CN111366855A - Battery equivalent circuit model disturbance-resistant parameterization method - Google Patents

Battery equivalent circuit model disturbance-resistant parameterization method Download PDF

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CN111366855A
CN111366855A CN202010194106.5A CN202010194106A CN111366855A CN 111366855 A CN111366855 A CN 111366855A CN 202010194106 A CN202010194106 A CN 202010194106A CN 111366855 A CN111366855 A CN 111366855A
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battery
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CN111366855B (en
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魏中宝
何洪文
曹万科
周稼铭
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Beijing Institute of Technology BIT
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/36—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/382—Arrangements for monitoring battery or accumulator variables, e.g. SoC
    • G01R31/3842—Arrangements for monitoring battery or accumulator variables, e.g. SoC combining voltage and current measurements
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/36—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/367—Software therefor, e.g. for battery testing using modelling or look-up tables

Abstract

The invention discloses a battery equivalent circuit model disturbance-resistant parameterization method, which comprises the following steps: s1, establishing an equivalent circuit model of a battery, determining parameters of the model to be identified, and determining a relational expression of a state of charge (SOC) and an open-circuit voltage (OCV) through fitting; s2, collecting the load current and terminal voltage at the moment k in real time; s3, calculating the SOC of the battery at the moment k, and calculating an OCV value; s4, establishing a discrete domain regression equation for model parameter identification, and updating model parameters on line by adopting a recursive least square method (RLS); s5, constructing a tool vector constraint condition, calculating the k-time current noise variance on line, and further calculating the k-time voltage noise variance according to a FrischScheme method; and S6, correcting the RLS result in the S4 according to the current and voltage variance estimation values to obtain an unbiased model parameter vector at the moment k. The invention can estimate the current and voltage measurement noise statistical characteristics on line, thereby compensating the model identification deviation under the noise interference environment and realizing unbiased model parameter identification.

Description

Battery equivalent circuit model disturbance-resistant parameterization method
Technical Field
The invention relates to battery model parameter identification, in particular to an online disturbance-resistant unbiased identification method for a battery equivalent circuit model.
Background
A Battery Management System (BMS) is an important guarantee for the safe and efficient operation of a Battery System, and needs to complete key tasks such as state estimation, fault diagnosis, life prediction, and charge control. The BMS strategy based on the equivalent circuit model has high precision and strong robustness, and the calculation complexity is moderate, so that the method is the most common method in the BMS field. However, the method has strong dependence on model accuracy, and the failure of estimation, diagnosis and control algorithms is caused by model misalignment, so that serious safety accidents are caused. Therefore, an accurate equivalent circuit model is one of the difficult problems of the BMS design.
The equivalent circuit model in the prior art is based on static parameters, namely, the parameters are calibrated in advance under standard working conditions and are assumed to be constant in use. However, model parameters of the equivalent circuit model are easily affected by the state of the battery (state of charge SOC, state of health SOH) and external conditions (temperature, charge-discharge rate), and the uncertainty is strong, so the accuracy of the traditional method is poor in actual operation. The online adaptive calibration of the equivalent circuit model parameters can effectively track the change of the model parameters in a complex environment, and the current methods are mostly based on a least square criterion, such as a recursive least square method (RLS), a multiple forgetting factor recursive least square Method (MFFRLS), a partial least square method (PLS) and the like. It should be noted that the existing parameter identification method depends on high-precision load current and terminal voltage sampling, however, in a real BMS application environment, due to the influence of sensor errors, electromagnetic interference and the like, a large amount of noise often exists in a measurement signal, which can cause the failure of the traditional online model parameter identification method, the model precision is remarkably reduced, and finally, the reliability of the equivalent circuit model-based BMS strategy is remarkably influenced.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide an equivalent circuit model disturbance-resistant parameterization method which can estimate the statistical characteristics of current and voltage measurement noise on line, thereby compensating the model identification deviation under the noise interference environment, realizing unbiased model parameter identification, effectively improving the precision of the equivalent circuit model and further improving the reliability of a management algorithm based on the equivalent circuit model; the statistical characteristics of the measurement noise of the online estimation can also be used in the fields of state estimation based on filtering and the like, and the estimation precision is improved.
The purpose of the invention is realized by the following technical scheme: a battery equivalent circuit model disturbance-resistant parameterization method comprises the following steps:
s1, establishing an equivalent circuit model of a battery, determining parameters of the model to be identified, and determining a relational expression of a state of charge (SOC) and an open-circuit voltage (OCV) through fitting; (ii) a
S2, acquiring load current and terminal voltage at the moment k in real time by adopting a voltage sensor and a current sensor;
s3, calculating the SOC of the battery at the moment k, and calculating an OCV value according to an SOC-OCV function relation expression;
s4, establishing a discrete domain regression equation for model identification, inputting terminal voltage values and current values at the moment k, and updating model parameters on line by adopting a recursive least square method (RLS);
s5, constructing a tool vector constraint condition, calculating the current noise variance at the k moment on line according to the model parameter at the k-1 moment and the voltage noise variance estimation value, and further calculating the voltage noise variance at the k moment according to a Frisch Scheme method;
and S6, correcting the RLS result in the S4 according to the current and voltage variance estimation value calculated in the S5 to obtain an unbiased model parameter vector at the k moment.
The invention has the beneficial effects that: the method can estimate the statistical characteristics of the current and voltage measurement noise on line, thereby compensating the model identification deviation in the noise interference environment, realizing unbiased model parameter identification, effectively improving the simulation precision of the equivalent circuit model, and the estimated statistical characteristics of the measurement noise can also be used for other BMS strategies.
Drawings
FIG. 1 is a flow chart of a method of the present invention;
FIG. 2 is a schematic circuit diagram of a first-order equivalent circuit model in the embodiment.
FIG. 3 is a graph of a five-time fit of the SOC-OCV function of the examples.
Fig. 4 is a current and terminal voltage curve for the mixed pulse condition in the example.
FIG. 5 is a result of online estimation of noise standard deviation of load current and terminal voltage under the mixed pulse condition in the embodiment.
FIG. 6 is a diagram illustrating an online identification result of a first-order equivalent circuit model under the mixed pulse condition in the embodiment.
Detailed Description
The technical solutions of the present invention are further described in detail below with reference to the accompanying drawings, but the scope of the present invention is not limited to the following.
As shown in fig. 1, a method for parameterizing the disturbance resistance of a battery equivalent circuit model includes the following steps:
s1, establishing an equivalent circuit model of a battery, determining parameters of the model to be identified, and determining a relational expression of a state of charge (SOC) and an open-circuit voltage (OCV) through fitting; the specific implementation steps are as follows:
s101, establishing an equivalent circuit model of the battery, determining model parameters to be identified, and adopting a first-order equivalent circuit model shown in figure 2 in the embodiment, wherein the mathematical description method comprises the following steps:
CpdVp(t)/dt+Vp(t)/Rp＝I(t)
Vt(t)＝Voc(t)-Vp(t)-I(t)Rs
Figure BDA0002416957200000021
where t is time, I is load current, and correspondingly I (t) is load current at time t, VpIs a polarization voltage, VtThe voltage of the line terminal is η coulombic efficiency of the battery, Q is rated capacity of the battery, and Rs、RpAnd CpModel parameters to be found, specifically: rsIs ohmic internal resistance, RpIs a polarization resistance, CpIs a polarization capacitance.
S102, charging the lithium ion battery under a rated working condition (room temperature) until the SOC reaches 100%, performing an intermittent discharge-standing experiment, and fitting to determine an SOC-OCV relational expression; in the embodiment of the present application, the SOC-OCV relational expression obtained by polynomial fitting is:
Figure BDA0002416957200000031
wherein VocThe battery open circuit voltage OCV; z is the battery SOC; n ispTo fit the polynomial order, n in this embodimentpIs 5, ciIs a fitting coefficient; the resulting SOC-OCV curve was fitted to the measured value pairs as shown in FIG. 3.
S2, acquiring load current and terminal voltage at the moment k in real time by adopting a voltage sensor and a current sensor;
s3, calculating the SOC of the battery at the moment k, and calculating the OCV value according to the SOC-OCV function relational expression, wherein the specific implementation steps are as follows:
in this embodiment, the battery SOC at the time k is estimated online using extended kalman filtering. First, the continuous state space equation of the system in S101 is discretized into:
Figure BDA0002416957200000032
Vt,k＝Voc,k-Vp,k-IkRs
where the subscript k denotes time k, tsIs the time interval (seconds) over which the parameter identification is performed. Definition x ═ Vpz]TFor the state vector of the system, the input u of the system is I, and the output of the system is VtThe general state space model of the extended kalman filter algorithm can be expressed as:
xk+1＝Akxk+Bkuk+wk
Vt,k＝f(zk)-Vp,k-Rsuk+vk
wherein, wkIs random process noise, vkFor measuring noise, the parameter matrix included is:
Figure BDA0002416957200000033
executing an extended Kalman filtering algorithm at the moment k, and specifically comprising the following steps:
and (3) updating state transition to obtain prior estimation of a state vector at the moment k:
Figure BDA0002416957200000034
error covariance matrix prior estimation:
Figure BDA0002416957200000035
calculating a Kalman gain:
Figure BDA0002416957200000036
wherein
Figure BDA0002416957200000037
State vector feedback correction and a posteriori estimation:
Figure BDA0002416957200000038
updating the error covariance matrix posteriori:
Figure BDA0002416957200000039
wherein Q iswAnd QvCovariance matrices of the equation of state and system measurements, respectively, model parameters included in each step
Figure BDA0002416957200000041
The parameter identification result obtained at the time S6 at the time k-1 is used. And estimating the battery SOC at the k moment in real time according to the algorithm flow of the extended Kalman filtering, and estimating the OCV at the k moment according to the SOC-OCV functional relation in S1.
S4, establishing an equivalent circuit model of the battery and a discrete domain regression equation for model identification, inputting terminal voltage values and current values at the moment k, and updating model parameters on line by adopting RLS (recursive least squares), wherein the specific implementation steps are as follows:
s401, according to the mathematical description method of the continuous time domain first-order equivalent circuit model in S01, constructing a discrete domain regression equation for model identification as follows. Definition of y ═ Vt–VocAnd performing Laplace transformation on the formula to obtain a transfer function:
Figure BDA0002416957200000042
in the formula: s represents the complex variable of the laplace transform, and the above formula is transformed bilinearly as follows:
Figure BDA0002416957200000043
wherein q is a shift operator, the following discrete domain transfer function is obtained:
y(q-1)/I(q-1)＝-(b0+b1q-1)/(1+a1q-1)
wherein q is-1Representing the reverse shift operator, the discrete domain regression equation is constructed as:
Figure BDA0002416957200000044
wherein, yk＝Vt,k–Voc,k，
Figure BDA0002416957200000045
Defining a parameter vector theta to be identifiedkIs the parameter vector to be identified.
S402, RLS online identification model parameter vectors are adopted, and the specific process is as follows:
initializing an error covariance matrix P0And RLS initial estimation
Figure BDA0002416957200000046
Collecting current and terminal voltage of battery on line, and determining input of k-time regression model
Figure BDA0002416957200000047
And output yk；
Calculating the RLS gain at time k:
Figure BDA0002416957200000048
calculating RLS parameter vector update at k time:
Figure BDA0002416957200000049
updating the error covariance matrix P at time kk：
Figure BDA00024169572000000410
In the above step, λ represents a forgetting factor, and λ takes a value of 0.99 in this embodiment.
S5, constructing a tool vector constraint condition, calculating the current noise variance at the k moment on line according to the model parameter at the k-1 moment and the voltage noise variance estimation value, and further calculating the voltage noise variance at the k moment according to a Frisch Scheme method; the FrischScheme method is early used for a static regression problem, can effectively process implicit relations among multiple uncertainties, is widely applied in the fields of statistics and finance, and comprises the following specific implementation steps:
s501, defining thetak＝[akbk T]TWherein a isk＝[a1,k]，bk＝[b0,kb1,k]T(ii) a Introducing a time lag current as a tool variable:
Figure BDA0002416957200000051
wherein
Figure BDA0002416957200000052
Number of input quantities for tool variables, in this embodiment
Figure BDA0002416957200000053
nbIs a vector bkNumber of elements of (1), n in this embodimentb＝2；
S502.k time current noise variance estimation generalizes to solve the minimization problem as follows:
Figure BDA0002416957200000054
Figure BDA0002416957200000055
wherein the content of the first and second substances,
Figure BDA0002416957200000056
is a vector
Figure BDA0002416957200000057
And vector
Figure BDA0002416957200000058
The covariance matrix of (a) is expected to be updated at time k,
Figure BDA0002416957200000059
is a vector
Figure BDA00024169572000000510
And the update of the covariance vector of the variable y at the time k is calculated according to the following formula:
Figure BDA00024169572000000511
Figure BDA00024169572000000512
in this embodiment, the above nonlinear minimization problem is solved by a numerical method with low computational complexity, and the specific steps are as follows:
Figure BDA00024169572000000513
Figure BDA00024169572000000514
Figure BDA00024169572000000515
Figure BDA00024169572000000516
Figure BDA00024169572000000517
s503, define
Figure BDA00024169572000000518
And (3) calculating the voltage noise variance estimation value at the time k by adopting the following minimum feature root extraction mode:
Figure BDA00024169572000000519
the minimum characteristic root extraction problem adopts a singular value decomposition method or other common numerical solution methods;
wherein the content of the first and second substances,
Figure BDA00024169572000000520
is a vector
Figure BDA00024169572000000521
The autocorrelation matrix of (a) is expected to be updated at time k,
Figure BDA00024169572000000522
is a vector
Figure BDA00024169572000000523
The autocorrelation matrix of (a) is expected to be updated at time k,
Figure BDA00024169572000000524
is a vector
Figure BDA00024169572000000525
And vector
Figure BDA00024169572000000526
The covariance matrix is expected to be updated at time k. The matrix calculation methods are as follows:
Figure BDA00024169572000000527
Figure BDA00024169572000000528
Figure BDA00024169572000000529
wherein the content of the first and second substances,
Figure BDA0002416957200000061
ρ0for the parameters to be initialized, 0 is selected in this embodiment.
And S6, correcting the RLS result in the S4 according to the current and voltage variance estimation value calculated in the S5 to obtain an unbiased model parameter vector at the k moment. The specific implementation steps are as follows:
s601, correcting the RLS result in S4 according to the current and voltage variance estimation values calculated in S5, wherein the calculation method comprises the following steps:
Figure BDA0002416957200000062
wherein the content of the first and second substances,
Figure BDA0002416957200000063
is the RLS-based model parameter solution obtained in S4,
Figure BDA0002416957200000064
is a vector
Figure BDA0002416957200000065
The autocorrelation matrix of (a) is expected to be updated at time k,
Figure BDA0002416957200000066
is equal to P in S401kThe symbol delta represents the noise term of a variable or vector,
Figure BDA0002416957200000067
is a vector
Figure BDA0002416957200000068
The autocorrelation matrix is expected to be updated at the time k, and the correlation matrix calculation method comprises the following steps:
Figure BDA0002416957200000069
Figure BDA00024169572000000610
s602, estimating the parameter vector according to the step S601
Figure BDA00024169572000000611
Determining
Figure BDA00024169572000000612
And
Figure BDA00024169572000000613
the model parameter R to be solveds、RpAnd CpUpdate as follows:
Figure BDA00024169572000000614
in the embodiment of the application, a mixed pulse working condition experiment is performed at room temperature by taking an NMC 18650 lithium ion battery with the nominal capacity of 2.2Ah as an example, and the method disclosed by the invention is adopted to identify the model parameters in real time. Obtained under mixed-pulse conditionsThe curves of the load current and the terminal voltage of the lithium ion battery are shown in the attached figure 4, the online estimation result of the noise standard deviation of the load current and the terminal voltage is shown in the figure 5, and the identification result of the first-order equivalent circuit model is shown in the figure 6 (ohmic internal resistance R is in sequence from top to bottom)sIdentification result of (1), polarization internal resistance RpThe identification result of (1), polarization capacitance CpThe recognition result of (1). Therefore, compared with the existing method, the method can accurately estimate the noise standard deviation contained in the load current and the terminal voltage, and maintains unbiased online parameter identification by compensating the noise effect.
In conclusion, the invention fully considers the uncertainty of the internal parameters of the battery caused by the actual environment change and the state change of the battery, and establishes an online self-adaptive mathematical model; the statistical characteristics of current and voltage noise are estimated on line and deviation compensation is carried out, so that the unbiased property of model parameter identification under noise interference is ensured, and the precision of the on-line self-adaptive model is improved. Compared with the traditional offline calibration-based battery model parameterization method, the method has the advantages that the robustness of the model to environmental changes and battery self-state changes is improved; compared with the traditional online adaptive model parameter identification method based on the least square criterion, the method has the advantages of obvious anti-interference and unbiased characteristics, and improves the identification precision of the model parameters.
The foregoing is a preferred embodiment of the present invention, it is to be understood that the invention is not limited to the form disclosed herein, but is not to be construed as excluding other embodiments, and is capable of other combinations, modifications, and environments and is capable of changes within the scope of the inventive concept as expressed herein, commensurate with the above teachings, or the skill or knowledge of the relevant art. And that modifications and variations may be effected by those skilled in the art without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (6)

1. A battery equivalent circuit model disturbance-resistant parameterization method is characterized by comprising the following steps: the method comprises the following steps:
s1, establishing an equivalent circuit model of a battery, determining parameters of a model to be identified, and determining a relational expression of a battery state of charge (SOC) and an open-circuit voltage (OCV) through fitting;
s2, acquiring load current and terminal voltage at the moment k in real time by adopting a voltage sensor and a current sensor;
s3, calculating the SOC of the battery at the moment k, and calculating an OCV value according to an SOC-OCV function relation expression;
s4, establishing a discrete domain regression equation for model parameter identification, inputting terminal voltage values and current values at the moment k, and updating model parameters on line by adopting a recursive least square method;
s5, constructing a tool vector constraint condition, and calculating the current noise variance at the k moment on line according to the model parameter at the k-1 moment and the voltage noise variance estimation value so as to calculate the voltage noise variance at the k moment;
and S6, correcting the update result of the recursive least square method obtained in the S4 according to the current and voltage variance estimation values obtained in the S5 to obtain the unbiased model parameter vector at the moment k.
2. The method according to claim 1, wherein the method comprises the following steps: the step S1 includes the following sub-steps:
s101, establishing an equivalent circuit model of the battery:
CpdVp(t)/dt+Vp(t)/Rp＝I(t)
Vt(t)＝Voc(t)-Vp(t)-I(t)Rs
Figure FDA0002416957190000011
where t is time, I is load current, and correspondingly I (t) is load current at time t, VpIs a polarization voltage, VtFor end-of-line voltage, η is the coulombic efficiency of the battery, Q is the rated capacity of the battery, Rs、RpAnd CpModel parameters to be found, specifically: rsIs ohmic internal resistance, RpIs a polarization resistance, CpIs a polarization capacitor;
s102, charging the battery to a state of charge (SOC) of 100% under a rated working condition, carrying out an intermittent discharge-standing experiment, obtaining a measured value of an open-circuit voltage (OCV) under a specific state of charge (SOC), and fitting the SOC-OCV corresponding relation by adopting the following expression:
Figure FDA0002416957190000012
wherein VocThe battery open circuit voltage OCV; z is the battery SOC; n ispTo fit the polynomial order, ciAre fitting coefficients.
3. The method according to claim 1, wherein the method comprises the following steps: in step S3, the calculation method of the battery SOC at the time k includes one of an ampere-hour integration method, a machine learning method, an extended kalman filter, a lunberg observer, an infinite kalman filter, a particle filter, and a slip film observer.
4. The method according to claim 1, wherein the method comprises the following steps: the step S4 includes the following sub-steps:
s401, constructing a discrete domain regression equation for model identification:
Figure FDA0002416957190000021
wherein, yk＝Vt,k–Voc,k，
Figure FDA0002416957190000022
Defining a parameter vector theta to be identifiedkIs a parameter vector to be identified, and a subscript k in the formula represents a time k;
s402, identifying model parameter vectors on line by adopting a recursive least square method, wherein the specific process is as follows:
initializing an error covariance matrix P0And RLS initial estimation
Figure FDA0002416957190000023
Collecting current and terminal voltage of battery on line, and determining input of k-time regression model
Figure FDA0002416957190000024
And output yk；
Calculating the RLS gain at time k:
Figure FDA0002416957190000025
calculating RLS parameter vector update at k time:
Figure FDA0002416957190000026
updating the error covariance matrix P at time kk：
Figure FDA0002416957190000027
Wherein, the lambda represents a forgetting factor and takes a value of 0.98-1.
5. The method according to claim 1, wherein the method comprises the following steps: the step S5 includes the following sub-steps:
s501, defining thetak＝[akbk T]TWherein a iskAnd bkIs defined as: a isk＝[a1,k]and bk＝[b0,kb1,k]T(ii) a Introducing a time lag current as a tool variable:
Figure FDA0002416957190000028
in the formula
Figure FDA0002416957190000029
Number of inputs, n, for tool variablesbIs a vector bkElement (2) ofCounting;
s502.k time current noise variance estimation generalizes to solve the minimization problem as follows:
Figure FDA00024169571900000210
Figure FDA00024169571900000211
wherein the content of the first and second substances,
Figure FDA00024169571900000212
is a vector
Figure FDA00024169571900000213
And vector
Figure FDA00024169571900000214
The covariance matrix of (a) is expected to be updated at time k,
Figure FDA00024169571900000215
is a vector
Figure FDA00024169571900000216
And the update of the covariance vector of the variable y at the time k is calculated according to the following formula:
Figure FDA00024169571900000217
Figure FDA00024169571900000218
the above minimization problem is solved by the following numerical solution:
Figure FDA0002416957190000031
S503.definition of
Figure FDA0002416957190000032
And (3) calculating the voltage noise variance estimation value at the time k by adopting the following minimum feature root extraction mode:
Figure FDA0002416957190000033
the minimum characteristic root extraction problem adopts a singular value decomposition method or other common numerical solution methods;
wherein the content of the first and second substances,
Figure FDA0002416957190000034
is a vector
Figure FDA0002416957190000035
The autocorrelation matrix of (a) is expected to be updated at time k,
Figure FDA0002416957190000036
is a vector
Figure FDA0002416957190000037
The autocorrelation matrix of (a) is expected to be updated at time k,
Figure FDA0002416957190000038
is a vector
Figure FDA0002416957190000039
And vector
Figure FDA00024169571900000310
The covariance matrix is expected to be updated at time k, and the matrix calculation methods are as follows:
Figure FDA00024169571900000311
Figure FDA00024169571900000312
Figure FDA00024169571900000313
wherein the content of the first and second substances,
Figure FDA00024169571900000314
ρ0are parameters to be initialized.
6. The method according to claim 1, wherein the method comprises the following steps: the step S6 includes the following sub-steps:
s601, correcting the update result of the recursive least square method in S4 according to the current and voltage variance estimation values calculated in S5, wherein the calculation method is as follows:
Figure FDA00024169571900000315
wherein the content of the first and second substances,
Figure FDA00024169571900000316
is the recursive least squares based model parameter solution obtained in S4,
Figure FDA00024169571900000317
is a vector
Figure FDA00024169571900000318
The autocorrelation matrix of (a) is expected to be updated at time k,
Figure FDA00024169571900000319
is equal to P in S402kAnd therefore need not be calculated again, the symbol delta represents the noise term of the variable or vector,
Figure FDA00024169571900000320
is a vector
Figure FDA00024169571900000321
The autocorrelation matrix is expected to be updated at the time k, and the correlation matrix calculation method comprises the following steps:
Figure FDA00024169571900000322
Figure FDA00024169571900000323
s602, estimating the parameter vector according to the step S601
Figure FDA0002416957190000041
Determining
Figure FDA0002416957190000042
And
Figure FDA0002416957190000043
the model parameter R to be solveds、RpAnd CpUpdate as follows:
Figure FDA0002416957190000044
wherein, tsIs the time interval for parameter identification in seconds.
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