CN109634240B - Dynamic process monitoring method based on novel dynamic principal component analysis - Google Patents

Dynamic process monitoring method based on novel dynamic principal component analysis Download PDF

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CN109634240B
CN109634240B CN201811577508.2A CN201811577508A CN109634240B CN 109634240 B CN109634240 B CN 109634240B CN 201811577508 A CN201811577508 A CN 201811577508A CN 109634240 B CN109634240 B CN 109634240B
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皇甫皓宁
童楚东
葛英辉
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Ningbo University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/418—Total factory control, i.e. centrally controlling a plurality of machines, e.g. direct or distributed numerical control [DNC], flexible manufacturing systems [FMS], integrated manufacturing systems [IMS], computer integrated manufacturing [CIM]
    • G05B19/41885—Total factory control, i.e. centrally controlling a plurality of machines, e.g. direct or distributed numerical control [DNC], flexible manufacturing systems [FMS], integrated manufacturing systems [IMS], computer integrated manufacturing [CIM] characterised by modeling, simulation of the manufacturing system
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
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Abstract

The invention discloses a dynamic process monitoring method based on novel dynamic principal component analysis, which aims to simultaneously dig out variance characteristics and autocorrelation characteristics in training data and implement dynamic process monitoring on the basis of the variance characteristics and the autocorrelation characteristics. Specifically, the method redefines the objective function of the traditional principal component analysis algorithm, so that the objective function of the novel principal component analysis algorithm related in the invention simultaneously covers the variance characteristic and the autocorrelation characteristic. The method simultaneously considers the variance characteristic of the data and the autocorrelation characteristic on the time sequence during data characteristic mining. Thus. The novel principal component analysis algorithm related in the method is not only a brand-new feature extraction algorithm, but also can mine more comprehensive features. It can be said that the method of the present invention is a more preferred dynamic process monitoring method.

Description

Dynamic process monitoring method based on novel dynamic principal component analysis
Technical Field
The invention relates to a data-driven process monitoring method, in particular to a dynamic process monitoring method based on novel dynamic principal component analysis.
Background
In recent years, in the industrial 'big data' hot tide, modern industrial processes gradually move towards digital management. The method is mainly benefited from the rapid development and wide application of the advanced instrument technology and the computing technology, and the production process object can store mass data off line and measure mass data on line. The data contains information which can reflect the running state of the production process, and the monitoring of the running state of the process by utilizing the sampling data is favored by more scholars. In the last decade, both academic and industrial fields, a great deal of manpower and material resources have been invested in the research of process monitoring methods with fault detection and diagnosis as core tasks. In the field of data-driven process monitoring research, statistical process monitoring is the most studied method, and the Principal Component Analysis (PCA) algorithm is the most popular implementation technique. The core essence of statistical process monitoring is that: and (3) carrying out feature mining on the sampled data under the normal working condition, and establishing a single-classification model by using the features to carry out monitoring. Therefore, feature mining or feature extraction plays an important role in the field of statistical process monitoring.
When a data-driven process monitoring model is established, the characteristics of the sampled data under normal working conditions need to be fully mined. Due to the short sampling time interval, the sampled data inevitably has autocorrelation in time sequence. Thus, the dynamic nature of the data autocorrelation is a problem that must be considered. For the research of the dynamic process monitoring problem, the most common idea is to use an augmentation matrix, confuse the autocorrelation and cross correlation of data, and then use the PCA algorithm to perform feature extraction. The dynamic PCA method is the most classical dynamic PCA method in scientific research documents, and on the basis of the classical dynamic PCA method, a plurality of novel modeling ideas are invented by scholars, so that the dynamic process monitoring performance of the dynamic PCA method is improved. However, the objective function of the PCA algorithm determines that the PCA model can only mine variance information in the training data, and cannot mine the correlation in time series. Of course, both the variance characteristic and the autocorrelation characteristic of normal data are potential characteristics of data, and one of the characteristics cannot be emphasized unilaterally.
Disclosure of Invention
The invention aims to solve the main technical problems that: and simultaneously excavating variance characteristics and autocorrelation characteristics in the training data, and carrying out dynamic process monitoring on the basis of the variance characteristics and the autocorrelation characteristics. Therefore, the invention discloses a novel dynamic principal component analysis algorithm which is applied to dynamic process monitoring. Specifically, the method redefines the objective function of the traditional PCA algorithm, so that the objective function simultaneously covers the variance characteristic and the autocorrelation characteristic.
The technical scheme adopted by the invention for solving the technical problems is as follows: a dynamic process monitoring method based on novel dynamic pivot analysis comprises the following steps:
(1) collecting samples under normal operation condition in production process to form a training data matrix X belonging to Rn×mAnd calculating the mean value mu of each column vector in the matrix X1，μ2，…，μmAnd standard deviation delta1，δ2，…，δmCorresponding component mean vector μ ═ μ1，μ2，…，μm]TAnd the standard deviation vector delta ═ delta1，δ2，…，δm]Wherein n is the number of training samples, m is the number of process measurement variables, R is the set of real numbers, R is the number of training samplesn×mRepresenting a matrix of real numbers in dimensions n x m, the upper index T representing the transpose of the matrix or vector.
(2) According to the formula
Figure BSA0000176381230000021
Normalizing the matrix X to obtain a matrix
Figure BSA0000176381230000022
Wherein U is E.Rn×mIs a matrix consisting of n identical mean vectors μ, i.e. U ═ μ, μ, …, μ]TThe diagonal elements in the diagonal matrix Φ consist of the standard deviation vector δ.
(3) Memory matrix
Figure BSA0000176381230000023
After setting the autocorrelation order to D, according to the formula Xd＝[xd，xd+1，…，xn-D+d-1]TConstruction matrix X1，X2，…，XD+1Wherein the subscript D ═ 1, 2, …, D +1, xiFor the normalized ith data sample, i is 1, 2, …, n.
To be provided with
Figure BSA0000176381230000024
For training data, the objective function of the conventional PCA algorithm is as follows:
Figure BSA0000176381230000025
in the above formula, the load vector p ∈ Rm×1Is the solution of the objective function of the PCA algorithm. As is apparent from equation (1), the PCA algorithm is aimed at maximizing the variance of the feature components when mining the latent features, and does not consider the correlation features in the time series.
On the basis of formula (1), the method defines an objective function of a novel principal component analysis algorithm as follows:
Figure BSA0000176381230000026
in the above formula, the vector β ═ β1，β2，…，βD]T，
Figure BSA0000176381230000027
The Kronecker product is expressed, and the specific calculation mode is as follows:
Figure BSA0000176381230000028
from the comparison between formula (2) and formula (1), it can be seen that the objective function defined by the method of the present invention is added with a term on the basis of the PCA algorithm objective function:
Figure BSA0000176381230000029
this term is primarily reflected in the correlation over time series. Therefore, when the load vector p is solved, the variance characteristic and the autocorrelation characteristic in the training data are considered, and more useful characteristics can be excavated compared with the traditional PCA algorithm.
The target function of the method can be solved by a Lagrange multiplier method, and a Lagrange function J shown as follows is firstly designed by utilizing a Lagrange multiplier lambda and a gamma:
Figure BSA00001763812300000210
next, the partial derivative of the function J with respect to the vector p and the vector β is calculated:
Figure BSA0000176381230000031
Figure BSA0000176381230000032
forcing the partial derivatives in equations (5) and (6) to be equal to zero yields the following equation:
Figure BSA0000176381230000033
Figure BSA0000176381230000034
it can be seen that equation (7) describes a eigenvalue problem and equation (8) describes the relationship between vector p and vector β. The load vector p is a feature vector corresponding to the maximum feature value of the feature value problem in equation (7). Since the vector β is needed for solving the eigenvalue problem in the formula (7) and the vector p is needed for solving the vector β in the formula (8), the vector p and the vector β can be solved through mutual iteration of the formula (7) and the formula (8) until convergence.
(4) After the number of the load vectors is set to be K, K load vectors p are solved by utilizing a novel principal component analysis algorithm1，p2，…，pKThe specific implementation procedure is as follows.
(4.1) initialization vector β ═ 1, 1, …, 1]TI.e. all elements in the vector beta are equal to 1.
(4.2) processing the vector β according to the formula β/| β | | | normalization, where the notation | | β | | | represents the length of the calculation vector β.
(4.3) calculating the eigenvector p corresponding to the maximum eigenvalue in the formula (7)kThis step requires a feature vector pkIs a unit length.
(4.4) according to the formula
Figure BSA0000176381230000035
The vector beta is updated.
(4.5) determine whether the vector β converges? The judgment standard is as follows: until each element in the vector beta is not changed any more, if not, returning to the step (4.2); if yes, a k load vector p is obtainedk。
(4.6) according to the formula
Figure BSA0000176381230000036
Calculating score vectors t respectivelykAnd the regression vectorwk。
(4.7) according to the formula
Figure BSA0000176381230000037
Updating a matrix
Figure BSA0000176381230000038
And according to formula Xd＝[xd，xd+1，…，xn-D+d-1]TConstruction matrix X1，X2，…，XD+1。
(4.8) repeating the steps (4.1) to (4.7) until K load vectors p are obtained1，p2，…，pKK regression vectors w1，w2，…，wKAnd K score vectors t1，t2，…，tK。
(5) According to the formula Θ ═ P (W)TP)-1Computing a projective transformation matrix Θ, wherein the matrix W ═ W1，w2，…，wK]The load matrix P ═ P1，p2，…，pK]。
(6) Calculating a scoring matrix T ═ T1，t2，…，tK]Is given by the covariance matrix Λ ═ TTT/(n-1), and calculates the control limit ψ as followslimAnd Qlim：
Figure BSA0000176381230000039
Figure BSA00001763812300000310
In the above two formulae, FK，n-K，αRepresenting the confidence coefficient as alpha and the degree of freedom as the corresponding values of the F distribution of K and n-K respectively,
Figure BSA0000176381230000041
the degree of freedom is h, the confidence coefficient is alpha, a and tau are respectivelyRespectively, the estimated mean and the estimated variance of the Q statistic.
The steps (1) to (6) are the off-line modeling stage of the method of the present invention, and the mean and standard deviation vectors in the step (1), the projection transformation matrix Θ and the load matrix P in the step (5), and the covariance matrix Λ and the upper control limit in the step (6) need to be retained for the following on-line monitoring process invocation.
(7) Collecting data sample x ∈ R at new sampling momentm×1According to the formula
Figure BSA0000176381230000042
Normalizing x to obtain
Figure BSA0000176381230000043
(8) The specific values of statistics ψ and Q are calculated according to the formula shown below:
Figure BSA0000176381230000044
Figure BSA0000176381230000045
(9) judging whether the condition psi is less than or equal to psilimAnd Q is less than or equal to QlimIs there a If yes, the current sample is collected from a normal working condition, and the step (7) is returned to continue monitoring the sample data at the next moment; if not, the current monitoring sample is collected from the fault working condition.
Compared with the traditional method, the method has the advantages that:
in the process of extracting the potential features, the method simultaneously considers the variance features of the data and the autocorrelation features on the time sequence. Thus. The novel principal component analysis algorithm related in the method can mine more comprehensive characteristics. Although the novel principal component analysis algorithm involved in the method is improved based on the classical PCA algorithm, the novel principal component analysis algorithm is a brand-new feature extraction algorithm.
Drawings
FIG. 1 is a flow chart of an embodiment of the method of the present invention.
FIG. 2 is a flow chart of the implementation of the novel principal component analysis algorithm in the method of the present invention.
Detailed Description
The method of the present invention is described in detail below with reference to the accompanying drawings and specific embodiments.
As shown in fig. 1, the present invention discloses a dynamic process monitoring method based on a novel dynamic principal component analysis, and the specific implementation manner is as follows.
Step (1): collecting samples under normal operation condition in production process to form a training data matrix X belonging to Rn×mAnd calculating the mean value mu of each column vector in the matrix X1，μ2，…，μmAnd standard deviation delta1，δ2，…，δmCorresponding component mean vector μ ═ μ1，μ2，…，μm]TAnd the standard deviation vector delta ═ delta1，δ2，…，δm]。
Step (2): according to the formula
Figure BSA0000176381230000047
Normalizing the matrix X to obtain a matrix
Figure BSA0000176381230000046
Wherein U is E.Rn×mIs a matrix consisting of n identical mean vectors μ, i.e. U ═ μ, μ, …, μ]TThe diagonal elements in the diagonal matrix Φ consist of the standard deviation vector δ.
And (3): memory matrix
Figure BSA0000176381230000051
After setting the autocorrelation order to D, according to the formula Xd＝[xd，xd+1，…，xn-D+d-1]TConstruction matrix X1，X2，…，XD+1Wherein the index D is 1, 2, …, D + 1.
And (4): after the number of the load vectors is set to be K, K load vectors p are solved by utilizing a novel principal component analysis algorithm1，p2，…，pK. As shown in fig. 2, a flowchart of an implementation of the novel principal component analysis algorithm according to the present invention includes steps (4.1) to (4.8).
Step (4.1): initialization vector β ═ 1, 1, …, 1]TI.e. all elements in the vector beta are equal to 1.
Step (4.2): the vector β is processed normalized according to the formula β/| β | |, where the notation | | β | | | represents the length of the calculation vector β.
Step (4.3): calculating the eigenvector p corresponding to the largest eigenvalue in the eigenvalue problem as shown belowkThis step requires a feature vector pkIs a unit length:
Figure BSA0000176381230000052
in the above formula, λ is a feature value, and p is a feature vector.
Step (4.4): according to the formula
Figure BSA0000176381230000053
The vector beta is calculated.
Step (4.5): determine whether vector β converges? If not, returning to the step (4.2); if yes, a k load vector p is obtainedk。
Step (4.6): according to the formula
Figure BSA0000176381230000054
And
Figure BSA0000176381230000055
calculating score vectors t respectivelykAnd the regression vector wk。
Step (4.7): according to the formula
Figure BSA0000176381230000056
Updating a matrix
Figure BSA0000176381230000057
And according to formula Xd＝[xd，xd+1，…，xn-D+d-1]TConstruction matrix X1，X2，…，XD+1。
Step (4.8): repeating the steps (4.1) to (4.7) until K load vectors p are obtained1，p2，…，pKK regression vectors w1，w2，…，wKAnd K score vectors t1，t2，…，tK。
And (5): according to the formula Θ ═ P (W)TP)-1Computing a projective transformation matrix Θ, wherein the matrix W ═ W1，w2，…，wK]The load matrix P ═ P1，p2，…，pK]；
And (6): calculating a scoring matrix T ═ T1，t2，…，tK]Is given by the covariance matrix Λ ═ TTT/(n-1), and calculates the control limit ψ as followslimAnd Qlim：
Figure BSA0000176381230000058
Figure BSA0000176381230000059
In the above two formulae, FK，n-K，αRepresenting the confidence coefficient as alpha and the degree of freedom as the corresponding values of the F distribution of K and n-K respectively,
Figure BSA00001763812300000510
representing the degree of freedom h and the confidence coefficient alpha as the corresponding values of chi-square distribution, and a and tau are the estimated mean value and the estimated variance of the Q statistic respectively;
and (7): collecting data sample x ∈ R at new sampling momentm×1According to the formula
Figure BSA00001763812300000511
Normalizing x to obtain
Figure BSA00001763812300000512
And (8): the specific values of statistics ψ and Q are calculated according to the formula shown below:
Figure BSA0000176381230000061
Figure BSA0000176381230000062
and (9): judging whether the condition psi is less than or equal to psilimAnd Q is less than or equal to QlimIs there a If yes, the current sample is collected from a normal working condition, and the step (7) is returned to continue monitoring the sample data at the next moment; if not, the current monitoring sample is collected from the fault working condition.
The above embodiments are merely illustrative of specific implementations of the present invention and are not intended to limit the present invention. Any modification of the present invention within the spirit of the present invention and the scope of the claims will fall within the scope of the present invention.

Claims (1)

1. A dynamic process monitoring method based on novel dynamic pivot analysis is characterized by comprising the following steps:
step (1): collecting samples under normal operation condition in production process to form a training data matrix X belonging to Rn×mAnd calculating the mean value mu of each column vector in the matrix X1，μ2，…，μmAnd standard deviation delta1，δ2，…，δmCorresponding component mean vector μ ═ μ1，μ2，…，μm]TAnd the standard deviation vector delta ═ delta1，δ2，…，δm]Wherein n is the number of training samples, m is the number of process measurement variables,r is a real number set, Rn×mA real number matrix of n × m dimensions is represented, and the upper label T represents the transpose of a matrix or a vector;
step (2): according to the formula
Figure FSB0000190183780000011
Normalizing the matrix X to obtain a matrix
Figure FSB0000190183780000012
Wherein U is E.Rn×mIs a matrix consisting of n identical mean vectors μ, i.e. U ═ μ, μ, …, μ]TThe elements on the diagonal in the diagonal matrix phi consist of a standard deviation vector delta;
and (3): memory matrix
Figure FSB0000190183780000013
After setting the autocorrelation order to D, according to the formula Xd＝[xd，xd+1，…，xn-D+d-1]TConstruction matrix X1，X2，…，XD+1Wherein the subscript D ═ 1, 2, …, D +1, xiThe normalized ith data sample is i ═ 1, 2, …, n;
and (4): setting the number of load vectors as K, and solving K load vectors p by using a novel principal component analysis algorithm1，p2，…，pKThe specific implementation process is shown as follows;
step (4.1): initialization vector β ═ 1, 1, …, 1]TI.e. all elements in the vector β are equal to 1;
step (4.2): processing the vector beta according to a formula beta/beta normalized, wherein the symbol beta represents the length of the calculation vector beta;
step (4.3): calculating the eigenvector p corresponding to the largest eigenvalue in the eigenvalue problem as shown belowkThis step requires a feature vector pkIs a unit length:
Figure FSB0000190183780000014
in the above formula, λ is a characteristic value, and p is a characteristic vector;
step (4.4): according to the formula
Figure FSB0000190183780000015
Calculating to obtain a vector beta;
step (4.5): judging whether the vector beta is converged; if not, returning to the step (4.2); if yes, a k load vector p is obtainedk；
Step (4.6): according to the formula
Figure FSB0000190183780000016
Calculating score vectors t respectivelykAnd the regression vector wk；
Step (4.7): according to the formula
Figure FSB0000190183780000017
Updating a matrix
Figure FSB0000190183780000018
And according to formula Xd＝[xd，xd+1，…，xn-D+d-1]TConstruction matrix X1，X2，…，XD+1；
Step (4.8): repeating the steps (4.1) to (4.7) until K load vectors p are obtained1，p2，…，pKK regression vectors w1，w2，…，wKAnd K score vectors t1，t2，…，tK；
And (5): according to the formula Θ ═ P (W)TP)-1Computing a projective transformation matrix Θ, wherein the matrix W ═ W1，w2，…，wK]The load matrix P ═ P1，p2，…，pK]；
And (6): calculating a scoring matrix T ═ T1，t2，…，tK]Is given by the covariance matrix Λ ═TTT/(n-1), and calculating the control limit psi according to the formulalimAnd Qlim：
Figure FSB0000190183780000021
Figure FSB0000190183780000022
In the above two formulae, FK，n-K，αRepresenting the confidence coefficient as alpha and the degree of freedom as the corresponding values of the F distribution of K and n-K respectively,
Figure FSB0000190183780000023
representing the degree of freedom h and the confidence coefficient alpha as the corresponding values of chi-square distribution, and a and tau are the estimated mean value and the estimated variance of the Q statistic respectively;
the steps (1) to (6) are an off-line modeling stage, and the mean value and standard deviation vector in the step (1), the projection transformation matrix Θ and the load matrix P in the step (5), and the covariance matrix Λ and the control upper limit in the step (6) need to be reserved for calling the on-line monitoring process shown in the following;
and (7): collecting data sample x ∈ R at new sampling momentm×1According to the formula
Figure FSB0000190183780000024
Normalizing x to obtain
Figure FSB0000190183780000025
And (8): the specific values of statistics ψ and Q are calculated according to the formula shown below:
Figure FSB0000190183780000026
Figure FSB0000190183780000027
and (9): judging whether the condition psi is less than or equal to psilimAnd Q is less than or equal to Qlim(ii) a If yes, the current sample is collected from a normal working condition, and the step (7) is returned to continue monitoring the sample data at the next moment; if not, the current monitoring sample is collected from the fault working condition.
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