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ISSN: 2157-7048
Journal of Chemical Engineering & Process Technology
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Statistical Fault Detection of Chemical Process - Comparative Studies

Majdi Mansouri1*, Mohammed ZS2, Raoudha Baklouti2,5, Mohamed Nounou4, Hazem Nounou5, Ahmed Ben Hamida2 and Nazmul Karim3
1Electrical and Computer Engineering Program, Texas A&M University at Qatar, Doha, Qatar
2Advanced Technologies for Medicine and Signals, National Engineering School of Sfax, Tunisia
3Chemical Engineering Department, Texas A&M University, College Station, TX 77843, USA
4Chemical Engineering Program, Texas A&M University at Qatar, Doha, Qatar
5Electrical and Computer Engineering Program, Texas A&M University at Qatar, Doha, Qatar
*Corresponding Author : Majdi Mansouri
Electrical and Computer Engineering Program
Texas A&M University at Qatar, Doha, Qatar
Tel: 97477734583
E-mail: [email protected]
Received February 16, 2016; Accepted February 28, 2016; Published February 29, 2016
Citation: Mansouri M, Mohammed ZS, Baklouti R, Nounou M, Nounou H, et al. (2016) Statistical Fault Detection of Chemical Process - Comparative Studies. J Chem Eng Process Technol 7:282. doi:10.4172/2157-7048.1000282
Copyright: © 2016 Mansouri M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

This paper addresses the statistical chemical process monitoring using improved principal component analysis (PCA). PCA-based fault-detection technique has been used successfully for monitoring systems with highly correlated variables. However, standard PCA-based detection charts, such as the Hotelling statistic, T2 and the sum of squared residuals, SPE, or Q statistic, are not able to detect small or moderate events since they use only data from the most recent measurements. Different fault detection (FD) charts, namely generalized likelihood ratio test (GLRT), shewhart control chart and exponentially weighted moving average chart (EWMA) control chart have been shown to be among the most effective univariate fault detection methods and more suitable for detection small faults. The objective of this work is to improve the PCA-based fault detection by using more sophisticated FD charts to achieve further improvements and widen the applicability of the process monitoring techniques in practice. The PCA presented here is investigated as modeling algorithm in the phase of fault detection. The fault detection problem is addressed so that the data are first modeled using the PCA algorithm and then the faults are detected using FD chart. The detection stage is related to the evaluation of detection charts, which are declares the presence of the fault. Those charts are computed using the PCA-based residual. The fault detection performance is illustrated through a simulated continuously stirred tank reactor (CSTR) data. The results demonstrate the effectiveness of the PCA-based FD chart methods for detecting the single and the multiple sensor faults.

Keywords
CSTR process; Fault detection; Generalized likelihood ratio test; Principal component analysis; Shewhart; Exponentially weighted moving average; Cumulative sum
Introduction
Effective operation of various engineering systems requires tight monitoring of some of their key process variables. For example, detection of anomalies in chemical systems is crucial for their efficient application on a controlled continuous stirred tank reactor (CSTR). Also, detecting aberrations in chemical data helps the diagnosis of various diseases. The fault detection problem is an important process in process monitoring. Abnormal faults management mainly depends on diagnosis of the process faults and accurate fault detection.
Various fault detection techniques have been developed and utilized in practice. For example, statistical fault detection techniques that are based on hypothesis testing, such the generalized likelihood ratio test (GLRT), have been shown to be among the most effective univariate fault detection methods. Most practical processes, however, are multivariate, i.e., involve many variables that need to be monitored at the same time. In a previous research effort, we have developed Principal Component Analysis (PCA) and kernel PCA (KPCA)-based GLRT fault detection schemes [1,2], in which PCA and KPCA have been used as a modeling framework for fault detection. We have also, developed a recursive PCA and KPCA methods for modeling and fault detection problems to processes where online fault detection is needed [3]. In this work, we will focus on fault detection problem based on more sophisticated statistical approaches. Different multivariate fault detection techniques have been proposed for process monitoring of such systems: such as chemical, environmental, power, etc. Faults detection has been performed manually using data visualization tools [4], but these tools are time consuming for real-time detection in streaming data. PCA is among the most popular statistical methods used for modeling and faults detection problems, however, it provides linear combinations of variables that demonstrate major trends in data set. In mathematical terms, PCA provides linear combinations of a set of measured variables that capture major trends in data set. Specifically, PCA yields orthogonal vectors of high energy contents in terms of the variance of the data.
The main indices used with PCA methods are Hotelling statistic, T2 and the sum of squared residuals, SPE, or Q statistic. The T2 statistic is a measure of the variation captured in the PCA model and the Q statistic is a measure of the percent variance not captured by the PCA model. In the current work, we address the fault detection problem, in which the data are modeled using the PCA method and the faults are identified using the fault detection charts. The FD charts include: Hotelling statistic, T2, Q statistic, generalized likelihood ratio test (GLRT), shewhart control chart and exponentially weighted moving average chart (EWMA) control chart. In fact, PCA model has been shown to be suitable to obtain an accurate principal component of a set of data. The PCA algorithm is applied to obtain the model and find the combinations of parameters that describe the major trends in a data set [5,6] and FD chart is used to detect the faults and both are applied to enhance the fault detection process. The Shewhart chart is a simple univariate monitoring chart that utilizes process data without the application of filters. The Shewhart chart is mainly able to detect fairly large fault. Other univariate charts, such as CUSUM and EWMA, through the application of filters are able to smaller faults [7]. The CUSUM statistic assumes each process observation is of equal weightage, while the EWMA statistic assigns an exponential weightage to consecutive observations [8]. The advantage of the CUSUM and EWMA charts in the detection of smaller faults can be attributed to their extensive process memory. Although, the CUSUM and EWMA charts may be able to better detect smaller faults than the Shewhart chart, they cannot be used to detect a wide range of fault sizes, as they often need to be tuned to detect faults of different sizes.
Therefore, a more robust chart, such as the GLRT chart might be required for fault detection. GLRT has been proposed in [9] in order to monitor an adaptive system, which reaches three important problems; estimation, fault detection and magnitude compensation of jumps. GLRT is proposed for fault detection of different applications: geophysical signal segmentation [6], signals and dynamic systems [5], incident fault detection on freeways [9], missiles trajectory [10]. Hence, in the current work, we propose to benefit from the advantages of the GLRT in order to improve the fault detection task in the cases where the process model is not available. The fault detection performance is illustrated through a simulated continuously stirred tank reactor (CSTR) data. The results show the performance of the PCA-based FD chart methods for detecting the single and the multiple anomalies.
The rest of the paper is organized as the following. In Section 2, an introduction to PCA method is given. Then, the FD charts descriptions are presented in Section 3. After that, the PCA-based FD chart method used for fault detection which integrates PCA modeling and FD control chart, is presented in Section 4. Next, in Section 5, the PCA-based FD chart performances are studied through a simulated continuously stirred tank reactor data. At the end, the conclusions are presented in Section 6.
Description of Principal Component Analysis Methods
PCA is a linear dimensionality reduction modeling technique, which is very helpful when dealing with data sets having a high degree of cross correlation among the variables [11]. Let Xi E Rm denotes the i-th sample vector representing m different variables or sensors. Also, assume there are n samples dedicated to each variable or sensor, and then the data can be represented as a matrix Xi E Rm, where each column corresponds to a variable and each row corresponds to a sample. After scaling each variable to have a zero mean and unit variance, the X matrix can be expressed as the multiplication of two matrices, a score matrix S and a loading matrix W, through singular value decomposition (SVD), i.e.,
X=SWT (1)
Where equation is a transformed variables matrix, equation are the score vectors or PCs, and equation is an orthogonal vectors matrix wi Î Rm which includes the eigenvectors associated with the covariance matrix of X, i.e., Σ , which is given by,
equation (2)
Where ∧ = diag (λ123,..,λm) is a diagonal matrix containing the eigenvalues related to the m PCs, λ1>,λ2>…> λm and IN is the identity matrix [12].
It is important to note that the PCA model yields same number of principal components (PCs) as the number of original variables (m). For collinear process variables, however, a smaller number of PCs (l) are required to capture most of the variations in the data. The effectiveness of the PCA model depends on the number of retained PCs. Several methods for determining the optimum number of PCs have been proposed, which include the Scree plot [13], the cumulative percent variance (CPV), the cross validation [14], and the profile likelihood [15-17]. In this study, the cumulative percent variance method is utilized to estimate the optimum number of retained PCs, which can be computed as follows:
equation (3)
After determining the number of PCs (l), the data matrix X can be written as,
equation (4)
Where equation and equation , are matrices of equation retained PCs and the equation ignored PCs, respectively, and the matrices equation and equation are matrices of l retained eigenvectors and the (m-l) ignored eigenvectors, respectively. Using Equation (4), the following can be written,
equation (5)
Where the matrix equation represents the modeled variation of X based on first l components, and the matrix R represents the residuals.
Fault Detection Charts
The PCA model is used for fault detection through one of the detection statistics (T2, Q, GLRT, Shewart and EWMA) which are presented next.
Hotelling’s T2 statistic
The Hotelling’s T2 statistic is a way of measuring the variation captured in the principal components at various time samples, and it is expressed as [17]:
equation (6)
where equation, is a diagonal matrix containing the eigenvalues related to the l retained PCs. For new real-time data, when the value of T2 statistic exceeds the threshold, T2l calculated as in [17], a fault is detected. The threshold number used for the T2 statistic is computed as [17]
equation (7)
where α is the level of significance (α usually between 1% and 5%), N is the number of samples in data set, l is the number of retained PCs, and Fl,N −l is the Fisher F distribution with l and (N-l) degrees of freedom. These thresholds are computed using faultless data. When the number of observations, N, is high, the T2 statistic threshold is approximated with a X2 distribution with l degrees of freedom, i.e., equation
Q statistic or squared prediction error (SPE)
Another fault detection index is the squared prediction error SPE or Q statistic, which is a measure of the amount of variations not captured by the PCA model [17]. It can be computed as the sum of squares of the residuals [18], i.e.,
equation (8)
Where, equation
The monitored system, meanwhile, is accepted to be in normal operation if,
Q ≤ Qα (9)
The threshold Qα used for the Q statistic can be computed as [12],
equation (10)
Where equation is level of confidence and Cα is the value of the normal distribution. For new data, the Q statistic is computed and compared to the threshold Qa [12]. When the confidence limit is violated, a fault is declared. It is important to note that the threshold value is computed based on the assumption that the measurements are independent and follow a multivariate normal distribution; therefore, the Q statistic is highly sensitive to modeling errors [19].
Shewhart chart
Walter Shewhart developed the Shewhart chart in the 1920s, while working for Bell Systems [20]. Shewhart believed continuous process monitoring carried out at different stages during a process could prove to be more economical and effective, as opposed to inspecting the final product [21]. The Shewhart chart is widely used in practice for process monitoring, mainly due to its relative simplicity, as opposed to other univariate control charts [22]. Shewhart charts have three distinct features [23]: Center Line (C), Upper Control Limit (UCL), and Lower Control Limit (LCL). The center line typically represents the targeted process mean. The Shewhart chart is designed based on the following main assumptions [24]: the presence of a moderate level of noise in the evaluated residuals, the residuals being independent (uncorrelated), and the fault-free residuals following a normal (Gaussian) distribution. Numerous variations of the Shewhart chart are available. However, the most popular chart monitors the sample mean equation. This chart is occasionally coupled with either the range (R) or standard deviation (S) chart, which increases the robustness of the Shewhart chart against the variability in observations collected from different sensors monitoring a particular process variable [25]. The (R) and (S) charts are able to spot features in the trend of the data, that might not necessarily come forth with the use of the equation chart only. If observations are collected from multiple sensors (or if sampling is carried out), the following equations can be used to compute the sample mean:
equation (11)
and
equation (12)
Where, n and k represent the subgroup size and the number of subgroups, respectively. Sub-grouping is generally carried out if observations from multiple sensors monitoring a particular process variable are available, or if sampling is carried out. The approach presented here assumes that only a single sensor provides measurements, and single readings are used, and hence the R and S charts are not required. Shewhart computed the limits for the control chart as follows [25]:
equation (13)
And
equation (14)
Where, equation is the targeted process mean and Ln is the control width computed by the following equation [25]:
equation (15)
Where, σ is the standard deviation of the fault-free data set and the constant can be computed using a nomogram [25]. The scale corresponds to the range where a given percentage of fault-free observations should lie. Analysis of a nomogram for different processes can be time-consuming, and therefore it is common practice to use the following equation to compute the control limits [26]:
equation (16)
And
equation (17)
For a fault-free data test that follows a normal distribution, the value of 3σ accounts for nearly 99.73% of all deviation, which makes it a popular choice for the limits of the Shewhart chart [27]. The conventional Shewhart chart is unable to detect relatively small faults, as it is only able to detect faults larger than three times the standard deviation of the fault-free data set [25]. This can be attributed to the fact that the Shewhart chart only considers the current process measurement when deciding the presence or absence of a fault, and thus has a very short memory as indicated in Figure 1a. The insensitivity of the Shewhart chart to faults with small magnitudes is even more evident when the data is contaminated with high levels of noise, as features get masked by measurement noise. Other univariate monitoring schemes through application of linear filters, possess a longer memory than the conventional Shewhart as they utilize additional information from previous observations. The linear filters help deal with the assumption of noise to an extent. The CUSUM chart takes into account all previous observations Figure 1b, while the EWMA chart applies an exponentially weighted average filter Figure 1c. Although, the other univariate charts may show an improved performance when compared the Shewhart chart, their performance is limited by the same assumptions. Hence, it is important to find an alternative that will help address these concerns.
The CUSUM chart is effective in detecting small faults in process data. However, the extensive memory of the CUSUM chart Figure 1b increases the possibility of false alarms as the CUSUM statistic takes extra observations to return to the fault-free steady state values. Therefore, an approach such as the EWMA chart, that utilizes an exponential filter might prove useful, and will be described next.
Exponentially weighted moving average (EWMA) chart
The exponentially weighted moving average chart (EWMA) chart was developed by Roberts in 1959 and was initially referred to as the Geometric Moving Average (GMA) chart [28]. Over time the GMA chart became popularly known as the EWMA chart [29]. Similar to the CUSUM chart, the EWMA chart is able to detect smaller faults shifts in the mean when compared to the Shewhart chart [30,31].
The EWMA statistic can be computed by [32]:
equation (18)
Where, λ is the smoothing parameter (exponential filter) that can be assigned a value between 0 and 1. The smoothing parameter controls the memory of the process, i.e., a value closer to 0 placing less emphasis on more recent observations, and vice versa. The control limits for the EWMA chart can be computed using [7]:
equation (19)
And
equation (20)
Where, L is defined as the control width of the EWMA chart. At steady state equation simplifies to unity, and the following steady state values are obtained [7]:
equation (21)
And
equation (22)
Although, the CUSUM and EWMA charts are better able to detect smaller faults, they are not capable of detecting a wide range of fault sizes. Hence, a more robust approach such as the GLRT chart might be required and will be described next.
Generalized likelihood ratio test (GLRT)
The GLRT is a hypothesis testing technique which has been utilized successfully in model-based fault detection [5,9,10]. Let, Y ∈RN be an observation vector formed by one of the two Gaussian distributions: equation or equation , where θ is the mean vector (which is the value of the fault) and σ2 >0 is the variance (assumed to be known in this problem). The hypothesis test can be expressed as,
equation (23)
Here, the GLRT method replaces the unknown parameter, θ, by its maximum likelihood estimate. This estimate is computed by maximizing the GLRT equation as follows,
equation (24)
Where equation is the maximum likelihood estimate of θ, the probability density function of Y is equation and equation represents the Euclidean norm. Because the GLRT utilizes the ratio of the distributions of the faulty and fault-free data, in the case of non-Gaussian variables, non-Gaussian distributions need to be used. It must be noted that, in the derivation shown above, maximizing the likelihood function is equivalent to maximizing its natural logarithm since the logarithmic function is a monotonic one. The GLRT then decides between the hypotheses equation and equationas follows,
equation (25)
Here, the distribution of the decision function equation under equation allows designing a statistical test with a desired false alarm rate, α, where the threshold tα is chosen to satisfy the following false alarm probability,
equation (26)
Where equation represent the probability of an event A when Y is distributed according to the null hypothesis H0 and α is the desired probability of the false alarm. Since Y is normally distributed (equation (23)), the statistics equation is distributed according to the X2 law with (m-l) degrees of freedom. This law is central under H0 and noncentral under H1 with a parameter of non-centrality equal to: equation. Also, the power function of δ can be calculated as,
equation (27)
To select an appropriate threshold for the GLRT statistic, its distribution needs to be determined. Since the noise is assumed to follow a Gaussian distribution, the test statistic will follow a chi-square distribution [33]. The normalized residual equation δ is distributed as:
equation (28)
where θ=0 under the null hypothesis (26). Then, the scaled test statistic is distributed as the non-central chi-square distribution as follows,
equation (29)
with N degrees of freedom. Since the GLRT is applied online, the norm used in its statistic is computed using only the current data sample, and thus, the GLRT statistic follows a chi-square distribution with a degree of freedom equal 1.
Fault Detection Using PCA-based FD Chart Method
In this section, PCA is combined with FD chart to develop new fault detection with more sensitivity to small data faults. The PCA method is investigated here as modeling framework in the task of fault detection. The residuals of the response variables from PCA model can be assigned control limits. The proposed scheme can be used to detect the existence or lack of faults [34]. Under normal operating conditions (no faults), the residual of the monitored model is zero or close to zero when modeling measurement noise and uncertainties. However, in the presence of a fault the residuals differ significantly from zero, showing the existence of a new state that can be clearly distinguished from the normal faultless working mode [35]. Here, FD chart is used to improve the process monitoring by using a more appropriate and accurate model. Due to the capacity of the FD chart to detect drifts with low severity in the data, this technique is appropriate for enhancing the detection of small or moderate faults. Thus, the PCA is used to create the model and find an accurate combinations of parameters which describe the major trends in a data set [5,6] and FD chart is used to detect the faults and both are utilized to improve faults detection process.
PCA-based FD chart process monitoring algorithm
Here, the FD chart is obtained using the residuals of the responses variables from the PCA model. Let the matrices X, equation and R be defined as follows: equation, equation and equation let Xj, equation and Rj be the j-th columns of the matrices, X, equation , and R, respectively. In the absence of a fault, the residual can be calculated as follows:
equation (30)
The difference between the observed value of the input variable, X, and the predicted value, equation, obtained from PCA model represent the residual of the input variable, equation which can be used as an indicator to detect a possible fault. Then, the FD chart decision function based on the residuals of the response variable can be computed using one of the fault detection chart described above using equation (6) for T2 statistic, equation (8) for Q statistic, equation (11) for Shewhart, equation (18) for EWMA and equation (24) for GLRT statistic.
The developed PCA-based FD chart fault detection method can be implemented as described in Algorithm 1, and its performance is illustrated in the next section through its application to monitor the operation of a chemical reactor.
Algorithm 1: PCA-based FD fault detection algorithm
Input: Nxm data matrix X, Confidence interval α
Output: FD statistic, FD threshold
• Data preprocessing step:
Standardize: computes data’s mean and standard deviation, and standardize it
• PCA running step:
Compute the covariance matrix
Calculate the eigenvalues and eigenvectors and sort the eigenvalues in decreasing order
Compute the optimal number of principal components to be used using the CPV method
• Compute the sum of approximate and residual matrices
PCA testing step
Standardize the new data
Compute the FD chart decision function
Compute the FD chart control limits
Compute the FD chart statistic for the new data
Declare a fault when the FD chart decision function, exceeds the control limits.
Next, we present the developed PCA-based FD chart process monitoring algorithm for fault detection of chemical process.
PCA-based Charts and Application to Fault Detection in Simulated CSTR Model
Next, the developed PCA-based FD chart algorithm presented is illustrated through its application on a controlled continuous stirred tank reactor (CSTR) in which a non-isothermal, irreversible first order reaction A→B takes place. Next, the CSTR model that is used for fault detection is described.
CSTR process description
The dynamic model for the non-isothermal CSTR can be given by [36,37],
equation (31)
Where ko is the reaction rate constant, E is the activation energy, CA is the concentration of “A” in the inlet stream, CB is the concentration of “B” in the exit stream, T is the temperature of the inlet stream, F is the flow rate in and out of the reactor, V is the reactor volume, Ti is the temperature of exit stream, Tj is the temperature of the cooling fluid in the jacket, ΔΗ is the heat of reaction, U is the overall heat transfer coefficient, A is the area through which heat transfers from the reactor to the cooling jacket, and ρ and Cp are the density and heat capacity of the reactor contents and of all streams. Assuming a stoichiometric proportion of compounds “A” and “B” in the feed, one can assume that equation. The outlet temperature (T) and the concentration (CA) are controlled using proportional integral (PI) controllers by manipulating the inlet coolant flow rate (FC) and the feed flow rate (F), respectively. The parameters of the PI controllers are as follows: KC1= -0.8 and τ1 = 0.1for the temperature controller, and KC2=2 and τ1 = 0.1 for the concentration controller.
Generation of dynamic data
In a practical setting, the data would be collected by changing the feed flow rate (which is chosen in this example to be the model input, i.e., F), and then measuring the state variables, i.e., the concentration and temperature as functions of time. Thus, the data are generated given some pre-defined model parameters.
The CSTR model parameters as well as other physical properties are shown in Table 1. The simulated CSTR is used to generate training and testing data sets by changing the set points of the concentration and temperature controllers in step-wise fashions. The process data used in training includes four variables, the coolant flow rate (FC), the feed flow rate (F), the outlet concentration CA, and the reactor outlet temperature T. Thus, the data matrix, which has 1000 rows and 4 columns, is used to construct the PCA model after scaling the variables.
Next, the performance of the developed PCA-based FD chart fault detection method is illustrated and compared to PCA through its two charts Q, T2. The comparison is assessed through three different cases studies representing three different types of faults. In the first case study, the sensor measuring the concentration of A (CA) is assumed to be faulty with single as well as multiple faults. In the second case study, similar faults (single and multiple) are introduced in the temperature of the reactor (T). In third case study, multiple faults are assumed to occur simultaneously in the concentration and temperature inside the reactor.
Case 1: Faults in the concentration CA
The testing data used to evaluate the fault detection performances, which consist of 500 samples, are generated using the CSTR model described earlier. To simulate a single fault in the state variable CA, an additive fault having a magnitude 20% of the total variation in CA is introduced between samples 100 and 150. The results using the PCAbased Q technique Figure 2a show that it could successfully detect this single fault but with some false alarms. While, the performance of the PCA-based T2, GLRT, Shewhart and EWMA methods, on the other hand, Figures 2b-2e, shows that they could detect this single fault without any false alarms.
In the presence of a multiple faults in CA, we can show from Figures 3a- 3e the results of the process monitoring of CSTR process using the PCA-based FD chart techniques. The PCA based Q technique cannot detect these faults Figure 3a. However, the PCA-based T2, Q, GLRT, Shewhart and EWMA methods can detect the faults effectively (as shown in Figures 3b-3e.
Case 2: Faults in the temperature T
In this case study, the sensor measuring the temperature T is assumed to be faulty with single as well as multiple faults. First, a single fault in the reactor temperature represented by a constant bias of amplitude equal 5% of the total variation in T is introduced between the sample numbers 100 and 150. Figures 4a-4e show the ability of the PCA-based T2, Q, GLRT, Shewhart and EWMA methods to detect this additive fault, while the PCA-based Q technique results in some missed detections Figure 4b. However, the PCA-based T2 method cannot detect this additive fault as shown in Figure 4b.
Case 3: Faults in the concentration CA and temperature T
In this case study, simultaneous faults are introduced in both the concentration and temperature (each of which is represented by a bias of magnitude equal 20% of the variation in its corresponding variable). The results using the PCA-based T2, Q, GLRT, Shewhart and EWMA techniques for these multiple faults are shown in Figures 5a-5e. These results show that the PCA-based Q and PCA-based T2 techniques could detect these multiple faults. The PCA-based GLRT, Shewhart and EWMA techniques, however, are capable to detect these faults without any false alarms as shown in Figures 5c-5e.
Conclusion
In this paper, principal component analysis (PCA)-based fault detection (FD) charts are used for fault detection. The FD charts include: Hotelling statistic, T2 and Q statistic, generalized likelihood ratio test (GLRT), Shewhart control chart and exponentially weighted moving average chart (EWMA) control chart. The fault detection problem was addressed in which the data are modeled using the PCA method and the faults are identified using the fault detection charts. The PCA method is applied here as modeling framework in the phase of fault detection. The idea is to improve the FD control chart performance introducing modeling of the data using the PCA. The PCA-based FD chart fault detection performances are assessed through a simulated continuously stirred tank reactor (CSTR) data. The results show the performance of the PCA-based FD chart techniques over the conventional PCA through its two charts T2 and Q for detecting a single and multiple sensor faults.
Acknowledgements
This work was made possible by NPRP grant NPRP7-1172-2-439 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.
References

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