Selecting the Number of Principal Components on the Basis of

Article pubs.acs.org/IECR

Selecting the Number of Principal Components on the Basis of Performance Optimization of Fault Detection and Identification Jiyang Xuan, Zhengguo Xu,* and Youxian Sun State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China ABSTRACT: A novel method for selecting the number of principal components (PCs) is proposed in this paper. This method adopts a new fault detection performance index and a new fault identification performance index to complete the selection. The new fault detection performance index is developed to represent the fault detection ability using the concept of detectable fault coverage, and the new fault identification performance index is proposed for the purpose of judging whether the fault identification result is correct or not. Furthermore, detailed mathematical analysis is given to explain the influence of the number of PCs on the fault identification performance index. The Monte Carlo numerical model and the quadruple-tank process model are used to demonstrate the advantages of our proposed indices and to illustrate the implementation of our proposed method.

1. INTRODUCTION Principal component analysis (PCA) has been widely applied in chemical processes for fault diagnosis in the past decades.1−4 This multivariate statistical method establishes its statistical model using data measured during normal operation conditions, and partitions the measurement space into two orthogonal subspaces: principal component subspace (PCS) and residual subspace (RS). The PCS indicates the normal variability with a control limit defined by the T2 statistic, while the RS mainly includes noise with a control limit defined by the squared prediction error (SPE). Fault diagnosis, including fault detection and fault identification, can be implemented utilizing the information in PCS and RS. Some statistic indices like T2 or SPE can be used for fault detection, and some methods like contribution plots can be used for fault identification. Although the PCA method has been extensively used in the process monitoring field, selecting the number of principal components (PCs) is still an open question. It is quite important to choose the adequate number of PCs to represent a system.5 There exist many known criteria for determining the number of PCs, such as cumulative percent variance (CPV),6,7 variance of reconstruction error (VRE),5,8 fault signal-to-noise ratio (Fault SNR),9,10 and so on. The CPV approach is to select the number of PCs such that the variance reaches a predetermined percentage, say 85%, which means that a substantial part of the total variance of the data is included. This method can express the main information on the original data, but it is rather subjective to decide the percentage. The VRE criterion is to determine the number of PCs based on the best reconstruction of the variables. The optimum number of PCs is obtained when the error of fault reconstruction comes to the minimum. The VRE criterion is preferable compared with the CPV criterion,5 yet it still does not consider the fault detection ability which relies on the number of PCs retained in the PC model. The Fault SNR method reflects the relationship between the number of PCs and the fault detection sensitivity that measures the performance of fault detection. It is easy to choose the number of PCs that gives the maximum sensitivity to a fault by examining the maximum value of Fault SNR. However, this © 2015 American Chemical Society

method is attractive only when a certain kind of faults occurs repeatedly, and is not suitable for general process monitoring. Almost all of the previous studies were based on only one selection criterion, and none of them took the performance of fault identification into consideration. In this paper, we introduce a new method that takes the performance of fault detection and identification into account together for selecting the number of PCs. To begin with, a new fault detection performance index is given as the first selection criterion. The concept of minimum detectable fault magnitude (MDFM) is proposed on the basis of the Fault SNR theory, and then the fault detection performance index indicating detectable fault coverage is formed utilizing MDFM. The optimum number of PCs will be chosen when maximum detectable fault coverage (MDFC) is achieved. Moreover, a new fault identification performance index is provided as the second selection criterion. On the basis of some conclusions about the contribution plots given by Alcala and Qin,11 the projection matrix to RS is modified and further used to evaluate the performance of fault identification. The fault identification performance index using the infinite norm of the modified matrix is put forward to judge whether false fault identification will happen or not. Meanwhile, detailed analysis is given to explain the influence of the number of PCs on the fault identification performance index mathematically. Finally, we combine these two aforementioned performance indices to select the optimum number of PCs, thus achieving the objectives of maximum detectable fault coverage and correct fault identification if such a PC model exists. The rest of the paper is organized as follows. Some background knowledge of PCA is first reviewed in section 2. In section 3, two performance indices about fault detection and identification are developed as two selection criteria, separately. Received: Revised: Accepted: Published: 3145

November 1, 2014 February 9, 2015 March 15, 2015 March 16, 2015 DOI: 10.1021/ie5043177 Ind. Eng. Chem. Res. 2015, 54, 3145−3153

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Industrial & Engineering Chemistry Research

3. PROPOSED METHOD 3.1. Selection Criterion with MDFM Based Fault Detection Performance Index. Fault SNR was proposed as an index of fault detection ability by Tamura and Tsujita.9 The Fault SNR for SPE is defined as follows:9,10

Then, a new method combining these two performance indices for selecting the number of PCs is designed. Section 4 gives the application results from the Monte Carlo numerical model and the quadruple-tank process model. These two models are also used to demonstrate the advantages of our proposed indices. Finally, conclusions are given in the last section.

2

2. PCA FOR PROCESS MONITORING PCA is a widely used statistical method for process monitoring, and it can transform correlated original data to uncorrelated data set that represents most of the information in the original data. Let x ∈ ℜm denote a sample vector of m sensors. Assuming that there are n samples from each sensor with the same sampling time, a data matrix X ∈ ℜn × m is composed with each row representing a sample vector. The original data matrix X is scaled to zero-mean and unit-variance, and then it is decomposed into a score matrix T ∈ ℜn × k and a loading matrix P ∈ ℜm × k X = TPT + E

SNR SPE =

(1)

(2)

where C is the projection matrix to the PCS. The projection on the RS is T

̃ x̃ = (I − PP )x = Cx

SPE limit

Cd , i = (fmd , i , +∞)

∑

Cd̅ , i = (0, fmd , i )

h0 = 1 −

(4)

fmd ̅ =

(10)

(11)

1 m

m

∑ fmd ,i i=1

(12)

is selected as the fault detection performance index that represents the fault detection ability. It is necessarily noted that all the different sensor faults are regarded as having the equal probability of occurrence in the process. Moreover, bias sensor faults are mainly discussed here, and those unknown faults, especially process faults, are not considered in our method. The value of this fault detection performance index varies with the number of PCs, and the maximum detectable fault coverage (MDFC), which corresponds to the minimum undetectable fault coverage or the minimum average MDFM, points to the optimum number of PCs, and it will lead to the optimum detection of a single sensor fault that can be from any sensor. 3.2. Selection Criterion with Projection Matrix Based Fault Identification Performance Index. Contribution plots

(5)

(6)

2θ1θ3 3θ22

(9)

Furthermore, the maximum detectable fault coverage corresponds to the minimum undetectable fault coverage, and the length of the undetectable fault coverage corresponding to ξi is equal to f md,i. Hence, the total undetectable fault coverage of all sensor faults (ξ1,...,ξm), which can be represented by the average MDFM

λji (i = 1, 2, 3)

j=k+1

(8)

and the undetectable fault coverage corresponding to ξi as

m

θi =

SPE limit

This MDFM means that if the fault direction is ξi, the fault with magnitude f i ∈ (f md,i, + ∞) can be detected. Therefore, we define the detectable fault coverage corresponding to ξi as

(3)

The control limit of the SPE statistic was developed by Jackson and Mudholkar12 ⎡ ⎤1/ h0 Cα 2θ2h02 θ2h0(h0 − 1) ⎥ ⎢ = θ1 +1+ ⎢ ⎥ θ1 θ12 ⎣ ⎦

|| fi ξĩ ||

SPE limit 2 || ξĩ ||

fmd , i =

where C̃ is the projection matrix to the RS. Bias sensor faults are mainly considered in our work, and the situation that normal variation gets too large is not included for assessing fault detection ability. Besides, our proposed method for selecting the number of PCs is based on only one statistic index, and the fault information is mostly in the residual subspace indicated by SPE. As Qin2 pointed out, it usually took a large fault magnitude to exceed the T2 control limit, while faults just with small or moderate magnitudes could exceed the SPE control limit. Thus, only the SPE statistic is considered in this paper. The SPE statistic is defined by ̃ ||2 SPE = || x̃ ||2 = || Cx

SPE limit

2

=

where f i is a scalar value that represents the fault magnitude of the i th sensor, ξi is the fault direction and ξĩ is the projection of the fault direction ξi onto the RS. Examining the maximum value of Fault SNR, the optimum number of PCs can be selected. However, this method is attractive only when a certain kind of faults appears repeatedly, because the method enables sensitively detecting a fault from its second appearance. On account of the drawback that the Fault SNR method aims at only one sensor fault, it is necessary to develop a new performance index for assessing the fault detection ability with respect to all sensor faults (ξ1,...,ξm) rather than one particular sensor fault (ξi). To show the basic idea of the new fault detection performance index, let SNRSPE in eq 8 be equal to 1, then the minimum detectable fault magnitude (MDFM), which is acquired when the SPE statistic reaches the control limit, can be expressed as

where E ∈ ℜn × m is the residual matrix and k is the number of PCs. A sample vector x ∈ ℜm can be projected onto the PCS and RS, respectively. The projection on the PCS is x̂ = PPTx = Cx

|| fi ξiΤ(I − PPT)||

(7)

where Cα is the upper fractile value of standard normal distribution with the level of significance α, and λj (j = 1,...,m) is the eigenvalue of the covariance matrix S = (1/(n − 1))XTX. When a fault occurs, the fault can make the SPE statistic exceed its control limit, leading to the detection of the fault. 3146

DOI: 10.1021/ie5043177 Ind. Eng. Chem. Res. 2015, 54, 3145−3153

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Industrial & Engineering Chemistry Research are well-known diagnostic tools for fault identification.13−16 Based on the idea that the variables with the largest contributions to the fault detection index are most likely the faulty variables, the contribution plots are constructed by calculating the contribution of each variable to the fault detection index. As stated in the previous section, SPE is selected as the fault detection index in this paper. The contribution to SPE is simply breaking down the summation of SPE into each element that corresponds to each variable:3 m

SPE =

demonstrate this phenomenon. If we want to know the quantitative relation between the number of PCs and the possibility of false fault identification, it is suggested that we should analyze it mathematically. In other words, it is necessary for us to find the relationship between the number of PCs and the variation of the fault identification performance index. The projection matrix C̃ can be expressed as C̃ = I − PPT = I − [ p1 p2 ⋯ pk ][ p1 p2 ⋯ pk ]T

m

∑ xĩ 2 =

∑ ciSPE

i=1

i=1

= I − p1p1T − p2p2T − ⋯ − pk pkT

(13)

where x̃i is the i th element of the residual vector x ̃ and

ciSPE

xĩ 2

=

where pi = [pi1 pi2... pim] is the eigenvector of the covariance matrix. The absolute value of the ith element on the primary diagonal is

(14)

is the contribution of the ith variable to SPE. The contribution plots are easy to calculate with no prior knowledge. However, a fault smearing effect exists in this method, which means that a fault in one variable smears to the contributions of other variables in contribution plots. Alcala and Qin11 analyzed whether or not the smearing effect led to false fault identification when using the method of contribution plots based on cSPE i . They examined the case where the fault sample vector was exactly in the fault direction ξi, that was x = ξifi

k

|ciĩ | = |1 −

l=1

cij̃ ciĩ

(16)

(17)

∀ i, j (18)

(19) C̅ = Ĉ − I ̂ where ĉij is in the ith row of the jth column of C, and C̅ is a matrix whose primary diagonal elements are all zeros. Then the new fault identification performance index Id is given in the form of the infinite norm of C̅ .

Id = || C̅ ||′∞ = max i , j( cij̅ )

(23)

It is difficult for us to confirm whether |c̃ij| increases or decreases with the increase of k, so it is also hard to find the variation of the exact relative size between |c̃ii| and |c̃ij| when k increases. Nevertheless, it is feasible to explain the influence of the number of PCs on the fault identification performance index, which can be inferred from Theorem 1. Theorem 1. For ∀i, j ∈ {1,..., m} and i ≠ j, define ei = minj(|c̃ii| − |c̃ij|) and k is the number of PCs selected. If 1 ≤ k < m, then ei ≥ 1 − 1.21k, and the greatest lower bound of ei decreases with the increase of k. ■ The proof of this theorem is given in the Appendix. This theorem indicates that the possibility of false fault identification (Id > 1) increases with the increase of k, and Id > 1 corresponds to ei < 0. This results because the greatest lower bound of ei is negative and decreases with the increase of k. The decrease of the greatest lower bound will increase the possibility of the event that ei gets a negative value. In a word, it is harmful to select too many PCs if contribution plots are used for fault identification. 3.3. Implementation of Selecting the Number of PCs. The fault detection performance index fmd ̅ and the fault identification performance index Id just consider the performance of fault detection and fault identification, respectively. Fault detection and fault identification, however, are both equally important in process monitoring. For this reason, these two tasks should be taken into account together. Thus, an algorithm that takes these two performance indices into consideration together for selecting the number of PCs is recommended in this paper. The main strategy in our algorithm is that correct fault identification, which means that Id ≤ 1, should be guaranteed, and under this constraint the maximum detectable fault coverage, which can be indicated by fmd ̅ , should be determined as large as possible. However, one exception is that Id > 1 when one PC is selected, and this situation does exist from the fact

On the basis of this judgment condition, we propose a new performance index for determining whether this judgment condition is satisfied or not. First of all, a modified matrix of the projection matrix C̃ is formed as follows

ciĵ =

(22)

k

|cij̃ | = |−∑ pli plj | , i ≠ j

where c̃ii is the ith primary diagonal element of the projection matrix C̃ , and c̃ij is in the ith row of the jth column of the projection matrix C̃ . Correct fault identification using cSPE is i guaranteed only if

ciĩ2 ≥ cij̃2

l=1

where k is the number of the PCs selected. It is easy to notice that |c̃ii| decreases with the increase of k. The absolute value of the element c̃ij, which is not on the primary diagonal but in the same row with c̃ii, is

(15)

⎧ c ̃2f 2 i ≠ j ⎪ ij i =⎨ ⎪ ciĩ2f 2 i = j ⎩ i

k

∑ pli2 | = 1 − ∑ pli2 l=1

This is equivalent to the case where the fault magnitude is sufficiently large. For this case, cSPE can be expressed as i ciSPE

(21)

T

(20)

If Id is less than or equal to 1, the judgment condition in eq 17 is satisfied, thus correct fault identification is guaranteed. Otherwise, false fault identification will take place. Lieftucht et al.17 once gave the conclusion that the contribution plot from the SPE statistic which was at a larger ratio (k/m) was more likely to produce an incorrect fault identification, but the authors just used a simple example to 3147


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Industrial & Engineering Chemistry Research that even though k is equal to 1, ei has the possibility of being negative. If the aforementioned situation happens, one PC has to be determined as the optimal selection, and correct fault identification is not guaranteed any more. The flowchart of our proposed algorithm is shown in Figure 1. As shown in the

of our selection algorithm is presented in the above two simulation models. 4.1. Monte Carlo Numerical Simulation. The Monte Carlo simulation model is frequently employed in the previous research.11,18,19 The model to be used in this paper is ⎡ x1 ⎤ ⎡−0.2310 − 0.0816 − 0.2662 ⎤ ⎢x ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢−0.3241 0.7055 − 0.2158 ⎥⎡ t ⎤ ⎢ x3 ⎥ ⎢−0.217 − 0.3056 − 0.5207 ⎥⎢ 1 ⎥ ⎢ ⎥=⎢ ⎥⎢ t 2 ⎥ + noise ⎢ x4 ⎥ ⎢−0.4089 − 0.3442 − 0.4501 ⎥⎢ ⎥ ⎢ x5 ⎥ ⎢−0.6408 0.3102 0.2372 ⎥⎣ t3 ⎦ ⎢ ⎥ ⎢ ⎥ ⎢⎣ x6 ⎥⎦ ⎣−0.4655 − 0.433 0.5938 ⎦ (24)

where t1 ∼ N(10, 0.1), t2 ∼ N(20, 0.2) and t3 ∼ N(30, 0.3) are normally distributed variables. The random noise included in this model is normally distributed with the mean of zero and the standard deviation of 0.01. One thousand historical samples under the normal condition are used to establish the PC model. The values of the fault detection and identification performance indices from k = 1 to k = 5 are calculated, as shown in Table 1. If our proposed fault detection performance Table 1. Fault Detection and Identification Performance Indices of the Monte Carlo Model

Figure 1. Flowchart of the proposed method for selecting the number of PCs.

flowchart, we first include all the m PCs, and then the fault detection performance index fmd ̅ is calculated to determine the number of PCs k1 that corresponds to the maximum detectable fault coverage. In the next step, the values of Id(k1) and k1 are checked to verify whether the terminating condition is satisfied or not. In the terminating condition, Id(k1) ≤ 1 means that correct fault identification is guaranteed; and k1 = 1 means that we have to select one PC and correct fault identification is not definitely guaranteed. Once the terminating condition is not satisfied, we will move to observe the top k1 − 1 PCs and repeat the procedure until the optimal number of PCs is achieved. It is noteworthy that if one PC is selected in our algorithm, the fact that whether correct fault identification is guaranteed or not is unknown for us, and the reason why we move to observe the top k1 − 1 PCs is that the possibility of Id ≤ 1 among the top k1 − 1 PCs is larger than that among any other number of PCs where the number changes from k1 + 1 to m. Finally, the number of PCs selected can make our PC model provide the capability of containing largest detectable fault coverage and identifying single sensor fault correctly if such a PC model exists.

k

fmd ̅

Id

1 2 3 4 5

2.8551 1.8742 0.3334 0.7676 1.6037

0.3231 1.0656 1.2099 14.8575 19.5296

index fmd ̅ is used to select the PCs, three PCs corresponding to the minimum fmd ̅ are determined as the optimal selection of PCs. According to the CPV criterion with a predetermined percentage of 85%, only two PCs are selected. Then we will compare the fault detection abilities of these two different criteria. The sensor faults added to the six different sensors are with the magnitudes of f1 = 0.1, f 2 = 0.2, f 3 = 0.1, f4 = 0.1, f5 = 0.2 and f6 = 0.2 when t = 501, respectively. The monitoring plots are shown in Figures 2−7. From Figures 2−7 we can see that the PC model using our proposed criterion can detect all sensor faults (ξ1,...,ξ6), while the PC model using the CPV criterion fails in these cases. It comes into the conclusion that the detectable fault coverage of our proposed criterion is larger than that of the CPV criterion. Next we will discuss the advantages of the fault identification performance index Id. As we can see from Table 1, Id is less than 1 only when one PC is selected, so the optimal number of PCs is one according to our selection criterion. Besides, the situation that two PCs are selected by the CPV criterion will cause the fault smearing effect. An example is used to demonstrate the difference between these two criteria. The sensor fault added to the second sensor is with the magnitude of f 2 = 1, and the contribution plots method is applied to identify the faulty sensor. The simulation results are illustrated in Figure 8. From Figure 8 we can find that the contribution plots using our proposed criterion can identify the second sensor as the faulty one correctly, but the contribution plots adopting the CPV criterion misidentify the sixth sensor as the faulty one.

4. CASE STUDIES In this section, a numerical model called Monte Carlo simulation model and the quadruple-tank process model are used to demonstrate the advantages of our proposed fault detection performance index fmd ̅ and fault identification performance index Id, respectively. Then, the implementation 3148


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Figure 2. Monitoring plots of the first sensor fault in Monte Carlo model with criteria: (a) our proposed fault detection performance index, (b) CPV.

Figure 4. Monitoring plots of the third sensor fault in Monte Carlo model with criteria: (a) our proposed fault detection performance index, (b) CPV.

Figure 3. Monitoring plots of the second sensor fault in Monte Carlo model with criteria: (a) our proposed fault detection performance index, (b) CPV.

Figure 5. Monitoring plots of the fourth sensor fault in Monte Carlo model with criteria: (a) our proposed fault detection performance index, (b) CPV.

diagram of the process can be found in Johansson’s paper, and the target is to control the levels of two lower tanks with two pumps. The process model is

Hence, the fault identification performance index Id can help the PC model identify the single sensor fault correctly. Now we will give the whole procedure of our proposed selection algorithm. First, the number of PCs is selected as k1 = 3 when the minimum fmd ̅ is achieved, and this situation corresponds to the maximum detectable fault coverage. Second, the value of the index Id with three PCs and k1 = 3 are checked to verify whether the terminating condition is satisfied or not, and unluckily the terminating condition is not satisfied. Hence, we have to find the minimum fmd ̅ from k = 1 to k = 2, and two PCs are kept. Nevertheless, the terminating condition is still not satisfied, so just one PC is selected finally for this Monte Carlo model to establish the PC model. 4.2. Quadruple-Tank Process Simulation. The quadruple-tank process is a multivariate laboratory process which has a mathematical model developed from physical data by Johansson.20 This process is composed of four interconnected water tanks, two pumps, and associated valves. A schematic

γ k1 a dh1 a = − 1 2gh1 + 3 2gh3 + 1 v1 dt A1 A1 A1 γ k2 a dh 2 a = − 2 2gh2 + 4 2gh4 + 2 v2 dt A2 A2 A2 (1 − γ2)k 2 dh 3 a = − 3 2gh3 + v2 dt A3 A3 (1 − γ1)k1 dh4 a = − 4 2gh4 + v1 dt A4 A4

(25)

where h1−h4 are the water levels, and v1−v2 are the voltages supplied to the pumps. The flow to each tank, f l, is adjusted using the associated valve coefficients γ1 and γ2. 3149


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Figure 6. Monitoring plots of the fifth sensor fault in Monte Carlo model with criteria: (a) our proposed fault detection performance index, (b) CPV.

Figure 8. Contribution plots of the second sensor fault in Monte Carlo model: (a) our proposed fault identification performance index, (b) CPV.

Table 2. Parameter Values of the Quadruple-Tank Process parameter A1 ; A3 A2; A4 a1 ; a3 a2 ; a4 k1 ; k2 v1 ; v2 γ1 ; γ2 g

unit

value

2

cm cm2 cm2 cm2 cm3/(V s) V _ cm/s2

28 32 0.071 0.057 3.33 ; 3.35 3.00 ; 3.00 0.7 ; 0.6 981

dh1 A 1 = − (h1 − h10) + 1 (h3 − h30) dt T1 A3T3 γ1k1 (v1 − v10) + A1 dh 2 A1 1 (h4 − h4 0) = − (h 2 − h 2 0 ) + dt T2 A4 T4 γ k2 + 2 (v2 − v2 0) A2

Figure 7. Monitoring plots of the sixth sensor fault in Monte Carlo model with criteria: (a) our proposed fault detection performance index, (b) CPV.

f1 = γ1k1v1

(1 − γ2)k 2 dh 3 1 (v2 − v2 0) = − (h 3 − h 3 0 ) + dt T3 A3

f2 = γ2k 2v2 f3 = (1 − γ2)k 2v2 f4 = (1 − γ1)k1v1

(1 − γ1)k1 dh4 1 (v1 − v10) = − (h4 − h4 0) + dt T4 A4

(26)

Ti =

The parameter values of this process are given in Table 2.21

Ai ai

2hi 0 , i = 1, 2, 3, 4 g

(27)

where h0 = [h10 h20 h30 h40] = [12.4 12.7 1.8 1.4] and v0 = [v10 v20] = [3.0 3.0] are the values when the process is at operating points. The data are generated by eqs 26 and 27, where the measured variables h1−h4 and f1−f4 are corrupted by the Gaussian white noise with zero mean and the standard deviation of 0.01, γl and vl are corrupted by the Gaussian white noise with zero mean and the standard deviation of 0.05.

Since the quadruple-tank process is nonlinear and our proposed performance indices are much more suitable for linear system, the process model in eq25 is linearized as20 3150


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Industrial & Engineering Chemistry Research One thousand samples under the normal condition are generated to train the PC model after the quadruple-tank process has been in its steady state. The values of the fault detection and identification performance indices from k = 1 to k = 7 are listed in Table 3. If our proposed fault detection performance index fmd ̅ is used Table 3. Fault Detection and Identification Performance Indices of the Quadruple-Tank Process k

fmd ̅

Id

1 2 3 4 5 6 7

3.7591 3.2824 1.5007 0.9609 1.0048 1.2878 35.9476

0.3487 0.4973 1.0001 1.2330 1.3213 3.4898 1034.1

Figure 10. Monitoring plots of the second sensor fault in quadrupletank model with criteria: (a) our proposed fault detection performance index, (b) CPV.

to select the PCs, four PCs corresponding to the minimum fmd ̅ are determined as the optimal selection of PCs. According to the CPV criterion with a predetermined percentage of 85%, four PCs are also selected. However, in order to show the difference between these two methods, six PCs are chosen using the CPV criterion with a predetermined percentage of 95%. Then we will compare the fault detection abilities of these two different criteria. The sensor faults added to the four different sensors indicating water levels are with the magnitudes of f1 = 0.2, f 2 = 0.2, f 3 = 0.1, and f4 = 0.2, when t = 501, respectively. The monitoring plots are shown in Figures 9−12.

Figure 11. Monitoring plots of the third sensor fault in quadruple-tank model with criteria: (a) our proposed fault detection performance index, (b) CPV.

that six PCs are selected by the CPV criterion will cause the fault smearing effect. An example is used to demonstrate the difference between these two criteria. The sensor fault added to the third sensor is with the magnitude of f 3 = 0.1, and the contribution plots method is applied to identify the faulty sensor. The simulation results are illustrated in Figure 13. From Figure 13 we can find that the contribution plots using our proposed criterion can identify the third sensor as the faulty one correctly, but the contribution plots adopting the CPV criterion misidentify the seventh sensor as the faulty one. Hence, the fault identification performance index Id can help the PC model identify the single sensor fault correctly. Now we will give the whole procedure of our proposed selection algorithm. Four PCs are first selected on the basis of the optimum performance of fault detection, and the values of Id corresponding to four PCs and k1 are checked. The result that Id is larger than 1 and k1 ≠ 1 leads us to find the suboptimum performance from k = 1 to k = 3. The situation of

Figure 9. Monitoring plots of the first sensor fault in quadruple-tank model with criteria: (a) our proposed fault detection performance index, (b) CPV.

From Figures 9−12 we can see that the PC model using our proposed criterion can detect all these four sensor faults, while the PC model using the CPV criterion fails in these cases. It comes to the same conclusion that the detectable fault coverage of our proposed criterion is larger than that of the CPV criterion. Next we will discuss the advantages of the fault identification performance index Id. As we can see from Table 3, Id is less than 1 when two PCs are selected, so the optimal number of PCs is two according to our selection criterion. Besides, the situation 3151


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5. CONCLUSIONS Selecting the number of PCs is an important but open problem in PCA for process monitoring. There are many known solutions, such as CPV, VRE, and Fault SNR. However, none of these methods takes the performance of fault detection and identification into consideration together. In this paper, a novel method based on two new performance indices of fault detection and identification is proposed for selecting the optimum number of PCs. The fault detection optimization related to the proposed fault detection performance index is to make the detectable fault coverage maximum, which means that the PC model can detect all single sensor faults (ξ1,...ξm) with the largest fault coverage. The selection strategy related to the proposed fault identification performance index is to ensure that the fault identification result is correct if this situation can be achieved. The selection algorithm that combines these two performance indices is applied to the Monte Carlo model and the quadruple-tank process. Also, the advantages of our proposed performance indices are illustrated in these two models. Furthermore, detailed analysis is given to explain the influence of the number of PCs on the fault identification performance index mathematically. The proposed fault identification performance index just considers the correct identification result of single sensor fault, so some new indices representing the performance of multiple sensor faults identification need to be researched in the future.

Figure 12. Monitoring plots of the fourth sensor fault in quadrupletank model with criteria: (a) our proposed fault detection performance index, (b) CPV.

■

APPENDIX: PROOF OF THEOREM 1 We consider the situation when 1 ≤ k < m, and assume that |c̃ia| = maxj,j≠i(|c̃ij|). The minimum difference between |c̃ii| and |c̃ij| is ei = min(|ciĩ | − |cij̃ |) = |ciĩ | − |ciã | j

k

=1−

∑ pli2

k

− |∑ pli pla |

l=1

l=1

k

>1−

∑ pli2 l=1

Figure 13. Contribution plots of the third sensor fault in quadrupletank model: (a) our proposed fault identification performance index, (b) CPV.

k

−

∑ |pli pla | l=1

k

=1−

∑ (pli2

+ |pli pla |)

l=1

≥ 1 − 1.21k

It should be noticed that the greatest lower bound 1−1.21k is the optimum solution of the nonlinear optimization under the constrained condition as follows:

three PCs still cannot meet the terminating condition, and finally two PCs are selected as the terminating condition is satisfied. From the results of our two simulation cases, it seems that if we choose the maximum number of PCs when Id(k1) < 1 is satisfied, the optimum number of PCs k1 is obtained. However, it is really difficult for us to prove that the average MDFM fmd ̅ is monotone decreasing from 1 PC to k1 PCs, and we cannot exclude the situation that the average MDFM fmd ̅ of k1 PCs is not minimum from 1 PC to k1 PCs. Moreover, the strategy of selecting the number of PCs using our proposed twoperformance indices is not one and only, and sometimes the priority of fault detectable coverage is higher than that of the correction of fault identification. In addition, we can find that the variation of Id in each simulation conforms to the description about Theorem 1, which validates our proposed theorem.

⎧ f (x , y) = 1 − k(x 2 + |xy|) ⎪ ⎪ min f (x , y) ⎨ ⎪ ⎪ s.t. x 2 + y 2 ≤ 1 ⎩

When x = 0.92 and y = 0.38, the optimum solution is obtained.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 3152


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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 61473254 and the 973 Program of China under Grant No. 2012CB720500.

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REFERENCES

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Selecting the Number of Principal Components on the Basis of

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