Detection, isolation and reconstruction of faulty sensors using principal component analysis +

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1 Indan Journal of Chemcal Technology Vol. 1, July 005, pp Detecton, solaton and reconstructon of faulty sensors usng prncpal component analyss + M S R K Bose, G Sathyendra Kumar & Ch Venkateswarlu * Process Dynamcs and Control Group, Chemcal Engneerng Scences Dvson, Indan Insttute of Chemcal Technology, Hyderabad , Inda Receved 0 November 003; revsed receved 10 May 005; accepted 0 May 005 A strategy based on prncpal component analyss (PCA) s presented for detecton, dentfcaton and reconstructon of faulty sensors. In ths strategy, sensor fault detecton s carred out by usng multvarate statstcs, faulty sensors are solated usng prncpal component score contrbutons and reconstructon of faulty sensors s accomplshed through the analyss of fault drecton vector. The performance of the strategy s evaluated by applyng to a closed-loop controlled CSTR system. The smulaton results demonstrate the ablty of the strategy for detecton, dentfcaton and reconstructon of sngle and multple faulty sensors. Keywords: Sensor, prncpal component analyss, controlled CSTR system IPC Code: G01N; G1C17/00 Performance montorng and fault dagnoss s an ntegral part of successful operaton of a process. Modern chemcal ndustres are well equpped wth measurement sensors, whch form a crucal lnk n provdng nformaton to controllers and operators. The sensor measurements are hghly correlated under normal operatng condtons, thus provdng redundancy for sensor fault detecton and dentfcaton. The correlatons n sensor measurements are mostly due to physcal and chemcal prncples governng the process operaton. Sensor faults can cause process dsturbances, loss of control performance, proft loss or even catastrophc accdents. Hard or catastrophc sensor falures can be detected easly, whereas noncatastrophc sensor degradatons are dffcult to detect and dentfy due to correlatve nteractons. Therefore, automatc detecton and dentfcaton of sensor faults s an asset for effcent montorng and control of the process. In recent years, varous technques have been reported for montorng and dagnoss of faulty sensors. Luo et al. 1 proposed a wavelet based sequental approach for real tme sensor fault detecton and appled for fault detecton n CSTR temperature sensor. Ths approach manly focuses on fault detecton based on +IICT Communcaton No *For correspondence (E-mal: chvenkat@ct.res.n; Fax: ) the analyss of a sngle sensor sgnal. Yng and Joseph proposed an approach of sensor fault detecton usng analyss of hgh frequency nose present n the measurements. Ths method has been appled for fault detecton n temperature sensor of a hot ar blower and pressure sensor of a tank system, wth sngle control loop n each system. Duna et al. 3 proposed a method based on prncpal component analyss for detecton and dentfcaton of faulty sensors and appled to a boler process. In ther method, squared predcton error (SPE) s used to test the breakdown of sensor correlatons, and sensor valdty ndex s used to determne the status of each sensor. Reconstructon of faulty sensors s carred out usng PCA based successve substtuton and optmsaton. Ths method concentrates on the dentfcaton of a sngle sensor assumng that one sensor fals at a tme. Doymaz et al. 4 proposed a method nvolvng T and SPE statstcs for sensor fault detecton, correlaton coeffcents for sensor fault solaton and calbraton model for reconstructon of faulty sensors, and appled to a lqud fed ceramc melter. Most of these technques have been evaluated by applyng to openloop systems or systems wth sngle control loop. In closed loop process operaton, the controller s effectveness depends on gettng relable data from the sensors. If the sensor s faulty, the controller acton may become neffectve and can lead to degraded process behavour. Then t s necessary that ts value

2 BOSE et al.: RECONSTRUCTION OF FAULTY SENSORS USING PRINCIPAL COMPONENT ANALYSIS 431 be reconstructed so as to send the correct nformaton to the controller. Moreover, t s also mportant to dagnose sensors when faults occur smultaneously n more than one sensor n processes wth mult-loop control operatons. In ths study, a strategy based on prncpal component analyss (PCA) s presented for detecton, dentfcaton and reconstructon of faulty sensors. The PCA model s constructed by transformng the hgh dmensonal orgnal data matrx nto a representaton space of sgnfcantly reduced dmenson. The PCA model so created based on the data of normal operatng condtons s then used as a bass of reference for montorng and dagnoss of process sensors. The PCA based strategy of ths study employs multvarate statstcs for sensor fault detecton, prncpal component score contrbutons for solaton of faulty sensors and analyss of fault drecton vector for reconstructon of faulty sensors. The performance of the proposed strategy s evaluated through smulaton by applyng to a closed-loop operated non-sothermal CSTR system. Prncpal component analyss Prncpal component analyss (PCA) s a wellestablshed multvarate statstcal process control technque, whch bulds low dmensonal models from the analyss and montorng of routne process operatons 5. PCA transforms a number of correlated varables nto a smaller number of uncorrelated varables called prncpal components (PCs), whch are lnear combnatons of the measured varables n the data set. The PCA based fault dagnostc approach s useful because t focuses attenton on a small number of PCs representng large number of process varables. The PCA model s establshed by the egen value decomposton of the covarance matrx of the orgnal data matrx. The data matrx can be denoted by X wth rows as samples and columns as varables. The covarance matrx of the normalsed data matrx X wth m rows and n columns s defned by Covarance (X) = (X T X) / (m-1) (1) The PCA model dvdes the montorng space nto two orthogonal subspaces, called the prncpal component (PC) subspace and the resdual subspace (RS). Out of these two subspaces, normal data varaton wll be captured by the PC subspace, and the abnormal varaton and random nose may be captured by the RS. Thus, the PCA model decomposes X as the sum of the product of n pars of vectors, each par consstng of a n 1 vector called the loadngs, P, and a m 1 vector called the scores, T plus a resdual matrx E: X = T 1 P 1 T + T P T T k P k T + E () where the resdual matrx E represents the nformaton not captured by the model. Each of the scores vectors T s smply the projecton of X onto the new bass vector P T = X P (3) Thus, the PCA forms latent varables (PCs or scores) whch are lnear combnatons of the nput varables and the loadngs. The frst few PCs capture the maxmum amount of varance n the measurements. Therefore, t s mportant to choose the adequate number of PCs to represent the system n an optmal way. Varous methods have been reported for selectng the number of PCs 6. In ths study, an average egen value method s used for selectng the optmum number of PCs. Strategy for detecton, dentfcaton and reconstructon of faulty sensors Fault detecton In PCA, latent varables encapsulate the major varaton n the process. Hence, montorng these varables usng varous statstcal measures can provde ndspensable nsght about the process characterstcs. In ths strategy, fault detecton s carred out for both PC subspace and RS. The PC subspace s montored through the use of Hotellng s τ statstc and RS s montored by usng Q statstc 7. These tests provde better understandng of unantcpated changes n each sensor characterstc. Hotellng s τ statstc Havng establshed the PCA model, process behavour can be referenced aganst the n-control model, whch s bult usng fault-free data. The new observatons can be projected onto the plane defned by loadng vectors to obtan ther scores. Hotellng s τ statstc determnes the measure of varaton wthn the PCA model based on the retaned prncpal components. τ s the sum of normalzed squares defned by τ = t λ k -1 t T = x P k λ k -1 P k T x T (4) If k s the retaned prncpal components, then P k and T k represent the correspondng loadngs and scores. In the above equaton, x s the observaton vector at th

3 43 INDIAN J. CHEM. TECHNOL., JULY 005 samplng nstant, t s the th row of the retaned prncpal component scores and λ k s the egen values correspondng to the retaned prncpal components. In general, the hgher the τ value, the more dstant s the observaton from the mean. The matrx λ -1 s a dagonal matrx contanng the nverse of egen values assocated wth the k th egen vector. The Hotellng s control chart then plots the sample τ test statstc on the vertcal axs and the subgroup dentfer on the horzontal axs. An upper control lmt on ths chart s obtaned usng F-dstrbuton 8. Q Statstc Q statstc s a measure of the amount of varaton not captured by the PCA model and t represents the sum of squared errors. The Q statstc s also referred as the squared predcton error (SPE). When a new type of event occurs, ths type of event can be detected by computng the SPE of the new observatons. Ths statstc s defned as Q = e e T (5) where e = (I - P k P k T ). Q s the sum of squares of each row of E. When the process s n-control, the SPE statc represents unstructured resduals that cannot be accounted for by the PCA model. When an unusual event occurs, t results a change n the process mean or varance structure whch wll be detected by a hgh value of ths statstc. The upper control lmts for ths statstc can be computed based on the reference data 9. Fault dentfcaton When sensor faults occur, there wll be a change n correlaton of the faulty sensor wth other sensors. Ths phenomenon can be used to fnd the dentfablty condtons for a fault. If m s the total number of prncpal components and k s the number of prncpal components retaned, then the dmenson of resdual subspace s m-k. Hence, the maxmum degree of freedom avalable for any observaton vector s m-k. It has been shown that the necessary condton 10 for dentfablty s m-k. Once a fault s detected by multvarable statstcal tests, the next task s to solate the faulty sensors. When these tests volate the confdence control lmts, the contrbuton of the ndvdual varables can be plotted and those varables havng large contrbutons are examned to dentfy the faulty sensors. Contrbuton plots can reveal the process varables whch make the hghest contrbuton to the PCA model or to the resdues. When the SPE statstc volates ts upper control lmt, the contrbuton of the ndvdual varables can be plotted and those varables havng larger contrbutons are examned to ndcate possble faults. Smlarly, f the varaton n the model space (τ ) becomes large, then contrbuton of each varable (x ) to a large value of the j th suspect PC score (t j ) s gven by Contrbuton (x ) = P j x (6) where P j s the weght of the th varable (x ) n the jth latent varable P j, and x s the observaton vector over the tme perod n consderaton. These values are plotted aganst varable number. More detals can be found n the lterature 11. Sensor fault reconstructon Once the faulty sensors are dentfed, the mmedate task s to rectfy them pror to sendng the sgnal to controllers or n estmatng the present state of the process. In ths work, fault free sensor nformaton s used for reconstructon of faulty sensors. The task of fault reconstructon s to best estmate x *, usng the PCA model and fault drecton ξ. The sample vector for normal operaton s denoted by x *, whch s unknown when a fault occurs. In the presence of a fault, the sample vector x can be represented usng a fault drecton vector ξ, whch characterses the effect of fault on the actual measurements, x = x * + ƒ ξ (7) where ξ I s the normalsed vector, and scalar f represents the magntude of the fault. The reconstructed vector x s obtaned by correctng x n the drecton ξ x = x - ƒ ξ (8) where ƒ s an estmate of the fault magntude ƒ whch measures the dsplacement n the drecton ξ. The dstance between x and the PC subspace s gven by the magntude of resdual vector, x%, or the SPE of x : SPE = x% = x% f ξ = x% f% % 0 ξ (9)

4 BOSE et al.: RECONSTRUCTION OF FAULTY SENSORS USING PRINCIPAL COMPONENT ANALYSIS ξ Here f% f and = % ξ % ξ =. The mnmzaton of ξ SPE leads to, or and ( x% f% ξ) d SPE = % ξ 0 % = 0 df% f% = % ξ x% = % ξ x f% % ξ x f = = % ξ % ξ The reconstructed normal vector s and x = x - f ξ ξξ% = I x % ξ x% = I x% 0 % ( ξ % ξ ) (10) (11) The feasblty of calculatng f assures the exstence of x, whch s the best estmate for x * by reconstructon n the drecton ξ. Further detals on faults reconstructon can be obtaned from lterature 10. Applcaton system The sensor dagnoss strategy s evaluated by applyng to a closed-loop controlled CSTR system, the schematc of whch s shown n Fg.1. The reactor has cross sectonal area A wth coolng surface area A c. A sngle feed stream contanng a reactant A wth flow rate of Q F and concentraton C AF enters the tank. The reactant A undergoes a frstorder exothermc reacton. A coolant flowng through the jacket around the tank cools the reactor contents. The dynamc model equatons for the reactor system are: dtc QC ( TC F TC ) UAC ( T TC ) = + dt VC ρ CC pc VC dh QF Cv h = dt A where k 0 s Arrhenus constant; E s the actvaton energy for the reacton, R s the unversal gas constant; C v s the valve constant; ΔH s the heat of reacton; U s the overall heat transfer coeffcent; and ρ, ρ c, and C p, C PC are denstes and specfc heats of reacton and coolng lqud, respectvely. The data used n the system s reported elsewhere 1. The system s controlled by proportonal ntegral (PI) controllers. The concentraton C A s controlled by adjustng the coolant flow rate Q C, the reactor temperature T s controlled by manpulatng the feed flow rate Q F, and the heght of the lqud n the reactor h s controlled by changng the flow rate of the output stream Q. The proportonal and ntegral constants used for the controllers are 0.05 and for concentraton controller, 0.05 and for reactor temperature controller, and.0 and 5.0 for the lqud level controller, respectvely. The system s smulated wth the help of fourth order Runge-Kutta method. In order to represent the actual data, random Gaussan nose of about 5% to the magntude of the varables s added to the smulated data. Results and Dscusson The ablty of the proposed strategy s evaluated through smulaton by applyng to a non-sothermal CSTR. The process operaton for a tme perod of 5 dc dt A QC C hc = ke C + Ah E/ RT F AF V A 0 A ( Δ ) E / RT ke C A H QF TF T dt 0 ( ) = + dt ρ C p Ah UAC( TC T ) + ρ C Ah p (1) Fg. 1 Jacketed CSTR wth control scheme

5 434 INDIAN J. CHEM. TECHNOL., JULY 005 mn s consdered for the analyss and dagnoss of sensors. Ths operaton perod corresponds to the treatment of 500 data sets, each data set nvolvng 4 sensor measurements, namely, reactant concentraton, reactor temperature, coolant temperature and lqud level n the reactor, respectvely. The frst 100 data sets of the operaton perod are consdered as fault free data, and used to buld the PCA model. The remanng data sets represent faulty operaton. Sngle and multple sensor faults are ntroduced n measurement sensors n the form of hgh percentage nose. Confdence lmts for normal operaton are obtaned assumng that the data s under normal dstrbuton. The measurements of normal operaton are corrupted usng random Gaussan nose of 5% to the magntude of the orgnal varables. The sensor measurements below 5% nose lmt s consdered to be under normal status. Sensors wth nosy dsturbance of above 5% random Gaussan nose are consdered to be faulty. The covarance matrx of the normal data s used to calculate egen values and egen vectors. The average egen value technque s used to evaluate the optmum number of PCs to be retaned. The multvarate control charts, namely, τ chart, Q/SPE chart are used for fault detecton. The results of τ statstc and Q statstc for normal status of the process are shown n Fg.. The squared predcton error to the Q statstc s calculated for every new observaton and plotted aganst the samplng tme. Confdence lmt for ths Q test s calculated to be Fg. (a) shows that the normal process operaton of the entre run (5mn) s below the lmt of Fg. (b) shows the results of Hotellng τ statstc for normal operaton. The calculated confdence lmt for ths test s 4. The results show that the plot s completely below the lmt of 4 ndcatng the normal behavour of the process. The results of plots shown n Fg. 3 ndcate the volaton of the confdence lmts from normal operaton due to sensor faults. Intally the process s under normal operaton, and after 5 mn of elapsed tme, faults are ntroduced nto dfferent sensors. A random Gaussan nose of 10% to the magntude of orgnal varables s consdered to be the case of sensor faults. The results of Q plot and τ plot shown n Fg. 3 clearly ndcate the faulty operaton. Thus, the τ and Q statstcs are able to successfully detect the sensor faults. Once the fault s detected, the ndvdual contrbutons of each sensor varable to the suspected prnc- Fg. Tests for normal operaton: (a) Q statstc, (b) τ statstc Fg. 3 Tests for faulty operaton: (a) Q statstc, (b) τ statstc pal components are computed to dentfy the faulty sensors. Isolaton of faulty sensors s carred out usng the procedure descrbed n the prevous secton. Fg. 4(a) shows the results of fault solaton when a fault s consdered as 0% nose n reactor temperature measurement sensor. Fault s also ntroduced n more than one sensor smultaneously n a sngle plant run. The

6 BOSE et al.: RECONSTRUCTION OF FAULTY SENSORS USING PRINCIPAL COMPONENT ANALYSIS 435 Fg. 4 Identfcaton of sensor faults: (a) Fault n reactor temperature sensor, (b) faults n reactor and coolant temperature sensors Sensor fault rectfcaton s carred out usng the reconstructon procedure based on the analyss of fault drecton vector as llustrated n the prevous secton. When reactor temperature sensor s faulty, ts reconstructon s carred out usng the fault drecton vector ξ as [ ]. When both the reactor temperature and coolant temperature sensors are faulty, the fault drecton vector used s [ ]. The sensors after reconstructon are further tested through fault detecton tests. The results of these tests shown n Fg. 5 ndcate the rectfcaton of sensor faults due to reconstructon. The nformaton on reconstructon s fed to the controllers to mantan the specfed process operaton. Prevously reported methods for sensor fault dagnoss 10,13 employ a procedure n whch sensor faults are solated after reconstructon of all the sensors n the process. The sensor fault dagnostc approach presented n ths study employs a procedure n whch faulty sensors are dagnosed pror to reconstructon. Thus, by ths strategy, only faulty sensors can be reconstructed nstead of reconstructng all the sensors. Dagnoss of sensor faults pror to reconstructon has the advantage that the reconstructon can be carred only f fault exsts n sensors. Concluson In ths study, a prncpal component analyss based strategy s presented for detecton, solaton and reconstructon of faulty sensors. The performance of the strategy s evaluated by applyng to a non-sothermal CSTR system. The results demonstrate the effectveness of the strategy for dentfcaton and reconstructon of sngle as well as multple sensor faults. Fg. 5 Fault detecton tests after reconstructon: (a) Q statstc, (b) τ statstc results of the strategy when faults occur as 0% nose n reactor and coolant temperature measurement sensors are shown n Fg. 4(b). These results show the effectveness of the dagnostc strategy for dentfcaton of sngle and multple sensor faults. References 1 Luo R, Msra M, Qn S, Barton J & Hmmelblau D, Ind Eng Chem Res, 37 (1998) 104. Yng C M & Joseph B, Ind Eng Chem Res, 37 (000) Duna R, Qn S J, Edgar T F & McAvoy T J, AIChE J, 4 (1996) Doymaz F, Romagnol J A & Palazoglu A, Chemometr Intlg Lab Sys, 55 (001) Tong H & Crowe C M, AIChE J, 41 (1995) Valle S, L W & Qn S J, Ind Eng Chem Res, 38 (1999) Russel E L, Chang L H & Braatz R D, Chemomet Intlg Lab Sys, 51 (000) MacGregor J F & Kourt T, Control Eng Pract, 3 (1995) Jackson J E & Mudholkar G S, Technometrcs, 1 (1979) Duna R & Qn S, Control Eng Pract, 6 (1998) Yoon S & MacGregor J F, AIChE J, 46 (000) Snghal A & Serborg D E, AIChE J, 46 (000) Duna R & Qn S J, Comput Chem Eng, (1998) 97.