A fault detection method and system based on low-rank decomposition and common trend decoupling

Abstract

This invention provides a fault detection method and system based on low-rank decomposition and common trend decoupling. The method first acquires historical process data under normal operating conditions of an industrial process as training data. Then, it decomposes the training data into clean data representing the main structural information and sparse noise data representing noise interference using low-rank decomposition. Next, it constructs a common trend learning model based on the clean data, reconstructs the non-stationary common trend, and uses the residual between the clean data and the non-stationary common trend as a stationary feature. Decoupling is achieved by applying stationarity constraints. Further, a fault detection model is obtained through joint optimization. In the detection stage, the process data to be detected is input into the model, and monitoring statistics based on the reconstruction error of the stationary feature and the common trend are constructed respectively. The result of the control limit comparison determines whether a fault has occurred, thereby improving the robustness of fault detection in non-stationary industrial processes.

Description

Technical Field

[0001] This invention relates to the field of industrial process monitoring technology, and in particular to a fault detection method and system based on low-rank decomposition and common trend decoupling. Background Technology

[0002] As industrial processes become increasingly complex, more and more industrial process data exhibit significant non-stationary characteristics, meaning that the mean, autocovariance, or both of the process data change over time. Such non-stationary trends can mask the true dependencies between variables, thereby increasing the difficulty of fault identification and reducing the accuracy of process monitoring.

[0003] While existing technologies such as stationary subspace analysis and trend analysis can extract structural information from non-stationary processes to some extent, these methods are usually based on linear mappings and are difficult to accurately characterize the nonlinear co-driving relationships in complex industrial processes, thus limiting their application in complex non-stationary industrial scenarios.

[0004] On the other hand, actual industrial environments are often accompanied by external interference, sensor malfunctions, and signal transmission failures, leading to noise in the process data. Noise not only masks non-stationary trends but also blurs the boundary between trend changes and faults, affecting the reliability of feature extraction results. Although low-rank decomposition methods can achieve a certain degree of noise reduction by decomposing data into low-rank structures and sparse components, they do not directly address the decoupling of common trends in non-stationary time series and cannot simultaneously achieve noise suppression, non-stationary trend extraction, and stationary feature preservation.

[0005] Therefore, how to effectively separate the non-stationary common trend, stationary characteristics and noise information in industrial process data when noise interference and non-stationarity coexist, and to construct a robust fault detection model accordingly, has become a technical problem that urgently needs to be solved in this field. Summary of the Invention

[0006] The purpose of this invention is to provide a fault detection method and system based on low-rank decomposition and common trend decoupling, which aims to solve the technical problem that it is difficult to accurately decouple trends and characteristics and achieve robust fault detection in non-stationary industrial processes under noise interference.

[0007] To achieve the above objectives, in a first aspect, the present invention provides a fault detection method based on low-rank decomposition and common trend decoupling, the steps of which include: S1. Obtain historical process data under normal operating conditions of the industrial process, and preprocess the historical process data to obtain training data; S2. Perform low-rank decomposition on the training data, decomposing the training data into clean data representing the main structural information and sparse noise data representing noise interference. S3. Construct a common trend learning model based on clean data. Obtain latent feature representation by nonlinear mapping of clean data through encoder, and reconstruct nonstationary common trends in clean data based on latent feature representation by decoder. S4. Determine the residuals between clean data and non-stationary common trends as stationary features, and impose stationarity constraints on the stationary features to decouple the non-stationary common trends and stationary features in the clean data. S5. Jointly optimize the low-rank decomposition, non-stationary common trend reconstruction, and stationarity constraints to obtain the fault detection model; S6. Acquire the process data to be detected and input the process data to be detected into the fault detection model to obtain the stationary features and non-stationary common trend reconstruction error corresponding to the process data to be detected; construct a first monitoring statistic based on the stationary features and construct a second monitoring statistic based on the non-stationary common trend reconstruction error; compare the first monitoring statistic and the second monitoring statistic with the corresponding control limits respectively; when at least one of the first monitoring statistic and the second monitoring statistic exceeds the corresponding control limit, it is determined that a fault has occurred in the industrial process.

[0008] As a further improvement to the above technical solution, in step S1, the preprocessing includes: normalizing the historical process data and constructing training samples for offline modeling in chronological order.

[0009] As a further improvement to the above technical solution, in step S2, the training data satisfies the following low-rank decomposition relation: ; in, Represents training data, Indicates clean data. Represents sparse, noisy data; The clean data consists of a non-stationary common trend and stationary features, satisfying the following: ; in, Indicates a non-stationary common trend. Indicates loading matrix, It indicates a stable characteristic.

[0010] As a further improvement to the above technical solution, in steps S3 and S4, the stationary feature is obtained through the following relationship: ; in, Indicates encoder mapping, Represents decoder mapping, Indicates stationary characteristics, This indicates clean data.

[0011] As a further improvement to the above technical solution, in step S4, the stability constraint satisfies: ; in, Represents a column vector consisting entirely of 1s. Represents the global mean vector. This represents the Frobenius norm.

[0012] As a further improvement to the above technical solution, in step S5, the clean data satisfies the low-rank subspace projection representation: ; in, Denotes the representation matrix in the low-rank subspace. Represents the projection matrix in the low-rank subspace; The projection matrix satisfies the orthogonality constraint: ; in, The dimension is The identity matrix.

[0013] As a further improvement to the above technical solution, in step S5, the joint optimization is solved using the alternating direction multiplier method. An augmented Lagrangian function is constructed by introducing auxiliary variables, and the low-rank decomposition subproblem, the non-stationary common trend reconstruction subproblem, and the stationarity constraint subproblem are alternately iterated and updated until the preset convergence condition is met or the preset number of iterations is reached.

[0014] As a further improvement to the above technical solution, in step S6, the first monitoring statistic is: ; in, Indicates stationary characteristics, The mean vector representing stationary features. The covariance matrix representing stationary features; The second monitoring statistic is: ; in, This represents the data from the process to be tested. This represents the L2 norm.

[0015] As a further improvement to the above technical solution, the control limits corresponding to the first monitoring statistic and the second monitoring statistic are constructed using the kernel density estimation method.

[0016] Secondly, the present invention also provides a fault detection system based on low-rank decomposition and common trend decoupling, comprising: The data acquisition and preprocessing module is used to acquire historical process data and process data to be detected under normal operating conditions of industrial processes, and to preprocess the historical process data and process data to be detected. The low-rank decomposition module is used to perform low-rank decomposition on the preprocessed historical process data, decomposing the historical process data into clean data and sparse noise data. A common trend learning module is used to perform nonlinear common trend learning based on the clean data in order to reconstruct the nonstationary common trend in the clean data. A stationary feature extraction module is used to determine the residual between the clean data and the non-stationary common trend as a stationary feature; The stationarity constraint module is used to apply stationarity constraints to the stationary features in order to decouple the non-stationary common trends from the stationary features; The joint optimization module is used to jointly optimize low-rank decomposition, non-stationary common trend reconstruction and stationarity constraints to obtain a fault detection model. The fault discrimination module is used to input the process data to be detected into the fault detection model, calculate the first monitoring statistic and the second monitoring statistic, and output the fault detection result based on the comparison result of the first monitoring statistic and the second monitoring statistic with the corresponding control limit.

[0017] As a further improvement to the above technical solution, the common trend learning module includes: An encoder unit is used to map clean data to a nonlinear feature representation in the latent space; Decoder unit, used to reconstruct nonstationary common trend based on the nonlinear feature representation; The fault discrimination module is used to construct Mahalanobis distance statistics based on stationary features and squared prediction error statistics based on non-stationary common trend reconstruction error, and to determine the corresponding control limits using kernel density estimation method, so as to realize online fault detection.

[0018] Thirdly, the present invention also provides an electronic device, including a processor, a memory, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it is used to implement the steps of the fault detection method based on low-rank decomposition and common trend decoupling described in the first aspect.

[0019] Fourthly, the present invention also provides a computer-readable storage medium storing a computer program, which, when executed by a processor, is used to implement the steps of the fault detection method based on low-rank decomposition and common trend decoupling described in the first aspect.

[0020] Because the present invention adopts the above technical solutions, the beneficial effects of the present invention are as follows: This invention provides a fault detection method based on low-rank decomposition and common trend decoupling. First, by performing low-rank decomposition on historical process data, the main structural information and sparse noise information are separated, which can reduce the adverse effects of noise on feature extraction and fault detection and improve the model's anti-interference ability.

[0021] Secondly, by using a common trend learning model based on encoders and decoders to nonlinearly reconstruct non-stationary common trends in clean data, we can more accurately characterize the common driving changes in complex industrial processes and improve the ability to model non-stationary trends.

[0022] Furthermore, by identifying the residuals between clean data and non-stationary common trends as stationary features and further imposing stationarity constraints, it is beneficial to effectively decouple non-stationary common trends from stationary features, thereby more accurately extracting key features that reflect the time-invariant dependencies of variables.

[0023] Then, by constructing a first monitoring statistic based on stationary characteristics and a second monitoring statistic based on the reconstruction error of non-stationary common trends, faults can be comprehensively judged from the perspectives of changes in stationary equilibrium relationships and deviations in trend reconstruction, thereby improving the accuracy of fault detection in complex non-stationary industrial processes.

[0024] This invention combines low-rank decomposition, common trend reconstruction, and stationarity constraints under a unified framework for joint optimization, thereby promoting denoising and trend decoupling and further enhancing the robustness and applicability of the model in complex industrial scenarios. Attached Figure Description

[0025] 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.

[0026] Figure 1 This is a flowchart illustrating a fault detection method based on low-rank decomposition and common trend decoupling disclosed in this invention. Figure 2This is a schematic diagram of the actual roasting process of a zinc smelter disclosed in this invention. Figure 3 This invention provides a performance comparison of different methods disclosed in the present invention during a real roasting process; wherein, Figure 3 (a) represents the SSA (Stationary Subspace Analysis) method. Monitoring statistics; Figure 3 (b) The WSSA (Wasserstein Stationary Subspace Analysis) method Monitoring statistics; Figure 3 (c) ASSA (Analytic Stationary Subspace Analysis) Monitoring statistics; Figure 3 (d) represents the method proposed in this invention. and Statistics.

[0027] The objectives, features, and advantages of the invention will be further explained in conjunction with the implementation methods and with reference to the accompanying drawings. Detailed Implementation

[0028] 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.

[0029] 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.

[0030] Example 1

[0031] See Figures 1-3 This invention provides a fault detection method based on low-rank decomposition and common trend decoupling, suitable for industrial process monitoring scenarios with non-stationary trends and noise interference. See also Figure 1 This method includes two stages: offline modeling and online monitoring.

[0032] During the offline modeling phase: First, historical process data under normal operating conditions of the industrial process is acquired, and this historical process data is preprocessed. The preprocessing may include normalization, preliminary outlier removal, and chronological organization of samples to obtain training data for offline modeling. This processing helps to mitigate the impact of differences in dimensions and magnitudes on the subsequent modeling process.

[0033] Then, the training data is subjected to low-rank decomposition, which separates the training data into clean data and sparse noise data. Since industrial process data is often affected by external disturbances, sensor malfunctions, and signal transmission errors, sparse noise information is frequently mixed into the process data. Low-rank decomposition preserves the main structural information in the training data and separates information deviating from the main structure into noise components, thus providing a more stable data foundation for subsequent common trend extraction.

[0034] After obtaining clean data, a common trend learning model is constructed based on the clean data. Preferably, the common trend learning model employs an autoencoder structure, where the encoder maps the clean data to a nonlinear feature representation in the latent space, and the decoder reconstructs the nonstationary common trend in the clean data based on the nonlinear feature representation. Since the driving relationships between variables in complex industrial processes are often nonlinearly coupled, the above structure can capture the common driving changes between variables better than linear trend analysis methods.

[0035] Furthermore, the residuals between the clean data and the reconstructed non-stationary common trend are identified as stationary features. To ensure the extracted stationary features have good stationarity, stationarity constraints are imposed on them. Preferably, a global mean term can be used to constrain the residuals over consecutive time periods, ensuring that the stationary features maintain similar means across different time periods. This approach improves the ability of the extracted stationary features to represent the time-invariant dependencies of variables.

[0036] Furthermore, joint optimization is performed on the low-rank decomposition, non-stationary common trend reconstruction, and stationarity constraints. Preferably, the joint optimization is solved using the alternating direction multiplier method, which involves introducing auxiliary variables to construct an augmented Lagrangian function, and then iteratively updating the low-rank decomposition subproblem, the common trend learning subproblem, and the stationarity constraint subproblem. Iteration stops when a preset convergence condition is met or a preset number of iterations is reached, resulting in the fault detection model. This joint optimization approach allows the low-rank denoising and trend decoupling processes to mutually reinforce each other, thereby improving the modeling effect and robustness.

[0037] Online fault detection process: First, acquire the data of the process to be detected, and input the data into the fault detection model trained offline to obtain the corresponding stationary features and non-stationary common trend reconstruction error. Under normal operating conditions, the stationary features usually remain within a relatively stable distribution range, and the reconstruction error of the non-stationary common trend is within the normal range; when a fault occurs in the industrial process, the stationary features will shift, and the reconstruction relationship between the data of the process to be detected and the non-stationary common trend under normal operating conditions will also change significantly.

[0038] A first monitoring statistic and a second monitoring statistic are then constructed. The first monitoring statistic, preferably based on Mahalanobis distance, characterizes the degree of shift in stationary features. The second monitoring statistic characterizes the magnitude of the error in the data of the process under test relative to the reconstructed trend, preferably a squared prediction error statistic. Furthermore, control limits corresponding to the two monitoring statistics can be established using kernel density estimation. During online monitoring, the first and second monitoring statistics are compared with their corresponding control limits. If at least one of the first and second monitoring statistics exceeds its corresponding control limit, the industrial process is determined to have malfunctioned; otherwise, the industrial process is considered to be in a normal state. By simultaneously utilizing information on changes in stationary relationships and anomalies in trend reconstruction, the ability to identify malfunctions in complex non-stationary industrial processes can be effectively improved.

[0039] As a preferred embodiment, in complex industrial scenarios, process data typically contains non-stationary common trends, stationary features, and noise information simultaneously. To effectively decouple this multi-layered information, a coupled representation model of the original process data is first established. Let the original multivariate time series data matrix be: ; in, Represents the original process data matrix; Indicates the number of process variables; Indicates the number of sample time points.

[0040] From the perspective of low-rank decomposition, the original process data can be represented as a superposition of non-stationary common trends, stationary features, and sparse noise, specifically: ; in, Represents the original process data matrix; This represents a non-stationary common trend matrix; Indicates the loading matrix; Represents stationary characteristic components; This represents sparse noise components.

[0041] Furthermore, Together they can constitute a clean data component. , that is The formula for the original process data matrix shows that there is coupling between non-stationary common trends, stationary features, and noise information in the original process data. Therefore, it is necessary to achieve automatic decoupling through unified modeling.

[0042] In a preferred embodiment, to extract the nonlinear common trend among variables, this embodiment embeds the idea of ​​common trend learning into an autoencoder structure. The encoder and decoder perform nonlinear mapping and trend reconstruction on the input data, and the reconstructed residual is used as a stationary feature, specifically expressed as follows: ; in, Represents stationary characteristic components; Indicates clean data components; This represents the encoder mapping, used to map input data to a nonlinear feature representation in the latent space; The decoder mapping represents the reconstruction of nonlinear features in the latent space into nonstationary common trends. Indicates the parameters of the autoencoder; This represents the input process data. Using the above formula, stationary feature components can be obtained while adaptively learning common trends.

[0043] As a preferred embodiment, to ensure that the extracted residual components are indeed stationary features, this embodiment further introduces a stationarity reconstruction constraint. First, the stationary residual is expressed as the sum of the global mean term and the zero-mean disturbance term: ; in, Represents stationary characteristic components; Represents the global mean vector; This represents a stationary disturbance term with zero mean.

[0044] Based on this, the stationarity constraint can be expressed as: ; in, Represent the objective function for stationarity constraints; Indicates stationary residuals; Denotes the Frobenius norm; Indicates clean data components; Represents a column vector consisting entirely of 1s; Represents the global mean vector; Represents decoder mapping; Indicates encoder mapping; This represents the input process data. The stationarity constraint formula enhances the stationarity of the extracted stationary features by reducing the difference between the residuals at different time points and the global mean, thus keeping the residuals similar in mean over continuous time periods.

[0045] As can be seen from the optimization objective of the above objective function, this term is the autoencoder's... Reconstruction constraints are used to solve the nonlinear common trend mapping under stationarity constraints. and The aim is to obtain a common trend representation of non-stationary features. By imposing a stationarity requirement on the extracted features, it is ensured that the extracted non-stationary features can capture the common driving information of the data, while stationary features represent the time-invariant dependencies between variables, thus achieving adaptive decoupling of stationary features and non-stationary components.

[0046] As a preferred embodiment, for the representation of low-rank clean data, this embodiment adopts a low-rank subspace projection method, which can be specifically represented as follows: ; in, Indicates clean data components; Represents the original process data matrix; Represents the representation matrix in the low-rank subspace; This represents the projection matrix in the low-rank subspace. Typically, if the projection space has a dimension of 1... Then there is ,in .

[0047] To ensure that the projection mapping has effective orthogonality, the representation matrix and projection matrix satisfy the following constraints: ; in, Represents the representation matrix in the low-rank subspace; Represents the projection matrix in the low-rank subspace; express 3D identity matrix; Let represent the dimension of the projected subspace. The above equation shows that... Satisfies the idempotency constraint, and It satisfies orthogonal constraints.

[0048] As a preferred embodiment, real-world industrial data often suffers from the coupling of multiple information elements, including low-rank clean information and noise interference, as well as non-stationary common trends and stationary features. Modeling these factors individually can easily lead to information separation errors. Therefore, this invention designs a unified framework to automatically decouple these multiple information elements. To simultaneously achieve low-rank denoising, common trend reconstruction, and stationary feature extraction, this embodiment incorporates the above processes into a unified optimization framework, constructing the following joint optimization objective function: ; in, Indicates the parameters of the autoencoder; Indicates clean data components; Represents sparse noise components; Represents the representation matrix in the low-rank subspace; Represents the projection matrix in the low-rank subspace; Indicates the weight coefficient of the lower-rank term; This represents the weighting coefficient of the sparse noise term; Represents the nuclear norm; express Norm; Denotes the Frobenius norm; Represents a column vector consisting entirely of 1s; Represents the global mean vector; Represents decoder mapping; Indicates encoder mapping; Represents the original process data matrix; express The unit matrix is ​​1.5. In the joint optimization objective function, the first term ensures that clean data has a low-rank structure, the second term encourages the sparsity of noise terms, the third term learns the low-rank subspace representation, and the fourth term applies a stationarity constraint and reconstructs the true non-stationary common trend.

[0049] In this embodiment, by incorporating low-rank decomposition and stationary feature reconstruction into the same joint optimization process, multiple types of information coupled together in industrial process data can be synergistically separated. Specifically, low-rank decomposition is used to extract the main structural information in the data and suppress noise interference; stationary feature reconstruction is used to weaken the influence of non-stationary components and improve the stationary representation capability. The two are not performed independently but cooperate and correct each other during the iterative optimization process: low-rank decomposition provides a less noisy data foundation for stationary feature extraction, while stationary feature reconstruction helps improve the accuracy of identifying the main structural information. Based on this synergistic mechanism, complex changing trends in process data can be captured more effectively, achieving reasonable decoupling of main structural information, noise components, and stationary features, thereby providing more stable and reliable data support for subsequent monitoring modeling and state identification.

[0050] As a preferred embodiment, since the aforementioned joint optimization objective function contains non-convex constraints, it is generally a non-convex optimization problem, and it is usually difficult to directly obtain the global optimum. However, the optimization problem also has a multi-convex structure, that is, when the other variables are fixed, the subproblems formed for any variable to be optimized are all convex problems. Based on this characteristic, this embodiment uses the alternating direction multiplier method to solve the optimization problem. Specifically, the alternating direction multiplier method (ADMM) decomposes the original joint optimization problem into multiple relatively independent subproblems, and achieves iterative solution through variable updates and constraint coordination among the subproblems.

[0051] Specifically, to simplify the objective function and improve separability, auxiliary variables are introduced. The original optimization problem can be rewritten as: ; in, Indicates an auxiliary variable, used for substitution. Participated in several optimizations of the low-rank kernel norm.

[0052] After introducing auxiliary variables, an augmented Lagrangian function can be constructed: ; in, Represents the augmented Lagrange function; , , Represents the Lagrange multipliers; , Indicates the penalty factor; This represents the matrix trace operation.

[0053] To facilitate subsequent variable alternation updates, the augmented Lagrangian function described above can be further written as: ; in, , , These represent the Lagrange multipliers corresponding to the constraint terms; , This represents the penalty factor.

[0054] Then, by solving for one variable and fixing the others, each parameter is finally given. The updated formula.

[0055] 1. Auxiliary variables The update formula, while keeping other variables fixed, optimizes the subproblem. The following can be updated: ; in, , ; Indicates auxiliary variables; Indicates the weight coefficient of the lower-rank term; Indicates the penalty factor; Indicates clean data components; Represents the Lagrange multipliers; This represents the singular value threshold parameter; Represents intermediate variables. Subproblems There exists a closed-form solution that can be found using the singular value threshold (SVT). Considering the rank is... of SVD: ; in, Indicates intermediate variables; This represents the orthogonal matrix formed by the left singular vectors; Represents a singular value diagonal matrix; This represents the orthogonal matrix formed by the right singular vectors; and They are and an orthogonal matrix, if The rank is ,but ,and and singular values It is positive. Regarding Soft threshold operator Defined as follows: ; in, Indicates the threshold is Soft threshold operator; , , These represent the results of singular value decomposition; Indicates the first One singular value; Indicates the threshold parameter; express The positive part, that is The operator pairs The singular values ​​are subject to a soft threshold rule, which encourages these singular values ​​to shrink toward zero.

[0056] For any positive value and data You can get Update formula: ; in, Indicates the first The auxiliary variables obtained from the next iteration; Indicates the threshold is Singular value threshold shrinkage operator; This represents the clean data component of the current iteration; Represents the Lagrange multipliers; This represents the penalty factor.

[0057] 2. The formula for updating a low-rank matrix, keeping other parameters constant. The optimization problem was proposed: ; in, Indicates clean data components; Represents the original process data matrix; , Let represent the representation matrix and projection matrix in the low-rank subspace, respectively; Represents a column vector consisting entirely of 1s; Represents the global mean vector; , These represent decoder mapping and encoder mapping, respectively. Represents sparse noise components; , Represents the Lagrange multipliers; This represents the penalty factor.

[0058] Solving the above equation involves taking the derivative and setting its value with respect to... The derivative is zero, resulting in clean data components. The update formula is: ; 3. In a fixed manner, independent of sparse noise components When other variables, The optimization subproblem is: ; in, , ; Represents sparse noise components; Indicates the weight coefficient of the sparse term; Indicates the penalty factor; Represents the original process data matrix; Indicates clean data components; Represents the Lagrange multipliers; Indicates intermediate variables; This represents the threshold parameter. Optimal solution. The i-th column is: ; in, Represents the optimal noise matrix The List; Representation matrix The List; Represents the L2 norm; Indicates the threshold parameter; This represents the zero vector. By shrinking this column, noise information that deviates from the low-rank structure can be separated.

[0059] 4. When other variables are fixed, the autoencoder parameters The optimization problem is: ; in, Indicates the parameters of the autoencoder; Indicates clean data components; Represents a column vector consisting entirely of 1s; Represents the global mean vector; and These represent decoder mapping and encoder mapping, respectively. This represents the input process data matrix. By calculating a stationarity-constrained optimization problem, the stationarity requirement is guaranteed. The autoencoder captures non-stationary common trend information to extract stationary features. The process involves solving for the autoencoder parameters. In optimization problems, fixed and Parameters, parameters It can be learned through the backpropagation algorithm.

[0060] 5. In a fixed projection matrix When other variables are not present, The optimization subproblem is: ; in, Represents the projection matrix in the low-rank subspace; Indicates clean data components; Represents the original process data matrix; Represents the representation matrix in the low-rank subspace; express 3D identity matrix; Represents the Lagrange multipliers; This represents the penalty factor.

[0061] Through the Taking the derivative and setting its gradient to zero, we get: ; in, Represents the representation matrix in the low-rank subspace; Represents the projection matrix in the low-rank subspace; Represents the original process data matrix; Indicates clean data components; Represents the Lagrange multipliers; Indicates the penalty factor; express The above equation forms the Sylvester equation, which can be solved using the Hesenberg-Schuur algorithm.

[0062] 6. Similarly, in a fixed division matrix representation... When other variables are not present, The optimization subproblem is: ; P represents the representation matrix in the low-rank subspace.

[0063] Through the Taking the derivative and setting it to zero, we get: ; in, Represents the projection matrix in the low-rank subspace; Represents the representation matrix in the low-rank subspace; Represents the original process data matrix; Indicates clean data components; Represents the Lagrange multipliers; Indicates the penalty factor; express 3D identity matrix.

[0064] When matrix When it is reversible, the above equation can be transformed into the standard Sylvester equation form: ; in, Representation matrix The inverse matrix; similarly, it can be solved using [method name missing]. The algorithm is used to solve it.

[0065] 7. Residual mean update: When other variables are fixed, the global mean vector... The optimization subproblem is: ; in, Represents the global mean vector; Indicates clean data components; Represents a column vector consisting entirely of 1s; and These represent decoder mapping and encoder mapping, respectively. This represents the input process data matrix.

[0066] Similarly, regarding the above formula... Taking the partial derivative and setting it to zero, we get: ; in, Represents a column vector consisting entirely of 1s; Indicates the first The global mean vector obtained from the next iteration; Indicates clean data components; and These represent decoder mapping and encoder mapping, respectively. This represents the input process data matrix. Because... ,in This represents the number of sample time points, and therefore we can further obtain... The closed-form update formula.

[0067] The specific update formula is as follows: ; in, Indicates the first The global mean vector of the next iteration; Indicates the number of sample time points; Represents a column vector consisting entirely of 1s; Indicates clean data components; and These represent decoder mapping and encoder mapping, respectively.

[0068] After all variables have been updated, the Lagrange multipliers and Update according to the following formula: ; in, , , They represent the first The three Lagrange multipliers in the next iteration; , , They represent the first The corresponding Lagrange multiplier for the next iteration; , Indicates the penalty factor; Represents the original process data matrix; This represents the updated clean data component; This represents the updated sparse noise components; This represents the updated auxiliary variable; This represents the updated orthogonal constraint term; express An identity matrix. Relevant adjustment parameters can be set empirically or determined through methods such as grid search.

[0069] When the iteration reaches the preset convergence condition or the number of iterations exceeds the limit, the above iteration process stops, thereby obtaining the fault detection model under normal operating conditions. Through the offline modeling process, it is possible to achieve joint modeling and automatic decoupling of non-stationary common trends, stationary features, and noise information in the presence of noise interference.

[0070] In a preferred embodiment, during the online monitoring phase, the normalized sample to be detected is input into the fault detection model to obtain the corresponding stationary equilibrium relationship components and reconstruction error. Since the stationary equilibrium relationship in normal samples has a stable mean and covariance structure, stationary feature statistics can be constructed based on Mahalanobis distance. ; in, This represents a stationary characteristic statistic; This represents the stationary equilibrium component or stationary characteristic residual; express The mean vector; express The covariance matrix; The inverse matrix of the covariance matrix; This represents the matrix transpose. This statistic measures the deviation of the test sample from the stationary characteristic distribution of the normal sample. Simultaneously, to measure the deviation between the test sample and the reconstruction results of the common trend under normal operating conditions, a squared prediction error statistic is constructed: ; in, This represents the squared prediction error statistic; Indicates the sample to be tested; Represents decoder mapping; Indicates encoder mapping; Indicates the parameters of the autoencoder; Represents a column vector consisting entirely of 1s; Represents the global mean vector; This represents the L2 norm. This statistic is used to reflect the degree of anomalousness in the reconstruction of non-stationary common trends in the sample under test.

[0071] Furthermore, the stationary characteristic statistics Sum of squared prediction error statistics The control limits can be determined by kernel density estimation methods, denoted as: ; in, Represents statistics At the significance level The corresponding control limit; Represents statistics At the significance level The corresponding control limit; Indicates the significance level.

[0072] During online fault diagnosis, if the sample to be tested meets the following conditions: ; If the condition is met, the industrial process is determined to be in a normal operating state; otherwise, the industrial process is determined to be in an abnormal operating state. This represents a stationary characteristic statistic; This represents the reconstruction error statistic; and These represent the corresponding control limits. By simultaneously considering the deviation from the stationary equilibrium relationship and the reconstruction deviation from the common trend of non-stationary processes, faults can be jointly identified from two different perspectives, thereby improving the accuracy of fault detection in complex non-stationary industrial processes.

[0073] In this embodiment, the method can be applied to online monitoring scenarios of continuous industrial processes such as metallurgy, chemical industry, and energy. For example, in the zinc smelting roasting process, due to factors such as fluctuations in operating conditions, changes in equipment status, and environmental disturbances, process data often exhibits both non-stationary trends and noise pollution. This invention, by combining low-rank decomposition with common trend learning, can effectively separate noise information, extract non-stationary common trends and stationary features, and identify abnormal states through dual statistics, thereby providing technical support for the safe and stable operation of industrial processes.

[0074] It should be noted that if the input data dimension, network structure layer number, threshold parameter or penalty factor value changes in the specific application scenario, as long as the basic concept of robust fault detection achieved by the present invention through low-rank decomposition, common trend decoupling and stationarity constraints is not changed, it shall fall within the protection scope of the present invention.

[0075] To verify the effectiveness of the monitoring method based on low-rank decomposition and common trend decoupling proposed in this invention in a real industrial process, the roasting process of a zinc smelter was selected as the application object. See [link to relevant documentation]. Figure 2This zinc smelter employs a hydrometallurgical zinc refining process, with roasting being a critical upstream step in its production flow. Specifically, zinc concentrate is fed into a fluidized bed roasting furnace via a feeding unit, while a blower unit supplies air to the furnace, ensuring thorough mixing and oxidation of the zinc concentrate and air within the furnace. During this process, metal sulfides are converted into oxides and other zinc compounds. The resulting roasted ore is cooled and then fed into subsequent processes such as ball milling and leaching. The flue gas enters a flue gas treatment unit, where it is cooled and treated before being introduced into the subsequent sulfuric acid production stage. Because the roasting process directly affects the continuity and stability of subsequent smelting processes such as leaching, timely and effective monitoring of the roasting process is crucial for maintaining high productivity and low energy consumption.

[0076] In this embodiment, the normal operating conditions of the roasting process include two categories: high-efficiency state and healthy state. Besides normal operating conditions, there are also three abnormal operating conditions: decomposition failure, oxidation failure, and bed deposition failure. In the healthy state, the roasting furnace functions normally, and there are no abnormal phenomena such as excessive sulfur dioxide emissions or reduced boiler pressure. The high-efficiency state is defined according to the existing on-site operating condition classification rules. Because the feed composition, feed rate, and blower volume fluctuate in actual production, process variables such as blower current, furnace temperature, and the temperatures of the upper, middle, and lower parts of the boiling layer collected under different normal operating conditions usually exhibit non-stationary characteristics. To more specifically verify the monitoring capability of the method of this invention in non-stationary industrial processes, this embodiment takes the over-decomposition failure under high-efficiency conditions as the research object, selecting eight-dimensional process variables such as outlet pressure, burner oil pressure, blower pressure, and cooler flow rate to construct a monitoring dataset.

[0077] In the offline modeling phase, 1000 sets of samples generated by the acquisition equipment under efficient and normal operating conditions were used as training samples to establish the fault detection model of this invention. In the online monitoring phase, 700 sets of test samples corresponding to this operating condition were selected for verification. Among these, over-decomposition faults occurred starting from the 301st sampling time and continued until the end of the test data. To ensure the comparability of the comparison results, uniform parameter settings were used for the comparison methods SSA, WSSA, and ASSA, and monitoring performance was evaluated under the same data conditions.

[0078] Combination Figure 3 It can be seen that after completing the training under efficient normal operating conditions, all methods can reflect the state changes of the roasting process to a certain extent, but there are significant differences in the false alarm rate and detection rate of different methods. Figure 3 Although the SSA method shown in (a) can generate a response after a fault occurs, it has a high false alarm rate and many misjudgments during the normal operating phase, indicating that it is difficult to effectively distinguish between normal and abnormal states. Figure 3 (b) and Figure 3While both the WSSA and ASSA methods shown in (c) possess some fault detection capability, the presence of noise interference in actual industrial process data leads to numerous instances of exceeding limits in the initial normal data, resulting in a still relatively high false alarm rate. In contrast, Figure 3 As shown in (d), the method provided by the present invention exhibits good monitoring performance under both types of monitoring statistics. Among them, the SPE statistic can identify faults earlier while maintaining a low false alarm rate. Statistics are highly sensitive to faults, thus they can accurately reflect the evolution of the roasting process from a normal state to an over-decomposition fault state.

[0079] Furthermore, from Figure 3 The results show that the false alarm rate (FAR) of the SSA method is 13.0%, and the fault detection rate (FDR) is 86.5%; the false alarm rate (FAR) of the WSSA method is 16.0%, and the fault detection rate (FDR) is 80.0%; the false alarm rate (FAR) of the ASSA method is 10.0%, and the fault detection rate (FDR) is 83.0%. In the method of this invention, the false alarm rate (FAR) of the SPE statistic is 2.0%, and the fault detection rate (FDR) is 98.5%. The false alarm rate (FAR) of the statistic is 12.3%, and the fault detection rate (FDR) is 99.5%. This demonstrates that, in actual zinc smelting and roasting scenarios, the method provided by this invention, compared to SSA, WSSA, and ASSA methods, can effectively reduce false alarms while maintaining a high fault detection rate. In particular, the SPE statistic exhibits superior overall monitoring performance in non-stationary scenarios with noise.

[0080] The reasons for the above differences are as follows: although SSA, WSSA, and ASSA all monitor non-stationary processes, their stationary feature extraction process mainly relies on the linear mapping assumption, making it difficult to accurately represent the nonlinear common driving relationships that objectively exist in actual industrial systems. Furthermore, these methods do not adequately consider the masking effect of noise on fault information, resulting in unclear boundaries between normal and abnormal operating conditions, thus affecting the stability and reliability of the monitoring results. The method provided in this invention first separates noise information through low-rank decomposition, and then combines common trend reconstruction and stationarity constraints to achieve joint modeling and decoupling of noise information, non-stationary common trends, and stationary features within a unified optimization framework. Therefore, it can extract key features that more closely resemble real operating conditions and improve the accuracy and robustness of fault detection.

[0081] In summary, during actual zinc smelting and roasting processes, the method proposed in this invention demonstrates superior monitoring performance compared to comparative methods in monitoring over-decomposition faults under high-efficiency operating conditions. This indicates that the method of this invention is not only applicable to industrial processes with significant non-stationary characteristics, but also achieves relatively effective online fault detection even in real-world scenarios with noise interference. This embodiment further verifies the feasibility and practicality of the method and system of this invention in monitoring complex industrial processes.

[0082] Example 2

[0083] This invention also provides a fault detection system based on low-rank decomposition and common trend decoupling, suitable for fault detection scenarios in industrial processes with non-stationary characteristics, complex variable coupling relationships, and susceptibility to noise interference. This system separates and collaboratively models the main structural information, non-stationary common trend information, and stationary characteristic information in industrial process data, reducing the impact of normal operating condition changes and noise disturbances on fault identification, thereby improving the stability and reliability of fault detection in non-stationary industrial processes.

[0084] In this embodiment, the fault detection system includes a data acquisition and preprocessing module, a low-rank decomposition module, a common trend learning module, a stationary feature extraction module, a stationarity constraint module, a joint optimization module, and a fault discrimination module.

[0085] The data acquisition and preprocessing module is used to acquire historical process data and process data to be detected under normal operating conditions of the industrial process, and to preprocess the historical process data and the process data to be detected. The historical process data can be obtained from industrial field sensors, control systems, or process databases, and the process data to be detected can be data collected in real time during the operation of the industrial process. The preprocessing can include one or more of the following: missing value handling, outlier removal, time alignment, normalization, and standardization. By preprocessing the input data, the impact of dimensional differences, sampling errors, and invalid outliers on subsequent model training and online detection can be reduced, providing a unified and stable data input foundation for each module.

[0086] The low-rank decomposition module is used to perform low-rank decomposition on the preprocessed historical process data, decomposing the historical process data into clean data and sparse noise data. The clean data is used to characterize the main structural information of the industrial process, while the sparse noise data is used to characterize random noise, impulse interference, or local anomalies. By first performing low-rank decomposition on the historical process data, noise components can be separated from the main process structural information, allowing subsequent modeling processes to focus more on the dynamic changes of the industrial process itself. This reduces the interference of noise on fault feature extraction and improves the model's adaptability to complex industrial environments.

[0087] The common trend learning module is used to perform nonlinear common trend learning based on the clean data to reconstruct the non-stationary common trend in the clean data. Industrial processes typically exhibit obvious non-stationary characteristics under the influence of factors such as load fluctuations, operating condition switching, and gradual equipment changes. If the raw process data is directly used for fault monitoring, normal trend changes can easily be misjudged as abnormalities. By learning and reconstructing the non-stationary common trend in the clean data, the dominant change patterns shared among multiple process variables can be extracted, thereby providing a basis for distinguishing between normal trend changes and abnormal disturbances, which helps to reduce the impact of non-stationarity on fault detection results.

[0088] Furthermore, the common trend learning module includes an encoder unit and a decoder unit. The encoder unit maps clean data to a nonlinear feature representation in the latent space. By performing nonlinear mapping on the clean data through the encoder unit, the potential correlation structure and dynamic coupling features between variables can be extracted, enhancing the system's ability to represent nonlinear relationships in complex industrial processes. The decoder unit reconstructs the nonstationary common trend based on the nonlinear feature representation. The decoder unit recovers the dominant trend changes in the clean data according to the nonlinear feature representation in the latent space, thereby obtaining the corresponding nonstationary common trend reconstruction result. The encoder unit and decoder unit work together to better represent complex nonstationary changes in industrial processes, providing a reliable foundation for subsequent residual separation.

[0089] The stationary feature extraction module is used to determine the residual between the clean data and the non-stationary common trend as a stationary feature. That is, after the common trend learning module reconstructs the non-stationary common trend in the clean data, the residual obtained by subtracting the non-stationary common trend from the clean data is used as a stationary feature. Since this stationary feature has eliminated the influence of the main trend term, it is more representative of abnormal disturbances in the industrial process than the original data, thus improving the identification accuracy of fault-related features.

[0090] The stationarity constraint module is used to apply stationarity constraints to the stationary features to decouple the non-stationary common trend from the stationary features. By introducing stationarity constraints, the model can, during training, classify trend changes as much as possible into the non-stationary common trend, while retaining relatively stable changes that are more sensitive to faults in the stationary features. This reduces the overlap between the non-stationary common trend and the stationary features, improves the separation effect between the two, and enhances the targeting of subsequent monitoring statistics construction.

[0091] The joint optimization module is used to jointly optimize low-rank decomposition, non-stationary common trend reconstruction, and stationarity constraints to obtain a fault detection model. Compared with implementing each processing step independently, joint optimization can coordinately adjust the low-rank decomposition results, common trend learning results, and stationary feature constraint results under a unified objective, thereby reducing the accumulation of errors between modules and local mismatch problems. This makes the obtained fault detection model more consistently reflect the data structure characteristics of industrial processes under normal operating conditions, and improves the overall stability and generalization ability of the model.

[0092] The fault discrimination module is used to input the process data to be detected into the fault detection model, calculate a first monitoring statistic and a second monitoring statistic, and output the fault detection result based on the comparison results of the first and second monitoring statistics with the corresponding control limits. Specifically, the process data to be detected is preprocessed and then input into the fault detection model to obtain the corresponding stationary features and non-stationary common trend reconstruction results; then, a first monitoring statistic is constructed based on the stationary features, and a second monitoring statistic is constructed based on the non-stationary common trend reconstruction error; when at least one of the first and second monitoring statistics exceeds the corresponding control limit, the industrial process is determined to have failed. By simultaneously utilizing stationary feature information and common trend reconstruction error information for joint discrimination, stationary anomalies and trend anomalies can be reflected respectively, thereby improving the detection capability of different types of faults and reducing the risk of missed detection.

[0093] In a preferred embodiment, the fault discrimination module is used to construct a Mahalanobis distance statistic based on stationary features and a squared prediction error statistic based on the reconstruction error of a non-stationary common trend, respectively, and uses a kernel density estimation method to determine the corresponding control limits to achieve online fault detection. The Mahalanobis distance statistic measures the degree of deviation of the sample under test from the center of the stationary feature distribution under normal operating conditions. Since the Mahalanobis distance statistic can comprehensively consider the correlation between multiple variables, it is suitable for reflecting the shift in the overall state of the industrial process. The squared prediction error statistic characterizes the magnitude of the reconstruction error of the sample under test under the common trend learning model. When the data of the process under test is inconsistent with the common trend pattern learned under normal operating conditions, the reconstruction error will increase, thus reflecting abnormal changes at the trend level. By simultaneously introducing the above two types of statistics, the operating status of the industrial process can be monitored from different perspectives, improving the comprehensiveness of fault detection.

[0094] Furthermore, the control limits corresponding to the Mahalanobis distance statistic and the squared prediction error statistic are determined using the kernel density estimation method. Considering that industrial process data typically does not strictly conform to a preset distribution in practical applications, directly determining the control limits based on a fixed distribution assumption can easily lead to threshold setting deviations. Using the kernel density estimation method, the control limits can be adaptively determined based on the actual statistical distribution of normal operating condition samples, thereby improving the rationality of the control limit setting and enhancing the reliability of online fault detection results.

[0095] The fault detection system based on low-rank decomposition and common trend decoupling provided in this embodiment achieves effective separation of main trend information and fault-sensitive information in non-stationary industrial process data through the cooperation of modules such as data acquisition and preprocessing, low-rank decomposition, common trend learning, stationary feature extraction, stationarity constraint, joint optimization, and fault discrimination. It also improves the robustness of fault detection through a dual-statistic joint discrimination mechanism, making it suitable for online monitoring of industrial processes with strong noise interference, significant operating condition fluctuations, and complex variable coupling relationships.

[0096] Example 3

[0097] The present invention also provides an electronic device, including a memory for storing computer program instructions and a processor for executing the computer program instructions, wherein when the computer program instructions are executed by the processor, the device is triggered to execute some or all of the steps in Embodiment 1; A processor may include one or more processing units, such as an application processor (AP), a modem processor, a graphics processing unit (GPU), an image signal processor (ISP), a controller, a video codec, a digital signal processor (DSP), a baseband processor, and / or a neural network processing unit (NPU). Different processing units may be independent devices or integrated into one or more processors.

[0098] The controller can serve as the nerve center and command center of an electronic device. Based on the instruction opcode and timing signals, the controller generates operation control signals to control the fetching and execution of instructions.

[0099] The processor may also include memory for storing instructions and data. In some embodiments, the memory in the processor is a cache memory. This memory can store instructions or data that the processor has just used or that are used repeatedly. If the processor needs to use the instruction or data again, it can retrieve it directly from the memory. This avoids repeated accesses, reduces processor waiting time, and thus improves system efficiency.

[0100] Example 4

[0101] The present invention also provides a storage medium having a computer program stored thereon, wherein, when the program is executed, it controls the device where the storage medium is located to perform some or all of the steps in Embodiment 1.

[0102] The storage medium may include high-speed RAM memory, and may also include nonvolatile memory, such as at least one disk storage device. It is understood that the storage medium can be any machine-readable medium capable of storing program code, such as random access memory (RAM), magnetic disk, hard disk, solid state disk (SSD), or nonvolatile memory.

[0103] Those skilled in the art will understand that embodiments of the present invention can be provided as methods or storage media. Therefore, embodiments of the present invention can take the form of entirely hardware embodiments, entirely software embodiments, or embodiments combining software and hardware aspects. Furthermore, embodiments of the present invention can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0104] The above are merely preferred embodiments of the present invention and do not limit the patent scope of the present invention. All 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 / indirect applications in other related technical fields, are included within the patent protection scope of the present invention.

Claims

1. A fault detection method based on low-rank decomposition and common trend decoupling, characterized in that, The steps include: S1. Obtain historical process data under normal operating conditions of the industrial process, and obtain training data after preprocessing; S2. Perform low-rank decomposition on the training data, decomposing the training data into clean data representing the main structural information and sparse noise data representing noise interference. S3. Construct a common trend learning model based on clean data. Obtain latent feature representation by nonlinear mapping of clean data through encoder, and reconstruct nonstationary common trends in clean data based on latent feature representation by decoder. S4. The residual between the clean data and the non-stationary common trend is determined as a stationary feature, and a stationarity constraint is applied to the stationary feature to achieve decoupling between the non-stationary common trend and the stationary feature in the clean data. S5. Jointly optimize the low-rank decomposition, non-stationary common trend reconstruction, and stationarity constraints to obtain the fault detection model; S6. Input the process data to be detected into the fault detection model to obtain the stationary features and non-stationary common trend reconstruction error corresponding to the process data to be detected; construct a first monitoring statistic based on the stationary features, construct a second monitoring statistic based on the non-stationary common trend reconstruction error, and compare the first monitoring statistic and the second monitoring statistic with the corresponding control limits respectively. When at least one of the first monitoring statistic and the second monitoring statistic exceeds the corresponding control limit, it is determined that the industrial process has failed.

2. The fault detection method according to claim 1, characterized in that, In step S1, the preprocessing includes: normalizing the historical process data and constructing training samples for offline modeling in chronological order.

3. The fault detection method according to claim 1, characterized in that, In step S2, the training data satisfies the following low-rank decomposition relation: ； in, Represents training data, Indicates clean data. Represents sparse, noisy data; The clean data consists of a non-stationary common trend and stationary features, satisfying the following: ； in, Indicates a non-stationary common trend. Indicates loading matrix, It indicates a stable characteristic.

4. The fault detection method according to claim 3, characterized in that, In steps S3 and S4, the stationary feature is obtained through the following relationship: ； in, Indicates encoder mapping, Represents decoder mapping, Indicates stationary characteristics, This indicates clean data.

5. The fault detection method according to claim 1, characterized in that, In step S4, the stability constraint satisfies: ； in, Represents a column vector consisting entirely of 1s. Represents the global mean vector. Denotes the Frobenius norm. Indicates clean data. Indicates encoder mapping, This represents the decoder mapping.

6. The fault detection method according to claim 1, characterized in that, In step S5, the clean data satisfies the low-rank subspace projection representation: ； in, Denotes the representation matrix in the low-rank subspace. Denotes the projection matrix in the low-rank subspace. Indicates clean data. Represents training data; The projection matrix satisfies the orthogonality constraint: ； in, The dimension is The identity matrix.

7. The fault detection method according to claim 1, characterized in that, In step S5, the joint optimization is solved using the alternating direction multiplier method. An augmented Lagrangian function is constructed by introducing auxiliary variables, and the low-rank decomposition subproblem, the non-stationary common trend reconstruction subproblem, and the stationarity constraint subproblem are alternately iterated and updated until the preset convergence condition is met or the preset number of iterations is reached.

8. The fault detection method according to claim 1, characterized in that, In step S6, the first monitoring statistic is: ； in, Indicates stationary characteristics, The mean vector representing stationary features. The covariance matrix representing stationary characteristics; The second monitoring statistic is: ； in, This represents the data from the process to be tested. This represents the L2 norm.

9. A fault detection system based on low-rank decomposition and common trend decoupling, characterized in that, include: The data acquisition and preprocessing module is used to acquire historical process data and process data to be detected under normal operating conditions of industrial processes, and to preprocess the historical process data and process data to be detected. The low-rank decomposition module is used to perform low-rank decomposition on the preprocessed historical process data, decomposing the historical process data into clean data and sparse noise data. A common trend learning module is used to perform nonlinear common trend learning based on the clean data in order to reconstruct the nonstationary common trend in the clean data. A stationary feature extraction module is used to determine the residual between the clean data and the non-stationary common trend as a stationary feature; The stationarity constraint module is used to apply stationarity constraints to the stationary features in order to decouple the non-stationary common trends from the stationary features; The joint optimization module is used to jointly optimize low-rank decomposition, non-stationary common trend reconstruction and stationarity constraints to obtain a fault detection model. The fault discrimination module is used to input the process data to be detected into the fault detection model, calculate the first monitoring statistic and the second monitoring statistic, and output the fault detection result based on the comparison result of the first monitoring statistic and the second monitoring statistic with the corresponding control limit.

10. An electronic device, characterized in that, It includes a processor, a memory, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the fault detection method based on low-rank decomposition and common trend decoupling as described in any one of claims 1-8.

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