Sensor Fault Identification in MSPM Using Reconstructed Monitoring

Ind. Eng. Chem. Res. 2004, 43, 4293-4304

4293

Sensor Fault Identification in MSPM Using Reconstructed Monitoring Statistics Changkyu Lee, Sang Wook Choi,† Jong-Min Lee, and In-Beum Lee* Department of Chemical Engineering, Pohang University of Science and Technology, San 31 Hyoja Dong, Pohang 790-784, Korea

Several reconstruction-based methods for fault isolation have been developed using missing value estimation to treat incomplete data obtained from modern chemical and environmental processes. This paper focuses on sensor fault identification through the reconstruction of each process variable using missing value estimations based on principal component analysis (PCA). We discuss two representative reconstruction methods: the method of projection onto the model plane and the method of known data regression. Through the theoretical analysis of sensor fault effects in the model and residual spaces, we propose two new sensor fault identification indices, FIIM and FIIR. Further, we point out a problem of reconstruction-based sensor fault identification in residual space and demonstrate that FIIM provides consistent fault identification regardless of the choice of reconstruction method. This application of the proposed indices is carried out for a simple five-variable system and a nonisothermal continuous stirred tank reactor. We construct two sensor failure situations and attempt to detect the sensor faults in both the T 2 and SPE charts. The proposed indices are then used to successfully identify the faulty sensors from these simulation results. 1. Introduction Multivariate statistical process monitoring (MSPM) has been developed as an alternative to theoretical model-based monitoring approaches for the analysis of data obtained from chemical processes, which are highly noisy, significantly correlated, and redundant. In particular, principal component analysis (PCA) is a powerful MSPM technique and has been widely used to monitor such modern chemical processes. PCA-based monitoring can be used to extract information from regularly obtained data, construct model and residual spaces, and determine the control limits of both spaces. After the construction of the PCA model, the on-line data are projected into the two subspaces. If abnormalities arise in the process operation, PCA monitoring charts such as the T 2 chart and the squared prediction error (SPE) chart can be used to detect them. After an abnormality is detected in a process operation, a reliable method is needed to identify the variables that cause this abnormality. MacGregor et al.1 proposed a fault identification method based on contribution plots. However, this approach occasionally results in inadequate fault identification, particularly when a faulty signal affects other signals that are significantly correlated with the faulty measurements. To avoid these problems, further reconstruction methods have been developed for use in sensor fault identification.2-11 Wise and Ricker2 proposed a faulty sensor isolation method using partial least-squares (PLS) analysis, and Dunia et al.3 developed a sensor fault identification method in residual space that is based on a PCA reconstruction method and uses the sensor validity index (SVI). This SVI method provides satisfactory sensor fault identification only if * To whom correspondence should be addressed. Tel.: 8254-279-2274. Fax: 82-54-279-3499. E-mail: [email protected]. † Present address: School of Chemical Engineering & Advanced Materials, University of Newcastle, Newcastle upon Tyne, NE1 7RU, UK.

the fault can be detected in residual space, as it considers only model residuals. In this respect, if a sensor fault can be detected only in model space, SVI cannot identify the faulty sensor. Therefore, a further sensor fault identification index is needed for fault isolation in model space. This paper focuses on how, by using reconstruction methods, faulty sensors can be identified not only in residual space, but also in model space. These reconstruction approaches are motivated by the need for missing value estimation in the treatment of the incomplete data obtained from modern chemical and environmental processes.12-19 The incompleteness of gathered data obstructs information extraction and makes the reconstruction of missing data one of the most important issues in MSPM. Many researchers have developed PCA-based reconstruction methods such as the trimmed scored method (TRI),13 the projection into model space (PMP) method,13,19 the known data regression (KDR) method,13 the conditional mean replacement (CMR) method,12,13,18 a method based on the minimization of the SPE,2,12,13 an iterative imputation method, and the trimmed score regression (TRS) method.13 Nelson et al.12 researched score calculation using PCA and PLS. They concluded that KDR is equivalent to CMR. Arteaga and Ferrer13 showed that the iterative imputation method, PMP, and the minimization of the SPE provide equivalent score vector estimations and commented that, if KDR is not faced with inversion problems, it is statistically superior to the other methods. In this paper, two reconstruction methods, PMP and KDR, are used for sensor fault identification. PMP has been used for reconstruction-based sensor fault identification by Dunia et al.,3 and KDR was found by Arteaga and Ferrer to be more effective than other reconstruction methods.13 We discuss how sensor faults can be identified using reconstruction methods when they can be detected in model space (T 2 chart). We also provide the mathematical derivations of the effects of faults on

10.1021/ie034246z CCC: $27.50 © 2004 American Chemical Society Published on Web 06/22/2004

4294 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004

the original T 2 and SPE, as well as on the reconstructed T 2 and SPE. Using these results, two fault identification indices, FIIM and FIIR, are proposed, and the problems of reconstruction-based sensor fault identification in residual space are discussed. FIIM is defined as the ratio of T 2 to reconstructed T 2, and FIIR is the ratio of the original to the reconstructed SPE. Through simulation studies, it is demonstrated that FIIM provides consistent identification performance regardless of the choice of reconstruction method. This paper is organized as follows: The missing value estimation algorithms based on KDR and PMP are discussed in section 2. In section 3, faulty sensor identification procedures using the two reconstruction methods are outlined, and the proposed indices, FIIM and FIIR, are explained. In section 4, we apply the proposed method to a simple five-variable model and to a model of a continuous stirred tank reactor (CSTR). Finally, we present our conclusions regarding the proposed fault identification strategy in section 5. In Appendix I, the notations employed in this paper are explained, and in Appendix II, more detailed background on the proposed sensor fault identification methods is provided. 2. Missing Value Estimation Methods: PMP and KDR In MSPM, the problem of reconstructing incomplete data is a critical issue,12-19 because monitoring models such as the PCA model are derived from historical process data obtained under normal operating conditions. Nelson et al.12 have researched missing data estimators using two different approaches: singlecomponent projection and conditional mean replacement (CMR). The former estimates the score vector from the observed measurements via the minimization of SPEs and produces results that are equivalent to those obtained from PMP. The latter replaces the missing measurements with conditional means, which are the maximum values of the likelihood function for given observed measurements. They commented that the method using minimization of SPEs is generally more stable than CMR because the former has the advantage of requiring the inverse of a much smaller matrix but, in extreme cases where there are no inversion problems and critical combinations of measurements are missing, CMR is superior to other approaches.12 Arteaga and Ferrer13 have discussed several missing value estimation methods. They showed that the PMP method is equivalent not only to the minimization of SPEs but also to the iterative imputation method and that the CMR method is equivalent to the KDR algorithm. They also claimed, through examples, that KDR provides better missing value estimation performance than other methods because of its use of all principal components. In this paper, we consider the use of two missing value estimators, KDR and PMP, as reconstruction methods. The PMP method has been used for sensor fault identification by Dunia et al.3 The score vectors calculated using PMP and KDR are # / T / / T τˆ 1:A ) (P1:A P1:A)-1P1:A z* ) Ψ#PMPz*

(1)

/ T (P*ΘP*T)-1z* ) Ψ#KDRz* τˆ 1:A ) Θ1:AP1:A

(2)

and

where / T / / T Ψ#PMP ) (P1:A P1:A)-1P1:A

and / T Ψ#KDR ) Θ1:AP1:A (P*ΘP*T)-1

are the score estimation matrices based on the PMP and KDR methods for obtaining τˆ 1:A and τˆ 1:A denotes an estimated principal score vector that is calculated without using the measurements corresponding to the missing position index vector #. To estimate the missing measurements, eqs 1 and 2 can easily be rearranged using the following relation between score vectors and measurements

[ ]

# / T z # T # P1:A τˆ 1:A ) Ψ#KDRz* ) P1:Azˆ ) [P1:A ] ˆ ) z* # T # / T zˆ + P1:A z* (3) P1:A # # T -1 # / T zˆ # ) (P1:A P1:A ) P1:A(Ψ - P1:A )z* ) Φ#z* (4) # # T -1 # / T P1:A ) P1:A(Ψ# - P1:A ) Φ ≡ (P1:A

(5)

where Φ denotes the variable reconstruction matrix for the missing variable, zˆ , and Ψ can be ΨPMP or ΨKDR. The variable reconstruction matrices of these reconstruction methods can be expressed as follows # # T -1 / T P1:A ) P1:AP1:A z* ) ΦPMPz* (6) zˆ # ) (I - P1:A # # T -1 # / T P1:A ) P1:A(Θ1:AP1:A (P*ΘP*T)-1 zˆ # ) (P1:A / T )z* ) ΦKDRz* (7) P1:A

where # # T -1 # / T P1:A ) P1:A P1:A Φ#PMP ≡ (I - P1:A

(8)

# # T -1 # / T P1:A ) P1:A(Θ1:AP1:A (P*ΘP*T)-1 Φ#KDR ≡ (P1:A / T ) (9) P1:A

In eq 6, the inversion lemma is used to obtain the simpler expression. 3. Sensor Fault Identification Using Reconstruction Algorithms The use of a contribution plot in faulty sensor identification can give erroneous results in cases that have highly correlated process variables.4 A reconstruction-based faulty sensor identification approach was developed by Wise and Ricker to eliminate this difficulty.2 They proposed a sensor fault identification method that uses PLS-based reconstruction. Dunia et al.3 proposed the use of an identification index, SVI, that is obtained with a PCA-based reconstruction method. In the PMP approach, the SVI is the ratio of the reconstructed and inherent SPEs. This index can be used to identify a faulty sensor only if the sensor fault can be detected in the residual space. However, some abnormalities caused by sensor failures can be detected in the model and residual spaces simultaneously or only in model space. Therefore, another index for identifying faulty sensors in model space is needed.

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To develop new reconstruction-based identification indices, we must consider the way in which traditional monitoring indices (T 2 and SPE) and the corresponding reconstructed monitoring indices (reconstructed T 2 and SPE) are affected by sensor faults, and then the differences between the inherent and reconstructed indices can be discussed. For this procedure, it is assumed that (i) an observed vector affected by a sensor fault can be divided into normal and abnormal parts and (ii) the abnormal part is not correlated with the normal part. These assumptions have previously been discussed by several researchers.3,4,6,7,10 The first assumption can be expressed as z ) znormal + zfault, where znormal and zfault denote the normal and abnormal parts of the vector, respectively. If only the kth sensor is faulty, all elements in zfault are zeros except the kth element, i.e., zfault ) [01:k-1T fk 0k+1:KT]T. For one measured vector with abnormality, two possible cases can be considered depending on whether the value obtained from a faulty sensor is reconstructed. Therefore, the measured vector can be expressed as follows

[ ] [] [ ]

# f# z# + fk z ) znormal + zfault ) z + k/ ) z* fk z* if # ) k (10)

[ ] [] [

Using this approach, two fault identification indices, FIIM and FIIR, are proposed. Each index is defined as the ratio of the monitoring statistics for the original sample z and those for the reconstructed sample zˆ

FII#M2 )

]

where znormal ≡ In eq 10, the kth element, fk, of zfault is permuted to the reconstructed position. Therefore, f#k becomes fk, and f/k consists of the other elements of zfault, which are zeros. In eq 11, another element is located in the reconstructed position (f#k * fk). Therefore, f#k becomes zero, and f/kconsists of zero elements and fk. Thus, the two kinds of reconstructed vectors can be calculated from an observed sample vector and expressed as [z#

z*T]T.

[ ] [ ]

# # zˆ ) zˆ ) Φ z* z* z*

or

[] [

# Φ#(z* + f/k) zˆ ) zˆ ) z* z* + f/k

if # ) k

]

if # * k

(12)

(13)

where Φ# can be Φ#PMP or Φ#KDR according to the choice of reconstruction method. Equation 12 shows that, if a variable affected by a sensor fault is reconstructed, the fault effect is eliminated. On the other hand, eq 13 shows that, if other normal variables that are unaffected by the sensor fault are reconstructed, the fault effect propagates to the reconstructed variables. Thus, when a corrupted measurement is selected for reconstruction, the measurements affected by the sensor fault are reconstructed using the others obtained under normal conditions, and the reconstructed measurements are equal to the estimated missing values under normal conditions. When a normal value obtained from a nonfaulty sensor is selected for reconstruction, the effect of the sensor fault is propagated into the reconstructed vector. Hence, the difference between the original and reconstructed measurements in the monitoring statistics provides a clue that helps identify the faulty sensor.

# ) 1, ..., K

eˆ #Teˆ # FII#R2 ) T e e

# ) 1, ..., K

(14)

where T 2 is the inherent monitoring index in model space, and the reconstructed T 2 is expressed as T ˆ #2, # # which is calculated from τˆ 1:A. e and eˆ denote the inherent and reconstructed residual vectors, respectively; the latter is estimated using zˆ . These two identification indices can be used to identify faulty sensors in the model and residual subspaces, respectively. Using the expectations for the inherent and reconstructed indices under the abnormal operating conditions caused by a faulty sensor, FIIM and FIIR can be expressed as

{

E(T ˆ #2)

f# z# z ) znormal + zfault ) z + k/ ) z* + f/k z* fk if # * k (11) #

T ˆ #2 T2

FII#M2 )

and

A ˆ# (# ) k) E(T 2) A + δk2 ˆ # + δk2 + δˆ #2 E(T ˆ #2) A k (# * k) ) 2 2 E(T ) A + δk

{

)

E(eˆ #Teˆ #)

R ˆ# (# ) k) E(eTe) R + υk2 #2 FIIR ) ˆ # + υˆ #2 E(eˆ #Teˆ #) R k (# * k) ) T E(e e) R + υk2 )

(15) (16)

(17) (18)

A, R, A ˆ #, and R ˆ # are the expectations under normal operating conditions for the inherent and reconstructed monitoring indices in the model and residual spaces; more details are provided in Appendix II. δk2, δˆ #k2, υk2, and υˆ #k2 denote the effects of a sensor fault on the original and reconstructed monitoring indices, as also explained in Appendix II. After a sensor fault occurs, eqs 15 and 16 for the model space indicate that the FIIM of the faulty sensor decreases and becomes smaller than those of the other sensors that are operating normally. Equations 17 and 18 for the residual space show that the FIIR of the faulty sensor is smaller than those of the other sensors under sensor fault conditions. Our procedure for sensor fault identification is based on these characteristics. For more convenient visualization and sensor fault identification, the identification indices # are modified to FII#M,lim/FII#M, and FIIR,lim /FII#R, where # # FIIM,lim and FIIR,lim denote the control limits of FIIM and FIIR, respectively, for the #th variable. After a process abnormality resulting from a sensor fault occurs, only the FIIM and FIIR indices of the faulty sensor lie below the control limits. Under these condi# /FII#R are greater than 1. tions, FII#M,lim/FII#M and FIIR,lim Therefore, for sensor fault identification, only the lower control limits of FII#M and FII#R are needed. The control limits of the identification indices can be obtained from


normal historical data though a heuristic approach, as previously described by Kano et al.20 This isolation procedure focuses on static processes. However, this identification method can be extended to dynamic processes using time-lagged PCA.10 One aspect of reconstruction-based sensor fault identification must be considered here; specifically the situation in which the reconstructed fault effects, δˆ #k2 and υˆ #k2, are very small. In this case, FIIM can still be used to identify the faulty sensor because the differences between FIIM of the faulty sensor and the indices of the other sensors are dominated by the effects of the fault, as indicated by eqs 15 and 16, i.e., if δˆ #k2 is very small or close to zero, the FIIM of the faulty sensor, A ˆ #/(A + 2 δk ), is smaller than the FIIMs of the other sensors, (A ˆ# + δk2)/(A + δk2). Therefore, we conclude that sensor fault identification can be achieved in model space regardless of the choice of reconstruction method, as identification performance is dominated by the conventional T 2. However, if the effect of the reconstructed fault in the residual space, υˆ #k2, is close to zero, the FIIR of the faulty sensor and the indices of the other sensors have the same value, R ˆ #/(R + υk2), and so cannot be used to identify the faulty sensor. If υˆ #k2 is not zero but very small, the control limit can be guaranteed when υˆ #k2 is approximately the same as or larger than υk2, so # FIIR,lim /FII#R can be at the lower control limit, as indicated by eqs 17 and 18. Therefore, there is a difficulty with reconstruction-based sensor fault identification in residual space. The simulation results of sensor fault identification using the two reconstruction methods are discussed in the next section. 4. Simulation Results In this section, the proposed methods are applied to two models with several sensor faults: a simple model and a model of a nonisothermal CSTR process.21-23 Four types of sensor fault are discussed by Dunia et al.3, namely, complete failure, precision degradation, bias, and drifting. For further details about these sensor fault types, we refer the reader to the study of Dunia et al.3 Below, we present data corrupted by these sensor faults and then discuss the results obtained when the proposed methods are applied to such data sets. For more sensitive identification using FIIM and FIIR, the exponential-weighted moving average (EWMA) with 0.2 forgetting factor was employed, and the confidence limits of the identification indices were obtained from the 95 percentile values of the identification indices obtained under normal conditions. 4.1. Application to a Simple Model. The simple five-variable model has the following structure

[

-0.07 -0.45 z ) As ) 0.97 0.65 -1.02

0.63 -1.05 0.11 0.26 0.19

-0.32 -1.08 -0.47 -0.24 -0.17

-0.72 -0.72 -0.62 0.96 0.13

]

-0.60 0.01 0.51 s -0.01 -0.03 (19)

where the elements of s are uncorrelated random signals with zero means and unity variances. One thousand training samples were used to build a PCA model. This model space consists of three principal components that capture about 90% of the covariance

Figure 1. (a) First set of test data for the simple model (solid line, first variable; dotted lines, the other variables) (b) Second set of test data for the simple model (solid line, fourth variable; dotted lines, the other variables).

of the training data. Two sets of test data consisting of 200 samples each were generated and corrupted with two different types of abnormalities. These test data sets consist of normal and abnormal parts. The normal part of each test data set consists of 100 sample vectors; these vectors were generated using eq 19 as a training sample. The normal part corresponds to data for normal operating conditions of a stationary static process. The abnormal part consists of 100 samples generated by adding a constant vector and a random vector to every sample in each test data set. The constant vector was [5 0 0 0 0]T, and the elements in each random vector were normally distributed with a zero-mean vector and a standard deviation of 5. The first test data set was constructed to simulate a bias fault in the first sensor, and the second test data set was constructed to simulate a precision degradation sensor fault in the fourth sensor. These test data sets are shown in parts a and b, respectively, of Figure 1. In Figure 1a,b, the solid lines indicate the faulty sensors, and the dotted lines indicate the normal sensors. For convenience of visualization, solid lines indicate faulty sensors, and dotted lines indicate normal sensors in all of the figures that follow. The PCA-based monitoring results for the first test data set are shown in Figure 2. Both monitoring indices, T 2 and SPE, catch the abnormalities of the test data in the model and residual spaces. Thus, sensor fault identification can be performed using the proposed indices, FIIM and FIIR. Figure 3 shows the KDR-based sensor fault identification results. In Figure 3a, the KDR-based FIIM provides satisfactory sensor fault identification. In Figure 3b, the KDR-based FIIR indices not only of the faulty sensor but also of two normal sensors are abnormal. These unwanted results for the two normal sensors are discussed in section 3. However, because the KDR approach reconstructs enough fault effects to identify the faulty sensor, we consider the sensor fault identification provided by FIIR in this instance satisfactory. Figure 4 shows the FIIM and FIIR indices calculated using PMP as the reconstruction method. PMPbased FIIM also provides satisfactory sensor fault identification (Figure 4a). On the other hand, Figure 4b shows the limitations of sensor fault identification using


Figure 2. Monitoring results for the first set of test data for the simple model.

Figure 3. KDR-based identification for the first set of test data: (a) KDR-based FIIM,lim/FIIM (solid line, first variable; dotted lines, the other variables), (b) KDR-based FIIR,lim/FIIR (solid line, first variable; dotted lines, the other variables).

PMP-based FIIR. In this case, PMP-based FIIR does not provide satisfactory sensor fault identification. However, PMP-based FIIR does enable sensor fault identification in some cases, as is verified by the identification results for the second test data set. Figure 5 shows the monitoring results for the second test data set. As for the first set, the abnormalities produced by the sensor fault can be detected using both monitoring indices. In this case, the KDR-based FIIM and FIIR indices enable satisfactory sensor fault identification (Figure 6a,b). The results for the PMP-based indices are presented in Figure 7. Figure 7b shows that the PMP-based FIIR enables the identification of faulty sensors in some cases. Although there are limitations in the sensor fault identification provided by FIIR in residual space in some cases, FIIM always enables satisfactory sensor fault identification regardless of the reconstruction method. In simple model cases, KDR-based FIIR are superior to PMP-based FIIR. However, KDR-based FIIR does not always provide adequate sensor fault identification, as was found for the results for the nonisothermal CSTR simulator.

Figure 4. PMP-based identification for the first set of test data: (a) PMP-based FIIM,lim/FIIM (solid line, first variable; dotted lines, the other variables), (b) PMP-based FIIR,lim/FIIR (solid line, first variable; dotted lines, the other variables).

Figure 5. Monitoring results for the second set of test data for the simple model.

4.2. Application to a First-Order Reaction in a CSTR. The process flow of a single, nonisothermal CSTR is depicted in Figure 8. The reactor model is based on three assumptions: perfect mixing, constant physical properties, and negligible shaft work. In the reactor, reactant A premixed with a solvent is converted into product B via a first-order reaction with rate r ) k0e-E/RTCA, where k0 is the preexponential kinetic constant, E is the activation energy of the reaction, R is the ideal gas constant, and CA is the reactant concentration. The dynamical behavior of the process is described by the mass balance of reactant A and the total energy balance for the reacting system, which can be expressed as

V VFCp

dC ) F(C - CAi) - Vr dt

(20)

dT ) FCpF(Ti - T) dt aFcb+1 (T - Tci) + (-∆Hr)Vr (21) Fc + aFcb/2FcCpc


Figure 8. Simulated nonisothermal CSTR process (F: flow rate; C: concentration; T: temperature/a: reactant; s: solute; c: coolant/i: inlet; o: outlet/TC: temperature controller; CC: concentration controller).

Figure 6. KDR-based identification for the second set of test data: (a) KDR-based FIIM,lim/FIIM (solid line, fourth variable; dotted lines, the other variables), (b) KDR-based FIIR,lim/FIIR (solid line, fourth variable; dotted lines, the other variables).

Table 1. Summary of the CSTR Process Variables and Parameters state variables controlled variables manipulated variables disturbances measured (monitored) variables

V F Fc Cp Cpc

Process Variables Ca, T Ca, T Fs, Fc Fi (or Fa, Fs), Ci (or Ca, Cs), Ti, Tci, Tco Tc, Ti, Cai, Cs, Fs, Fa, Fc, Ca, T

1 m3 106 g/m3 106 g/m3 1 cal/(g K) 1 cal/(g K)

Parameters k0 a b ∆Hr

1010 min-1 1.678 × 106 cal/min 0.5 -1.3 × 107 cal/kmol

Table 2. Measurement Noise and Disturbances for the CSTR Processa meansurement noise, σm2

Figure 7. PMP-based identification for the second set of test data: (a) PMP-based FIIM,lim/FIIM (solid line, fourth variable; dotted lines, the other variables), (b) PMP-based FIIR,lim/FIIR (solid line, fourth variable; dotted lines, the other variables).

T CA Fc Tc Ti CAi Fa Cs Fs a1 a2 a

where V is the volume of the reacting mixture in the tank; F is the density of the reacting mixture in the inlet stream; Fc is the density of the coolant; Cp and Cpc are the specific heat capacities of the reacting mixture and coolant, respectively; ∆Hr is the heat of reaction; and F and Fc are the reacting mixture and coolant flow rates, respectively. The process variables and parameters used in the mathematical modeling and monitoring of the CSTR are listed in Table 1. All process disturbances were generated by a first-order autoregressive model. In addition, all measured variables were disrupted by Gaussian white noise with different variances. The process disturbance model and measurement noise characteristics are presented in Table 2. The temperature and the concentration of reactant A in the outlet stream are controlled by manipulating the coolant flow rate in the cooling jacket and the solute flow rate in the inlet stream. The simulation parameters, initial conditions, and controller information are those provided by

4.0 × 10-4 2.5 × 10-5 1.0 × 10-2 2.5 × 10-3 2.5 × 10-3 1.0 × 10-2 4.0 × 10-6 2.5 × 10-5 4.0 × 10-6 ×10×10-

process noise,σe2

AR coefficient

0.475 × 10-1 0.475 × 10-1 0.475 × 10-1

0.9 0.9 0.9

1.875 × 10-3 0.19 × 10-2 0.19 × 10-2 0.0975 × 10-2

0.5 0.9 0.9 0.95

xt ) φxt-1 + σeetd, xt,meas ) xt + σmmt; et ∼ N(0,1), mt ∼ N(0,1).

Yoon and MacGregor.21 A more detailed description of the process is given by Yoon and MacGregor21 and by Choi et al.22 To build the PCA model, a total of 500 samples were generated under normal operating conditions; the number of principal components was then determined by considering the accumulated percent variance, resulting in capture by the PCA model of over 90% of the total variable covariance. Two abnormal operating conditions consisting of 400 samples were simulated. In the first simulation, the process was operated under normal conditions up to the 199th sample time. From the 200th sample time until the end of the simulation, the values measured at the third sensor had a constant value of 6. This simulated condition indicates that the observations of the third sensor have completely failed. In the second simulation, the first 99 samples were obtained under normal operating conditions. Further samples were then


Figure 9. (a) First set of test data for the CSTR process (solid line, third variable; dotted lines, the other variables) (b) Second set of test data for the CSTR process (solid line, third variable; dotted lines, the other variables).

Figure 10. Monitoring results for the first set of test data of the CSTR process.

corrupted by a drifting type of sensor fault. From the 100th sample onward, the value obtained from the sixth sensor was increased by 0.002 as the sample time increased. The sixth sensor measures the inlet flow rate of the pure reactant A solute. The data for these two test cases are shown in parts a and b, respectively, of Figure 9. The monitoring results for the first case are shown in Figure 10. The use of the monitoring indices in the model and residual spaces enables the detection of the abnormality. To identify the faulty sensor (the third sensor), the proposed indices were calculated, and the results are shown in Figures 11 and 12. KDRbased FIIM and FIIR provide satisfactory sensor fault identification, as is clear from Figure 11a,b. Figure 12a also shows that PMP-based FIIM can be used to identify the faulty sensor. However, PMP-based FIIR cannot be used to identify the faulty sensor, as shown in Figure 12b. Thus, KDR-based FIIM and FIIR and PMPbased FIIM are useful for identification of faulty sensors, but PMP-based FIIR is sometimes inadequate for use in sensor fault identification. As previously discussed,

Figure 11. KDR-based identification for the first set of CSTR test data: (a) KDR-based FIIM,lim/FIIM (solid line, third variable; dotted lines, the other variables), (b) KDR-based FIIR,lim/FIIR (solid line, third variable; dotted lines, the other variables).

Figure 12. PMP-based identification for the first set of CSTR test data: (a) PMP-based FIIM,lim/FIIM (solid line, third variable; dotted lines, the other variables), (b) PMP-based FIIR,lim/FIIR (solid line, third variable; dotted lines, the other variables).

the reason for the failure of this index is that PMPbased reconstruction does not provide sufficient magnitude to the reconstructed fault effect, υˆ #k2, to enable the identification of the faulty sensor. Another example of this problem arises in the second CSTR test. The monitoring results for the second case are shown in Figure 13. The KDR- and PMP-based FIIM indices in model space enable the identification of the faulty sensor, the sixth sensor, as shown in Figures 14a and 15a. Figures 14a and 15a also show the limitations of reconstruction-based sensor fault identification in residual space. FIIR calculated with either reconstruction method cannot be used to identify the faulty sensor because the reconstructed fault effects for the eighth sensor, υˆ #k2 (# ) 8), are insufficiently large (Figure 16). The eighth sensor monitors the outlet concentration of reactant A. The results of the two simulators for the simple fivevariable model and the CSTR benchmark show that reconstruction-based FIIM provides satisfactory sensor


Figure 13. Monitoring results for the second set of CSTR test data (solid line, sixth variable; dotted lines, the other variables).

Figure 14. KDR-based identification for the second set of CSTR test data: (a) KDR-based FIIM,lim/FIIM (solid line, sixth variable; dotted lines, the other variables), (b) KDR-based FIIR,lim/FIIR (solid line, sixth variable; dotted lines, the other variables).

fault identification and that the reconstruction-based approach in residual space has limitations. 5. Conclusion and Future Work This paper focused on reconstruction-based faulty sensor identification using two identification indices. The PMP and KDR methods were selected as the reconstruction methods. We have shown that, in the absence of inversion problems, the KDR approach is more effective than the PMP method. We have proposed two identification indices, FIIM and FIIR, that can be calculated using either reconstruction method. With these indices, we were able to identify sensor faults not only in residual space but also in model space. We found that FIIM provides similar sensor fault identification performance regardless of the choice of reconstruction method. In residual space, the reconstruction approach occasionally fails to find a faulty sensor because of the weak reconstruction of fault effects within SPE. Nev-

Figure 15. PMP-based identification for the second set of CSTR test data: (a) PMP-based FIIM,lim/FIIM (solid line, sixth variable; dotted lines, the other variables), (b) PMP-based FIIR,lim/FIIR (solid line, sixth variable; dotted lines, the other variables).

Figure 16. KDR-based FIIR and PMP-based FIIR for the 250th sample in the second set of CSTR test data.

ertheless, the KDR-based FIIR provides better fault identification than the PMP-based FIIR. Future work on sensor fault identification to overcome these limitations can be carried out in two different directions. One is to verify the sensor fault identification performances of the existing reconstruction methods in residual space and to propose more effective reconstruction methods. The other is the development of different approaches to sensor fault identification. Acknowledgment This work was supported by Grant (R01-2002-00000007-0) from the Korea Science & Engineering Foundation. Nomenclature z ) measured vector τ ) score vector e ) residual vector Φ ) reconstruction matrix

Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4301 Ψ ) score vector estimator P ) loading vector with a full rank Θ ) diagonal matrix which consists of squared eigenvalues A ) expectation for the inherent T 2 (which is actually equal to the number of principal components) A ˆ # ) expectation for the reconstructed T 2 R ) expectation for the inherent SPE R ˆ # ) expectation for the reconstructed SPE δk2 ) expectation for the fault effect in the inherent T 2 δˆ #k2) expectation for the fault effect in the reconstructed T2 2 υk ) expectation for the fault effect in the inherent SPE υˆ #k2) expectation for the fault effect in the reconstructed SPE Superscripts T ) transpose ∧ ) reconstructed or estimated data * ) remaining position index vector # ) removed/missing position index vector

Appendix I. Notation Matrices and vectors are denoted by uppercase and lowercase bold capital letters, respectively. The training data matrix has dimensions N × K and is meancentered and autoscaled, where N and K denote the number of variables and the number of samples, respectively. The symbol A denotes the selected number of principal components, and z denotes a measured vector. P and τ are used to denote the K × K loading matrix and the K × 1 latent vector, respectively. The matrix Pa:b contains columns a-b of matrix P, and τa:b contains elements a-b of vector τ. Θ denotes a diagonal matrix whose diagonal elements are the squared eigenvalues of the training matrix in decreasing order, and Θa:b is a diagonal matrix of eigenvalues a-b in decreasing order along the diagonal. The superscripts # (sharp) and * (asterisk) of z denote the removed/missing position index vector and the remaining position index vector, respectively. P# and P* denote the submatrices containing the columns of P corresponding to z# and z*, respectively. For example, consider a measured vector z with five variables [z1 z2 z3 z4 z5]T and a 5 × 5 loading matrix P; the element in row i and column j is pij. If the missing variables are the second and fourth variables, the missing position index vector, #, is [2 4], so z is [z2 z4]T; the remaining position index vector, *, is [1 3 5]; and z* is [z1 z3 z5]T. The matrix P# contains pij for i ) 2 and 4 and j ) 1-5. P* contains pij for i ) 1, 3, and 5 and j ) 1-5. Using this notation, the covariance matrix can be expressed as

[

][

[ ] ][

# T / T # # P1:A P1:A P1:A PA+1:K Θ1:A 0 #T / / / T ΘA+1:K P P1:A PA + 1:K 0 PA+1:K A+1:K

This paper focuses on the identification of single faulty sensors. However, this appendix describes the identification of multiple faulty sensors, which is more general. We assume that measurements corrupted by faulty sensors can be divided into two parts, the normal terms and the corrupted terms. These terms can be expressed as

[]

a:b ) columns/elements a-b of a matrix/vector k ) position index vector of faulty sensors PMP ) PMP method used for reconstruction/estimation KDR ) KDR method used for reconstruction/estimation

cov(z) ) PΘPT ) [P1:A

Appendix II. Strategy for Identification of Faulty Sensors

[]

# f# # znormal ) z , zfault ) k/ z* fk

Subscripts

P T PA+1:K ]Θ 1:A T ) PA+1:K

For convenience of expression, permutated matrices and vectors are employed. These consist of the preceding part and the following part. The preceding part consists of the removed variables or loading coefficients related to the removed variables. The other variables or the loading coefficients corresponding to these variables compose the following part.

]

(A-1)

where k is a set of elements indicating the positions of the faulty sensors. Parts I and II below discuss our sensor fault identification strategy using the proposed indices. Part I. Identification Strategy Using FIIM. Case 1. Under Normal Operating Conditions. If measurements are obtained without sensor failures, the measurements and reconstructed measurements can be written as

[ ]

[ ] [ ]

# # # # z ) znormal ) z , zˆ ) zˆ normal ) zˆ ) Φ z* z* z* z*

(A-2)

The expectations for T 2 and the reconstructed T 2 and T ˆ #2 under normal operating conditions can be estimated using

E(T 2) ) Ε(τ1:ATΘ1:A-1τ1:A) ) tr(E(τ1:Aτ1:AT)Θ1:A-1) ) tr(Θ1:AΘ1:A-1) ) A # T # # T # E(T ˆ #2) ) Ε(τˆ 1:A Θ1:A-1τˆ 1:A ) ) tr(E(τˆ 1:A τˆ 1:A )Θ1:A-1) ) # )Θ1:A-1) ) A ˆ # (A-3) tr(var(τˆ 1:A

where A ˆ # is A ˆ #PMP or A ˆ #KDR. The expectations for T 2 and T ˆ #2 can be decomposed as follows

E(T 2) ) (τ1:ATΘ1:A-1τ1:A) ) E(znormalTP1:AΘ1:A-1P1:ATznormal)

([ ] [ ] [ ] [ ]) ([ ] [

# )E z z*

T

# )E z z*

T

)A

# P1:A P# Θ1:A-1 1:A / / P1:A P1:A

T

z# z*

(A-4)

][ ])

# # T # / T P1:A Θ1:A-1P1:A P1:A Θ1:A-1P1:A z# / -1 # T / -1 / T P1:AΘ1:A P1:A P1:AΘ1:A P1:A z*


E(T ˆ #2)

δk2 ≡ E

# T # ) E(τˆ 1:A Θ1:A-1τˆ 1:A )

([ ] [ f#k f/k

T

][ ])

# # T # / T P1:A Θ1:A-1P1:A P1:A Θ1:A-1P1:A f#k / -1 # T / -1 / T P1:AΘ1:A P1:A P1:AΘ1:A P1:A f/k

(A-11)

) E(zˆ normalTP1:AΘ1:A-1P1:ATzˆ normal)

([ ] [ ] [ ] [ ]) ( [ ][ ][ ] ) T

# ) E Φ z* z*

# ) E z* Φ I T

# # P1:A -1 P1:A Θ 1:A / / P1:A P1:A

T

T

Φ#z* z*

(A-5)

# # T # / T P1:A Θ1:A-1P1:A P1:A Θ1:A-1P1:A / -1 # T / -1 / T P1:AΘ1:A P1:A P1:AΘ1:A P1:A

Φ# z* I

)A ˆ

Case 2. Under Abnormal Operating Conditions Caused by Faulty Sensors. If the measurements are corrupted by sensor faults, the measured and reconstructed vectors can be expressed as follows

[] [] ] [ ]

# f# z ) znormal + zfault ) z + k/ z* fk

[

(A-6)

The expectation for T ˆ #2 can be expressed as

E(T ˆ #2)

( [ ] [ ] ) ( [ ] [ ]) []) ( [ ] ( [ ] [ ]) T

# # ) E (z* + f/k)T Φ P1:AΘ1:A-1P1:AT Φ (z* + f/k) I I T

# # ) E z*T Φ P1:AΘ1:A-1P1:AT Φ z* + I I (A-12) # T T Φ -1 T Φ / P1:AΘ1:A P1:A 2E z* f + I k I T

# # E f/kT Φ P1:AΘ1:A-1P1:AT Φ f/k I I

where

( [ ]

[ ])

T

# # ˆ E z*T Φ P1:AΘ1:A-1P1:AT Φ z* ) A I I

Φ#(z* + f/k) Φ# (z* + f/ ) ) zˆ ) k I z* + f/k

(A-7)

from eq A-12 and

( [ ]

[ ])

T

# # E z*T Φ P1:AΘ1:A-1P1:AT Φ f/k ) 0 I I

The expectation for T 2 can be expressed as

E(T 2) ) E((znormal + zfault)TP1:AΘ1:A-1P1:AT(znormal + zfault)) )E

(([ ] [ ]) ([ ] [ ])) [ ]) ([ ] ([ ] [ ]) ([ ] [ ]) z# + fk f/k z*

T

# f# P1:AΘ1:A-1P1:AT z + k/ z* fk

T

# # ) E z P1:AΘ1:A-1P1:AT z z* z*

+

T

([

f#kz#T f/kz#T f#kz*T f/kz*T

])

([ ]

T

#

)E z z*

-1

P1:AΘ1:A P1:A E

([ ] [

)A+E

f#k f/k

T

([ ]

[ ])

T

#

z z*

) 0 (A-9)

/ T # P1:A Θ1:A-1P1:A

/ / T P1:A Θ1:A-1P1:A

+

[ ])

f#k T P1:AΘ1:A-1P1:AT f/k

f#k f/k

][ ] ) Φ# f/ k I

# # T / # T # where ΦTP1:A Θ1:A-1P1:A Φ and P1:A Θ1:A-1P1:A Φ + Φ#T # / T -1 P1:AΘ1:A P1:A are symmetric matrices. Two cases can be considered, depending on whether the sensor fault exists in the reconstructed measurement. If the sensor fault exists in the reconstructed positions, the other positions do not have the fault effect. That is, f#k * 0 and f/k) 0 in the abnormal part. Thus, the expected value in eq A-13 is zero. Therefore, the expectation for T ˆ #2 can be expressed as

E(T ˆ #2) ) A ˆ#

(A-14)

If the sensor fault does not exist in the reconstructed positions, f#k ) 0 and f/k * 0. In this case (A-10)

][ ])

# # T # / T P1:A Θ1:A-1P1:A P1:A Θ1:A-1P1:A f#k / -1 # T / -1 / T P1:AΘ1:A P1:A P1:AΘ1:A P1:A f/k

)

A + δk2 where

-1 # T # T P# Θ 1:A 1:A P1:A f/kT Φ / # T I P1:A Θ1:A-1P1:A

/ / T / E(f/kTP1:A Θ1:A-1P1:A fk) (A-13)

Thus, the expectation for T 2 can be expressed as E(T 2)

[ ])

# P1:AΘ1:A-1P1:AT Φ f/k I

/ T / # Θ1:A-1P1:A )fk) + Φ#TP1:A

From assumption i in section 3

Ε(zfaultznormalT) ) Ε

)E

T

/ # # T # # T # ) E(f/kT(Φ#TP1:A Θ1:A-1P1:A Φ + P1:A Θ1:A-1P1:A Φ +

+

f#k T f# P1:AΘ1:A-1P1:AT k/ / fk fk

E

( [ ] ( [ ][

# E f/kT Φ I

(A-8)

# f# 2E z P1:AΘ1:A-1P1:AT k/ z* fk

from eq A-9. The last term in eq A-12 provides the expectation for the fault effect in T ˆ #2. It can be decomposed as

/ -1 / T / 2 E(f/T k P1:AΘ1:A P1:A fk) ) δk

from eq A-11. The expectation for the reconstructed fault effect, δˆ #k2, is defined as #T # /T -1 # T # δˆ #2 k ≡ E(fk (Φ P1:AΘ1:A P1:A Φ + / # T #T # / T / Θ1:A-1P1:A Φ + Φ#TP1:A Θ1:A-1P1:A )fk) (A-15) P1:A


Thus, from the same observed vector, the other type of expectation for T ˆ 2 can be calculated as

E(T ˆ

#2

#

2

))A ˆ + δk +

δˆ #2 k

(A-16)

When the abnormality caused by the faulty sensor is detected in model space, the proposed identification index, FIIM, can be expressed as the ratio of two ˆ #2 different forms using the expectations for T 2 and T for one observed vector

{

E(T ˆ

#2

)

(A-17)

T

(A-21)

where

Further, the expectation for the reconstructed SPE for the faulty measurements can be expressed as

E(eˆ #Teˆ #) ) E((z* + f/k)T(Φ#TΦ# + I - Ψ#TΨ#)(z* + f/k)) ) E(z*T(Φ#TΦ + I - Ψ#TΨ#)z*) + ˆ + υˆ k2 (A-23) E(f/kT(Φ#TΦ# + I - Ψ#TΨ#)f/k) ) R

([ ] [ ]) ([ ] [ ][ ] [ ]) T P# 1:A / P1:A

# T P1:A / P1:A

z# z*

(A-18)

# # T # P1:A z + ) E(z#Tz# + z*Tz*) - E(z#TP1:A / / T # / T P1:A z* + 2z#TP1:A P1:A z*) z*TP1:A

)R

As for T 2, if the corrupted measurements are located in the reconstructed positions, f#k * 0 and f/k ) 0. For this case, the expectation for the reconstructed SPE can be expressed as

ˆ# E(eˆ #Teˆ #) ) E(z*T(Φ#TΦ# + I - Ψ#TΨ)z*) ) R (A-25) On the other hand, if all other measurements except the reconstructed measurements are corrupted by the sensor fault, f#k ) 0 and f/k * 0, which gives

where

e ) (I - P1:AP1:AT)z

E(eˆ #Teˆ #) ) E((z* + f/k)T(Φ#TΦ# + I - Ψ#TΨ#)(z* + ˆ # + υˆ #2 f/k)) ) R k (A-26)

and

E(eˆ #Teˆ #) # T # ) E(zˆ Tzˆ ) - E(τˆ 1:A τˆ 1:A)

( [ ][ ] )

Thus, when the sensor fault is detected in residual space, the proposed index, FIIR, can be expressed as the ratio of two different forms using the expectations for eTe and eˆ #Teˆ # for one observed vector

T

# Φ# z* - E(z*TΨ#TΨz*) ) E z*T Φ I I ) E(z*T(Φ#TΦ# + I - Ψ#TΨ#)z*)

(A-19)

)R ˆ where eˆ ) (I - P1:AP1:AT)zˆ . R ˆ # can be R ˆ #KDR or R ˆ #PMP and (Φ#TΦ + I - Ψ#TΨ#) is a positive definite matrix. Case 2. Under Abnormal Operating Conditions Caused by Faulty Sensors. The expectation for eTe for the corrupted measurements can be expressed as

(([ ] [ ]) ([ ] [ ])) (([ ] [ ]) [ ][ ] ([ ] [ ]))

E(eTe) ) E E

(A-24)

T

) E(znormal znormal) - E(znormal P1:AP1:A znormal) # -E z z*

) R + υk2

#T # #T # / /T υˆ #2 k ≡ E(fk (Φ Φ + I - Ψ Ψ )fk)

T

z# z*

# / T / 2f#T k P1:AP1:A fk)

where υˆ #k2 denotes the reconstructed fault effect in residual space and is defined as

E(eTe) ) E(zTz) - E(τ1:ATτ1:A)

T

# # T # /T / / T / f/kTf/k) - E(f#T k P1:AP1:A fk + fk P1:AP1:A fk+

/ / T / # / T / f/kTP1:A P1:A fk + 2f#T k P1:AP1:A fk) (A-22)

Part II. Identification Strategy Using FIIR. Case 1. Under Normal Operating Conditions. The inherent SPE, eTe, and the reconstructed SPE, eˆ Teˆ , can be expressed under normal conditions as

# )E z z*

/ / T # / T # P1:A z* + 2z#TP1:A P1:A z*) + E(f#T z*TP1:A k fk +

# /T / # T # #T # υk2 ) E(f#T k fk + fk fk) - E(fk P1:AP1:A fk +

#

A ˆ (# ) k) E(T ) A + δk2 #2 FIIM ) ˆ # + δk2 + δˆ #2 E(T ˆ #2) A k (# * k) ) 2 2 E(T ) A + δk )

2

# # T E(eTe) ) E(z#Tz# + z*Tz*) - E(z#TP1:A P1:A z+

# z# + fk z* f/k

# z# + fk z* f/k

T

T

# z # + fk z* f/k

# # P1:A P1:A / / P1:A P1:A

From eqs A-9 and A-18

T

-

# z# + fk z* f/k

(A-20)

{

E(eˆ #Teˆ #)

R ˆ# (# ) k) E(e e) R + υk2 #2 (A-27) FIIR ) ˆ # + υˆ #2 E(eˆ #Teˆ #) R k (# * k) ) E(eTe) R + υk2 T

)

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Received for review November 13, 2003 Revised manuscript received May 10, 2004 Accepted May 17, 2004 IE034246Z

Sensor Fault Identification in MSPM Using Reconstructed Monitoring

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