A non-stationary industrial process monitoring method and system

Abstract

The application provides a non-stationary industrial process monitoring method and system, and the method comprises an offline training stage: a time Laplacian matrix and a space Laplacian matrix are calculated based on a historical data matrix; a target function of a stationary subspace analysis method is constructed, and a time constraint term of the time Laplacian matrix and a space constraint term of the space Laplacian matrix are added to the target function; the target function is solved to obtain a stationary projection matrix; stationary components and monitoring indexes of each sample in the data matrix X are calculated in turn based on the data matrix X; a control limit is determined by using a kernel density estimation method; and an online monitoring stage: based on real-time running data x, stationary components of the real-time running data x and corresponding real-time monitoring indexes are calculated according to the stationary projection matrix, and if the real-time monitoring indexes are greater than the control limit, it is judged that a non-stationary process operation has a fault; and the application can improve monitoring accuracy.

Description

Technical Field

[0001] This invention relates to the field of industrial process monitoring technology, and in particular to a method and system for monitoring non-stationary industrial processes. Background Technology

[0002] Actual industrial processes often exhibit strong non-stationary characteristics due to factors such as changing operating conditions, equipment aging, and external interference. Specifically, the mean and variance of data collected by sensors change over time. This characteristic hinders the application of traditional monitoring methods and easily leads to two types of problems:

[0003] 1. Real faults are easily masked by non-stationary trends, which will lead to the failure to detect real faults in a timely manner, resulting in a large number of missed faults, posing potential safety hazards to production equipment and operators, and hindering the normal operation of industrial production systems.

[0004] 2. Traditional monitoring methods cannot track actual non-stationary trends, which can lead to normally operating equipment being incorrectly identified as faulty equipment, resulting in unacceptable false alarms, adding extra inspection and maintenance tasks for maintenance personnel, and disrupting normal industrial production processes.

[0005] In recent years, many scholars have conducted in-depth research on the monitoring of non-stationary processes, which can be mainly divided into four aspects;

[0006] Adaptive modeling: Some studies have introduced recursive strategies to optimize traditional principal component analysis, forming recursive principal component analysis to adapt to dynamic changes in the process; other methods use sliding windows combined with kernel principal component analysis to achieve real-time monitoring of non-stationary processes.

[0007] Cointegration analysis: There are technical solutions that combine alternating conditional expectation with cointegration analysis for monitoring non-stationary sequences; for multimodal data generated by multiple operating conditions, an adaptive cointegration analysis method with continuous learning capability has been developed; in addition, sparse cointegration analysis is used to achieve complete decomposition of non-stationary sequences and extract stationary features for monitoring.

[0008] Stationary subspace analysis: An analytical stationary subspace analysis method is proposed to effectively reduce computational complexity; further, it is combined with the exponential analytical method to improve the monitoring efficiency of non-stationary processes; for process uncertainty, probabilistic stationary subspace analysis is adopted to separate the uncertain components in non-stationary trends and enhance the early fault detection capability.

[0009] Trend analysis: The method based on cointegration and common trend analysis is applied to the monitoring of non-stationary processes, and the process state is identified by extracting common trend features.

[0010] The methods described above, from different perspectives, have been well applied in monitoring various non-stationary industrial processes. However, while eliminating the negative impact of non-stationarity, these methods do not consider the local spatiotemporal correlations between process data. Considering only global stationarity while ignoring the local spatiotemporal correlations between process data may lead to erroneous monitoring results or make it difficult to achieve accurate detection in the early stages of a fault, thus affecting the performance of the monitoring algorithm and failing to effectively monitor actual industrial processes.

[0011] Therefore, there is an urgent need for a non-stationary industrial process monitoring method and system that considers the local spatiotemporal correlation between process data, which can improve monitoring accuracy and effectively monitor industrial processes. Summary of the Invention

[0012] The purpose of this invention is to provide a method and system for monitoring non-stationary industrial processes, aiming to solve the technical problem that traditional non-stationary process monitoring methods suffer from local spatiotemporal information loss and cannot accurately and effectively monitor industrial processes.

[0013] To achieve the above objectives, in a first aspect, the present invention provides a method for monitoring non-stationary industrial processes, the steps of which include an offline training phase and an online monitoring phase;

[0014] During the offline training phase: Historical data matrix X is collected under normal operation during non-stationary dynamic processes, and the time Laplacian matrix is ​​calculated based on the data matrix X. And space Laplace matrix ,in n is the number of samples, and m is the number of dimensions. Represents the real number field;

[0015] Construct an objective function for a stationary subspace analysis method, and incorporate the time Laplacian matrix into the objective function. Time constraints and space-based Laplace matrix Spatial constraints;

[0016] The objective function is solved using generalized eigenvalue decomposition to obtain the stationary projection matrix. ;

[0017] based on For each sample in the data matrix X, calculate its stationary component and monitoring index sequentially. Stable component ,in, , express The sample points corresponding to the dimensionality reduction, where d is the number of stationary components;

[0018] Determining control limits using kernel density estimation methods ;

[0019] Online monitoring phase: Based on the real-time running data x of the non-stationary process, and the stationary projection matrix obtained during the offline training phase... Calculate the stationary component of real-time running data x and calculation If the corresponding real-time monitoring indicator is greater than the control limit, If the non-stationary process is faulty, then the corresponding sample point is the fault sample point.

[0020] As a further improvement to the above scheme, the objective function of the stationary subspace analysis method is specifically shown in the following equation:

[0021] ;

[0022] in, For the stationary projected component to be solved, the S matrix is ​​used as a stationarity metric matrix to reflect the degree of shift in the statistical characteristics of non-stationary data among different subsequences. The S matrix is ​​shown in the following equation:

[0023] ;

[0024] and These are the mean vector and covariance matrix of each subsequence group, respectively. , , , These are empirical coefficients used to balance stationary features and local spatiotemporal similarity, where N is the total number of subsequences divided when segmenting the historical data sequence. It represents a d-dimensional identity matrix.

[0025] As a further improvement to the above scheme, the objective function The time-based Laplace matrix The steps to obtain the time constraint are as follows:

[0026] Constructing a temporal adjacency matrix for non-stationary data Preferably, the temporal adjacency matrix is ​​established using a Gaussian kernel function and the K-nearest neighbor (KNN) method, and the temporal adjacency matrix... The specific formula is as follows:

[0027] ;

[0028] in, Represents sample points The set of K nearest neighbors, This represents an adjustable heat kernel parameter, a scale factor used to control the rate of decay of similarity between samples;

[0029] The result of dimensionality reduction and To minimize the distance between the data after dimensionality reduction Keep the original data The local time structure is used to obtain the objective function of the time constraint term. The specific formula is as follows:

[0030] ;

[0031] in, Let be the time-degree matrix, and let be a diagonal matrix, with each element on the diagonal being... The diagonal elements reflect the total correlation strength between a sample point and its neighbors in the time domain; the matrix Let be the time Laplace matrix.

[0032] As a further improvement to the above scheme, the objective function Based on the spatial Laplace matrix The steps for obtaining spatial constraint terms are as follows:

[0033] Constructing a spatial adjacency matrix for nonstationary data (Preferredly, the spatial adjacency matrix is ​​established using a Gaussian kernel function and the K-nearest neighbor (KNN) method), and the spatial adjacency matrix... The specific formula is as follows:

[0034] ;

[0035] in, Representing dimensions The set of K nearest neighbors, This indicates adjustable thermonuclear parameters;

[0036] The local spatial relationships in the original data are integrated into a linear representation of the original dimension on the stationary dimension. Specifically, , Then the stationary component It can be represented as: ;

[0037] Dimensionally reduced data Keep the original data The local spatial structure minimizes the loss of local spatial information in the original data during the dimensionality reduction process;

[0038] make and Minimize the distance between them to minimize the loss of local spatial information in the original data during dimensionality reduction; obtain the objective function of the spatial constraint term. The specific formula is as follows:

[0039] ;

[0040] in, Let be the spatial degree matrix, and let be a diagonal matrix, where each element on the diagonal is . ,matrix Let be the space Laplace matrix.

[0041] As a further improvement to the above scheme, the steps for solving the objective function are as follows:

[0042] For the objective function Using the Lagrange multiplier method, the corresponding Lagrange function is obtained, as shown in the following equation:

[0043] ;

[0044] in It is a Lagrange multiplier matrix;

[0045] The Lagrange function with respect to The first-order partial derivatives, after merging and simplification, are shown in the following equation:

[0046] (1);

[0047] Setting equation (1) to zero transforms the problem of solving the spatiotemporally stationary subspace analysis method into a generalized eigenvalue decomposition problem.

[0048] The solution of the spatiotemporally stationary subspace analysis method is shown in the following equation:

[0049] (2);

[0050] The eigenvalues ​​obtained according to equation (2) represent the strength of nonstationarity of different features; the larger the eigenvalue, the stronger the nonstationarity. The obtained eigenvalues ​​are arranged in ascending order. From the front The matrix composed of the eigenvectors corresponding to the eigenvalues ​​is the stationary projection matrix to be solved. The details are as follows:

[0051] ;

[0052] in, For generalized eigenvectors, it represents the projection direction when projecting the original high-dimensional industrial data onto a stationary subspace. Each Each corresponds to a feature value This vector allows the raw observation data from multiple sensors to be linearly combined into a feature component with a specific level of "stationarity". Let d be the d-th stationary projection basis vector, representing the last basis vector that constitutes the "stationary subspace".

[0053] As a further improvement to the above scheme, monitoring indicators for whether a fault has occurred can be obtained, including but not limited to those based on Mahalanobis distance (such as squared prediction error and Hotelling statistic).

[0054] The monitoring indicators obtained based on Mahalanobis distance are shown in the following formula:

[0055] ;

[0056] in, For the stationary components of the data points, and Refers to the stationary component The mean and covariance.

[0057] As a further improvement to the above scheme, control limits The result is obtained from equation (3), as shown in the following equation:

[0058] (3);

[0059] in, For confidence level, Let be the probability density function. , where n represents the total number of data points and h represents the bandwidth value.

[0060] Secondly, the present invention also provides a non-stationary industrial process monitoring system, including a memory and a processor, wherein the memory stores a computer program, and when the computer program is executed by the processor, the processor enables the processor to implement a non-stationary industrial process monitoring method as provided in the first aspect.

[0061] Because the present invention adopts the above technical solutions, the beneficial effects of this application are as follows:

[0062] This invention provides a method for monitoring non-stationary industrial processes. By adding a Laplace matrix constraint term to the objective function of the analytical stationary subspace analysis method, global stationarity and neighborhood relationships between data are preserved. Specifically, traditional stationary subspace analysis methods are prone to losing the temporal continuity and spatial correlation of data during dimensionality reduction, resulting in incomplete characterization of the data characteristics of non-stationary processes. This invention achieves precise preservation of spatiotemporal information in data by introducing dual constraint terms of a temporal Laplace matrix and a spatial Laplace matrix into the objective function. On the one hand, the temporal Laplace matrix is ​​constructed based on the temporal correlation of the data, and its constraint term forces the stationary components after dimensionality reduction to maintain the time-dependent structure of the original data, effectively capturing the implicit global stationary features in non-stationary processes, such as long-term trends and periodic fluctuations. On the other hand, the spatial Laplace matrix is ​​constructed through similarity measures of neighboring variables, and its constraint term ensures that the spatial adjacency relationships of neighboring variables after dimensionality reduction are aligned with the original data, preserving local spatial correlations, such as propagation delays between sensors and regional coupling effects. The synergistic effect of these two constraints ensures that the dimensionality-reduced data retains both the global dynamic evolution law and fully transmits local neighborhood interaction information, significantly improving the information utilization rate and feature expression capability of process data.

[0063] To address the prevalent spatiotemporal correlations in non-stationary processes, such as the gradual change in material concentration over time and the gradient distribution of spatial location in a continuous stirred tank reactor, this invention overcomes the limitations of traditional methods that only focus on a single dimension by employing a spatiotemporal constraint mechanism based on Laplace eigenmaps. Specifically, the time constraint term aligns the temporal correlations before and after dimensionality reduction through generalized eigenvalue decomposition, ensuring that the stationary components not only reflect the stationarity of a single variable time series but also capture the temporal synchronicity between multiple variables. The spatial constraint term, by restricting the adjacency of neighboring variables in the dimensionality-reduced space, preserves the cooperative change characteristics caused by spatial proximity in the original data, such as the temperature linkage between different measuring points in the same reactor. The stationary components extracted by this method can simultaneously characterize global stationary trends (such as steady-state fluctuations in reaction temperature) and local spatiotemporal anomalies (such as gradient mutations caused by local hotspots), providing more comprehensive feature inputs for subsequent fault monitoring.

[0064] In step fault monitoring and multiplicative fault monitoring scenarios, traditional methods often face the contradiction of "insufficient sensitivity" and "high false alarm rate": over-focusing on local anomalies makes them susceptible to interference from normal production disturbances, while relaxing the threshold may lead to missed fault detection. This invention achieves synergistic optimization of both by extracting stationary components under a spatiotemporal constraint mechanism: on the one hand, the preservation of global stationary features enables the method to identify early weak faults and improve detection timeliness; on the other hand, the constraint of local neighborhood relationships reduces the probability of random disturbances being misjudged as faults. Experimental data show that in the simulation model of a continuous stirred tank reactor, compared with traditional fault monitoring methods, the method provided by this invention can improve the early fault detection rate while maintaining a low false alarm rate.

[0065] This invention, through the deep integration of Laplace eigenmaps and analytical stationary subspace analysis, breaks through the limitations of traditional methods in preserving spatiotemporal information and extracting stationary features. It significantly improves the accuracy, timeliness, and engineering applicability of monitoring non-stationary industrial processes, and has significant technological innovation value and promising industrial application prospects. Attached Figure Description

[0066] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0067] Figure 1 This is a schematic flowchart of a non-stationary industrial process monitoring method disclosed in this invention;

[0068] Figure 2 This is a schematic diagram illustrating the monitoring performance of step faults introduced by the stationary subspace analysis method disclosed in this invention.

[0069] Figure 3 This is a schematic diagram illustrating the monitoring performance of step faults introduced by the cointegration analysis method disclosed in this invention.

[0070] Figure 4 This is a schematic diagram illustrating the monitoring performance of step faults introduced by the spatiotemporally stable subspace analysis method proposed in this invention.

[0071] Figure 5 This is a schematic diagram illustrating the monitoring performance of step faults introduced by the analytical stationary subspace analysis method disclosed in this invention.

[0072] Figure 6 This is a schematic diagram illustrating the monitoring performance of the stationary subspace analysis method disclosed in this invention, which introduces multiplicative faults.

[0073] Figure 7 This is a schematic diagram illustrating the monitoring performance of multiplicative faults introduced by the cointegration analysis method disclosed in this invention.

[0074] Figure 8 This is a schematic diagram illustrating the monitoring performance of the spatiotemporally stationary subspace analysis method disclosed in this invention, which introduces multiplicative faults.

[0075] Figure 9 This is a schematic diagram illustrating the monitoring performance of the analytical stationary subspace analysis method disclosed in this invention, which introduces multiplicative faults.

[0076] The realization of the objective, functional characteristics and advantages of the present invention will be further explained in conjunction with the embodiments and with reference to the accompanying drawings. Detailed Implementation

[0077] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0078] It should be noted that the technical solutions of the various embodiments of the present invention can be combined with each other, but only if they are based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0079] Example 1

[0080] Traditional stationary subspace analysis methods can extract stationary components from non-stationary data from a global perspective, thus describing the inherent stationary long-term equilibrium relationship of non-stationary processes. However, traditional stationary subspace analysis methods do not consider the spatiotemporal information hidden in the data from a local perspective, thereby ignoring the local similarity between non-stationary data samples. The local structure in non-stationary data represents the similarity between different sample points in the time dimension and the similarity between different variables in the spatial dimension, which is the inherent geometric structure of non-stationary data. Local temporal structure refers to the correlation of the same measured variable at different time points, that is, the correlation between the current value of the measured variable and its historical and future values; this is determined by the characteristics of industrial systems themselves. Local spatial structure refers to the correlation between different measured variables. For example, multiple reactant concentration measurement points on the same floor of an industrial device reflect the reactant concentrations at different locations on the same floor of the device, and these measurement point data have strong spatial correlations.

[0081] Local structure contains local details of the process, such as information on noise and outliers, and the gradual variation effect of fault amplitude. Local structure can expand and supplement global information, thereby further revealing the microscopic mechanism of non-stationary processes. If only global stationarity is considered while ignoring the local spatiotemporal correlations between process data, it may lead to erroneous monitoring results or make it difficult to achieve accurate detection in the early stages of fault occurrence, affecting the performance of monitoring algorithms. Therefore, in the monitoring of non-stationary processes, accurately capturing the local structure of data is crucial. It can effectively improve the fault detection capability of the model, thereby ensuring the safe operation of industrial processes and the safety of operators. Simultaneously considering both global and local information and finding a suitable balance between the two will have a beneficial impact on improving the monitoring performance of the model.

[0082] Based on this inventive motivation, see [link / reference] Figure 1 This invention provides a method for monitoring non-stationary industrial processes, which is divided into an offline training stage and an online monitoring stage. The specific steps are as follows:

[0083] S1. In the offline training phase - model building and parameter calibration:

[0084] S11. Historical data collection and preprocessing;

[0085] Collect historical data matrices under normal operation during non-stationary dynamic processes, and construct a data matrix. , Where n is the number of samples, i.e., the multivariate observations at one time point for each sample, and m is the number of dimensions, such as the number of sensors. Represents the real number field;

[0086] Data preprocessing includes:

[0087] Outlier removal: Use box-line method or Z-score method to remove obviously outlier observation points;

[0088] Normalization: In actual production activities, the units of measurement and range of variation of different observed variables may differ. This difference may cause the trend of low-amplitude characteristics to be masked by high-amplitude characteristics. At the same time, excessively large data scales can also lead to excessive computational costs for monitoring algorithms. Normalization can eliminate these adverse effects, giving the data similar scale and distribution characteristics. Therefore, before applying spatiotemporal stationary subspace analysis methods and systems, it is necessary to normalize the training and test datasets. The corresponding normalization process can be represented as follows:

[0089] ;

[0090] in, This represents the raw data to be processed. This represents the mean vector composed of the average values ​​of variables in each dimension of the training data. It is a diagonal matrix and vector composed of the standard deviations of the dimensional variables in the training data. It consists of 1 element, and the length of the vector is the same as the number of samples.

[0091] S12, Construction of the spacetime Laplace matrix;

[0092] To capture the temporal correlation and spatial coupling of process dynamics, the time Laplacian matrix was calculated separately. With the space Laplace matrix ;

[0093] Specifically, the time Laplace matrix The calculation process is as follows:

[0094] Assuming the adjacent sampling interval of the time series is Δt, construct the time adjacency matrix for non-stationary data. (Preferredly, the temporal adjacency matrix is ​​constructed using a Gaussian kernel function and the K-nearest neighbor (KNN) method), and the temporal adjacency matrix... The specific formula is as follows:

[0095] ;

[0096] in, Represents sample points The set of K nearest neighbors, This indicates adjustable thermonuclear parameters;

[0097] Time Matrix It is a diagonal matrix, and each element on the diagonal is... Then the time Laplace matrix is ;

[0098] Spatial Laplace Matrix The calculation process is as follows:

[0099] Constructing a spatial adjacency matrix for nonstationary data (Preferredly, the spatial adjacency matrix is ​​established using a Gaussian kernel function and the K-nearest neighbor (KNN) method), and the spatial adjacency matrix... The specific formula is as follows:

[0100] ;

[0101] in, Representing dimensions The set of K nearest neighbors, This indicates adjustable thermonuclear parameters;

[0102] Spatial degree matrix It is a diagonal matrix, and each element on the diagonal is... Then the space Laplace matrix is ;

[0103] S13. Objective Function Construction and Optimization

[0104] The objective of traditional stationary subspace analysis is to extract stationary components from the original data. This invention introduces a spatiotemporal constraint term to enhance the discriminative power of stationary components and constructs an improved objective function. The specific formula is as follows:

[0105] ;

[0106] in, The stationary projective component to be solved is represented by the matrix shown below:

[0107] ;

[0108] and These are the mean vector and covariance matrix of each subsequence group, respectively. , ; , All are greater than 0, representing spatiotemporal constraint weight parameters used to balance stationary features and local spatiotemporal similarity; Represents the trace of a matrix;

[0109] Improve the objective function The first term is the reconstruction error term, which forces the projected data to retain key information; the second term (time constraint) is determined by... Penalizing abrupt changes in the time dimension promotes temporal smoothness; the third term (spatial constraint) is achieved through... Punishing outliers in spatial dimensions promotes consistency among sensors.

[0110] S14. Solving for the stationary projection matrix:

[0111] The improved objective function is solved by generalized eigenvalue decomposition. The objective function is transformed into a generalized eigenvalue problem:

[0112] Sort the obtained feature values ​​in ascending order. From the front The matrix composed of the eigenvectors corresponding to the eigenvalues ​​is the stationary projection matrix to be solved. .

[0113] S15. Calculation of stationary components, calculation of monitoring indicators, and determination of control limits:

[0114] Stationary component extraction for standardized data matrices Projection yields the stationary component matrix of each sample. , , where y i ∈R d Let i be the stationary component of the i-th sample;

[0115] The monitoring indicators are calculated using, but are not limited to, indicators based on Mahalanobis distance (such as squared prediction error and Hotelling statistic) to determine whether a fault has occurred. In this embodiment, Mahalanobis distance is used to establish the monitoring indicators for the sequence. Mahalanobis distance is a commonly used distance metric in statistics and pattern recognition. Unlike Euclidean distance, Mahalanobis distance uses a covariance matrix in its calculation, taking into account the correlation between data features, and can well reflect the distance between a stationary source and its center.

[0116] The monitoring indicators obtained based on Mahalanobis distance are shown in the following formula:

[0117] ;

[0118] in, For the stationary components of the data points, and Refers to the stationary component The mean and covariance;

[0119] After determining the control limits, the probability density function of the stationary component is fitted using the kernel density estimation method. Specifically, kernel density estimation is a non-parametric probability density method. Its basic idea is to superimpose a set of kernel functions on each data point to smooth the dataset and estimate its probability density, as shown in the following formula:

[0120] ;

[0121] Where n represents the total number of data, h represents the bandwidth value, and K(•) represents the kernel function. In this embodiment, the Gaussian kernel function is used as its kernel function.

[0122] Give the specific confidence level Then, the estimated control limits can be obtained from the estimated probability density curve. Control Limits The specific formula is as follows:

[0123] .

[0124] S2, Online Monitoring Phase - Real-time Fault Detection:

[0125] S21. Real-time data acquisition and preprocessing:

[0126] Obtain real-time running data x∈R of a non-stationary processm (Single-time-point multivariate observations) were obtained using the same standardization method as the offline phase. .

[0127] S22. Real-time stationary component calculation:

[0128] Using the stationary projection matrix obtained through offline training Calculate the stationary components of real-time data:

[0129] = ∈R d ;

[0130] S23. Real-time monitoring indicator calculation:

[0131] Mahalanobis distance was used as a real-time monitoring indicator. :

[0132] ;

[0133] in and This refers to training steady components. The mean and covariance;

[0134] S24. Fault diagnosis:

[0135] like > If the current real-time data is not found to be faulty, the corresponding sample point is determined to be a faulty sample point, and a fault alarm is output; otherwise, it is determined to be in a normal state.

[0136] This invention preserves global stationarity and neighborhood relationships between data by adding a Laplace matrix constraint term to the objective function of the analytical stationary subspace analysis method. Specifically, traditional stationary subspace analysis methods tend to lose the temporal continuity and spatial correlation of data during dimensionality reduction, resulting in incomplete representation of the dynamic characteristics of non-stationary processes. This invention achieves precise preservation of spatiotemporal information of data by introducing dual constraint terms of a temporal Laplace matrix and a spatial Laplace matrix into the objective function: On the one hand, the temporal Laplace matrix is ​​constructed based on the temporal series correlation of the data, and its constraint term forces the stationary components after dimensionality reduction to maintain the temporal dependence structure of the original data, effectively capturing the implicit global stationary characteristics in non-stationary processes, such as long-term trends and periodic fluctuations; on the other hand, the spatial Laplace matrix is ​​constructed through the similarity measurement of neighboring variables, and its constraint term ensures that the spatial adjacency relationship of neighboring variables after dimensionality reduction is aligned with the original data, preserving local spatial correlations, such as propagation delay between sensors and regional coupling effects; the synergistic effect of the two means that the data after dimensionality reduction not only retains the global dynamic evolution law but also fully transmits local neighborhood interaction information, improving the information utilization rate and feature expression ability of process data.

[0137] To address the prevalent spatiotemporal correlations in non-stationary processes, such as the gradual change in material concentration over time and the gradient distribution of spatial location in a continuous stirred tank reactor, this invention overcomes the limitations of traditional methods that only focus on a single dimension through the spatiotemporal constraint mechanism of Laplace eigenmaps. Specifically, the time constraint term aligns the temporal correlations before and after dimensionality reduction through generalized eigenvalue decomposition, ensuring that the stationary components not only reflect the stationarity of a single variable time series but also capture the temporal synchronicity between multiple variables. The spatial constraint term, by restricting the adjacency of neighboring variables in the dimensionality-reduced space, preserves the cooperative change characteristics caused by spatial proximity in the original data, such as the temperature linkage between different measuring points in the same reactor. Experimental verification shows that the stationary components extracted by this method can simultaneously characterize global stationary trends and local spatiotemporal anomalies, such as gradient mutations caused by local hotspots, providing more comprehensive feature inputs for subsequent fault monitoring. This invention, through the deep integration of Laplace eigenmaps and analytical stationary subspace analysis methods, overcomes the limitations of traditional methods in preserving spatiotemporal information and extracting stationary features, improving the accuracy, timeliness, and engineering applicability of non-stationary industrial process monitoring, and possesses significant technological innovation value and promising industrial application prospects.

[0138] As a preferred embodiment, the objective function The time-based Laplace matrix The steps to obtain the time constraint are as follows:

[0139] Constructing a temporal adjacency matrix for non-stationary data Preferably, the temporal adjacency matrix is ​​established using a Gaussian kernel function and the K-nearest neighbor (KNN) method, and the temporal adjacency matrix... The specific formula is as follows:

[0140] ;

[0141] in, Represents sample points The set of K nearest neighbors, This indicates adjustable thermonuclear parameters;

[0142] If two sample points in the original data and If the distance between them is small, they are considered to have a high degree of similarity. The similarity is relatively large; this invention aims to preserve this similarity, so that the dimensionality-reduced data can maintain the local temporal structure of the original data, hence the dimensionality reduction results in... and The distance should be kept relatively small to a certain extent; in order to ensure that the projected stationary components still retain the local temporal correlation between the original data, the objective function of the time constraint term is obtained. The specific formula is as follows:

[0143] ;

[0144] in, Let be the time-degree matrix, and let be a diagonal matrix, with each element on the diagonal being... ,matrix Let be the time Laplace matrix.

[0145] As a further improvement to the above scheme, the objective function Based on the spatial Laplace matrix The steps for obtaining spatial constraint terms are as follows:

[0146] Constructing a spatial adjacency matrix for nonstationary data Preferably, the spatial adjacency matrix is ​​established using a Gaussian kernel function and the K-nearest neighbor (KNN) method, and the spatial adjacency matrix... The specific formula is as follows:

[0147] ;

[0148] in, Representing dimensions The set of K nearest neighbors, This indicates adjustable thermonuclear parameters;

[0149] Accordingly, to preserve the local spatial correlations between the original data in the projected stationary components, new constraint terms need to be introduced. Specifically, due to dimensionality reduction, the dimension of the stationary data is less than that of the original data, which makes it difficult to preserve local spatial correlations using the original Laplacian eigenmap. Therefore, this invention extends the original Laplacian eigenmap, designs new constraint terms, and selects constraints on the dimensionality reduction matrix. Constraints are applied to integrate the local spatial relationships in the original data into a linear representation of the original dimension on a stationary dimension; specifically, , Then the stationary component It can be represented as: ;

[0150] Dimensionally reduced data Keep the original data The local spatial structure minimizes the loss of local spatial information in the original data during dimensionality reduction; based on the above objectives, if and The similarity is high, that is If it is relatively large, then its corresponding and It still has a high degree of similarity; this requires and The objective function of the spatial constraint term is obtained by minimizing the distance between them. The specific formula is as follows:

[0151] ;

[0152] in, Let be the spatial degree matrix, and let be a diagonal matrix, where each element on the diagonal is . ,matrix Let be the space Laplace matrix.

[0153] It should be noted that the adjacency matrix uses Euclidean distance as the metric to calculate the similarity between sample points and between dimensions, and uses a Gaussian kernel function to map the similarity between sample points into intervals. Weight values ​​within; sample points and The closer the distance, the larger the corresponding weight value, indicating a higher similarity between the two sample points. Correspondingly, the distance between their corresponding points in the low-dimensional space is also closer. and ,like and The weight values ​​are set to 0; the weight calculation rules between different dimensions are consistent with the weight calculation rules between different sample points.

[0154] As a preferred embodiment, the steps for solving the objective function are as follows:

[0155] For the objective function Using the Lagrange multiplier method, the corresponding Lagrange function is obtained, as shown in the following equation:

[0156] ;

[0157] in It is a Lagrange multiplier matrix;

[0158] The Lagrange function with respect to Find the first-order partial derivative.

[0159] ;

[0160] Among them, matrix , , , and Since it is a symmetric matrix, the above equation can be combined and simplified as shown in the following equation:

[0161] (1);

[0162] Setting equation (1) to zero transforms the problem of solving the spatiotemporal stationary subspace analysis method into a generalized eigenvalue decomposition problem.

[0163] The solution of the spatiotemporally stationary subspace analysis method is shown in the following equation:

[0164] (2);

[0165] The eigenvalues ​​obtained according to equation (2) represent the strength of nonstationarity of different features; the larger the eigenvalue, the stronger the nonstationarity. The obtained eigenvalues ​​are arranged in ascending order. From the front The matrix composed of the eigenvectors corresponding to the eigenvalues ​​is the stationary projection matrix to be solved. The details are as follows:

[0166] .

[0167] To illustrate the present invention in more detail and to visually demonstrate the monitoring effect of the non-stationary industrial process monitoring method provided by the present invention, an example is provided below; the data for this example comes from a simulation model of a continuous stirred tank reactor with a drift term introduced; step faults and multiplicative faults are introduced to simulate the occurrence of faults in actual industrial processes; the specific settings are shown in Table 1.

[0168] Table 1 Fault Settings for Simulation Model of Continuous Stirred Tank Reactor

[0169] .

[0170] During the offline modeling phase, this example continuously collected 1200 normal samples as training data; each sample consisted of 7 process variables, namely... During the online monitoring phase, this example collected an additional 1200 samples, and introduced the two types of faults mentioned above starting from the 601st sample. Four methods were implemented in the experiment: a cointegration-based method, a stationary subspace analysis method, an analytical stationary subspace analysis method, and the spatiotemporal stationary subspace analysis method proposed in this invention. The cointegration order in the cointegration-based method was set to r=5. In the stationary subspace analysis method, the analytical stationary subspace analysis method, and the spatiotemporal stationary subspace analysis method, the number of stationary components was set to 5. Furthermore, the non-stationary dataset was divided into 40 groups for training the stationary subspace analysis method, the analytical stationary subspace analysis method, and the spatiotemporal stationary subspace analysis method. Regarding the spatiotemporal stationary subspace analysis method proposed in this invention… =0.5, =0.5, given confidence level =0.99, and the control limits for all methods were determined by kernel density estimation.

[0171] In this example, false alarm rate and detection rate are used as the criteria for evaluating the monitoring performance of each method; Figure 2 , Figure 3 , Figure 4 and Figure 5 The performance of four methods for monitoring introduced step faults is demonstrated; from Figure 2 , Figure 3 and Figure 5 As can be seen, the monitoring results of the stationary subspace analysis method, the cointegration-based method, and the analytical stationary subspace analysis method are similar, which is related to their monitoring strategies. The analytical stationary subspace analysis method is a variant of the stationary subspace analysis method, which obtains a stationary subspace similar to that of the stationary subspace analysis method by solving the generalized eigenvalue problem. The cointegration relation extracted by the cointegration-based method is similar to the stationary subspace of the stationary subspace analysis method and the analytical stationary subspace analysis method, so the monitoring results are also quite similar. Figure 2 , Figure 3 and Figure 5 It can also be seen that the Mahalanobis distance statistics of these three methods have similar trends; although the false alarm rate of these three methods is low, only 0.3%, their highest detection rate is only 79.2%, which reflects their poor monitoring performance; the monitoring results of the spatiotemporal stationary subspace analysis method show that the data generated by the continuous stirred tank reactor simulation model contains usable spatiotemporal information; from Figure 4 As can be seen, compared with the other three methods, the spatiotemporal stationary subspace analysis method provided by this invention achieves the highest detection rate of 95.7% and a low false alarm rate of 0.3%. The good monitoring performance of the spatiotemporal stationary subspace analysis method provided by this invention is attributed to its effective use of the local spatiotemporal correlation in the original data. The results show that in non-stationary processes, the local spatiotemporal correlation in the original data contains important information that cannot be ignored.

[0172] Figure 6 , Figure 7 , Figure 8 and Figure 9The monitoring performance of the four methods described above for introduced multiplicative faults is demonstrated. Due to the nature of the fault, it cannot be effectively detected in its early stages. Therefore, the stationary subspace analysis method, the cointegration-based method, the analytical stationary subspace analysis method, and the spatiotemporal stationary subspace analysis method provided by this invention all exhibit varying degrees of missed detection. However, by comparing the Mahalanobis distance trend and detection rate of the four methods, the spatiotemporal stationary subspace analysis method provided by this invention has the highest detection rate of 92.3%, indicating that the detection delay of the spatiotemporal stationary subspace analysis method provided by this invention is the shortest. The good monitoring performance of the spatiotemporal stationary subspace analysis method provided by this invention is attributed to its effective preservation of global and local information. In the initial stage of a multiplicative fault, the data distribution can be considered constant. Therefore, the spatiotemporal stationary subspace analysis method retains the local spatiotemporal correlation between fault samples and nearby normal samples, which may lead to it misclassifying these normal samples as fault samples; however, this makes the spatiotemporal stationary subspace analysis method more sensitive to faults with slowly changing amplitudes; the other three methods can only accurately identify faults when the fault amplitude reaches a certain value; in summary, the spatiotemporal stationary subspace analysis method provided by this invention can effectively monitor slowly changing multiplicative faults.

[0173] Example 2

[0174] The present invention also provides a non-stationary industrial process monitoring system, including a memory and a processor. The memory stores a computer program, and when the computer program is executed by the processor, the processor implements a non-stationary industrial process monitoring method as described in Embodiment 1.

[0175] The above are merely preferred embodiments of the present invention and do not limit the patent scope of the present invention. Any equivalent structural transformations made using the contents of the present invention's specification and drawings under the inventive concept of the present invention, or direct or indirect applications in other related technical fields, are within the patent protection scope of the present invention.

Claims

1. A method for monitoring non-stationary industrial processes, characterized in that, The process includes an offline training phase and an online monitoring phase; During the offline training phase: Historical data matrix X is collected under normal operation during non-stationary dynamic processes, and the time Laplacian matrix is ​​calculated based on the data matrix X. And space Laplace matrix ,in n is the number of samples, and m is the number of dimensions. Represents the real number field; Construct an objective function for the stationary subspace analysis method, and incorporate the time Laplacian matrix into the objective function. Time constraints and space-based Laplace matrix Spatial constraints; The objective function of the stationary subspace analysis method is shown in the following equation: ； in, The stationary projective component to be solved is represented by the following matrix: ； and These are the mean vector and covariance matrix of each subsequence group, respectively. , , , Here, is the empirical coefficient, and N is the total number of subsequences. Represents a d-dimensional identity matrix; The objective function The time-based Laplace matrix The steps to obtain the time constraint are as follows: Constructing a temporal adjacency matrix for non-stationary data And the time adjacency matrix The specific formula is as follows: ； in, Represents sample points The set of K nearest neighbors, This indicates adjustable thermonuclear parameters; The result of dimensionality reduction and To minimize the distance between the data after dimensionality reduction Keep the original data The local time structure is used to obtain the objective function of the time constraint term. The specific formula is as follows: ； in, Let be the time-degree matrix, and let be a diagonal matrix, with each element on the diagonal being... ,matrix The time Laplace matrix; The objective function Based on the spatial Laplace matrix The steps for obtaining spatial constraint terms are as follows: Constructing a spatial adjacency matrix in non-stationary data And the spatial adjacency matrix The specific formula is as follows: ； in, Representing dimensions The set of K nearest neighbors, This indicates adjustable thermonuclear parameters; By incorporating the local spatial relationships in the original data into a linear representation of the original dimension to the stationary dimension, the stationary components are obtained. It can be represented as: ; make and Minimize the distance between them; obtain the objective function of the spatial constraint terms. The specific formula is as follows: ； in, Let be the spatial degree matrix, and each element on the diagonal be... ,matrix It is the space Laplace matrix; The objective function is solved using generalized eigenvalue decomposition to obtain the stationary projection matrix. ; based on For each sample in the data matrix X, calculate its stationary component and monitoring index sequentially. ,in, d is the number of stationary components; Determining control limits using kernel density estimation methods ; Online monitoring phase: Based on the real-time running data x of the non-stationary process and the stationary projection matrix obtained during the offline training phase. Calculate the stationary component of real-time running data x and calculation If the corresponding real-time monitoring indicator is greater than the control limit If a non-stationary process is found to have a fault, the corresponding sample point is the fault sample point.

2. The method for monitoring a non-stationary industrial process according to claim 1, characterized in that, The steps for solving the objective function are as follows: For the objective function Using the Lagrange multiplier method, the corresponding Lagrange function is obtained, as shown in the following equation: ； in It is a Lagrange multiplier matrix; For the Lagrange function with respect to The first-order partial derivatives, after merging and simplification, are shown in the following equation: ； Setting the above equation to zero transforms the problem of solving the spatiotemporally stationary subspace analysis method into a generalized eigenvalue decomposition problem.

3. The method for monitoring a non-stationary industrial process according to claim 2, characterized in that, The solution of the spatiotemporally stationary subspace analysis method is shown in the following equation: ； in, As a generalized eigenvector, the obtained eigenvalues ​​are arranged in ascending order. From the front The matrix composed of the eigenvectors corresponding to the eigenvalues ​​is the stationary projection matrix to be solved. The details are as follows: 。 4. A method for monitoring non-stationary industrial processes according to any one of claims 1-3, characterized in that, Monitoring indicators for whether a fault has occurred are obtained based on Mahalanobis distance. The specific formula is as follows: ； in, For the stationary components of the data points, and Refers to the stationary component The mean and covariance.

5. The method for monitoring a non-stationary industrial process according to claim 4, characterized in that, Control Limits It is calculated by the following formula, as shown below: ； in, Confidence level Let be the probability density function. , where n represents the total number of data points and h represents the bandwidth value.

6. A non-stationary industrial process monitoring system, comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the computer program is executed by the processor, the processor enables the processor to implement a non-stationary industrial process monitoring method as described in any one of claims 1-5.

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