CN109143995B - Quality-related slow characteristic decomposition closed-loop system fine operation state monitoring method - Google Patents

Quality-related slow characteristic decomposition closed-loop system fine operation state monitoring method Download PDF

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CN109143995B
CN109143995B CN201810771664.6A CN201810771664A CN109143995B CN 109143995 B CN109143995 B CN 109143995B CN 201810771664 A CN201810771664 A CN 201810771664A CN 109143995 B CN109143995 B CN 109143995B
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赵春晖
郑嘉乐
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Zhejiang University ZJU
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Abstract

The invention discloses a closed loop system fine operation state monitoring method based on quality-related slow characteristic full decomposition. The method comprises the steps of extracting static correlation of a process variable and a quality variable, decomposing a closed-loop large-scale process variable into two spaces which are related to the quality and are unrelated to the quality, extracting static and dynamic information of the process in each space to model the space, respectively establishing static indexes and dynamic indexes in the two spaces, and carrying out cooperative fault detection on the two spaces. The method can monitor the dynamic abnormal conditions and the working condition changes of the closed-loop system, effectively identify the real faults in the closed-loop fault system, further realize the related monitoring of the product quality of the closed-loop system by performing the quality related decomposition on the variable space, and judge the specific space where the faults and the abnormalities occur and whether the product quality is influenced.

Description

Quality-related slow characteristic decomposition closed-loop system fine operation state monitoring method
Technical Field
The invention belongs to the field of industrial system process monitoring, and particularly relates to a method for monitoring a fine running state related to the product quality of a closed loop system.
Background
With the development of scientific technology, the scale and complexity of modern industrial systems are increasing day by day. Once an abnormality occurs in a complex large system, significant property loss and casualties can be brought about. The device abnormity is detected in advance by monitoring the daily running state of the system device, so that the device is maintained timely, the maintenance period is prolonged, the maintenance cost is greatly reduced, the maintenance is more targeted, the service life of the device is prolonged, and serious safety accidents are avoided. Therefore, in order to ensure the safety and reliability of the system in operation, find and process abnormal conditions in the system operation in time, reduce potential safety hazards in production, and improve the service cycle of equipment, it is necessary to adopt effective means to perform real-time monitoring and fault detection on the system. Meanwhile, with the development of the sensing technology, it becomes easier to obtain data in an industrial field, a large amount of process information is contained in the process data, and state monitoring and fault monitoring based on the data gradually become a research hotspot.
In the past decades, process monitoring and fault detection technologies have been extensively studied and developed, and a large number of research results have been published, and the predecessors have made corresponding studies on data-based fault detection and fault diagnosis. Multivariate statistical analysis methods such as Principal Component Analysis (PCA), Partial Least Squares (PLS), and Fisher Discriminant Analysis (FDA) have been widely used in the field of data-based process monitoring. However, most of the existing process monitoring and fault detection methods are designed for open-loop systems, and the influence of a closed-loop control law is not considered. However, in a real industrial system, in order to meet the requirements of stability, rapidity and accuracy and achieve the actual production target and ensure that the product quality is maintained at a reasonable level, closed-loop feedback control needs to be applied to the system, and a large number of closed-loop control laws, such as PID control, optimal control and the like, are widely applied to the actual production system. The introduction of the closed-loop control law enables the relation between the input and output relations and system variables to be changed, the closed-loop control law enables the system to be more robust to external interference, and the influence caused by the fact that a fault is in an early stage or when the amplitude is small is submerged in the system and cannot be timely diagnosed through a residual signal, so that false alarm and false alarm are caused. These limit the practical application of previously developed diagnostic methods, and therefore it is necessary to develop system condition monitoring and fault detection methods on a closed-loop basis to address practical production safety issues. At present, the research results of closed-loop system monitoring and fault detection based on data are very limited, and the closed-loop system monitoring and fault detection based on data is still in a preliminary exploration stage and needs further deep research.
The invention provides an algorithm for monitoring the quality-related precise running state of a closed-loop system by combining slow characteristic analysis and typical correlation analysis aiming at the process monitoring and fault detection of the closed-loop system and from the perspective of the product quality of an industrial system. The method can realize the full decomposition of the process data space by extracting the correlation between the process data and the product quality variable, divide the process data space into two spaces which are closely related to the quality and are unrelated to the quality, and establish a model to monitor the dynamic and static information of the two spaces.
Disclosure of Invention
The invention aims to provide a closed loop system fine operation state monitoring method based on quality-related slow characteristic full decomposition aiming at industrial closed loop system process fault detection and paying attention to process product quality.
The purpose of the invention is realized by the following technical scheme: a closed-loop system fine operation state monitoring method based on quality-related slow feature full decomposition comprises the following steps:
(1) selecting normal process measurement data and concerned product quality data of an industrial closed loop system, wherein the operation process of the industrial closed loop system comprises m measurable process variables, and sampling at the time t can obtain a vector x (t) ═ x (x) of 1 × m1(t),x2(t),…,xm(t)), after N times of sampling, obtaining a data matrix X of process measurement variables in the normal processM＝(x(t),x(t+1),…,x(t+N))TSelecting k concerned product quality variables according to the industrial process, and obtaining a data matrix Y of the quality variables after N times of samplingk(∈N*k)。
(2) Data normalization: respectively to process data matrix XMAnd quality variable data matrix YkThe mean of the column is subtracted from the column and divided by the standard deviation of the column for normalization.
(3) And (3) carrying out slow feature analysis modeling on the process data matrix X after standardization in the step (2) to obtain initial slow features.
Considering here linear slow feature analysis, each slow feature can be considered as a linear combination of process variables, so from the process data matrix X to the slow feature s (t) ═ s1(t),s2(t),…,sm(t)]Is represented as:
S(t)＝W×X (24)
wherein W ═ W1,w2,…,wm]TIs a coefficient matrix to be optimized by SFA, m is the number of slow features and is corresponding to measurable process variablesThe same is true. The problem of solving the slow feature is to make the obtained slow feature change rate Δ(s)i) At a minimum, solving the slow eigenvalue problem translates to solving the following generalized eigenvalue problem:
AW＝BWΩ (25)
where A represents a first order difference matrix of an input process data matrix X
Figure GDA0002570881790000031
Is expressed as a covariance matrix of
Figure GDA0002570881790000032
B represents the covariance matrix of the X matrix<XXT>tCorresponding, W ═ W1,w2,…,wm]TThe number of variables of the m process input data is the feature matrix formed by the feature vectors; and omega is a diagonal matrix formed by corresponding generalized eigenvalues. The slow features are solved by using two-step Singular Value Decomposition (SVD), and the specific process is as follows:
(3.1) the matrix X is first whitened, i.e. the singular value decomposition of the B matrix. The covariance matrix of matrix X may be expressed as<XXT>tAnd performing Singular Value Decomposition (SVD) on the obtained product to obtain:
<XXT>t＝UΛUT(26)
where U is the covariance matrix<XXT>Λ is a diagonal matrix, and the element on each diagonal is an eigenvalue, the whitened data matrix can be expressed as:
Z＝Λ-1/2UTX＝QX (27)
the covariance of the whitened data matrix Z is satisfied<ZZT>t＝Q<XXT>tQT。
(3.2) solving the linear SFA problem is equivalent to finding a matrix P ═ WQ-1Making S ═ P × Z, and making S satisfy<SST>t＝P<ZZT>tPT＝I。
The matrix P may be formed byDifference matrix of Z pairs
Figure GDA0002570881790000036
Covariance matrix of
Figure GDA0002570881790000033
Singular value decomposition is carried out to obtain:
Figure GDA0002570881790000034
from this, the initial slow eigen coefficient matrix can be calculated as:
W＝PΛ-1/2UT(29)
the initial slow characteristic is:
S＝WX＝PΛ-1/2UTX (30)
(4) using the initial slow feature S and the quality variable Y obtained in equation (30)kPerforming a typical correlation analysis (CCA) to obtain a typical variable, wherein the specific process is as follows:
for the initial slow feature S and the quality variable YkDefining two variables of u and v, wherein u and v are the start-slow characteristic S and the quality variable YkThe linear combination of (a) is as follows:
Figure GDA0002570881790000035
therein Ψ1＝[a1,a2,…,am]T，Ψ2＝[b1,b2,…,bk]T. Solving the typical variable u, v is to solve the coefficient matrix Ψ1，Ψ2The Pearson coefficient between u, v is maximized. The following optimization problem solution is converted:
Figure GDA0002570881790000041
Subjectto：
Figure GDA0002570881790000042
where corr (u, v) represents the Pearson coefficient between u, v, cov (u, v) is the covariance of u, v, and Var (u) and Var (v) are the variances of u, v, respectively. This problem is solved by constructing Lagrangian equations, solving two typical variables u, v corresponding to the maximum Pearson coefficient.
Similarly, on the basis, the typical variable u corresponding to the second large Pearson coefficient is continuously solved and searched2And v2：
Figure GDA0002570881790000043
In total, n pairs of typical variables can be obtained, where n is min (m, k), and the corresponding Pearson coefficients decrease in sequence, i.e., the correlation between the typical variables is smaller and smaller. And selecting the first p typical correlation variable pairs corresponding to the larger Pearson coefficients according to the obtained Pearson coefficients, and discarding to leave (n-p) typical correlation variable pairs corresponding to the smaller Pearson coefficients. Using the first p pairs of canonical correlation variables, using the derived canonical variables u for the linear combination of the slow features Si(i ═ 1,2, …, p), combined into a new matrix Uy＝[u1,u2,…,up]Wherein u isi(i ═ 1,2, …, p) the Pearson coefficients for the decrease in order, and uiAre orthogonal to each other.
(5) Combining the matrix U obtained in (4)yAnd (3) reconstructing the process data X space by using a partial least square formula, wherein the reconstruction formula is as follows:
Figure GDA0002570881790000044
Eofor reconstructing residual errors, i.e. with UyAn extraneous portion. Whereby the X space can be divided intokRelated part UyAnd with YkExtraneous moiety XoAnd two parts, the quality of the X space is fully decomposed.
X＝Uy+Xo(35)
(6) To X againoPerforming slow feature analysis to obtain slow features, and arranging the slow features in the order of the change speed from small to large：
So＝WoXo＝[So1,So2,…,Som](36)
XoIs the original process data minus the mass variable YkThe related partial data is obtained, so that the obtained slow characteristics are not all meaningful, and the selection is satisfied
Figure GDA0002570881790000045
G slow features of (1), E { } is the operator of expectation, I is the unit matrix;
according to the change speed of the obtained slow features, q features with slower change speed form a matrix Sod＝[so1,so2,…,soq]And the remaining (g-q) features with faster change speed form another matrix: soe＝[so(q+1),so(q+2),…,sog]。
(7) And establishing a static and dynamic index system.
Considering that for a real industrial system, the product quality variable of interest may be one or more than two, for both cases, different index systems are established to monitor the system.
(7.1) for the case of a single quality variable
Since there is only one quality variable, and therefore only one representative variable is extracted, the representative variable U is directly utilized for the quality-related spaceyAnd (5) monitoring. Wherein the static indexes are as follows:
Figure GDA0002570881790000051
the dynamic indexes are as follows:
Figure GDA0002570881790000052
monitoring the dynamic and static indexes by adopting a shewhart control chart method, and for the static indexes:
Figure GDA0002570881790000053
wherein UTH1Is composed of
Figure GDA0002570881790000054
Upper limit of control of, LTH1Is composed of
Figure GDA0002570881790000055
Lower limit of control, μ1Is composed of
Figure GDA0002570881790000056
The average value of (a) of (b),
Figure GDA0002570881790000057
as its standard deviation, b1The control map is the threshold width. For the dynamic index:
Figure GDA0002570881790000058
wherein UTH2Is composed of
Figure GDA0002570881790000059
Upper limit of control of, LTH2Is composed of
Figure GDA00025708817900000510
Lower limit of control, μ2Is composed of
Figure GDA00025708817900000511
The average value of (a) of (b),
Figure GDA00025708817900000512
as its standard deviation, b2The control map is the threshold width.
For a space independent of quality variables, the static index is:
Figure GDA00025708817900000513
Figure GDA00025708817900000514
the dynamic indexes are as follows:
Figure GDA00025708817900000515
Figure GDA0002570881790000061
Figure GDA0002570881790000062
denotes SodThe inverse of the diagonal matrix formed by the covariance matrix eigenvalues of (a),
Figure GDA0002570881790000063
denotes SoeThe inverse of the diagonal matrix formed by the covariance matrix eigenvalues, the control limits of the indexes are all obtained by the kernel density function under the confidence level of 0.95, which are respectively
Figure GDA0002570881790000064
Are respectively as
Figure GDA0002570881790000065
The inverse of the diagonal matrix formed by the eigenvalues of the covariance matrix, the control limits corresponding to the two dynamic indexes are calculated by the kernel density function under the confidence level of 0.95, respectively
Figure GDA0002570881790000066
(7.2) for cases with more than two quality variables
The typical variables obtained are not unique, so it is necessary to compute statistics to divide U by twoyMonitoring is carried out, and the static indexes are calculated as follows:
Figure GDA0002570881790000067
wherein
Figure GDA0002570881790000068
Represents UyThe control limits of the indexes are obtained by a kernel density function under the confidence level of 0.95 and are respectively
Figure GDA0002570881790000069
To UyPerforming a difference to obtain
Figure GDA00025708817900000610
The dynamic indexes are as follows:
Figure GDA00025708817900000611
Figure GDA00025708817900000612
is composed of
Figure GDA00025708817900000613
The eigenvalues of the covariance matrix form the inverse of the diagonal matrix,
Figure GDA00025708817900000614
and calculating the control limits corresponding to the three dynamic indexes by using a kernel density function under the confidence level of 0.95. The spatial index independent of the quality variable is calculated in the same way as in the case of the single quality variable in step 7.1.
(8) The dynamic and static indexes can be used for respectively monitoring the process dynamic and steady-state operation conditions of the closed-loop system.
For the space related to the quality variable:
1) static index
Figure GDA00025708817900000615
Overrun, dynamic index
Figure GDA00025708817900000616
After exceeding the limit, the control is recoveredThe space related to the quality variable is adjusted to a new working condition through a control action, and the quality variable is recovered to a set level after transient fluctuation;
2) static index
Figure GDA00025708817900000617
Overrun, dynamic index
Figure GDA00025708817900000618
The quality variable is always out of limit, which indicates that the space related to the quality variable has a fault, the controller fails to adjust, and the product quality is damaged;
3) static index
Figure GDA00025708817900000619
Dynamic index
Figure GDA00025708817900000620
If the quality variable does not exceed the set value, the space related to the quality variable is in a good operation state, and the product quality is maintained near the set value.
For a space independent of the quality variable:
1) quality independent static indicators
Figure GDA0002570881790000071
And dynamic index
Figure GDA0002570881790000072
Returning to the control limit after short time overrun, indicating that short time abnormality occurs in the space irrelevant to the quality variable, and then recovering the space to operate under the same steady-state working condition as before the abnormality occurs through the adjustment of the closed-loop control action;
2) static indicators of quality-independent spaces
Figure GDA0002570881790000073
Always out of limit, dynamic index
Figure GDA0002570881790000074
Go back to after a short overrunThe space irrelevant to the quality variable is indicated to have transient abnormality below the control limit, but the space is regulated by the closed-loop control action to be recovered to run under a different steady-state working condition before the abnormality occurs, the static performance is changed, but the dynamic performance is still maintained to be normal;
3) quality independent static indicators
Figure GDA0002570881790000075
And dynamic index
Figure GDA0002570881790000076
The data are all out of limit, which indicates that the space has a fault, and the space correlation controller fails to regulate, and the static and dynamic performances are damaged;
4) quality independent static indicators
Figure GDA0002570881790000077
And dynamic index
Figure GDA0002570881790000078
All remained normal, indicating that the space remained in a good operating condition.
The invention has the beneficial effects that: the invention can divide the industrial closed-loop system into two spaces which are related to the product quality and unrelated to the product quality, can effectively distinguish which space the fault specifically occurs in, further judge whether the fault affects the product quality, can effectively identify the change of the working condition and the real fault by monitoring the dynamic and static information of the system, and realize the fine operation state monitoring of the closed-loop system.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a process flow diagram of a closed loop TE process with particular application of the present method;
FIG. 3 is a failure class presentation diagram provided by the TE process;
FIG. 4 shows the result of monitoring the static indicator when the method is applied to the TE process failure 1;
FIG. 5 shows the dynamic index monitoring result when the method is applied to the TE process failure 1;
FIG. 6 is a graph comparing product quality A for TE process fault 1 versus product quality A for normal conditions;
FIG. 7 is a graph comparing product quality B for TE process fault 1 versus normal product quality B;
FIG. 8 is a graph comparing product quality C for TE process fault 1 versus normal product quality C;
FIG. 9 shows the result of monitoring the static indicator when the method is applied to the TE process failure 4;
FIG. 10 shows the dynamic index monitoring result of the TE process failure 4 applied in the present method;
FIG. 11 is a graph comparing product quality A for TE process fault 4 versus normal product quality A;
FIG. 12 is a graph comparing product quality A for TE process fault 4 versus normal product quality A;
fig. 13 is a graph comparing the corresponding product quality a in the case of TE process failure 4 with the product quality a in the normal case.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and specific examples.
The Tennessee Eastman (TE) process is a simulation system provided by downloads et al based on a certain practical chemical production process of the Tennessee Eastman chemical company, and in the research of the process system engineering field, the TE process is a common standard problem (Benchmark problem) which well simulates many typical characteristics of a practical complex industrial process system, so that the TE process is taken as a simulation example to be widely applied to the research of control, optimization, process monitoring and fault diagnosis. In this study, the closed-loop TE process of the plant wide control strategy proposed by Lyman and Georgakis was used. The TE process comprises four gaseous feeds A, C, D and E, two liquid products G and H, and also comprises by-product F and inert gas B.
The TE process includes five main units: the reactor, condenser, compressor, separator and stripper, as shown in fig. 2, included a total of 41 measured variables and 12 control variables. As in fig. 3, the TE process provides 21 fault classes available for use.
As shown in fig. 1, the present invention comprises the steps of:
(1) selecting 33 variables of process measurement data XMEAS (1-22) and XMV (1-11) in normal operation of the TE process as process input variables X, taking product quantity variable Y concerned by three variables of XMEAS (29-31), and obtaining a vector X (t) ═ X (X) by sampling each time1(t),x2(t),…,xm(t)) dimension 1 × m, and after N samplings, the process measurement variable X ═ X (t), X (t +1), …, X (t + N))TObtaining a data matrix Y of the quality variable after N times of samplingk(∈N*k)。
(2) Data normalization: for process data matrix X and quality variable data matrix Y respectivelykNormalization was performed by column mean removal and division by standard deviation.
(3) And (3) carrying out slow feature analysis modeling on the process data matrix X after standardization in the step (2) to obtain initial slow features.
Considering linear slow feature analysis here, each slow feature can be viewed as a linear combination of process variables, so from the process data matrix to the slow feature s (t) ═ s1(t),s2(t),…,sm(t)]The mapping of (d) can be expressed as:
S(t)＝W×X
wherein W ═ W1,w2,…,wm]TThe coefficient matrix is needed to be optimized by the SFA, and m is the number of slow features and is the same as the process variable number. The problem of solving the slow feature is to make the obtained slow feature change rate Δ(s)i) At a minimum, solving the slow eigenvalue problem can be converted to solving the following generalized eigenvalue problem:
AW＝BWΩ
wherein A represents a first order difference matrix of input data X
Figure GDA0002570881790000091
Can be expressed as
Figure GDA0002570881790000092
B represents the covariance matrix of the X matrix<XXT>t，W＝[w1,w2,…,wm]TThe feature matrix is formed by the feature vectors, m is the number of variables of the process input data, and omega is a diagonal matrix formed by corresponding generalized eigenvalues. The slow features can also be solved by using two-step Singular Value Decomposition (SVD), which comprises the following steps:
(3.1) the matrix X is first whitened, i.e. the singular value decomposition of the B matrix. The covariance matrix of matrix X may be expressed as<XXT>tAnd performing Singular Value Decomposition (SVD) on the obtained product to obtain:
<XXT>t＝UΛUT
where U is the covariance matrix<XXT>tΛ is a diagonal matrix, and the element on each diagonal is an eigenvalue, the whitened data matrix can be expressed as:
Z＝Λ-1/2UTX＝QX
the covariance of the whitened data matrix Z is satisfied<ZZT>t＝Q<XXT>tQT。
(4.2) solving the linear SFA problem is equivalent to finding a matrix P ═ WQ-1Making S ═ P × Z, and making S satisfy<SST>t＝P<ZZT>tPT＝I。
The matrix P may be represented by a difference matrix of pairs Z
Figure GDA0002570881790000093
Covariance matrix of
Figure GDA0002570881790000094
Singular value decomposition is carried out to obtain:
Figure GDA0002570881790000095
from this, the initial slow eigen coefficient matrix can be calculated as:
W＝PΛ-1/2UT
the initial slow characteristic is:
S＝WX＝PΛ-1/2UTX
(5) utilizing the initial slow characteristic S and the quality variable Y obtained in the step (4)kPerforming a typical correlation analysis (CCA) to obtain a typical variable, wherein the specific process is as follows:
(5.1) for the initial slow feature S and the quality variable YkDefining two variables of u and v, wherein u and v are the start-slow characteristic S and the quality variable YkThe linear combination of (a) is as follows:
Figure GDA0002570881790000101
solving the typical variable u, v is to solve the coefficient matrix a, B so that the Pearson coefficient between u and v is the maximum.
The following optimization problem solution can be converted:
Maximizeu2
Subjectto：
Figure GDA0002570881790000102
where corr (u, v) denotes the Pearson coefficient, cov (u, v) is the covariance of u, v, Var (u) and Var (v) are the variances of u, v, respectively. This problem can be solved by constructing Lagrangian equations, solving the two typical variables u, v corresponding to the maximum Pearson coefficient.
(5.1) similarly, the solution can be continued to find the typical variable u corresponding to the second largest Pearson coefficient2And v2：
Figure GDA0002570881790000103
In total, n pairs of typical variables can be obtained, where n is min (m, k), and the corresponding Pearson coefficients decrease in sequence, that is, the correlation between the typical variables is smaller and smaller. Before selection according to the obtained Pearson coefficientp pairs of representative correlation variables corresponding to the largest correlation coefficients, leaving (n-p) pairs of representative correlation variables corresponding to the smaller Pearson coefficients. Combining the first p typical correlation variable pairs into a new matrix U by using the typical variables corresponding to the linear combination of the slow features Sy＝[u1,u2,…,up]，ui(i-1, 2, …, p) are typical variables found, the corresponding Pearson coefficients decrease in order, and u isiAre not related to each other.
(6) Subjecting the typical variable U obtained in (5)yReconstructing back X using partial least squaresdSpatially, the reconstruction formula is as follows:
X＝UyPT+E
PT＝(Uy TUy)-1Uy TXd
and divides the X space intokRelated part UyAnd with YkExtraneous moiety XoAnd two parts, the quality of the X space is fully decomposed.
X＝Uy+Xo
(7) To X againoAnd (3) carrying out slow feature analysis to obtain slow features, and arranging the slow features according to the sequence of the change speed from small to large:
So＝WoXo＝[So1,So2,…,Som]
Xois the original process data minus the mass variable YkThe related partial data is obtained, so that the obtained slow characteristics are not all meaningful, and the selection is satisfied
Figure GDA0002570881790000111
G slow features. And according to the change speed of the obtained slow characteristics, selecting q characteristics with slower change speed to form Sod＝[so1,so2,…,soq]Leaving (g-q) features S with faster change speedoe＝[so(q+1),so(q+2),…,sog]。
(8) And establishing a static and dynamic index system.
Considering that for a real industrial system, the product quality variable of interest may be one or more, for both cases, different index systems are established to monitor the system.
(8.1) for the case of a single quality variable
Since there is only one quality variable, and therefore only one representative variable is extracted, the representative variable U is directly utilized for the quality-related spaceyAnd (5) monitoring. Wherein the static indexes are as follows:
Figure GDA0002570881790000112
the dynamic indexes are as follows:
Figure GDA0002570881790000113
monitoring dynamic and static indexes by adopting a method of a shewhart control chart, taking the static indexes as an example:
Figure GDA0002570881790000114
wherein UTH is
Figure GDA0002570881790000115
LTH is a lower control limit, μ is
Figure GDA0002570881790000116
The average value of (a) of (b),
Figure GDA0002570881790000117
b is the control map threshold width for its standard deviation.
For a space independent of quality variables, the static index is:
Figure GDA0002570881790000118
Figure GDA0002570881790000119
the dynamic indexes are as follows:
Figure GDA00025708817900001110
Figure GDA00025708817900001111
Figure GDA00025708817900001112
denotes SodThe inverse of the diagonal matrix formed by the covariance matrix eigenvalues of (a),
Figure GDA00025708817900001113
denotes SoeThe control limits of the indexes are obtained by a kernel density function under the confidence level of 0.95 and are respectively
Figure GDA0002570881790000121
Are respectively as
Figure GDA0002570881790000122
The eigenvalues of the covariance matrix form the inverse of the diagonal matrix,
Figure GDA0002570881790000123
the control limits corresponding to the two dynamic indexes are respectively obtained by calculating the kernel density function under the confidence level of 0.95.
(8.2) for cases with multiple quality variables
The typical variables obtained are not unique, so it is necessary to compute statistics to divide U by twoyMonitoring is carried out, and the static indexes are calculated as follows:
Figure GDA0002570881790000124
wherein
Figure GDA0002570881790000125
Represents UyThe control limits of the indexes are obtained by a kernel density function under the confidence level of 0.95 and are respectively
Figure GDA0002570881790000126
To UyPerforming a difference to obtain
Figure GDA0002570881790000127
The dynamic indexes are as follows:
Figure GDA0002570881790000128
Figure GDA0002570881790000129
is composed of
Figure GDA00025708817900001210
The eigenvalues of the covariance matrix form the inverse of the diagonal matrix,
Figure GDA00025708817900001211
the control limits corresponding to the three dynamic indexes are obtained by calculating the kernel density function under the confidence level of 0.95,
the spatial index calculation independent of the quality variable is the same as above.
The dynamic and static indexes can be used for respectively monitoring the process dynamic and steady-state operation conditions of the closed-loop system. For the space related to the quality variable:
1) static index
Figure GDA00025708817900001212
Overrun, dynamic index
Figure GDA00025708817900001213
After exceeding the limit, the temperature returns to the control limit, indicating the qualityThe space related to the variable is regulated to a new working condition through a control action, and the quality variable is recovered to a set level after transient fluctuation;
2) static index
Figure GDA00025708817900001214
Overrun, dynamic index
Figure GDA00025708817900001215
The quality variable is always out of limit, which indicates that the space related to the quality variable has a fault, the controller fails to adjust, and the product quality is damaged;
3) static index
Figure GDA00025708817900001216
Dynamic index
Figure GDA00025708817900001217
If the quality of the product is not exceeded, the space related to the quality variable is in a good operation state, and the quality of the product is maintained near a reset value.
For a space independent of the quality variable:
1) quality independent static indicators
Figure GDA00025708817900001218
And dynamic index
Figure GDA00025708817900001219
Returning to the control limit after short time overrun, indicating that short time abnormality occurs in the space irrelevant to the quality variable, and then recovering the space to operate under the same steady-state working condition as before the abnormality occurs through the adjustment of the closed-loop control action;
2) static indicators of quality-independent spaces
Figure GDA0002570881790000131
Always out of limit, dynamic index
Figure GDA0002570881790000132
A short overrun back to below the control limit indicates an overrideThe space irrelevant to the quality variable is temporarily abnormal, but the space is adjusted by the closed-loop control action to be recovered to the operation under a different steady-state working condition before the abnormality occurs, the static performance is changed, but the dynamic performance is still maintained to be normal;
3) quality independent static indicators
Figure GDA0002570881790000133
And dynamic index
Figure GDA0002570881790000134
The data are all out of limit, which indicates that the space has a fault, and the space correlation controller fails to regulate, and the static and dynamic performances are damaged;
4) quality independent static indicators
Figure GDA0002570881790000135
And dynamic index
Figure GDA0002570881790000136
All remained normal, indicating that the space remained in a good operating condition.
The two spaces are independent from each other, and the corresponding dynamic and static indexes respectively indicate the change of the dynamic and static information of the respective spaces, so that the method can help people to judge in which space the fault and the abnormality specifically occur, and whether the product quality is influenced.
Fig. 4 and 5 illustrate the method for monitoring TE process fault 1 data, and fig. 6, 7 and 8 illustrate the comparison between normal changes in selected three product quality variables and changes in the case of fault 1. According to the graph, the space related to the product quality y, the static index and the dynamic index are recovered below the control limit after exceeding the limit for a period of time, the space is indicated to be temporarily interfered by faults, the static and dynamic characteristics are recovered to the initial level after being adjusted by the controller for a period of time, and the product quality is also recovered to the initial set value; and in the space irrelevant to the product quality, the static index always exceeds the limit but tends to be stable, and the dynamic index is recovered below the control limit after exceeding the limit for a period of time, which indicates that the space is recovered to a new steady-state working condition through regulation after being interfered. Fig. 9 and 10 are monitoring results of the TE process fault 4 data, and it can be seen from the diagrams that none of the spatial static and dynamic indexes related to the product quality y exceeds the limit, which indicates that the space operates normally and the product quality does not change, and it can also be seen from the comparison of the changes in the normal case and the four cases of the fault of the product quality in fig. 11, 12 and 13 that the judgment of the monitoring results is correct; in the space irrelevant to the product quality, the static index always exceeds the limit, and the dynamic index is recovered below the control limit after short time exceeding, so that the fault is shown to occur in a closed loop irrelevant to the product quality, and a new steady-state working condition is achieved through the adjustment of the controller. The traditional slow characteristic analysis can only distinguish the abnormal working condition and the fault, and the method can further distinguish the space where the fault occurs and whether the product quality is influenced.

Claims (1)

1. A closed loop system fine operation state monitoring method based on quality-related slow feature full decomposition is characterized by comprising the following steps:
(1) selecting normal process measurement data and concerned product quality data of an industrial closed loop system, wherein the operation process of the industrial closed loop system comprises m measurable process variables, and sampling at the time t can obtain a vector x (t) ═ x (x) of 1 × m1(t),x2(t),…,xm(t)), obtaining a data matrix X of the process variable in the normal process after N times of samplingM＝(x(t),x(t+1),…,x(t+N))TSelecting k concerned product quality variables according to the industrial process, and obtaining a data matrix Y of the quality variables after N times of samplingk∈N*k；
(2) Data normalization: respectively to process data matrix XMAnd quality variable data matrix YkSubtracting the mean of the column by column and dividing by the standard deviation of the column for normalization;
(3) performing slow feature analysis modeling on the process data matrix X normalized in the step (2) to obtain initial slow features;
considering linear slow signature analysis here, each slow signature can be viewed as a linear combination of process variables, so from the process data matrixX to slow feature s (t) ═ s1(t),s2(t),…,sm(t)]Is represented as:
S(t)＝W×X (1)
wherein W ═ W1,w2,…,wm]TIs a coefficient matrix to be optimized by the SFA, and m is the number of slow features and is the same as the measurable process variable; the problem of solving the slow feature is to make the obtained slow feature change rate Δ(s)i) At a minimum, solving the slow eigenvalue problem translates to solving the following generalized eigenvalue problem:
AW＝BWΩ (2)
where A represents a first order difference matrix of an input process data matrix X
Figure FDA0002503452140000011
Is expressed as a covariance matrix of
Figure FDA0002503452140000012
B represents the covariance matrix of the X matrix<XXT>tCorresponding, W ═ W1,w2,…,wm]TThe number of variables of the m process input data is the feature matrix formed by the feature vectors; omega is a diagonal matrix formed by corresponding generalized eigenvalues; the slow features are solved by using two-step Singular Value Decomposition (SVD), and the specific process is as follows:
(3.1) whitening the matrix X, namely decomposing singular values of a B matrix; the covariance matrix of matrix X may be expressed as<XXT>And carrying out Singular Value Decomposition (SVD) on the obtained product to obtain:
<XXT>t＝UΛUT(3)
where U is the covariance matrix<XXT>tΛ is a diagonal matrix, and the element on each diagonal is an eigenvalue, the whitened data matrix can be expressed as:
Z＝Λ-1/2UTX＝QX (4)
the covariance of the whitened data matrix Z is satisfied<ZZT>t＝Q<XXT>tQT；
(3.2) solving the linear SFA problem is equivalent to finding a matrix P ═ WQ-1Making S ═ P × Z, and making S satisfy<SST>t＝P<ZZT>tPT＝I；
The matrix P may be represented by a difference matrix of pairs Z
Figure FDA0002503452140000021
Covariance matrix of
Figure FDA0002503452140000022
Singular value decomposition is carried out to obtain:
Figure FDA0002503452140000023
from this, the initial slow eigen coefficient matrix can be calculated as:
W＝PΛ-1/2UT(6)
the initial slow characteristic is:
S＝WX＝PΛ-1/2UTX (7)
(4) using the initial slow characteristic S and the quality variable Y obtained in (7)kPerforming typical correlation analysis CCA to acquire typical variables, wherein the specific process is as follows:
for the initial slow feature S and the quality variable YkDefining two variables of u and v, wherein u and v are the initial slow characteristic S and the quality variable YkThe linear combination of (a) is as follows:
Figure FDA0002503452140000024
therein Ψ1＝[a1,a2,…,am]T，Ψ2＝[b1,b2,…,bk]T(ii) a Solving the typical variable u, v is to solve the coefficient matrix Ψ1，Ψ2Maximizing the Pearson coefficient between u and v; is converted intoSolving the following optimization problem:
Figure FDA0002503452140000025
Figure FDA0002503452140000026
wherein corr (u, v) represents the Pearson coefficient between u, v, cov (u, v) is the covariance of u, v, Var (u) and Var (v) are the variances of u, v, respectively; solving the problem by constructing a Lagrangian equation, and solving two typical variables u, v corresponding to the maximum Pearson coefficient;
similarly, on the basis, the typical variable u corresponding to the second large Pearson coefficient is continuously solved and searched2And v2：
Figure FDA0002503452140000027
N pairs of typical variables can be obtained, wherein n is min (m, k), and the corresponding Pearson coefficients are sequentially reduced, namely, the correlation among the typical variables is smaller and smaller; selecting the first p typical relevant variable pairs corresponding to the larger Pearson coefficient according to the obtained Pearson coefficient, and eliminating the n-p typical relevant variable pairs corresponding to the smaller Pearson coefficient; using the first p pairs of canonical correlation variables, using the derived canonical variables u for the linear combination of the slow features SiWhere i is 1,2, …, p, combined into a new matrix Uy＝[u1,u2,…,up]Wherein u isi(i ═ 1,2, …, p) the Pearson coefficients for the decrease in order, and uiAre mutually orthogonal;
(5) the matrix U obtained in the step (4) is processedyAnd (3) reconstructing the process data X space by using a partial least square formula, wherein the reconstruction formula is as follows:
Figure FDA0002503452140000031
Eofor reconstitutionResidual error, i.e. with UyAn extraneous portion; whereby the X space can be divided intokRelated part UyAnd with YkExtraneous moiety XoTwo parts, which realize the sufficient decomposition of the quality of the X space;
X＝Uy+Xo(12)
(6) to X againoAnd (3) carrying out slow feature analysis to obtain slow features, and arranging the slow features according to the sequence of the change speed from small to large:
So＝WoXo＝[So1,So2,…,Som](13)
Xois the original process data minus the mass variable YkThe related partial data is obtained, so that the obtained slow characteristics are not all meaningful, and the selection is satisfied
Figure FDA0002503452140000032
G slow features of (1), E { } is the operator of expectation, I is the unit matrix;
according to the change speed of the obtained slow features, q features with slower change speed form a matrix Sod＝[so1,so2,…,soq]And g-q characteristics with higher change speed are left to form another matrix: soe＝[so(q+1),so(q+2),…,sog]；
(7) Establishing a static and dynamic index system;
considering that the concerned product quality variable may be one or more than two for an actual industrial system, different index systems are established for monitoring the system for the two cases;
(7.1) for the case of a single quality variable
Since there is only one quality variable, and therefore only one representative variable is extracted, the representative variable U is directly utilized for the quality-related spaceyMonitoring is carried out; wherein the static indexes are as follows:
Figure FDA0002503452140000033
the dynamic indexes are as follows:
Figure FDA0002503452140000041
monitoring the dynamic and static indexes by adopting a shewhart control chart method, and for the static indexes:
Figure FDA0002503452140000042
wherein UTH1Is composed of
Figure FDA0002503452140000043
Upper limit of control of, LTH1Is composed of
Figure FDA0002503452140000044
Lower limit of control, μ1Is composed of
Figure FDA0002503452140000045
The average value of (a) of (b),
Figure FDA0002503452140000046
as its standard deviation, b1Controlling the graph threshold width; for the dynamic index:
Figure FDA0002503452140000047
wherein UTH2Is composed of
Figure FDA0002503452140000048
Upper limit of control of, LTH2Is composed of
Figure FDA0002503452140000049
Lower limit of control, μ2Is composed of
Figure FDA00025034521400000410
The average value of (a) of (b),
Figure FDA00025034521400000411
as its standard deviation, b2Controlling the graph threshold width;
for a space independent of quality variables, the static index is:
Figure FDA00025034521400000412
Figure FDA00025034521400000413
the dynamic indexes are as follows:
Figure FDA00025034521400000414
Figure FDA00025034521400000415
Figure FDA00025034521400000416
denotes SodThe inverse of the diagonal matrix formed by the covariance matrix eigenvalues of (a),
Figure FDA00025034521400000417
denotes SoeThe inverse of the diagonal matrix formed by the covariance matrix eigenvalues, the control limits of the indexes are all obtained by the kernel density function under the confidence level of 0.95, which are respectively
Figure FDA00025034521400000418
Figure FDA00025034521400000419
Are respectively as
Figure FDA00025034521400000420
The inverse of the diagonal matrix formed by the eigenvalues of the covariance matrix, the control limits corresponding to the two dynamic indexes are calculated by the kernel density function under the confidence level of 0.95, respectively
Figure FDA00025034521400000421
(7.2) for cases with more than two quality variables
The typical variables obtained are not unique, so it is necessary to compute statistics to divide U by twoyMonitoring is carried out, and the static indexes are calculated as follows:
Figure FDA00025034521400000422
wherein
Figure FDA00025034521400000423
Represents UyThe control limits of the indexes are obtained by a kernel density function under the confidence level of 0.95 and are respectively
Figure FDA0002503452140000051
To UyPerforming a difference to obtain
Figure FDA0002503452140000052
The dynamic indexes are as follows:
Figure FDA0002503452140000053
Figure FDA0002503452140000054
is composed of
Figure FDA0002503452140000055
The eigenvalues of the covariance matrix form the inverse of the diagonal matrix,
Figure FDA0002503452140000056
calculating the control limits corresponding to the three dynamic indexes of the three dynamic indexes by a kernel density function under the confidence level of 0.95; the way of calculating the spatial index independent of the quality variable is the same as the case of the single quality variable in step 7.1;
(8) the dynamic and static indexes can be used for respectively monitoring the process dynamic and steady-state operation conditions of the closed-loop system;
for the space related to the quality variable:
1) static index
Figure FDA0002503452140000057
Overrun, dynamic index
Figure FDA0002503452140000058
The space related to the quality variable is adjusted to a new working condition through the control action, and the quality variable is restored to the set level after transient fluctuation;
2) static index
Figure FDA0002503452140000059
Overrun, dynamic index
Figure FDA00025034521400000510
The quality variable is always out of limit, which indicates that the space related to the quality variable has a fault, the controller fails to adjust, and the product quality is damaged;
3) static index
Figure FDA00025034521400000511
Dynamic index
Figure FDA00025034521400000512
If the quality of the product is not exceeded, indicating that the space related to the quality variable is in a good operation state, and maintaining the product quality near a set value;
for a space independent of the quality variable:
1) quality independent static indicators
Figure FDA00025034521400000513
And dynamic index
Figure FDA00025034521400000514
Returning to the control limit after short time overrun, indicating that short time abnormality occurs in the space irrelevant to the quality variable, and then recovering the space to operate under the same steady-state working condition as before the abnormality occurs through the adjustment of the closed-loop control action;
2) static indicators of quality-independent spaces
Figure FDA00025034521400000515
Always out of limit, dynamic index
Figure FDA00025034521400000516
Returning to the position below the control limit after the transient overrun indicates that the space irrelevant to the quality variable is transient abnormal, but the space is adjusted by the closed-loop control action to recover to operate under a different steady-state working condition before the abnormality occurs, the static performance is changed, and the dynamic performance is still maintained to be normal;
3) quality independent static indicators
Figure FDA00025034521400000517
And dynamic index
Figure FDA00025034521400000518
The data are all out of limit, which indicates that the space has a fault, and the space correlation controller fails to regulate, and the static and dynamic performances are damaged;
4) quality independent static indicators
Figure FDA00025034521400000519
And dynamic index
Figure FDA00025034521400000520
All remained normal, indicating that the space remained in a good operating condition.
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