CN111665725B - Robust parameter identification method for electromechanical positioning system for measuring random data loss - Google Patents

Robust parameter identification method for electromechanical positioning system for measuring random data loss Download PDF

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CN111665725B
CN111665725B CN202010591443.8A CN202010591443A CN111665725B CN 111665725 B CN111665725 B CN 111665725B CN 202010591443 A CN202010591443 A CN 202010591443A CN 111665725 B CN111665725 B CN 111665725B
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杨宪强
刘新鹏
高会军
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Harbin Institute of Technology
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Abstract

A robust parameter identification method for an electromechanical positioning system with random missing measurement data belongs to the field of industrial automation and model parameter identification. The invention aims to solve the problem of low parameter identification precision caused by the fact that abnormal interference data existing in a system is not considered in the existing parameter identification method of the electromechanical positioning system. The method comprises the following steps: acquiring a position signal of a motor load end, and acquiring the speed and the acceleration of a load based on the position signal and a center difference method; establishing an inverse dynamics model of the positioning system; under a probability framework, establishing a robust identification model of the electromechanical positioning system when part of measured data is randomly missing; and deducing to obtain an iterative updating formula of the parameter to be identified and the offset based on the variational Bayesian framework, and obtaining estimated values of the parameter to be identified and the offset when the termination condition of iterative updating is met. The method is used for parameter identification of the electromechanical positioning system.

Description

Robust parameter identification method for electromechanical positioning system for measuring random data loss
Technical Field
The invention relates to a robust parameter identification method for an electromechanical positioning system for measuring random data loss, belonging to the field of industrial automation and model parameter identification.
Background
With the rapid development of integrated circuits, semiconductor industry and manufacturing industry, the modern industry has higher and higher requirements on the precision, speed, stability and the like of a positioning platform. An electromechanical positioning system is one of standard components of a driving mechanism, and is widely applied to practical industrial systems, such as train traction, aircraft launching, industrial gantry and the like.
Due to the influence of factors such as sensor faults and external interference, the problems of outliers, data loss and the like often exist in identification data collected by a positioning platform. In the existing identification method of the positioning system, the bias of parameter identification is caused by directly deleting abnormal data; however, the influence of outliers and lost data on the recognition result is not considered, which results in the degradation of the system recognition accuracy.
Disclosure of Invention
The invention provides a robust parameter identification method of an electromechanical positioning system, which aims to solve the problem of low parameter identification precision caused by the fact that abnormal interference data existing in the system is not considered in the existing parameter identification method of the electromechanical positioning system.
The invention relates to a robust parameter identification method of an electromechanical positioning system for measuring random loss of data, which comprises the following steps,
the method comprises the following steps: acquiring a position signal of a motor load end, and acquiring the speed and the acceleration of a load based on the position signal and a center difference method; establishing an inverse dynamics model of the positioning system;
step two: under a probability framework, establishing a robust identification model of the electromechanical positioning system when part of measured data is randomly missing;
step three: and deducing to obtain an iterative updating formula of the parameter to be identified and the offset based on the variational Bayesian framework, and obtaining estimated values of the parameter to be identified and the offset when the termination condition of iterative updating is met.
The robust parameter identification method for the electromechanical positioning system for measuring the random loss of data comprises the following steps of measuring a position signal qkEstimating velocity
Figure BDA0002556283810000011
And acceleration
Figure BDA0002556283810000012
The process comprises the following steps:
firstly, filtering the position measurement data of the motor load end to obtain a position signal qk；
From the position signal qkAnd estimating the speed and the acceleration of the load by adopting a center difference method:
Figure BDA0002556283810000013
Figure BDA0002556283810000021
k is sampling time, k is 1,2, and N is total sampling times;
wherein f issThe system sampling frequency.
According to the robust parameter identification method for the electromechanical positioning system with the random missing measured data, the inverse dynamics model comprises the following steps:
Figure BDA0002556283810000022
in the formula xkThe system control force at sampling time k;
m is the inertia parameter of the system to be identified, FvTo identify viscous friction force, FcFor coulomb friction to be identified, offset is the deflection term;
converting the inverse dynamics model into a linear form expression:
Figure BDA0002556283810000023
φkis an observation matrix of the inverse dynamics model,
Figure BDA0002556283810000024
theta is the parameter to be identified and the offset,
θ＝[M Fv Fc offset]T。
according to the method for identifying the robust parameters of the electromechanical positioning system with the random missing measured data, the establishment process of the robust identification model of the positioning system comprises the following steps:
the noise distribution characteristics in the linear form expression are described by a heavy end student-t distribution,
ek～St(ek|0,λ,v)， (5)
in the formula ekIn order to consider the heavy tail distribution noise, lambda is the precision parameter of student-t distribution, and v is the degree of freedom parameter;
the student-t distribution is represented as follows:
Figure BDA0002556283810000025
zkis an introduced implicit variable;
substituting the formula (4) into the formula (6) to obtain a likelihood function of the process output, wherein the likelihood function obeys student-t distribution,
Figure BDA0002556283810000031
decomposing the student-t distribution into a form of gaussian scale mixture, equation (7) is decomposed into:
Figure BDA0002556283810000032
the problem of random missing of partial measurement data is described:
yk＝Hkxk+νk， (9)
in the formula ykFor measuring output data, HkTo indicate the identity variable lost or not, HkWhen the value is 1, the measured data is not lost; when H is presentkWhen the value is 0, the measured data is lost; v iskFor measuring the noise, v, of the output datakObeying Gaussian distribution with the mean value of 0 and the precision of S;
further obtaining measurement output data ykThe conditional distribution expression of (1):
Figure BDA0002556283810000033
selecting prior distribution of robust identification model parameters of the positioning system, and assuming that the model parameters obey Gaussian distribution for forming conjugate prior:
Figure BDA0002556283810000034
wherein δ is a hyperparameter; the hyper-parameter delta follows Gamma distribution; d is the order of the model parameter θ;
Figure BDA0002556283810000035
in the formula
Figure BDA0002556283810000036
Is a shape parameter obeying the hyper-parameter delta of the Gamma distribution,
Figure BDA0002556283810000037
is a rate parameter of the hyper-parameter δ obeying the Gamma distribution;
the precision parameters and the degree of freedom parameters of the student-t distribution also follow the Gamma distribution:
Figure BDA0002556283810000038
Figure BDA0002556283810000039
in the formula
Figure BDA00025562838100000310
Is a shape parameter of lambda following the Gamma distribution,
Figure BDA00025562838100000311
is the rate parameter of λ obeying the Gamma distribution;
Figure BDA00025562838100000312
is a shape parameter of v following a Gamma distribution,
Figure BDA00025562838100000313
is the rate parameter of v obeying the Gamma distribution;
therefore, a robust identification model of the positioning system is established.
The invention has the beneficial effects that: the method is used for estimating the identification parameters of the electromechanical system when the observation data of the positioning system is lost and outliers exist, and can effectively solve the problem of biased estimation in the existing method, thereby improving the parameter identification precision.
The method comprises the steps of firstly establishing an inverse dynamics model of the electromechanical system, modeling measurement loss and outlier problems under a probability framework, then carrying out identification algorithm derivation on the established probability model based on a variational Bayes algorithm, and further estimating unknown parameters of the inverse dynamics model of the electromechanical system. The method can effectively ensure the efficiency and the precision of parameter estimation, and has important application value in the identification theory and the actual industrial process of an electromechanical system.
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FIG. 1 is a flow chart of a robust parameter identification method for an electromechanical positioning system with random missing measured data according to the present invention;
FIG. 2 is a comparison graph of self-verification output estimation for identifying parameters by using least square method (IDIM-LS) based on inverse dynamics model and the method of the present invention (RM-VB) when SNR is 5dB, random loss ratio of output is 20%, and abnormal value ratio of output is 20%;
FIG. 3 is a cross-validation output estimation comparison graph for identifying parameters by using least square method (IDIM-LS) based on inverse dynamics model and the method of the present invention (RM-VB) when SNR is 5dB, random loss ratio of output is 20%, and abnormal value ratio of output is 20%;
fig. 4 is a graph showing a change in the lower boundary value of variation.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict.
The invention is further described with reference to the following drawings and specific examples, which are not intended to be limiting.
In a first embodiment, referring to fig. 1, the present invention provides a robust parameter identification method for an electromechanical positioning system with random missing measurement data, which includes,
the method comprises the following steps: acquiring a position signal of a motor load end, and acquiring the speed and the acceleration of a load based on the position signal and a center difference method; establishing an inverse dynamics model of the positioning system;
step two: under a probability framework, establishing a robust identification model of the electromechanical positioning system when part of measured data is randomly missing;
step three: and deducing to obtain an iterative updating formula of the parameter to be identified and the offset based on the variational Bayesian framework, and obtaining estimated values of the parameter to be identified and the offset when the termination condition of iterative updating is met.
Further, by the position signal qkEstimating velocity
Figure BDA0002556283810000051
And acceleration
Figure BDA0002556283810000052
The process comprises the following steps:
firstly, filtering the position measurement data of the motor load end to obtain a position signal qk；
From the position signal qkAnd estimating the speed and the acceleration of the load by adopting a center difference method:
Figure BDA0002556283810000053
Figure BDA0002556283810000054
k is sampling time, k is 1,2, and N is total sampling times;
wherein f issThe system sampling frequency.
In this embodiment, the filtering of the position measurement data of the load end of the motor may be implemented by using a butterworth filter, and the filtered data may be approximately differentiated by using a center difference method. The filter cut-off frequency is chosen according to the rule of thumb as omegacf≥5ωdynWherein ω iscfCut-off frequency, omega, of filtersdynIs the natural frequency of the motor system.
Still further, the inverse dynamics model is:
Figure BDA0002556283810000055
in the formula xkThe system control force at sampling time k;
m is the inertia parameter of the system to be identified, FvTo identify viscous friction force, FcFor coulomb friction to be identified, offset is the deflection term;
converting the inverse dynamics model into a linear form expression:
Figure BDA0002556283810000056
φkis an observation matrix of the inverse dynamics model,
Figure BDA0002556283810000057
theta is the parameter to be identified and the offset,
θ＝[M Fv Fc offset]T。
still further, the process of establishing the robust identification model of the positioning system comprises:
in consideration of the outlier problem which may exist in the output data, the noise distribution characteristic in the linear form expression is described by a heavy end student-t distribution,
ek～St(ek|0,λ,v)， (5)
in the formula ekIn order to consider the heavy tail distribution noise, lambda is the precision parameter of student-t distribution, and v is the degree of freedom parameter;
the student-t distribution is represented as follows:
Figure BDA0002556283810000061
zkis an introduced implicit variable;
substituting the formula (4) into the formula (6) to obtain a likelihood function of the process output, wherein the likelihood function obeys student-t distribution,
Figure BDA0002556283810000062
as can be seen from equation (6), by introducing an additional hidden variable zkThe student-t distribution can be decomposed into the form of Gaussian Scale Mixture (Gaussian Scale Mixture), and then equation (7) is decomposed into:
Figure BDA0002556283810000063
the problem of random missing of partial measurement data is described:
yk＝Hkxk+νk， (9)
in the formula ykFor measuring output data, HkTo indicate whether the identity variable is lost or not, is a known quantity, HkWhen the value is 1, the measured data is not lost; when H is presentkWhen the value is 0, the measured data is lost; to build a probabilistic model, v is introducedk，νkFor measuring the noise, v, of the output datakObeying Gaussian distribution with the mean value of 0 and the precision of S; the precision parameter S tends to be infinite;
further obtaining measurement output data ykThe conditional distribution expression of (1):
Figure BDA0002556283810000064
selecting prior distribution of robust identification model parameters of the positioning system, and assuming that the model parameters obey Gaussian distribution for forming conjugate prior:
Figure BDA0002556283810000065
wherein δ is a hyperparameter; the hyper-parameter delta follows Gamma distribution; d is the order of the model parameter θ;
Figure BDA0002556283810000066
in the formula
Figure BDA0002556283810000071
Is a shape parameter obeying the hyper-parameter delta of the Gamma distribution,
Figure BDA0002556283810000072
is a rate parameter of the hyper-parameter δ obeying the Gamma distribution;
the precision parameters and the degree of freedom parameters of the student-t distribution also follow the Gamma distribution:
Figure BDA0002556283810000073
Figure BDA0002556283810000074
in the formula
Figure BDA0002556283810000075
Is a shape parameter of lambda following the Gamma distribution,
Figure BDA0002556283810000076
is the rate parameter of λ obeying the Gamma distribution;
Figure BDA0002556283810000077
is a shape parameter of v following a Gamma distribution,
Figure BDA0002556283810000078
is the rate parameter of v obeying the Gamma distribution;
therefore, based on student-t distribution, a robust identification model of the positioning system is established.
In the method, based on a variational Bayes framework, model parameters of the electromechanical system are obtained through robust identification according to an observation data set W ═ { phi, y, H }.
Based on a variational Bayesian framework, the specific process of deducing the iterative update formula of the parameters to be estimated is as follows:
the joint probability distribution p (W, X, Z, omega) of the robust identification model of the positioning system is as follows:
Figure BDA0002556283810000079
where W is known identification data (observation data set), and Y is { Y ═ Yk}k＝1:N,H＝{Hk}k＝1:N,X＝{xk}k＝1:NZ is a hidden variable, Z ═ Zk}k＝1:NΩ is an unknown parameter, Ω ═ θ, δ, λ, v }, C is a constant term, and C ═ p (Φ, H) is a constant term.
And further, updating the parameters to be identified according to a variational Bayesian formula:
lnqi(hi)＝<lnp(W,X,Z,Ω)>j≠i+const， (16)
wherein q isi(hi) In order to introduce a distribution of the variation,<·>i≠jis shown with respect to qj(hj) To find the expectation (i ≠ j),
according to equation (16), the hidden variable and the parameter to be identified are updated as follows:
q(xk) Obeying a gaussian distribution:
Figure BDA00025562838100000710
wherein the content of the first and second substances,
Figure BDA0002556283810000081
q(zk) Subject to the Gamma distribution,
Figure BDA0002556283810000082
wherein:
Figure BDA0002556283810000083
q (θ) follows a gaussian distribution:
Figure BDA0002556283810000084
wherein the content of the first and second substances,
Figure BDA0002556283810000085
q(δi) Obeying a Gamma distribution:
Figure BDA0002556283810000086
wherein
Figure BDA0002556283810000087
tr (-) denotes the trace of the matrix;
q (λ) obeys a Gamma distribution:
Figure BDA0002556283810000088
wherein the content of the first and second substances,
Figure BDA0002556283810000091
q (v) obeys a Gamma distribution:
Figure BDA0002556283810000092
wherein
Figure BDA0002556283810000093
The relative expected values of the above variables are as follows
Figure BDA0002556283810000094
Setting the termination condition of the iterative update of the variational Bayesian formula as follows:
Figure BDA0002556283810000095
wherein
Figure BDA0002556283810000096
Is the variation lower bound of the ith iteration, and epsilon is a preset iteration termination threshold;
the lower bound of variation is defined as:
Figure BDA0002556283810000097
wherein
Figure BDA0002556283810000098
When iteration is terminated, estimating to obtain a parameter to be identified and an offset theta*＝θi。
According to the embodiment, the variation distribution is introduced to approximate the posterior probability density function which is difficult to solve, the lower bound of the variation is optimized, the system unknown parameters are obtained through iterative estimation, and the efficiency and the precision of parameter estimation can be effectively guaranteed.
The following examples are used to demonstrate the beneficial effects of the present invention:
the specific embodiment is as follows: the effectiveness of the invention is verified by adopting an EMPS (Electro-Mechanical Positioning System) data set provided by French aerospace research institute A.Janot et al, a built experimental platform comprises a direct current motor, an incremental encoder, a high-precision low-friction ball screw transmission Positioning device, a load, an accelerometer and the like, the devices are connected with a dSPACE digital controller, Matlab and simulation software are used for acquiring experimental data, and the sampling frequency is 1 kHz.
In order to verify the effectiveness of the method, Gaussian white noise with the signal-to-noise ratio of 5dB is added to the output data (noiseless output) of the EMPS data set, 20% of the output value is randomly lost, and 20% of the output value is added and uniformly distributed in the range of-10, 10]The outlier of (a). The two figures also show the comparison of Absolute Error (AE) as shown in conjunction with figures 2 and 3. As can be seen from FIGS. 2 and 3, the output estimation relative error of parameter identification using the above two methods
Figure BDA0002556283810000101
The results are shown in Table 1;
table 1 output estimated relative error results
Figure BDA0002556283810000102
Table 1 shows that the method can better avoid the influence of loss, outliers and the like in the output data on the identification algorithm, and improves the robustness of the identification algorithm. The inverse dynamics model parameters estimated by the present invention are shown in table 2.
TABLE 2 inverse dynamics model parameter estimation results
Figure BDA0002556283810000103
In addition, the variation situation of the lower boundary value of the variation is shown in fig. 4, and it can be seen that the method can smoothly converge to the final estimation value after several steps of iteration, so that the method has good convergence and high estimation efficiency.
In summary, according to the present application, for load position data acquired by an electromechanical positioning system, speed and acceleration information of a load are obtained based on a center difference method, and an identification data set of an inverse dynamic model of the electromechanical positioning system is formed in combination with system control force data; then, converting the inverse dynamics model into a linear form, and comprehensively considering the parameter identification problem when uncertainty and abnormal interference (outlier and lost data) exist on the basis to form a robust identification model of the electromechanical positioning system; and then, based on the variational Bayes framework, iteratively updating unknown parameters in the inverse dynamics model until a variational lower bound value is converged to obtain a final estimation value of the required inverse dynamics model parameters.
The method is suitable for the positioning systems of the rotating motor and the linear motor, and has the parameter identification when uncertainty and abnormal interference (outlier and lost data) exist.
Although the invention herein has been described with reference to particular embodiments, it is to be understood that these embodiments are merely illustrative of the principles and applications of the present invention. It is therefore to be understood that numerous modifications may be made to the illustrative embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that features described in different dependent claims and herein may be combined in ways different from those described in the original claims. It is also to be understood that features described in connection with individual embodiments may be used in other described embodiments.

Claims (3)

1. A robust parameter identification method for an electromechanical positioning system for measuring random loss of data is characterized by comprising the following steps,
the method comprises the following steps: acquiring a position signal of a motor load end, and acquiring the speed and the acceleration of a load based on the position signal and a center difference method; establishing an inverse dynamics model of the positioning system;
step two: under a probability framework, establishing a robust identification model of the electromechanical positioning system when part of measured data is randomly missing;
step three: deriving an iterative updating formula of the parameter to be identified and the offset based on a variational Bayesian framework, and obtaining estimated values of the parameter to be identified and the offset when a termination condition of iterative updating is met;
from position signals qkEstimating velocity
Figure FDA0003028383120000011
And acceleration
Figure FDA0003028383120000012
The process comprises the following steps:
firstly, filtering the position measurement data of the motor load end to obtain a position signal qk；
From the position signal qkAnd estimating the speed and the acceleration of the load by adopting a center difference method:
Figure FDA0003028383120000013
Figure FDA0003028383120000014
k is sampling time, k is 1,2, and N is total sampling times;
wherein f issThe system sampling frequency;
the inverse dynamics model is as follows:
Figure FDA0003028383120000015
in the formula xkTo adoptThe system control force at sample time k;
m is the inertia parameter of the system to be identified, FvTo identify viscous friction force, FcThe offset is the offset for the coulomb friction to be identified;
converting the inverse dynamics model into a linear form expression:
Figure FDA0003028383120000016
φkis an observation matrix of the inverse dynamics model,
Figure FDA0003028383120000017
theta includes the parameter to be identified M, Fv、FcAnd an offset amount offset, which is set,
θ＝[M Fv Fc offset]T；
the establishment process of the robust identification model of the positioning system comprises the following steps:
the noise distribution characteristics in the linear form expression are described by a heavy end student-t distribution,
ek～St(ek|0,λ,v)， (5)
in the formula ekIn order to consider the heavy tail distribution noise, lambda is the precision parameter of student-t distribution, and v is the degree of freedom parameter;
the student-t distribution is represented as follows:
Figure FDA0003028383120000021
zkis an introduced implicit variable;
Figure FDA0003028383120000022
which represents a gaussian distribution of the intensity of the light,
Figure FDA0003028383120000023
representing a gamma distribution, Γ (·) representing a gamma function;
substituting the formula (4) into the formula (6) to obtain a likelihood function of the process output, wherein the likelihood function obeys student-t distribution,
Figure FDA0003028383120000024
p (-) is a unified expression of the probability density function, and the specific form is determined by the distribution form on the right side of the function equation;
decomposing the student-t distribution into a form of gaussian scale mixture, equation (7) is decomposed into:
Figure FDA0003028383120000025
the problem of random missing of partial measurement data is described:
yk＝Hkxk+νk， (9)
in the formula ykFor measuring output data, HkTo indicate the identity variable lost or not, HkWhen the value is 1, the measured data is not lost; when H is presentkWhen the value is 0, the measured data is lost; v iskFor measuring the noise, v, of the output datakObeying Gaussian distribution with the mean value of 0 and the precision of S;
further obtaining measurement output data ykThe conditional distribution expression of (1):
Figure FDA0003028383120000026
selecting prior distribution of robust identification model parameters of the positioning system, and assuming that the model parameters obey Gaussian distribution for forming conjugate prior:
Figure FDA0003028383120000027
wherein δ is a hyperparameter; the hyper-parameter delta follows Gamma distribution; d is the order of the model parameter θ;
Figure FDA0003028383120000028
in the formula
Figure FDA0003028383120000031
Is a shape parameter obeying the hyper-parameter delta of the Gamma distribution,
Figure FDA0003028383120000032
is a rate parameter of the hyper-parameter δ obeying the Gamma distribution;
the precision parameters and the degree of freedom parameters of the student-t distribution also follow the Gamma distribution:
Figure FDA0003028383120000033
Figure FDA0003028383120000034
in the formula
Figure FDA0003028383120000035
Is a shape parameter of lambda following the Gamma distribution,
Figure FDA0003028383120000036
is the rate parameter of λ obeying the Gamma distribution;
Figure FDA0003028383120000037
is a shape parameter of v following a Gamma distribution,
Figure FDA0003028383120000038
is the rate parameter of v obeying the Gamma distribution;
therefore, a robust identification model of the positioning system is established.
2. The method of claim 1, wherein the method for identifying robust parameters of an electromechanical positioning system with random missing metrology data,
the joint probability distribution p (W, X, Z, omega) of the robust identification model of the positioning system is as follows:
Figure FDA0003028383120000039
wherein W is known identification data, and Y is known identification datak}k＝1:N,H＝{Hk}k＝1:N,X＝{xk}k＝1:NZ is a hidden variable, Z ═ Zk}k＝1:NΩ is an unknown parameter, { θ, δ, λ, v }, C is a constant term, and C is p (Φ, H), where Φ is defined by ΦkForming a matrix.
3. The method of claim 2, wherein the method for identifying robust parameters of an electromechanical positioning system with random missing metrology data,
updating the parameters to be identified according to a variational Bayesian formula:
lnqi(hi)＝<lnp(W,X,Z,Ω)>j≠i+const， (16)
wherein q isi(hi) In order to introduce a distribution of the variation,<·>i≠jis shown with respect to qj(hj) Obtaining expectation (i ≠ j), const is a constant term;
according to equation (16), the hidden variable and the parameter to be identified are updated as follows:
q(xk) Obeying a gaussian distribution:
Figure FDA0003028383120000041
wherein the content of the first and second substances,
Figure FDA0003028383120000042
where < > represents the expected value of the corresponding variable,
q(zk) Subject to the Gamma distribution,
Figure FDA0003028383120000043
wherein:
Figure FDA0003028383120000044
q (θ) follows a gaussian distribution:
Figure FDA0003028383120000045
wherein the content of the first and second substances,
Figure FDA0003028383120000046
diag (< δ >) represents the expansion of the expected value < δ > into a square matrix whose main diagonal elements are given by < δ >;
q(δi) Obeying a Gamma distribution:
Figure FDA0003028383120000047
wherein
Figure FDA0003028383120000048
tr (-) denotes the trace of the matrix;
q (λ) obeys a Gamma distribution:
Figure FDA0003028383120000051
wherein the content of the first and second substances,
Figure FDA0003028383120000052
q (v) obeys a Gamma distribution:
Figure FDA0003028383120000053
wherein
Figure FDA0003028383120000054
The relative expected values of the above variables are as follows
Figure FDA0003028383120000055
Setting the termination condition of the iterative update of the variational Bayesian formula as follows:
Figure FDA0003028383120000056
wherein
Figure FDA0003028383120000057
Is the variation lower bound of the ith iteration, and epsilon is a preset iteration termination threshold;
the lower bound of variation is defined as:
Figure FDA0003028383120000058
wherein
Figure FDA0003028383120000061
When the iteration is terminated, the estimate is θ*＝θiIn the formula [ theta ]iRepresenting the estimate of theta, for the ith iteration*Representing the final estimate of theta.
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