We suggest in this article a dynamic reduced algorithm in order to enhance the monitoring abilities of nonlinear processes. Dynamic fault detection using data-driven methods is among the key technologies, which shows its ability to improve the performance of dynamic systems. Among the data-driven techniques, we find the kernel partial least squares (KPLS) which is presented as an interesting method for fault detection and monitoring in industrial systems. The dynamic reduced KPLS method is proposed for the fault detection procedure in order to use the advantages of the reduced KPLS models in online mode. Furthermore, the suggested method is developed to monitor the time-varying dynamic system and also update the model of reduced reference. The reduced model is used to minimize the computational cost and time and also to choose a reduced set of kernel functions. Indeed, the dynamic reduced KPLS allows adaptation of the reduced model, observation by observation, without the risk of losing or deleting important information. For each observation, the update of the model is available if and only if a further normal observation that contains new pertinent information is present. The general principle is to take only the normal and the important new observation in the feature space. Then the reduced set is built for the fault detection in the online phase based on a quadratic prediction error chart. Thereafter, the Tennessee Eastman process and air quality are used to precise the performances of the suggested methods. The simulation results of the dynamic reduced KPLS method are compared with the standard one.
In general, the requirements of the industrial world nowadays are to guarantee the health and safety of people and to maintain our healthy environment. For this reason, system monitoring gives in the industrial production process an important role to ensure the safety and at the same time the reliability of industrial processes. However, the development and monitoring of modern industry are becoming increasingly difficult and complex. The chemical and industrial processes are frequently dynamic, changeable over time, and contain thousands of measurements every day.
In the literature work, machine learning methods have become one of the most productive areas in practice and research especially to fault detection (FD) and also fault diagnosis for production results and industrial process operations. Process monitoring, in general, includes four important tasks: fault detection, fault identification, fault reconstruction, and product quality control and monitoring. Several works over the last decades on FD in many important industrial processes have been suggested [
Modern industrial processes are nowadays equipped with control systems. In fact, the data collected on their operation are stored in a database. In this context, the data-driven modeling methods are more desirable and usable for industrial applications. The data-driven techniques are characterized by minimal process knowledge and easy implementation using the process historians for model development. Many successes of the data-driven FD are found in several industries, such as chemical industry [
The most known data-driven methods are mainly based on multivariate statistical techniques: principal component analysis (PCA), independent component analysis (ICA), and also partial least squares (PLS) [
The main idea of the PCA method is to make an orthogonal representation of the multivariable data, using linear combinations of the original variables. PCA method is based on summarizing high-dimensional data using a smaller and low number of transformed variables. On the other hand, the PLS method is an extension of the PCA method. The PLS, known by projection to latent structures, builds a linear relationship between the input and output data matrices. This method presents an ability to analyse data with many collinear and noisy variables in both input
For the purpose of process monitoring, extension methods based on kernel methods have been proposed for the nonlinear process. In this context, the kernel PLS (KPLS) and kernel PCA (KPCA) have been developed, respectively, in [ (i)The computation time and the memory size, which rise with the training data number of the large-scale systems The dynamic side of industrial systems. In this paper, we focus on PLS theory.
Furthermore, the classical KPLS method requires all process data available to build the model later. In this context, a reduced method named reduced KPLS (RKPLS) is used to better improve the detection phase [
On the other side, the most real industrial processes are often dynamic over time. Static methods, such as KPLS, KPCA, and also RKPLS, cannot follow the changes in systems of the monitoring process. Then, the dynamic methods use essentially the dynamic nature of the monitored process and analyse the autocorrelation and cross-correlation. Several dynamic methods, which are developed in the next section, have been developed for monitoring the dynamic process. Indeed, the dynamic characteristic is achieved by a time-variant model and also by introducing time-lagged variables into the matrices of data.
To overcome the difficulties stated above, a dynamic reduced KPLS (DRKPLS) method is suggested. The main purpose is based on the adaptive model creation. This suggested method includes adding a new observation according to both conditions without removing the old or the important observation. The proposed DRKPLS allows controlling and monitoring the reduced model observation by observation, depending on data availability. DRKPLS fault detection includes updating the RKPLS model if and only if a new normal sample presents useful and important information about the monitored system.
The online proposed approach DRKPLS is tested on the Tennessee Eastman process (TEP) and the air quality process. Afterwards, the FD performances of the suggested method are illustrated in terms of good detection rate (GDR), false alarm rate (FAR), and computation time (CT). In this article, a comparative study of data-driven fault detection and monitoring methods was performed between the proposed method DRKPLS, moving window RKPLS (MW-RKPLS) [ (i)Firstly, we handle the FD problem by a reduced method which consists in selecting the significant components, with an optimized statistic version We then use the dynamic DRKPLS that aims to update the reduced model observation by observation, without the risk of losing or deleting important information about the monitored system We use only the observations rich in information, which improves the FD performances in the dynamic version The proposed dynamic method is evaluated by using a real dataset
The remainder of this paper is presented as follows. Section
Industrial systems are regularly needed for special supervision thanks to technological developments. In recent years, research works for the diagnostics process are widely used in different fields. Several studies have been conducted to achieve a profitable solution for the diagnosis and monitoring of nonlinear dynamic systems; in the literature, kernel functions such as KPLS and KPCA are used.To achieve and to have the best monitoring and detection performance, Li and Yan [
However, the authors in [
The online reduced rank KPCA (ORRKPCA) method has been used in [
Variable moving window KPCA and moving window RRKPCA (MW-RRKPCA) have been proposed to take into account all changes in the dynamic process in [
In the other part, the data-driven methods based on KPLS prove their efficiency for fault detection [
In [
Dynamic methods using the PLS algorithm consider the dynamic nature of the monitored system and analyse cross-correlation and autocorrelation. Indeed, the dynamic methods are especially suitable for all changes over time for the real and continuous processes.
Over the years, the dynamic PLS method which is based on the monitoring system is developed by Komulainen et al. in [
In [
The conventional moving window algorithm has been used in [
A dynamic total PLS model has been proposed by Li et al. [
For the nonlinear dynamic process, a new FD method using a slow feature analysis for the dynamic kernel has been proposed by Zhang et al. [
For complex chemical process, the moving window technique presents good effectiveness compared to other methods [
Among the existing work, a dynamic KPLS method has been suggested in [
To conclude, we present in this paper many existing methods which prove their effectiveness for fault detection. To show the efficiency and performance of the proposed method, a comparative study of data-driven fault detection and monitoring methods was performed between (1) KPLS method which is the basic method of our work, (2) RKPLS which is the reduced method in the static mode, (3) MW-RRKPCA which is the dynamic method using the moving window, (4) ORRKPCA which is the online method based on the RRKPCA model, and (5) MW-RKPLS which is the dynamic method using the moving window based on the RKPLS method.
In this part, we introduce many online methods based on the kernel principle. Encouraged by these studies, the objective of this article is to propose a dynamic method with reduced complexity based on the reduced KPLS method, named DRKPLS, for nonlinear systems varying over time. However, we focus on an effective online method to follow changes in the dynamic system without losing important information and eliminating old and important observations. The DRKPLS method is proposed in this paper to update the reduced KPLS model observation by observation.
A lot of research studies have been presented and suggested for the FD procedure based on the kernel functions and more precisely the KPLS method for process monitoring. KPLS method was determined to resolve the limitation of the linear PLS which is a popular input/output latent variable method.
Consider an input matrix
Afterwards, it must be mentioned that the two matrices of input and output
Then the main objective of the KPLS method is to model using a nonlinear structure the process data. First of all, the KPLS method changes and transforms the basic nonlinear data, in a high-dimensional feature space, to a linear piece. Then its idea is to perform PLS in that space. This feature space is denoted by
In that case, the transformation from the space of input (
The input matrix, after the nonlinear map, deviates in the form of the feature matrix, as follows:
Then, in the space of feature
All
The zero mean of
The mapping function
Thus, the inner product can be determined and calculated using the element
Let us, in this step, give a kernel matrix
In the literature, many kernel functions
One of the most used and more elegant kernel functions is the radial basis function (RBF). The RBF kernel may present advantages owing to its flexibility in choosing the associated parameter, which is more detailed in the next section. The RBF kernel can be presented as follows:
Generally, the mean centering in this step must be realized in the Gram matrix, as indicated in the following equation:
Then by substituting the kernel Gram function, we can reformulate the score matrix (
For the KPLS method, the deflation step is determined using
For the KPLS method, the prediction of the response output variable on the training
After a detailed presentation of the essential principle of the KPLS method, it is necessary to regroup all settings. Algorithm
Calculate and determine the kernel matrix and next center; Set Randomly initialize If Deflate Repeat steps 3 to 6 to extract more latent variables Obtain the cumulative matrices.
The basic steps of the system diagnosis are firstly fault detection (FD) and then fault isolation and finally the fault identification. The FD step basically includes mentioning the presence of faults in the process. Indeed, the procedure of FD using KPLS is almost the same by using PLS. This step is based on the residuals that were evaluated and determined from the KPLS model.
The quadratic prediction error (SPE) is one of the most frequently exploited indices which used the information obtained through the KPLS model [
The SPE control limit can be mentioned to control all faults even the faults with small magnitudes. Therefore, the system is considered normal if
Several research works have used the optimization problem from different points of view: philosophies and objectives. Several optimization methods have been cited in the literature [
Among the most used methods, we find, for example, the multiobjective optimization method and the tabu search (TS) method. In our work, we are interested in the TS method.
The tabu search method is an optimization algorithm to control an integrated heuristic technique. TS is an iterative metaheuristic process qualified as a local search in the broad sense. It determines in a flexible way a compromise between the solution quality and the computation time.
However, the idea of TS is to explore the neighborhood from a given position and to choose the position in this neighborhood which minimizes the objective function [
A solution
In this case, we have to estimate the optimal parameter
The suggested DRKPLS is based on the reduced KPLS and the update of this reduced model. Thus, we determine the efficiency, ability, and precision of the proposed method studied to update the implicit KPLS model.
The training data for kernel methods, used for monitoring and modeling, must be stored in memory. More precisely, monitoring techniques based on kernel methods such as KPLS method suffer from the complexity of computation. This complexity is due to the learning time which is slow and the memory size which increases rapidly following the observation number.
The memory and calculation problems are present when the number of observations becomes large, mainly when complex processes are monitored.
Although the KPLS method solves the problem of nonlinearity, it is limited essentially in terms of computation time because of the dimension of kernel matrix
The RKPLS method chooses a reduced number of observations data among the
The retained data
To select the reduced matrix, we can project all transformed data vectors from the latent variables to get the most loaded samples in terms of information
We can at this step get the matrix of reduced data
Finally, the detection performance is based on the reduced set of data rich in information.
The monitoring RKPLS model presents a very important limitation from another point of view. This technique cannot update the reduced model as normal and new observations are collected. Indeed, the monitoring of the dynamic processes can be difficult.
This method updates the model according to new conditions or modifications. It allows the RKPLS model adaptation, observation by observation. At the first step, the identification of the reduced reference model took place to describe the status of normal operation. The second step is the acquired online phase and the model of reduced KPLS is updated and adapted if and only if a normal and new observation which presents useful and important information about the studied system is available. Consequently, the suggested DRKPLS method satisfies the following conditions: A normal observation An observation rich in information
The proposed monitoring process procedure includes two phases: Offline RKPLS: model identification Online FD: model update
The initial reduced data matrix, at first, is represented by
The kernel matrix is defined, in the initial state, as follows:
The update of the RKPLS model, observation by observation, is carried out by the online phase.
The update of the DRKPLS model includes two important steps as follows.
If this observation is considered as a faultless observation, we pass in this case to the second condition in equation (
The main steps of the suggested online method DRKPLS are presented in Algorithm
Initialize Acquire a new observation Calculate the SPE index using equation ( If If the condition presented by equation ( Update the matrix of reduced data; Update the reduced Gram matrix; Update the SPE index; Update the LVs and return to step 2;
To sum up, the flowchart of the proposed DRKPLS method, which describes their different stages of FD technique, is illustrated in Figure
Diagram of DRKPLS algorithm.
In Table
Cost of KPLS, RKPLS, and DRKPLS with adaptive model.
Method | Iterations | Cost |
---|---|---|
KPLS | Initialize training data | |
Calculate the matrix of kernel | ||
Calculate the number of LVs | ||
Calculate the SPE limit | ||
Obtain the new observation | ||
Compute the kernel vector | ||
Calculate the estimated output | ||
Evaluate SPE index | ||
Total: | ||
RKPLS | Initialize training data | |
Calculate the matrix of kernel | ||
Compute the reduced number of LVs | ||
Calculate the SPE limit | ||
Obtain the new observation | ||
Compute the vector of kernel | ||
Calculate the estimated output | ||
Evaluate SPE index | ||
Total: | ||
DRKPLS | Initialize training data | |
Compute the matrix of kernel | ||
Compute the reduced number of LVs | ||
Calculate the SPE limit | ||
Obtain the new observation | ||
Calculate the kernel vector | ||
Update kernel matrix | ||
If the condition | ||
If the condition presented by equation ( | ||
Update the LVs number | ||
Evaluate SPE index | ||
Total: |
In this case, we notice that the FD process using KPLS method consumes
In this part, a comparative study between the conventional KPLS, the static RKPLS, the MW-RKPLS, MW-RRKPCA, ORRKPCA, and the suggested DRKPLS method is carried out. The performance of these developed methods was evaluated in terms of FAR, GDR, and also CT.
The FAR can be determined as follows, which represents, in the nonfaulty region, the ratio betwixt the overall incorrect faulty declarations.
The GDR can be expressed by the following equation, which represents the total observations that are determined and specified in the faulty region:
To show the efficiency of the proposed DRKPLS method, we use the Tennessee Eastman process (TEP) presented in the next section.
The Tennessee Eastman process (TEP) is a highly nonlinear process used for conducting chemical reactions. It is widely used by the scientific community to assess process control and the performance of control and diagnostic algorithms. Indeed, the TEP is a large chemical reactor, which is widely described in the literature [
Flow diagram of TEP [
The TEP contains a total of 53 variables. Among them, 22 variables are measured continuously. For this reason, the data input matrix contains only these 22 variables.
On the other side, the TEP presents a challenge to the control, identification, and also monitoring of scientific community systems. A number of faults have been generalized, as given in Table
List of monitoring variables in TEP [
Fault number | Process variable | Type |
---|---|---|
IDV (1) | Step | |
IDV (2) | Step | |
IDV (3) | Step | |
IDV (4) | Reactor cooling water inlet temperature | Step |
IDV (5) | Condenser cooling water inlet temperature | Step |
IDV (6) | Step | |
IDV (7) | Step | |
IDV (8) | Random variation | |
IDV (9) | Random variation | |
IDV (10) | Random variation | |
IDV (11) | Reactor cooling water inlet temperature | Random variation |
IDV (12) | Condenser cooling water inlet temperature | Random variation |
IDV (13) | Reaction kinetics | Slow drift |
IDV (14) | Reactor cooling water valve | Sticking |
IDV (15) | Condenser cooling water valve | Sticking |
IDV (16) | Unknown | Unknown |
IDV (17) | Unknown | Unknown |
IDV (18) | Unknown | Unknown |
IDV (19) | Unknown | Unknown |
IDV (20) | Valve fixed at steady-state position | Constant position |
IDV (21) | Step |
The AIRLOR (air quality monitoring network) is used in this part, operating in Lorraine, France. AIRLOR is a network which contains twenty stations spread out over several sites: urban, periurban, and rural [
In fact, each station has controlled the acquisition of the air pollution, using a set of sensors: Nitrogen oxides ( Ozone Carbon monoxide Sulfur
In this case, six stations are used for recording additional metrological parameters. The main purpose of these stations is to detect and measure the faults of each sensor, which determine the concentration of ozone
Thus, monitoring of air quality is becoming increasingly essential and important to protect public health and the environment. The observation vector represents 18 controlled variables, corresponding to the concentration of ozone, nitrogen oxide, and nitrogen dioxide, respectively, in each station.
The input data matrix contains 18 variables, which contain the ozone concentration
In this part, we demonstrate the performances of the FD related to the proposed DRKPLS method. To evaluate the yield of our proposed FD, the optimal kernel parameter
The optimal kernel parameter
Optimization method/process | TEP | Air quality |
---|---|---|
Tabu search |
Nevertheless, the TS optimization algorithm adds in the search process a flexible memory which is based on a more intelligent search in the space of solutions. In our work, we are interested in the TS algorithm, which helps to reach our objective with a simple and flexible operation.
In this section, the application of the suggested DRKPLS was evaluated firstly by the TEP for the fault detection operation. The FD performances of the suggested DRKPLS method based on adaptive model are determined and demonstrated with the conventional KPLS, RKPLS method with fixed model, MW-RRKPCA, ORRKPCA, and MW-RKPLS methods.
Figure
Monitoring TEP IDV (1) fault using (a) KPLS, (b) RKPLS, (c) MW-RRKPCA, (d) ORRKPCA, (e) MW-RKPLS, and (f) the proposed DRKPLS techniques.
We can mention that the DRKPLS proved FD correctly using the SPE threshold in 95%. Thus, Figure
According to Figure
Table
Performances of KPLS, RKPLS, MW-RRKPCA, ORRKPCA, MW-RKPLS, and DRKPLS for online FD for TEP.
Default | KPLS | RKPLS | MW-RRKPCA | ORRKPCA | MW-RKPLS | DRKPLS | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
FAR (%) | GDR (%) | FAR (%) | GDR (%) | FAR (%) | GDR (%) | FAR (%) | GDR (%) | FAR (%) | GDR (%) | FAR (%) | GDR (%) | |
IDV (1) | 6.25 | 95.02 | 1.32 | 99.9 | 14.7 | 100 | 0.89 | 96 | 0.571 | 98 | 0.4464 | 97.5825 |
IDV (2) | 49.8 | 20.03 | 29 | 45 | 6.69 | 98.9 | 2.67 | 99.6 | 1.33 | 96.26 | 0.8929 | 97.8763 |
IDV (3) | 62.08 | 50 | 34 | 35.6 | 21.87 | 29.38 | 2.67 | 80.79 | 2.23 | 67.092 | 0.4434 | 79.77 |
IDV (4) | 22 | 56.66 | 17 | 43.33 | 14.28 | 32.6 | 1.33 | 62.6 | 0.23 | 62.75 | 0.177 | 78.863 |
IDV (5) | 1.05 | 19.84 | 0.91 | 98 | 14.28 | 63.40 | 1.33 | 32.73 | 0.23 | 89.32 | 0.7743 | 86.7010 |
IDV (6) | 14.44 | 84.05 | 2.2 | 97.33 | 4.9 | 100 | 8.03 | 100 | 1.78 | 99.097 | 0.9766 | 100 |
IDV (7) | 0.44 | 77.46 | 0.02 | 99.61 | 20.09 | 83.76 | 16.96 | 83.37 | 2.23 | 68.55 | 1.647 | 89.3299 |
IDV (8) | 0.448 | 96 | 4.05 | 99.23 | 42.41 | 100 | 17.85 | 100 | 0.01 | 97.06 | 0.234 | 96.456 |
IDV (9) | 21.34 | 56.55 | 0.11 | 99.33 | 37.94 | 29.5 | 4.1 | 93.58 | 0.23 | 95.12 | 0.6444 | 94.677 |
IDV (10) | 23.11 | 68.8 | 0.02 | 67.64 | 17.41 | 80.54 | 8.48 | 94.20 | 1.23 | 86.18 | 0.768 | 88.8745 |
IDV (11) | 38.7 | 44.09 | 21.01 | 96.64 | 30.35 | 55.92 | 2.14 | 1.23 | 2.99 | 77.91 | 2.5437 | 84.3499 |
IDV (12) | 9.55 | 97.29 | 0.01 | 97.33 | 21.87 | 100 | 16.96 | 100 | 0.44 | 99.03 | 0.6644 | 99.4793 |
IDV (13) | 38.82 | 90 | 5.31 | 96.64 | 12.94 | 100 | 0 | 97.80 | 3.12 | 95.74 | 1.22 | 96.6606 |
IDV (14) | 34.4 | 92.78 | 0.9 | 97.39 | 3.25 | 100 | 1.39 | 100 | 0.89 | 77.75 | 0.6883 | 86.909 |
IDV (15) | 22.22 | 47.4 | 1.73 | 89.06 | 14.73 | 38.27 | 0.89 | 83.11 | 0.03 | 85.67 | 0.6457 | 93.3033 |
IDV (16) | 19.98 | 45.2 | 10.05 | 100 | 49.10 | 85.05 | 15.62 | 70.87 | 0.44 | 100 | 0.231 | 99 |
IDV (17) | 43.09 | 56 | 27 | 47 | 44.64 | 96.13 | 2.35 | 97.29 | 1.33 | 61.57 | 0.678 | 100 |
IDV (18) | 22.67 | 44.4 | 13 | 46 | 10.28 | 94.32 | 5.37 | 94.58 | 3.12 | 88.40 | 1.345 | 95.001 |
IDV (19) | 27.09 | 43.34 | 10.71 | 99 | 22.76 | 86.59 | 2.67 | 89.04 | 6.25 | 99.49 | 1.233 | 100 |
IDV (20) | 19.36 | 98 | 10.71 | 100 | 7.14 | 74.74 | 3.57 | 90.07 | 6.78 | 96.56 | 0.765 | 100 |
IDV (21) | 33 | 53 | 27.45 | 55 | 7.85 | 75.38 | 1.16 | 71.57 | 2.67 | 62.24 | 1.022 | 78.534 |
According to this table, although the RKPLS method has shown its efficiency and performance, in terms of detection quality and CT, it cannot be able to detect several faults correctly, as (IDV (2), IDV (3), IDV (5),
Furthermore, in general, the dynamic methods prove its ability to minimize the FAR percentage and to improve and also increase the percentage of GDR compared to the static methods. Then, the adaptive model using the suggested DRKPLS proves their ability, for several simulation cases using TEP, especially to minimize the FAR percentage, compared to other methods (the static methods: KPLS and RKPLS and the online methods: MW-RRKPCA, ORRKPCA, and MW-RKPLS).
Compared to the presented methods, the online proposed method DRKPLS shows acceptable results and also good results of FD in many cases, in terms of GDR.
The suggested DRKPLS has less computation time compared to the other techniques. In addition, the evolution of the model and the SPE index over time are validated. The small detection delay for chemical process monitoring is among the most important setting. The dynamic proposed DRKPLS method was upgraded when a new normal observation is available. A good detection performance has been proven by Figure
We use the air quality process, for the purpose of the simulations, provided with Matlab. We have used, in this part, the RBF kernel value, and the optimal parameter of this kernel is chosen using the TS algorithm.
In the following, 500 samples were collected from the AIRLOR process to prove the performance of the proposed online reduced method. Two bias faults in different stations and different apparition times are introduced. Fault 1 is an additive fault by adding 30% of the standard Fault 2 is an additive fault by adding 30% of the
The SPE index is evaluated using the KPLS, RKPLS, MW-RRKPCA, ORRKPCA, MW-RKPLS, and DRKPLS methods. Figure
Monitoring faults in
The application result to the AIRLOR process is demonstrated in Figure
Using Table
Summary of FAR, GDR, and CT for AIRLOR data.
Chart/fault detection metric | Fault 1 | Fault 2 | ||||
---|---|---|---|---|---|---|
FAR (%) | GDR (%) | CT (s) | FAR (%) | GDR (%) | CT (s) | |
KPLS | 17.67 | 94 | 0.544 | 19.89 | 83.44 | 0.426 |
RKPLS | 10.55 | 97.9 | 0.274 | 13.3 | 92.6 | 0.24 |
MW-RRKPCA | 7.71 | 98 | 2.66 | 14.12 | 75 | 2.82 |
ORRKPCA | 16.42 | 100 | 4.31 | 11.75 | 98 | 4.15 |
MW-RKPLS | 7.55 | 99.09 | 0.456 | 11.34 | 87 | 0.444 |
DRKPLS | 5.75 | 96.3 | 0.318 | 9.08 | 89.02 | 0.313 |
In order to examine FD performances of the suggested method, a study simulates two types of faults. Figure
Table
Finally, the proposed DRKPLS method has proved its detection performance and is much less expensive in terms of computation complexity.
The concept of this article is to handle a reduced kernel method for FD in the online version characterized by the less expensive method in terms of computation complexity and also computation time. The developed DRKPLS method has shown improved fault detection at the level of GDR and the CT, mostly, when compared to the static method and also the methods based on moving window. Firstly, the GDR is improved thanks to the choice of the optimal kernel parameter and also the choice of data rich in information. Secondly, the CT is a very important factor of the fault detection structure. Using the reduced matrix and reduced data, we obtain a minimum computation time.
A dynamic reduced KPLS (DRKPLS) is applied for FD of dynamic systems. We have used a new online FD based on reduced form to get only the important and normal observation to monitor the dynamic process. Furthermore, the proposed DRKPLS is to monitor observation by observation the system operation. Then we control and take in this case the normal observation and at the same time data rich in information.
The DRKPLS method performances are assessed and compared to those of the classical KPLS, the static RKPLS, and the online MW-RRKPCA, ORRKPCA, and MW-RKPLS methods. The simulation results have demonstrated the developed method performances in terms of good detection rate, false alarm rate, and computation time compared with the conventional KPLS and RKPLS method.
Compared to the online method, the proposed dynamic method presents a false alarm rate more less and an interesting good detection rate. The dynamic DRKPLS technique has been tested on highly dynamic systems. Indeed, the relevance of the suggested FD methods was destined for monitoring using a TEP system and an air quality network data.
This paper improved RKPLS for fault detection in the dynamic phase. In our situation, the suggested method provided acceptable and good results to design a real-time monitoring strategy compared with the other methods.
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.