgrades:

There is sampled point to upgrade

Calculating measurement according to observation equation upgrades

Upgrade measuring

Be weighted, calculate to measure and estimate

Calculate to measure and estimate

Variance

Calculate

With

Cross covariance

Calculate kalman gain K _k: $K_k=P_{z_k{y}_k}{P}_{{\tilde{y}}_k}^{-1}$

Computing mode is upgraded

${\hat{z}}_k=z_k^{-}+K_k(y_k-{\hat{y}}_k^{-})$

Computing mode is upgraded

Variance

$P_{x_k}=P_{x_k}^{-}-K_k{P}_{{\tilde{y}}_k}{K}_k^T$

The resulting state updating value of recursion

Be the estimated battery dump energy that obtains of current time k.The whole circulation recursive process is online finishing, and promptly finishes each estimation of battery dump energy constantly in the battery practical work process synchronously.

Claims (1)

1. estimation method of battery dump energy based on combined sampling point Kalman filtering is characterized in that the concrete steps of this method are:

Step (1) records the battery terminal voltage y constantly at k by metering circuit _kWith the powered battery current i _k, k=1,2,3,

Step (2) is represented each state-of-charge constantly, wherein state equation of battery with state equation and observation equation:

Observation equation:

Z is the state-of-charge of battery, i.e. dump energy, and z=100% represents that battery electric quantity is in full state, z=0% represents that battery electric quantity is in spent condition; η _iDischarge scale-up factor for battery; Q _nBe battery getable specified total electric weight under 25 ℃ of conditions of room temperature, when discharging with the discharge rate of 1/30 times of rated current; R is the internal resistance of battery; K ₀, K ₁, K ₂, K ₃, K ₄Be constant, p=[K ₀R K ₁K ₂K ₃K ₄] ^T, p is the parameter of battery observation model, is a column vector, they are constant to battery of the same type; Δ t is a measuring intervals of TIME, and u is for handling noise, and v is an observation noise, and subscript k is for measuring constantly; Wherein

Discharge scale-up factor η _iDefinite step be:

(a) will be full of the battery of electricity fully with different discharge rate C _iConstant-current discharge N time, the total electric weight Q of battery under the corresponding discharge rate is calculated in N＞10 _i, 1≤i≤N;

(b) simulate Q according to least square method _iWith C _iBetween quafric curve relation, promptly under minimum mean square error criterion, obtain simultaneously and satisfy Optimal coefficient a, b, c;

(c) be i at discharge current _kThe time, corresponding discharge scale-up factor η _iFor:

${\mathrm{\η}}_i=\frac{Q_n}{{\mathrm{ai}}_k^2+{\mathrm{bi}}_k+c}$

Battery observation model parameter p adopts equation of the ecentre sampling point Kalman filtering algorithm to determine that concrete steps are:

(d) under 25 ℃ of conditions of room temperature, with 1/30 times of rated current the battery that is full of electricity is carried out steady current discharge and exhausts until electric weight;

(e) in discharge process, measure battery at s terminal voltage y constantly with time interval Δ t by metering circuit _s, s=0,1,2 ... M, the initial discharging time after wherein the corresponding battery of s=0 is full of, the termination that the corresponding battery electric quantity of s=M exhausts is constantly;

(f) calculate s dump energy z constantly _s, z _s=1-s/M;

(g) an optional initial parameter

Setting its square root mean square deviation matrix is

I wherein ₆It is 6 * 6 unit matrix; Choose proportionality constant h, h＞1; Set variable Set weighting coefficient

I=1,2 ..., 12;

To s=1,2 ..., M, (h)～(j) carries out successive iteration as follows:

(h) computing time area update:

The estimated value of counting cell observation model parameter

The estimated value of the square root mean square deviation matrix of counting cell observation model parameter

Wherein,

Diag{} is the column vector that the diagonal entry of corresponding matrix constitutes;

(i) calculate

The sampled point sequence

Be 6 * 1 column vectors, Be 6 * 6 matrixes, so

Be 6 * 13 matrixes;

(j) measure renewal by following various calculating:

The observation sequence of calculating sampling point

Be 6 * 13 matrixes;

The calculating observation sequence

Estimated value

For

I row;

The calculating observation sequence

Square root mean square deviation matrix

Calculate covariance matrix

Calculate kalman gain K _s:

Calculating parameter upgrades

Calculate temporary variable U:

The renewal of the square root mean square deviation matrix of counting cell observation model parameter

$S_{p_s}=\mathrm{cholupdate}\{S_{p_s}^{-},U,-1\};$

Wherein qr{} represents to ask the ORTHOGONAL TRIANGULAR DECOMPOSITION of matrix, and returns the upper triangular matrix that obtains; () ^TBe the transpose of a matrix operation;

Matrix is asked in expression

Cholesky decompose;

By above-mentioned steps, final iteration obtains

Be the estimated battery observation model parameter that obtains;

Step (3) adopts standard sample point Kalman filtering to carry out the estimation of battery dump energy, specifically:

1. carry out following initialization procedure:

Initial state

And variance P ₀Be respectively:

${\hat{z}}_0={\mathrm{SOC}}_0=100\%,$ P ₀＝var(z ₀)＝10 ^-2，

Handle noise variance R _wWith observation noise variance R _vBe respectively:

R _w＝10 ^-5，R _v＝10 ^-2

State vector after the expansion

And covariance

For:

${\hat{z}}_0^a={\left[\begin{array}{ccc}{z}_0& 0& 0\end{array}\right]}^T,$ $P_0^a=\left[\begin{array}{ccc}{P}_0& 0& 0\\ 0& R_w& 0\\ 0& 0& R_v\end{array}\right]$

Scale parameter γ is:

$\mathrm{\γ}=\sqrt{3}$

The average weighting coefficient

I=0,1,2 ..., 6 and the variance weighted coefficient

I=0,1,2 ..., 6 are respectively:

$w_0^{\left(m\right)}=0,$ $w_0^{\left(c\right)}=2,$ $w_i^{\left(m\right)}=w_i^{\left(c\right)}=1/6,$ 1≤i≤6

2. adopt the standard sample point Kalman filtering algorithm recursion that circulates:

Measuring k=1 constantly, 2,3 ..., the battery terminal voltage y in the real work that records according to metering circuit _kAnd the supply current i of battery _k, calculate by the following various recursion of carrying out:

(l) according to k-1 extended mode vector constantly

And covariance

Calculate all sampled point sequences in this moment

(m) carrying out time domain according to state equation upgrades:

By the sampled point sequence

Upgrade according to state equation calculating sampling point

Sampled point is upgraded

Be weighted, computing mode is estimated

Computing mode is estimated

Variance

(n) finishing measurement according to observation equation according to following formula upgrades:

Upgrade by sampled point

Calculating measurement according to observation equation upgrades

Upgrade measuring

Be weighted, calculate to measure and estimate

Calculate to measure and estimate

Variance

Calculate

With

Cross covariance

Calculate kalman gain K _k:

Computing mode is upgraded

Computing mode is upgraded Variance

The resulting state updating value of recursion

Be the estimated battery dump energy that obtains of current time k.