A new usage of reconstruction based contributions for ... - IEEE Xplore
A new usage of reconstruction based contributions for quality-related fault diagnosis Gang Li1 , Kaixiang Peng2 , Jie Ma3 ,Donghua Zhou1 1. department of Automation, TNList, Tsinghua University, Beijing 100084, P. R. China E-mail: [email protected] E-mail: [email protected] 2. Key Laboratory for Advanced Control of Iron and Steel Process, School of Automation and Electrical Engineering, University of Science and Technology of Beijing, Beijing, 100083, P. R. China E-mail: [email protected] 3. Automation College, Beijing Information Science and Technology University ,Beijing 100192, P.R. China E-mail: [email protected]
Abstract: Quality-related fault detection attracted more and more attention in quality control and process monitoring. In recent literature, reconstruction based contributions (RBC) are used for isolating faulty variables which affect product quality. If datasets of known faults are available, fault-specific RBCs are used to identify fault types. Otherwise, variable RBCs are used to isolate faulty variables. However, the existing generalized RBC are quite improvable. On one side, fault-specific RBC can not tell faulty variables, which makes it hard to locate fault root. On the other side, it is well known that the variable RBC suffers a lot from the smearing effect, which provides too many candidates of faulty variables. In the present work, a new usage of RBC is derived to select faulty variables without smearing effect on nonfaulty variables. The benchmark examples of Tennessee Eastman (TE) process is used to demonstrate the efficiency of the proposed approach for quality-related fault diagnosis. Key Words: fault diagnosis and isolation, multidimensional reconstruction based contribution, quality-related faults, smearing effect, total projection to latent structures
1
INTRODUCTION
Quality control and monitoring has become a significant issue in modern manufacturing industries, such as semiconductors, Biomedical engineering, and chemical engineering. With the huge amount of data collected from industrial processes by distributed control systems, it is more proper to use data-driven techniques for detecting abnormal situations and isolating faulty components, rather than to use techniques based on an assumed model or artificial knowledge. Data-driven Statistical process monitoring (SPM) applies multivariate projection models, such as principal component analysis (PCA) and projection to latent structures (PLS), and machine learning approaches to fault detection and diagnosis, which has been one of the most fruitful areas in both practice and research[1, 2]. For many applications, PCA is a popular and useful tool to capture the inner correlation among thousands of measured variables. A measurement sample can be decomposed into principal part and residual by a PCA model, and monitored in two subspaces with different statistics. However, the detection results by PCA is quality-unrelated. That is to say, one cannot tell whether a fault detected by PCA method will affect the quality index or not. PLS model constructs This work is partially supported by the national 973 project under grants 2010CB731800 and 2009CB320602 and the National Nature Science Foundation under Grant 61074085 and 61273173, and Beijing Natural Science Foundation under grant 4122029.
c 978-1-4673-5534-6/13/$31.00 2013 IEEE
the relations among measured variables (or called process variables) and quality variables, and divides the measured space into two parts with respect to quality variables. For a long time, PLS model has not been proper used for detecting quality-related faults until Li et al. first revealed the geometric properties of PLS for the purpose of process monitoring [3]. Then Zhou et al. proposed total PLS model to solve the problems in the PLS based monitoring [4]. In T-PLS model, the process space is decomposed into four parts according to quality index, while two parts are quality-related and other two part are quality-unrelated. It is more proper to detect quality-related faults in the corresponding subspaces with specific statistics, rather than in the whole process variable space with a global statistic. After a fault is detected, a diagnosis tool should be used for isolating faulty variables or identifying fault types. There are several approaches for identifying fault types when historical faulty data are available, such as discriminant analysis [5], structure residual method[6], pattern recognition method[7] and reconstruction based method[8]. However, these approaches are not practical for real process, because the known fault list may not exist. The counterpart is to isolate faulty variables without a prior knowledge, which is categorized as an unsupervised method. The most popular method is to use contribution plot and its variants, which does not need known faulty data or information [9, 10, 11]. However, contribution plots suffer a lot from smearing effect in many cases, which leads to many false 4297
diagnosis results[10, 12]. Recently, Alcala and Qin proposed reconstruction-based contributions for process monitoring, which combines contribution plots and reconstruction methods together. It is strictly guaranteed that the correct variable will be isolated for an univariate sensor fault[12]. Furthermore, Li et al. extended RBC method to multidimensional fault including sensor faults and process faults, which is called generalized RBC method[13]. However, generalized RBC still needs historical faulty information for identifying fault types. More recently, kariwala et al. combined the branch and bound method with missing variable approach of probabilistic PCA (PPCA) to locate faulty variables[14][8]. However, the known event data sets are not needed. Liu proposed a similar policy for contribution plots based on missing data analysis of traditional PCA model[15], which eliminates the smearing effect on non-faulty variables greatly. However, the method proposed by Liu lacks theoretical analysis and its relationship with reconstruction based contribution is interesting. In the presented work, a new usage of reconstruction based contributions is proposed to extend RBC to multidimensional faults without known fault list. In the case of unknown fault datasets, the combination of faulty variables is searched for fault isolation. At last, the Tennessee eastman process is used for demonstrating the efficiency of proposed method[16].
2
Quality-related fault detection and diagnosis
2.1
T-PLS model
Denote X ∈ Rn×m as input matrix consisting of n samples with m process variables, and Y ∈ RN ×l as output matrix with l quality variables per sample. Generally, columns of X and Y are centered to zero mean and scaled to unit variance. PLS captures the relation of X and Y as follows [9]: X = TPT + E (1) Y = TQT + F where T ∈ Rn×A is the score matrix, P ∈ Rm×A and Q ∈ Rp×A are the loading matrices for X and Y, respectively. A is the number of PLS components, which is usually determined by cross validation. E and F are the residual matrices of X and Y, respectively. However, it is misleading to use standard PLS model for detecting qualityrelated faults with T 2 statistic and quality-unrelated faults with Q statistic. To improve PLS based monitoring, Zhou et al. proposed the T-PLS algorithm to decompose PLS model further [4]: T T X = Ty PT y + To Po + Tr Pr + Er (2) T Y = Ty Qy + F where Ty ∈ Rn×Ay , To ∈ Rn×(A−Ay ) , Tr ∈ Rn×Ar are the score matrices, and Py ∈ Rm×Ay , Po ∈ Rm×(A−Ay ) , Pr ∈ Rm×Ar are the corresponding loading matrices. Qy ∈ Rl×Ay is the new loading matrix for Y. Er is the new residual matrix. In the T-PLS model, Ty reflects the quality-related variation in T, To reflects the variation orthogonal to Y in T, Tr means the principal part of E, and Er means the residual part that is not excited in normal
4298
situation. T-PLS based methods can increase the detection rate and reduce the false alarm rate for quality-related faults [4]. 2.2
fault detection indices
Traditional PLS based methods monitor quality-related variations with T 2 type of indices and quality-unrelated variations with Q type of indices, respectively[9]. However, T 2 still monitors Y-unrelated variations, and the residual E still contains a large variability. Instead, T-PLS based methods can monitor quality-related faults more efficiently. As quality variables are infrequently measured and usually with a significant time delay, process samples should be considered only. Denote a new sample as xnew , then the scores and the residual are calculated as[4]: ty to tr ˜r x
= = = =
RT y xnew T T PT o (PR − Py Ry )xnew T T Pr (I − PR )xnew T (I − Pr PT r )(I − PR )xnew
(3)
where Ry = RQT Qy
(4)
In T-PLS based methods, ty , to and tr contain the systematic process variation and are suitable to monitor with ˜r represents the ultimate residual part, it T 2 statistic. As x is suitable to use the Q statistic. Table 1 lists calculations of fault detection indices and their control limits, where 1 1 1 T T TT Λy = n−1 y Ty , Λo = n−1 To To , Λr = n−1 Tr Tr , S is the sample variance of Qr and μ is the sample mean of Qr , α is the significance level of χ2 and F distribution. Table 1: Fault detection indices and control limits Statistic
Calculation
Control limit
Ty2
−1 tT y Λy ty
To2
−1 tT o Λo to
Tr2 Qr
−1 tT r Λr tr ˜ xr 2
Ay (n2 −1) FAy ,n−Ay ,α n(n−Ay ) (A−Ay )(n2 −1) FA−Ay ,n−A+Ay ,α n(n−A+Ay ) Ar (n2 −1) FAr ,n−Ar ,α n(n−Ar ) δr2 = (S/2μ)χ22μ2 /S,α
δy2 =
The abnormal situations in subspaces Sy and Srr affect output data. Therefore, Ty2 and Qr need to be monitored simultaneously. However, one may prefer to observe one index rather than two indices for fault detection. Li et al. followed Yue and Qin [8] to propose a combined index for monitoring quality-related faults as follows, which incorporates Ty2 and Qr in a balanced way [13]: ϕ(x) =
Ty2 Qr + 2 = xT Φx δy2 δr
(5)
where Φ
=
−1 T RT y Λy Ry 2 δy
+
T (I−RPT )(I−Pr PT r )(I−PR ) δr2
(6)
From (5), ϕ(x) is a quadratic function of x, thus the control limit for ϕ(x) can be calculated as follows: ζ 2 = gχ2h,α
2013 25th Chinese Control and Decision Conference (CCDC)
(7)
Φ) , h = [tr(SΦ)] and S = 1 XT X is where g = tr(S n−1 tr(SΦ) tr(SΦ)2 the estimated covariance of input variables. In this paper, ϕ is used for detecting quality-related faults. 2
2.3
2
diagnosis based on contribution methods
Once a fault is detected, a diagnosis tool is required to determine the fault type or identify the faulty variables. Contribution plot is a very popular tool to isolate the faulty variables, which does not require known fault lists. Based on T-PLS model, Li et al. proposed contribution plots to Qr and Ty2 , Tr2 , To2 in a unified form [17]. Contributions to the index can be expressed as: 1
= (γ j x)2 = (ξjT Γ 2 x)2 ContIndex j
(8)
where Index is a fault detection index such as Qr or Ty2 , denotes Γ represents the corresponding matrix, ContIndex j the contribution of variable j to Index, γj is the j th row 1 of Γ 2 and ξj is the j th column of the identity matrix Im . Table 2 lists the matrix Γ for different indices based on TPLS model. Table 2: Γ for various detection index Detection Γ index T Qr (I − RPT )(I − Pr PT r )(I − PR ) T Ry Λ−1 R Ty2 y y −1 T T To2 (RPT − Ry PT PT y )Po Λo o (PR − Py Ry ) 2 T −1 T T (I − RP )Pr Λr Pr (I − PR ) Tr ϕ Φ
The method of contributions assumes variable with large contribution to fault detection index are most likely faulty variables. However, this assumption does not have a solid theoretical basis and may cause misleading diagnosis result in most cases. Even for a sensor fault, contribution plots may give a misleading result due to the smearing effect [7]. To overcome the problem, Alcala and Qin proposed a reconstruction based contribution method to diagnose faulty variables[12]. The method combines contribution analysis and reconstruction based identification together, which gives improved diagnosis results. The RBC for variable j is defined as = xT Γξj (ξjT Γξj )−1 ξjT Γx = RBCIndex j
(ξjT Γx)2 ξjT Γξj
(9)
where all parameters are the same as in (8). It can be seen that RBC and Cont are different in calculation with the same information.
3
where (·)+ denotes the Moore-Penrose pseudoinverse of a matrix, Ξ is the fault direction matrix consisting of several faulty sensors or extracted from fault datasets. If the actual fault direction Ξ is stored in the known fault sets, then RBC method can tell the correct fault type according to two properties of RBC[13]. However, for many cases, there is not known fault sets for multidimensional faults. Kariwala et al. firstly proposed the idea of using all different combinations of sensors as missing variables based on Probabilistic PCA and used a branch and bound algorithm to search the minimum of reconstructed index[14]. Liu transplanted the idea into traditional PCA and simplified the search program[15]. His work made use of combined index (RCI), and chose the faulty variables via maximizing the RCI. In the presented work, the similar strategy will be introduced into reconstruction based contributions with T-PLS model for fault-related diagnosis. The difference and similarity of proposed method with aforementioned approaches will be discussed in this section. As RBC has been extended to multidimensional fault with known direction matrix, it can deal with arbitrary given direction matrix. The proposed approach first calculate RBC with each ξi (i = 1, ..., m) and inserts the variable with the maximum RBC into a faulty variable set Xf . The selected faulty variable set Xf can eliminate fault effect on ϕ(x) most efficiently. Next, the RBC are recalculated with all selected faulty variables and one variable in non-faulty variables. The non-faulty variable with the maximum RBC will be added to Xf . The steps are repeated until the reconstructed ϕ(x) is under its control limit. The whole algorithm is as follows: 1. set k=0 and Xf = ∅. 2. for i=1:m-k, construct the fault direction Ξi = [ξi , ξf 1 , ...ξf k ], where xf i ∈ Xf . Calculate RBC with Ξi . 3. Insert the variable xi with the maximum RBC into Xf , and k = k + 1. 4. Go back to step 2, until the RBC is over ϕ(x) − ζ 2 . After this simple strategy, Xf consists of minimum of faulty variables which can describe the fault effect thoroughly. According to the property of RBC, the RBC for selected faulty direction increases with the growing number of Xf . Therefore, there must exist limited k (usually small) faulty variables to satisfy the condition in step 4. After Step 4, all selected faulty variables can be sorted according to the contributions to RBC as follows: 1
Proposed approach
Although RBC has a better performance than traditional contribution, it can hardly tell faulty variables for simultaneous multiple faults. In literature [13], Li et al. proposed generalized RBC with known multidimensional fault direction as follows: RBC = xT ΦΞ(ΞT ΦΞ)+ ΞT Φx
(10)
= {ξjT [(ΞT ΦΞ)+ ] 2 ΞT Φx}2 (j = 1, ..., k) contRBC j (11) Note that j is the number in Xf , and ξj is jth column of identity matrix with k dimensions. Remark 1 The proposed approach can be applied to other statistic models such as PCA, ICA and so on. However, the diagnosis results may be different as the index are not consistent. RBC for a given index is responsible for only this
2013 25th Chinese Control and Decision Conference (CCDC)
4299
index, which should be explained together with detection results. In this paper, only quality-related faults are concerned, therefore the selected faulty variables will not only contribute to fault detection index, but also affect quality index significantly. Remark 2 The proposed approach (10) and (11) are equivalent to Eq. (14) and (15)in [15], in the case that ΞT ΦΞ is full-ranked. The proof of the equivalence is omitted due to page limitation. Although Φ may be singular, ΞT ΦΞ may be non-singular. However, the approach in [15] has not defined RCI when ξT Φξ is not invertible, while in this approach, RBC for arbitrary fault direction is defined consistently. Remark 3 The search strategy in above algorithm hides a fact that the variables with maximum RBC in a loop will still stay in Xf in the next loop. Apparently, it sounds reasonable. However, it may be wrong. According to case study on TEP, the search for all the combination such as Ckn is more accurate. However, it is still a good approximation for the whole optimal solution, which can save a large amount of computation load.
4
Case study on TEP
The Tennessee Eastman process was a realistic simulation model of industrial process, generated by the Eastman Chemical Company to evaluate different process control and monitoring methods[16]. The gaseous reactants A, C, D, and E and the inert B were fed to the reactor, while the liquid products G and H are formed there. The by-product of the reactions was species F. The process used here was operated under a closed-loop control. TEP has been widely used as a benchmark process for evaluating the process diagnosis methods such as PCA, multi-way PCA, fisher discriminant analysis (FDA), and PLS[3, 13]. T-PLS based methods were also applied to TEP[4]. Chiang et al. reviewed fault detection and diagnosis methods of multivariate statistics process monitoring based on the case study of TEP[18]. Kariwala et al. developed a branch and bound method that incorporated missing variable analysis to isolate the faulty variables in TE process [14]. Liu proposed a simplified search algorithm with missing variable analysis based on PCA to select faulty variables in TE process[15]. TEP contains two blocks of variables, including 12 manipulated variables and 41 measured variables. Process measurements are sampled with interval of 3 minutes. Nineteen composition measurements are measured with time delays that vary from six minutes to fifteen minutes. This time delay has a critical impact on product quality possibly. It implies that the fault effect on product quality can not be detected until the next output sample comes. During this time, the products are produced with lower quality. T-PLS based monitoring methods can detect the fault using process input data, thus it is able to detect quality-related faults before quality data are measured. In this study, the composition of G in stream 9, i.e. MEAS(35), is chosen as the output variable y with a time delay of 6 minutes. Twenty two process measurements
4300
and eleven manipulated variables, i.e. MEAS1 through MEAS22 and MV1 through MV11, are chosen as input data x. The detailed information of x and y can refer to [13]. Firstly, 480 normal sample pairs of x and y are centered to zero mean and scaled to unit variance, which are used to built a T-PLS model with A = 6, Ay = 1, and Ar =19. There are 15 types of known faults in TEP, which are represented as IDV (1) - (15). However, only eight of them can be seen as quality-related faults, which are IDV 1,2,5,6,8,10,12 and 13 [4]. Figs. 3-4 in [13] show output y and its prediction error in the normal and faulty cases, which indicates these faults affect y significantly. Here, IDV (1) is taken as an example to illustrate the efficiency of new usage of RBC. Fault 1 was a step type abnormal situation in which the composition of A of steam 4 was changed from 48.5 mol% to 45.5 mol%, meanwhile, the composition of C was changed from 51 mol% to 54 mol%. This leaded to a decrease in the A feed in the recycle stream 5 and control loop reacted to increase the A feed in Stream 1. These two reactions counteracted each other over time. In Figure 1, subplot (a) used the proposed algorithm to select faulty variables and sorted them according to normalized contributions to RBC, while subplot (b) just selected faulty variables with RBC exceeding the control limits. The results showed that RBC gave too many faulty variables which are actually non-faulty, making the fault diagnosis confusing. As shown in subplot (b) of Figure 2, detection index removing faulty variables according to variable RBC is far below the control limit. However, the new usage of RBC can reduce the number of faulty variables to the minimum that can just reconstruct measurement thoroughly in subplot (a) of Figure 2. Figure 3 reflected the efficiency of the fault isolation. Variable RBC selected more 15 faulty variables for every sample while the proposed usage of RBC selected no more than 3 faulty variables for each sample. According to the proposed method, variables 10,11,19,20 are selected as chief faulty variables. Figure 4 indicated the trends of these faulty variables, which were consistent with the analysis. Notice that the diagnosis result is quality-related, which means the selected faulty variables contribute a lot to the violation in quality index. This is the chief difference from PCA based methods.
5
conclusion
In the presented work, a new usage of reconstruction based contributions is proposed to isolate quality-related faulty variables based on T-PLS model. In order to select faulty variables without smearing effect on non-faulty variables, a simplified search algorithm is used to find the fault direction combination which just increase RBC above to a lower limitation. The selected faulty variables can be responsible for the violation in detection index and describe the abnormal situation sufficiently. The case study on Tennessee Eastman process demonstrates the new usage of RBC can isolate faulty variables more efficiently and accurately than the traditional usage of RBC.
2013 25th Chinese Control and Decision Conference (CCDC)
32
32
30
30
28
28
80%
26
26
24
24 22
variable index
variable index
22 40%
20 18 16 14
20%
20
10%
18 16 14
12
12
10
10
8
8
10%
6
6
4
4
2
2 100
200
300
400 500 600 samples number
700
800
5%
900
100
(a) normalized contribution plot to RBC
200
300
400 500 600 samples number
700
800
5%
900
(b) normalized variable RBC with control limit
Figure 1: The detection and isolation result for Fault 1 2
5
10
10 φ removing FVs control limit
φ removing fault variables control limit
1
0
10
0
10
φ
φ
10
−1
−10
10
10
−2
10
−5
10
−15
0
100
200
300
400 500 600 samples number
700
800
10
900
0
(a) faulty variables are selected by new usage
200
400 600 samples number
800
1000
(b) faulty variables are selected by variable RBC
Figure 2: Detection index removing faulty variables
1000
100
500
100 90
frequency accumlating percentage
400
80
90
500
70
80
frequency
frequency
frequency accumlating percentage 300
60 50
200
40 30
70 100
20 10
0
0
1
2
3
4 5 6 7 8 number of fault variables
9
(a) minimum number of faulty sensor
10
60 12
0
0
5
10
15 20 25 number of fault variables
30
0 35
(b) number of faulty sensor given by RBC
Figure 3: The efficiency of fault isolation 2013 25th Chinese Control and Decision Conference (CCDC)
4301
purge rate
separator temp
0.4
80
x
x
11
82
10
0.5
0.3
0
500 sample number stripper steam flow
76
1000
400
360
350
350
x20
x
19
0.2
78
300 250 200
0
500 sample number compressor work
1000
0
500 sample number
1000
340 330
0
500 sample number
1000
320
Figure 4: Trends of faulty variables for Fault 1
REFERENCES [1] S. J. Qin, statistical process monitoring: basic and beyond, J. chemometrics, Vol. 17, No. 8-9, 480-502,2003. [2] S. J. Qin, survey on data-driven industrial process monitoring and diagnosis, Annual reviews in control, Vol. 36, No. 2, 220234, 2012. [3] G. Li, S. J. Qin, D. H. Zhou, Geometric properties of partial least squares for process monitoring, Automatica, Vol. 46, No. 1, 204-210, 2010. [4] D. H. Zhou, G. Li, S. J. Qin, Total projectoin to latent structures for process monitoring, AIChE Journal, Vol. 56, No. 1, 168-178, 2010. [5] A. Raich and A. Cinar, statistical process monitoring and disturbance diagnosis in multivarible coninuous processes, AIChE Journal, Vol. 42, No. 4, 995-1009, 1996. [6] S. J. Qin, W. Li, Detection, indetification and reconstruction of faulty sensors with maximized sensitivity, AIChE journal, Vol. 45, no. 9, 1963-1976, 1999. [7] S. Yoon, J. MacGregor, Fault diagnosis with multivairate statistical models part I: Using steady state fault signatures, J. Process control, Vol. 11, No. 4, 387-400, 2001. [8] H. Yue, S. J. Qin, reconstruction-based fault identification using a combined index, Ind. Eng. Chem. Res., Vol. 40, No. 20, 4403-4414, 2001. [9] J. F. MacGregor, C. Jaeckle, C. Kiparissides, and M. Koutoudi, Process monitoring and diagnosis by multiblock PLS methods, AIChE Journal, Vol. 40, No.5,826-838,1994. [10] J. A. Westerhuis, S. P. Gurder, and A. K. Smilde, generalized contribution plot in multivarate statisttical process monitoirng, chemometrics. Intell. Lab. Syst.,Vol. 51, No. 1, 95114, 2000.
4302
[11] S. W. Choi and I. B. Lee, multiblock PLS -based localized process diagnosis, J. Process control, Vol. 15, No. 3, 295306, 2005 [12] C. F. Alcala, S. Qin, reconstruction-based contribution for process monitoring, Automatica, Vol. 45, No. 7, 1593-1600, 2009 [13] G. Li, C. F. Alcala, S. J. Qin, D. H. Zhou, Generalized reconstruction-based contributions for output-relevant fault diagnosis with application to the tennessee eastman process IEEE Trans. congtrol system technology, Vol. 19, No. 5, 1114-1127, 2011 [14] V. Kariwala, P.E. Odiowei, Y. Cao, T. Chen, A branch and bound method for isolation of faulty variables through missing variable analysis, Vol. 20, No. 10, 2010. [15] J. Liu, Fault diagnosis using contribution plots without smearing effect on non-faulty variables, J. Process control, http://dx.doi.org/10.1016/j.jprocont.2012.06.016, 2012. [16] J. J. Downs, E. F. Vogel, A plant-wide industrial process control problem, Computer Chemical Engineering, Vol. 17, No. 3, 245-255, 1993. [17] G. Li, S. J. Qin, Y. D. Ji, D. H. Zhou. Total PLS based contribution plots for fault diagnosis. ACTA Automatica Sinica. 2009, 35(6): 759-765 [18] L.H. Chiang, E.L. Russell, and R.D. Braatz, fault diagnosis in chemical processes using Fisher discriminant analysis, discriminant partial least squares, and principal component analysis, chemometrics intell. lab. syst., Vol. 50, No. 2, 243252, 2000.
2013 25th Chinese Control and Decision Conference (CCDC)
Abstract: Quality-related fault detection attracted more and more attention in quality control and process monitoring. In recent literature, reconstruction based contributions (RBC) are used for isolating faulty variables which affect product quality. If datasets of known faults are available, fault-specific RBCs are used to identify fault types. Otherwise, variable RBCs are used to isolate faulty variables. However, the existing generalized RBC are quite improvable. On one side, fault-specific RBC can not tell faulty variables, which makes it hard to locate fault root. On the other side, it is well known that the variable RBC suffers a lot from the smearing effect, which provides too many candidates of faulty variables. In the present work, a new usage of RBC is derived to select faulty variables without smearing effect on nonfaulty variables. The benchmark examples of Tennessee Eastman (TE) process is used to demonstrate the efficiency of the proposed approach for quality-related fault diagnosis. Key Words: fault diagnosis and isolation, multidimensional reconstruction based contribution, quality-related faults, smearing effect, total projection to latent structures
1
INTRODUCTION
Quality control and monitoring has become a significant issue in modern manufacturing industries, such as semiconductors, Biomedical engineering, and chemical engineering. With the huge amount of data collected from industrial processes by distributed control systems, it is more proper to use data-driven techniques for detecting abnormal situations and isolating faulty components, rather than to use techniques based on an assumed model or artificial knowledge. Data-driven Statistical process monitoring (SPM) applies multivariate projection models, such as principal component analysis (PCA) and projection to latent structures (PLS), and machine learning approaches to fault detection and diagnosis, which has been one of the most fruitful areas in both practice and research[1, 2]. For many applications, PCA is a popular and useful tool to capture the inner correlation among thousands of measured variables. A measurement sample can be decomposed into principal part and residual by a PCA model, and monitored in two subspaces with different statistics. However, the detection results by PCA is quality-unrelated. That is to say, one cannot tell whether a fault detected by PCA method will affect the quality index or not. PLS model constructs This work is partially supported by the national 973 project under grants 2010CB731800 and 2009CB320602 and the National Nature Science Foundation under Grant 61074085 and 61273173, and Beijing Natural Science Foundation under grant 4122029.
c 978-1-4673-5534-6/13/$31.00 2013 IEEE
the relations among measured variables (or called process variables) and quality variables, and divides the measured space into two parts with respect to quality variables. For a long time, PLS model has not been proper used for detecting quality-related faults until Li et al. first revealed the geometric properties of PLS for the purpose of process monitoring [3]. Then Zhou et al. proposed total PLS model to solve the problems in the PLS based monitoring [4]. In T-PLS model, the process space is decomposed into four parts according to quality index, while two parts are quality-related and other two part are quality-unrelated. It is more proper to detect quality-related faults in the corresponding subspaces with specific statistics, rather than in the whole process variable space with a global statistic. After a fault is detected, a diagnosis tool should be used for isolating faulty variables or identifying fault types. There are several approaches for identifying fault types when historical faulty data are available, such as discriminant analysis [5], structure residual method[6], pattern recognition method[7] and reconstruction based method[8]. However, these approaches are not practical for real process, because the known fault list may not exist. The counterpart is to isolate faulty variables without a prior knowledge, which is categorized as an unsupervised method. The most popular method is to use contribution plot and its variants, which does not need known faulty data or information [9, 10, 11]. However, contribution plots suffer a lot from smearing effect in many cases, which leads to many false 4297
diagnosis results[10, 12]. Recently, Alcala and Qin proposed reconstruction-based contributions for process monitoring, which combines contribution plots and reconstruction methods together. It is strictly guaranteed that the correct variable will be isolated for an univariate sensor fault[12]. Furthermore, Li et al. extended RBC method to multidimensional fault including sensor faults and process faults, which is called generalized RBC method[13]. However, generalized RBC still needs historical faulty information for identifying fault types. More recently, kariwala et al. combined the branch and bound method with missing variable approach of probabilistic PCA (PPCA) to locate faulty variables[14][8]. However, the known event data sets are not needed. Liu proposed a similar policy for contribution plots based on missing data analysis of traditional PCA model[15], which eliminates the smearing effect on non-faulty variables greatly. However, the method proposed by Liu lacks theoretical analysis and its relationship with reconstruction based contribution is interesting. In the presented work, a new usage of reconstruction based contributions is proposed to extend RBC to multidimensional faults without known fault list. In the case of unknown fault datasets, the combination of faulty variables is searched for fault isolation. At last, the Tennessee eastman process is used for demonstrating the efficiency of proposed method[16].
2
Quality-related fault detection and diagnosis
2.1
T-PLS model
Denote X ∈ Rn×m as input matrix consisting of n samples with m process variables, and Y ∈ RN ×l as output matrix with l quality variables per sample. Generally, columns of X and Y are centered to zero mean and scaled to unit variance. PLS captures the relation of X and Y as follows [9]: X = TPT + E (1) Y = TQT + F where T ∈ Rn×A is the score matrix, P ∈ Rm×A and Q ∈ Rp×A are the loading matrices for X and Y, respectively. A is the number of PLS components, which is usually determined by cross validation. E and F are the residual matrices of X and Y, respectively. However, it is misleading to use standard PLS model for detecting qualityrelated faults with T 2 statistic and quality-unrelated faults with Q statistic. To improve PLS based monitoring, Zhou et al. proposed the T-PLS algorithm to decompose PLS model further [4]: T T X = Ty PT y + To Po + Tr Pr + Er (2) T Y = Ty Qy + F where Ty ∈ Rn×Ay , To ∈ Rn×(A−Ay ) , Tr ∈ Rn×Ar are the score matrices, and Py ∈ Rm×Ay , Po ∈ Rm×(A−Ay ) , Pr ∈ Rm×Ar are the corresponding loading matrices. Qy ∈ Rl×Ay is the new loading matrix for Y. Er is the new residual matrix. In the T-PLS model, Ty reflects the quality-related variation in T, To reflects the variation orthogonal to Y in T, Tr means the principal part of E, and Er means the residual part that is not excited in normal
4298
situation. T-PLS based methods can increase the detection rate and reduce the false alarm rate for quality-related faults [4]. 2.2
fault detection indices
Traditional PLS based methods monitor quality-related variations with T 2 type of indices and quality-unrelated variations with Q type of indices, respectively[9]. However, T 2 still monitors Y-unrelated variations, and the residual E still contains a large variability. Instead, T-PLS based methods can monitor quality-related faults more efficiently. As quality variables are infrequently measured and usually with a significant time delay, process samples should be considered only. Denote a new sample as xnew , then the scores and the residual are calculated as[4]: ty to tr ˜r x
= = = =
RT y xnew T T PT o (PR − Py Ry )xnew T T Pr (I − PR )xnew T (I − Pr PT r )(I − PR )xnew
(3)
where Ry = RQT Qy
(4)
In T-PLS based methods, ty , to and tr contain the systematic process variation and are suitable to monitor with ˜r represents the ultimate residual part, it T 2 statistic. As x is suitable to use the Q statistic. Table 1 lists calculations of fault detection indices and their control limits, where 1 1 1 T T TT Λy = n−1 y Ty , Λo = n−1 To To , Λr = n−1 Tr Tr , S is the sample variance of Qr and μ is the sample mean of Qr , α is the significance level of χ2 and F distribution. Table 1: Fault detection indices and control limits Statistic
Calculation
Control limit
Ty2
−1 tT y Λy ty
To2
−1 tT o Λo to
Tr2 Qr
−1 tT r Λr tr ˜ xr 2
Ay (n2 −1) FAy ,n−Ay ,α n(n−Ay ) (A−Ay )(n2 −1) FA−Ay ,n−A+Ay ,α n(n−A+Ay ) Ar (n2 −1) FAr ,n−Ar ,α n(n−Ar ) δr2 = (S/2μ)χ22μ2 /S,α
δy2 =
The abnormal situations in subspaces Sy and Srr affect output data. Therefore, Ty2 and Qr need to be monitored simultaneously. However, one may prefer to observe one index rather than two indices for fault detection. Li et al. followed Yue and Qin [8] to propose a combined index for monitoring quality-related faults as follows, which incorporates Ty2 and Qr in a balanced way [13]: ϕ(x) =
Ty2 Qr + 2 = xT Φx δy2 δr
(5)
where Φ
=
−1 T RT y Λy Ry 2 δy
+
T (I−RPT )(I−Pr PT r )(I−PR ) δr2
(6)
From (5), ϕ(x) is a quadratic function of x, thus the control limit for ϕ(x) can be calculated as follows: ζ 2 = gχ2h,α
2013 25th Chinese Control and Decision Conference (CCDC)
(7)
Φ) , h = [tr(SΦ)] and S = 1 XT X is where g = tr(S n−1 tr(SΦ) tr(SΦ)2 the estimated covariance of input variables. In this paper, ϕ is used for detecting quality-related faults. 2
2.3
2
diagnosis based on contribution methods
Once a fault is detected, a diagnosis tool is required to determine the fault type or identify the faulty variables. Contribution plot is a very popular tool to isolate the faulty variables, which does not require known fault lists. Based on T-PLS model, Li et al. proposed contribution plots to Qr and Ty2 , Tr2 , To2 in a unified form [17]. Contributions to the index can be expressed as: 1
= (γ j x)2 = (ξjT Γ 2 x)2 ContIndex j
(8)
where Index is a fault detection index such as Qr or Ty2 , denotes Γ represents the corresponding matrix, ContIndex j the contribution of variable j to Index, γj is the j th row 1 of Γ 2 and ξj is the j th column of the identity matrix Im . Table 2 lists the matrix Γ for different indices based on TPLS model. Table 2: Γ for various detection index Detection Γ index T Qr (I − RPT )(I − Pr PT r )(I − PR ) T Ry Λ−1 R Ty2 y y −1 T T To2 (RPT − Ry PT PT y )Po Λo o (PR − Py Ry ) 2 T −1 T T (I − RP )Pr Λr Pr (I − PR ) Tr ϕ Φ
The method of contributions assumes variable with large contribution to fault detection index are most likely faulty variables. However, this assumption does not have a solid theoretical basis and may cause misleading diagnosis result in most cases. Even for a sensor fault, contribution plots may give a misleading result due to the smearing effect [7]. To overcome the problem, Alcala and Qin proposed a reconstruction based contribution method to diagnose faulty variables[12]. The method combines contribution analysis and reconstruction based identification together, which gives improved diagnosis results. The RBC for variable j is defined as = xT Γξj (ξjT Γξj )−1 ξjT Γx = RBCIndex j
(ξjT Γx)2 ξjT Γξj
(9)
where all parameters are the same as in (8). It can be seen that RBC and Cont are different in calculation with the same information.
3
where (·)+ denotes the Moore-Penrose pseudoinverse of a matrix, Ξ is the fault direction matrix consisting of several faulty sensors or extracted from fault datasets. If the actual fault direction Ξ is stored in the known fault sets, then RBC method can tell the correct fault type according to two properties of RBC[13]. However, for many cases, there is not known fault sets for multidimensional faults. Kariwala et al. firstly proposed the idea of using all different combinations of sensors as missing variables based on Probabilistic PCA and used a branch and bound algorithm to search the minimum of reconstructed index[14]. Liu transplanted the idea into traditional PCA and simplified the search program[15]. His work made use of combined index (RCI), and chose the faulty variables via maximizing the RCI. In the presented work, the similar strategy will be introduced into reconstruction based contributions with T-PLS model for fault-related diagnosis. The difference and similarity of proposed method with aforementioned approaches will be discussed in this section. As RBC has been extended to multidimensional fault with known direction matrix, it can deal with arbitrary given direction matrix. The proposed approach first calculate RBC with each ξi (i = 1, ..., m) and inserts the variable with the maximum RBC into a faulty variable set Xf . The selected faulty variable set Xf can eliminate fault effect on ϕ(x) most efficiently. Next, the RBC are recalculated with all selected faulty variables and one variable in non-faulty variables. The non-faulty variable with the maximum RBC will be added to Xf . The steps are repeated until the reconstructed ϕ(x) is under its control limit. The whole algorithm is as follows: 1. set k=0 and Xf = ∅. 2. for i=1:m-k, construct the fault direction Ξi = [ξi , ξf 1 , ...ξf k ], where xf i ∈ Xf . Calculate RBC with Ξi . 3. Insert the variable xi with the maximum RBC into Xf , and k = k + 1. 4. Go back to step 2, until the RBC is over ϕ(x) − ζ 2 . After this simple strategy, Xf consists of minimum of faulty variables which can describe the fault effect thoroughly. According to the property of RBC, the RBC for selected faulty direction increases with the growing number of Xf . Therefore, there must exist limited k (usually small) faulty variables to satisfy the condition in step 4. After Step 4, all selected faulty variables can be sorted according to the contributions to RBC as follows: 1
Proposed approach
Although RBC has a better performance than traditional contribution, it can hardly tell faulty variables for simultaneous multiple faults. In literature [13], Li et al. proposed generalized RBC with known multidimensional fault direction as follows: RBC = xT ΦΞ(ΞT ΦΞ)+ ΞT Φx
(10)
= {ξjT [(ΞT ΦΞ)+ ] 2 ΞT Φx}2 (j = 1, ..., k) contRBC j (11) Note that j is the number in Xf , and ξj is jth column of identity matrix with k dimensions. Remark 1 The proposed approach can be applied to other statistic models such as PCA, ICA and so on. However, the diagnosis results may be different as the index are not consistent. RBC for a given index is responsible for only this
2013 25th Chinese Control and Decision Conference (CCDC)
4299
index, which should be explained together with detection results. In this paper, only quality-related faults are concerned, therefore the selected faulty variables will not only contribute to fault detection index, but also affect quality index significantly. Remark 2 The proposed approach (10) and (11) are equivalent to Eq. (14) and (15)in [15], in the case that ΞT ΦΞ is full-ranked. The proof of the equivalence is omitted due to page limitation. Although Φ may be singular, ΞT ΦΞ may be non-singular. However, the approach in [15] has not defined RCI when ξT Φξ is not invertible, while in this approach, RBC for arbitrary fault direction is defined consistently. Remark 3 The search strategy in above algorithm hides a fact that the variables with maximum RBC in a loop will still stay in Xf in the next loop. Apparently, it sounds reasonable. However, it may be wrong. According to case study on TEP, the search for all the combination such as Ckn is more accurate. However, it is still a good approximation for the whole optimal solution, which can save a large amount of computation load.
4
Case study on TEP
The Tennessee Eastman process was a realistic simulation model of industrial process, generated by the Eastman Chemical Company to evaluate different process control and monitoring methods[16]. The gaseous reactants A, C, D, and E and the inert B were fed to the reactor, while the liquid products G and H are formed there. The by-product of the reactions was species F. The process used here was operated under a closed-loop control. TEP has been widely used as a benchmark process for evaluating the process diagnosis methods such as PCA, multi-way PCA, fisher discriminant analysis (FDA), and PLS[3, 13]. T-PLS based methods were also applied to TEP[4]. Chiang et al. reviewed fault detection and diagnosis methods of multivariate statistics process monitoring based on the case study of TEP[18]. Kariwala et al. developed a branch and bound method that incorporated missing variable analysis to isolate the faulty variables in TE process [14]. Liu proposed a simplified search algorithm with missing variable analysis based on PCA to select faulty variables in TE process[15]. TEP contains two blocks of variables, including 12 manipulated variables and 41 measured variables. Process measurements are sampled with interval of 3 minutes. Nineteen composition measurements are measured with time delays that vary from six minutes to fifteen minutes. This time delay has a critical impact on product quality possibly. It implies that the fault effect on product quality can not be detected until the next output sample comes. During this time, the products are produced with lower quality. T-PLS based monitoring methods can detect the fault using process input data, thus it is able to detect quality-related faults before quality data are measured. In this study, the composition of G in stream 9, i.e. MEAS(35), is chosen as the output variable y with a time delay of 6 minutes. Twenty two process measurements
4300
and eleven manipulated variables, i.e. MEAS1 through MEAS22 and MV1 through MV11, are chosen as input data x. The detailed information of x and y can refer to [13]. Firstly, 480 normal sample pairs of x and y are centered to zero mean and scaled to unit variance, which are used to built a T-PLS model with A = 6, Ay = 1, and Ar =19. There are 15 types of known faults in TEP, which are represented as IDV (1) - (15). However, only eight of them can be seen as quality-related faults, which are IDV 1,2,5,6,8,10,12 and 13 [4]. Figs. 3-4 in [13] show output y and its prediction error in the normal and faulty cases, which indicates these faults affect y significantly. Here, IDV (1) is taken as an example to illustrate the efficiency of new usage of RBC. Fault 1 was a step type abnormal situation in which the composition of A of steam 4 was changed from 48.5 mol% to 45.5 mol%, meanwhile, the composition of C was changed from 51 mol% to 54 mol%. This leaded to a decrease in the A feed in the recycle stream 5 and control loop reacted to increase the A feed in Stream 1. These two reactions counteracted each other over time. In Figure 1, subplot (a) used the proposed algorithm to select faulty variables and sorted them according to normalized contributions to RBC, while subplot (b) just selected faulty variables with RBC exceeding the control limits. The results showed that RBC gave too many faulty variables which are actually non-faulty, making the fault diagnosis confusing. As shown in subplot (b) of Figure 2, detection index removing faulty variables according to variable RBC is far below the control limit. However, the new usage of RBC can reduce the number of faulty variables to the minimum that can just reconstruct measurement thoroughly in subplot (a) of Figure 2. Figure 3 reflected the efficiency of the fault isolation. Variable RBC selected more 15 faulty variables for every sample while the proposed usage of RBC selected no more than 3 faulty variables for each sample. According to the proposed method, variables 10,11,19,20 are selected as chief faulty variables. Figure 4 indicated the trends of these faulty variables, which were consistent with the analysis. Notice that the diagnosis result is quality-related, which means the selected faulty variables contribute a lot to the violation in quality index. This is the chief difference from PCA based methods.
5
conclusion
In the presented work, a new usage of reconstruction based contributions is proposed to isolate quality-related faulty variables based on T-PLS model. In order to select faulty variables without smearing effect on non-faulty variables, a simplified search algorithm is used to find the fault direction combination which just increase RBC above to a lower limitation. The selected faulty variables can be responsible for the violation in detection index and describe the abnormal situation sufficiently. The case study on Tennessee Eastman process demonstrates the new usage of RBC can isolate faulty variables more efficiently and accurately than the traditional usage of RBC.
2013 25th Chinese Control and Decision Conference (CCDC)
32
32
30
30
28
28
80%
26
26
24
24 22
variable index
variable index
22 40%
20 18 16 14
20%
20
10%
18 16 14
12
12
10
10
8
8
10%
6
6
4
4
2
2 100
200
300
400 500 600 samples number
700
800
5%
900
100
(a) normalized contribution plot to RBC
200
300
400 500 600 samples number
700
800
5%
900
(b) normalized variable RBC with control limit
Figure 1: The detection and isolation result for Fault 1 2
5
10
10 φ removing FVs control limit
φ removing fault variables control limit
1
0
10
0
10
φ
φ
10
−1
−10
10
10
−2
10
−5
10
−15
0
100
200
300
400 500 600 samples number
700
800
10
900
0
(a) faulty variables are selected by new usage
200
400 600 samples number
800
1000
(b) faulty variables are selected by variable RBC
Figure 2: Detection index removing faulty variables
1000
100
500
100 90
frequency accumlating percentage
400
80
90
500
70
80
frequency
frequency
frequency accumlating percentage 300
60 50
200
40 30
70 100
20 10
0
0
1
2
3
4 5 6 7 8 number of fault variables
9
(a) minimum number of faulty sensor
10
60 12
0
0
5
10
15 20 25 number of fault variables
30
0 35
(b) number of faulty sensor given by RBC
Figure 3: The efficiency of fault isolation 2013 25th Chinese Control and Decision Conference (CCDC)
4301
purge rate
separator temp
0.4
80
x
x
11
82
10
0.5
0.3
0
500 sample number stripper steam flow
76
1000
400
360
350
350
x20
x
19
0.2
78
300 250 200
0
500 sample number compressor work
1000
0
500 sample number
1000
340 330
0
500 sample number
1000
320
Figure 4: Trends of faulty variables for Fault 1
REFERENCES [1] S. J. Qin, statistical process monitoring: basic and beyond, J. chemometrics, Vol. 17, No. 8-9, 480-502,2003. [2] S. J. Qin, survey on data-driven industrial process monitoring and diagnosis, Annual reviews in control, Vol. 36, No. 2, 220234, 2012. [3] G. Li, S. J. Qin, D. H. Zhou, Geometric properties of partial least squares for process monitoring, Automatica, Vol. 46, No. 1, 204-210, 2010. [4] D. H. Zhou, G. Li, S. J. Qin, Total projectoin to latent structures for process monitoring, AIChE Journal, Vol. 56, No. 1, 168-178, 2010. [5] A. Raich and A. Cinar, statistical process monitoring and disturbance diagnosis in multivarible coninuous processes, AIChE Journal, Vol. 42, No. 4, 995-1009, 1996. [6] S. J. Qin, W. Li, Detection, indetification and reconstruction of faulty sensors with maximized sensitivity, AIChE journal, Vol. 45, no. 9, 1963-1976, 1999. [7] S. Yoon, J. MacGregor, Fault diagnosis with multivairate statistical models part I: Using steady state fault signatures, J. Process control, Vol. 11, No. 4, 387-400, 2001. [8] H. Yue, S. J. Qin, reconstruction-based fault identification using a combined index, Ind. Eng. Chem. Res., Vol. 40, No. 20, 4403-4414, 2001. [9] J. F. MacGregor, C. Jaeckle, C. Kiparissides, and M. Koutoudi, Process monitoring and diagnosis by multiblock PLS methods, AIChE Journal, Vol. 40, No.5,826-838,1994. [10] J. A. Westerhuis, S. P. Gurder, and A. K. Smilde, generalized contribution plot in multivarate statisttical process monitoirng, chemometrics. Intell. Lab. Syst.,Vol. 51, No. 1, 95114, 2000.
4302
[11] S. W. Choi and I. B. Lee, multiblock PLS -based localized process diagnosis, J. Process control, Vol. 15, No. 3, 295306, 2005 [12] C. F. Alcala, S. Qin, reconstruction-based contribution for process monitoring, Automatica, Vol. 45, No. 7, 1593-1600, 2009 [13] G. Li, C. F. Alcala, S. J. Qin, D. H. Zhou, Generalized reconstruction-based contributions for output-relevant fault diagnosis with application to the tennessee eastman process IEEE Trans. congtrol system technology, Vol. 19, No. 5, 1114-1127, 2011 [14] V. Kariwala, P.E. Odiowei, Y. Cao, T. Chen, A branch and bound method for isolation of faulty variables through missing variable analysis, Vol. 20, No. 10, 2010. [15] J. Liu, Fault diagnosis using contribution plots without smearing effect on non-faulty variables, J. Process control, http://dx.doi.org/10.1016/j.jprocont.2012.06.016, 2012. [16] J. J. Downs, E. F. Vogel, A plant-wide industrial process control problem, Computer Chemical Engineering, Vol. 17, No. 3, 245-255, 1993. [17] G. Li, S. J. Qin, Y. D. Ji, D. H. Zhou. Total PLS based contribution plots for fault diagnosis. ACTA Automatica Sinica. 2009, 35(6): 759-765 [18] L.H. Chiang, E.L. Russell, and R.D. Braatz, fault diagnosis in chemical processes using Fisher discriminant analysis, discriminant partial least squares, and principal component analysis, chemometrics intell. lab. syst., Vol. 50, No. 2, 243252, 2000.
2013 25th Chinese Control and Decision Conference (CCDC)
