Recursive Fault Detection and Isolation Approaches of Time-Varying

Article pubs.acs.org/IECR

Recursive Fault Detection and Isolation Approaches of Time-Varying Processes Lamiaa M. Elshenawy* and Hamdi. A. Awad Department of Industrial Electronics and Control Engineering, Faculty of Electronic Engineering, Menoufiya University, 32952 Menouf, Egypt ABSTRACT: Recursive principal component analysis (RPCA) has gained significant attention as a monitoring tool for timevarying systems in recent years. The contribution of this article is the development of numerically efficient and memory-saving recursive fault detection and isolation (FDI) approaches for time-varying processes. The proposed approaches incorporate a recursive PCA based on a first-order perturbation (RPCA-FOP) analysis procedure and two recursive fault isolation methods. The proposed recursive fault isolation methods are the (i) recursive partial decomposition contribution (RPDC) and (ii) recursive diagonal contribution (RDC) methods. Four types of sensor faults, including bias, drifting, precision degradation, and complete failure, are simulated to test the proposed approaches. The utility of the proposed FDI approaches is demonstrated using a nonisothermal continuous stirred tank reactor (CSTR) system. subspace tracking applications. The first approach is based on first-order perturbation (FOP) theory, which is a rank-one update of the eigenpairs of the data covariance matrix.21,22 The second is based on the data projection method, which serves as a simple and reliable approach for adaptive subspace tracking.23 These approaches significantly reduce the online computational cost compared to other known approaches, and their results have demonstrated the efficiency of these approaches compared to conventional PCA for process monitoring. After detecting a fault, it is important to identify the root cause of this fault. Although much work has been reported in fault detection using data-correlation-based models, only a few methods are available for fault isolation methods.24 The most popular method is the contribution plots that were originally proposed by Miller et al.25 for the Q statistic and extended for the T2 index by Wise et al.26 The assumption behind the contribution plot method is that faulty variables make high contributions to fault detection indices. Other methods of fault isolation include (i) reconstruction methods,27−34 (ii) discrimination by angles,35,36 (iii) pattern matching methods by calculating similarity and dissimilarity factors that compare current and historical data,37−39 (iv) structured residuals based on multivariate statistical methods,40−42 (v) cumulative-sum(CUSUM-) based PCA.43 In general, the fault isolation methods stem from different backgrounds and considerations and are suitable for time-invariant processes. Most industrial processes are time-varying,44 so efficient adaptive fault isolation methods are required. The main objective of this article is to introduce numerically efficient adaptive FDI approaches for time-varying industrial processes. The proposed adaptive FDI approaches utilize the FOP analysis technique that has lower complexity [O(m2)]

1. INTRODUCTION With the advent of instrumentation and automation, industrial processes now produce large amounts of information. Early detection and isolation of faults can help to avoid major breakdowns and incidents. Fault detection and isolation (FDI) deals with the timely detection, isolation, and correction of abnormalities in a process. In the literature, FDI approaches range from analytical methods to knowledge-based and datadriven approaches.1−5 Data-driven approaches use process data collected during normal conditions to detect and isolate process faults. These approaches are classified into univariate and multivariate statistical techniques. The former techniques ignore the correlation between variables, so they are not adequate for most modern chemical processes and result in significant numbers of false alarms. The latter techniques have been developed to address this problem in a straightforward manner.6,7 Principal component analysis (PCA) is one of the most commonly used techniques for monitoring multivariate chemical processes. Two indices have been widely used for fault detection based on PCA: Hotelling’s T2 statistic, which measures the variation in the PCA model, and the squared prediction error (SPE) or Q statistic, which determines the deviation of measurements from the PCA model.8−10 Although the PCA technique has been successfully applied to industrial process monitoring, the stationarity assumption is challenged in PCA, because normal process changes and process drifts are frequently reflected in the process variables. Therefore, the monitoring of such processes requires the adaptation of the PCA model to accommodate this behavior.11 Several recursive approaches for adaptive process monitoring have been reported in the literature.12−19 The most important challenge faced by recursive adaptations of the PCA model is the high computational costs, because of repeated eigenvalue decomposition (EVD) or singular value decomposition (SVD). Recently, Elshenawy et al.20 introduced two recursive principal component analysis (RPCA) approaches, motivated by developments in the field of signal processing, especially for © 2012 American Chemical Society

Received: Revised: Accepted: Published: 9812

January 9, 2012 June 9, 2012 June 20, 2012 June 20, 2012 dx.doi.org/10.1021/ie300072q | Ind. Eng. Chem. Res. 2012, 51, 9812−9824

Industrial & Engineering Chemistry Research

Article

where 0 ≤ α < 1 is a forgetting factor. The updated correlation matrix according to the FOP analysis technique is given by21

than other known adaptive techniques, such as the inverse iteration approach [O(m 3 )], 45 the Lanczos approach [O(m2lk)],12 fast moving-window PCA (MWPCA) [O(m3)],44 and MWPCA using the incremental method [O(m2lk)],19 where m and lk are the numbers of process variables and retained principal components (PCs), respectively. This is motivated by the conceptual simplicity of the FOP analysis technique. The eigenpairs of the data covariance matrix can be derived from perturbation expansions of the eigenvalue decomposition (EVD)21 and singular value decomposition (SVD).22 The fault detection step depends on recursively calculating the monitoring indices T2 and Q of the PCA model and updating the mean data value and variance, the retained PCs, and the confidence limits. The adaptive fault isolation methods proposed in this article are the (i) recursive partial decomposition contribution (RPDC) and (ii) recursive diagonal contribution (RDC) methods. The contribution indices for the T2 and Q statistics of the proposed fault isolation methods are updated corresponding to the adaptive eigendecomposition of data correlation matrix. Four types of sensor faults, including bias, drifting, precision degradation, and complete failure, are simulated to test the proposed approaches. This article is organized as follows: Section 2 provides a brief review of the RPCA approach based on the FOP analysis technique and provides the necessary background of the fault detection step. The development of the recursive fault isolation methods are given in section 3. This is followed by a description of the proposed recursive FDI approaches in section 4. Section 5 demonstrates the effectiveness of the proposed approaches by application to a simulated chemical process. Finally, section 6 concludes the topics discussed in this article and proposes some future research points.

R k = R k − 1 + ε(xkxk T − R k − 1)

where ε is a small positive number (ε → 0). Next, an eigendecomposition of Rk is derived from perturbation expansions λk , i = (1 − ε)λk − 1, i + fi 2

(3)

m

vk , i = vk − 1, i +

∑ bjivk− 1,j (4)

j=1

with fi = vk − 1, i Txk

(5)

⎧0 j=i ⎪ ⎪ f j fi bji = ⎨ j ≠ i , i , j = 1, ..., m ⎪− bij = ⎪ (λk − 1, i − λk − 1, j) ⎩

(6)

where λk,i and vk,i represent the kth eigenvalue−eigenvector pair of Rk. A stabilization mechanism is necessary to prevent an undesirable behavior of the algorithm, which can occur when the difference between two successive eigenvalues is very small.21,22 Fault detection is usually the first step in multivariate process monitoring. An important issue for RPCA-based process monitoring is to update the process monitoring statistics, namely, Hotelling’s Tk2 = xkTP̂kΛ̂k−1P̂ kTxk and Qk = xkTP̃ kTxk, which are used to represent the variability in the principal component subspace (PCS) and the residual subspace (RS), respectively. Each of the previous statistics can be plotted against time, and their control limits (Tδk2 and Qδk) can be updated as detailed in Li et al.12 The complete FOP-analysisbased recursive PCA is given in Elshenawy et al.20

2. PRELIMINARIES Because a PCA model is structured from the data correlation matrix of the scaled process data, its adaptation can be achieved efficiently by calculating the data correlation matrix recursively. For the original data matrix X ∈ 9 n × m that records n samples of m variables, Rk = [1/(n − 1)XTX] = (P̂ kΛ̂kP̂ kT + P̃kΛ̃kP̃ kT) ∈ 9 m × m is the data correlation matrix. P̂k ∈ 9 m × lk and P̃k ∈ 9 m × (m − lk) are the principal and residual loadings matrices, respectively. Λ̂k and Λ̃k are diagonal matrices of the first lk largest eigenvalues and m − lk smallest eigenvalues of Rk, respectively, in descending order. The number of PCs, lk, determines the acceptable degree of compressibility without losing much information. Because the number of PCs can vary over time,12 it is necessary to determine this number adaptively. There are numerous methods for determining lk including cumulative percent variance (CPV), average eigenvalue (AE), imbedded error function (IEF), and variance of the reconstruction error (VRE). Good overviews on methods for selecting this number can be found in refs 9, 12, 46, and 47. To adapt the process monitoring model online, both the data mean mk ∈ 9 m and the variance σk,i2 ∈ 9 of a new sample data vector xk ∈ 9 m are required to update

3. RECURSIVE FAULT ISOLATION This section details the two proposed recursive fault isolation methods. The first is the recursive partial decomposition contribution (RPDC) method, and the second is the recursive diagonal contributions (RDC) method. The main idea of the proposed fault isolation methods is based on the recursive calculated eigendecomposition of the sample correlation matrix to build the recursive PCA monitoring model. The proposed RPDC and RDC methods identify the faulty variables that make high contributions to the corresponding fault detection index with different measures, as discussed later in subsections 3.1 and 3.2. For simplicity, The Hotelling’s T2 and Q statistics can be rewritten as Tk 2 = xk TAk xk

(7)

and Q k = xk TBk xk

P̂kΛ̂k−1P̂kT

(8)

P̃ kP̃ kT.

where Ak = and Bk = Let ζi be the ith column of the identity matrix and represent the direction of xi, for example, ζi = [0 0 ··· 1 ··· 0]T. This article introduces two adaptive contribution indices corresponding to the Hotelling’s T2 and Q statistics of the RPDC and RDC isolation methods. Based on these contribution indices, a fault can be isolated in an effective manner.

mk = αmk − 1 + (1 − α)xk , σk , i 2 = ασk − 1, i 2 + (1 − α)(xk , i − mk , i)2 ,

(2)

i = 1, ..., m (1) 9813

dx.doi.org/10.1021/ie300072q | Ind. Eng. Chem. Res. 2012, 51, 9812−9824


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3.1. Recursive Partial Decomposition Contribution Method. The traditional partial decomposition contribution (PDC) method was originally proposed by Nomikos and MacGregor48 for the T2 index and further generalized for the Q index by Alcala and Qin.24 It partially decomposes a fault detection index as the summation of variable contributions. This yields what are called contribution indices. The main difference between the RPDC method developed in this article and the traditional PDC is that the former depends on recursive calculated eigenpairs of Rk. In other words, the adaptive contribution of the T2 statistic can be calculated as

4. RECURSIVE FAULT DETECTION AND ISOLATION This section structures complete adaptive fault detection and isolation (FDI) approaches. With every new data sample xk, the eigenpairs of the recursively calculated correlation matrix Rk, λk,i and vk,i, are directly determined using eqs 3−6. Once the eigenpairs are updated, the contribution indices of the proposed fault isolation methods can be calculated according to eqs 10 and 12 in the case of the RPDC method and eqs 14 and 16 in the case of the RDC method. The proposed recursive FDI approaches can be divided into two stages: (i) training the monitoring model off-line and (ii) updating the monitoring model online. The application of the proposed recursive FDI approaches can be summarized by the following steps: Off-line Training Procedure (1) Collect the training data matrix X ∈ 9 n × m and normalize the data to zero mean mk−1 and unit variance σ2k−1,i. (2) Compute the data correlation matrix Rk−1 ∈ 9 m × m and the corresponding eigenpairs λk−1,i, vk−1,i. (3) Construct the PCA model. (4) Calculate the number of retained PCs lk−1 by using a method such as CPV. (5) Calculate the control limits of the fault detection indices, T2δk−1 and Qδk−1. Online Monitoring Procedure (1) Collect a new data vector xk and normalize it to zero mean and unit variance using eq 1. (2) Calculate the monitoring indices Tk2 and Qk.12 (3) If the monitoring indices do not exceed their control limits for the last samples, then the process is in normal mode. The updating procedure can be continued with the following steps: (i) Update mk and σki using eq 1. (ii) Calculate the updated eigenpairs λk,i and vk,i according to eqs 3−6. (iii) Calculate the retained PCs, lk. (iv) Calculate the control limits, Tδk2 and Qδk. (v) Go to step 1. Otherwise, go to the next step. (4) The hypothesis that the process is out of statistical control is accepted. The updating algorithm stops, and the proposed adaptive fault isolation methods (RPDC and RDC) work to identify the process variable responsible for the detected fault. (i) Calculate the contribution indices of the RPDC method using eqs 10 and 12. (ii) Calculate the contribution indices of the RDC method using eqs 14 and 16. (iii) Check the process variables with largest contributions. The complete strategy of the proposed adaptive FDI approaches is depicted in Figures 1−3.

m

Tk 2 = xk TAk xk = xk TAk Ixk = xk TAk (∑ ζiζi T)xk i=1 m

=

∑ xk TAk ζiζi Txk

(9)

i=1

Hence, the recursive contribution of each variable in the T2 index is 2

T C PD = xkAk ζiζi Txk k ,i

(10)

where I ∈ 9 m × m is the identity matrix, with I = ∑mi=1ζiζiT. Similarly to the Tk2 index, the contribution index of the Q statistic can be determined as m

Q k = xk TBk xk = xk TBk Ixk = xk TBk (∑ ζiζi T)xk i=1 m

=

∑ xk TBk ζiζi Txk

(11)

i=1

and the contribution of the ith variable is calculated recursively as Q C PD = xk TBk ζiζi Txk k ,i

(12)

3.2. Recursive Diagonal Contribution Method. The proposed recursive diagonal contribution (RDC) method is described as follows: Starting with the recursive contribution of the T2 statistic based on the diagonal contribution method m 2

T

T

m

T

Tk = xk Ak xk = xk IAk Ixk = xk (∑ ζiζi)Ak (∑ ζiζi)xk i=1

i=1

m

=

∑ xTζiζi TAk ζiζi Txk

(13)

i=1 2

the contribution of each ith variable in the T statistic is 2

C DTk ,i = x Tζiζi TAk ζiζi Txk

(14)

The definition of the contribution in the Q statistic is calculated according to the relation m

m

Q k = xk TBk xk = xk TIBk Ixk = xk T(∑ ζiζi T)Bk (∑ ζiζi T)xk i=1

5. SIMULATION RESULTS This section describes the application of the proposed adaptive FDI approaches to a simulated chemical process. This process is a nonisothermal continuous stirred tank reactor (CSTR).15,36 In addition, it is shown that the conventional PCA model interprets the normal changes in the process, such as variations in parameters and shifts in the operating point, as faults, resulting in numerous false alarms. Moreover, the traditional

i=1

m

=

∑x

T

T

T

ζiζi Bk ζiζi xk

i=1

(15)

Finally, the ith variable contribution based on the diagonal contribution method is C DQk ,i = x Tζiζi TBk ζiζi Txk

(16) 9814



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Figure 1. Off-line training procedure of the proposed adaptive FDI approaches.

Figure 4. Schematic diagram of the nonisothermal CSTR process.

Table 1. Process Variables of the CSTR 1 2 3 4 5 6 7 8 9

variable

symbol

units

coolant temperature reactant mixture temperature solute concentration solvent concentration solvent flow rate solute flow rate coolant flow rate outlet concentration outlet temperature

Tc Ti Ca Cs Fs Fa Fc C T

K K kmol/m3 kmol/m3 m3/min m3/min m3/min kmol/m3 K

Table 2. Process Variable Variations

Figure 2. Outline of the online FDI strategy of the proposed RPDC method.

description

magnitude

slow-varying parameter

βrk = βrk − r1(k − 300) + r2(k − 700)

set-point change

Ck = Ck + r3

r1 = 10−3 r2 = 10−3 r3 =.05

Table 3. Sensor Fault Simulation case 1 2 3 4

bias drift precision degradation complete failure

faulty sensor

fault description

fault time

Ti Fs Ti T

f k,1 ≈ U(2,3) f k,2 = 0.05(k − kf) f k,3 ∼ N(0,5) T = 366

700 701 698 701

contribution methods (PDC and DC) cannot correctly isolate the root cause of the faults. A detailed description of this process is given next, followed by analyses of two types of natural changes in this process and a number of sensor faults to demonstrate the benefits of the proposed FDI approaches. 5.1. Process Description. A schematic diagram of the CSTR process is shown in Figure 4. The process presents a first-order chemical reaction (A → B), in which reactant A, premixed with a solvent, converts into product B at a rate of r = βr k 0e−E /(RT )C

(17)

The dynamic behavior of this process is described by V

dC = F(C − C i) − Vr dt

VρCp Figure 3. Outline of the online FDI strategy of the proposed RDC method.

dT UA = ρCpF(Ti − T ) − (T − Tc) dt 1 + UA/2Fcρc Cpc + ( −ΔHr)Vr

9815

(18)

(19)



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Figure 5. Time series plot and updated mean values of Fa, Fc, and Ti (case 1).

Figure 6. Fault detection results of the conventional PCA and RPCA-FOP approaches for a bias fault in the reactant-mixture temperature sensor Ti (case 1).

where UA is the heat-transfer coefficient, which is related to the coolant flow rate by the empirical relation UA = βUAaFcb. The inlet reactant concentration Ci obtained from the two feed streams, the solute and the solvent, is calculated as Ci =

FaCa + FC s s Fa + Fs

xk , i = ϕxk − 1, i + vk

or

xk , i = d sin(wk) + vk

(21)

where the process noise is vk ∼ (0, σv2). ϕ, w, and d ∈ 9 are deterministic variables. In addition, all measured variables are contaminated with white Gaussian noise ek ∼ (0, σe2). The process variables are recorded for process monitoring purposes at a sampling time interval of 1 s. A more detailed description of this process can be found in Elshenawy et al.20 5.2. Fault Detection and Isolation Results. To build the normal PCA model, 200 samples were generated for training purposes. The data were scaled to zero mean and unit variance. Another 800 samples were generated as test data to simulate the time-varying characteristics and abnormal behavior. The timevarying characteristics were represented by two types of natural changes in this process: (i) slow parameter variation (i.e., a slow drift in process parameter βr) and (ii) an abrupt change in one of the process set points (i.e., the outlet concentration C changed from 0.8 to 0.85 kmol/m3 at sampling time 300). The details of these changes are listed in Table 2. The abnormal behavior was simulated by one of the sensor faults, bias, drift, precision degradation, or complete failure, that was superimposed on the process variables xk,i to produce xk,if = xk,i + f k,i. The sensor faults

(20)

The outlet temperature (T) and concentration (C) are under closed-loop control, provided by proportional−integral controllers. The manipulated variables, the coolant flow rate (Fc) and the solute flow rate (Fa), are used to control the outlet temperature (T) and concentration (C), respectively. Other process variables, which are listed in Table 1, are recorded in an open-loop operation mode. Two stochastic disturbances that influence the performance of the process are caused by poisoning of the reaction and fouling of the cooling coils. These stochastic disturbances are represented in the parameters βr and βUA, respectively. The process variables and disturbances are simulated using first-order autoregressive models or sinusoidal signals 9816



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Figure 7. Fault isolation results of the traditional PDC and proposed RPDC methods for a bias fault in the reactant-mixture temperature sensor Ti (case 1).

Figure 8. Fault isolation results of the traditional DC and proposed RDC methods for a bias fault in the reactant-mixture temperature sensor Ti (case 1).

and (iv) for complete failure, xk,if = c, where f k,i represents the fault magnitude in the ith sensor and the parameters a, b, c, and σ are constants.49 In this study, sensor faults were simulated in the reactant mixture temperature Ti, solvent flow rate Fs, and outlet temperature T, as listed in Table 3. In each simulation case, the time-varying characteristics and one of the sensor faults were considered. The number of retained PCs, lk, was calculated using the CPV method such that the variance explained was approximately 99% of the total variance.47 The resulting monitoring statistics and the charts detailing contributions to the T2 and Q statistics are detailed in the following subsections. 5.2.1. Case 1: Bias Sensor Fault. In the first simulation case, a slow parameter variation in process parameter βr with a slope of −10−3 s−1 was considered as a natural change in this process. The change of this parameter started at k = 301 and continued to k = 700. According to eq 17, the reaction rate decreased as βr

Figure 9. Time series plot and updated mean value of Fs (case 2).

have the following representations: (i) for bias, f k,i = a; (ii) for drift, f k,i = b(k − kf); (iii) for precision degradation, f k,i ∼ N(0, σ2); 9817



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Figure 10. Fault detection results of the conventional PCA and RPCA-FOP approaches for a drift fault in the solvent flow rate sensor Fs (case 2).

Figure 11. Fault isolation results of the traditional PDC and proposed RPDC methods for a drift fault in the solvent flow rate sensor Fs (case 2).

decreased. Consequently, the coolant flow rate, Fc, and the solute flow rate, Fa, decreased to maintain the outlet temperature T and the outlet concentration C, respectively. Time series plots of the process variables, Fa and Fc, are shown in the two upper plots in Figure 5. In addition to this time-varying characteristic, a bias fault following a uniform distribution within the interval2,3 was superimposed on the reactant mixture temperature sensor Ti at k = 700 until the end of the simulation, as shown in the lower plot in Figure 5. The results for the monitoring indices obtained by the conventional PCA and RPCA-FOP approaches are shown in Figure 6. It appears that the monitoring statistics of the RPCAFOP approach can effectively handle the time-varying characteristic, so the false alarm rate decreases dramatically, as shown in the two right plots in Figure 6. In contrarst, the monitoring statistics of the conventional PCA approach give a large number of violations

when the process variables begin to change, as shown in the two left plots in Figure 6. By calculating the contribution indices of each process variable, the contribution charts for the T2 and Q statistics of the traditional PDC method were determined, as shown on the left side of Figure 7. It appears that the Q statistic is more sensitive for isolating this fault, but there are still large contributions from another variable, namely, Fc. In addition, the contribution charts for the T2 statistic show large contributions from the process variable Fc. This causes some confusion for the process operator about the root cause of the fault. In contrast, in the contribution charts for the T2 and Q statistics of the proposed RPDC method shown on the right side in Figure 7, the main contributions are due to the process variable Ti. These results provide a clear indication about the correct faulty sensor. 9818



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Figure 12. Fault isolation results of the traditional DC and proposed RDC methods for a drift fault in the solvent flow rate sensor Fs (case 2).

the two right plots in Figure 8. Furthermore, the two left plots in Figure 8 for the traditional DC method were not able to provide accurate knowledge regarding the cause of this fault. 5.2.2. Case 2: Drift Sensor Fault. A data set of 800 samples was considered in case 2, including a time-varying characteristic and a drift sensor fault that occurred after each other. The timevarying characteristic was the same as in the first simulation case. The sensor fault was a drift in the solvent flow rate sensor Fs, as shown in Figure 9. It started at k = 701 and continued until the end of the simulation. The results for the monitoring indices of the conventional PCA approach are shown in Figure 10a. The monitoring charts violated these control limits after the natural change in the process variables began, which increased the number of type I errors. On the other hand, the monitoring statistics of the RPCA-FOP approach were more credible in the parameter

Figure 13. Time series plot and updated mean value of Ti (case 3).

The proposed RDC method was also applied to this fault scenario. It was able to isolate the faulty sensor Ti as making large contributions to both the T2 and Q statistics, as shown in

Figure 14. Fault detection results of the conventional PCA and RPCA-FOP approaches for a precision degradation fault in the reactant mixture temperature sensor Ti (case 3). 9819



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Figure 15. Fault isolation results of the traditional PDC and proposed RPDC methods for a precision degradation fault in the reactant mixture temperature sensor Ti (case 3).

Figure 16. Fault isolation results of the traditional DC and proposed RDC methods for a precision degradation fault in the reactant mixture temperature sensor Ti (case 3).

contributors, Fs and Fc, are the main influence for this fault type. Recursively calculating the contribution indices for the proposed RDC method improved the fault isolation results as presented in the two right plots in Figure 12. 5.2.3. Case 3: Precision Degradation Sensor Fault. Again, a slow variation in the process parameter βr with the details given above was considered. A sensor fault of the precision degradation type was injected into the process variable reactant mixture temperature Ti as indicated in Figure 13. The monitoring statistics for the time-varying characteristic and this sensor fault are shown starting at k = 300 in Figure 14. At the time when the conventional PCA approach gave false alarms in the range of parameter variation, the RPCA-FOP approach handled the natural change and dramatically decreased the number of type I errors. At the same time, the

variation range, as shown in Figure 10b. Once the fault occurred, the monitoring charts detected it, but with some delay. This is because the fault was a type of drift, which is difficult to allocate early because of its minor impact. The contribution charts for the traditional PDC method are presented in Figure 11a. In this case, large contributions from both the solvent flow rate Fs and the coolant flow rate Fc can be noticed. This leads to the conclusion that there are abnormal events in these two process variables but cannot provide precise fault isolation. In contrast, the contribution charts for the proposed RPDC presented in Figure 11b indicate that the process variable Fs is the main influence of the fault in case 2. The two left plots in Figure 12 show the corresponding T2 and Q contribution charts for the traditional DC method. These charts indicate that the two variables that are the main 9820



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Figure 17. Time series plot and updated mean values of C, T, Fa, and Fc (case 4).

Figure 18. Fault detection results of the conventional PCA and RPCA-FOP approaches for a complete failure fault in the outlet temperature sensor T (case 4).

Following the time-varying characteristic in this simulation scenario, the monitoring statistics for the conventional PCA approach still violate the corresponding confidence limits, as shown in Figure 18a. The improvement of the monitoring process performance by recursive updating of the eigenstructure using the FOP-RPCA approach leads to a decrease in the rate of false alarms, as depicted in Figure 18b. Once the fault was detected, the updating process was terminated according to the proposed adaptive FDI approaches described in section 4. Because the outlet temperature T is a controlled variable, its effect is propagated into other variables. The propagated effects of the fault are noticed in the manipulated variable coolant flow rate Fc, so there are trends of the contributions of T and Fc to Hotelling’s T2 and Q statistics. The contribution charts for the traditional PDC and DC methods are presented in Figures 19a and 20a, respectively. To identify the root cause of this fault, the contribution charts for the T2 and Q

RPCA-FOP approach could detect this sensor fault without delay. In contrast to the traditional PDC and DC methods shown in Figures 15a and 16a, respectively, the proposed RPDC and RDC methods could isolate the faulty sensor correctly. The contribution charts, given in Figures 15b and 16b, indicate that the main contributing variable is the reactant mixture temperature Ti, which describes the fault in this sensor. 5.2.4. Case 4: Complete Failure Sensor Fault. In this simulation scenario, an abrupt change in the outlet concentration C occurred, starting at k = 300. The manipulated variable Fa consequently increased to regulate the outlet concentration. The increase in the solute flow rate led to an increase in the process outlet temperature T, which had an impact on the coolant flow rate Fc, as shown in Figure 17a,c,d. A complete failure of the outlet temperature sensor T is presented in Figure 17b, starting at k = 701. 9821



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Figure 19. Fault isolation results of the traditional PDC and proposed RPDC methods for a complete failure fault in the outlet temperature sensor T (case 4).

Figure 20. Fault isolation results of the traditional DC and proposed RDC methods for a complete failure fault in the outlet temperature sensor T (case 4).

statistics of the proposed RPDC and RDC methods, shown in Figures 19b and 20b, respectively, are difficult to interpret because this fault is considered to be complex. Hence, no conclusive identification of the root cause of this fault could be offered. Instead, the fault isolation problem was only narrowed down to two variables. To measure the capability of the adaptive FDI approaches, two indices were used. The first is the false alarm rate (FAR), which is considered a type I error. The second is the correct fault isolation rate (IR). The false alarm rate is measured by calculating the number of violated samples compared to the number of normal data points as FAR =

number of violated samples × 100% number of normal data points

The isolation rate is determined by calculating the number of data points of the faulty variable contribution (FVC) that can exceed the corresponding monitoring control limits according to the relation IR =

FVC × 100% number of faulty data points

(23)

Table 4 summarizes the false alarm rates of the conventional PCA and RPCA-FOP approaches. The results show the efficiency of the RPCA-FOP approach in terms of decreasing the false alarm rate. Furthermore, the isolation rates of the traditional fault isolation methods (PDC and DC) and the proposed recursive fault isolation methods (RPDC and RDC) are listed in Table 5. The correct fault isolation rates increase in the case of proposed recursive fault isolation methods (RPDC and RDC) compared to the traditional methods (PDC and

(22) 9822



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monitoring model online including (i) adaptation of the data mean and variance, (ii) adaptation of the retained PCs and confidence limits, (iii) approximation of the eigendecomposition of the correlation matrix based on FOP analysis procedure with low computational cost O(m2),20 and (iv) adaptation of the contribution charts for the T2 and Q statistics of two fault isolation methods. To evaluate the utility of the proposed recursive FDI approaches, this article presented simulation data from a nonisothermal continuous stirred tank reactor (CSTR) chemical process. Two types of natural changes occurring in this process were considered to implement time-varying characteristics. Four types of sensor faults, namely, bias, drift, precision degradation, and complete failure, were superimposed on the process sensor variables. This simulation study confirmed that the proposed recursive FDI approaches are effective in adapting time-varying process behavior, while also being able to detect and successfully isolate abnormal events. In the case where the T2 and Q statistics were used simultaneously for fault isolation, the simulation results showed that the Q statistic in both of the proposed fault isolation methods gave better isolation results than T2. This difference arises from the fact that the T2 and Q statistics measure different variabilities in the data. Future work will concentrate on how to improve the adaptive FDI scheme in isolating complex faults. Further, it is proposed to study and develop schemes for fault detection and isolation in complex nonlinear and time-varying processes.

Table 4. False Alarm Rates of the Conventional PCA and RPCA-FOP Approaches in a Simulated CSTR Process conventional PCA

RPCA-FOP

case

T2

Q

T2

Q

1 2 3 4

40.5714 33 40.5714 10.7143

34.7143 30.2857 34.7143 4.4286

1.7143 2.4286 1.7143 0.4286

1.4286 4.4286 1.4286 0.8571

DC) through the simulation scenarios. These improvements were due to the adaptive nature of the proposed methods. Table 5. Fault Isolation Rates of the Traditional (PDC and DC) and the Proposed (RPDC and RDC) Methods in a Simulated CSTR Process PDC 2

RPDC 2

case

T

Q

T

1 2 3 4

0 0 0 91

100 86 74 96

97 42 47 92

DC 2

RDC 2

Q

T

Q

T

Q

100 86 77 98

0 0 0 0

100 82 73 100

99 41 48 0

100 88 76 100

Moreover, the Q statistics were more sensitive to the faults than the T2 statistics. This can be interpreted from the increase in the corresponding isolation rates. In summary, the proposed recursive FDI approaches could handle the time-varying characteristics and increase the monitoring process reliability, in contrast to the conventional PCA approach. Although the first three sensor faults discussed in this article were simple faults (other process variables were unaffected), there were large contributions from more than one process variable in the case of the traditional contribution methods (PDC and DC). This implies that, even with these simple faults, the traditional contribution charts did not provide a completely unambiguous isolation. In contrast, the proposed recursive contribution methods (RPDC and RDC) could isolate most sensor faults successfully without ambiguity.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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6. CONCLUSIONS This article has studied the detection and isolation of abnormal process events using conventional PCA and two traditional contribution methods, namely, the (i) partial decomposition contribution (PDC) and (ii) diagonal contribution (DC) methods. Through the simulation of time-varying process, three problems can be encountered: (i) an increase in the false alarm rate (type I error) that can threaten the credibility of the process monitoring procedure; (ii) an inability to correctly isolate, without ambiguity, even simple sensor faults; and (iii) the high computational complexity of adaptive PCA approaches, such as inverse iteration [O(m3)]45 and the Lanczos technique [O(m2lk)],12 that can be used to adapt the eigenstructure of the correlation matrix to build the PCA model and the contribution charts. To address these problems, this article proposed two recursive fault detection and isolation approaches. The fault detection method is based on a numerically efficient and memory saving RPCA-FOP procedure. The fault isolation methods are the (i) recursive partial decomposition contribution (RPDC) and (ii) recursive diagonal contribution (RDC) methods. This article proposes an adaptation of the process 9823



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Recursive Fault Detection and Isolation Approaches of Time-Varying

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