CN109145256B - Process monitoring method based on normative variable nonlinear principal component analysis - Google Patents
Process monitoring method based on normative variable nonlinear principal component analysis Download PDFInfo
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Abstract
The invention discloses a nonlinear dynamic process monitoring method based on normalized variable nonlinear principal component analysis, which comprises the following steps: acquiring a data matrix Y, and pre-specifying a numerical value of p and a system order n; combining Hankel matrixes of past and future observation values according to a formula; calculating covariance and mutual variance matrices of past and future observations; performing singular value decomposition on the H matrix; calculating a state vector and a residual vector; projecting the state vector to a high-dimensional feature space through explicit second-order polynomial mapping; determining a front k pivot element through characteristic value decomposition in principal component analysis; finally, calculate T 2 Statistic, combination statistic Q c And its corresponding control limits. The method provided by the invention is used for monitoring the faults of three different types in the Tennessman chemical process, and simulation results show that compared with KPCA and NDPCA, the CV-NPCA method provided by the invention has higher fault detection rate and relatively lower fault false alarm rate.
Description
Technical Field
The invention relates to a nonlinear dynamic process monitoring method in the technical field of data driving, in particular to a nonlinear dynamic process monitoring method based on explicit polynomial mapping.
Background
The traditional multivariate statistical process monitoring method is limited by the assumption that linear and measured variables conform to normal distribution, such as principal component analysis and normative variable analysis, and when the method is used for monitoring a nonlinear dynamic industrial process, higher false alarm rate of faults and lower fault detection rate can be generated. Nuclear principal component analysis based on radial basis functions has found application in many non-linear industrial processes. However, infinite-dimensional non-linear mapping based on an uncertain kernel function is inefficient and redundant.
Disclosure of Invention
The purpose of the invention is as follows: the invention aims to solve the problems of uncertainty and redundancy based on a radial basis kernel function, and provides a process monitoring method based on normative variable nonlinear principal component analysis and a nonlinear dynamic process monitoring method combining the advantages of normative variable analysis and principal component analysis.
The technical scheme is as follows: the invention discloses a process monitoring method based on normative variable nonlinear principal component analysis, which comprises the following steps:
1) obtaining a data matrix Y ∈ R m×l Pre-specifying a numerical value of p and a system order n;
2) hankel rectangle Y for combining past observation values according to formula p Hankel matrix Y of future observations f ；Y p ＝[y p,p+1 y p,p+2 … y p,p+N ]∈R mp×N And Y f ＝[y f,p+1 y f,p+2 … y f,p+N ]∈R mp×N The N moment is the column number of a Hankel matrix;
3) calculating covariance of Hankel matrix of past and future observations
Sum and mutual variance matrix
4) Singular value decomposition of Hankel matrix H
5) Computing state vectors
Sum residual vector
6) Projecting state vectors to high dimensions by explicit second-order polynomial mappingCharacteristic space G ═ G 1 ,g 2 ,…,g N ] T ；
7) Determining front k pivot by eigenvalue decomposition in principal component analysis
8) Calculating T 2 Statistic, combination statistic Q c And its corresponding control limit; if it is not
Or Q c >Q UCL And (α), indicating that a fault was detected.
Further, the past observation vector y in step 1) p,r And a future observation vector y f,r The measurement values of past and future p sampling moments in the data matrix Y are combined to form:
wherein r is p +1, p +2, …, p + N, y r Are measurements from a training data set.
Further, the step 6) is specifically as follows:
the state vector is mapped to the high dimensional feature space by displaying a second order as follows:
for a two-dimensional vector, the dimensionality of the mapped state vector is increased to 5; for a vector of dimension D, the dimension of the mapped state vector can be determined by the following equation:
an m-dimensional state vector x r ∈R m After the display binomial mapping, the change is g r ∈R (m(m+3)/2) ；
In a high-dimensional feature space, a matrix G epsilon R consisting of the first n state vectors can be obtained (N×(m(m+3)/2) With less non-linear behavior.
Further, the step 7) is specifically as follows:
reducing the dimensionality of the mapped state vector through conventional principal component analysis, and obtaining the most important l-dimensional latent variable and D-l residual vector:
by performing eigenvalue decomposition on the matrix S, a load vector matrix V and a diagonal matrix Lambda can be obtained:
S＝G T G/(N-1)＝VΛV T
wherein Λ ═ diag (λ) 1 ,λ 2 ,…,λ k )；
The matrix G can be decomposed as the product of the load vector and the score vector:
wherein p is i Is a load vector t i And E represents a residual matrix, and the number k of the principal elements can be obtained by accumulating the contribution rate.
Further, the method also comprises online monitoring, and comprises the following specific steps:
1) after the observed value of the 2p sampling moment is obtained, the past observed vector y is assembled pp ；
2) Calculating monitoring statistics
And combined statistics
3) The monitoring statistics do not exceed their corresponding control limits, indicating that the process is operating normally; if it is used
Or Q c >Q UCL And (α), indicating that a fault was detected.
Has the advantages that: the invention carries out standardization preprocessing on the training data through the standardization variable analysis to reduce the influence of the dynamic characteristics of the data. The state vector is then projected into a high dimensional feature space using the determined second order polynomial mapping. Simultaneously considering the linear and nonlinear changes of the monitoring process, and providing a combined statistic Q c And its control limit is determined using kernel density function estimation. The method is used for monitoring three different types of faults in the chemical process of Tennessee. Simulation results show that compared with KPCA and NDPCA, the proposed CV-NPCA method has higher fault detection rate and relatively lower fault false alarm rate.
Drawings
FIG. 1 is a schematic diagram of the monitoring method of the present invention;
FIG. 2 is a simulation comparison monitoring diagram of TE process fault 2 of the present invention;
FIG. 3 is a simulation versus monitoring diagram of the TE process fault 10 of the present invention;
figure 4 is a simulation versus monitoring graph of a TE process fault 19 of the present invention.
Detailed Description
The technical solution of the present invention will be further described in detail with reference to the following specific examples.
FIG. 1 is a schematic diagram of the monitoring method of the present invention. The proposed nonlinear dynamic process monitoring method mainly comprises three stages. Stage one, reducing the influence of data dynamic characteristics by using a CVA; step two, mapping the state vector to a high-dimensional feature space through display polynomial mapping; determining front k principal components and residual error by using PCA, and calculating T 2 And Q c Statistics are obtained.
1) Dynamic data pre-processing
The canonical variable analysis is a linear dimension reduction method based on multivariate statistical analysis, and past observation vector y p,r And a future observation vector y f,r From past and future p samples in the data matrix YThe measured values of the scales are combined to form:
wherein, y r Are measurements from a training data set.
Hankel matrix Y setting r ═ p +1, p +2, …, p + N, past and future observations p And Y f The definition is as follows:
Y p ＝[y p,p+1 y p,p+2 …y p,p+N ]∈R mp×N
Y f ＝[y f,p+1 y f,p+2 …y f,p+N ]∈R mp×N
the Hankel matrix contains N ═ l-2p +1 columns. The covariance and cross variance matrices of past and future observations may be calculated using the following equations:
the solution for the optimal linear combination can be obtained by performing singular value decomposition on the Hankel matrix H:
the state vector is a subset of the estimated canonical variables, defined as follows: state vector
Wherein, V x First n columns containing V(n needs to be specified in advance).
The prediction error is as follows: residual vector
2) State vector mapping
The state vector is mapped to the high dimensional feature space by displaying a second order as follows:
for a two-dimensional vector, the dimensionality of the mapped state vector is increased to 5; for a vector of dimension D, the dimension of the mapped state vector can be determined by the following equation:
an m-dimensional state vector x r ∈R m After the display binomial mapping, the change is g r ∈R (m(m+3)/2) 。
G＝[g 1 ,g 2 ,…,g N ] T 。
3) Performing principal component analysis
And then reducing the dimensionality of the mapped state vector through conventional principal component analysis, and acquiring the most important l-dimensional latent variable and D-l residual vector.
By performing eigenvalue decomposition on the matrix S, a load vector matrix V and a diagonal matrix Lambda can be obtained:
S＝G T G/(N-1)＝VΛV T
wherein Λ ═ diag (λ) 1 ,λ 2 ,…,λ k )。
The matrix G can be decomposed as the product of the load vector and the score vector:
wherein p is i Is a load vector t i The number of principal elements k can be obtained by accumulating the contribution rates.
Based on the monitoring method, the specific monitoring step of the invention can be divided into two aspects of off-line training and on-line monitoring, and specifically comprises the following steps:
A. off-line training
1) Obtaining a data matrix
Pre-appointing a numerical value of p and a system order n;
2) hankel rectangle Y for combining past observation values according to formula p Hankel matrix Y of future observations f ；Y p ＝[y p,p+1 y p,p+2 …y p,p+N ]∈R mp×N And Y f ＝[y f,p+1 y f,p+2 …y f,p+N ]∈R mp×N The N moment is the column number of a Hankel matrix;
3) computing covariance of Hankel matrix of past and future observations
Sum and mutual variance matrix
4) Singular value decomposition of H matrix
5) ComputingState vector
Sum residual vector
6) Projecting a state vector to a high-dimensional feature space G-G through explicit second-order polynomial mapping 1 ,g 2 ,…,g N ] T ；
7) Determining top-k pivot by eigenvalue decomposition in principal component analysis
8) Calculating T 2 Statistic, combination statistic Q c And its corresponding control limits. If it is not
Or Q c >Q UCL And (α), indicating that a fault was detected.
B. On-line monitoring
1) After the observed value of the 2p sampling moment is obtained, the past observed vector y is assembled pp ；
2) Calculating monitoring statistics
And combined statistics
3) The monitoring statistic does not exceed its corresponding control limit, indicating that the process is operating properly. If it is not
Or Q c >Q UCL And (α), indicating that a fault was detected.
Simulation verification
Three different types of fault data sets are generated by utilizing a Tiannaxi Iseman chemical process simulation platform, and the effectiveness and the performance of the algorithm are verified. The Tennessee Eastman (TE) chemical process simulation platform in Tennessman of Tennessee can simulate the characteristics of nonlinearity, non-Gaussian, time-varying, multi-mode and the like of a process, and provides a standard simulation model for verifying various process modeling and control methods and fault monitoring and diagnosis methods. The TE process includes five main operating units: a reactor, a condenser, a compressor, a separator and a stripper; and also comprises 4 gas feeds, 2 main products generated by 2 gas-liquid exothermic reactions and 2 by-products generated by two derived exothermic reactions; the process mechanism is complex, and the variables are more, including multiple data fault types such as steps, random changes, slow drift, viscosity and constant positions. The process contained 41 measured variables and 12 controlled variables. Each measured variable is superimposed with additive noise to simulate the noise in an actual industrial process. Each data set has 52 variables, and the sampling time of most variables is 3 minutes; the sampling time for 14 variables was 6 minutes and for 5 variables was 15 minutes. The training and testing data set for the TE process may be downloaded from the following website: the http:// web. mit. edu/braatzgroup/TE _ process. zip data set includes 1(Fault 0) normal operation data and 20(Fault 1-Fault 20) failure operation data. Three types of test data, namely step, random variation and position fault, are adopted in the simulation example.
In order to compare the monitoring performance of the KPCA, NDPCA and CV-NPCA under the same condition, the length p of past and future observation windows of NDPCA and CV-NPCA are set to be 2; the number of principal elements for KPCA, NDPCA, and CV-NPCA are all set to 7. The width parameter c of the radial basis kernel function is chosen by reference as c 500 × D.
FIGS. 2(a) (b), 3(a) (b) and 4(a) (b) show the T generated by KPCA and NDPCA, respectively 2 And a monitoring map of the Q statistics. FIG. 2(c), FIG. 3(c) and FIG. 4(c) show the CV-NPCA generated T, respectively 2 Statistics sum Q c A monitoring graph of the combined statistics. It can be seen from the figure that NDPCA and CV-NPCA have higher sensitivity and lower perturbation in the case of fault 10. However, in the case of faults 2, 19, it is not easy to observe that the monitoring performance of that method is better.
For more accurate comparison of monitoring performance, two indexes of fault detection rate and false alarm rate are adopted in the example. The specific calculation formula is as follows:
wherein N is f Representing the number of samples exceeding the control limit; n is a radical of ab Total number of samples indicating abnormal operation state; n is a radical of no Representing the total number of samples in normal operation.
TABLE 1
TABLE 2
Statistics T for three fault types are given in tables 1 and 2, respectively 2 And Q (Q) c ) And comparing the fault detection rate with the false alarm rate. On the one hand, as can be seen from the table, the proposed CV-NPCA method has the highest statistic T in comparison with KPCA and NDPCA 2 And Q c And (4) fault detection rate. That is, the proposed CV-NPCA can account for more data variance variation in principal and residual spaces than the other two methods. On the other hand, as can be seen from Table 2, CV-NPCA achieves the lowest T of the three types of faults 2 And (4) fault false alarm rate. In the case of fault 2, KPCA and NDPCA have the same Q fault false alarm rate. The proposed CV-NPCA method achieves the lowest T only in the case of a failure of 10 2 And Q c The false alarm rate of (2). Yu's NDPCA method achieves the lowest Q false alarm rate only in the case of fault 2. Simulation result comparison shows that the proposed CV-NPCA methodNot only has relatively high fault detection sensitivity, but also has the same fault false alarm rate as KPCA and NDPCA.
Although the present invention has been described with reference to the preferred embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the present invention.
Claims (3)
1. A nonlinear dynamic process monitoring method based on normalized variable nonlinear principal component analysis is characterized in that: the method comprises the following steps:
1) collecting normal operation data of Tensai chemical process of Tennessman to obtain data matrix Y ∈ R m×l Pre-specifying a numerical value of p and a system order n;
2) hankel rectangle Y for combining past observation values according to formula p Hankel matrix Y of future observations f ；
3) calculating covariance of Hankel matrix of past and future observations
Sum and mutual variance matrix
4) Performing singular value decomposition on a Hankel matrix H to obtain:
5) computing state vectors
Sum residual vector
6) Projecting a state vector to a high-dimensional feature space G-G through explicit second-order polynomial mapping 1 ,g 2 ,…,g N ] T (ii) a The method comprises the following specific steps:
the state vector is mapped to the high dimensional feature space by displaying a second order as follows:
for a two-dimensional vector, the dimensionality of the mapped state vector is increased to 5; for a vector of dimension D, the dimension of the mapped state vector is determined by the following equation:
an m-dimensional state vector x r ∈R m After the display binomial mapping, the change is g r ∈R (m(m+3)/2) ；
In the high-dimensional feature space, obtaining a matrix G epsilon R consisting of the first n state vectors N×(m(m+3)/2) ；
7) Determining top-k pivot by eigenvalue decomposition in principal component analysis
The method comprises the following specific steps:
reducing the dimensionality of the mapped state vector through conventional principal component analysis, and acquiring an l-dimensional latent variable and a D-l residual vector:
and (3) decomposing the characteristic value of the matrix S to obtain a load vector matrix V and a diagonal matrix Lambda:
S＝G T G/(N-1)＝VΛV T
wherein Λ ═ diag (λ) 1 ,λ 2 ,…,λ k )；
The matrix G is decomposed as the product of the load vector and the score vector:
wherein p is i Is a load vector t i Scoring vectors, wherein E represents a residual matrix, and the number k of principal elements is obtained by accumulating the contribution rate;
8) calculating T 2 Statistic, combination statistic Q c And its corresponding control limit; if it is not
Or Q c >Q UCL And (α), indicating that a fault was detected.
2. The nonlinear dynamic process monitoring method based on normative variable nonlinear principal component analysis according to claim 1, wherein the method comprises the following steps: past observation vector y in step 2) p,r And a future observation vector y f,r The measurement values of past and future p sampling moments in the data matrix Y are combined to form:
wherein r is p +1, p +2, …, p + N, y r Are measurements from a training data set.
3. The nonlinear dynamic process monitoring method based on normative variable nonlinear principal component analysis according to claim 1, wherein the method comprises the following steps: the method also comprises the following steps of online monitoring:
1) after the observed value of the 2p sampling moment is obtained, the past observed vector y is assembled pp ；
2) Calculating monitoring statistics
Sum combined statistics
3) The monitoring statistics do not exceed their corresponding control limits, indicating that the process is operating normally; if it is not
Or Q c >Q UCL And (α), indicating that a fault was detected.
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