CN113254874A - Uncertainty non-stationary industrial process oriented anomaly monitoring method - Google Patents
Uncertainty non-stationary industrial process oriented anomaly monitoring method Download PDFInfo
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Abstract
The invention discloses an abnormality monitoring method for an uncertain non-stationary industrial process, and particularly relates to the field of industrial process abnormality monitoring. The invention provides a probabilistic stationary subspace analysis method based on the stationary subspace analysis method by considering the process uncertainty. The method models the uncertainty explicitly and effectively separates non-stationary trends from process uncertainty. In consideration of mutual coupling among model parameters, the method utilizes an expectation maximization algorithm to conduct parameter decoupling and deduces a closed-form solution of iterative updating. Based on the model, two detection indexes are provided for anomaly monitoring under a probability framework. Compared with the existing abnormal monitoring method for the non-stationary process, the method provided by the invention eliminates the influence of uncertainty of the process and improves the detection capability of tiny faults in the non-stationary process; and the problem of overfitting of model parameters can be avoided, and a more accurate generative model is established for non-stationary data.
Description
Technical Field
The invention belongs to the field of industrial process abnormity monitoring, and particularly relates to an abnormity monitoring method for an uncertain non-stable industrial process.
Background
Anomaly monitoring is important for ensuring normal and efficient operation of industrial processes and equipment. Practical industrial processes often exhibit significant non-stationary characteristics, namely: the statistical properties of the process data may change over time. Non-stationary characteristics may be caused by a variety of factors, including raw material variations, load fluctuations, and equipment aging. The non-stationary characteristic affects the application of traditional anomaly monitoring methods such as principal component analysis and the like, and two types of errors often occur in the conventional anomaly monitoring methods. The first type of error is false alarm, that is, the influence of fault is covered by non-stationary trend, thereby resulting in a large amount of false alarm; the second category of errors is false positives, because conventional methods have difficulty tracking changes that are not stationary trends, resulting in higher false positives.
Over the past decade, non-stationary industrial process anomaly monitoring methods have grown in length. To our knowledge, these methods can be mainly divided into four categories, namely: the self-adaptive updating method is based on a method of co-integration analysis, a method of subspace decomposition and a method of trend analysis. Taking recursive principal component analysis as an example, the self-adaptive updating method establishes a data model according to training data, and then updates the parameters of the model by using normal test data. Second, the method based on co-integration analysis wants to find its stationary linear combination among the non-stationary variables. Subspace decomposition-based methods typically separate the stationary subspace from the full space, of which stationary subspace analysis methods are representative. Finally, trend analysis based methods typically extract trend information from non-stationary processes to accomplish the monitoring task.
The above methods are well applicable in anomaly monitoring of non-stationary industrial processes, however they do not take into account the uncertainty issues that exist in non-stationary processes. Actual industrial processes often suffer from various uncertainties, which may be caused by factors such as random noise and unknown disturbances. Process uncertainty presents a challenge to anomaly monitoring of non-stationary processes. On the one hand, the actual non-stationary trend is partially masked by process uncertainties, which may degrade the monitoring performance of the algorithm, especially for minor faults. On the other hand, it is difficult to distinguish true non-stationary trends from changes due to uncertainty, which leads to over-fitting problems of the model parameters. Probabilistic models provide a new view to these challenges, which strikes a good balance between the actual trends of real interest and the changes due to uncertainty. In fact, the measurement data of an industrial process is naturally presented in a statistical form, not in a deterministic manner. Currently, there are some probabilistic latent variable models that have been applied to anomaly monitoring of industrial processes, but they focus on stationary processes, not non-stationary processes.
Disclosure of Invention
In order to solve the problems, the invention provides an abnormity monitoring method for an uncertain non-stationary industrial process, which eliminates the influence of process uncertainty and improves the detection capability of tiny faults in the non-stationary process.
The technical scheme of the invention is as follows:
an abnormity monitoring method facing to uncertain non-stationary industrial process comprises an off-line training stage and an on-line monitoring stage; wherein,
A. the off-line training stage comprises the following steps:
A1. collecting historical data of equipment operation in normal working condition of non-stationary processWherein N is the number of samples in the historical data set, and m is the number of measurement variables;
A2. the equipment operation historical data is standardized to enable each variable to be in a zero-mean and unit variance standardized form, and the standardization method is shown as a formula (1):
where Λ is a diagonal matrix, the diagonal elements are made up of the standard deviations of m variables, and 1 is a column direction made up of m 1 sThe amount of the compound (A) is,is a sample mean vector of historical data;
A3. carrying out Johansen test on the standardized data matrix X, and determining the number d of stable components, wherein d is more than or equal to 1 and less than or equal to m-1;
A4. the k-th sample X (k) in the normalized data matrix X is decomposed into a probabilistic stationary subspace analysis model shown in equation (2):
wherein,is a mixing matrix that is reversible and that,consists of the preceding d columns of A,consists of the rear m-d columns of A,is a stationary component, sn(k) Is a non-stationary component and is,representing process uncertainty, which is independent of s (k), ImIs an identity matrix;
A5. the samples in the normalized data matrix X are divided into n consecutive, non-overlapping data segments, each data segment being denoted asThe number of samples of each data segment is recorded as NiAnd is provided with
A6. Setting a stationary component ss(k) And non-stationary component sn(k) Is not related, and ss(k) Gaussian distribution obeying a d dimensionIs invariant on a time scale, sn(k) Gaussian distribution obeying an m-d dimensionIt varies across data segments;
A7. combining a4 and a6, the prior distribution of s (k) is:
then, the probability density function for x (k) is:
A8. estimating parameters of a probabilistic stationary subspace model using an expectation-maximization algorithm
A9. For the kth sample x (k), its corresponding local component is calculated by equation (5):
wherein Hi=I-Σi(σ2I+ATAΣi)-1ATA,i=1,2,…,n;
A10. Calculating the similarity between the sample x (k) and the sample belonging to the ith data segment, as shown in equation (6):
A11. the estimated contribution from samples x (k) is calculated by equation (7):
A12. combining equations (2) and (7), the estimated process uncertainty is shown as equation (8):
A13. due to the fact thatIs a stationary time series, and is therefore designedStatistics monitor changes in process uncertainty, sample x (k) corresponds toThe statistic is shown as equation (9):
A14. for estimated local componentsThe mean vector and covariance matrix are calculated by equations (10) and (11), respectively:
wherein R isi=AΣiAT+σ2Im,i=1,2,…,n;
A15. Will be provided withIs recorded asThenThe mean vector and covariance matrix of the corresponding stationary components are calculated by equations (12) and (13), respectively:
wherein, W1From an identity matrix ImThe first d columns of (1);
A17. designing based on the weighted Mahalanobis distance as shown in equation (15)Statistics to monitor the change in stationary components:
A18. given a confidence level α, statistics are determined using a kernel density estimation methodAndare respectively recorded asAnd
B. the on-line monitoring stage comprises the following steps:
B1. acquiring real-time operation data y (t) of a non-stationary process of the equipment, and standardizing y (t):
wherein, yiAnd xiThe ith variable of the real-time data before and after normalization respectively,is the mean of the ith variable and is,is the ith variableStandard deviation of (d);
B2. for the real-time sample x (t) after normalization, its corresponding local component is calculated by equation (17):
wherein Hi=I-Σi(σ2I+ATAΣi)-1ATA,i=1,2,…,n;
B3. Calculating the similarity between the sample x (t) and the samples belonging to the ith data segment, as shown in equation (18):
B4. the estimated contribution from samples x (t) is calculated by equation (19):
B5. combining equations (2) and (19), the estimated process uncertainty is shown as equation (20):
B9. will make statistics ofAndrespectively in their control limitsAndby comparison, ifAnd isJudging that the industrial process is currently in a normal operation condition, otherwise, judging that the industrial process is in a normal operation conditionThe operation is abnormal.
Preferably, step A8 specifically includes:
A802. calculating the first moment and the second moment corresponding to s (k) according to the parameter value theta obtained in the last step:
〈s(k)s(k)T〉=HiΣi+〈s(k)〉〈s(k)〉T (25)
wherein Hi=I-Σi(σ2I+ATAΣi)-1ATA,i=1,2,…,n;
A803. The model parameters Θ are updated using equations (24) and (25):
wherein, W1From an identity matrix ImThe first d columns of (A) constitute (W)2From an identity matrix ImThe last m-d columns of (A);
A804. if the absolute value of the difference between the norm values of theta obtained by two iterations is less than 10-5Stopping iteration and outputting the estimated value of the model parameter theta, otherwise returning to the step A802 for next iteration.
The invention has the following beneficial technical effects:
the method provided by the invention eliminates the influence of process uncertainty and improves the detection capability of tiny faults in a non-stationary process; and the problem of overfitting of model parameters can be avoided, and a more accurate generative model is established for non-stationary data.
Drawings
FIG. 1 is a flow chart of an anomaly monitoring method for an uncertain non-stationary industrial process according to the present invention;
FIG. 2 is a schematic diagram of a closed-loop controlled continuous stirred tank reactor according to an embodiment of the invention;
FIG. 3 shows a synergistic analysis-T in example 1 of the present invention2A graph of the monitored results of the statistics;
FIG. 4 is a graph showing the results of monitoring the Q statistic of the recursive principal component analysis in example 1 of the present invention;
FIG. 5 shows T of recursive principal component analysis in example 1 of the present invention2A graph of the monitored results of the statistics;
FIG. 6 is a graph showing the results of stationary subspace analysis-Mahalanobis distance monitoring in example 1 of the present invention;
FIG. 7 is T of probability stationary subspace analysis in embodiment 1 of the present inventione 2A graph of the monitored results of the statistics;
FIG. 8 is T of probability stationary subspace analysis in embodiment 1 of the present inventions 2A graph of the monitored results of the statistics;
FIG. 9 shows a co-integration analysis-T in example 2 of the present invention2Of statisticsA monitoring result graph;
FIG. 10 is a graph showing the results of monitoring the Q statistic of the recursive principal component analysis in example 2 of the present invention;
FIG. 11 shows T of recursive principal component analysis in embodiment 2 of the present invention2A graph of the monitored results of the statistics;
FIG. 12 is a graph showing the results of stationary subspace analysis-Mahalanobis distance monitoring in example 2 of the present invention;
FIG. 13 is T of probability stationary subspace analysis in embodiment 2 of the present inventione 2A graph of the monitored results of the statistics;
FIG. 14 is T of probability stationary subspace analysis in embodiment 2 of the present inventions 2And (5) a monitoring result graph of the statistics.
Detailed Description
The invention is described in further detail below with reference to the following figures and detailed description:
as shown in fig. 1, an anomaly monitoring method for uncertain non-stationary industrial processes includes an offline training phase and an online monitoring phase; wherein,
A. the off-line training stage comprises the following steps:
A1. collecting historical data of equipment operation in normal working condition of non-stationary processWherein N is the number of samples in the historical data set, and m is the number of measurement variables;
A2. the equipment operation historical data is standardized to enable each variable to be in a zero-mean and unit variance standardized form, and the standardization method is shown as a formula (1):
where Λ is a diagonal matrix, the diagonal elements consist of the standard deviations of m variables, 1 is a column vector consisting of m 1's,is a sample mean vector of historical data;
A3. carrying out Johansen test on the standardized data matrix X, and determining the number d of stable components, wherein d is more than or equal to 1 and less than or equal to m-1;
A4. it is assumed that the kth sample X (k) in the normalized data matrix X can be decomposed into the form shown in equation (2):
this model is called a probabilistic stationary subspace analysis model, where,is a mixing matrix that is reversible and that,consists of the preceding d columns of A,consists of the rear m-d columns of A,is a stationary component, sn(k) Is a non-stationary component and is,representing process uncertainty, which is independent of s (k), ImIs an identity matrix;
A5. the samples in the normalized data matrix X are divided into n consecutive, non-overlapping data segments, each data segment being denoted asThe number of samples of each data segment is recorded as NiAnd is provided with
A6. Hypothesis of stationarityIngredient ss(k) And non-stationary component sn(k) Is not related, and ss(k) Gaussian distribution obeying a d dimensionIs invariant on a time scale, sn(k) Gaussian distribution obeying an m-d dimensionIt varies across data segments;
A7. the prior distribution of s (k) is known from the joint assumptions in A4 and A6 as:
then, the probability density function for x (k) is:
A8. estimating parameters of a probabilistic stationary subspace model using an expectation-maximization algorithm
A9. For the kth sample x (k), its corresponding local component is calculated by equation (5):
wherein Hi=I-Σi(σ2I+ATAΣi)-1ATA,i=1,2,…,n;
A10. Calculating the similarity between the sample x (k) and the sample belonging to the ith data segment, as shown in equation (6):
A11. the estimated contribution from samples x (k) can be calculated by equation (7):
A12. combining equations (2) and (7), the estimated process uncertainty is shown as equation (8):
A13. due to the fact thatIs a smooth time sequence and can be designed accordinglyStatistics monitor changes in process uncertainty, sample x (k) corresponds toThe statistic is shown as equation (9):
A14. for estimated local componentsThe mean vector and covariance matrix can be calculated by equations (10) and (11), respectively:
wherein R isi=AΣiAT+σ2Im,i=1,2,…,n;
A15. Will be provided withIs recorded asThenThe mean vector and covariance matrix of the corresponding stationary components can be calculated by equations (12) and (13), respectively:
wherein, W1From an identity matrix ImThe first d columns of (1);
A17. designing based on the weighted Mahalanobis distance as shown in equation (15)Statistics to monitor the change in stationary components:
A18. given a confidence level α, statistics are determined using a kernel density estimation methodAndare respectively recorded asAnd
B. the on-line monitoring stage comprises the following steps:
B1. acquiring real-time operation data y (t) of a non-stationary process of the equipment, and standardizing y (t):
wherein, yiAnd xiThe ith variable of the real-time data before and after normalization respectively,is the mean of the ith variable and is,is the ith variableStandard deviation of (d);
B2. for the real-time sample x (t) after normalization, its corresponding local component is calculated by equation (17):
wherein Hi=I-Σi(σ2I+ATAΣi)-1ATA,i=1,2,…,n;
B3. Calculating the similarity between the sample x (t) and the samples belonging to the ith data segment, as shown in equation (18):
B4. the estimated contribution from samples x (t) can be calculated by equation (19):
B5. combining equations (2) and (19), the estimated process uncertainty is shown as equation (20):
B9. will make statistics ofAndrespectively in their control limitsAndby comparison, ifAnd isJudging that the industrial process is currently in a normal operation condition, otherwise judgingAn operational anomaly is identified.
The specific content of the step A8 includes:
A802. calculating the first moment and the second moment corresponding to s (k) according to the parameter value theta obtained in the last step:
<s(k)s(k)T>=HiΣi+<s(k)〉〈s(k)>T (25)
wherein Hi=I-Σi(σ2I+ATAΣi)-1ATA,i=1,2,…,n;
A803. The model parameters Θ are updated using equations (24) and (25):
wherein, W1From an identity matrix ImThe first d columns of (A) constitute (W)2From an identity matrix ImThe last m-d columns of (A);
A804. if the absolute value of the difference between the norm values of theta obtained by two iterations is less than 10-5Stopping iteration and outputting the estimated value of the model parameter theta, otherwise returning to the step A802 for next iteration.
In order to help understand the invention and simultaneously visually show the effect of the method for monitoring the abnormal state of the uncertain non-stationary industrial process, the following description is based on two embodiments. The data for both examples was derived from a closed loop controlled continuous stirred tank reactor which provides a library of standard models widely used in the field of monitoring of industrial process anomalies. A schematic of the reactor is shown in fig. 2, where a first order exothermic reaction is performed. Its dynamic model is shown in equation (32):
wherein C is the reaction concentration, T is the reaction temperature, TcIs the temperature of cold water, QcIs the cold water flow, viIs measuring noise, Ci、TiAnd TciConstituting the system input.
Since the original continuous stirred tank reactor is a stationary process, system inputs need to be modified in order to simulate non-stationary characteristics. In the modified version, TciIs switched to a steady state, CiAnd TiAdding a non-stationary drift term lambda:
wherein,andrespectively represent CiAnd TiThe nominal value of (a) is,andrepresenting two gaussian random variables. Thus, the continuous stirred tank reactor exhibits non-stationary characteristics.
In the off-line training phase, 1200 normal samples were collected with a1 minute sampling interval. Each sample consists of 7 measured variables, i.e.: x ═ Ci,Ti,C,T,Qc,Tci,Tc]T. In addition to the method proposed by the present invention, three other monitoring algorithms are also added for comparison, respectively, a method based on co-integration analysis, a recursive principal component analysis method and a method based on stationary subspace analysis. Wherein the method based on the co-integration analysis combines the co-integration analysis with T2The statistics are combined, and the co-integration order is selected as r-5. For the recursive pivot analysis method, the pivot is selected based on an accumulated contribution rate of 85%. For stationary subspace analysis and probabilistic stationary subspace analysis, the number of stationary components is set to d-5, and the number of data segment partitions is set to n-19. In addition, the iteration error of the probabilistic stationary subspace analysis method is assumed to be ∈ 1 × 10-6. Given a confidence level α of 0.99, the control limits for all monitoring statistics are determined by the kernel density estimation method.
TABLE 1 two typical failures in a continuous stirred tank reactor
In both embodiments of the invention, two typical failures in a continuous stirred tank reactor were considered separately, including multiplicative and additive failures, as shown in table 1. For both faults, 1200 samples were collected for performance comparison of the algorithm. The fault is introduced starting from the 601 st sample and continuing until the end.
Example 1
The heat transfer coefficient degradation failure was tested in example 1 and the monitoring results of the different algorithms are summarized in fig. 3-8. The fault is a typical tiny fault, and in the early stage of the fault, three methods based on the synergistic analysis, the stationary subspace analysis and the probabilistic stationary subspace analysis all show the false negatives in different degrees, wherein the probabilistic stationary subspace analysisThe statistic has the shortest detection delay as shown in fig. 7.The statistic has the advantage of containing non-stationary information. In addition, as shown in fig. 4 and 5, the recursive principal component analysis method alarms early in the early stage of a fault, but leaves many false alarms as the magnitude of the fault increases. This reflects the failure of the recursive principal component analysis method to discern subtle faults from non-stationary trends.
Example 2
An additive sensor constant offset fault was tested in example 2, which occurred in magnitude 1 in the reaction temperature variation. This is a minor fault of small magnitude compared to the nominal value of the reaction temperature of 430 c. In this example, the monitoring results of the four different methods are shown in fig. 9-14. As can be seen from fig. 9 and 12, the method based on both the co-integration analysis and the stationary subspace analysis can only partially detect the fault, but leaves a certain degree of false negative. Neither of these methods models non-stationary components, thus resulting in unsatisfactory monitoring performance for minor faults. In addition, the recursive principal component analysis method can hardly detect the fault (see fig. 10 and 11) because the fault magnitude is too small to be masked by the non-stationary trend. For probabilistic stationary subspace analysis, itThe detection effect of statistics is similar to that of the co-integration analysis and stationary subspace analysis, andthe statistics are able to effectively detect the fault at a detection rate of 97.8%. This reflects that the true stationary trend is separated from the process uncertainty, which can effectively improve the algorithm's detection capability for minor faults.
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 (2)
1. An abnormity monitoring method facing to uncertain non-stationary industrial process is characterized by comprising an off-line training stage and an on-line monitoring stage; wherein,
A. the off-line training stage comprises the following steps:
A1. collecting historical data of equipment operation in normal working condition of non-stationary processWherein N is the number of samples in the historical data set, and m is the number of measurement variables;
A2. the equipment operation historical data is standardized to enable each variable to be in a zero-mean and unit variance standardized form, and the standardization method is shown as a formula (1):
where Λ is a diagonal matrix, the diagonal elements consist of the standard deviations of m variables, 1 is a column vector consisting of m 1's,is a sample mean vector of historical data;
A3. carrying out Johansen test on the standardized data matrix X, and determining the number d of stable components, wherein d is more than or equal to 1 and less than or equal to m-1;
A4. the k-th sample X (k) in the normalized data matrix X is decomposed into a probabilistic stationary subspace analysis model shown in equation (2):
wherein,is a mixing matrix that is reversible and that,consists of the preceding d columns of A,consists of the rear m-d columns of A,is a stationary component, sn(k) Is a non-stationary component and is,representing process uncertainty, which is independent of s (k), ImIs an identity matrix;
A5. the samples in the normalized data matrix X are divided into n consecutive, non-overlapping data segments, each data segment being denoted asThe number of samples of each data segment is recorded as NiAnd is provided with
A6. Setting a stationary component ss(k) And non-stationary component sn(k) Is not related, and ss(k) Gaussian distribution obeying a d dimensionIs invariant on a time scale, sn(k) Gaussian distribution obeying an m-d dimensionIt varies across data segments;
A7. combining a4 and a6, the prior distribution of s (k) is:
then, the probability density function for x (k) is:
A8. estimating parameters of a probabilistic stationary subspace model using an expectation-maximization algorithm
A9. For the kth sample x (k), its corresponding local component is calculated by equation (5):
wherein Hi=I-Σi(σ2I+ATAΣi)-1ATA,i=1,2,…,n;
A10. Calculating the similarity between the sample x (k) and the sample belonging to the ith data segment, as shown in equation (6):
A11. the estimated contribution from samples x (k) is calculated by equation (7):
A12. combining equations (2) and (7), the estimated process uncertainty is shown as equation (8):
A13. due to the fact thatIs a stationary time series, and is therefore designedStatistics monitor changes in process uncertainty, sample x (k) corresponds toThe statistic is shown as equation (9):
A14. for estimated local componentsThe mean vector and covariance matrix are calculated by equations (10) and (11), respectively:
wherein R isi=AΣiAT+σ2Im,i=1,2,…,n;
A15. Will be provided withIs recorded asThenThe mean vector and covariance matrix of the corresponding stationary components are calculated by equations (12) and (13), respectively:
wherein, W1From an identity matrix ImThe first d columns of (1);
A17. designing based on the weighted Mahalanobis distance as shown in equation (15)Statistics to monitor the change in stationary components:
A18. given a confidence level α, statistics are determined using a kernel density estimation methodAndare respectively recorded asAnd
B. the on-line monitoring stage comprises the following steps:
B1. acquiring real-time operation data y (t) of a non-stationary process of the equipment, and standardizing y (t):
wherein, yiAnd xiThe ith variable of the real-time data before and after normalization respectively,is the mean of the ith variable and is,is the standard deviation of the ith variable;
B2. for the real-time sample x (t) after normalization, its corresponding local component is calculated by equation (17):
wherein Hi=I-Σi(σ2I+ATAΣi)-1ATA,i=1,2,…,n;
B3. Calculating the similarity between the sample x (t) and the samples belonging to the ith data segment, as shown in equation (18):
B4. the estimated contribution from samples x (t) is calculated by equation (19):
B5. combining equations (2) and (19), the estimated process uncertainty is shown as equation (20):
2. The anomaly monitoring method oriented to the uncertain non-stationary industrial processes as claimed in claim 1, wherein said step A8 specifically comprises:
A802. calculating the first moment and the second moment corresponding to s (k) according to the parameter value theta obtained in the last step:
<s(k)s(k)T>=HiΣi+<s(k)><s(k)>T (25)
wherein Hi=I-Σi(σ2I+ATAΣi)-1ATA,i=1,2,…,n;
A803. The model parameters Θ are updated using equations (24) and (25):
wherein, W1From an identity matrix ImThe first d columns of (A) constitute (W)2From an identity matrix ImThe last m-d columns of (A);
A804. if the absolute value of the difference between the norm values of theta obtained by two iterations is less than 10-5Stopping iteration and outputting the estimated value of the model parameter theta, otherwise returning to the step A802 for next iteration.
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