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Incipient sensor fault diagnosis using moving window reconstruction-based contribution Hongquan Ji, Xiao He, Jun Shang, and Donghua Zhou Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b03944 • Publication Date (Web): 24 Feb 2016 Downloaded from http://pubs.acs.org on February 25, 2016

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Industrial & Engineering Chemistry Research

Incipient sensor fault diagnosis using moving window reconstruction-based contribution Hongquan Ji,† Xiao He,† Jun Shang,† and Donghua Zhou∗,‡,† †Tsinghua National Laboratory for Information Science and Technology (TNList) and Department of Automation, Tsinghua University, Beijing, 100084, P.R. China ‡College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, 266590, P.R. China E-mail: [email protected] Abstract Reconstruction-based contribution (RBC) is widely used for fault isolation and estimation in conjunction with principal component analysis (PCA) based fault detection. Correct isolation can be guaranteed by RBC for single sensor faults with large magnitudes. However, the incipient sensor fault diagnosis problem is not well handled by traditional PCA and RBC methods. In this paper, the limitations of traditional PCA and RBC methods for incipient sensor fault diagnosis are illustrated and analyzed. By introducing a moving window, a new strategy based on the PCA model is presented for incipient fault detection. Regarding incipient fault isolation and estimation, a new contribution analysis method called moving window RBC is proposed to enhance the isolation performance and estimation accuracy. Rigorous fault detectability and isolability analyses of the proposed methods are provided. Besides, effects of the window width on fault detection, isolation, and estimation are discussed. Simulation studies on a numerical example and a continuous stirred tank reactor (CSTR) process are used to demonstrate the effectiveness of the proposed methods.

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1

Introduction

Statistical process monitoring (SPM) has received considerably increasing attention in research and has been applied to the monitoring of industrial processes successfully. 1–3 Several important tasks in SPM include fault detection, fault identification or isolation, and fault estimation. 4 Principal component analysis (PCA) and its variants have been widely used for the fault detection task, which are capable of handling high-dimensional and noisy data. 5 In PCA-based methods, measurements are typically partitioned into the principal component subspace (PCS) and the residual subspace (RS). Fault detection can then be implemented in the subspaces by comparing the detection indices against their corresponding thresholds. Three commonly used fault detection indices are Hotelling’s T 2 statistic, squared prediction error (SPE), and the combined index. 6 As the downstream step of fault detection, fault isolation aims to determine the faulty variables or identify the true fault from a set of candidates. To date a great number of fault isolation methods have been reported in the process monitoring area, 7–12 among which the contribution plot approach 7,13 is the most popular one. Usually the contribution plot approach is used with the PCA model, and identifies faulty variables according to the calculated contribution of each variable. One advantage of this approach is that it requires no a priori knowledge about the fault except for a normal PCA model. However, this approach suffers from the smearing effect that can lead to incorrect isolation in many cases. 2,14 To improve the fault isolation performance, reconstruction-based contribution (RBC) was proposed. 15 Correct isolation can always be guaranteed by the RBC method in the case of single sensor faults with large magnitudes. 15,16 Furthermore, fault estimation can be accomplished by RBC, as the reconstruction process leads to the estimation of the fault magnitude along the reconstructed direction. 4 In this paper, the terminology fault diagnosis is used to represent the ensemble of fault detection, isolation, and estimation. Though a large number of fault detection and isolation (FDI) methods have been proposed, most of them consider the monitoring of serious faults. In practice, however, the FDI 2 ACS Paragon Plus Environment

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task for incipient faults is also of vital importance so as to ensure safe and optimal process operation. Since the incipient fault usually has a small magnitude, it is more difficult to be detected and further isolated than the serious one. Recently, some novel approaches have been presented to address the problem of incipient FDI. For a class of nonlinear distributed processes with incipient faults, a robust detection and accommodation scheme was proposed. 17 However, the mathematical model of the monitored process needs to be known in this method. Combining signed directed graph and qualitative trend analysis, a new method 18 was proposed for incipient fault isolation. A modified support vector machine method using continuous decision function was presented so as to meet the characteristics of incipient faults and further isolate them effectively. 19 Both of the two methods require a priori knowledge of the fault, e.g., the historical datasets of various types of faults. Harmouche et al. 20 employed the Kullback-Leibler divergence to discriminate the probability densities of latent scores and then detect incipient faults within the PCS. A novel monitoring strategy combining feature extraction and residual evaluation was proposed, 21 yet only the fault detection task was involved in this method. The PCA and RBC approaches, as mentioned previously, have several elegant properties when used for fault diagnosis. They do not require the knowledge of rigorous process model or the historical faulty data. For the RBC method, the contribution of each variable is easy to calculate, and correct isolation can be guaranteed for simple sensor faults. In addition, the fault isolation and fault estimation tasks can be accomplished simultaneously. Since the proposal of conventional RBC, it has also been extended and modified 16,22–24 recently. Nevertheless, the incipient sensor fault diagnosis problem cannot be well handled by the PCA and RBC methods. On the one hand, traditional fault detection indices utilized in PCA are not sensitive to the incipient fault. This is because the fault magnitude is relatively small and thus does not satisfy the sufficient detectability condition. On the other hand, for single sensor faults correct isolation is only guaranteed by the RBC method if the fault magnitude is large enough. 15 When incipient sensor faults are considered, traditional RBC



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may suffer from the possibility of wrong isolation. Moreover, the fault estimate using RBC will be of low precision due to the randomness in the process data. The objective of this paper is to provide improved methods for the incipient sensor fault detection, isolation, and estimation problems. At present only the incipient single sensor fault is involved in this work. First, a motivational example is employed to illustrate the limitations of traditional PCA and RBC methods for incipient fault diagnosis. Then, a new fault detection strategy based on the PCA model is presented. Fault detectability analysis of the new strategy is provided. Regarding fault isolation and estimation, the reasons why incorrect isolation may occur and fault estimate is of low accuracy using traditional RBC are analyzed theoretically. By introducing a moving window, an improved RBC method termed as moving window RBC (MWRBC) is proposed to enhance the fault isolation performance and fault estimation accuracy. Fault isolability analysis of the proposed MWRBC method is given. Meanwhile, the estimation accuracy is improved in terms of variance of the estimated fault magnitude. Besides, influences of the window widths on fault detection, isolation, and estimation, including the advantages and disadvantages, are discussed and summarized. The remainder of this paper is organized as follows. Basic theories are reviewed briefly in Section 2. Section 3 includes the main results. Specifically, a numerical example is first used to demonstrate the limitations of conventional PCA and RBC methods for incipient fault diagnosis. Then, by employing a moving window, a new fault detection strategy based on conventional PCA, and the MWRBC method based on conventional RBC are proposed. Rigorous fault detectability and isolability analyses of the proposed methods are provided. Besides, effects of the window sizes for incipient fault diagnosis are evaluated. Section 4 presents simulations and comparison studies to illustrate the effectiveness of the proposed methods, followed by concluding remarks in Section 5.


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2

Preliminaries

2.1

PCA Modeling

Suppose a sample vector x ∈ Rm consists of m sensors and there are N samples for each variable. A data matrix denoted by X ∈ RN ×m can be formed with each row representing a sample. X is then scaled to zero mean and unit variance for correlation-based PCA modeling. The covariance of the normalized measurements x is approximated by the sample covariance S, which can be eigen-decomposed as 15

S=

1 ˜Λ ˜P ˜T XT X = PΛPT + P N −1

(1)

˜ ∈ Rm×(m−l) are the principal and residual loading matrices, respecHere, P ∈ Rm×l and P tively. The parameter l represents the number of principal components (PCs) retained in the model. 25 PCA model partitions the measurement space into the PCS and the RS, which are orthogonal and complementary. 4 Consequently, a sample vector x ∈ Rm can be decomposed as 2 ˆ = PPT x = Pt x

(2)

˜P ˜ Tx ˜ = (I − PPT )x = P x

(3)

where t = PT x ∈ Rl is the score vector of x.

2.2

Fault Detection

In PCA, three statistical indices, i.e., SPE, T 2 , and a combination of the two, are commonly used for fault detection. SPE measures variations in the RS, which is defined as the squared



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˜ norm of the residual vector x

˜P ˜ T x = xT Cx ˜ SPE = ∥˜ x∥2 = xT P

(4)

˜ ,P ˜P ˜ T . The T 2 index is a measure of variations in the PCS, which is defined as where C T 2 = tT Λ−1 t = xT PΛ−1 PT x = xT Dx

(5)

where D , PΛ−1 PT . The combined index ϕ, proposed by Yue and Qin, 6 integrates the SPE and T 2 indices in a balanced way

ϕ=

SPE T 2 + 2 = xT Φx δ2 τ

(6)

˜ 2 + D/τ 2 , and δ 2 , τ 2 are the control limits of SPE and T 2 , respectively. where Φ , C/δ Denote ζ 2 as the control limit of the combined index ϕ. The control limits of these fault detection indices can be obtained using the χ2 approximation based on the results of Box. 26 Notice that all the fault detection indices given by (4)-(6) can be expressed as the general quadratic form 15

Index(x) = xT Mx = ||x||2M

(7)

˜ D, and Φ for SPE, T 2 , and ϕ respectively. Under the assumption that where M denotes C, x is multi-normal, the χ2 upper control limit of Index(x), is calculated as 4,13

η 2 = g Index χ2α (hIndex )

(8)

where g

Index

tr(SM)2 = , tr(SM)

h

Index

[tr(SM)]2 = tr(SM)2


(9)

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2.3

RBC for Fault Isolation and Estimation

The RBC approach for fault isolation and fault estimation is reviewed briefly in this subsection according to Alcala and Qin. 15 The following fault model is assumed x = x∗ + ξj f

(10)

which indicates an additive fault occurring in sensor xj . Here, x∗ denotes the normal part, ξj is the jth column of the identity matrix, and f denotes the fault magnitude. RBC makes reconstruction along all variable directions ξi , i = 1, . . . , m in turn and determines the possible faulty variable according to their reconstruction contributions. The reconstructed sample along direction ξi is z i = x − ξ i fi

(11)

where fi is the estimated fault magnitude. The fault detection index of the reconstructed sample zi is Index(zi ) = ||zi ||2M = ||x − ξi fi ||2M

(12)

which should be minimized. The least squares solution for fi is fi = (ξiT Mξi )−1 ξiT Mx

(13)

Then, the RBC of variable xi to the fault detection index is defined as RBCIndex = ||ξi fi ||2M i

(14)

which is the amount of reconstruction along the ith variable direction that minimizes the fault detection index. The substitution of (13) in (14) gives RBCIndex = xT Mξi (ξiT Mξi )−1 ξiT Mx = i


(ξiT Mx)2 ξiT Mξi

(15)


˜ D, Φ in (15), The RBCs for the SPE, T 2 , and ϕ indices can be obtained by setting M = C, respectively. The control limits for the RBCs can also be determined using the results of Box, 26 since they have the quadratic form as well. Due to the smearing effect in the RBCs, nevertheless, the control limits cannot be utilized to identify which variable is the cause of the fault. The fault isolation task can only be achieved based on the magnitudes of the RBCs. In the case of single sensor faults, variable xi with the largest RBCIndex is identified as the most likely i faulty variable. 15

3

Proposed Methods

The main results are presented in this section. First, a motivational example is used to demonstrate the limitations of traditional methods for incipient sensor fault diagnosis. Then, by introducing a moving window, new approaches are proposed to accomplish the incipient sensor fault detection, isolation, and estimation tasks. In addition, influences of the window size on fault diagnosis are analyzed theoretically, which can guide the selection of the window width. FAR:1.5% FDR:20.33% SPE

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Figure 1: Fault detection using PCA in the numerical example.


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3.1

A Motivational Example

A simulated multivariate process model is employed to illustrate that, for the incipient single sensor fault which is considered in this work, traditional PCA for detection and traditional RBC for isolation and estimation cannot provide satisfying results. The process model to be used is 15,22

   x1  −0.2310    x  −0.3241  2        x3  −0.2170  =    x4  −0.4089       x  −0.6408  5     −0.4655 x6

−0.0816 0.7055 −0.3056 −0.3442 0.3102 −0.4330

 −0.2662    −0.2158   t1   −0.5207    t  + noise   2   −0.4501   t3 0.2372    0.5938

(16)

where t1 , t2 and t3 are zero-mean Gaussian variables with standard deviations of 1.0, 0.8 and 0.6, respectively. The noise included in (16) is zero-mean with a standard deviation of 0.1 and normally distributed. 1000 normal samples are generated according to the process model and the data are scaled to zero mean and unit variance to build the PCA model. In this paper, the cumulative percent variance (CPV) criterion 25 is utilized to choose the number of PCs. The threshold of CPV is set as 90% and three PCs are retained. In addition, test data comprising 1000 samples are generated according to (16) and a step change of x2 by 0.6 is introduced starting from sample 401. The SPE, T 2 , and ϕ monitoring charts of the test data are shown in Fig. 1. Moreover, the false alarm rate (FAR) and fault detection rate (FDR) for each statistic are calculated and displayed. The T 2 statistic fails to detect this sensor bias fault. Though the SPE and ϕ statistics in Fig. 1 increase after sample 400, which may indicate a possible fault, their FDRs are two low. This is because the fault magnitude imposed on x2 is relatively small, and the sufficient conditions of fault detectability 27,28 for all three statistics are not satisfied. Thus, the incipient sensor fault cannot be effectively detected by traditional PCA-based fault detection strategy. 9 ACS Paragon Plus Environment


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4 3 2 1 0 400

600

800

1000

samples (b)

Figure 2: Fault isolation using RBC in the numerical example. (a) RBC values for all variables; (b) Variable with largest RBC value. As we can see in Fig. 1, the SPE index has a little higher FDR than the combined index ϕ. However, the combined index is preferred in fault isolation as ϕ-based RBC could always give a higher rate of correct isolation. 15 Supposing that the fault has been successfully detected, we apply traditional RBC to perform fault isolation. Fig. 2 shows the fault isolation results obtained with ϕ-based RBC. For each sample in the test dataset, the ϕ-based RBC value of each variable is calculated. After that, they are normalized and visualized in Fig. 2 (a). It can be observed that the main candidates of faulty variables are x2 , x5 , and x6 , yet the true faulty variable is x2 . To clearly show the faulty variable identified by the RBC method, for each faulty sample, the variable with the largest contribution value is shown in Fig. 2 (b). The rate of correct fault isolation, i.e., the percentage of variable x2 having the largest contribution among all variables after sample 400, is 58.2%. That is, for more than 40 percent of the faulty samples, the fault isolation result using traditional RBC method is incorrect. Under the assumption that the single sensor fault is isolated correctly, fault estimation can then be accomplished according to (13) directly. Fig. 3 plots the estimation result of the


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1.2 estimated value true value

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0.4 0.2 0 −0.2 −0.4 −0.6

0

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samples

Figure 3: Fault estimation of f2 using RBC in the numerical example. fault magnitude imposed on x2 , and the true value is also marked with the red line. As seen in Fig. 3, due to the randomness in the normal portion x∗ , the estimated fault magnitude varies in a random manner accordingly. Though an unbiased estimate of the fault magnitude can be obtained using (13), the randomness in x∗ reduces the accuracy of fault estimation, especially for incipient faults.

3.2

New Fault Detection Strategy

By introducing a moving window, a new strategy for fault detection is presented first in this subsection. Then, through theoretical analysis, it is pointed out that given an appropriate window width, the proposed detection strategy can guarantee the detectability of the incipient sensor fault. As illustrated in Fig. 1, traditional PCA-based fault detection indices are not sensitive to the incipient fault. To improve the fault detection ability, we propose to use the following fault detection index ¯ Tk M¯ xk = ||¯ xk ||2M Indexw (xk ) = x

(17)

where xk is the kth test sample and M is the same as in (7). The subscript w in (17) implies



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¯ k is defined as the use of a moving window, and the averaged sample x 1 ¯k = x W

k ∑

xw

(18)

w=k−W +1

where W represents the window width for fault detection. Comparing (17) with (7), we have the following relation Indexw (xk ) = Index(¯ xk )

(19)

That is, the proposed fault detection statistic of a sample xk is just the traditional fault ¯k. detection statistic of the averaged sample x To accomplish the fault detection task, the upper control limit of the new fault detection index under normal conditions should be determined. As we can see from (17), the new fault detection index has the quadratic form as well. Thus, similar to the control limit of Index(x) as expressed in (8), the control limit of Indexw (x) = Index(¯ x) can also be ¯ = E(¯ ¯ ∗T ) respectively calculated using the χ2 approximation. Denote S = E(x∗ x∗T ) and S x∗ x as the covariance matrices of the original and averaged samples under normal conditions. Under the assumption that the normal samples are i.i.d., we have var(¯ xi ) = var(xi )/W and cov(¯ xi , x¯j ) = cov(xi , xj )/W theoretically, leading to the following result ¯= 1S S W

(20)

Denote ηw2 as the control limit of Indexw (x). By integrating (9) and (20), it can be easily verified that ηw2 =

1 2 η W

(21)

That is, the new threshold for fault detection is reduced by the introduction of a moving window. This point is not difficult to understand since the use of moving average reduces the normal variations in x∗ . Before analyzing the performance of the new fault detection strategy, the concept of 12 ACS Paragon Plus Environment

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traditional PCA-based fault detectability is reviewed briefly. Dunia and Qin 27 first derived the sufficient condition for fault detectability using the SPE index, and later the condition was extended to the T 2 and ϕ indices. 6,28 For traditional PCA-based fault detection strategy, the sufficient condition is ||M1/2 ξj f || > 2η

(22)

If the sufficient condition is not satisfied, the fault cannot be guaranteed detectable. This is the reason why traditional PCA-based fault detection strategy fails to effectively detect the incipient sensor fault, as illustrated in Fig. 1. For the incipient sensor fault with a constant bias, indeed the new fault detection strategy is more effective than the traditional one. Define the critical window width for fault detection as

( WFD =

2η 1/2 ||M ξj || · |f |

)2 (23)

As for the detectability of the new fault detection strategy, we have the following theorem. Theorem 1. With the proposed fault detection strategy (17) and (21), the incipient sensor fault as expressed in (10) can be guaranteed to be detectable if W > WFD . The proof of this theorem is given in the Supporting Information. This theorem indicates the effectiveness of using the moving window technique for the incipient sensor fault detection task. It can be observed from (23) that the critical window width depends on the original PCA model (M, η) and the imposed sensor fault (ξj , f ). Besides, it is noted that the smaller the fault magnitude (f ) is, the larger the critical window width (WFD ) is required. If the critical window width calculated is less than one, then their is no need to use a moving window technique since in this case the sufficient condition (22) is naturally satisfied.

3.3

MWRBC for Isolation and Estimation

In this subsection, the limitations of traditional RBC for incipient sensor fault isolation and estimation are analyzed. To obtain enhanced fault isolation and estimation results, the 13 ACS Paragon Plus Environment


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MWRBC method is proposed by employing a moving window. Besides, influences of the window width on fault isolation and estimation are quantitatively analyzed. For incipient sensor fault isolation, traditional RBC suffers from the chance of incorrect identification, as illustrated in Fig. 2. This is due to the influence of the normal variations in the process, which weaken the impact of the faulty portion in a faulty sample. Substituting the faulty sample x in (15) with (10), we have

RBCIndex = i

[ξiT M(x∗ + ξj f )]2 (ξiT Mx∗ + ξiT Mξj f )2 = ξiT Mξi ξiT Mξi

(24)

Correct isolation using RBC is guaranteed only if RBCIndex ≥ RBCIndex , ∀i ̸= j holds, i.e., j i (ξjT Mx∗ + ξjT Mξj f )2 (ξiT Mx∗ + ξiT Mξj f )2 ≥ ξjT Mξj ξiT Mξi

(25)

|ξjT Mx∗ + ξjT Mξj f | |ξiT Mx∗ + ξiT Mξj f | ≥ (ξjT Mξj )1/2 (ξiT Mξi )1/2

(26)

which is equivalent to

Owing to the positive semi-definiteness of M, the following relation is always guaranteed 15 |ξjT Mξj f | |ξiT Mξj f | ≥ (ξjT Mξj )1/2 (ξiT Mξi )1/2

(27)

Nevertheless, due to the randomness in x∗ , the inequality (26) does not always hold. Therefore, RBCIndex ≥ RBCIndex , ∀i ̸= j cannot be guaranteed all the time, especially for the fault j i with small magnitude f . Based on the above analysis, it can be concluded that reducing the impact of the normal portion (x∗ ) so that the fault impact (ξj f ) is emphasized in the faulty sample (x) is of vital importance. To improve the fault isolation performance, a novel contribution analysis method called MWRBC is proposed. Denote RBCIndex as the RBC of variable xi to the fault i,k


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detection index for the kth sample xk , i.e., RBCIndex = i,k

(ξiT Mxk )2 (ξiT Mx∗k + ξiT Mξj f )2 = ξiT Mξi ξiT Mξi

(28)

By introducing a moving window, the MWRBC is defined as ( MWRBCIndex = i,k

1 W

√ RBCIndex i,w

k ∑

)2 =

w=k−W +1

(ξiT M¯ xk )2 ξiT Mξi

(29)

¯ k is the same as in (18). In the case where W is the window width for fault isolation, and x that all samples within a time window are faulty, we can further express (29) as follows

MWRBCIndex = i,k ¯ ∗k = where x

1 W

∑k w=k−W +1

(ξiT M¯ x∗k + ξiT Mξj f )2 ξiT Mξi

(30)

x∗w is the mean value of the normal portions of the faulty samples.

Comparing (30) with (28), we can find that the MWRBC method weakens the randomness in the normal portion. Thus, improved fault isolation performance using MWRBC is expected. Before analyzing the fault isolation performance of the proposed MWRBC method quantitatively, the concept of isolability for traditional RBC is first introduced. This concept provides the requirement on the fault magnitude so that only the true fault can be isolated as the candidate uniquely. In other words, when the actual fault Fj happens, any other fault Fi , ∀i ̸= j cannot be identified as the “true” fault. For the reconstruction-based fault identification method, Dunia and Qin 27 first presented the sufficient condition for fault isolability using the SPE index. Recently, this condition was extended to the fault detection index of general quadratic form. 28 The sufficient isolability condition for the reconstruction-based identification method is ||(Im − M1/2 ξi (ξiT Mξi )−1 ξiT M1/2 )M1/2 ξj f || > 2η


(31)


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Owing to the close relation between the reconstruction-based identification and RBC methods, the condition imposed by (31) can also act as a sufficient isolability condition for traditional RBC method, as described in Theorem 2. Theorem 2. The condition expressed in (31) is a sufficient isolability condition for the RBC method. The proof of this theorem is given in the Supporting Information. For incipient fault with a small fault magnitude, usually the sufficient isolability condition is not satisfied. As a result, the fault cannot be guaranteed to be isolated correctly. This is the reason why traditional RBC suffers from the chance of incorrect isolation when utilized for incipient sensor fault isolation, as illustrated in Fig. 2. Employing the moving average technique, the proposed MWRBC method is able to provide satisfying results for incipient sensor fault isolation. It is shown that given an appropriate window width, correct fault isolation can be guaranteed by the MWRBC method. Define the critical window width for fault isolation as ( WFI =

2η T 1/2 ||(Im − M ξi (ξi Mξi )−1 ξiT M1/2 )M1/2 ξj || · |f |

)2 (32)

Theorem 3 reveals the effectiveness of using a moving window for incipient sensor fault isolation explicitly. Theorem 3. With the MWRBC method, the incipient sensor fault as expressed in (10) can be guaranteed to be isolable, i.e., MWRBCIndex > MWRBCIndex , ∀i ̸= j, if W > WFI . j i The proof of this theorem is given in the Supporting Information. It is noted from (32) that the critical window width WFI is influenced not only by the PCA model (M, η) and the imposed sensor fault (ξj , f ), but also by other fault candidates (ξi , ∀i ̸= j). Moreover, the smaller the fault magnitude (f ) is, the larger the critical window width for fault isolation (WFI ) is required. 16 ACS Paragon Plus Environment

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Following fault isolation, fault estimation needs to be implemented so as to assess the fault magnitude. The influence of the normal variations in the process data on fault estimation is apparent if we substitute x in (13) with (10) and calculate fj as follows fj = (ξjT Mξj )−1 ξjT Mx = (ξjT Mξj )−1 ξjT M(x∗ + ξj f ) = (ξjT Mξj )−1 ξjT Mx∗ + f

(33)

It can be noted from (33) that, though fj is an unbiased estimate of f since E(x∗ ) = 0, the fault estimation accuracy is reduced by the randomness in x∗ . This mechanism has been vividly illustrated by Fig. 3 in Subsection 3.1. To improve the fault estimation accuracy, the moving average technique is applied to reduce the influence of randomness in x∗ . Rewrite (33) as fj,k = (ξjT Mξj )−1 ξjT Mxk = (ξjT Mξj )−1 ξjT Mx∗k + f

(34)

where the subscript k means the kth sampling time. The proposed fault estimation strategy is 1 f¯j,k = W

k ∑

fj,w = (ξjT Mξj )−1 ξjT M¯ xk

(35)

w=k−W +1

¯ k is the same as in (18). If all where W is the window width for fault estimation, and x ¯ k can be calculated as x ¯k = x ¯ ∗k + ξj f . Then, (35) samples within a time window are faulty, x can further be simplified as f¯j,k = (ξjT Mξj )−1 ξjT M¯ x∗k + f

(36)

By comparing (36) with (34), it is noted that the influence of randomness in x∗ on the estimated magnitude is reduced, while f¯j,k is still an unbiased estimate of f . Thus, an improvement of the fault estimation accuracy provided by the proposed strategy is expected. This improvement is quantitatively evaluated by Theorem 4, whose proof is given in the Supporting Information. 17 ACS Paragon Plus Environment


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Theorem 4. The variance of the estimated fault magnitude using the proposed estimation strategy (36) is inversely proportional to the window width. x2 x1

- original normal region - new normal region x1*

x2

z1 z2

x*2 x3

0

x1

S

x*3

PC

z3

Figure 4: Randomness of the fault estimate and comparison of estimation strategies between RBC and MWRBC. To geometrically illustrate the randomness of fj,k caused by the variations in x∗ , as well as the superiority of f¯j,k over fj,k for fault estimation, a simple two-variable system with one PC is considered, as shown in Fig. 4. Without loss of generality, the SPE statistic is utilized for fault reconstruction and estimation. The rectangular area within black dashed lines represents the normal variation region for x∗ . This region is determined by the upper control limits (δ 2 , τ 2 ) of the SPE and T 2 indices. Three normal samples x∗k , k = 1, 2, 3, denoted by △, lie within the normal variation region. A sensor bias fault along ξ2 = [0 1]T is imposed on x∗k , resulting in the faulty sample xk . Fault reconstruction along ξ2 leads to the reconstructed sample zk denoted by ◦. In this example, zk lies exactly on the PCS and thus has a SPE statistic equal to zero. Due to the normal variations in x∗ , the reconstructed sample zk is not necessarily the same as x∗k , and their difference represents the inaccuracy of fault estimation. The inaccuracy can be described by the variance of the estimated fault magnitude fj,k , which depends directly on the range of the normal variations in x∗ . When we ¯ ∗ instead of x∗ use the proposed fault estimation strategy f¯j,k , the averaged normal sample x 18 ACS Paragon Plus Environment

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¯ ∗ is significantly is involved. Based on the result in (21), the new normal variation region for x shrunk, as illustrated by the rectangular area within red dash-dotted lines. Consequently, the fault estimation accuracy using f¯j,k is enhanced. It should be pointed out that the proposed fault estimation strategy (35) in this work is similar to but more general than the estimation strategy utilized by Xuan et al. 22 In their work, the estimated fault magnitudes obtained by the SPE-based fault reconstruction were averaged within a sampling window, so as to reduce “the effect of random noises”. In fact, as we can see from (35) and (36), the moving window technique for fault estimation aims to reduce the effect of randomness in the normal portion x∗ . To accomplish fault isolation, the average residual-difference reconstruction contribution plot (ARdR-CP) was proposed in the literature. 22 The residual-difference contribution plot for the ith variable, zi , is defined as 22 ˜ n || = ||˜ xf − x 2

m ∑

zi

(37)

i=1

˜ n and x ˜ f represent the residuals under normal and possible faulty ˜ n = Cx ˜ f = Cx where x ˜ and further, averaged conditions. Then, zi is divided by c˜ii (the ith diagonal element of C) within a time window, resulting in the ARdR-CP for the ith variable. Notice that, in the ARdR-CP method, xn is used to calculate the new contribution for online fault isolation. By contrast, the proposed MWRBC method in this paper, as defined by (29), uses the averaged test sample for online fault isolation. The normal sample xn is not involved in the calculation of MWRBC. As for incipient sensor fault detection, the work of Xuan et al. 22 accomplished this task by noting that the contribution values of the fault-free and faulty conditions are not the same order of magnitude. This strategy, however, without using a valid threshold, may be somewhat subjective. By contrast, the detection strategy proposed in this paper makes use of modified fault detection indices based on traditional PCA. Thus, though the fault estimation strategy utilized in this paper can be viewed as an extension of the one used in the literature, 22 their fault detection and isolation strategies are fundamentally different.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

˜ D, or Φ, the proposed detection, isolation, Besides, since M in (17), (29), and (35) can be C, and estimation methods can be applied not only to the SPE, but also to the T 2 and ϕ indices. More importantly, the rationality of using a moving window and quantitative evaluation are provided in this paper, as illustrated by Theorems 1, 3, and 4.

3.4

Selection of the Window Width

In this paper, the moving window technique is combined with traditional PCA and RBC methods to accomplish incipient sensor fault diagnosis. The window width, as a key parameter, has a direct influence on incipient sensor fault detection, isolation, and estimation. This subsection analyzes and summarizes the benefits and drawbacks of using a moving window, which can provide some suggestions on the selection of the window width. The advantages of using a moving window for incipient sensor fault diagnosis are apparent, as discussed in the previous subsections. For fault detection, we can conclude from (23) and Theorem 1 that, the larger the window width is, the smaller the fault magnitude can be guaranteed detectable. Similarly, it is noted from (32) and Theorem 3 that a larger window width for fault isolation can guarantee the isolability of a smaller fault magnitude. Regarding fault estimation, the proposed fault estimation strategy by introducing a moving window does not change the unbiasedness of the fault estimate. Moreover, it can be concluded from Theorem 4 that a larger window width implies a higher precision for fault estimation. Nevertheless, two disadvantages of using too large window width should also be considered when we attempt to choose an appropriate window width. First, it can be observed from (17), (29), and (35) that the introduction of a moving window increases the amount of computation for fault detection, isolation, and estimation. Second, the timeliness of fault detection, correct isolation, and unbiased estimation will be restricted if the window width is too large. The time delay for fault detection is derived first, followed by those for isolation and estimation. To guarantee the detectability of the incipient fault, the window width for fault 20 ACS Paragon Plus Environment

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detection should be larger than the critical window width WFD . According to the proof of Theorem 1, the fault is guaranteed detectable if all samples within a moving window are faulty. To derive the time delay, we analyze the initial stage of the fault, that is, the samples within a time window are normal or faulty. Denote W (W > WFD ) as the window width for detection, and (LFD − 1) as the time delay for detection. For the time window that contains LFD faulty samples and (W − LFD ) normal samples, the averaged sample within this window is ¯=x ¯∗ + x

LFD ξj f W

(38)

Using the same technique as in the proof of Theorem 1, we can obtain the sufficient de¯ expressed in (38) as tectability condition for x

||M1/2 ξj ·

Solving for LFD gives LFD

√ LFD f || > 2η/ W W

√ √ 2η · W > = W · WFD ||M1/2 ξj || · |f |

(39)

(40)

Thus, the delayed time for fault detection is approximately proportional to the square root of the selected window width. In other words, too large window width may incur serious time delay. The time delay for fault isolation when the proposed MWRBC method is utilized can be derived similarly. Denote (LFI − 1) as the time delay for correct fault isolation. Using (38) where LFI replaces LFD and the proof of Theorem 3, we give the following conclusion without detailed derivation LFI >

√ W · WFI

(41)

This conclusion indicates that a larger window width for fault isolation may cause a more serious time delay for correct isolation, which is very similar to the fault detection case. As for fault estimation, a large window width implies a large delay to obtain the unbiased estimate of the fault. It should be noted that the proposed fault estimation strategy (36) holds if all



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the samples within a time window are faulty. For the case that only LFE (LFE < W ) samples are contained in a window, (36) should be rewritten as LFE f¯j,k = (ξjT Mξj )−1 ξjT M¯ x∗k + f W

(42)

resulting in a biased estimate since E(f¯j,k ) = LFE f /W ̸= f . If LFE = W , the unbiased estimate can then be obtained. Therefore, the delayed time for an unbiased estimate is just equal to the window width for fault estimation minus one. Table 1: Influence of using a large window width on incipient sensor fault diagnosis aspect

advantages

disadvantages

fault detection

the smaller fault can be guar- the amount of computation anteed detectable increases; a large time delay for fault detection

fault isolation

the smaller fault can be guar- the amount of computation anteed isolable increases; a large time delay for correct isolation

fault estimation

fault estimation with a higher precision can be obtained

the amount of computation increases; a large time delay for unbiased fault estimate

To sum up, Table 1 lists the advantages and disadvantages of using a large window width for incipient sensor fault detection, isolation, and estimation. Note that the window sizes for fault detection, isolation, and estimation are not necessarily the same.

4

Simulation Studies

In this section, simulation studies on a numerical example and a continuous stirred tank reactor (CSTR) process are carried out to demonstrate the effectiveness of the proposed fault detection, isolation, and estimation strategies. Regarding fault isolation, the MWRBC method is compared with not only traditional RBC, but also its extensions or modifications available in the literature. 16,22,24 22 ACS Paragon Plus Environment

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4.1

Numerical Simulation

The motivational example described in Subsection 3.1 is employed here to compare the fault diagnosis results given by traditional and proposed methods. The simulation parameters, as well as the introduced fault scenario, remain unchanged.

SPEw

FAR:1.25% FDR:99.33% 0.5

T2w

0

0

200

0

200

0

200

400 600 samples FAR:0% FDR:91.17%

800

1000

800

1000

800

1000

0.5 0

400

600 samples FAR:1% FDR:99.17%

1 φw

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5 0

400

600 samples

Figure 5: Fault detection using new strategy in the numerical example. Fault detection is the first step of process monitoring. The fault detection results using the proposed detection index (17), together with its corresponding control limit (21), are shown in Fig. 5. The window width employed in the proposed detection strategy is a key parameter, since it has a direct effect on the monitoring results. According to Theorem 1, the window width should be at least larger than the critical value WFD so as to guarantee the detection of the incipient fault. However, as analyzed in Subsection 3.4, there may be an undesirable time delay if the window width is too large. In this example, the critical window widths for the SPEw , Tw2 , and ϕw indices are 8, 133, and 11 respectively, which can be calculated according to (23). The monitoring results shown in Fig. 5 are obtained with window widths equal to 8, 100, and 8 respectively for the SPEw , Tw2 , and ϕw indices. The incipient sensor fault is successfully detected by the SPEw and ϕw indices after a delay of a few samples. The Tw2 index can also detect the fault after a relatively large time delay. By comparing Fig. 5 with Fig. 1, the superior detection performance provided by using a



moving window can be observed obviously, which illustrates the effectiveness of the proposed fault detection strategy. 1 6 0.8

variables

5 4

0.6

3

0.4

2

0.2

1 200

400

600 samples (a)

800

1000

0

6 5 variables

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4 3 2 1 0 400

600

800

1000

samples (b)

Figure 6: Fault isolation using MWRBC in the numerical example. (a) MWRBC values for all variables; (b) Variable with largest MWRBC value. Following fault detection, fault isolation should be carried out to determine which variable is faulty. The proposed MWRBC approach (29) is applied to this numerical example for improved fault isolation. The window width for the ϕ-based MWRBC is set as 50, which satisfies the sufficient isolability condition according to Theorem 3. For each sample in the test dataset, the ϕ-based MWRBC value of each variable is calculated. Then, they are normalized and visualized in Fig. 6 (a). It can be observed that, though the true faulty variable is x2 , variables x5 and x6 possess large contribution values as well. This phenomenon indicates that the fault smearing effect 15 exists in the MWRBC method as well. Nevertheless, it should be noted that the logic of fault isolation using MWRBC is that, the variable with the largest contribution is designated as the faulty variable. For each faulty sample in the test dataset (i.e., sample 401 to sample 1000), Fig 6 (b) shows the variable with the largest ϕ-based MWRBC value. As we can see from Fig. 6 (b), incorrect isolation happens only at the initial stage of the fault, where variable x1 or x5 is identified as the faulty variable


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due to smearing. With the sliding of the moving window, more and more faulty samples are incorporated within a time window, and the fault impact is emphasized. Starting from sample 425, correct isolation is guaranteed all the time by the MWRBC approach. The rate of correct isolation using the ϕ-based MWRBC is 96.0%, which is much higher than that using traditional RBC approach. 1 6 0.8

5 variables

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4

0.6

3

0.4

2

0.2

1 200

400

600

800

1000

0

samples (a) 6 5 variables

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4 3 2 1 0 400

600

samples (b)

800

1000

Figure 7: Fault isolation using ARdR-CP in the numerical example. (a) ARdR-CP values for all variables; (b) Variable with largest ARdR-CP value. In addition to the proposed MWRBC, several extended RBC methods 16,22,24 are also applied to this numerical example so as to investigate their fault isolation capacities for incipient sensor fault. In the ARdR-CP approach, 22 as we can see from (37), xn is involved in the calculation of the new contribution for fault isolation. Due to the permitted normal variations in xn , the fault isolation performance will be affected by the selection of xn . The fault isolation results using the ARdR-CP method with an arbitrarily selected xn are shown in Fig. 7. The meaning of Fig. 7 is similar to that of Fig. 6. The correct fault isolation rate using ARdR-CP is 72.2%. The weighted RBC (WRBC) method 24 was proposed to reduce the smearing of the fault coefficient estimates and, therefore, improve isolation accuracy. In this method, α is a key parameter utilized in the weighting matrix. Fig. 8 shows the



1 6 0.8

variables

5 4

0.6

3

0.4

2

0.2

1 200

400

600 samples (a)

800

1000

6 5 variables

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4 3 2 1 0 400

600

800

1000

samples (b)

Figure 8: Fault isolation using WRBC in the numerical example. (a) WRBC values for all variables; (b) Variable with largest WRBC value. fault isolation results provided by the ϕ-based WRBC method with α = 0.3. The meaning of the two sub-figures in Fig. 8 is similar to those in Fig. 6. The rate of correct isolation, i.e., the percentage of variable x2 having the largest WRBC value in Fig. 8 (b), is 41.5%. More recently, a new fault isolation method called RBC ratio (RBCR) 16 was proposed. The isolability analysis of the RBCR method reveals that, the variable with a RBCR value less than or equal to one is identified as the true faulty variable, whereas the variable with a RBCR value greater than one is not the faulty variable. For an arbitrary faulty sample numbered sample 499, the fault isolation result using the ϕ-based RBCR method is illustrated in Fig. 9. It can be concluded that, for the incipient sensor fault imposed actually on x2 , the RBCR incorrectly identifies x2 , x5 , and x6 as the faulty variables simultaneously. Table 2 shows a comparison of correct isolation rate among these fault isolation methods. The traditional RBC, MWRBC, ARdR-CP, RBCR, as well as WRBC with different α values, based on the SPE, T 2 , and ϕ indices, are involved. As analyzed previously, the chance of incorrect isolation for traditional RBC is because the normal variations in x∗ weaken the impact of the incipient fault. Though the ARdR-CP method provides an acceptable fault isolation performance, it is only applicable to the SPE index. Moreover, the correct isolation 26 ACS Paragon Plus Environment

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1.5

1 RBCR

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0.5

0

1

2

3

4

5

6

variables

Figure 9: Fault isolation using RBCR for test sample 499 in the numerical example. Table 2: A comparison of correct isolation rate among different methods in the numerical example method

SPE

T2

ϕ

RBC

47.3%

23.8%

58.2%

MWRBC

94.7% 93.5%

96.0%

ARdR-CP

72.2%

N/A

N/A

RBCR

1.3%

0.0%

0.7%

WRBC: α = 0.1

34.3%

40.7%

34.5%

α = 0.3

41.0%

38.3%

41.5%

α = 0.5

45.3%

33.5%

46.0%

α = 0.7

49.5%

30.3%

49.7%

α = 0.9

57.5%

25.7%

56.8%

rate using ARdR-CP depends considerably on the selection of xn , making its isolation performance unstable. It can be observed from Table 2 that, the RBCR method provides very low isolation rate. This is because almost for all the faulty samples, more than one variable is identified as faulty. In other words, the RBCR values for several variables are simultaneously less than one, as illustrated by Fig. 9. For the WRBC method, different weighting factors (α) lead to different fault isolation performances. However, as seen from Table 2, the



WRBC method cannot provide very satisfying fault isolation results. Therefore, the RBCR and WRBC methods are not suitable for incipient sensor fault isolation. For the MWRBC method, critical window widths for fault isolation using various fault detection indices can be calculated according to (32). The correct isolation rates of the SPE-, T 2 -, and ϕ-based MWRBC methods shown in Table 2 are obtained with window widths equal to 50, 150, and 50 respectively. Obviously, the proposed MWRBC method provides superior fault isolation performance.

estimated value true value

0.8

fault magnitude

0.6

0.4

0.2

0

−0.2

0

200

400

600

800

1000

samples

Figure 10: Fault estimation of f2 using MWRBC in the numerical example.

0.2 fault estimate on x1 fault estimate on x5 0

fault magnitude

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

−0.2

−0.4

−0.6

0

200

400

600

800

1000

samples

Figure 11: Fault estimation of f1 and f5 using MWRBC in the numerical example. As the downstream step of fault isolation, fault estimation needs to be carried out to assess the fault magnitude. It is known that due to the smearing effect, 15,24 the estimated 28 ACS Paragon Plus Environment

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fault magnitudes on non-faulty variables are not necessarily equal to zero. This phenomenon is illustrated by Fig. 10 and Fig. 11 intuitively. Nevertheless, as we can see from (34) and (36), only the estimated fault magnitude on the true faulty variable is unbiased. Thus, the reliability of fault isolation is of vital importance, which determines whether the fault estimation is accurate or not. Notice that at present only single sensor faults are involved in this paper. According to Fig. 6 (b) and the corresponding analysis, it is concluded that only variable x2 is identified as the faulty variable by the proposed MWRBC method. Fault estimation can then be implemented on variable x2 . Fig. 10 shows the estimated fault magnitude on x2 using the proposed estimation strategy (35), where the combined index is utilized and the window width is set as 50. In addition, the true fault magnitude superimposed on variable x2 is also shown in this figure with the red line. By comparing Fig. 10 with Fig. 3, it can be observed obviously that the accuracy of fault estimation is enhanced using the proposed estimation strategy. Meanwhile, the time delay for an unbiased estimate can also be observed from Fig. 10, which is equal to the window width. Table 3: Influence of the window width on estimation in the numerical example W

10

20

50

100

150

var(fj )

3.37e-2

3.35e-2

3.37e-2

3.40e-2

3.41e-2

var(f¯j )

3.31e-3

1.55e-3

5.63e-4

2.82e-4

1.63e-4

21.6

59.8

120.6

209.2

var(fj )/var(f¯j ) 10.2

According to Theorem 4, the relationship between the variances of fj and f¯j can be measured quantitatively. Here fj and f¯j , as depicted in (34) and (36), represent the fault estimates using traditional and proposed fault estimation strategies, respectively. Monte Carlo simulations are carried out to demonstrate the conclusion that var(fj )/var(f¯j ) = W . For each window size, 100 times Monte Carlo experiments are performed. Then, the averaged values of var(fj ) and var(f¯j ), as well as the ratio var(fj )/var(f¯j ) are calculated. Table 3 shows the results. It can be observed that the calculated ratio var(fj )/var(f¯j ) obtained by the Monte Carlo simulations is almost consistent with the theoretical one, i.e. the window 29 ACS Paragon Plus Environment


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width. However, the difference between the calculated ratio and the theoretical one indeed exists, especially when the window size is large. The difference is caused by the imbalance of samples utilized to calculate var(f¯j ) due to the introduction of a moving window.

4.2

Case Study on CSTR

The CSTR process can be described by the component material balance on the reactant and the energy balance as follows: 29,30 ) ( dCA q E CA + v1 = (CAf − CA ) − k0 exp − dt V RT

(43)

) ( q dT −∆H UA E = (Tf − T ) + k0 exp − CA + (Tc − T ) + v2 dt V ρCp RT V ρCp

(44)

where CA is the outlet concentration, T is the reaction temperature, Tc is the temperature of cooling water, q is the feed flow rate, CAf is the input reactant concentration, Tf is the input reactant temperature, and v1 , v2 are independent process noises. Other variables in (43)-(44) are constant parameters for the process. In the simulation, [CA , T ]T are the controlled variables with nominal values, and [q, Tc ]T are chosen as manipulated variables with feedback from control errors. The measurement sample vector consists of six variables, i.e., x = [CA , T, Tc , q, CAf , Tf ]T , to which independent measurement noises are added. All the simulation parameters and conditions, as well as the controller information, are the same as in Li et al. 29 The sampling interval is 1s and 2000 samples are collected under normal operating conditions in order to train the PCA model. After normalizing the collected data, the PCA model is built with four PCs retained according to the CPV criterion. The test dataset comprising 2000 samples is generated based on the simulated process as well. But this time a sensor bias fault with magnitude ∆q = 5 L/min is added to the fourth variable q starting from sample 1001. Note that the fault introduced here is a single sensor fault which affects only the observations of one variable. The fault magnitude (5 L/min) is even less than the 30 ACS Paragon Plus Environment

FAR:0.7% FDR:29% SPE

4 2 0

0

500


1500

2000

0

500


1500

2000

0

500

1000 samples

1500

2000

T2

20 10 0

φ

4 2 0

Figure 12: Fault detection using PCA in the CSTR example. standard deviation (5.6 L/min) of variable q under normal conditions. FAR:5.4% FDR:99% SPEw

1.5 1 0.5 0

T2w

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

500


1500

2000

0

500


1500

2000

0

500

1000 samples

1500

2000

0.2 0.1 0

2 φw

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1 0

Figure 13: Fault detection using new strategy in the CSTR example. For the test dataset containing an incipient sensor fault, the monitoring results using traditional PCA-based detection strategy are shown in Fig. 12. The T 2 index is less sensitive to incipient faults and thus fails to detect the fault in this example. Instead, the SPE and ϕ statistics indicate a potential fault after sample 1000. However, their detection rates are very low, implying the inefficiency of traditional PCA for incipient sensor fault detection. Fig. 13 shows the fault detection results using the proposed strategy, i.e., the new detection index (17) and its corresponding threshold (21). The window width utilized in (17) directly



affects the monitoring results. Similar to the numerical example, the critical window widths for various fault detection indices can be calculated according to (23) as well. On the other hand, an undesirable time delay may be incurred if the window width is too large. In Fig. 13, the window widths for the SPEw , Tw2 , and ϕw indices are set as 4, 200, and 4 respectively. It is obvious that the incipient sensor fault is successfully detected by the proposed fault detection strategy. 1 6 0.8

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Figure 14: Fault isolation using RBC in the CSTR example. (a) RBC values for all variables; (b) Variable with largest RBC value. Fault isolation follows the detection of a fault and aims to find the true faulty variable among all variable candidates. Traditional RBC method is first applied to the test dataset. Fig. 14 shows the fault isolation results using the ϕ-based RBC. For each test sample, the ϕ-based RBC of each variable is calculated. Afterwards, they are normalized and visualized in Fig. 14 (a). It can be observed that due to the smearing effect in RBC, normal variables (such as x3 ) can also have large contribution values. Sometimes the smearing effect leads to incorrect isolation, especially for incipient faults. Notice that the faulty variable identified by the RBC method is the one with the largest RBC. For each faulty sample in the test dataset (i.e., sample 1001 to sample 2000), Fig. 14 (b) depicts the variable with the largest RBC value.


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Figure 15: Fault isolation using MWRBC in the CSTR example. (a) MWRBC values for all variables; (b) Variable with largest MWRBC value. As we can see from Fig. 14 (b), occasionally the normal variable x3 is incorrectly identified as the faulty variable by the RBC method. Fig. 15 shows the fault isolation results using the ϕ-based MWRBC method with the window width equal to 100. The smearing effect is not eliminated by the MWRBC, as illustrated by Fig. 15 (a). However, the MWRBC method indeed provides improved fault isolation performance. As seen from Fig. 15 (b), variable x4 , i.e., the true faulty variable, is correctly isolated for all faulty samples by the MWRBC method. Table 4 lists a comparison of correct isolation rate among different methods. These methods include the proposed MWRBC, conventional RBC, 15 ARdR-CP, 22 RBCR, 16 and WRBC. 24 The window widths used by the SPE-, T 2 -, and ϕ-based MWRBC methods are 100, 300, and 100 respectively. For the RBC and MWRBC methods, the ϕ index provides a higher isolation rate than the SPE and T 2 indices. Thus, the combined index is usually preferred for isolation in practice. As mentioned previously, the ARdR-CP method can be applied only to the SPE index. Meanwhile, how to choose the normal sample xn will affect its isolation performance. The RBCR method fails to isolate the incipient sensor fault since



Table 4: A comparison of correct isolation rate among different methods in the CSTR example method

SPE

T2

ϕ

RBC

53.3%

9.2%

80.5%

MWRBC

82.7% 51.7%

100.0%

ARdR-CP

69.8%

N/A

N/A

RBCR

0.0%

0.0%

6.4%

WRBC: α = 0.1 59.2%

11.4%

58.6%

α = 0.3

62.4%

12.8%

61.9%

α = 0.5

67.5%

14.7%

66.2%

α = 0.7

74.5%

16.1%

72.0%

α = 0.9

74.8%

17.1%

79.5%

the fault magnitude does not meet the sufficient isolability condition of RBCR. As seen in Table 4, the isolation rate of WRBC varies along with the change of α. Nevertheless, the WRBC method is even not superior to the ϕ-based RBC method. The proposed MWRBC method, by contrast, provides satisfying fault isolation results. 10 estimated value true value

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Figure 16: Fault estimation of f4 using RBC in the CSTR example. Fault estimation follows correct fault isolation and aims to assess the fault magnitude. For the test dataset containing a sensor fault on x4 , Figs. 16∼18 show respectively, the fault estimate on x4 using RBC, the fault estimate on x4 using MWRBC, and the fault estimate on 34 ACS Paragon Plus Environment

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x3 using MWRBC. It should be noted that due to the smearing effect, which exists in both the RBC and MWRBC methods, fault estimates on normal variables are not necessarily equal to zero. This phenomenon is intuitively illustrated by Fig. 18, where a fault “appears” after sample 1000; in fact, no fault has been added to x3 . Despite the smearing, the proposed MWRBC method successfully isolates x4 as the unique faulty variable, as illustrated by Fig. 15 (b). Thus, only the estimation of f4 , i.e., the fault estimate on x4 , makes sense. Comparing Fig. 17 with Fig. 16, we can conclude that, though unbiased estimates can be obtained by both two methods, the MWRBC reduces the variance of fault estimate and thus improves the estimation accuracy. At the same time, a time delay can be observed in Fig. 17 due to the introduction of a moving window. 10 estimated value true value

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Figure 17: Fault estimation of f4 using MWRBC in the CSTR example.

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Concluding Remarks

This paper has involved the incipient sensor fault detection, isolation, and estimation problems in the framework of SPM. By introducing a moving window, new fault detection indices based on traditional PCA model, together with their corresponding control limits, are presented for incipient fault detection. The new detection indices are more sensitive than traditional PCA-based ones to the incipient sensor fault. Concerning the isolation and estimation tasks for incipient sensor fault, a new contribution analysis method termed as MWRBC is 35 ACS Paragon Plus Environment


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Figure 18: Fault estimation of f3 using MWRBC in the CSTR example. proposed, which utilizes the moving window technique as well. The proposed MWRBC method overcomes the drawbacks of traditional RBC for incipient sensor fault isolation and estimation. Rigorous fault detectability and isolability analyses of the proposed methods are provided. It is pointed out that, with the proposed methods, the incipient sensor fault can be guaranteed detectable (isolable) if the window width is larger than the critical window width for detection (isolation). The estimation accuracy using MWRBC is improved in a way that the variance of the fault estimate is reduced while the unbiasedness property remains unchanged. Despite the merits provided by the moving window, two disadvantages of using too large window width for incipient sensor fault diagnosis are also discussed in detail. Two simulated experiments have illustrated the effectiveness of the proposed fault detection strategy and the MWRBC method. A comparison study of the fault isolation performance among different methods is implemented. In the future work, fault diagnosis for more complicated faults, such as process faults or multiple sensor faults, even with small fault magnitudes will be considered.

Acknowledgement This work was supported by the National Natural Science Foundation of China (61490701, 61290324, 61473163), Tsinghua University Initiative Scientific Research Program, and Re36 ACS Paragon Plus Environment

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search Fund for the Taishan Scholar Project of Shandong Province of China.

Supporting Information Available Proofs of the theorems. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Graphical TOC Entry comparison of detection

comparison of isolation

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rates

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comparison of estimation

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