llnearitv of model-free fault diagnosis in ...llnearitv of model-free fault diagnosis in...
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LlNEARITV OF MODEL-FREE FAULT DIAGNOSIS IN STEADV-STATEThomasw. Rauber*, Luís Gomes** and Adolfo S. Steiger Garção***Departamento de Informática - Universidade Federal do Espírito Santo
Av. F. Ferrari s/n, 29065-900 Vitória - ES, [email protected]
**Universidade Nova de Lisboa - Faculdade de Ciências e Tecnologia - Departamento de Electrotécnica2825 Monte de Caparica - Portugal - {lugo,asg}@uninova.pt
Resumo: Mostra-se nesse texto que a detecção e diagnóstico defalhas em ausência de um modelo analítico do processo érealizável exclusivamente por técnicas lineares dereconhecimento de padrões. Esse fato normalmente não seconsidera na literatura onde em geral se usam métodos nãolineares de classificação, por exemplo o perceptron multi-camada da área das redes neurais artificiais.
Palavras Chaves: Diagnóstico de falhas, Controle,Reconhecimento de Padrões.
Abstract: It is shown in this paper that model -free faultdetection and diagnosis for steady states in a controlled processcan be realized exc1usively by linear pattem recognitiontechniques. This fact is normally not considered in the literaturewhere nonlinear c1assifiers are usually employed for this task,e.g. multilayer perceptrons from the field of artificial neuralnetworks.
Keywords: Fault detection, control, pattem recognition
1 INTRODUCTIONModel-based fault diagnosis of a process assumes a sufficientlyreliable mathematical model of the dynarnic and static behaviorof the process, see e.g . Isermann (1993). The faults are definedwithin this model as non-permitted deviations of the processparameters or process states from their nominal values. Thedrawback of this approach is the limited class of processeswhich allow to derive the process model.
Model-free fault diagnosis of a process is based on training datawhich is used to generalize about the states of the processo Theadvantage of this approach is the absence of a mathematicalmodel. The knowledge about the process has to be generalizedfrom an appropriate training set. The drawback of this approachis the usually limited training set, i.e . it can happen that duringprocess operation, faults occur for which there never had beenany training data . Hence the classifier which diagnoses thefaults has not been trained for those situations and one canexpect unreliable results. Application of model-free faultdiagnosis mainly based onneural nets are e.g . Sorsa and Koivo(1993), Zhang and Morris (1994), Zhang et alii (1996) or Sidhuet alii (1998).
For the normal situation and for the fault classes data isaccumulated which serves as the basis to train a classifier. If the
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process is controlled then process states are mainly steady-state,both the normal situations and the faults. Usually nonlinearc1assifier models are used to realize the fault detection. Fromthe area of artificial neural nets the perceptron with one hiddenlayer is the most popular architecture encountered. oEven if thec1assification is only made in steady-states, generally nonlinearc1assifiers are proposed. This approach was e.g. pursued inSorsa et alii (1991) , Venkatasubrarnanian et alii (1990) orKavuri and Venkatasubramanian (1994).
This paper suggest that linear classifiers are absolutelysufficient for fault detection and diagnosis if only steady statesare considered in the context of controlled processes. This factis due to the linear relationship among the different classescaused by the multidimensional constant offset from oneprocess situation to the other. This is true even for highlynonlinear relationships among the involved variables o (inputvariables, state variables, output variables, physicalparameters). First the theoretical concepts about linearclassification are outlined. An illustrative example is given forfault detection of a system consisting of a pump, a pipeline anda valve . In this system the material flow is controlled and keptconstant. A more complex example is a simulated chemicalprocess, consisting of a continuously stirred tank reactor, a heatexchanger and a pump/pipe system. Also in this case it is shownthat linear pattem recognition is sufficient for fault diagnosis.
2 CONTROLLED PROCESSES INSTEADVSTATE
Using the terminology o of (1993) we consider adynarnic nonlinear model of a controlled processo In steadystate we can set the time derivative of the state vector as the nullvector. This means that the involved input variables, outputvariables, state variables or any other non-constant quantitieshave an algebraic relationship among themselves. Thisrelationship is generally nonlinear. (It is assumed that periodicsignals, for instance rotational positions and angular velocitieshave appropriately been transformed to a static representation,e.g . from the time domain into the frequency domain).
The following extremely simple process c.f, Isermann (1984)serves as an iIlustrative example to explain the main idea of thepaper. Consider a circular pipeline with a pump and a valve Iikein Figure 1. For the sake of simplicity no controller, input or
outlet pipes are shown. The objective of the closed loop controlis to keep the flow rate of the material M in the pipelineconstant. In order to reach that goal (within certain limits) thevalve can be opened or closed. We have five quantities involved,the pressure-difference of the pump àp , the sectional area of thevalve A, the flow rate M of the liquid, the constant Cv of thevalve and the constant cR of the material. In steady state M isconstant and we have the relationship
(1)
Steady state a is expressed in a finite nurnber n of non-dynamic(algebraic) relationships among the involved variables x aj as .
= O, k = 1, ... , n . To reach another steady state b wehave to add the offset to the state vector, i.e,
= + =O, k = 1, ... •n.
It should be mentioned that the normal operation states of theprocess for different values of the involved variables are alIconsidered as different classes. The final decision if the processis in normal state can then be made on a higher leveI but it isavoided that the normal class stretches along a line in patternspace or hasa multimodal distribution.
It is assumed that only two of the five quantities can vary, àpand A, since M is kept constant by a controller and Cv and <«
constants a priori.
A (O.OOlm'l654
16
PumpAp
. 2--Pipe - ..- · -1
I
Normal and Faulty States
Figure I - A system with a pipe, pump and valve
3lfwe consider the relation between A and t1p in (1) we have a2-Dspace which characterizes the current steady-state of theprocess by the two variables A and Sp . For a constant(controlled) value of M the relationship between A and ãp isnonlinear, namely t1p cc A-2. In model-based approaches wecould consider this 2-D space as the state space. In theterminology of pattern recognition this 2-D space is the featurespace. For the noise-free analytical case, a normal behavior ofthe process in steady-state is a point (t1Pn' An) in this 2-Dspace, a fault another point (t1Pf.Af) on the image of therelation. A fault for instance is a decreasing pump pressuredifference t1pf which has to be compensated for by the openingof the valve, increasing the sectional area of the valve to Af ' (Innoisy pattern classification these two points are the expectedvalues (centers) of member patterns belonging to the respectiveclass.) Although the two points lie on a nonlinear curve definedby the nonlinear relationship (1) between t1p and A, there is aconstant hence linear offset (d Sp, dA) which transforms thenormal class into the faulty class as follows:(t1Pn' An) + (dt1p . dA) = (t1pf' Af) . In Figure 2 these facts areoutlined.
Similarly one could imagine other situations, i.e, other valuepairs (t1p. A) , each with a different meaning (different faults ordifferent normal situations) . Important however is the followingfact:
Although the relationship among the involved variablesis in general nonlinear there exists a constant hence
linear offset to change the state of the process from onesteady state into the other, regarding the vector space of
alI involved varíables.
In order to use the terminology of pattern recognition we willmerge alI involved variables xj of the process into amultidimensional vector -!.
Even if the number of variables is much higher and even if thenonlinear relationships among them are extremely complex,there is always a constant offset from one state into the other.
Figure 2 - Relationship between the two variables A and àpin steady state. The following values were used in (1):
-4 3 2Cv = 6.7x 10 (m /kg) , CR = 5Pa/(kgls) ,
M = lOkgls . A may vary between the two extremes2 2Amin = 0.002m and Amax = 0.008111 . Two different
process situations are shown.
The consequence of the linear transition from one class to theother in the variable (=feature) space is that linear classificationis sufficient to separate the two classes, or in general amongnormal and different fault states , see Figure 3.
In higher dimensional feature spaces the class separators arehyperplanes. Figure 3 shows the noise-free case. In a real-worlddiagnostic task the states would be represented by a scatter ofdata points around their expected value. This is due to theintrinsic noisy nature of sensorial data acquisition and externaland internal disturbances of the processo
t1p = X2
\ Non linear rclatiooshlpV\. "Normal·2"... X',\/ -0,
"Fault-2"
..--_.! __o ••Figure 3 - Linear separation of the process classes, eachclass representing a steady state. As an example the 2-Dfeature space of the pipeline example is used. First featureXl = A , second feature x2 = Sp . The definition of whichstate is normal or faulty is arbitrary.
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4 KNOWLEDGEACQUISITION 6.1 Steady-State Transition of PatternsDifferent steady states wilI be considered which represent theprocess in one of the normal or faulty situations. The situationslisted in Table I were implemented in the simulation of theprocesso
For instance the fault SI "Input pipe partially blocked" wasimplemented as a reduction of the input flow from 2.5kgIsto2kgl s , After one or more faults have been activated. thecontrollers of the process take care of the disturbances and leadthe process into another steady state, however with different(constant) values of the involved variables. Figure 4 shows theevolution of the 13 variables of the process that define its state.They are the features .f = (XI' . .. . X13/ in the tenninology ofpattern recognition. One can observe that after an event occurs,i.e. a fault or the modification of the setpoint of the normaloperation, the variables are stabilized to a new steady statesituation by the controIlers.
Let us assume that we want to classify the process situationsthat appear in Figure 4. We have to acquire training data for 4different classes, So. S6' S4' and (S4 and S8)' Since the faultdetection is assumed to be performed onIy in steady state wehave to coIlect the example data after the stabilization of thevariables. The values of Table 1 were taken immediately beforea change of the state and should represent the noiseless case inwhich each class can be modeled by a single data vector. Thevectors are the representatives of the process classes (states) inthe sense ofFigure 3.
In model-free fault diagnosis all knowledge must be derivedfrom training data . This data is composed be a finite number oftraining patterns T = {(.fp' C1>p)}; = I , each pair consisting ofthe feature vector representing a steady state and the class labelrepresenting a process situation. When nonlinear classifiers arepresented as a solution to the fauIt diagnosis problem they areusually trained by iterative, non-deterministic methods, forinstance the error backpropagation algorithm if the multilayerperceptron is the classifier. The essence about the training of theclassifier from the previous ideas is that because of the linearseparability of the classes an iterative, usual ly time-consumingtraining phase is not necessary.
As a linear classifier one could e.g. employ the ADALINE(Widrow-Hoff, 1960) together with a deterministic trainingmethod using the pseudoinverse, c.f. Annex 2. This reduces theknowledge acquisition phase of the fault diagnosis system to anextremely fast step.
5 PRINCIPALCOMPONENT ANALYSISThe intrinsic linear relationship of variables in steady states ofcontroIled processes also explain the popularity of PrincipalComponent Analysis (PCA) (JoIliffe, 1986) as a linear tool toeliminate redundant information in model-free fault diagnosis.Since there is a linear offset frorn state to state in the variablespace and since furthermore not alI variables are usualIyaffected by a state change, a highly linearly correlatedredundancy be expected among the variables. ConsequentlyPCA achieves a high compression rate of the original variablespace by projecting the variables onto their eigenaxes thuseliminating linear correlation. The use of PCA can for instancebe found in Zhang et alii (1996) or Dunia and Qin (1998).EmpiricaIly it can be observed that only a few non-zeroeigenvariables remain from the original set of variables after thePCA analysis. In future work it wilI be tried to discover theformal mathematical relationships between the originalvariables and the principal components in a steady state model-free fault diagnosis.
6 EXPERIMENTAL RESULTS
Table 1 - Process statesState
NormalInput pipe partiaIly blockedInput concentration of A highRecycle flow set point highFouled heat exchangerTemperature control valve stuck highLeak flow in reactorRecycle flowmeter stuck highMalfunction in pump
Syrnbol
Figure 4 - A typical scenario for the control of the reactorprocesso Starting from a normal situation SI' the first faultS6 occurs after SOOs, then retuming to SI again . At 3S00sfauIt S4appears , then at SOOOs fault S8. causing a situationof multiple faults . Finally at 6000s retum to normality SI-
000sec]
I f'.--
I 3 4 7
Ú" NDTm4' "- l[IN.mw>l FotlkáMD' dif""
ur1tm&&U NomtlJl
VariabJe (=fealUre)
a2acBcAFPFWTMLMTRFRcÀOTOFO
A more complex example to ilIustrate the linearity properties ofsteady state fault diagnosis.is a process simuIation that has beenused extens ively in control and fault deteetion and diagnosis(Oyeleye and Kramer, 1988), (Sorsa et alii, 1991), (Sorsa andKoivo, 1993), (Zhang and Morris, 1994), (Zhang et alii, 1996).A similar process is described in Venkatasubramanian et alii(1990). The continuously stirred tank reactor is the object ofstudy that shows typical qualities of an industrial processoDetails can be found in Annex 1. In Rauber and Munaro (1997)the process was also used with an emphasis on general purposepattem recognition applied to fauIt diagnosis. Different fauIt. situations were simulated in the process and the behavior of thevariables in steady and transient states were observed. It wasobserved that in the steady state linear pattern recognitiontechniques, Iike Linear Discrirninant Analysis and PCA wereabsolutely sufficient to classify the faults , thus motivating themain ideas of this paper. In the foIlowing experimental resultswill be presented which empirically show the validity of theconcepts outlined here.
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Figure 5 - Sammon plot of the 13-D process statesvectors.
. Table 2 - Process states as classes and their representativevariable vector (feature vector)
Since there exist more than two classes to be separated we mustuse a piecewise linear classifier, for instance a Linear Machine(Duda and Hart, 1973), implemented by an ADALINE (Widrowand Hoff, 1960) andoshown as the dashed lines in Figure 5which represent the projected piecewise hyperplanes of theoriginal 13-D space, Nevertheless the classification is stiUlinear.
OOOsec]
I r--
r--
,II
1:0-"1 2 t 4
j,,,,,.,plp< 1[/Foldtdlt tal
i u chanstr lu.> I blocUd Normal
<H> <H>' <H>' <H>'
Figure 6 - Multiple faults
The Sammon plot of the 200 training samples can be observedin Figure 7. Obviously the noise in the training data has causeda dispersion of the data points around an expected value. In thecase of the two classes So and S4 the dispersion has caused anoverlapping of the two classes. Nevertheless a linear separationis by no means worse to distinguish the classes since anonlinear classifier cannot improve the classification result.This affirmation is based on the fact that the covariance matricesof the classes have zero elements outside the main diagonal duetothe uncorrelated Gaussian noise and therefore the decisionboundaries are linear (Duda and Hart, 1973) .
This step seems necessary due to the extremely different scalesof the involved variables in order to democratize the influenceof each variable.
Another possible fault scenario can be observed in Figure 6. Atl0000s the fault event "Fouled heat exchanger" S4was caUed,lowering the surface area of the heat exchanger from 40m2 to5m2
• at 30000s the fault "Leak" S6' introducing a leak flow oflkgls, at 50000s the "fault "Input pipe partially blocked" SIwhich lowers the input flow rate Fo from 2.5kg/s to 2kg/s .The sampling intervals were defined each as l00s before a newevent occurred with a resolution of 2s resulting in 50 samplesfor each of the four classes. Observe that the first class is So, thesecond S4' the third S4and S6. and finally the fourth S4 and S6and S( thus partiaUy representing multiple faults .
Varíable (=feature)
a2alcBcAFPFWTMLMTRFRcAOTOFO
Sompling Intervals
Class
42
Steady state vector
/I.>:
Píecewíse linear separator íII,
/
! ...... . .., ............ A .// .
..........;.-..I )C \
- lO ·8 -6 -4 -2 OMappeddimcnsion111
Normal •Fouled heat exchanger •
Fouled heat exchanger + Malfunctioning in pump D
Malfuncliollillgin pllf>lP x
SAMMONPLOTofprocess classes: 13·D mapped to 2-D
6c
4E:a 21o. O
-2
(2.5 30 12007 37.53 3 58 3.33 2.5 20.16118035.48 0.15W(2.5301200 7 37.803 58 3.291.520.16 1I80 35.84 0.087)T(2.5301200 7 37.62 3 58 3.88 2.5 20.16118035.49 0.I49)T(2.5 30 1200 7 37.77 3 58 3.87 2.5 20.16118051.83 O.I40l
Although the dimension of the variable space has increased to13, their is stiU the same relationship among the classes as inthe 2-D space of Figure 3, namely there is a constant offsetwhich transforms one class into the other. The Sammon plot of "the 13-dimensional state vector is a mapping which conservesthe reiative .Euclidean distances among the data vectors(Sammon, 1969), see Figure 5 for the mapping of the fourvectors of Table 1. It can be neatly seen that there is a linearseparability among the classes .
Table 3 - Estimated error using Leave-One-Out errorestímatíon with different nonlinear classifiers compared to a
Linear Machine implemented by an ADALINE.
An empírical comparison of the performanceof differentnonlinear classifiers to an ADALINE "trained" by adeterministic pseudoinverse corroborates this. For this purposean error estimation using a Leave-One-Out error countingmethod (Devijver and Kittler, i982) was used to determine theerror rate of the different classifiers. In Table 3 one can observethat nonlinear classification techniques show no qualitativeimprovement over the Linear Machine. (The slight advantage of.the Multilayer Perceptron can be explained by overfitting tonoise.
6.2 Simulation of Noisy Acquisition ofSensor Data
The foregoing experiment helped to explain the situation in thevariable space in the case where no. external or internaldisturbances influence the values of the 13 involved variables.In reality we must reckon with an addition of random noise thatis due to the imperfections of the acquisition process andnatural stochastic phenomena of the process environment.Besides classification techniques in most cases are based on theassumption that noise is contained in the measured data.Artificial neural networks and more traditional patternclassification techniques are based on the stochastic nature ofthe processo
For the following experiment white Gaussian noise with astandard deviation of (J = 0.05 was added to the variables toachieve the stochastic behavior of the measured values.Additionally the values of alI variables were previously scaledto the range near unity by dividing them by their default values.
Linear MachineQuadratic Gaussian
Multilayer Perceptron
5.306.378.334.90
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. SAMMONPLOTof process trainingdata: 13-D mappedto 2-D
-0.8 -0.6 -0.4 ·0.2 O 0.2 0.4 0.6Mapped dimension #I
were made in order to complete or modify process values andequations supposedly not appropriate for the simulation:TR=36°C, Tou/=57.9°C, 0,=35.5 ,Amax =0.001, bracketing (RTM) always, settingF =Fp+FR,LM = (lI(pAM»(Fo-Fp-Fleak) , using Fw directly as the
controlled variable of controller CT and substituting the PI-controllers by a PID-controller for CT with TD = 200s. Theother parameters for the controllers are (using the terrninologyof Isermann (1989»: K = -1 for all controllers, TI = lOs forCL and TI = 15s for CT and CR •
Figure 8 - The continuously stirred tank reactor
-0.4
Normal o
Fouled heat exchanger •Fouled heat exchanger + Leak o
Fouled hea; exchanger + Leak+ Input pip e partial/y blocked •
0.4c
0.2
O
::;: ' -0.2
It has to be mentioned that the observations of the similarclassification performance have been made in this particularcase in which white Gaussian noise was added to the simulatedprocess variables. Further studies especially in the context of areal world process could clarify the facts stated so faro
Annex 2: Classification by Linear Machinesand Training by OeterministicPseudoinverse Method
The sirnulator was coded in C, the soIution of the ordinarydifferential equations was done by a fourth-order Runge-Kuttamethod and the nonlinear equations for the flow parameters FpandFR were soIved by bisection (Press et alii, 1988).
with dimensions (nx(D+1»«D+1)Xc) = (nxc) . Pre-multiplying (3) with XT yields '(,TXW = XTT. Pre-rnultiplyingagain by the inverse of XTX which for this overdeterrnined
T -1 T T -I Tsystem usually exists gives (X X) X xw = (X; X) X T orequivalently W = X*T where x- = (XTxf XT is thepseudoinverse of X.
There are c classes roi' i = 1, ... , c, each meaning a processstate . Each class roi is represented by a linear decision functiondi (:f ) = !!!;:f with the class-specific weight vector !i' i as its freeparameter. The augmented (i.e. always 1 at position O) (D + 1) -dimensional vector :f is the actual state of the processoA LinearMachine is implemented by the following decision rule:
d(:f) = roi' if di (:f) >d/:f) , for i. j = l...c , i* j (2)
This role divides the D -dimensional state space into convexpiecewise linear decision regions .
AlI training samples T = H :fp ' ropn; =, consist of the featurevector x, and the class labeI rop ' pe {l, .. .,c}. The classmembership of a training sample is coded in the c-dimensionaltarget vector l = (tI' .. ., tc{ , using the l -out-of-c coding, i.e.:f e roi t =.(0, O, .... O. 1, O. ... , O, O{ with 1 at the r-thposition such that !!
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