CN110942258A - Performance-driven industrial process anomaly monitoring method - Google Patents

Performance-driven industrial process anomaly monitoring method Download PDF

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CN110942258A
CN110942258A CN201911254450.2A CN201911254450A CN110942258A CN 110942258 A CN110942258 A CN 110942258A CN 201911254450 A CN201911254450 A CN 201911254450A CN 110942258 A CN110942258 A CN 110942258A
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CN110942258B (en
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周东华
陈茂银
吴德浩
纪洪泉
高明
朱继峰
闫飞
郑水明
郭恩陶
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Shandong University of Science and Technology
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Abstract

The invention discloses a performance-driven industrial process abnormity monitoring method, and particularly relates to the technical field of industrial process monitoring. The method defines a detection performance index on the basis of fault detectability analysis, and selects components according to the index; the component selection problem is then described as a random optimization problem, and the optimal solution to the problem is given in a statistical sense. The method comprises two parts of off-line training and on-line monitoring. In the off-line training stage, historical data under normal working conditions are utilized to calculate the detection statistics of each sample, and the monitoring control limit is determined under the given significance level; in the on-line monitoring stage, the detection statistic is calculated according to the new sample, and if the detection statistic exceeds the control limit, the process is considered to be abnormal. Compared with a principal component analysis method based on the CPV criterion, the method selects principal components according to the detection performance, weakens sufficient conditions of detectability, and can generally obtain better monitoring performance.

Description

Performance-driven industrial process anomaly monitoring method
Technical Field
The invention belongs to the technical field of industrial process monitoring, and particularly relates to a performance-driven industrial process abnormity monitoring method.
Background
Principal Component Analysis (PCA) is a classic data dimension reduction technique, and is widely applied in many fields such as image processing and signal processing. In recent years, PCA and various improvements thereof have become effective technical means in the field of industrial process monitoring.
One of the key issues in building an industrial process data model using PCA is the selection of an appropriate number of principal elements, which is relevant to the monitoring performance of the built model. Too many principal elements may contain measurement noise and too few principal elements may lose critical information and fail to reflect some changes in the process. For the pivot selection problem, researchers have successively proposed many criteria or methods, such as eigenvalue limits, cross validation, CPV criteria, VRE criteria, and the like. However, the monitoring performance is not considered in these methods, for example, the eigenvalue limit and the CPV criterion consider that the principal element corresponding to the small eigenvalue is measurement noise, and both of them select the principal element with the starting point of retaining the maximum information amount; the VRE criterion takes into account the reconstruction performance and it is desirable that the selected pivot have the smallest reconstruction error. Thus, a pivot selected in the manner described above may be insensitive to anomalies.
Currently, there are only a few methods to consider monitoring performance when selecting a pivot. These methods all have several common drawbacks. Firstly, the method depends heavily on the fault direction, so that the method is only limited to detecting the fault of a sensor, and the fault direction needs to be estimated by using abnormal data for complex process faults, but in practice, the fault data is usually difficult to obtain and the fault direction is difficult to estimate accurately; secondly, for unknown faults without prior information, a parallel monitoring scheme for simultaneously monitoring a plurality of models is provided by a scholars, but huge calculation complexity is caused, and real-time monitoring on the process is not facilitated; thirdly, the methods determine the number of the pivot elements off line and keep the number of the pivot elements unchanged in the on-line monitoring stage, so that the monitoring performance of unknown faults is not ideal.
Disclosure of Invention
The invention aims to provide a performance-driven industrial process anomaly monitoring method which dynamically selects pivot elements according to detection performance indexes, does not need abnormal data and does not need to estimate the direction and the amplitude of a fault.
The invention specifically adopts the following technical scheme:
a performance-driven industrial process anomaly monitoring method comprises an offline training stage and an online monitoring stage;
the off-line training phase comprises the following steps:
1.1, collecting historical data under normal working conditions to obtain a training data set
Figure BDA0002309844270000011
Where N is the number of samples and m is the number of measurement variables;
1.2 calculating the sample mean μ by equation (1)xAnd calculating a sample covariance matrix sigma using equation (2)x：
Figure BDA0002309844270000021
Figure BDA0002309844270000022
Wherein the content of the first and second substances,
Figure BDA0002309844270000023
1.3 covariance matrix on samples ∑xPerforming eigenvalue decomposition to obtain formula (3):
Σx＝QΛQT(3)
where Q is an orthogonal matrix, Λ ═ diag (λ)[1],λ[2],…,λ[m]) Is a diagonal matrix and has a[1]≥λ[2]≥…≥λ[m]；
1.4, for the kth sample in the training data set X, k is more than or equal to 1 and less than or equal to N,
Figure BDA0002309844270000024
the component vector y is calculated from equation (4)k：
yk＝QT(xk-μx) (4)；
1.5, calculating a component vector y according to the definition of the detection performance index PkCorresponding detection performance index
Figure BDA0002309844270000025
1.6 based on the selection matrix W ∈ {0,1}m×dObtaining the component vector after selection
Figure BDA0002309844270000026
And calculating corresponding detection performance index
Figure BDA0002309844270000027
1.7, traversing the dimension d from 1 to m in sequence to obtain the index of the detection performance
Figure BDA0002309844270000028
Maximum optimal selection matrix
Figure BDA0002309844270000029
1.8, calculating the detection statistic D of the kth sample according to the selected optimal component subsetkAs shown in formula (5):
Figure BDA00023098442700000210
wherein the content of the first and second substances,
Figure BDA00023098442700000211
representing a degree of freedom of
Figure BDA00023098442700000212
α quantites of chi-square distribution of (1);
1.9 given significance level α, detection control limit η was determined empiricallyα；
The on-line monitoring phase comprises the following steps:
2.1, for a real-time sample x, its component vector y is calculated according to equation (6):
y＝QT(x-μx) (6)；
2.2, according to the definition of the detection performance index P, calculating the detection performance index P corresponding to the component vector yy；
2.3 based on the selection matrix W ∈ {0,1}m×dObtaining the component vector after selection
Figure BDA0002309844270000031
And calculating corresponding detection performance index
Figure BDA0002309844270000032
2.4, traversing the dimension d from 1 to m in sequence to obtain the index of the detection performance
Figure BDA0002309844270000033
Maximum optimal selection matrix
Figure BDA0002309844270000034
2.5, calculating a detection statistic D of the real-time sample x according to the selected optimal component subset, wherein the formula (7) is as follows:
Figure BDA0002309844270000035
wherein the content of the first and second substances,
Figure BDA0002309844270000036
α quantites representing a chi-square distribution with d degrees of freedom;
2.6, detecting statistic D and control limit ηαBy comparison, if D is greater than ηαIf so, the process is considered to be abnormal, otherwise, the process is in a normal state.
Preferably, steps 1.5 and 2.2 are in particular:
mean value μ of component vectorsySum covariance matrix ΣyAs shown in formulas (8) and (9), respectively:
Figure BDA0002309844270000037
Figure BDA0002309844270000038
based on an additive fault model shown in equation (10):
x＝x*+Ξif (10)
where x is the failure sample, x*Is a corresponding normal sample, xiiIs a failure
Figure BDA0002309844270000039
The direction matrix of (1), wherein, | f | | | represents the amplitude of the fault;
the mahalanobis distance corresponding to the component vector y is as shown in equation (11):
Figure BDA00023098442700000310
wherein, y*＝QT(x*-μx)；
From the trigonometric inequality of the vector, equation (12) is obtained:
Figure BDA00023098442700000311
in view of
Figure BDA00023098442700000312
To ensure failure
Figure BDA00023098442700000313
Is sufficiently detected that
Figure BDA00023098442700000314
Then, a sufficient condition for the failure to be detectable is as shown in equation (13):
||Λ-1/2QTΞif||＞2χα(m) (13)；
the detection performance index corresponding to the component vector y is defined as formula (14):
Figure BDA0002309844270000041
preferably, steps 1.6 and 2.3 are in particular:
the definition selection matrix W is as shown in equation (15):
Figure BDA0002309844270000042
wherein d is not more than m, and
Figure BDA0002309844270000043
is a except for
Figure BDA00023098442700000413
A column vector having 1 as one element and 0 as the remaining elements;
for the component vector y, the subset of components y after selectionsGiven by equation (16):
Figure BDA0002309844270000044
having a mean value of
Figure BDA0002309844270000045
The covariance is as shown in equation (17):
Figure BDA0002309844270000046
subset of components ysThe corresponding mahalanobis distance is shown as equation (18):
Figure BDA0002309844270000047
similarly, a sufficient condition for failure to be detectable is obtained, i.e., equation (19):
||(WTΛW)-1/2WTQTΞif||＞2χα(d) (19)；
thus, the subset of components ysCorresponding performance index
Figure BDA0002309844270000048
As shown in equation (20):
Figure BDA0002309844270000049
preferably, steps 1.7 and 2.4 are in particular:
let the selection matrix defined in equation (15) be
Figure BDA00023098442700000410
Then the performance index is detected
Figure BDA00023098442700000411
The maximum optimal selection matrix W is given by equation (21):
Figure BDA00023098442700000412
given d (1. ltoreq. d. ltoreq.m), formula (21) is converted to formula (22):
Figure BDA0002309844270000051
due to xiiThe value of f is unknown, and equation (22) is really a random optimization problem;
according to the fault model shown in the formula (10), XIif＝x-x*Hence xiif can be regarded as a random variable subject to a Gaussian distribution, i.e. having
Figure BDA0002309844270000052
It is expressed as a form shown in formula (23):
Ξif＝x-μx+e (23)
wherein the content of the first and second substances,
Figure BDA0002309844270000053
representing measurement noise;
substituting formula (23) into F (W) to obtain formula (24):
F(W)＝(y+g)TW(WTΛW)-1WT(y+g) (24)
wherein the content of the first and second substances,
Figure BDA0002309844270000054
is a gaussian random variable;
in consideration of (W)TΛW)-1＝WTΛ-1W, to yield formula (25) in a statistical sense:
Figure BDA0002309844270000055
due to WWTIs a diagonal matrix with diagonal elements all being 0 or 1, so the two terms to the right of the equal sign of equation (25) are respectively formulated into the forms shown in equations (26) and (27):
Figure BDA0002309844270000056
Figure BDA0002309844270000057
wherein the content of the first and second substances,
Figure BDA0002309844270000058
formula (28) can thus be further obtained:
Figure BDA0002309844270000059
given d, in order to maximize
Figure BDA00023098442700000510
Should be reduced toiD values of medium maximum are added, assuming σ[1]≥σ[2]≥…≥σ[m]Optimal selection matrix
Figure BDA00023098442700000511
Given by equation (29):
Figure BDA00023098442700000512
sequentially traversing the dimension d from 1 to m to obtain a globally optimal selection matrix W*As shown in equation (30):
Figure BDA0002309844270000061
the invention has the following beneficial effects:
the method dynamically selects the pivot element according to the detection performance index, and the selected pivot element is sensitive to abnormality, so that better monitoring performance can be obtained; the method does not need abnormal data, does not need to estimate the direction and the amplitude of the fault, and can obtain better monitoring performance for unknown abnormal without prior information; and the online calculation complexity is low, thereby being beneficial to monitoring the industrial process in real time.
Drawings
FIG. 1 is a flow chart of the present invention for off-line training and on-line monitoring;
FIG. 2 is a graph showing the results of Mahalanobis distance based monitoring;
FIG. 3 is a graphical illustration of the results of monitoring of a PCA-based Q statistic;
FIG. 4 is PCA-based T2A schematic diagram of the monitoring result of the statistic;
FIG. 5 is a graph showing the monitoring result according to the method of the present invention;
FIG. 6 is a schematic diagram illustrating the number of pivot elements selected based on the method of the present invention
Detailed Description
With reference to fig. 1, a performance-driven industrial process anomaly monitoring method includes an offline training phase and an online monitoring phase;
the off-line training phase comprises the following steps:
1.1, collecting historical data under normal working conditions to obtain a training data set
Figure BDA0002309844270000062
Where N is the number of samples and m is the number of measurement variables;
1.2 calculating the sample mean μ by equation (1)xAnd calculating a sample covariance matrix sigma using equation (2)x：
Figure BDA0002309844270000063
Figure BDA0002309844270000064
Wherein the content of the first and second substances,
Figure BDA0002309844270000065
1.3 covariance matrix on samples ∑xPerforming eigenvalue decomposition to obtain formula (3):
Σx＝QΛQT(3)
where Q is an orthogonal matrix, Λ ═ diag (λ)[1],λ[2],…,λ[m]) Is a diagonal matrix and has a[1]≥λ[2]≥…≥λ[m]；
1.4, for the kth sample in the training data set X, k is more than or equal to 1 and less than or equal to N,
Figure BDA0002309844270000066
the component vector y is calculated from equation (4)k：
yk＝QT(xk-μx) (4)；
1.5, calculating a component vector y according to the definition of the detection performance index PkCorresponding detection performance index
Figure BDA0002309844270000071
1.6 based on the selection matrix W ∈ {0,1}m×dObtaining the component vector after selection
Figure BDA0002309844270000072
And calculating corresponding detection performance index
Figure BDA0002309844270000073
1.7, traversing the dimension d from 1 to m in sequence to obtain the index of the detection performance
Figure BDA0002309844270000074
Maximum optimal selection matrix
Figure BDA0002309844270000075
1.8, calculating the detection statistic D of the kth sample according to the selected optimal component subsetkAs shown in formula (5):
Figure BDA0002309844270000076
wherein the content of the first and second substances,
Figure BDA0002309844270000077
representing a degree of freedom of
Figure BDA0002309844270000078
α quantites of chi-square distribution of (1);
1.9 given significance level α, detection control limit η was determined empiricallyα。
The on-line monitoring phase comprises the following steps:
2.1, for a real-time sample x, its component vector y is calculated according to equation (6):
y＝QT(x-μx) (6)；
2.2, according to the definition of the detection performance index P, calculating the detection performance index P corresponding to the component vector yy；
2.3 based on the selection matrix W ∈ {0,1}m×dObtaining the component vector after selection
Figure BDA0002309844270000079
And calculating corresponding detection performance index
Figure BDA00023098442700000710
2.4, traversing the dimension d from 1 to m in sequence to obtain the index of the detection performance
Figure BDA00023098442700000711
Maximum optimal selection matrix
Figure BDA00023098442700000712
2.5, calculating a detection statistic D of the real-time sample x according to the selected optimal component subset, wherein the formula (7) is as follows:
Figure BDA00023098442700000713
wherein the content of the first and second substances,
Figure BDA00023098442700000714
represents a degree of freedom of d*α quantites of chi-square distribution of (1);
2.6、the detection statistic D is compared with the control limit ηαBy comparison, if D is greater than ηαIf so, the process is considered to be abnormal, otherwise, the process is in a normal state.
Steps 1.5 and 2.2 specifically include the following processes:
mean value μ of component vectorsySum covariance matrix ΣyAs shown in formulas (8) and (9), respectively:
Figure BDA0002309844270000081
Figure BDA0002309844270000082
based on an additive fault model shown in equation (10):
x＝x*+Ξif (10)
where x is the failure sample, x*Is a corresponding normal sample, xiiIs a failure
Figure BDA0002309844270000083
The direction matrix of (1), wherein, | f | | | represents the amplitude of the fault;
the mahalanobis distance corresponding to the component vector y is as shown in equation (11):
Figure BDA0002309844270000084
wherein, y*＝QT(x*-μx)；
From the trigonometric inequality of the vector, equation (12) is obtained:
Figure BDA0002309844270000085
in view of
Figure BDA0002309844270000086
To ensure failure
Figure BDA0002309844270000087
Is sufficiently detected that
Figure BDA0002309844270000088
Then, a sufficient condition for the failure to be detectable is as shown in equation (13):
||Λ-1/2QTΞif||＞2χα(m) (13)；
the detection performance index corresponding to the component vector y is defined as formula (14):
Figure BDA0002309844270000089
steps 1.6 and 2.3 specifically include the following processes:
the definition selection matrix W is as shown in equation (15):
Figure BDA00023098442700000810
wherein d is not more than m, and
Figure BDA00023098442700000811
is a except for
Figure BDA00023098442700000812
A column vector having 1 as one element and 0 as the remaining elements;
for the component vector y, the subset of components y after selectionsGiven by equation (16):
Figure BDA0002309844270000091
having a mean value of
Figure BDA0002309844270000092
The covariance is as shown in equation (17):
Figure BDA0002309844270000093
subset of components ysThe corresponding mahalanobis distance is shown as equation (18):
Figure BDA0002309844270000094
similarly, a sufficient condition for failure to be detectable is obtained, i.e., equation (19):
||(WTΛW)-1/2WTQTΞif||＞2χα(d) (19)；
thus, the subset of components ysCorresponding performance index
Figure BDA0002309844270000095
As shown in equation (20):
Figure BDA0002309844270000096
the specific processes of the steps 1.7 and 2.4 are as follows:
let the selection matrix defined in equation (15) be
Figure BDA0002309844270000097
Then the performance index is detected
Figure BDA0002309844270000098
The maximum optimal selection matrix W is given by equation (21):
Figure BDA0002309844270000099
given d (1. ltoreq. d. ltoreq.m), formula (21) is converted to formula (22):
Figure BDA00023098442700000910
due to xiiThe value of f is unknown, and equation (22) is really a random optimization problem;
according to the fault model shown in the formula (10), XIif＝x-x*Hence xiif can be regarded as a random variable subject to a Gaussian distribution, i.e. having
Figure BDA00023098442700000911
It is expressed as a form shown in formula (23):
Ξif＝x-μx+e (23)
wherein the content of the first and second substances,
Figure BDA00023098442700000912
representing measurement noise;
substituting formula (23) into F (W) to obtain formula (24):
F(W)＝(y+g)TW(WTΛW)-1WT(y+g) (24)
wherein the content of the first and second substances,
Figure BDA0002309844270000101
is a gaussian random variable;
in consideration of (W)TΛW)-1＝WTΛ-1W, to yield formula (25) in a statistical sense:
Figure BDA0002309844270000102
due to WWTIs a diagonal matrix with diagonal elements all being 0 or 1, so the two terms to the right of the equal sign of equation (25) are respectively formulated into the forms shown in equations (26) and (27):
Figure BDA0002309844270000103
Figure BDA0002309844270000104
wherein the content of the first and second substances,
Figure BDA0002309844270000105
formula (28) can thus be further obtained:
Figure BDA0002309844270000106
given d, in order to maximize
Figure BDA0002309844270000107
Should be reduced toiD values of medium maximum are added, assuming σ[1]≥σ[2]≥…≥σ[m]Optimal selection matrix
Figure BDA0002309844270000108
Given by equation (29):
Figure BDA0002309844270000109
and traversing the dimension d from 1 to m in sequence to obtain a globally optimal selection matrix W as shown in a formula (30):
Figure BDA00023098442700001010
the monitoring method is validated below on the basis of a Continuous Stirred Tank Heater (CSTH), a process that provides a standard model library that is widely studied in the field of industrial process monitoring.
In this process, hot and cold water are mixed and heated by steam, where temperature and level are the controlled variables with nominal values. The process model has three PI controllers for controlling temperature, liquid level and cold water flow separately. Without loss of generality, the process is operated in the laboratory in the first operating condition. The measuring sample consists of the measured values of three sensors of liquid level, cold water flow and temperature and the output values of three controllers, namely x ═ L, F, T and CL,CF,CT]T。
In the off-line training phase, 2000 samples under normal conditions are collected for off-line modeling, the significance level is selected to be α ═ 0.01, and the control limit is determined empirically, 1500 samples are generated in the on-line monitoring phase, wherein the first 500 samples are normal, an abnormal condition is introduced starting from the 501 th sample and continuing until the end, the abnormality is a constant deviation fault with an amplitude of +0.02 applied to the liquid level sensor.
In order to show the advantages of the method of the present invention more clearly, mahalanobis distance and PCA methods are used for comparison. FIG. 2 shows a graph of Mahalanobis distance based monitoring results, and FIGS. 3 and 4 show the PCA-based Q statistic and T, respectively2The monitoring result of the statistic is shown schematically, and fig. 5 shows the monitoring result based on the method of the invention. Comparing these four figures, one can conclude that: PCA has poor monitoring results in this scheme, either Q statistic or T statistic2Statistics, anomalies can hardly be detected because the amplitude of the fault is small and easily masked by noise; the Mahalanobis distance can partially monitor the abnormity, but the report missing rate is still more than 20%; the method provided by the invention has an optimal monitoring result, and can monitor the abnormality at a detection rate of more than 90%. In addition, fig. 6 shows a schematic diagram of the number of pivot elements selected based on the method of the present invention. As can be seen from fig. 6, there is a large variation in the number of selected pivot elements before an anomaly occurs because the process is operating normally and measurement noise is present; but after the exception occurs, the method can quickly locate the pivot element subset which is most sensitive to the exception, so that better monitoring performance can be obtained.
It is to be understood that the above description is not intended to limit the present invention, and the present invention is not limited to the above examples, and those skilled in the art may make modifications, alterations, additions or substitutions within the spirit and scope of the present invention.

Claims (4)

1. A performance-driven industrial process anomaly monitoring method is characterized by comprising an off-line training stage and an on-line monitoring stage;
the off-line training phase comprises the following steps:
1.1, collecting historical data under normal working conditions to obtain a training data set
Figure FDA0002309844260000011
Where N is the number of samples and m is the number of measurement variables;
1.2 calculating the sample mean μ by equation (1)xAnd calculating a sample covariance matrix sigma using equation (2)x：
Figure FDA0002309844260000012
Figure FDA0002309844260000013
Wherein the content of the first and second substances,
Figure FDA0002309844260000014
1.3 covariance matrix on samples ∑xPerforming eigenvalue decomposition to obtain formula (3):
Σx＝QΛQT(3)
where Q is an orthogonal matrix, Λ ═ diag (λ)[1],λ[2],…,λ[m]) Is a diagonal matrix and has a[1]≥λ[2]≥…≥λ[m]；
1.4, for the kth sample in the training data set X, k is more than or equal to 1 and less than or equal to N,
Figure FDA0002309844260000015
the component vector y is calculated from equation (4)k：
yk＝QT(xk-μx) (4)；
1.5, calculating a component vector y according to the definition of the detection performance index PkCorresponding detection performance index
Figure FDA0002309844260000016
1.6 based on the selection matrix W ∈ {0,1}m×dObtaining the component vector after selection
Figure FDA0002309844260000017
And calculating corresponding detection performance index
Figure FDA0002309844260000018
1.7, traversing the dimension d from 1 to m in sequence to obtain the index of the detection performance
Figure FDA0002309844260000019
Maximum optimal selection matrix
Figure FDA00023098442600000110
1.8, calculating the detection statistic D of the kth sample according to the selected optimal component subsetkAs shown in formula (5):
Figure FDA00023098442600000111
wherein the content of the first and second substances,
Figure FDA00023098442600000112
representing a degree of freedom of
Figure FDA00023098442600000113
α quantites of chi-square distribution of (1);
1.9 given significance level α, detection control limit η was determined empiricallyα；
The on-line monitoring phase comprises the following steps:
2.1, for a real-time sample x, its component vector y is calculated according to equation (6):
y＝QT(x-μx) (6)；
2.2, according to the definition of the detection performance index P, calculating the detection performance index P corresponding to the component vector yy；
2.3 based on the selection matrix W ∈ {0,1}m×dObtaining the component directions after selectionMeasurement of
Figure FDA0002309844260000021
And calculating corresponding detection performance index
Figure FDA0002309844260000022
2.4, traversing the dimension d from 1 to m in sequence to obtain the index of the detection performance
Figure FDA0002309844260000023
Maximum optimal selection matrix
Figure FDA0002309844260000024
2.5, calculating a detection statistic D of the real-time sample x according to the selected optimal component subset, wherein the formula (7) is as follows:
Figure FDA0002309844260000025
wherein the content of the first and second substances,
Figure FDA0002309844260000026
represents a degree of freedom of d*α quantites of chi-square distribution of (1);
2.6, detecting statistic D and control limit ηαBy comparison, if D is greater than ηαIf so, the process is considered to be abnormal, otherwise, the process is in a normal state.
2. A performance driven industrial process anomaly monitoring method according to claim 1, characterized in that 1.5 and 2.2 are in particular:
mean value μ of component vectorsySum covariance matrix ΣyAs shown in formulas (8) and (9), respectively:
Figure FDA0002309844260000027
Figure FDA0002309844260000028
based on an additive fault model shown in equation (10):
x＝x*+Ξif (10)
where x is the failure sample, x*Is a corresponding normal sample, xiiIs a failure
Figure FDA0002309844260000029
The direction matrix of (1), wherein, | f | | | represents the amplitude of the fault;
the mahalanobis distance corresponding to the component vector y is as shown in equation (11):
Figure FDA0002309844260000031
wherein, y*＝QT(x*-μx)；
From the trigonometric inequality of the vector, equation (12) is obtained:
Figure FDA0002309844260000032
in view of
Figure FDA0002309844260000033
To ensure failure
Figure FDA0002309844260000034
Is sufficiently detected that
Figure FDA0002309844260000035
Then, a sufficient condition for the failure to be detectable is as shown in equation (13):
||Λ-1/2QTΞif||＞2χα(m) (13)；
the detection performance index corresponding to the component vector y is defined as formula (14):
Figure FDA0002309844260000036
3. a performance driven industrial process anomaly monitoring method according to claim 1, characterized in that 1.6 and 2.3 are in particular:
the definition selection matrix W is as shown in equation (15):
Figure FDA0002309844260000037
wherein d is not more than m, and
Figure FDA0002309844260000038
is a except for
Figure FDA0002309844260000039
A column vector having 1 as one element and 0 as the remaining elements;
for the component vector y, the subset of components y after selectionsGiven by equation (16):
Figure FDA00023098442600000310
having a mean value of
Figure FDA00023098442600000311
The covariance is as shown in equation (17):
Figure FDA00023098442600000312
subset of components ysThe corresponding mahalanobis distance is shown as equation (18):
Figure FDA00023098442600000313
similarly, a sufficient condition for failure to be detectable is obtained, i.e., equation (19):
||(WTΛW)-1/2WTQTΞif||＞2χα(d) (19)；
thus, the subset of components ysCorresponding performance index
Figure FDA0002309844260000041
As shown in equation (20):
Figure FDA0002309844260000042
4. a performance driven industrial process anomaly monitoring method according to claim 3, characterized in that 1.7 and 2.4 are in particular:
let the selection matrix defined in equation (15) be
Figure FDA0002309844260000043
Then the performance index is detected
Figure FDA0002309844260000044
Maximum optimal selection matrix W*Given by equation (21):
Figure FDA0002309844260000045
given d (1. ltoreq. d. ltoreq.m), formula (21) is converted to formula (22):
Figure FDA0002309844260000046
due to xiiThe value of f is unknown, and equation (22) is really a random optimization problem;
according to the fault model shown in the formula (10), XIif＝x-x*Hence xiif can be viewed as a Gaussian-compliant distributionRandom variables of, i.e. having
Figure FDA0002309844260000047
It is expressed as a form shown in formula (23):
Ξif＝x-μx+e (23)
wherein the content of the first and second substances,
Figure FDA0002309844260000048
representing measurement noise;
substituting formula (23) into F (W) to obtain formula (24):
F(W)＝(y+g)TW(WTΛW)-1WT(y+g) (24)
wherein the content of the first and second substances,
Figure FDA0002309844260000049
is a gaussian random variable;
in consideration of (W)TΛW)-1＝WTΛ-1W, to yield formula (25) in a statistical sense:
Figure FDA00023098442600000410
due to WWTIs a diagonal matrix with diagonal elements all being 0 or 1, so the two terms to the right of the equal sign of equation (25) are respectively formulated into the forms shown in equations (26) and (27):
Figure FDA00023098442600000411
Figure FDA0002309844260000051
wherein the content of the first and second substances,
Figure FDA0002309844260000052
formula (28) can thus be further obtained:
Figure FDA0002309844260000053
given d, in order to maximize
Figure FDA0002309844260000054
Should be reduced toiD values of medium maximum are added, assuming σ[1]≥σ[2]≥…≥σ[m]Optimal selection matrix
Figure FDA0002309844260000055
Given by equation (29):
Figure FDA0002309844260000056
sequentially traversing the dimension d from 1 to m to obtain a globally optimal selection matrix W*As shown in equation (30):
Figure FDA0002309844260000057
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