CN116186464A - Nonlinear input/output system parameter identification method based on high-order least square method - Google Patents

Nonlinear input/output system parameter identification method based on high-order least square method Download PDF

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CN116186464A
CN116186464A CN202310464689.2A CN202310464689A CN116186464A CN 116186464 A CN116186464 A CN 116186464A CN 202310464689 A CN202310464689 A CN 202310464689A CN 116186464 A CN116186464 A CN 116186464A
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CN116186464B (en
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文成林
李雅会
郑锦辉
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Guangdong University of Petrochemical Technology
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Abstract

The invention discloses a nonlinear input/output system parameter identification method based on a high-order least square method, which belongs to the technical field of system identification and comprises the following specific steps of: determining parameters of a nonlinear input/output system and a system to be identified; step S2: converting the system parameters to be identified into minimum values of residual functions to be identified; step S3: performing high-order Taylor expansion on the residual function to be identified; step S4: according to the type of the nonlinear input and output system, adopting a least square method to solve the update value of the parameter to be identified of the system; step S5: and solving the parameters to be identified of the system by using an iteration formula. By adopting the nonlinear input/output system parameter identification method based on the high-order least square method, the accuracy of the parameters to be identified of the system model is improved by adopting high-order expansion of the objective function, so that the introduced error is reduced.

Description

Nonlinear input/output system parameter identification method based on high-order least square method
Technical Field
The invention relates to the technical field of system identification, in particular to a nonlinear input/output system parameter identification method based on a high-order least square method.
Background
The least square method is widely applied to various fields such as error estimation of a radar system, curve fitting, parameter identification of a time-varying system, orbit prediction and the like, the identification is to discuss the parameter estimation problem of a model under the condition that the state is measurable, the parameter identification of the traditional least square method is biased estimation, and the linear model parameter can be well identified. When using the traditional least square method for parameter identification, each observation point in the obtained sample data is treated equally. In this case, the dependent variable changes with its own sample value, and thus the condition of homogeneity of the residual variance is not satisfied. In the prior art, when the least square method is adopted to identify the model parameters, a first-order linear solution of the model is generally adopted. However, both linear and nonlinear models introduce errors in the process of solving, resulting in divergence of the system model parameters to be solved. For most of the existing system models, a large amount of errors are introduced when the system parameters are identified through linearization models, namely, through Taylor expansion. When the nonlinearity of the system model is stronger, the model parameters to be solved are more seriously diverged, so that the model parameters to be solved are inaccurate.
Disclosure of Invention
The invention aims to provide a nonlinear input/output system parameter identification method based on a high-order least square method, which reduces introduced errors and improves the accuracy of parameters to be identified of a model.
In order to achieve the above purpose, the present invention provides a nonlinear input/output system parameter identification method based on a high-order least square method, which comprises the following specific steps:
step S1: determining parameters of a nonlinear input/output system and a system to be identified;
step S2: converting the system parameters to be identified into minimum values of residual functions to be identified;
step S3: performing high-order Taylor expansion on the residual function to be identified;
step S4: according to the type of the nonlinear input and output system, adopting a least square method to solve the update value of the parameter to be identified of the system;
step S5: and solving the parameters to be identified of the system by using an iteration formula.
Preferably, in step S1, the nonlinear input-output system is a strong nonlinear system of multidimensional input and multidimensional output, and the expression is as follows:
Figure SMS_1
（1）
wherein ,
Figure SMS_20
is a known input and output value, < >>
Figure SMS_22
Represents->
Figure SMS_26
Is->
Figure SMS_3
Set of dimension real numbers->
Figure SMS_6
Represents->
Figure SMS_12
Is->
Figure SMS_17
Set of dimension real numbers->
Figure SMS_15
and />
Figure SMS_21
Dimension of input value and output value, respectively, +.>
Figure SMS_25
Is a time series; />
Figure SMS_27
Is about->
Figure SMS_19
Is a continuously differentiable multidimensional nonlinear function, < >>
Figure SMS_24
and />
Figure SMS_28
Respectively a linear parameter and a non-linear parameter, +.>
Figure SMS_30
and />
Figure SMS_5
For the system parameters to be identified, < > for>
Figure SMS_8
Represents->
Figure SMS_10
Is->
Figure SMS_11
Set of dimension real numbers->
Figure SMS_2
Represents->
Figure SMS_7
Is->
Figure SMS_13
Set of dimension real numbers->
Figure SMS_16
and />
Figure SMS_4
Parameters->
Figure SMS_9
and />
Figure SMS_14
Is a dimension of (2); />
Figure SMS_18
Is Gaussian white noise, obeys the expectation that 0 variance is +.>
Figure SMS_23
Has statistical properties->
Figure SMS_29
；
And (3) making:
Figure SMS_31
（2）
wherein ,
Figure SMS_32
is about->
Figure SMS_33
Is a continuously differentiable multidimensional nonlinear function; />
Figure SMS_34
Representing a transpose of the matrix;
combining equations (1) and (2) yields:
Figure SMS_35
（3）。
preferably, in step S2,
order the
Figure SMS_36
Identifying solving parameters +.>
Figure SMS_37
The transformation is as follows:
Figure SMS_38
；
wherein ,
Figure SMS_39
is a 2-norm;
the residual function to be identified is:
Figure SMS_40
（4）。
preferably, in step S3, the residual function to be identified is subjected to a higher-order taylor expansion,
assuming that the first is obtained
Figure SMS_41
Solving value +.>
Figure SMS_42
Then is provided with->
Figure SMS_43
Second to->
Figure SMS_44
The next iteration formula is as follows:
Figure SMS_45
（5）
will be
Figure SMS_46
At->
Figure SMS_47
The high-order Taylor expansion is carried out at the position, and the expansion process is as follows:
Figure SMS_48
（6）
wherein ,
Figure SMS_50
,/>
Figure SMS_54
in the form of a jacobian matrix,
Figure SMS_58
is the partial derivative; />
Figure SMS_51
Is the order of expansion; vector function->
Figure SMS_53
Is about->
Figure SMS_55
Vitamin variable->
Figure SMS_57
Jacobian matrix of (a); />
Figure SMS_49
Is->
Figure SMS_52
Kronnecker product of (a); />
Figure SMS_56
For the variables->
Figure SMS_59
Is used for the estimation of the estimated value of (a).
Preferably, in step S4,
the types of the nonlinear input-output system comprise an over-measurement system and an under-measurement system, and the over-measurement system and the under-measurement system respectively correspond to an over-measurement equation solution and an under-measurement equation solution.
Preferably, in step S4, the solving of the residual function to be identified is converted into the solving of the identification parameter to be updated
Figure SMS_60
I.e.:
Figure SMS_61
（7）
Figure SMS_62
is equivalent to the objective function of:
Figure SMS_63
（8）
the equivalent of equation (8) is rewritten as:
Figure SMS_64
（9）
and (3) recording:
Figure SMS_65
（10）
when (when)
Figure SMS_66
When the equation (9) is an overdetering equation, the overdetering equation solution can be obtained by adopting a least square method to solve the overdetering equation:
Figure SMS_67
（12）
when (when)
Figure SMS_68
When the equation (9) is an under-measurement equation, the solution of the under-measurement equation is obtained by adopting a least square method to solve the solution of the under-measurement equation:
Figure SMS_69
（14）。
therefore, the nonlinear input/output system parameter identification method based on the high-order least square method is adopted, and when the objective function is linearized, the introduction of errors is reduced by utilizing the high-order terms expanded by Taylor, so that the precision of the parameter to be estimated is improved.
The technical scheme of the invention is further described in detail through the drawings and the embodiments.
Drawings
FIG. 1 is a flow chart of a nonlinear input/output system parameter identification method based on a high-order least square method;
FIG. 2 is a parameter of simulation 1
Figure SMS_70
A plot of the true value and the estimated value;
FIG. 3 is a parameter of simulation 1
Figure SMS_71
A plot of the true value and the estimated value;
FIG. 4 is a parameter of simulation 1
Figure SMS_72
Is a graph of the estimated error of (2);
FIG. 5 is a parameter of simulation 1
Figure SMS_73
Is a graph of the estimated error of (2);
FIG. 6 is a parameter of simulation 1
Figure SMS_74
A plot of the true value and the estimated value;
FIG. 7 is a parameter of simulation 1
Figure SMS_75
A plot of the true value and the estimated value; />
FIG. 8 is a parameter of simulation 1
Figure SMS_76
Is a graph of the estimated error of (2);
FIG. 9 is a parameter of simulation 1
Figure SMS_77
Is a graph of the estimated error of (2);
FIG. 10 is a parameter of simulation 2
Figure SMS_78
A plot of the true value and the estimated value;
FIG. 11 is a parameter of simulation 2
Figure SMS_79
A plot of the true value and the estimated value;
FIG. 12 is a parameter of simulation 2
Figure SMS_80
A plot of the true value and the estimated value;
FIG. 13 is a parameter of simulation 2
Figure SMS_81
Is a graph of the estimated error of (2);
FIG. 14 is a parameter of simulation 2
Figure SMS_82
Is a graph of the estimated error of (2);
FIG. 15 is a parameter of simulation 2
Figure SMS_83
Is a graph of the estimated error of (2);
FIG. 16 is a parameter of simulation 2
Figure SMS_84
A plot of the true value and the estimated value;
FIG. 17 is a parameter of simulation 2
Figure SMS_85
A plot of the true value and the estimated value;
FIG. 18 is a parameter of simulation 2
Figure SMS_86
A plot of the true value and the estimated value;
FIG. 19 is a parameter of simulation 2
Figure SMS_87
A plot of the true value and the estimated value;
FIG. 20 is a parameter of simulation 2
Figure SMS_88
Is a graph of the estimated error of (2);
FIG. 21 is a parameter of simulation 2
Figure SMS_89
Is a graph of the estimated error of (2);
FIG. 22 is a parameter of simulation 2
Figure SMS_90
Is a graph of the estimated error of (2);
FIG. 23 is a parameter of simulation 2
Figure SMS_91
Is a graph of the estimated error of (a).
Detailed Description
Examples
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments of the present invention. The components of the embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations.
Thus, the following detailed description of the embodiments of the invention, as presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
Referring to fig. 1, a nonlinear input/output system parameter identification method based on a high-order least square method specifically comprises the following steps:
step S1: and determining parameters of the nonlinear input/output system and the system to be identified.
The nonlinear input-output system is a strong nonlinear system with multidimensional input and multidimensional output, and the expression is as follows:
Figure SMS_92
（1）
wherein ,
Figure SMS_110
is a known input and output value, < >>
Figure SMS_112
Represents->
Figure SMS_116
Is->
Figure SMS_93
Set of dimension real numbers->
Figure SMS_98
Represents->
Figure SMS_103
Is->
Figure SMS_111
Set of dimension real numbers->
Figure SMS_94
and />
Figure SMS_97
Dimension of input value and output value, respectively, +.>
Figure SMS_101
Is a time series; />
Figure SMS_105
Is about->
Figure SMS_95
Is a continuously differentiable multidimensional nonlinear function, < >>
Figure SMS_99
and />
Figure SMS_102
Respectively a linear parameter and a non-linear parameter, +.>
Figure SMS_107
and />
Figure SMS_109
For the system parameters to be identified, < > for>
Figure SMS_114
Represents->
Figure SMS_115
Is->
Figure SMS_119
Set of dimension real numbers->
Figure SMS_96
Represents->
Figure SMS_100
Is->
Figure SMS_104
Set of dimension real numbers->
Figure SMS_106
and />
Figure SMS_108
Parameters->
Figure SMS_113
And
Figure SMS_117
is a dimension of (2); />
Figure SMS_120
Is Gaussian white noise, obeys the expectation that 0 variance is +.>
Figure SMS_118
Has statistical properties->
Figure SMS_121
；
And (3) making:
Figure SMS_122
（2）
wherein ,
Figure SMS_123
is about->
Figure SMS_124
Is a continuously differentiable multidimensional nonlinear function; />
Figure SMS_125
Representing a transpose of the matrix;
combining equations (1) and (2) yields:
Figure SMS_126
（3）。
step S2: and converting the system parameters to be identified into minimum values of residual functions to be identified.
Order the
Figure SMS_127
Identifying solving parameters +.>
Figure SMS_128
The transformation is as follows:
Figure SMS_129
wherein->
Figure SMS_130
Is a 2-norm;
the residual function to be identified is:
Figure SMS_131
（4）。
step S3: and performing high-order Taylor expansion on the residual function to be identified.
Assuming that the first is obtained
Figure SMS_132
Solving value +.>
Figure SMS_133
Then is provided with->
Figure SMS_134
Second to->
Figure SMS_135
The next iteration formula is as follows:
Figure SMS_136
（5）
will be
Figure SMS_137
At->
Figure SMS_138
The high-order Taylor expansion is carried out at the position, and the expansion process is as follows:
Figure SMS_139
（6）
wherein ,
Figure SMS_141
,/>
Figure SMS_145
is Jacobian matrix->
Figure SMS_147
Is the partial derivative; />
Figure SMS_142
Is the order of expansion. Vector function->
Figure SMS_144
Is about->
Figure SMS_148
Vitamin variable->
Figure SMS_149
Jacobian matrix of (a); />
Figure SMS_140
Is->
Figure SMS_143
Kronnecker product of (a); />
Figure SMS_146
For the variables->
Figure SMS_150
Is used for the estimation of the estimated value of (a).
Step S4: and adopting a least square method to solve the parameter updating value to be identified of the system according to the type of the nonlinear input and output system.
The types of the nonlinear input-output system comprise an over-measurement system and an under-measurement system, and the over-measurement system and the under-measurement system respectively correspond to an over-measurement equation solution and an under-measurement equation solution.
A simple linear model is expressed as follows:
Figure SMS_151
wherein ,
Figure SMS_153
called state value->
Figure SMS_157
In which case the system is an overdetering system. When->
Figure SMS_158
Called state value->
Figure SMS_154
In which case the system is an under-measurement system.
Figure SMS_155
Is a measurement matrix; />
Figure SMS_159
The normal distribution is met; />
Figure SMS_160
To model errors conforming to Gaussian distribution, obeying the expectation that 0 variance is +.>
Figure SMS_152
And has statistical properties->
Figure SMS_156
。
Least squares method for over-measurement system state estimation.
When the system model is an overdetermined model, the linear model can be equivalently deformed into:
Figure SMS_161
has the following components
Figure SMS_162
The solution of (2) is as follows:
Figure SMS_163
mathematical expectations for the above equation are:
Figure SMS_164
with respect to state estimation values
Figure SMS_165
Error analysis of (c):
Figure SMS_166
；
the estimation error covariance matrix is as follows:
Figure SMS_167
。
least squares method for under-measured system state estimation.
In a time-dependent linear model
Figure SMS_168
When the system is called an under-measurement system. Introducing intermediate variables
Figure SMS_169
The method comprises the following steps:
Figure SMS_170
substituting the above into a linear model to obtain:
Figure SMS_171
/>
the equivalent deformation is carried out to obtain the following components:
Figure SMS_172
when there is
Figure SMS_173
When we have->
Figure SMS_174
Is solved by:
Figure SMS_175
for the mathematical expectation calculation, there are
Figure SMS_176
。
For a pair of
Figure SMS_177
After mathematical expectation operation is carried out on two sides, the formula is substituted into the formula to obtain:
Figure SMS_178
with respect to the status
Figure SMS_179
The estimation error of (2) is:
Figure SMS_180
the corresponding estimation error covariance matrix is:
Figure SMS_181
in step S4, the residual function to be identified is solved and converted into the identification parameter to be updated
Figure SMS_182
I.e.:
Figure SMS_183
（7）
Figure SMS_184
is equivalent to the objective function of:
Figure SMS_185
（8）/>
the equivalent of equation (8) is rewritten as:
Figure SMS_186
（9）
and (3) recording:
Figure SMS_187
（10）
when (when)
Figure SMS_188
When the equation (9) is an overdetering equation, the overdetering equation solution can be obtained by adopting a least square method to solve the overdetering equation:
Figure SMS_189
（12）
when (when)
Figure SMS_190
When the equation (9) is an under-measurement equation, the solution of the under-measurement equation is obtained by adopting a least square method to solve the solution of the under-measurement equation:
Figure SMS_191
（14）。
step S5: and solving the parameters to be identified of the system by using an iteration formula.
To verify the effectiveness of the present invention, simulation experiments were performed and analyzed.
Simulation 1:
for a strong nonlinear multiple input multiple output system of the type:
Figure SMS_192
wherein ,
Figure SMS_193
the model accords with normal distribution, and is an under-measurement equation. />
Figure SMS_194
Is white noise with a mean value of zero and a variance of 0.2; />
Figure SMS_195
The linear parameter and the nonlinear parameter, respectively. Wherein->
Figure SMS_196
，
Figure SMS_197
. Initial data of the experiment are->
Figure SMS_198
，/>
Figure SMS_199
. Simulation results of the 20 Monte Carlo simulation tests are shown in FIGS. 2-9. The following table shows a comparison of the performance of the different algorithms.
Table 1 the performance of the different algorithms is compared as follows:
Figure SMS_200
table 1 shows the average square error of the simulation results. From table 1, it can be seen that the second order least squares method (2 LS) improves the recognition accuracy of the linear parameters by 16.85% and 21.69% and the recognition accuracy of the nonlinear parameters by 0.38% and 15.78% relative to the least squares method (LS). Compared with a third-order least square method (3 LS) and a fourth-order least square method (4 LS), the identification precision of the least square method (LS) to linear parameters is improved by 19.92% -24.61% and 24.61% -30.31%, and the identification precision to nonlinear parameters is improved by 17.76% -48.01% and 28.51% -56.19%. Therefore, the less errors are introduced, and the identification accuracy of the parameters can be improved more obviously.
Simulation 2:
aiming at a stronger nonlinear model, the simulation test is formed by conforming a plurality of exponential functions:
Figure SMS_201
wherein ,
Figure SMS_202
the model accords with normal distribution, and is an under-measurement equation. />
Figure SMS_203
Is white noise with a mean value of zero and a variance of 0.2; />
Figure SMS_204
The linear parameter and the nonlinear parameter, respectively. Wherein the method comprises the steps of
Figure SMS_205
，/>
Figure SMS_206
. Initial data of the experiment are->
Figure SMS_207
，
Figure SMS_208
. The curves in FIGS. 10-23 all pass through 20 Monte CarloThe experimental results of the simulation are shown in Table 2, which is a comparison of the performance of different algorithms.
Table 2 the performance of the different algorithms is compared as follows:
Figure SMS_209
table 2 shows the average square error of the simulation results. As can be seen from the data in table 2, the algorithm mentioned in this document has significantly improved accuracy in identifying parameters compared to the original method. Table 2 can be derived: compared with the least square method (LS), the second-order least square method (2 LS) improves the identification accuracy of the linear parameters by 55.87% on average, and improves the identification accuracy of the nonlinear parameters by 63.98% on average. Compared with the least square method (LS), the identification accuracy of the three-order least square method (3 LS) and the four-order least square method (4 LS) is averagely improved by 63.41% -69.70%, and the identification accuracy of the nonlinear parameters is averagely improved by 77.44% -89.91%.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention and not for limiting it, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that: the technical scheme of the invention can be modified or replaced by the same, and the modified technical scheme cannot deviate from the spirit and scope of the technical scheme of the invention.

Claims (6)

1. A nonlinear input/output system parameter identification method based on a high-order least square method is characterized by comprising the following specific steps:
step S1: determining parameters of a nonlinear input/output system and a system to be identified;
step S2: converting the system parameters to be identified into minimum values of residual functions to be identified;
step S3: performing high-order Taylor expansion on the residual function to be identified;
step S4: according to the type of the nonlinear input and output system, adopting a least square method to solve the update value of the parameter to be identified of the system;
step S5: and solving the parameters to be identified of the system by using an iteration formula.
2. The nonlinear input/output system parameter identification method based on the higher-order least square method as claimed in claim 1, wherein the method is characterized by comprising the following steps: in step S1, the nonlinear input-output system is a strong nonlinear system with multidimensional input and multidimensional output, and the expression is as follows:
Figure QLYQS_1
（1）
wherein ,
Figure QLYQS_24
is a known input and output value, < >>
Figure QLYQS_28
Represents->
Figure QLYQS_30
Is->
Figure QLYQS_3
Set of dimension real numbers->
Figure QLYQS_11
Represents->
Figure QLYQS_16
Is->
Figure QLYQS_19
Set of dimension real numbers->
Figure QLYQS_18
and />
Figure QLYQS_22
Dimension of input value and output value, respectively, +.>
Figure QLYQS_26
Is a time series; />
Figure QLYQS_29
Is about->
Figure QLYQS_21
Is a continuously differentiable multidimensional nonlinear function, < >>
Figure QLYQS_23
and />
Figure QLYQS_25
Respectively a linear parameter and a non-linear parameter, +.>
Figure QLYQS_27
and />
Figure QLYQS_10
For the system parameters to be identified, < > for>
Figure QLYQS_13
Represents->
Figure QLYQS_17
Is->
Figure QLYQS_20
Set of dimension real numbers->
Figure QLYQS_2
Represents->
Figure QLYQS_7
Is->
Figure QLYQS_12
Set of dimension real numbers->
Figure QLYQS_15
and />
Figure QLYQS_4
Parameters->
Figure QLYQS_6
and />
Figure QLYQS_9
Is a dimension of (2); />
Figure QLYQS_14
Is Gaussian white noise, obeys the expectation that 0 variance is +.>
Figure QLYQS_5
Has statistical properties->
Figure QLYQS_8
；
And (3) making:
Figure QLYQS_31
（2）
wherein ,
Figure QLYQS_32
is about->
Figure QLYQS_33
Is a continuously differentiable multidimensional nonlinear function; />
Figure QLYQS_34
Representing a transpose of the matrix;
combining equations (1) and (2) yields:
Figure QLYQS_35
（3）。
3. the nonlinear input/output system parameter identification method based on the higher-order least square method as claimed in claim 2, wherein the method is characterized by comprising the following steps: in the step S2 of the process of the present invention,
order the
Figure QLYQS_36
Identifying solving parameters +.>
Figure QLYQS_37
The transformation is as follows:
Figure QLYQS_38
, wherein ,/>
Figure QLYQS_39
Is a 2-norm;
the residual function to be identified is:
Figure QLYQS_40
（4）。/>
4. the nonlinear input/output system parameter identification method based on the higher-order least square method as claimed in claim 3, wherein the method is characterized by comprising the following steps: in step S3, the residual function to be identified is subjected to a higher-order taylor expansion,
assuming that the first is obtained
Figure QLYQS_41
Solving value +.>
Figure QLYQS_42
Then is provided with->
Figure QLYQS_43
Second to->
Figure QLYQS_44
The next iteration formula is as follows:
Figure QLYQS_45
（5）
will be
Figure QLYQS_46
At->
Figure QLYQS_47
The high-order Taylor expansion is carried out at the position, and the expansion process is as follows:
Figure QLYQS_48
（6）
wherein ,
Figure QLYQS_51
,/>
Figure QLYQS_53
is Jacobian matrix->
Figure QLYQS_56
Is the partial derivative; />
Figure QLYQS_50
Is the order of expansion; vector function->
Figure QLYQS_54
Is about->
Figure QLYQS_55
Vitamin variable->
Figure QLYQS_59
Jacobian matrix of (a); />
Figure QLYQS_49
Is->
Figure QLYQS_52
Kro of (5)The nnecker product; />
Figure QLYQS_57
For the variables->
Figure QLYQS_58
Is used for the estimation of the estimated value of (a).
5. The nonlinear input/output system parameter identification method based on the higher-order least square method as claimed in claim 4, wherein the method is characterized by comprising the following steps: in the step S4 of the process of the present invention,
the types of the nonlinear input-output system comprise an over-measurement system and an under-measurement system, and the over-measurement system and the under-measurement system respectively correspond to an over-measurement equation solution and an under-measurement equation solution.
6. The nonlinear input/output system parameter identification method based on the higher-order least square method as claimed in claim 5, wherein the method is characterized by comprising the following steps: in step S4, the residual function to be identified is solved and converted into the identification parameter to be updated
Figure QLYQS_60
I.e.:
Figure QLYQS_61
（7）
Figure QLYQS_62
is equivalent to the objective function of:
Figure QLYQS_63
（8）
the equivalent of equation (8) is rewritten as:
Figure QLYQS_64
（9）
and (3) recording:
Figure QLYQS_65
（10）
when (when)
Figure QLYQS_66
When the equation (9) is an overdetering equation, the overdetering equation solution can be obtained by adopting a least square method to solve the overdetering equation:
Figure QLYQS_67
（12）
when (when)
Figure QLYQS_68
When the equation (9) is an under-measurement equation, the solution of the under-measurement equation is obtained by adopting a least square method to solve the solution of the under-measurement equation:
Figure QLYQS_69
（14）。/>
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