Detection of abrupt change and applications in sensor decalibration monitoring

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ISATRANSACTIONS®

ISA Transactions 41~2002! 155–166

Detection of abrupt change and applications in sensordecalibration monitoring

Ashish Malhotra,* Biao Huang†

Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6

~Received 12 July 2000; accepted 1 March 2001!

Abstract

The primary objective of fault detection is to detect abrupt undesirable changes in a process at an early staearly detection has a potential of preventing loss of production and equipment damage due to these undchanges, thus reducing process downtime. This paper details the implementation of some parametric fault dtechniques for sensor decalibration monitoring. A parametric fault detection approach that is handled in depthpaper is the local approach. This approach developed by Benveniste, Basseville, and Moustakides@Benveniste, A.,Basseville, M., and Moustakides, G., The asymptotic local approach to change detection and model validatioTrans. Autom. ControlAC-32 ~7!, 583–592~1987!# offers a computationally inexpensive way to attain the objectivemonitoring changes in model parameters. However, the algorithm in its original formulation is not applicable toprocesses such as sensors. Therefore, the local approach is coupled with other estimation algorithms such asindependent Kalman filter to derive a robust sensor decalibration monitoring algorithm. The proposed fault dealgorithm is applied to a pilot scale process for evaluation of its performance. © 2002 ISA—The InstrumenSystems, and Automation Society.

Keywords: Fault detection; Local approach; Sensor decalibration; Kalman filtering

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1. Introduction

Fault detection and isolation aims at the detetion of undesirable process changes or malfutions @1–5#. If this objective of early detection cabe achieved inexpensively and timely, then it cbe easily applied to real time process monitorinThe successful implementation of such a continous online approach can provide a means to desmall deviations at an early stage. This can factate timely action and prevent damage or dowtime to the system.

*E-mail address: [email protected]†Corresponding author. Tel.:11-780-492-9016; fax:11-

780-492-2881.E-mail address: [email protected]

0019-0578/2002/$ - see front matter © 2002 ISA—The Instru

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In the literature cited above, faults~or malfunc-tions, undesirable process changes! are broadlyclassified as measurement faults, additive fauand nonadditive or multiplicative faults. Whilmeasurement faults and additive faults can shup as a difference between the actual processput and the measured output the multiplicativenonadditive faults are difficult to detect by normmeans of inspection. These faults can be visuized as changes acting through the processhence affecting the ‘‘generating mechanism’’ itse~like changes in the state transition matrix! @6#.These nonadditive faults can generally be repsented as changes of plant parameters from tnominal or ‘‘safe’’ value.

The actual computation strategy of most faudetection algorithms is generally split into tw

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156 Ashish Malhotra, Biao Huang / ISA Transactions 41 (2002) 155–166

parts, namely, generation of residuals and residevaluation@6#. Ideally the residuals generated fthe purpose of fault diagnosis should be closezero for normal plant operation and nonzero ffaulty operation. The residuals should have favable properties like minimal sensitivity to distubances and maximum sensitivity to faults. Rsidual evaluation is making decisions basedthese residuals. A statistical approach calledlocal approach@7#, provides ways in which thisobjective ~of residual generation and evaluatio!can be attained in a neat, computationally inexpsive manner. This approach aims at early detecand isolation of small undesirable nonadditichanges in a process.

Sensor fault is one of the most common procefaults in industry and its detection has beenarea of active research. Most of the existing motoring algorithms have relied on causal modesuch as parity relation or statistical models suchprincipal components analysis/partial least squa~PCA/PLS!. Thus, these approaches depend onavailability of measurements of all related vaables needed by the model. If some variablesthe causal model are not available, then theseproaches may not be applied. The main stumblblock to the sensor monitoring problem is oftethat there is no direct knowledge of the inputsthe sensor other than the sensor output anddecision on whether the sensor is at fault or nodone using only the outputs from the sensor. Thsensor monitoring is a typical example of monitoing problem where the input information is likelunknown.

Compared to systems where the inputsknown there is little work done on monitoring osystems where there is no knowledge of theputs. Sensors are critical elements in process ctrol and automation. A sensor giving wrong infomation to a controller can have a snowballineffect on the rest of the process as undesiredunnecessary control moves may be implemenbased on these incorrect signals. Sensor faultscause deterioration of productivity due to pocontroller performance even though the controlmay be optimally tuned. This can trigger mancontrol problems.

To understand this issue and the inherent prlem in sensor monitoring as compared to procor controller monitoring let us consider the illustration in Fig. 1. Here a sensor is treated as a s

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tem. The output of this system is the measuremof its input that is an unknown quantity and wibe the same as the output if there is no fault anoise. For a healthy sensor, the output follows tinput closely, subject to measurement noisdisturbances. If there is a change in the unknoinput, then the output should track the chanquickly. On the other hand, a change in the outpcan be attributed to either a change in the inputfault in the sensor. It is not possible to distinguithem by simply monitoring the change in the ouput. This ‘‘unknown’’ nature of the input signamakes sensor monitoring fundamentally differefrom regular process or control loop monitorinTherefore, new techniques for sensor monitorihave to be considered.

There are many techniques available for procand controller monitoring. There are relatively femethods available for sensor monitoring withousing certain information of the inputs. Somethe available techniques in sensor monitorinwithout using input knowledge, include the appcation of multiscale dynamic PCA analysis@8#,PCA @9#, and the local approach@10#.

In this paper a sensor decalibration detectimethod proposed in@10# that uses the local approach will be further investigated. This methouses autoregressive integrated moving aver~ARIMA ! modeling, which can well describeslow drifting time series. Hence this method is aplicable to systems with slow changing inputHowever, for a truly robust faulty sensor detectioalgorithm it is important that the detection algorithm should also be robust to fast changing iputs. In view of this, an alternate method is prposed that is applicable to sensor monitoriproblems where the input is unknown and canbitrarily change. Work has been done in the fie

Fig. 1. The inherent difficulty in sensor monitoring as compared to process or control loop monitoring.

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157Ashish Malhotra, Biao Huang / ISA Transactions 41 (2002) 155–166

of Kalman filter design in which knowledge oinputs is not required@11,12#. In this paper theKalman filter algorithm in the above-mentionereferences is combined with the local approa@7,6,10,13# to derive a sensor decalibration algrithm that is independent of inputs. The algorithis implemented upon a simulation example andpilot scale stirred tank process.

The remainder of this paper is organized as flows. Section 2 gives a detailed overview of tlocal approach algorithm for fault detection, froan engineering perspective. In Section 3, the loapproach is coupled with the input-independeKalman filter to derive a sensor decalibratiomonitoring algorithm, followed by conclusions iSection 4.

2. The local approach

2.1. The local approach algorithm

Basseville and co-workers started work omonitoring plant parametric changes@7,14# fromwhich they developed a procedure for the dettion and isolation of abrupt change in the paraeters of a process~multiplicative or nonadditivefaults!. This approach provides tools by which thdetection problem of monitoring the change in tparameters of a stochastic process can be simfied to the problem of monitoring the mean ofGaussian vector@7,15,16#.

A fair amount of work has been done in thtopic of fault detection using the local approaever since. Robustness issues for the localproach are explored by Zhanget al. @17#. In thiswork issues such as effect of model plant mmatch and underparametrization on the detectalgorithm are discussed. It is shown in this wothat this approach has a wide applicability as it cdeal with reduced-order models and bias resultfrom identification. This approach has also beapplied to monitoring of nonlinear models@13#.Monitoring of nonlinear models based on engneering principals, which are usually represenby differential algebraic equations, is targetedthis work.

The local approach has been applied to freaching and challenging fields like monitoringgas turbines, oil rig vibration, emission from autmobile catalytic converters@6#, bioreactors@13#,etc. A variety of objectives can be attained in t

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framework of this approach—such as model vadation@17#, detection of sensor decalibration@10#,and process monitoring among other applicatio

The formal problem statement for the local aproach, can be written as a hypothesis test@7#:

H0 :u5u0 ~1!

against

H1 :u5u01h

AN. ~2!

Here the system is parametrized by the vectou,with u0 representing the nominal value identifieover a nominal or ‘‘safe’’ run of the system andNis the length of the data used in the test. This hpothesis test means that if there is no fault tpresent parameter vector will be the same asnominal parameter vector, and if there is a fathe present parameter vector will be different frothe nominal parameter vector whereh is an arbi-trary unknown vector having the same dimensiasu, which signifies that the change of the systeparameter from the nominal parameter can bean arbitrary direction, i.e., along any elementthe parameter vectoru.

The local approach deals with the problemchange detection hypothesis test@Eqs.~1! and~2!#by transforming them to the problem of monitoing the mean of a Gaussian vector. The monitoralgorithm using the local approach can be summrized as follows.

A statisticjN(u0) given by

jN~u0!51

AN(t51

N

H~zt ,u0!

is defined as a normalized residual if

E@H~zt ,u0!#50 for u5u0 ~3!

and

E@H~zt ,u0!#Þ0 for uPv~u0!\u0 , ~4!

wherezt is a function ofu;v(u0)\u0 reads as aneighborhood ofu0 exclusive ofu0 . H(zt ,u0) isdefined as the primary residual, wherezt

5@u(t),y(t)#T. Note that the residuals definehere are different from the regular prediction rsiduals, and are asymptotically sufficient statist@6#. The empirical version of Eqs.~3! and~4! maybe written as

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1

N (t51

N

H~zt ,u0!50 for u5u0 ~5!

and

1

N (t51

N

H~zt ,u0!Þ0 for uPv~u0!\u0 . ~6!

If we can find such residuals, then for sufciently largeN, the following results hold:

jN~u0!;N„0,S~u0!… under H0 ,

jN~u0!;N„2M ~u0!h,S~u0!… under H1 ,

where

M ~u0!5S ]

]uH~zt ,u!U

u5u0

D , ~7!

S~u0!5 (t52`

`

Cov„H~z1 ,u0!,H~zt ,u0!….

~8!

In practice, the empirical version of Eq.~7! maybe written as

M ~u0!']

]u S 1

N (t51

N

H~zt ,u! Du5u0

~9!

andS(u0) may be approximated by@13#

S~u0!'1

N (t51

N

H~zt ,u0!HT~zt ,u0!

1(i 51

I 1

N2 i (t51

N2 i

@H~zt ,u0!HT~zt1 i ,u0!

1H~zt1 i ,u0!HT~zt ,u0!#, ~10!

where the valueI should be properly selected acording to the correlation of the signals. In pratice, one can gradually increase the value ofI untilS(u0) converges. With these results, detectionsmall changes in the parameteru is asymptoticallyequivalent to the detection of the changes inmean of a Gaussian vector. The generalized lilihood ratio ~GLR! test detecting unknownchanges in the mean of a Gaussian vector is ax2

test. It is shown that the GLR test ofH1 againstH0 can be written as@6#

xglobal2 5jN~u0!TS21~u0!M ~u0!

3@MT~u0!S21~u0!M ~u0!#21MT~u0!

3S21~u0!jN~u0!. ~11!

If M (u0) is a square matrix, then this test canfurther simplified to

xglobal2 5jN~u0!TS21~u0!jN~u0!. ~12!

xglobal2 has a centralx2 distribution underH0 , and

a noncentralx2 distribution underH1 . The degreeof freedom ofxglobal

2 is the row dimension ofu. Athreshold valuexa

2 can be found from ax2 table,where a is the false alarm rate specified by thusers. If xglobal

2 is found to be larger than thethreshold value, then a change in the parametedetected.

2.2. Robustness issues

Some of the restrictions on the initial formulation of the local approach@7# were that thereshould be no plant-model mismatch in the nominparameters. Zhanget al. @17# show that these constraints can be relaxed and the applicability of tlocal approach can be increased. A simple undstanding of this work is presented in this sectio

The initial requirement for the primary residuais given by

EH~zk ,u0!50 as u0→Q, ~13!

wherezk is the function ofQ, Q is the actual plantparameter,u0 is the nominal model parameter, anzk is the input-output measurements or the regrsor. This relation will not hold in the case of sombiased identification algorithm, i.e., whenu0→” Q. This requirement must be relaxed forwider applicability of the local approach.

To work around this problem for the case ththere is a mismatch between the modelu and theplant Q a functionF is introduced such that

u→F~Q! as N→`. ~14!

F is called the bias function and defines a relatioship between the actual system~Q! and the iden-tified model~u!. This means that as the numberdata points tend to infinity the model parametewill approach some functional form of the originasystem parameters because of the model-p

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mismatch. If there is no model-plant mismatch tmodel parameters will approach the actual systparameters.

Using this notation of the bias function, the hpothesis test of parameter change will be

H0 : F~Q!5u0 ,

H1 : F~Q!5u01u

AN,

whereu is the actual parameter,u0 is the nominalidentified parameter, andu is some unknownchange andF is the bias function of the primaryresidual.

Here the model is being checked to see iffunction is still in agreement with a new set odata. From this hypothesis test the detection alrithm can be derived in the similar lines as havibeen discussed in the previous sections by takinto account Eq.~14!. It is also important to notethat the primary residual will not have a zero medue to the model-plant mismatch but can be mato have a zero mean by a bias correction term:

H0~zk ,u0!,H~zk ,u0!2h0 ,

where the bias term can be approximated as

h0'h0,1

K (k51

K

H~zk ,u0!,

whereK is the number of data used for the traiing. The normalized residual is now defined as

jN~u0!,(k51

N

@H~zk ,u0!2h0#.

This normalized residual has the propertiesbeing zero mean for no change in its functionparameters and nonzero mean otherwise. Hethe detection problem reduces to monitoring of tmean of this normalized residual.

Here the changes in the bias functionF are to bedetected and the changes inQ that are not cap-tured by the bias function cannot be detected.the local approach can still be used for chandetection despite the mismatch in the plant andmodel, the only drawback being that the changin the plant not captured by the bias function wnot be detected. Hence there may be a loss offormation due to the mismatch between the mo

e

and the plant. An important thing to note is thknowledge of the bias functionF is not needed forthe detection algorithm.

3. Local approach for sensor decalibrationmonitoring

In this section, the local approach is appliedmonitoring of sensor decalibration. We will firsdiscuss an existing method in the literature apoint out its limitations. Then a new methodproposed and its performance is evaluatedsimulation and a pilot scale experiment.

3.1. Sensor decalibration monitoring for slow-changing inputs

3.1.1. Sensor decalibration monitoring usinglocal approach and ARIMA modeling

In this subsection, we will further investigatsensor decalibration monitoring problem usingscheme proposed by O’Reilly@10# and point outits limitations in application. Although the algorithm targets sensor decalibration monitoring, tresult can be applied to any gain monitoring prolem with unknown inputs. In this particular implementation of the local approach the sensor outy is modeled by an ARIMA model@10#:

yk5C

ADek , ~15!

where ek is a white noise series,k is the timeinstant,D is the ‘‘difference’’ operator that is equato 12z21, and A and C are polynomials in thebackshift operator. To explain why the ARIMAmodel is used here, let us transfer the equation

Dyk5C

Aek .

If the input is a constant, then thisD operatoreliminates the effect of the input whose magnituis unknown. If the input is a slow-changing signathen thisD operator can approximately play thsame role. This model structure can, therefore,used to describe slow drifting time series.

Eq. ~15! can be rewritten in terms of the prediction error:

ek5AD

Cyk . ~16!

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Assuming linear decalibration, the prediction erris

ek5AD

C~pyk!, ~17!

where p is a variable scalar quantity, havingvalue of p051 under normal operation and othevalues under faulty operation. This can be viewas a decalibration factor or a representation ofsensor~or process! gain. Eq.~17! can also be writ-ten as

ek5p(i 50

q

ai@yk2 i2yk2 i 21#2(i 51

r

ciek2 i ,

~18!

wherek is the time index such thatt5kT ~T is thesampling time!, q andr are the orders ofA andCpolynomials, andai andci are the coefficients oA andC polynomials. The primary residual can bwritten as the product ofek and its derivative:

H~yk ,u0!5S ]ek

]u D T

ek . ~19!

~This can be derived from the identification objetive function that minimizes the square of the prdiction errorek .! The derivative can be calculatefrom Eq. ~18! as

S ]ek

]p D5(i 50

n

ai@yk2 i2yk2 i 21#2(i 51

r

ci S ]ek2 i

]p D~20!

or using the transfer function notation, the derivtive can be written as

S ]ek

]p D5AD

Cyk5~1/p!ek . ~21!

In this casep is u which is being monitored withother parameters being assumed to be consThus, the primary residual in this case is given

H~yk ,u0!5ek2, ~22!

where u05p051. Clearly, this primary residuadoes not satisfy the conditionEH(yk ,u0)50.Thus, a bias correction of the primary residualdiscussed in Section 2 has to be used.

The normalized residual~after correction of thebias term! is given by

t.

jN~u0!51

AN(k51

N

~ek22h0!, ~23!

where

h051

K (k51

K

ek2

andK is the number of data used for training. Thx2 test is given by

xn25jN

T~u0!S21~u0!jN~u0!. ~24!

This is calculated over the testing data for eveN sampling instances. Then degrees of freedomin this case correspond to the dimension of tmonitored vectorp, which has dimension oneThe calculatedx2 test value is then compared tthe threshold value, obtained from thex2 table. Acalculated value greater than this threshold incates a possible sensor decalibration or fault.

3.1.2. Simulation example for slow-changinginput

The primary objective of this algorithm is to detect slow sensor decalibration that would notdetected easily by visual observation, using thecal approach. Analysis is carried on a simulatexample to gauge the potential of the method asearly indicator of decalibration and abrupt faultA fault would be indicated if the calculated valupersistently exceeds the threshold. Results osimulation are given as follows.

A bio reactor@18#, is taken as the example:

x15ax1x2

x21b2x1u,

x25dax1x2

x21b1~c2x2!u, ~25!

y5x2 ,

wherex1 : microbial concentration in the process,x2 : substrate concentration in the process,u: dilution rate,a: maximum growth rate,b: saturation parameter,c: inlet substrate concentration,d: yield factor.

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The output from this process is assumed to pthrough a sensor. The transfer function of this ssor is simply taken to be a gain of one. Inputtaken to be a ramp function of slope 0.01.simulate a decalibration the gain is changed. Rsults of two such simulations are shown in twfigures. Fig. 2 is the calculatedx2 plot of a 20%change in the sensor gain at around 1000 sampinstances~a total of 5000 data points are tested!.The fault is detected and can be seen as violaof the threshold. The threshold taken for this cais 3.84, with 5% false alarm rate. Fig. 3 is thcalculatedx2 plot of a 40% change in the sensgain at around 1000 sampling instances. Herefault is detected clearly.

Fig. 2. Calculatedx2 plot of a 20% change in the sensogain at around 1000 sampling instances.

Fig. 3. Calculatedx2 plot of a 40% change in the sensogain at around 1000 sampling instances.

3.2. Sensor decalibration monitoring for fast-changing input

3.2.1. Monitoring algorithm using inputindependent Kalman filter

With the primary objective of developing an agorithm to detect sensor decalibration in the preence of fast changing input, a method is evolvthat consists of using some results in the inpindependent Kalman filter design@11#. The ideahere is to model the sensor as a system which gunknown inputs~Fig. 1! and to devise a fault detection strategy for the system without knowledof the inputs. A gain change would signify a fauin the sensor. Monitoring of this gain change isthe prime interest of this paper.

The method proposed by Darouach and Zaszinski @11# to get unbiased minimum variancstate estimations without the knowledge of inpuis presented in this section. The traditional aproach for the Kalman filter design requires asumptions on the unknown inputs. There has bework done in this field by making assumptions ohow the unknown input varies in time@19#. Insome cases the assumptions are on the naturthe unknown inputs, such as assuming it to bestochastic process with known mean and covaance@20–22#. This new approach aims to relax thassumptions on the inputs and directly estimathe state without the knowledge of the inputs.

The system structure assumed is given by E~26!:

xk115Fxk1Gkdk1wk ,

yk5Hkxk1vk , ~26!

wherexkPRn is the state,dkPRp is the unknowninput, andykPRm is the output.vk andwk are thezero mean, white noise sequences that are unrelated and have non-negative definite covarianRk and Qk , respectively.Fk , Gk , and Hk areknown matrices of appropriate dimensions suthat rank(Hk)5m and rank(Gk)5p (m>p). Themethod proposed by Darouach and Zasadzin@11# uses the filter formulation as proposed by Ktanidis @12#. This linear filter for minimum vari-ance unbiased estimation is given by Eq.~27!:

xk11/k115F xk/k1Lk11~yk112Hk11Fkxk/k!.~27!

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Since this filter must be unbiased the followinconstraint equation must hold for the filter:

E@ xk11/k112xk11#50. ~28!

By using Eqs.~26! and ~27!, Eq. ~28! can bewritten as

Lk11Hk11Gk2Gk50. ~29!

The covariance matrix for the prediction errorgiven as

Pk115E@~ xk11/k112xk11!~ xk11/k112xk11!T#.~30!

By using Eqs.~26! and~27!, Eq. ~30! can be writ-ten as

Pk115~ I 2Lk11Hk11!~FkPkFk1Qk!

3~ I 2Lk11Hk11!T1Lk11Rk11Lk11T .

~31!

The method proposed by Darouach and Zaszinski @11# to get the gainLk11 consists of param-etrization of the solution of Eq.~29! in the form ofEq. ~32! and then deriving the minimum variancestimation using the parametrization ofLk11 .

Eq. ~29! has a solution if rank(Hk11Gk)5rank(Gk)5p. The solution is given by Eq.~32!@11#:

Lk115GkPk1Kk11Tk ~32!

with Pk5(Hk11Gk)1PRp3m and Tk5ak@ I

2(Hk11Gk)(Hk11Gk)1#PR(m2p)3m, where ak

is an arbitrary matrix that must be chosen suthat Tk is a full-row rank matrix and the matrix

F Tk

PkG ~33!

is nonsingular.A1 represents the pseudoinvermatrix of A. Substituting this parametrization@Eq.~32!# in the variance equation~31!, the minimumvariance estimate problem is reduced to findithe matrixKk11 @refer to Eq.~32!#, which mini-mizes the trace of the covariance matrix@Eq.~31!#.

Darouach and Zasadzinski@11# have given thefollowing expression for estimating the states fotime invariant system:

xk115Fxk1GPyk111Kk11~Tyk112b xk!,

-

where

Kk115~FPkbT1S!~bPkb

T1Q!21,

Pk115~F2Kk11b!Pk~F2Kk11b!T1Q1

1Kk11QKk11T 2Kk11ST2SKk11

T ,

~34!

and

Q15SQST1GPRPTGT,

S5I 2GPH,

F5SF,

b5THF,

S5SQHTTT2GPRTT,

Q5T~HQHT1R!TT,

P5~HG!1,

T5a@ I 2~HG!~HG!1#. ~35!

The state estimate can be used to find the pretion errors or residuals:

yk5Hxk ,

ek5 yk2yk . ~36!

Using the earlier result on sensor decalibrati~refer to Section 3!, the primary residual can bdefined as the square of the prediction residua

H~yk ,u0!5ek2. ~37!

Recall that in the above derivation, an unbiasstate estimation is guaranteed. Therefore, the pdiction residual should have a zero mean. Hoever, unlike the traditional Kalman filter, the residual obtained here may not be white. To obtathe true whiteness, we may need to cascadeother whitening filter to the residualek . This caneasily be done by fittingek with a time seriesmodel. Note that in order to obtain a solutionthe input-independent state estimator, i.e., thelution of Eq. ~29! to exist, rank(HG)5rank(G).Also it is assumedrank(H)>rank(G) and bothHand G be full column rank matrices. In additionin order to apply this technique, a model relatinthe input to the output is required. This model cbe estimated through the identification of the sesor in the off-line calibration stage when it is ithe healthy mode.

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3.2.2. Simulation and pilot plant case studyFor the simulation a two-input two-output sy

tem is taken:

xk115F 0.9512 0 0 0

0 0.9048 0.0669 0.0227

0 0 0.8825 0

0 0 0 0.9048

G xk

1F 0.4877 0.4877

21.1895 3.5690

0 0

0 0

G uk1wk ,

yk5F1 0 0 0

0 1 0 0G xk1vk , ~38!

wherevk andwk are white noise with covariancmatricesQ andR, respectively, given by

Q510223F 0 0 0 0

0 0.004886 0 0

0 0 0.2212 0

0 0 0 0.7252

G ,

R510253F0.1 0

0 0.6G .It is assumed that the outputyk passes through a

sensor. It is also assumed in the following fadetection simulation that the inputuk is unknown.This state space model@Eq. ~38!# is used to gen-erate the outputs of the system. The input is takas a random number with zero mean and unit vaance, and the noise terms are also taken as rannumbers with zero means and variance taken frthe respective covariance matrices. No plamodel mismatch is assumed and the same mois used for the design of the input independeKalman filter. The input independent Kalman filtbased sensor decalibration algorithm is run ondata generated. The sensor fault or decalibratiogenerated by changing the sensor gain. The resover a 100 simulation runs using different noiseeds are shown in Table 1. Here the calculatedx2

is shown for each case of 10%, 20%, and 30

l

s

change in the sensor gain. The threshold in tcase is 3.84 with a 5% false alarm rate.

The simulation results show that the algorithmable to detect the simulated sensor fault, i.e.,change in the sensor gain. This is shown byviolation of the threshold by the calculatedx2

value.The pilot plant in the Computer Process Contr

Laboratory in the Department of Chemical anMaterials Engineering, University of Alberta, iconsidered. A schematic of the process is shownFig. 4. The setup is a tank with one input and twoutputs. Cold water flows continuously into thtank, and the level of water in the tank is measurby a level transmitter~shown as LT in the figure!.The water is heated by means of steam passthrough a heating coil in the tank. The outlet temperature of the water is measured by a thermcouple in the outlet pipe from the tank, as showin the schematic diagram~shown as TT1 to TT4 inthe figure!. The flow of steam is kept as constanWater to the tank can be varied by changing tvalve on the inlet water line. The system is intefaced to a computer through a real time SimulinMatlab interface.

The sampling time is taken to be 5 s. The inpto the process is the water valve position; the twoutputs are the water temperature and level.state space model identified using persistentlycited input signal is given by

Fig. 4. Schematic of the pilot plant setup.

Table 1Results of sensor decalibration algorithm on simulated d

No change 10% Change 20% Change 30% Chan

0.0032 17.8857 63.1454 187.2420

e

ca--ta

ine

tateisde-steromgeatut

inep-gehe

thedsuts

at


xk11

5F 0.9103 20.038320.048620.0016

20.5879 0.431920.1669 0.1842

20.652320.2223 0.192220.0801

20.4360 0.036020.712520.0896

G xk

1F 0.054120.04280.55410.8699

G uk1wk ,

yk5F 0.9275 20.056820.035820.0212

20.056320.0105 0.0031 0.0010G xk

1vk , ~39!

wherevk andwk are white noise with covariancmatricesQ andR, respectively, given by

Q5F 0.0001 20.000620.0006 0.0016

20.0006 0.0085 0.009620.0126

20.0006 0.0096 0.013020.0151

0.0016 20.012620.0151 0.0252

G ,

R510233F 0.1058 20.0359

20.0359 0.1746G . ~40!

The input independent Kalman filter based delibration detection algorithm is run on two different sets of data, first over the identification dathat is persistently excited and then over a rout

Fig. 5. Plot of the outputs along with the estimates frominput independent Kalman filter. The solid line corresponto actual outputs and the dotted line the estimated outp

operation data, i.e., data around the steady swith few step changes. The cold water inputassumed to be unknown in both cases for thesign of the Kalman filter. A decalibration fault ivisualized as a change of the gain in the wavalve. The actual outputs and the estimates frthe identification data set for the case of no chanin the gain are given in Fig. 5. It is seen here tha good estimate of the outputs is obtained withothe knowledge of the inputs.

A 10%, 20%, and 30% change in the valve gais generated at the 200th sampling instance. A rresentative plot for actual outputs with no chanand outputs with 20% change in the gain at t

.

Fig. 6. Plot of the actual outputs~solid line! along with theoutputs with 20% gain change~dotted line! at the 200thsampling instance.

Fig. 7. Plot of the outputs with a 20% change in the gainthe 200th sampling instance~solid line! along with the es-timates from the input independent Kalman filter~dottedline!.

hehe

hforgeatlot

ithin-heare

re-ted

ofa

ngeal-ec-,o

na

in

ex-

reis

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ing

the


200th sampling instance is shown in Fig. 6. Testimate for the 20% gain change along with tactual outputs are shown in Fig. 7.

Similarly, the analysis is carried out for a batcof routine data. The output and estimate plotsthe case of no fault are given in Fig. 8. A chanof 10%, 20%, and 30% in the gain is generatedthe 200th sampling instance. A representative pfor actual outputs with no change and output w20% change in the gain at the 200th samplingstance is shown in Fig. 9. The estimate for t20% gain change along with the actual outputsshown in Fig. 10.

In each case the prediction residuals are pwhitened to ensure that the residuals being tes

Fig. 8. Plot of the outputs~solid line! along with the esti-mates from the input independent Kalman filter~dottedline! for routine data.

Fig. 9. Plot of the outputs~solid line! along with the out-puts with 20% change in the gain at the 200th samplinstance~dotted line!.

are white noise. For this purpose the first 300the prediction residuals are trained to estimatesecond order AR model and then all the remainiresiduals are filtered by this AR model to get thwhitened residuals. The primary residuals are cculated from these whitened residuals. The dettion algorithm is run over the data taking 10%20%, and 30% change in the gain over the twdata sets. Thex2 values for each case is given iTable 2. The threshold in this case is 3.84 with5% false alarm rate.

The algorithm is found to be sensitive to gachange~except for the case of10% gain changeusing routine operating data where the externalcitation is weak! while being independent of thechanges in inputs.

4. Conclusions

A statistical approach available in the literatufor fault detection known as the local approachdiscussed in detail in this paper. Work done in tliterature to increase its applicability is also e

Fig. 10. Plot of the outputs~solid line! with a 20% changein the gain at the 200th sampling instance along withestimates from the input independent Kalman filter~dottedline!.

Table 2Results of the gain decalibration monitoring.

Nochange

10%Change

20%Change

30%Change

Identificationdata

0.0042 14.0730 61.8706 152.8949

Routinedata

0.0023 0.8169 7.4957 30.3215

tothe

torutofarenp-the

thehisinuts

e

del-

iont.

nd

us-a

onu-

.,ionol

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ults:

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.,tos.


plained. This approach is found to be sensitiveabrupt process changes that may be small inmagnitude.

The paper has also discussed the gain moniing problem for a class of systems where the inpis not known. We have focused on the casesensor monitoring since the inputs to a sensorlikely unknown. A detection algorithm based oinput independent Kalman filtering and local aproach is proposed and found to be sensitive tochange in sensor~or process! gain and indepen-dent to changes in the input, as observed fromsimulation and pilot plant case study results. Talgorithm can be used to monitor the changeprocess gain in general for cases where the inpare not known.

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Detection of abrupt change and applications in sensor decalibration monitoring

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