CN113111505A - Variable forgetting factor recursive least square method and system based on nonlinear Hammerstein system - Google Patents

Variable forgetting factor recursive least square method and system based on nonlinear Hammerstein system Download PDF

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CN113111505A
CN113111505A CN202110376632.8A CN202110376632A CN113111505A CN 113111505 A CN113111505 A CN 113111505A CN 202110376632 A CN202110376632 A CN 202110376632A CN 113111505 A CN113111505 A CN 113111505A
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forgetting factor
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赵旭楷
刘兆霆
陆敏杰
王亚峰
鲍辉明
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Hangzhou Dianzi University
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Abstract

The invention provides a variable forgetting factor recursive least square method and a variable forgetting factor recursive least square system based on a nonlinear Hammerstein system.

Description

Variable forgetting factor recursive least square method and system based on nonlinear Hammerstein system
Technical Field
The invention belongs to the technical field of nonlinear system identification, relates to a nonlinear system identification technology, and particularly relates to a forgetting factor-changing recursive least square method and a forgetting factor-changing recursive least square system based on a nonlinear Hammerstein system.
Background
At present, the research of experts and scholars at home and abroad on the nonlinear system identification has been greatly developed. The Hammerstein model is used for representing most practical nonlinear systems, and a plurality of novel methods based on the Hammerstein model are provided, but the methods can solve the problem of nonlinear system identification, but have some defects. For example, only considering reducing errors and improving the performance of the method, the method is complex and has large calculation amount, which causes difficulty in operation; the adopted method has a certain limit range on parameters and a system, and cannot be suitable for other conditions.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a forgetting factor-changing recursive least square method and a forgetting factor-changing least square system based on a nonlinear Hammerstein system.
The invention adopts the following technical scheme:
a variable forgetting factor recursive least square (VFF-RLS) method based on a nonlinear Hammerstein system specifically comprises the following steps:
step 1: firstly, establishing a nonlinear system model based on a Hammerstein system, wherein the relation between the output y (k) and the input u (k) of the system can be expressed as
Figure BDA0003011326890000011
Where x (k) is an intermediate variable, v (k) is white noise with a mean of zero, k is a formula variable, ai、b0And bjExpressed as the parameter to be estimated, m and n are known constants and bm＝1。
Step 2: since the Hammerstein system has a nonlinear characteristic, the variable forgetting factor recursive least square method cannot be applied to the model, and therefore the nonlinear system model in the step 1 is converted into a linear system model. The linear system model can be expressed as: y (k) ═ wTh (k) + v (k), where w is the parameter vector to be estimated, h (k) is the input signal, and T is the matrix transposed symbol.
In the step 2, a parameter mapping method is utilized to approximately convert the nonlinear Hammerstein system into a linear system, so that the method can be suitable for a variable forgetting factor recursion least square method.
And step 3: calculating a cost function of the adaptive filter: solving the prior error of the adaptive filter by using the linear system model obtained in the step 2e (k), calculating the cost function of
Figure BDA0003011326890000021
Wherein, λ (0)<Lambda.ltoreq.1) is the forgetting factor of the VFF-RLS method.
And 4, step 4: in order to obtain the estimated value of the optimal parameter w to be estimated, the cost function in the step 3 is subjected to derivation to obtain a regular equation of the VFF-RLS method, and the regular equation is converted to obtain the estimated value
Figure BDA0003011326890000022
Expression (2)
Figure BDA0003011326890000023
Where Φ (k) is the autocorrelation function matrix of the input signal and θ (k) is the cross-correlation function vector between the output signal and the input signal.
And 5: setting p (k) as the inverse of the input correlation matrix phi (k), calculating p (k) by using phi (k) obtained in step 4, and obtaining a gain vector g (k).
Step 6: initialization: setting initial values a of an input signal and a parameter to be estimated0And b0Initial parameters such as an inverse matrix p (0) and the number of iterations are initialized.
And 7: setting the posterior error epsilon (k) of the adaptive filter, utilizing the relational expression of the prior error and the posterior error in the step 3, and introducing power estimation to obtain the expression of the forgetting factor
Figure BDA0003011326890000024
Wherein q (k) is set to hT(k) p (k-1) h (k) is an intermediate variable,
Figure BDA0003011326890000031
is the power of q (k) and,
Figure BDA0003011326890000032
is the power of the a priori error,
Figure BDA0003011326890000033
is the power of the system noise; .
And 8: according to step 7, setting a forgetting factor discriminant: when in use
Figure BDA0003011326890000034
The forgetting factor is estimated as λ (k) ═ λmaxWherein λ ismaxIs a set constant; when in use
Figure BDA0003011326890000035
The forgetting factor of the proposed VFF-RLS method is estimated as
Figure BDA0003011326890000036
Wherein 1 is<Gamma 2 is a constant and xi is a very small normal number.
And step 9: calculating parameter estimation values by a VFF-RLS method
Figure BDA0003011326890000037
Substituting e (k) obtained in step 3, p (k) and g (k) obtained by calculation in step 5 and the forgetting factor value obtained by judgment in step 8 into the formula in step 4, and calculating an updated formula of the parameter w to be estimated
Figure BDA0003011326890000038
Then calculating an estimated value through iterative operation
Figure BDA0003011326890000039
Step 10: expression obtained by mapping parameters in step 2
Figure BDA00030113268900000310
And the estimated value obtained in step 9
Figure BDA00030113268900000311
The estimated value of the parameter to be estimated of the nonlinear Hammerstein system model can be calculated
Figure BDA00030113268900000312
And
Figure BDA00030113268900000313
preferably, in step 1, the nonlinear system is composed of a static nonlinear subsystem and a dynamic linear subsystem.
Preferably, in step 2, mapping transformation is performed on parameters of the system model in step 1, and the nonlinear system model is converted into a linear system model.
The invention also discloses a variable forgetting factor recursion least square system based on the nonlinear Hammerstein system, which comprises the following modules:
a modeling module: establishing a nonlinear system model based on a Hammerstein system, wherein the relation between the output y (k) and the input u (k) of the nonlinear system is expressed as
Figure BDA00030113268900000314
Where x (k) is an intermediate variable, v (k) is white noise with a mean of zero, k is a formula variable, ai、b0And bjExpressed as the parameter to be estimated, m and n are known constants and bm＝1；
A model conversion module: converting the nonlinear system model in the modeling module into a linear system model, wherein the linear system model is expressed as: y (k) ═ wTh (k) + v (k), where w is the parameter vector to be estimated, h (k) is the input signal, and T is the matrix transposition symbol;
a cost function solving module: the prior error e (k) of the adaptive filter is calculated by using a linear system model obtained by a model conversion module, and a cost function is calculated according to a minimum square error criterion
Figure BDA0003011326890000041
Wherein, λ (0)<Lambda is less than or equal to 1) is a forgetting factor of the VFF-RLS method;
a derivation and conversion module: the cost function in the cost function solving module is derived to obtain a regular equation of the VFF-RLS method, and the regular equation is converted to obtain an estimated value
Figure BDA0003011326890000042
Expression (2)
Figure BDA0003011326890000043
Where Φ (k) is an autocorrelation function matrix of the input signal, θ (k) is a cross-correlation function vector between the output signal and the input signal;
a gain vector calculation module: setting p (k) as the inverse of the input correlation matrix phi (k), and calculating p (k) by using phi (k) obtained by the derivation and conversion module and obtaining a gain vector g (k);
an initial module: setting initial values a of an input signal and a parameter to be estimated0And b0Initializing an inverse matrix p (0) and an initial parameter of iteration times;
a forgetting factor obtaining module: setting a posterior error epsilon (k) of the adaptive filter, introducing power estimation by utilizing a relational expression of the prior error and the posterior error in the cost function solving module, and obtaining an expression of a forgetting factor
Figure BDA0003011326890000044
Wherein q (k) is set to hT(k) p (k-1) h (k) is an intermediate variable,
Figure BDA0003011326890000045
is the power of q (k) and,
Figure BDA0003011326890000046
is the power of the a priori error,
Figure BDA0003011326890000047
is the power of the system noise;
a forgetting factor estimation module: setting a forgetting factor discriminant formula according to a forgetting factor acquisition module: when in use
Figure BDA0003011326890000048
The forgetting factor is estimated as λ (k) ═ λmax(ii) a When in use
Figure BDA0003011326890000049
The forgetting factor is estimated as
Figure BDA0003011326890000051
Wherein 1 is<Gamma is less than or equal to 2 and is a constant, and xi is a tiny normal number;
an estimated value calculation module: substituting e (k) obtained by solving the cost function module, the forgetting factor value obtained by distinguishing by the forgetting factor estimation module and p (k) and g (k) obtained by calculating the gain vector module into the formula of the derivation and conversion module, and calculating to obtain an update formula of the parameter w to be estimated
Figure BDA0003011326890000052
Calculating the estimated value through iterative operation
Figure BDA0003011326890000053
A parameter to be estimated calculation module: expression obtained by model conversion module
Figure BDA0003011326890000054
And the estimated value obtained by the estimated value calculation module
Figure BDA0003011326890000055
Calculating the estimated value of the parameter to be estimated of the nonlinear system model
Figure BDA0003011326890000056
And
Figure BDA0003011326890000057
preferably: in the modeling module, the nonlinear system consists of a static nonlinear subsystem and a dynamic linear subsystem.
Preferably: and in the model conversion module, mapping conversion is carried out on the parameters of the system model in the modeling module, and the nonlinear system model is converted into a linear system model.
The invention researches the identification problem of a single-input single-output nonlinear Hammerstein system, and provides a variable forgetting factor recursion least square method and system based on the nonlinear Hammerstein system. Due to the nonlinear characteristics of a Hammerstein system model, the traditional recursive least square method cannot be directly used for solving the identification problem of the system. Therefore, the Hammerstein system parameter is subjected to mapping transformation, so that the transformed system parameter and the input and the output of the Hammerstein system form a linear relation, and the system can be suitable for a recursive least square method. In addition, in order to solve the problem of convergence speed reduction caused by parameter mapping, the technical scheme provided by the invention has higher estimation precision and better tracking and convergence capabilities compared with the traditional recursive least square method.
The forgetting factor-changing recursive least square method and the system based on the nonlinear Hammerstein system have the following technical effects that:
1. the method utilizes parameter mapping transformation, so that the variable forgetting factor recursive least square method can be suitable for a nonlinear Hammerstein system, and the complexity of the method is reduced.
2. The method adopts a variable forgetting factor recursive least square method to carry out parameter estimation, and compared with the traditional recursive least square method, the method has higher convergence rate.
3. The method uses the variable forgetting factor, so that the forgetting factor is continuously changed along with the change of the convergence of the method, the forgetting factor can meet the requirements of the method at different moments, and the method is ensured to have higher estimation precision and convergence capability.
4. The method has better estimation on the time-varying parameters; when the real value of the parameter changes, the estimated value of the parameter can also track the change quickly, thereby ensuring that the estimated value has smaller estimation error with the real parameter and embodying the good tracking performance of the method of the invention.
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FIG. 1 is a detailed flow chart of a preferred method of the present invention.
FIG. 2 is a block diagram of a preferred system of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples for the purpose of facilitating understanding and practicing the invention by those of ordinary skill in the art, it being understood that the examples described herein are for the purpose of illustration and explanation, and are not to be construed as limiting the invention.
In the research on the identification of the nonlinear Hammerstein system, because the Hammerstein system has nonlinear characteristics, the traditional recursive least square method cannot be directly used for solving the identification problem of the system. Therefore, the invention adopts a parameter mapping method to map and transform the parameters of the Hammerstein system, so that the Hammerstein system can be suitable for a recursive least square method. In addition, in order to solve the problem of convergence rate reduction caused by parameter mapping, the invention provides a variable forgetting factor recursion least square method. The method of the invention utilizes the changed forgetting factor to carry out self-adaptive estimation on the parameters, improves the identification precision of the system and obtains better convergence performance.
Referring to fig. 1, a preferred embodiment of the present invention is a variable forgetting factor recursive least square method based on a nonlinear Hammerstein system, which includes the following specific steps:
step 1: firstly, establishing a nonlinear system model based on a Hammerstein system, wherein the relation between the output y (k) and the input u (k) of the system can be expressed as
Figure BDA0003011326890000071
Where x (k) is an intermediate variable, v (k) is white noise with a mean of zero, k is a formula variable, ai、b0And bjExpressed as the parameter to be estimated, m and n are known constants and bm＝1。
Step 2: since the Hammerstein system has a nonlinear characteristic, the variable forgetting factor recursive least square method cannot be applied to the model, and therefore the nonlinear system model in the step 1 is converted into a linear system model. The linear system model can be expressed as: y (k) ═ wTh (k) + v (k), where w is the parameter vector to be estimated, h (k) is the input signal, and T is the matrix transposed symbol。
In the step 2, a parameter mapping method is utilized to approximately convert the nonlinear Hammerstein system into a linear system, so that the method can be suitable for a variable forgetting factor recursion least square method.
And step 3: calculating a cost function of the adaptive filter: calculating the prior error e (k) of the adaptive filter by using the linear system model obtained in the step 2, and calculating a cost function of
Figure BDA0003011326890000072
Wherein, λ (0)<Lambda.ltoreq.1) is the forgetting factor of the VFF-RLS method.
And 4, step 4: in order to obtain the estimated value of the optimal parameter w to be estimated, the cost function in the step 3 is subjected to derivation to obtain a regular equation of the VFF-RLS method, and the regular equation is converted to obtain the estimated value
Figure BDA0003011326890000073
Expression (2)
Figure BDA0003011326890000074
Where Φ (k) is the autocorrelation function matrix of the input signal and θ (k) is the cross-correlation function vector between the output signal and the input signal.
And 5: setting p (k) as the inverse of the input correlation matrix phi (k), calculating p (k) by using phi (k) obtained in step 4, and obtaining a gain vector g (k).
Step 6: initialization: setting initial values a of an input signal and a parameter to be estimated0And b0Initial parameters such as an inverse matrix p (0) and the number of iterations are initialized.
And 7: setting the posterior error epsilon (k) of the adaptive filter, utilizing the relational expression of the prior error and the posterior error in the step 3, and introducing power estimation to obtain the expression of the forgetting factor
Figure BDA0003011326890000081
Wherein q (k) is set to hT(k) p (k-1) h (k) is an intermediate variable,
Figure BDA0003011326890000082
is the power of q (k) and,
Figure BDA0003011326890000083
is the power of the a priori error,
Figure BDA0003011326890000084
is the power of the system noise.
And 8: according to step 7, setting a forgetting factor discriminant: when in use
Figure BDA0003011326890000085
The forgetting factor is estimated as λ (k) ═ λmaxWherein λ ismaxIs a set constant; when in use
Figure BDA0003011326890000086
The forgetting factor of the proposed VFF-RLS method is estimated as
Figure BDA0003011326890000087
Wherein 1 is<Gamma 2 is a constant and xi is a very small normal number.
And step 9: calculating parameter estimation values by a VFF-RLS method
Figure BDA0003011326890000088
Substituting e (k) obtained in step 3, p (k) and g (k) obtained by calculation in step 5 and the forgetting factor value obtained by judgment in step 8 into the formula in step 4, and calculating an updated formula of the parameter w to be estimated
Figure BDA0003011326890000089
Then calculating an estimated value through iterative operation
Figure BDA00030113268900000810
Step 10: expression obtained by mapping parameters in step 2
Figure BDA00030113268900000811
And the estimated value obtained in step 9
Figure BDA00030113268900000812
The estimated value of the parameter to be estimated of the nonlinear Hammerstein system model can be calculated
Figure BDA00030113268900000813
And
Figure BDA00030113268900000814
in designing the present invention, specific values were substituted to verify the performance of the present invention: setting the true value a of the parameter to be estimated of the nonlinear Hammerstein system0＝[0.9,1.1,1.3]T，b0＝[0.8,0.5,1.2,1]T. The invention is applied to carry out simulation experiments in MATLAB to obtain parameter estimation values
Figure BDA00030113268900000815
The error between the true value and the estimated value is small, and the method has good estimation precision.
As shown in fig. 2, the present embodiment discloses a variable forgetting factor recursive least square system based on a nonlinear Hammerstein system, which includes the following modules:
a modeling module: establishing a nonlinear system model based on a Hammerstein system, wherein the relation between the output y (k) and the input u (k) of the nonlinear system is expressed as
Figure BDA0003011326890000091
Where x (k) is an intermediate variable, v (k) is white noise with a mean of zero, k is a formula variable, ai、b0And bjExpressed as the parameter to be estimated, m and n are known constants and bm1 is ═ 1; the nonlinear system consists of a static nonlinear subsystem and a dynamic linear subsystem.
A model conversion module: converting the nonlinear system model in the modeling module into a linear system model, wherein the linear system model is expressed as: y (k) ═ wTh(k)+v(k) Wherein, w is a parameter vector to be estimated, h (k) is an input signal, and T is a matrix transposition symbol; in this embodiment, the parameters of the system model in the modeling module are mapped and transformed, and the nonlinear system model is converted into a linear system model.
A cost function solving module: the prior error e (k) of the adaptive filter is calculated by using a linear system model obtained by a model conversion module, and a cost function is calculated according to a minimum square error criterion
Figure BDA0003011326890000092
Wherein, λ (0)<Lambda is less than or equal to 1) is a forgetting factor of the VFF-RLS method;
a derivation and conversion module: the cost function in the cost function solving module is derived to obtain a regular equation of the VFF-RLS method, and the regular equation is converted to obtain an estimated value
Figure BDA0003011326890000093
Expression (2)
Figure BDA0003011326890000094
Where Φ (k) is an autocorrelation function matrix of the input signal, θ (k) is a cross-correlation function vector between the output signal and the input signal;
a gain vector calculation module: setting p (k) as the inverse of the input correlation matrix phi (k), and calculating p (k) by using phi (k) obtained by the derivation and conversion module and obtaining a gain vector g (k);
an initial module: setting initial values a of an input signal and a parameter to be estimated0And b0Initializing an inverse matrix p (0) and an initial parameter of iteration times;
a forgetting factor obtaining module: setting a posterior error epsilon (k) of the adaptive filter, introducing power estimation by utilizing a relational expression of the prior error and the posterior error in the cost function solving module, and obtaining an expression of a forgetting factor
Figure BDA0003011326890000101
Wherein q (k) is set to hT(k) p (k-1) h (k) is an intermediate variable,
Figure BDA0003011326890000102
is the power of q (k) and,
Figure BDA0003011326890000103
is the power of the a priori error,
Figure BDA0003011326890000104
is the power of the system noise;
a forgetting factor estimation module: setting a forgetting factor discriminant formula according to a forgetting factor acquisition module: when in use
Figure BDA0003011326890000105
The forgetting factor is estimated as λ (k) ═ λmax(ii) a When in use
Figure BDA0003011326890000106
The forgetting factor is estimated as
Figure BDA0003011326890000107
Wherein 1 is<Gamma is less than or equal to 2 and is a constant, and xi is a tiny normal number;
an estimated value calculation module: substituting e (k) obtained by solving the cost function module, the forgetting factor value obtained by distinguishing by the forgetting factor estimation module and p (k) and g (k) obtained by calculating the gain vector module into the formula of the derivation and conversion module, and calculating to obtain an update formula of the parameter w to be estimated
Figure BDA0003011326890000108
Calculating the estimated value through iterative operation
Figure BDA0003011326890000109
A parameter to be estimated calculation module: expression obtained by model conversion module
Figure BDA00030113268900001010
And the estimated value obtained by the estimated value calculation module
Figure BDA00030113268900001011
Calculating the estimated value of the parameter to be estimated of the nonlinear system model
Figure BDA00030113268900001012
And
Figure BDA00030113268900001013
the variable forgetting factor recursive least square method is applied to the nonlinear Hammerstein system model, firstly, the nonlinear Hammerstein system is approximately converted into the linear system model by using parameter mapping transformation, then the identification problem of the linear system is solved by the variable forgetting factor recursive least square method, and the estimation value obtained by iteration through the method has better estimation precision and convergence performance.
It should be understood that the above description of the preferred embodiments is given for clearness of understanding and no unnecessary limitations are to be understood therefrom, and all changes and modifications that come within the spirit of the invention may be made by those skilled in the art without departing from the scope of the invention as defined by the appended claims.

Claims (6)

1. The variable forgetting factor recursive least square method based on the nonlinear Hammerstein system is characterized by comprising the following steps of:
step 1, establishing a nonlinear system model based on a Hammerstein system, wherein the relation between the output y (k) and the input u (k) of the nonlinear system is expressed as
Figure FDA0003011326880000011
Where x (k) is an intermediate variable, v (k) is white noise with a mean of zero, k is a formula variable, ai、b0And bjExpressed as the parameter to be estimated, m and n are known constants and bm＝1；
Step 2, adding the compound of step 1The linear system model is converted into a linear system model, which is expressed as: y (k) ═ wTh (k) + v (k), where w is the parameter vector to be estimated, h (k) is the input signal, and T is the matrix transposition symbol;
step 3, solving the prior error e (k) of the adaptive filter by using the linear system model obtained in the step 2, and calculating a cost function of
Figure FDA0003011326880000012
Wherein, λ (0)<Lambda is less than or equal to 1) is a forgetting factor of the VFF-RLS method;
step 4, carrying out derivation on the cost function in the step 3 to obtain a regular equation of the VFF-RLS method, and converting the regular equation to obtain an estimated value
Figure FDA0003011326880000013
Expression (2)
Figure FDA0003011326880000014
Where Φ (k) is an autocorrelation function matrix of the input signal, θ (k) is a cross-correlation function vector between the output signal and the input signal;
step 5, setting p (k) as the inverse of the input correlation matrix phi (k), calculating p (k) by using the phi (k) obtained in the step 4, and obtaining a gain vector g (k);
step 6, setting the initial value a of the input signal and the parameter to be estimated0And b0Initializing an inverse matrix p (0) and an initial parameter of iteration times;
step 7, setting the posterior error epsilon (k) of the adaptive filter, introducing power estimation by using the relational expression of the prior error and the posterior error in the step 3, and obtaining the expression of the forgetting factor
Figure FDA0003011326880000021
Wherein q (k) is set to hT(k) p (k-1) h (k) is an intermediate variable,
Figure FDA0003011326880000022
is the power of q (k) and,
Figure FDA0003011326880000023
is the power of the a priori error,
Figure FDA0003011326880000024
is the power of the system noise;
and 8, setting a forgetting factor discriminant formula according to the step 7: when in use
Figure FDA0003011326880000025
The forgetting factor is estimated as λ (k) ═ λmax(ii) a When in use
Figure FDA0003011326880000026
The forgetting factor is estimated as
Figure FDA0003011326880000027
Wherein 1 is<Gamma is less than or equal to 2 and is a constant, and xi is a tiny normal number;
step 9, substituting the values of the forgetting factor obtained in the step 3 and the values of the forgetting factor obtained in the step 8, the values of p (k) and g (k) obtained in the step 5 into the formula in the step 4, and calculating an updated formula of the parameter w to be estimated
Figure FDA0003011326880000028
Calculating the estimated value through iterative operation
Figure FDA0003011326880000029
Step 10, utilizing the expression obtained in step 2
Figure FDA00030113268800000210
And the estimated value obtained in step 9
Figure FDA00030113268800000211
Calculating the estimation of the parameters to be estimated of the nonlinear system modelEvaluating value
Figure FDA00030113268800000212
And
Figure FDA00030113268800000213
2. the recursive least square method of variable forgetting factors based on the nonlinear Hammerstein system according to claim 1, characterized in that: in step 1, the nonlinear system consists of a static nonlinear subsystem and a dynamic linear subsystem.
3. The recursive least square method of variable forgetting factors based on the nonlinear Hammerstein system according to claim 1 or 2, characterized in that: and 2, mapping and transforming the parameters of the system model in the step 1, and converting the nonlinear system model into a linear system model.
4. A variable forgetting factor recursion least square system based on a nonlinear Hammerstein system is characterized by comprising the following modules:
a modeling module: establishing a nonlinear system model based on a Hammerstein system, wherein the relation between the output y (k) and the input u (k) of the nonlinear system is expressed as
Figure FDA0003011326880000031
Where x (k) is an intermediate variable, v (k) is white noise with a mean of zero, k is a formula variable, ai、b0And bjExpressed as the parameter to be estimated, m and n are known constants and bm＝1；
A model conversion module: converting the nonlinear system model in the modeling module into a linear system model, wherein the linear system model is expressed as: y (k) ═ wTh (k) + v (k), where w is the parameter vector to be estimated, h (k) is the input signal, and T is the matrix transposition symbol;
a cost function solving module: using model rotationSolving the prior error e (k) of the adaptive filter by the linear system model obtained by the conversion module, and calculating a cost function of
Figure FDA0003011326880000032
Wherein, λ (0)<Lambda is less than or equal to 1) is a forgetting factor of the VFF-RLS method;
a derivation and conversion module: the cost function in the cost function solving module is derived to obtain a regular equation of the VFF-RLS method, and the regular equation is converted to obtain an estimated value
Figure FDA0003011326880000041
Expression (2)
Figure FDA0003011326880000042
Where Φ (k) is an autocorrelation function matrix of the input signal, θ (k) is a cross-correlation function vector between the output signal and the input signal;
a gain vector calculation module: setting p (k) as the inverse of the input correlation matrix phi (k), and calculating p (k) by using phi (k) obtained by the derivation and conversion module and obtaining a gain vector g (k);
an initial module: setting initial values a of an input signal and a parameter to be estimated0And b0Initializing an inverse matrix p (0) and an initial parameter of iteration times;
a forgetting factor obtaining module: setting a posterior error epsilon (k) of the adaptive filter, introducing power estimation by utilizing a relational expression of the prior error and the posterior error in the cost function solving module, and obtaining an expression of a forgetting factor
Figure FDA0003011326880000043
Wherein is provided with
q(k)＝hT(k) p (k-1) h (k) is an intermediate variable,
Figure FDA0003011326880000044
is the power of q (k) and,
Figure FDA0003011326880000045
is the power of the a priori error,
Figure FDA0003011326880000046
is the power of the system noise;
a forgetting factor estimation module: setting a forgetting factor discriminant formula according to a forgetting factor acquisition module: when in use
Figure FDA0003011326880000047
The forgetting factor is estimated as λ (k) ═ λmax(ii) a When in use
Figure FDA0003011326880000048
The forgetting factor is estimated as
Figure FDA0003011326880000049
Wherein 1 is<Gamma is less than or equal to 2 and is a constant, and xi is a tiny normal number;
an estimated value calculation module: substituting e (k) obtained by solving the cost function module, the forgetting factor value obtained by distinguishing by the forgetting factor estimation module and p (k) and g (k) obtained by calculating the gain vector module into the formula of the derivation and conversion module, and calculating to obtain an update formula of the parameter w to be estimated
Figure FDA00030113268800000410
Calculating the estimated value through iterative operation
Figure FDA00030113268800000411
A parameter to be estimated calculation module: expression obtained by model conversion module
Figure FDA0003011326880000051
And the estimated value obtained by the estimated value calculation module
Figure FDA0003011326880000052
Calculating the estimated value of the parameter to be estimated of the nonlinear system model
Figure FDA0003011326880000053
And
Figure FDA0003011326880000054
5. the variable forgetting factor recursive least squares system based on the nonlinear Hammerstein system of claim 4, wherein: in the modeling module, the nonlinear system consists of a static nonlinear subsystem and a dynamic linear subsystem.
6. The recursive least square system with variable forgetting factor based on nonlinear Hammerstein system as claimed in claim 4 or 5, wherein: and in the model conversion module, mapping conversion is carried out on the parameters of the system model in the modeling module, and the nonlinear system model is converted into a linear system model.
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CN113655277A (en) * 2021-08-13 2021-11-16 海南师范大学 Insulation resistance value detection method during disconnection of power battery of electric automobile
CN116400596A (en) * 2023-04-03 2023-07-07 中国矿业大学 Industrial time-varying system two-dimensional recursion identification modeling method for dead zone with input

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CN113655277A (en) * 2021-08-13 2021-11-16 海南师范大学 Insulation resistance value detection method during disconnection of power battery of electric automobile
CN113655277B (en) * 2021-08-13 2023-09-26 海南师范大学 Insulation resistance value detection method during disconnection of electric automobile power battery
CN116400596A (en) * 2023-04-03 2023-07-07 中国矿业大学 Industrial time-varying system two-dimensional recursion identification modeling method for dead zone with input
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