Exponential smoothing reconstruction approach for incipient fault

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Process Systems Engineering

Exponential smoothing reconstruction approach for incipient fault isolation Hongquan Ji, Xiao He, Jun Shang, and Donghua Zhou Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00478 • Publication Date (Web): 16 Apr 2018 Downloaded from http://pubs.acs.org on April 16, 2018

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Industrial & Engineering Chemistry Research

Exponential smoothing reconstruction approach for incipient fault isolation Hongquan Ji,† Xiao He,‡ Jun Shang,‡ and Donghua Zhou∗,†,‡ †College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, 266590, P.R. China ‡Department of Automation, Tsinghua University, Beijing, 100084, P.R. China E-mail: [email protected]

Abstract Accurate detection and isolation of an incipient fault is crucial for efficient and optimal operation of industrial processes. In the area of statistical process monitoring (SPM), reconstruction-based methods are widely used for fault isolation purposes. Though possessing several merits, most existing conventional reconstruction methods are not efficient in isolating incipient faults that have small magnitudes. This paper proposes a new method called exponential smoothing reconstruction (ESR) for incipient fault isolation. The proposed ESR introduces the exponentially weighted moving average (EWMA) technique to conventional reconstruction methods. Fault isolability analysis of the ESR is provided, which demonstrates its efficiency in isolating incipient faults, compared with conventional reconstruction-based methods. Several remarks on the use of ESR, such as selection of the smoothing parameter and some practical issues, are discussed as well. Finally, simulation studies on a numerical example and a continuous stirred tank heater (CSTH) process are used to demonstrate the effectiveness of the proposed ESR method, in comparison with conventional approaches.

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1

Introduction

The past several decades have witnessed the rapid development of statistical process monitoring (SPM). 1 Commonly used conventional SPM methods include principal component analysis (PCA), partial least squares or projection to latent structures (PLS), Fisher discriminant analysis (FDA), etc. Since the pioneering work 2 which introduces multivariate statistical models to process monitoring, a great many novel and modified methods have been proposed for advanced process monitoring, such as dynamic PCA, total PLS, canonical correlation analysis (CCA), slow feature analysis (SFA), as well as their variants. 3–7 For detection purposes, within each specific SPM method, multivariate statistics together with their control limits ought to be established. Typical fault detection statistics include the Hotelling’s T 2 index, squared prediction error (SPE), combined index, as well as global Mahalanobis distance, all of which can be expressed as quadratic forms of the sample vector. 8,9 Although widely used, the aforementioned fault detection methods or indices may not be sensitive to incipient faults, which have small magnitudes. To guarantee that the fault can be detected successfully, certain detectability conditions should be satisfied. If the fault magnitude is too small to meet the condition, then the fault may be left undetected. Nevertheless, detection of an incipient fault is of vital importance. In practice, many harmful abnormal conditions originate and gradually evolve from incipient faults. Successful detection and subsequent proper proposal of incipient faults can prevent serious faults from happening and ensure a safe and optimal operating status of the monitored process. To cope with the incipient fault detection problem, some efforts have been made in the SPM area. 10–13 Chen et al. 10 proposed to apply PCA to data handled by multivariate exponentially weighted moving average (EWMA) and multivariate s-term sum, respectively. It was demonstrated that new T 2 and SPE statistics are more efficient than conventional ones in detecting incipient faults. Harmouche et al. 11 proposed an incipient fault detection method based on a probability distribution measure, which applies Kullback-Leibler divergence to scores obtained via PCA. More recently, a hybrid strategy combining wavelet analysis with 2


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residual evaluation was presented for incipient fault detection. 12 Shang et al. 13 proposed a new method called recursive transformed component statistical analysis for the purpose of incipient fault detection and compared it with many conventional methods. In our previous work, 8 two smoothing techniques including moving average (MA) and EWMA were introduced to SPM, resulting in two incipient fault detection strategies. The strategies are based on a generic fault detection index, thus can be applied to various SPM models in addition to PCA. Following fault detection, fault isolation aims to determine the kind and location of the detected fault. 14 After the fault is correctly isolated, efficient component maintenance or replacement can then be accomplished, thus reducing costly maintenance activities to a great extent. Consequently, fault isolation is as important as fault detection. In the field of SPM, the main purpose of fault isolation is to find out the faulty variables if no a priori fault information is available or identify the actual fault among a set of fault candidates. 3 Though some work, as mentioned above, was proposed to deal with incipient fault detection, the incipient fault isolation task is relatively rarely considered. To achieve efficient isolation for incipient faults, novel methods or improvements of conventional methods are required so as to cater for the characteristics of incipient faults. For example, considering the feature that incipient faults progress over time, Namdari and Jazayeri-Rad 15 proposed to perform incipient fault diagnosis using SVM by monitoring continuous decision functions instead of discrete ones. This method is actually a classification method, thus historical datasets of various fault types are needed. Alternatively, contribution analysis based methods are also commonly used for fault isolation purposes. 16–19 Different from fault classification methods such as FDA, SVM, and neural networks related methods, contribution analysis methods usually do not need the historical data of different faults. The pioneering work using contribution analysis for fault isolation is the traditional contribution plot (TCP). 20 However, this method suffers from the smearing effect which may result in incorrect isolation in many cases. 3 Later, the reconstruction-based

3



contribution (RBC) method, which originates from the fault isolation via reconstruction method, 21–23 was proposed for improved fault isolation. 9 The RBC combines the concepts of fault reconstruction and contribution plot. It is pointed out that, though suffering from the smearing effect as well, RBC can guarantee correct isolation in the case of single sensor faults with a large magnitude. Based on the conventional RBC method, a large number of modifications and improvements have been proposed in the literature. 24–26 Nevertheless, these reconstruction-based methods may not be sensitive to incipient faults. To guarantee correct isolation, certain isolability conditions should be satisfied by the occurred fault. 22,27 If the fault magnitude is small, these methods tend to give incorrect isolation results due to the smearing effect. The objective of this paper is to provide a novel method for incipient fault isolation. First, conventional reconstruction-based methods for fault isolation are briefly reviewed and summarized. The fault isolability analysis for these methods is provided, which demonstrates that these methods may be inefficient in isolating faults with small magnitudes. Also, this point is illustrated by a motivational example. Then, based on conventional reconstruction methods, a novel method called exponential smoothing reconstruction (ESR) is proposed for incipient fault isolation. The proposal introduces the EWMA technique to conventional reconstruction-based fault isolation methods. Through fault isolability analysis, it is concluded that the isolability condition of ESR, compared with that of conventional methods, is much easier to satisfy by the fault. Thus, the proposed ESR method is more efficient for incipient fault isolation. The remainder of this paper is organized as follows. Section 2 briefly reviews and summarizes conventional reconstruction-based methods for fault isolation. In Section 3, a motivational example is first provided to demonstrate the inefficiency of conventional methods for incipient fault isolation. Then, a method called ESR is proposed, whose efficiency is illustrated by the fault isolability analysis. In addition, several practical issues when using ESR are noted. Case studies on a numerical example in the literature and the continuous stirred

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tank heater (CSTH) process are carried out in Section 4, followed by concluding remarks in Section 5.

2

Reconstruction-based methods for fault isolation and isolability analysis

In this section, many reconstruction-based methods in the literature for fault isolation purposes 9,21–23,26,27 are briefly reviewed and summarized. These methods can be generally divided into two categories, i.e., the isolation via reconstruction 22 and RBC 9 methods. The isolation via reconstruction method is first reviewed, followed by its isolability analysis. Then, a brief introduction to the RBC method is provided. It is pointed out that, in terms of usage and isolability property, the RBC method is in fact equivalent to the isolation via reconstruction method, because the fault reconstruction idea is utilized in both methods.

2.1

Fault isolation via reconstruction

In SPM, the isolation via reconstruction method, proposed by Dunia et al., 21,22,28,29 was first applied to the SPE statistic of PCA. Later, it was extended to be used with the combined index of PCA. 23 Recently, Mnassri et al. 27 generalized this method to various fault detection statistics of quadratic form within the PCA framework. In fact, in addition to PCA, this method can also be used with many other SPM methods such as PLS, provided that the detection statistic is of a quadratic form. The following generic fault detection index is considered 8,9,27

Υ (x) = xT Mx = kM1/2 xk2

(1)

where x ∈ Rm is a measurement sample and M ∈ Rm×m is a symmetric and positive definite or semi-definite matrix. Alcala and Qin 9 first used Υ (x) to uniformly represent the SPE, T 2 , 5



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and combined index in the PCA model. Later, Mnassri et al. 27 expressed the Hawkins’ TH2 statistic within PCA by Υ (x) as well. In fact, besides the PCA model, Υ (x) can also represent many fault detection indices within other SPM models such as PLS, factor analysis (FA), independent component analysis (ICA), and so on. 8 For different fault detection indices, the kernel matrix M varies correspondingly. Note for detection purposes, the control limit of Υ (x) should be determined in advance. By using the results reported in Box 30 and assuming x is multi-normal, the control limit can be calculated as follows

η 2 = g Υ χ2α (hΥ )

(2)

where gΥ =

tr[(SM)2 ] , tr(SM)

hΥ =

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

(3)

The matrix S represents the variance-covariance of the measurement x under normal conditions. For a new test sample xt collected online, the process is deemed as faulty if Υ (xt ) > η 2 ; otherwise, the process is fault-free. Once a fault is detected, the next step is to determine the fault type or faulty variables, i.e., fault isolation. In the literature different kinds of faults are usually characterized by fault directions, along with which normal samples may deviate. 22,31 Assume that there are in total I faults {Fi , i = 1, 2, . . . , I} with fault directions {Ξi , i = 1, 2, . . . , I}. The actual fault is designated as Fj with corresponding fault direction Ξj , where the subscript j is one certain value among {i = 1, 2, . . . , I} but unknown. The measurement sample becomes 22,27

x = x∗ + Ξ j f

(4)

where x∗ denotes the fault-free portion, implying Υ (x∗ ) ≤ η 2 , and Ξj f denotes the faulty part. In this work, the fault displacement f is a vector, 22,27 whose dimension depends on the column number of Ξj . If Ξj is one-dimensional, then f reduces to a scalar. This model (4)

6


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can express various kinds of faults. For example, simple single and multiple sensor faults can be represented if we choose Ξj as specific columns of the identity matrix. For complex faults such as process faults or sensor faults under feedback control, the fault direction Ξj may be obtained via priori knowledge or identified from historical faulty data. 23,31 As illustrated in Section 4, several cases including one-dimensional and multi-dimensional faults are involved, where f can be a scalar or vector. For a faulty sample x, the fault isolation task is to determine its actual fault type Fj among all possible fault candidates {Fi , i = 1, 2, . . . , I}. The isolation via reconstruction method aims to determine the actual fault type via fault reconstruction along all fault direction candidates in turn. Fault reconstruction along a specific direction eliminates the effect of this direction on the faulty sample as far as possible. The reconstructed sample along an arbitrary direction Ξi is

xi = x − Ξi fi

(5)

The estimated magnitude fi should minimize the detection index of the reconstructed sample, i.e., fi = arg min Υ (xi )

(6)

−1 T fi = (ΞT i MΞi ) Ξi Mx

(7)

fi

Solving for fi , we have

By integrating (1), (5), and (7), Υ (xi ) can be obtained as follows after simple manipulation 1/2 −1 T Υ (xi ) = k(I − M1/2 Ξi (ΞT )M1/2 xk2 = kRi M1/2 xk2 i MΞi ) Ξi M

(8)

−1 T 1/2 where Ri , I − M1/2 Ξi (ΞT . The substitution of (4) into (8) leads to i MΞi ) Ξi M

2

Υ (xi ) = Ri M1/2 x∗ + Ri M1/2 Ξj f | {z } | {z } A1

A2

7


(9)


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When the actual fault is assumed, i.e., variable i turns to j, we have A2 = Rj M1/2 Ξj f = 1/2 −1 T ]M1/2 Ξj f = [M1/2 Ξj − M1/2 Ξj ]f = 0, yielding Υ (xj ) = [I − M1/2 Ξj (ΞT j MΞj ) Ξj M

kRj M1/2 x∗ k2 . It can be easily verified that the matrix Rj is idempotent (i.e., R2j = Rj ), whose eigenvalues are either 0 or 1. 32 This further leads to

Υ (xj ) = kRj M1/2 x∗ k2 ≤ kM1/2 x∗ k2 = Υ (x∗ ) ≤ η 2

(10)

In other words, the detection index of the reconstructed sample along the true fault direction Ξj is less than its control limit, regardless of the fault magnitude. Nevertheless, the conclusion that Υ (xi ) ≤ η 2 , ∀i 6= j cannot be guaranteed due to the existence of term A2 in (9). Therefore, the actual fault type can be selected as the one which makes the detection index of the reconstructed sample less than the control limit.

2.2

Fault isolability analysis

Fault isolability tries to investigate whether another fault Fi , except for the actual fault Fj , can be mistakenly identified as the "true" fault. In practice, it is very likely that more than one fault is wrongly isolated even though there is only one actual fault, especially when the fault magnitude is small. To guarantee that only the actual fault Fj is identified and other faults {Fi , ∀i 6= j} are excluded, the occurred fault should satisfy certain fault isolability conditions. Within the PCA framework, fault isolability analysis based on the SPE, T 2 , and combined index has been provided in the literature. 27,29 For the generic fault detection index as expressed in (1), we have the following theorem regarding the sufficient isolability condition for unique and correct isolation. Theorem 1. Given the fault detection index (1) and the additive fault form (4), a sufficient isolability condition for the isolation via reconstruction method is

kRi M1/2 Ξj f k > 2η, 8

∀i 6= j


(11)

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This inequality is obtained by making Υ (xi ) > η 2 , ∀i 6= j. That is, the faulty sample cannot be reconstructed properly if a wrong fault is assumed. From (9), we know that Υ (xi ) > η 2 is equivalent to kRi M1/2 x∗ + Ri M1/2 Ξj f k > η

(12)

The triangle inequality for vector norm gives

kRi M1/2 x∗ + Ri M1/2 Ξj f k ≥ kRi M1/2 Ξj f k − kRi M1/2 x∗ k

(13)

From (10), we have kRi M1/2 x∗ k2 ≤ η 2 , or equivalently, kRi M1/2 x∗ k ≤ η. As a consequence, the condition (11) can guarantee Υ (xi ) > η 2 , ∀i 6= j. In other words, (11) is sufficient to distinguish the actual fault Fj from other candidates {Fi , ∀i 6= j}, because under this circumstance Υ (xj ) ≤ η 2 while Υ (xi ) > η 2 , ∀i 6= j. If (11) is not met, then correct isolation may not be achieved.

2.3

Generalized reconstruction-based contribution

The RBC method 9 was first proposed for single sensor fault isolation, in conjunction with PCA based fault detection. It was proven that the RBC method overcomes the smearing effect of traditional contribution plots 20 to a large extent, and can guarantee correct isolation provided that the fault magnitude is sufficiently large. Recently, the conventional RBC method was extended to explicitly handle more complex faults such as multiple sensor faults and process faults. 26 In addition, various modifications have been presented so as to apply RBC to other SPM models or improve the isolation accuracy. 24–26,33,34 The RBC based on the generic fault detection index Υ (x) for fault Fi is defined as −1 T RBCΥi = kM1/2 Ξi fi k2 = xT MΞi (ΞT i MΞi ) Ξi Mx

9


(14)


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Commonly used fault isolation logic with the generalized RBC method (14) is

Fj = arg

max

{Fi ,i=1,2,...,I}

RBCΥi

(15)

Note that for fault isolation purposes, the RBC method is in fact equivalent to the isolation via reconstruction method, though it seems that they are a little different. 26 First, the fault direction information {Ξi , i = 1, 2, . . . , I} is needed by both methods. If accurate fault information is unavailable, then fault reconstruction can only be carried out along variable directions, i.e., columns of the identity matrix. Second, the fault isolability condition (11) applies to the RBC method as well. From (1), (8), and (14), we have

Υ (x) = Υ (xi ) + RBCΥi

(16)

which links the two isolation methods closely. That is, the summation of Υ (xi ) and RBCΥi is constant, no matter for which fault among {Fi , i = 1, 2, . . . , I}. If (11) is satisfied, we know from Section 2.2 that Υ (xj ) < Υ (xi ), ∀i 6= j because Υ (xj ) ≤ η 2 but Υ (xi ) > η 2 , ∀i 6= j. Thus, in this case the conclusion RBCΥj > RBCΥi , ∀i 6= j holds. In brief, Theorem 1 guarantees correct isolation of the RBC method, too. In consideration of the equivalency between the two methods, in the following the RBC method (14) and (15) will be used to represent the reconstruction-based methods due to its relative simplicity in form.

3 3.1

Proposed Method A motivational example

A synthetic example, which was used in ref, 8 is employed here to demonstrate that conventional reconstruction-based methods are not effective in isolating incipient fault. The

10


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example consists of two correlated Gaussian variables as follows 











 x1   6   3 2.6  x=  ∼ N (µ, Σ), µ =   , Σ =   x2 4 2.6 4

(17)

The Mahalanobis distance, denoted by D, is chosen as the detection index. According to (17), 500 samples are collected to estimate the mean and covariance of x, which are used to calculate D. Test dataset, independent of the training dataset, consists of 500 samples as well. They are first generated according to (17) and a sensor bias fault with fault magnitude f = 2.0 is imposed on x2 since sample 201. The RBC method, as described in Section 2.3, is employed for fault isolation. Only single sensor faults are assumed, thus the two fault D directions are Ξ1 = [1, 0]T and Ξ2 = [0, 1]T . For all 300 faulty samples, RBCD 1 and RBC2 ,

which represent the RBC values for the two sensor faults respectively, are calculated. Correct isolation rate (CIR) is equal to 60.33%. In other words, about 40% faulty samples are not correctly isolated by the conventional RBC method. This is due to the fact that the imposed fault magnitude is relatively small. From (11) we know that, to guarantee correct isolation of the RBC method, a sufficient condition for the fault magnitude in this example is f > 12.8. However, the imposed fault is far from satisfying this condition. Therefore, the conventional RBC method cannot isolate the incipient fault efficiently. x2

x

x1

Ξ2 f *

x x2

x1

O 99% Control Limit

Figure 1: An illustration of reconstruction-based fault isolation. A corresponding geometric illustration is provided for this example. In Figure 1, a normal 11



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sample is denoted by x∗ , which is unknown when a fault occurs, and the sensor fault imposed on x2 is indicated by a vertical displacement Ξ2 f , jointly generating the fault sample x. To determine which sensor fault the fault sample x suffers from, x ought to be reconstructed along Ξ1 and Ξ2 respectively. As mentioned in Section 2, the objective of fault reconstruction is to minimize the detection index of reconstructed samples D(xi ). Geometrically, a specific D value corresponds to a specific ellipse among a set of concentric ellipses, and the smallest ellipse on which the reconstructed sample lies is expected. Thus, the objective ellipse should be tangent to the sliding direction of x along Ξi , and the point of tangency is just the reconstructed sample xi , as shown in Figure 1. Because x1 lies on a smaller ellipse, we D have D(x1 ) < D(x2 ). From (16), RBCD 1 > RBC2 . That is to say, though the sensor fault

occurs in x2 , the RBC method mistakenly locates it to x1 . Improved methods are needed for incipient fault isolation.

3.2

Exponential smoothing reconstruction (ESR)

Considering that exponential smoothing, or EWMA 35–37 is a commonly used tool which is sensitive to small shifts, it is thus incorporated to conventional reconstruction-based methods in this work. Notice that in the literature Dunia and his co-authors 21 have incorporated EWMA into the PCA framework. In their contribution, the EWMA technique was introduced to the residuals and fault identification index so as to favor the identification of different sensor fault types and avoid false alarms. In the present work, by contrast, the EWMA is applied to successively collected samples so as to deal with the incipient fault isolation problem within a generic SPM model. The proposed fault isolation method is introduced as follows. First, samples {x(k), k = 1, 2, . . . , N }, in which k represents the sampling time, are handled by the EWMA technique,

ˇ (k) = λx(k) + (1 − λ)ˇ x x(k − 1)

12


(18)

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ˇ (0) , 0. 35 Substituting recursively for where 0 < λ ≤ 1 is the weighting factor and x ˇ (k), k = 2, 3, . . . , we have x

ˇ (k) = λ x

k X

(1 − λ)k−w x(w)

(19)

w=1

Then, we propose a new fault isolation method, called exponential smoothing reconstruction (ESR), as follows −1 T ˇ T (k)MΞi (ΞT ESRΥi (k) = x x(k) i MΞi ) Ξi Mˇ

(20)

Comparing (20) with (14), we can conclude that the proposal ESR is essentially the RBC ˇ instead of original sample x. Similarly, the ESR method which uses the EWMA statistic x is also defined based on the generic detection index Υ , thus it can be applied to PCA, PLS, as well as some other SPM models. The fault isolation logic with ESR is

Fj = arg

max

{Fi ,i=1,2,...,I}

ESRΥi

(21)

Obviously, like the RBC method, the fault direction information is required by the ESR method as well. If the fault direction is unknown, then only one faulty variable can be identified by the proposed ESR. Before analyzing the isolability property of ESR, the incipient fault detection strategy should be first introduced. This is because fault detection is the first step in process monitoring and the conventional detection index Υ (x) as expressed in (1) may not be effective in detecting incipient faults. The incipient fault detection index reported in our previous work 8 is employed here, which has the following expression

ˇ T (k)Mˇ Υˇ (x(k)) = Υ (ˇ x(k)) = x x(k)

(22)

Because the new detection index (22) is different from the original one (1), its control limit has to be re-established. Control limit for (22) can also be approximated by a χ2 distribution 13



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using the results in Box, 30 similar to (2) and (3). Denote its control limit as ηˇ2 . The relationship between ηˇ2 and η 2 is ηˇ2 =

λ 2 η 2−λ

(23)

which can be obtained by analyzing the relationship between covariance matrices of x and ˇ. 8 x Regarding the fault isolability property of ESR, we have the following theorem. Theorem 2. Given the fault detection index (22) and the additive fault form (4), a sufficient isolability condition for the ESR method is

kRi M1/2 Ξj f k > 2ˇ η,

∀i 6= j

(24)

Note that certain preconditions are required to guarantee the correctness of Theorem 2, which will be demonstrated below. Substituting (4) into (19) and supposing that the fault occurs since time T , we have

ˇ (k) = λ x

k X

k−w ∗

(1 − λ)

x (w) + λ

w=1

=λ

k X

k X

(1 − λ)k−w Ξj f

w=T

(25)

(1 − λ)k−w x∗ (w) + [1 − (1 − λ)k−T +1 ]Ξj f

w=1

Shortly after the fault occurrence, we can assume that (1 − λ)k−T +1 → 0. It is a reasonable assumption because 0 ≤ 1 − λ < 1 and as k (k ≥ T ) gets larger, (1 − λ)k−T +1 rapidly approaches zero. In this case, (25) reduces to

ˇ (k) = x ˇ ∗ (k) + Ξj f x

ˇ ∗ (k) , λ where x

Pk

w=1 (1

(26)

− λ)k−w x∗ (w). Noting the similarity between (22) and (1), as well

as (26) and (4), we are able to obtain a sufficient isolability condition for the ESR method, as described by Theorem 2. The detailed theoretical derivation, which can be learned from 14


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the proof of Theorem 1, is not presented here. However, there will be some problems in the above analysis when the initial stage of the fault is considered, that is, when (1 − λ)k−T +1 cannot be regarded as 0. In this situation, a sufficient isolability condition for the ESR method changes to

kRi M1/2 Ξj f k > 2ˇ η /[1 − (1 − λ)k−T +1 ],

∀i 6= j

(27)

In fact, (24) is the special case of (27) when the exponential smoothing measurement sample ˇ (k) reaches its steady state after a fault occurs. In the following, as we mainly consider the x steady-state property of the ESR method, (24) will be used for comparison and analysis. Usually, the condition (11) is difficult to satisfy for faults with small magnitudes. Thus, the conventional RBC method cannot guarantee correct isolation for incipient fault, as illustrated by the motivational example in Section 3.1. By contrast, the proposed ESR method enhances the isolability performance for incipient faults. From (23), we have ηˇ = p λ/(2 − λ)η. Besides, 0 < λ ≤ 1, which further results in 0 < ηˇ ≤ η. Therefore, compared with (11), (24) is easier to fulfill. In other words, the proposed ESR method has a better isolation performance than the RBC method. Defining the critical weighting factor λc as follows λc =

min

i=1,2,...,I,i6=j

λic

(28)

where λic =

2 (2η/kRi

M1/2 Ξ

j f k)

2

+1

(29)

we have the following theorem. Theorem 3. With the ESR method, the incipient fault as expressed in (4) can be guaranteed to be isolable, i.e., ESRΥj > ESRΥi , ∀i 6= j, if λ < λc . This theorem can be proven directly by substituting (23) into (24). It can be observed from (28) and (29) that the critical parameter λc is affected by not only the fault detection model 15



(M, η), but also by all fault candidates (Ξi , i = 1, 2, . . . , I). In addition, the smaller the fault magnitude f is, the smaller the critical parameter λc is required. 3 normal data faulty data control limit

2

2

x

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

1

0

−1 −1

−0.5

0 x1

0.5

1

Figure 2: Scatter plot of the exponentially smoothed normal and faulty data (λ = 0.04). For the motivational example used in Section 3.1, the ESR method with weighting factor λ selected as 0.04 is applied for fault isolation. For all 300 faulty samples, only 4 samples, at the initial stage of the fault, are not correctly isolated. That is, CIR of the ESR method is equal to 98.67%, much higher than that of conventional RBC method. The superiority of the ESR method for incipient fault isolation, in contrast with RBC, can also be interpreted in an intuitive manner, in addition to the fault isolability analysis aforementioned. Comparing ˇ and x is the same, the (26) with (4), we note that, though the additive fault term Ξj f in x ˇ ∗ and x∗ are different from each other. Compared with x∗ , the variation normal portions x ˇ ∗ is shrunk, as indicated by (23). Hence, the fault in exponentially smoothed range of x ˇ is more prominent than in original data x, although its absolute magnitude does not data x ˇ is more change. Consequently, the ESR method which performs fault reconstruction on x sensitive to incipient faults than the RBC method which performs fault reconstruction on x. Figure 2 shows the scatter plot of 300 normal and 200 faulty samples handled by the EWMA technique with λ = 0.04 in the motivational example. It is observed that the exponentially smoothed normal and faulty data are well-separated and obviously the fault occurs in x2 rather than x1 . The situation described in Figure 1 will no longer exist because in the ESR method the incipient fault is "enlarged", thus correct isolation can usually be achieved.

16


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3.3

Some Remarks

Several notes should be mentioned when using the ESR method for incipient fault isolation. ˇ is used in both the detection statistic Υˇ (22) and the isolation method ESR First, though x (20), their weighting factors λ are not necessarily the same. The sufficient detectability condition for Υˇ is 8 kM1/2 Ξj f k > 2ˇ η

(30)

which is different from (24). Due to the fact that Ri in (24) is idempotent, usually a smaller λ is required by the ESR method than the detection statistic Υˇ . Second, as we can deduce from (28), (29), and Theorem 3, a smaller weighting factor λ can guarantee correct isolation of a smaller fault. Nevertheless, too small λ may cause a serious time delay for correct isolation. By comparing (27) with (24), we can observe that the isolability condition of the ESR method at the initial stage of a fault is more difficult to meet. Recall that in the motivational example there are indeed 4 samples at the initial stage of the fault that are not correctly isolated. This is because for the 4 initial faulty samples, their isolability condition (27) is not satisfied. For a smaller λ, more samples (a larger k) are needed so as to reasonably regard (1 − λ)k−T +1 as 0. Consequently, to avoid serious time delay for correct isolation, the weighting parameter λ should not be too small. Regarding the selection of an appropriate λ, some common settings have been presented in the literature 38 when considering the EWMA chart based statistical quality control problem. In the proposed ESR method, λ can be chosen as any value in the interval (0, 1], with the aim to isolate the incipient fault effectively. Recall that the fault direction information is required by the ESR. Therefore, given the fault information and the adopted detection method, an appropriate λ can be determined according to (28), (29), and Theorem 3. When the fault information is unavailable, only one fault variable can be identified. In this case, the idea presented in ref 39 can be incorporated in order to extend the ESR method to handle the case of identifying multiple unknown faulty variables.

17



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Moreover, the theoretical analysis presented in above sections requires that samples should be independent and identically Gaussian distributed. This assumption ensures (2) and (23) are valid and accurate. For some steady processes whose sampling frenquency is low, this assumption can be roughly satisfied. However, if some kinds of practical industrial processes cannot satisfy this assumption but (2) and (23) are still used directly, overmany false alarms may occur. In this case, the control limits η 2 and ηˇ2 should be adjusted, and can be determined by the empirical method which uses only historical data collected under normal conditions. 40 That is, given a significance level α, the control limit is set as a value below which (1 − α)100% percent of all normal samples’ detection statistics are located.

4

Simulation Studies

In this section, two simulation examples are used to demonstrate the efficiency of the proposed ESR method, in comparison with conventional isolation approaches, including TCP, 20 RBC, 9 RBC ratio (RBCR), 26 and weighted RBC (WRBC). 33 In the numerical example, a process fault and a sensor fault are respectively considered. In the CSTH benchmark process, a sensor fault under closed-loop control is involved.

4.1

A numerical example

The following synthetic process model is adopted: 8,41   xk = Azk + ek   yk = Cxk + vk

18


(31)

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where

zk ∈ R3 , zk,i ∼ U[0, 1],  1 3 4 4  A= 3 0 1 4  1 1 3 0

i = 1, 2, 3 T 0  1   0

ek ∈ R5 , ek,j ∼ N (0, 0.052 ), j = 1, . . . , 5 C = 2 2 1 1 0 , vk ∼ N (0, 0.12 ) The total projection to latent structures (T-PLS), 41 which was proposed for quality-related process monitoring, is employed here for data modeling. T-PLS is an improvement of the well-known partial least squares (PLS) method. In T-PLS, the X-space is decomposed into four subspaces denoted by Sy , So , Srp , and Srr , respectively, where four statistics Ty2 , To2 , Tr2 , and Qr , as well as their corresponding control limits are established for fault detection purposes. 500 normal samples produced according to (31) are used to train a T-PLS model, and the model parameter matrices obtained in this simulation are shown in Table 1. To simplify the fault detection task, one may prefer to observe one index rather than several indices. 23,25 Thus, we propose to use a new combined index which integrates Ty2 , To2 , Tr2 , and Qr together as follows Ty2 To2 Tr2 Qr φ(x) = 2 + 2 + 2 + 2 = xT Φx τy τo τr δr

(32)

where τy2 , τo2 , τr2 , and δr2 represent control limits of four statistics Ty2 , To2 , Tr2 , and Qr , respectively. One can refer to ref 41 for the calculation of these statistics. They are all quadratic forms of sample x, whose kernel matrices can be expressed via the model parameters listed in Table 1. Thus, for φ, its kernel matrix Φ can also be calculated using the T-PLS model parameters. Besides, Φ is symmetric and positive definite, thus the control limit of φ can

19



be obtained via the χ2 distribution. 30 Two testing datasets, both consisting of 200 samples, contain a process fault and a sensor fault, respectively. The fault is imposed since sample 101. Faulty samples are constructed via (4), in which the normal part x∗ is also produced via (31), but independent of the training samples. Table 1: T-PLS model parameter matrices. 8 py

Po

0.1292 0.1315 0.2282 0.2505 0.0246

Pr

-0.2388 0.2636 0.6218 -0.6779 -0.1655

-0.7298 0.5738 0.2321 -0.0990 -0.2727

˜r P -0.5047 -0.6961 0.4283 0.2192 0.1710

0.1406 -0.2682 0.0460 0.1191 -0.9445

FAR:0% FDR:5% 3

φ

2

1

0 0

50

100 samples

150

200

Figure 3: Fault detection of the process fault using φ in the numerical example. FAR:0% FDR:97%

0.6

0.4

φˇ

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 0

50

100 samples

150

200

Figure 4: Fault detection of the process fault using φˇ (λ = 0.12) in the numerical example. For the process fault, its direction is set as Ξj = Po and its magnitude is set as f = 1.5. 8 Figure 3 shows the fault detection result using φ. False alarm rate (FAR) and fault detection 20


Page 20 of 35

rate (FDR) are calculated and marked in each detection figure. It is observed that the φ index, whose FDR is only 5%, fails to detect the process fault effectively. However, the φˇ index with λ = 0.12 is capable of detecting the incipient fault efficiently, as shown in ˇ Figure 4. The calculation of φˇ can be learned from (22) and (32) directly, that is, φ(x(k)) = ˇ T (k)Φˇ x x(k). Refer to ref 8 for more details about the incipient fault detection index. 0.8

0.6

RBC

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


0.4

0.2

0

1

2

3 fault number

4

5

Figure 5: Fault isolation of the process fault for sample 171 using RBC in the numerical example.

1.2 1 0.8 RBCR

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0.6 0.4 0.2 0

1

2

3 fault number

4

5

Figure 6: Fault isolation of the process fault for sample 171 using RBCR in the numerical example. Then, we turn to investigate the fault isolation aspect of the incipient fault. Three fault isolation methods including RBC, RBCR, and the proposed ESR are involved and compared with each other. Fault isolation is based on the φ index, as well. As aforementioned, the fault direction information is required for fault isolation. We assume that the fault candidates

21



0.4

0.3

ESR

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

Page 22 of 35

0.2

0.1

0

1

2

3 fault number

4

5

Figure 7: Fault isolation of the process fault for sample 171 using ESR in the numerical example. which contain the actual fault Po is composed of the following directions ˜ r (:, 1), P ˜ r (:, 2)} {py , Po , Pr , P

(33)

consecutively numbered as 1 to 5. Take sample 171 (faulty sample) for illustration. Figure 5 shows the RBC values of the five fault candidates. According to the fault isolation logic (15), the first fault with direction py , which has the largest contribution value, is incorrectly determined as the true fault by RBC. Whereas, the second fault is the true fault. Thus, for incipient fault, the RBC method cannot guarantee correct isolation. The fault isolation logic of the RBCR method is that a fault whose RBCR value is less than or equal to 1 is identified as the true fault while a fault whose RBCR value is greater than 1 is not the true fault. From Figure 6, we observe that the RBCR method fails to provide correct isolation result as well since all fault candidates have a RBCR value less than 1. By contrast, the proposed ESR method correctly identify the second fault with direction Po as the true fault, as illustrated by Figure 7. The smoothing parameter λ used in ESR is set as 0.12, the same ˇ For all the 100 faulty samples, Table 2 lists the CIRs of as that in the detection index φ. these isolation methods, which further demonstrates the superiority of the proposed ESR method for incipient fault isolation. The sensor fault is applied to the second measurement x2 , with magnitude f = 0.14.

22


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Table 2: CIR provided by RBC, RBCR, and ESR for the process fault in the numerical example.

CIR (%)

RBC

RBCR

ESR

56.0

5.0

100.0

FAR:0% FDR:2%

φ

2

1

0 0

50

100 samples

150

200

Figure 8: Fault detection of the sensor fault using φ in the numerical example. FAR:1% FDR:97% 0.8

0.6

φˇ

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


0.4

0.2

0 0

50

100 samples

150

200

Figure 9: Fault detection of the sensor fault using φˇ (λ = 0.1) in the numerical example. As shown in Figure 8, the φ index completely fails to detect this fault since its FDR is almost equal to the significance level α (set as 1% throughout this paper). This is due to the fact that the imposed fault magnitude is small and far from satisfying the sufficient detectability condition of φ. However, the incipient sensor fault can be guaranteed detectable by φˇ with an appropriate smoothing parameter λ, as illustrated by Figure 9. In addition, it is observed from Figure 9 that the FAR of φˇ equals the significance level α, which verifies the reasonableness of the way to determine the control limit as expressed by (23) in this example. Regarding fault isolation of the incipient sensor fault, five methods including TCP, RBC, 23



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Table 3: CIR provided by TCP, RBC, RBCR, WRBC, and ESR for the sensor fault in the numerical example.

CIR (%)

TCP RBC RBCR

WRBC γ = 0.3 γ = 0.5 γ = 0.7

γ = 0.9

44.0

43.0

43.0

33.0

0

45.0

38.0

ESR 84.0

RBCR, WRBC, and the proposed ESR are involved. Considering that the imposed fault type is a single sensor fault, we limit all fault direction candidates to the columns of the five-dimensional identity matrix. The actual fault direction is the second column of the identity matrix. These methods’ CIRs are tabulated in Table 3. For the WRBC method, 0 ≤ γ < 1 is a key parameter utilized in the weighting matrix, 33 and four representative values, i.e. 0.3, 0.5, 0.7, and 0.9 are selected in this simulation. The smoothing parameter λ utilized in the ESR is 0.01, which is different from that in the φˇ index. It is noted that except for the ESR approach, all other methods cannot isolate the incipient sensor fault efficiently, whose CIRs are less than 50%. For the TCP method, it had been revealed that, due to the smearing effect, even the single sensor fault with sufficiently large fault magnitude cannot be guaranteed isolable. 9 Although the RBCR and WRBC methods were claimed as an alternative or improvement of the RBC method, they are not proposed to explicitly tackle the incipient fault isolation problem. Thus, their isolation performance is also not satisfying. The ESR method, by contrast, has a CIR equal to 84% and only 16 samples are not correctly isolated. As analyzed in Section 3.3, correct isolation cannot be guaranteed at the initial stage of the fault. Indeed, in this simulation incorrect isolation (i.e., the 16 samples) occurs at this stage. For the remaining 84 faulty samples, the ESR provides correct isolation stably.

4.2

The CSTH process

In this section, the CSTH simulation model, 42 established based on the CSTH pilot plant at the University of Alberta, is employed to demonstrate the effectiveness of the proposed ESR method. The nonlinearity of the process is carefully considered, and instrument, actuator,

24


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Hot water

Cold water

flow control

level control Steam

temperature control

Figure 10: The CSTH process. 42 and real disturbances are measured from the process rather than simulated to drive a first principles model, which make the simulation model more realistic. Figure 10 shows a brief configuration diagram of the CSTH process. The simulation programs of the CSTH process have been provided in the CSTH Website by Thornhill et al., corresponding to their article. 42 The level and temperature are controlled variables with nominal values. In total three PI controllers are involved in the simulation: one controller is used in the temperature control loop, where the steam valve serves as the manipulated variable; two other controllers are used in the level control loop, where the output of the first controller is the setpoint of the second controller and the cold water flow valve serves as the final manipulated variable. There are two operating conditions presented in Thornhill et al., 42 and only the first one is involved in this paper. Due to the introduction of feedback loops in the CSTH process, the phenomenon of fault propagation is usually inevitable. For example, if a bias fault occurs in the level sensor, then closed-loop control will work to eliminate the bias via changing manipulated variables because the level is a controlled variable with nominal value. When reaching a new steady state, the level sensor measurement reverts to its nominal value, but the cold water flow and outputs of all three controllers have been changed to new steady-state values. In other words, the fault occurring in the level sensor is propagated to the cold water flow and three controllers’ outputs. Therefore, to facilitate data-driven process monitoring, the measurement sample

25



consists of not only three sensor measurements (i.e., level, flow, and temperature), but also the outputs of all three controllers. For convenience, the measurement sample is denoted as x = [L, F, T, CL , CF , CT ]. The unit of all the six measurements is milliampere (mA) because they represent standard electronic signals on 4∼20 mA scale, as in the real plant.

T

10.7

10.5

10.3 0

500

1000

1500

2000

500

1000 samples

1500

2000

CT

13

12.5

12 0

Figure 11: Trends of T and CT in the presence of a sensor fault imposed on T since sample 501.

FAR:3% FDR:26.27%

D

40

20

0 0

500

1000 samples

1500

2000

Figure 12: Fault detection of the sensor fault using D in the CSTH process.

FAR:0% FDR:97.4% 16

12

ˇ D

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

8

4

0 0

500

1000 samples

1500

2000

ˇ (λ = 0.018) in the CSTH process. Figure 13: Fault detection of the sensor fault using D 26


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Based on the simulation programs, 42 2000 normal samples are first collected for data modeling. The Mahalanobis distance D is utilized for fault detection. The data of the CSTH process are not independent and identically Gaussian distributed, thus, as discussed in Section 3.3, the theoretical way to calculate the control limit as expressed by (2) may be inaccurate. The empirical method 40 is employed instead. The testing dataset consists of 2000 samples as well, and a fault is imposed since sample 501. The fault involved is a bias fault imposed on the temperature sensor T , with fault magnitude equal to 0.05 mA. This fault is in the temperature control loop but does not affect the level control loop. Thus, only two measurements T and CT are influenced by this fault: T is changed suddenly since the occurrence of the fault but reverts to its nominal value due to closed-loop control; CT reaches a new steady-state value due to the existence of temperature sensor bias fault. Nevertheless, the fault is not obvious because the fault magnitude is small and there exist disturbances in the process, as shown in Figure 11. As a consequence, the fault detection result, as shown in Figure 12, is not satisfying. The FDR is only 26.27% and most of the faulty samples are ˇ index can detect the incipient fault efficiently. Note that the missed. By contrast, the D ˇ is determined by the empirical method as well, instead of the theoretical control limit of D ˇ way (23). As we can see from Figure 13, the incipient fault is successfully detected by D ˇ is 0%, which shows that the control limit is after a short time delay. Besides, the FAR of D determined reasonably. Table 4: CIR provided by RBC, RBCR, and ESR for the sensor fault in the CSTH process.

CIR (%)

RBC

RBCR

ESR

55.0

0.87

98.93

Then, fault isolation of the incipient fault is analyzed. Though in the CSTH example the fault is imposed on a single sensor, due to fault propagation, the fault direction is no longer the corresponding column of the identity matrix, as in the numerical example. The fault directions of different faults in this example can be obtained by analyzing the process

27



model qualitatively. For simplicity, we only consider the fault direction of a fault when a new steady state after the fault occurs is reached and ignore the short transient process. As analyzed before, the sensor fault imposed on T only influences the measurement CT , thus the actual fault direction is Ξ1 = [0 0 0 0 0 1]T . Through process knowledge and analysis, we can also conclude that the fault direction of the sensor fault imposed on F is Ξ2 = [0 1 0 0 0 0; 0 0 0 1 0 0]T . In other words, the sensor fault on F affects the F itself and CL simultaneously. The sensor fault imposed on L influences F , CL , CF , and CT . Thus, its fault direction, denoted as Ξ3 is composed of four columns of the identity matrix. Since Ξ1 and Ξ2 are subspaces of Ξ3 , Ξ3 cannot be listed as a fault candidate; otherwise, Ξ1 and Ξ2 can never be identified as the true fault even if they indeed are. That is to say, in the CSTH simulation, the proposed ESR method cannot deal with the case of a fault occurring in the level sensor. Besides, two other fault directions, i.e., Ξ4 = [1 0 0 0 0 0]T and Ξ5 = [0 0 1 0 0 0]T are also included as interference terms. The two faults Ξ4 and Ξ5 can be regarded as simple single sensor faults applied respectively to L and T without feedback control. The fault isolation task is thus to identify the true fault Ξ1 from four candidates {Ξ1 , Ξ2 , Ξ4 , Ξ5 }. For comparison purposes, three methods including the RBC, RBCR, and proposed ESR are successively applied for incipient fault isolation. Table 4 shows their CIRs among all 1500 faulty samples. Similar to the numerical example, the RBCR method can hardly isolate the incipient fault in the CSTH process. As for RBC, the fault magnitude is far from satisfying its sufficient isolability condition (11). Thus, about half of the faulty samples are not correctly isolated by the RBC method. The ESR method with λ = 0.01, by contrast, provides a very high CIR equal to 98.93%, and only 16 faulty samples are not correctly isolated. The 16 samples occur at the initial stage of the fault (i.e., sample 501 to sample 516), and all subsequent faulty samples are correctly isolated. Therefore, compared with conventional methods, the proposed ESR is more effective in isolating incipient faults. The reason for incorrect isolation of the ESR method is twofold: first, at the initial stage of the fault, the

28


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fault direction may not be Ξ1 since a new steady state has not been reached; second, as analyzed in Section 3.3, correct isolation cannot be guaranteed by ESR at the initial stage of the fault.

5

Conclusions

In this work, the incipient fault isolation task within the SPM framework has been involved. Following the previous work 8 which handles the incipient fault detection problem, the present work is based on the generic detection index as well, so the proposal can be applied to various SPM models. First, the commonly used reconstruction-based methods for fault isolation are briefly reviewed and summarized, followed by their isolability analysis. It is pointed out that the isolation via reconstruction method is equivalent to the RBC method in terms of usage and isolability properties. Then, based on conventional reconstruction-based methods, a new method called ESR is proposed for incipient fault isolation. The ESR incorporates the EWMA technique into conventional reconstruction methods. Through fault isolability analysis, it is demonstrated that, compared with conventional methods, the ESR is more sensitive to incipient faults. Several notes about the use of ESR are discussed. To demonstrate the effectiveness of the proposed ESR, case studies on a numerical example and the CSTH benchmark process are carried out. In the simulations, several kinds of faults including a single sensor fault, a process fault, as well as a sensor fault under closed-loop control are considered. The proposed ESR is also compared with several conventional methods such as TCP, RBC, RBCR, and WRBC. Simulation results illustrate that, conventional methods cannot perform well, but the proposed ESR approach is capable of isolating the imposed incipient faults efficiently given an appropriate smoothing parameter. One limitation of the proposed ESR is that usually a time delay is required for correct isolation.

29



Acknowledgement This work was supported by the National Natural Science Foundation of China (61490701, 61751307, 61473163, 61522309), and Research Fund for the Taishan Scholar Project of Shandong Province of China (LZB2015-162).

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Graphical TOC Entry Traditional method

Proposed method

10

6

3 normal data faulty data control limit

normal data faulty data control limit

1.5

x2

2

x2

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


-2 0 -6

Fault direction is apparent (x2 )

Fault direction is unclear -10 -6

-4

-2

0

2

4

-1.5 -1

6

x1

-0.5

0

x1

35


0.5

1

Exponential smoothing reconstruction approach for incipient fault

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