Missing Value Estimation and Sensor Fault Identification using Multivariate Statistical Analysis

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1 Korean Chem. Eng. Res., Vol. 45, No. 1, February, 2007, pp l mk ~ m m i~ m m my Çmm e q 31 ( o 31p r, o 9p }ˆ) Missing Value Estimation and Sensor Fault Identification using Multivariate Statistical Analysis Changkyu LeeGand In-Beum Lee Department of Chemical Engineering, POSTECH, San 31, Hyoja-dong, Nam-gu, Pohang , Korea (Received 31 October 2006; accepted 9 January 2007) k rp p p v v o r edšp p r edš kl p t p p. rp llv p rp l on r r p rp rll n. p rp orp p m r p v pl r p rp p dv k. p r p o p edš lv lm l q p t (principal component analysis, PCA) qd (partial least squares, PLS) p p r r l rp p., p p v rl rn o p p m l. l l r t (dynamic PCA) ƒ (canonical variate analysis)p pn p p m r p op mq p Ž l l q. h Abstract Recently, developments of process monitoring system in order to detect and diagnose process abnormalities has got the spotlight in process systems engineering. Normal data obtained from processes provide available information of process characteristics to be used for modeling, monitoring, and control. Since modern chemical and environmental processes have high dimensionality, strong correlation, severe dynamics and nonlinearity, it is not easy to analyze a process through model-based approach. To overcome limitations of model-based approach, lots of system engineers and academic researchers have focused on statistical approach combined with multivariable analysis such as principal component analysis (PCA), partial least squares (PLS), and so on. Several multivariate analysis methods have been modified to apply it to a chemical process with specific characteristics such as dynamics, nonlinearity, and so on. This paper discusses about missing value estimation and sensor fault identification based on process variable reconstruction using dynamic PCA and canonical variate analysis. Key words: Process Monitoring, Multivariate Analysis, Missing Value Estimation, Sensor Fault Identification 1. rp p o rp s p p lv q lp, rp nr l pl o, m, k, s p p r p p r rp v p. p orp o l sp r pep p v To whom correspondence should be addressed. iblee@postech.ac.kr l v v pn l r v p e. p r o r p ee p pn l rp op p r l rp p o Ž v r edšp rk l. p edšp p r p op p r ee p r op p l n rp p Ž edšp n m. p edšp rp r r ppp 87

2 88 p} Ëpp e k r r p v p p r edšp p l [1]. o r p p o t (principal component analysis, PCA)p qd (partial least squares, PLS) p p p eq m p p r rp p v l v rl rn o p p p l m [2-4]. p p p rp p p v v k rp p p p r p p v rl p p l r o p r m. l l PCAm CVA p l p sq r p l } l r edšl on p }k vm ee rp r p o p l rp r q p p o p l l q. 2. ~ m z o l m p p p l r edš p l q tn r r r p p p. v er r p v rl o kp n p p p p p pl n. r p p } o l p r p nl v l v v p r p pn r p rk l. v rl m p l r p rl p q l e p po p p p } p l t v l m. pr l [5]l, p p r r~ p p 20Í p p on rn. p rp p p } rp, o EM(expectation-maximization) k vp Fig. 1. PCA and CVA based EM algorithm to deal with incomplete data. o45 o k vp ~ p f p p }k. E (m )l p p p, M ( )l r p pn l r e dš l n p p. Fig. 1p PCAm CVA l EM k vp pn mr p p m p p. rp l l p rneˆ l l m p }ˆ l mr p }ˆ l v p lv. r edšp p rp p PCA, PLS, ƒ (canonical variate analysis, CVA)[6, 7] p pp, p p ˆp EM k vp r t M m l. p p m l p ˆl vv, p rp rp eˆ p q sp p k rp. p p p p l pl s(rank deficiency)p p l r(inversion problem) p l r n l rp v p [8]. p l rn q (residual space)p rp qpp p p eˆp f l r p p m p p. pnl l r p pv, l l p vp p tp mr p p m p q PCA mk jn m m z rp p o p PCA ˆ m, rl m p p p m p v p. ~ w r p p s (conditional expectation)p p p m p [5]. s p p m p n l s (conditional mean replacement, CMR)p p p pn l PCA pn l p p rn v p s p p m. m p e PCA p l Fig. 1l rp l p l p }. p p k p p pn l p PCAl KDR(known data regression)p [8]. k p p p r pn m p l qrp pv, k p p p p m p r p kt nl l m p l r p rp v p. w p p rn q p l l r p }k p. p q p r p p sq r l p p. p p m l CMR lvv, CMRl l r p qrp v p. q p l llv er p l Œm(projection) l llv p l p p Œm (projection to the model plane, PMP) p [8].

3 p pn p p m p p 89 Table 1. Comparison of missing data treatment methods according to analysis strategy PCA CVA model projected vector and variance to principal components state space model reconstruction method KDR linear regression of score vector and measured data CMR linear regression of state and measured data PMP minimization in residual space NM minimization in noise space SRM minimization in state residual space NM : noise minimization, SRM: state residual minimization 2-2. CVA mk jn m m z p rp DPCA CVAp p }ˆ p ˆ. CVA edš p(system identification) p p l l p p r l pn l p p rk lp rp p v p l DPCA ˆo l l p [6, 7, 9]. DPCAm v CVA pn l p m p p. CVA pn p } le PCAm p v p. n ~w k p p p l ˆ (state space model)p (state)p m p [10]. p p PCA p CMR p o p r p, k p p p n l p l r p rp v p. wp r PCAp r o pv eˆ r PCAm v p. CVAl PCAl p q } (state)p ˆ v v ql ˆ (residual state space) k p m e p m l qp (noise space)p sq. v, PCA q p p p s q v, CVA ql ˆ k qp l p v p sq. p v p s p. Table 1p PCAm CVA p p p p l p. 3. o kl m m rp nr l p p opp q (actuator), p e p n l v p. r edšl rp p p v tn rpv p p p l v Ž r tn rp. sp rp l p p rk p l (contribution chart)p. p p rp p l r p l e p rp p p p p, rp p rž p p p r p pe l rp re v rp v. p o n rp k p rp p v p. rp p p l v l v [11]. ~ w p (simple fault) p. p p rp p o l ˆ o l rp p p r rž p lvv k ˆp rp p (sensor fault)p p. w l p p p rž ˆp p p (complex fault)p. m l q m p rp r~l m p n l p p m p n, rp p p q l v rp l n l m p. m p p rl m p r nr r l m p r m l p l p p p p p ˆ. p l o p rž p p Ž p q l š. p p p o p p rp rk p Ž (pattern classification) SDG(signed directed graph)m p p rk l. p p p l rž n l l p p pp r np. p p p p l o l e p p n tp (multiple fault). l r edšl q rp rk p rp r nr p p p, mq p pp p s p seˆ o l l rp pp p. l l l p p rp np mq l ee p p r ol mq p l l q. r op pn mq p pp k l p p m p p n. p } rl E m M p v, ˆ l rp l M. v, ee r p pn l l rp r l m p q r p l p p p p p. p rp p p ee p o r r p r pq(process monitoring indices) pn. o p p e p rp ep pn l o pv, r p v p p rp e (empirical reference distribution, ERD)[12]p pn p p p. Fig. 2 r p p op p v edšp s p. 4. o m kl m m io l l l r p op p p p p r e p p rn k. rn rp 1 pp p lv CSTR(continuous stirred tank reactor) r p o r 9 p n m. rp s m r p Fig. 3 Table 2 Korean Chem. Eng. Res., Vol. 45, No. 1, February, 2007

information V=1 m 3 ; ρ=106 g/m 3 ; ρ c =106 g/m 3 ; C p =1 cal/g/k; C pc =1 cal/g/k; k 0 = 1,010 min 1 ; a=1.678 106 cal/min; b=0.5; H r =-1.3 107 cal/kmol Fig. 3. The simulated CSTR process. l l p.

4 90 p} Ëpp Fig. 2. Structure of sensor fault identification system based on process variable reconstruction. Table 2. Information of CSTR process process variable state variable : C, T control variable : C, T manipulating variable : F s, F c measured variable : T c, T i, C a, C s, F s, F a, F c, C, T CSTR model information V=1 m 3 ; ρ=106 g/m 3 ; ρ c =106 g/m 3 ; C p =1 cal/g/k; C pc =1 cal/g/k; k 0 = 1,010 min 1 ; a= cal/min; b=0.5; H r = cal/kmol Fig. 3. The simulated CSTR process. l l p. r e p l q p l m [11, 13]. p p s p n r pv m r rp v n(completely broken), fouling l p qd rp r lv (precision degradation), e p v l rr er l lv p ˆ n(drifting), er l t p r (bias) 4 v o45 o k r p p p. mrl p CVA ˆ mp, p o vl e (lag time) 4, r r nr l v 1,000 p p p p l r~ p 90Í p p p p ˆ m [10]. p p e m r rp q r r p m. r p o p CMR ˆ mp, o p p e p v p tp 95Íp o r m. Fig. 4[9] r p m p pn l rp p p v p p p l t p. Fig. 4(a) e 200 p l o p r

5 p pn p p m p p 91 Fig. 4. Results of sensor fault detection and identificationu r r v ppp l t p. Fig. 4(b) CVA l r p vp l t p. Fig. 4(b)p ~ w p l p r p v, w p qp l p rp p v p l e 200 p l ˆ rp p p v p. op p p v Fig. 4(d)l e l p. l p op p p p Fig. 4(d) p ~ w l l pp, r rp r p (e )m mq (r ) p. Fig. 4(d) p w l qp l p p v le r p p l v ppp l t p. sl rk CVAl l Fig. 4(c)l e m. p r p p p o (r ) l. op rp p v sp rk p p p spp p p. CVA r p l vp p l ˆo p v p p p pp pr l k p [9]. 5. r edšp l pl q t ne tp p rl p p o p o p pn l mq p p p l l l l p. rp p PCAm CVA p p } p EM k vl l rn vl l mp, p pn l v p p o l o l l m. p p o p m q pl pn o l l m. rp p l sl rk l l [15]p vl rp l p ˆ }ˆ p qr rp m p rl m rn š rp kr m k vp r p p r edšp pp p. Korean Chem. Eng. Res., Vol. 45, No. 1, February, 2007

6 92 p} Ëpp l Brain Korea 21 vop lp vol. y 1. MacGregor, J. F. and Kourti, T., Statistical Process Control of Multivariate Processes, Cont. Eng. Prac., 3(3), (1995). 2. Choi, S. W., Lee, C., Lee, J.-M., Park, J. H. and Lee, I.-B., Fault Detection and Identification of Nonlinear Processes Based on Kernel PCA, Chem. Int. Lab. Sys., 75, 55-67(2005). 3. Cho, J.-H., Choi, S. W., Lee, D. and Lee, I.-B., Fault Identification for Process Monitoring Using Kernel Principal Component Analysis, Chem. Eng. Sci., 60, (2005). 4. Ku, W., Storer, R. H. and Georgakis, C., Disturbance Detection and Isolation by Dynamic Principal Component Analysis, Chem. Int. Lab. Sys., 30, (1995). 5. Nelson, P. R. C., Taylor, P. A. and MacGregor, J. F., Missing Data Methods in PCA and PLS; Score Calculations with Incomplete Observations, Chem. Int. Lab. Sys., 35, 45-65(1996). 6. Russell, E. L., Chiang, L. H. and Braatz, R. D., Fault Detection in Industrial Processes Using Canonical Variate Analysis and Dynamic Principal Component Analysis, Chem. Int. Lab. Sys., 51, (2000). 7. Negiz, A. and Cinar, A., Statistical Monitoring of Multivariable Dynamic Processes with State Space Model, AIChE J., 43, (1997). 8. Arteaga, F. and Ferrer, A., Dealing with Missing Data in MSPC: Several Method, Different Interpretations, Some Examples, J. Chemometrics, 16, (2002). 9. Lee, C., Choi, S. W. and Lee, I.-B., Variable Reconstruction and Sensor Fault Identification Using Canonical Variate Analysis, J. Process Control, 16, (2006). 10. Larimore, W. E., Canonical Variate Analysis in Identification, Filtering, and Adaptive Control, Proceedings of IEEE Conference on Decision and Control, Honolulu, Hawaii, (1990). 11. Yoon, S. and MacGregor, J. F., Fault Diagnosis with Multivariate Statistical Models Part I: Using Steady State Fault Signatures, J. Process Control, 11, (2001). 12. Willemain, T. R. and Runger, G. C., Designing Control Charts Using an Empirical Reference Distribution, J. Quality Technology, 28, 31-38(1996). 13. Marlin, T. E., Process Control, McGraw-Hill, New York(1995). 14. Conference report, Abnormal Situation Detection and Projection Methods-industrial Applications, Chem. Int. Lab. Sys., 76, (2005). 15. Lee, C., Choi, S. W., Lee, J.-M. and Lee, I.-B., Sensor Fault Identification in MSPM Using Reconstructed Monitoring Statistics, Ind. Eng. Chem. Res., 43(15), (2004). o45 o k

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