Sensor fault detection and localization using decorrelation methods

Sensors and Actuators A, 25-27 (1991) 43-50

43

Sensor Fault Detection and Localization Using Decorrelation Methods* K KROSCHEL and A WERNZ lnslrtut fir Automalron and Robortk, Umvers~tat Karlsruhe, Fraunhofer-Instrtut fur Informatrons- und Datenverarbertung (IITB),

Karlsruhe (FR G)

Abstract

This paper presents a computer-based fault supervision method for the detection, locahzatlon and ldentlficatlon of ‘small’ sensor fallures hke biases or dnfts, independent of parameter vanatlons m the process to be controlled The algorithm 1sbased upon analytic redundancy methods and uses a mathematical model of the process to compute estimates of the measured quantities These are compared with the measurements, so that the resulting residuals contam the Interesting faults of the sensors, changes of process parameters and parts caused by an inaccurate model On the assumption that the latter have slmllar or correlated effects on all or many of the residuals, the model faults are suppressed by an adaptive decorrelatlon method Statist+ cal tests ehmmate bad data and gve an alarm if a faulty instrument 1s detected A recursive least-squares algonthm estimates the stze of the sensor fault and specifies it as a constant bias, time-variable dnft or scale-factor devlation Some results of applications are gven in this paper for the supervlslon of temperature measurements of a steam generator used m power stations 1. Introduction

Algorithms for the control and supervlslon of complex technical systems reqmre exact measurements of the system output A general problem m this field 1sthe detectlon, locahzatlon and separation of faults caused by the *Dedxated to Professor Dr K H Hardtl on the occasion of his 60th blrthday 0924-4247/91/$3 50

system itself and by faults of the sensors For example, time-variant changes m the process parameters have to be dlstmgmshed from sensor failures to maintain normal system operation In the hterature, this problem 1s often called FDI (failure detection and ldentlfication) or IFD (instrument fault detectlon)

111 Sensor faults can be divided mto so-called ‘hard fsulures’and ‘soft failures’ The first refer to suddenly occurrmg defects or total failures, while the latter are rather slowly changmg biases, dnfts or scale-factor devlatlons It 1s obvious that hard failures have to be Identified as promptly as possible to accommodate for the malfunction and to restore normal system operation Such failures can be easily detected by checking the lmut of the measuring range or by analysmg the trend of the readings, e g , using the fact that a temperature signal cannot Jump for physical reasons Therefore this paper focuses on the more difficult detectlon of soft failures If the sensor signals provide feedback mformatlon, these dnfts or biases may also lead to an unacceptable behavlour of the process to be controlled The whole IFD problem consists of the followmg three steps (1) Detection, gves alarm m case of a sensor failure (2) Isolation, Identifiesthe faulty sensor (3) Speclficatlon, estimates the size and type of the fault The estimation of the failure size and type allows the readings to be corrected by computmg the measured quantity from the (faulty) measurements and the estimated faults In ad&tlon, the failure can be classified as a constant bias, time-vanant dnft or scale-factor deviation 0 Elsewer Sequoia/Pnnted m The Netherlands

44

One traditional strategy to detect and NOlate sensor failuresand to separate them from system faults 1sto implement more than one sensor for each measured quantity, this 1s known as hardware redundancy Redundant outputs of equal measurements are then checked for consistency,e g , by usmg a maJonty-voting scheme For clear lsolatlon of the faulty instrument, at least three equal sensors are needed The disadvantagesof this approach are the cost and the additional space it implies A more soptistlcated approach 1s based on the fact that often the readings from different measurements are correlated wrth each other Thus a parameter or a state descnbmg the system Influencesnot only one output but more than one Therefore faults can be detected by comparmg the signals from dlssmular sensors, using the so-called analytic or functional redundancy, see, e g , [1] A prerequisite for such an algonthm 1s knowledge about the normal system behavlour m the form of a mathematicalmodel to compute on-line estimates 9 of the measured quantltles y, Fig 1 Slgmficantdevlatlons of the estimates j from the readings yr m the error signals e indicate faults of the sensors,the system or the model, that have to be detected and dlstmgulshedm the declslon process System faults are malfunctions of components or subsystems and time-vanant changes of parameters Model faults result from simplificationsto allow on-line computation to take place, non-hneantles not taken mto account or uncertain knowledge of the input u The problem of separating these faults 1s often called the robuslness of the IFD scheme Severalcontnbutlons m this field use a bank of robust or unknown input observers

Rg I Instrument fault detection system

[2], or Kalman filters, that are dnven by a subset of measurements and are designed to detect only specificfailures, e g , a malfunction of sensor I or component1 The observed system has to be at least partially observable from each subset, and the state estimators are designed usmg a hnear, time-mvanant state space model Model faults or changes m parameters are treated as unknown inputs to this linear model Another approach to this problem, called the decorrelatron method, IS presented m this paper It 1sassumed that the systemor model faults influencemore than one error signal e, so that these are correlated, while a sensor fault occurs only m one of them at the same time Then the systemor model faults may be predicted and suppressed by an adaptive decorrelatlon algonthm Smce the sensor faults are unpredictable, they should remam nearly unchanged m the decorrelated signals To the authors’ knowledge, this idea was ongmated by Appel [3], who proposed a method to detect and dlcnmmate hard fallures m the sensorsand the supervisedprocess by linear prediction of each sensor signal yf from the others and by checkmg the resultmg residual power Ha approach does not require any a pm knowledge about the process parameters nor a mathematical model the analytic redundancy between different measurements 1s estimated However, soft failures cannot be detected, because the algonthm adapts to such slowly occurrmg faults without changmg the residual power sigmficantly This idea has been extended to detect soft failures, too, by predicting the error signals resulting from comparison with a mathematlcal model, instead of predlctmg the sensor signals themselves The analytic redundancy between different measurements 1scomputed by the model and the redundancy between the error signals 1sestimated by the decorrelation algonthm The paper 1sorgamzed as follows Section 2 describesthe declslon process that detects, estimates and identifies the sensor faults First of all it 1sassumed that the error signals

45

contam only unknown faults plus noise, so that the declslon process IS designed lgnormg the system and model faults The decorrelation algorithm IS addressed m Section 3 and some results using real process data from a steam generator are gven m Section 4

Assummg a Gaussian probablhty density f,(n) urlth E[n(k)J = 0, the two hypotheses

H-I fcW

1a-11 = 1/t& x

4

exp( - e(k) 2/2a2) (no fault f)

(2a)

H+I fcW) 1H+I) = l/(fia,) x exp( -MN

2. DecisionProcess

(fault f) (2b)

The declaon process consists of three statlstlcal tests to detect a fault plus an estlmator to compute its size and type, Fig 2 A simple threshold test quickly detects hard failures If the error slgnal exceeds a predetermmed threshold T, e g , one half of the measurement range In such a case the other tests are not carned out To avoid false alarms because of sporadic pulse-type noise, a test for ‘bad data’ IS lmplemented Such bad data points may result from disturbances on the pnmary element or mterfermg voltage on the measurmg line On the assumption that the error srgnal e(k) contams the Interesting fault f(k) plus noise n(k), e(k) = f(k) + n(k)

-S2/2d)

(1)

and by estimating the mean C(k)= E[e(k)] and the vanance a:(k) = E[(e(k) -@k))‘], the test signal t(k) = le(k) - t?(k)1can be computed The mean and vanance are estimated recursively wth exponential welghtmg of past data to include slow vanatlons of Z(k) and o:(k) A bad data pomt IS detected when t(k) > au,(k), with the constant a, e g , a = 3 if n IS assumed to be Gaussian At these sample points the followmg recursively computed tests are not updated The tngger test detects slowly changmg soft failures and starts or resets the estimator

can be stated The log hkehhood ratio over the observation space {e,}, 1= k, IS k-J/+1, e(k -I) _j

LJH+,

3

(fault)

< \ H_, (no fault)

(3)

Thus IS tested agamst a threshold T to determme the more likely hypothesis For eqn (3) a white noise sequence (n, } IS assumed The (unknown!) fault f m eqn (3) may be replaced by its maxlmum hkehhood estimate, so that the test becomes a generalized hkehhood ratlo test (GLR) [4] Slmulatlons showed that this GLR IS too sensltlve, 1 e , Dves false alarms, if the n, are not mdependent and Gaussian A better way IS to state a mnumum fault f.that should be detected Smce the sign of the fault IS not known a prtor~, two tests L+(k) and L-(k) have to be camed out, one for + PO1and another for -If01 Another problem m computmg eqn (3) 1s the choice of N and T It can be avoided by processmg L(k) recursively usmg the so-called sequential probahhty ratio test SPRT [ 51 The two log hkehhood ratios L+(k+l)=L+(k)+y(e(k+I)-F)

Fig 2 Decomposltlon of the decwon process

>A reset, Ht 1 (pos fault) else H,+ (7, N:=N + 1)
and L-(k+l)=L-(k)-!$(e(ktl)+$J)

c> A reset, H;

1

H,


(neg fault) (7, N

=N t 1)

(no fault) (4b)

are checked against two thresholds A and B that can be determmed from the false and missed alarm probablhtles PF, P,,, If neither A nor B 1s reached, no decision 1smade (HO) and the number N of samples 1sIncreased In the other cases, H_, or H+, 1s accepted for the two mmlmal faults + lfolor - lfoland the sum (integral) L + or L - is reset to zero, respectively For the estimation of a bias-type soft fallure the error signal 1sconsldered to be e,(k) = bot bl t,(k) t n(k)

W

with the constant bias bO, the dnft term &t,(k) and the falure time t& as the point when the trigger test starts the estimator A scale factor deviation sf ISrepresented by e,fW

=

sfi4 + 44

(5b)

with the output j from the model, Fig 1 Two recursive least-squares algonthms es& mate bO,bl and sf by mmmuzmg the exponentlally weighted mean-squared errors E,(k) = i

(e(z) -

i$(~))~iZ~-

(64

(e(z) -

c?sf(z))2~“-’

(W

with q sensor failures1; and noise n, Each e, ISpredicted or decorrelated from one or more other referencesignals e, with I #I A sensor falluref, m e, 1s(on the assumption that these faults do not occur at the same time) not contained m the other error signals and thus cannot be predicted However, model or system faults will mfluence all or many of the components of e and may be suppressed by decorrelatlon For a bnef development of this idea, consider only one reference slgnal el for the predlctlon of e2 In the general case mutual pairs e, and e, of the error signals have to be compared Figure 3 shows a model for the generation and decorrelatlon of e, and e2, eqn (7) It can be consldered as a slmphficabon of Ag 1 mth the process model replaced by an Ideal, 1e , exact model plus system or model faults Am The shapmg filters G,(z), with white noise input w, and G2(z) generate AmI and Am,, respectively For a clear separation of a faulty sensor, it has to be ensured that fi and fi are dlstmgmshable, because unlike the declslon process, e, and e, now are computed together Furthermore, the algorithm should suppress only model faults but not sensor failures It can be shown [6] that these two reqmrements can be fulfilledby using an AR model for G,(z) and a MA model for G2(z),both of order p

1=0

E,,(k) = i 1=0

with the decline factor I < 1 The failure type 1s determined by comparmg the two estlmatlon errors If E,(k) < E,(k), the formulation (Sa) 1s more likely than (5b) so that a biastype fault 1sassumed, and vice versa

G,(z) e1(4 = i %el (k - m) + w(k) (AR) m=l @a)

G(z) e2M= i kel@ - 4 m=l

3. DecorrelationMethod To suppress system or model faults Am, suppose the error signals e, are given by

Ag 3 Model for the error signals e

(MA) WI

41

The prediction of e,(k) from e, (k - m) with 1 < m

(9)

On the condltlon that sensor failures and system or model faults are uncorrelated, the residual g2(k) equals f*(k) and thus can be used as an estunated value for falures m sensor 2 The prediction algorithm includes the estimation of G,(z), G*(z)and adapts on falluresf, as well as on model faults that are described by eqns (8) [6] Observing eqns (8) It 1s clear that the algonthm adapts onfi but not on j& because e, IS whitened (decorrelated from its own past values) whde e2 1s decorrelated only from past values of el To detect failures m sensor 1, el (k) 1s predicted from e,(k - m) Instead of the panwise decorrelation of two error signals, multi-dimensional filters, based on vector AR and MA models, can be designed to make use of the mformatlon about the model faults contamed m more than one reference signal The propeties of the described one-dlmenslonal decorrelatlon method are maintained and a better suppression of model faults can be expected, but the computational costs increase with the number of reference signals The decorrelatlon filters are implemented as (m&l-dimensional) Jomt lattice filters [ 71 In comparison to the direct lmplementatlon by mmlmlzmg the mean-squared predlctlon errors and solving the normal equations, no numencally cntical matnx-mveraons are reqmred 4. Results The decision process IS tested wth slmulated error signals by adding appropnate faults to whte Gausslan noise with mean p = 0 and vanance u* = 10 For four dlfferent ‘sensor failures’, eqns (5) are generated, see Table 1 The parameters of the statistical tests, described m Section 2, are set to a = 3, fo=l, A = -B=46 (P,,,=Pf=OO1) and 1 = 0 995 In Table 1, the difference &,,(estl-

mated) -tfi, (simulated) 1s the detection time required for the tngger test, and the estimated bias and dnft values equal the sum & + 6, (t - 6,) and F,, respectively The hypothesis H indicates the estnnated failure type 1 refers to a blas/dnft and 2 refers to a scale-factor deviation sf It can be seen that m all cases the fadure size 1s estlmated quite well and the type 1s determmed correctly Figure 4(a) shows the simulated and estlmated bias fault No 4 and Fig 4(b) the hypothesis H for fault No 3 Two bad data points are detected and the nght declslon about the failure type 1s made about 20 mm after its detection Further slmulatlons showed that the correct dlvlslon mto bias- or sf-typefaults depends mainly on the dynamICSof the measured process When there are no dynamics, the two compared meansquared errors (eqns (6)) do not differ slgmficantly, which becomes clear when looking at eqns (5) To test the proposed decorrelation method, real temperature measurement data from an mdustnal fossil-heated steam generator were recorded at different operating points The sampling time was 10 s Figure 5(a, b) shows the resultmg error signals for four temperature measurements after dlfferent heat exchangers The non-linear process model used 1sof the order n = 150 and works with a fixed parameter set for the different operating points [8] It can be seen that the error signals are correlated due to dynamic model faults, Fig 5(a), as well as due to stationary model faults, Fig 5(b) The peak m the latter plot results from turning off one coal rmll Multi-dimensional decorrelation filters of order p = 5 with three reference signals generate the residuals gven m eqn (9), see Fig 6(a, b) The model faults are slgmficantly suppressed m both cases Sensor failures are then simulated by adding a bias to one of the error signals, Fig 7(a, b) They are estimated qmte well m Fig 7(a) with dynanuc model faults only, while m Rg 7(b) the filter adapts to them, too The stationary model faults Hrlth

TABLE

1 Simulated and estunated faults Estimated fault

Error sIgna

Simulated fault

No

tfio Wn)

bo

b, MM

sf

[fin (ml@

bias

dnft On@

sf

H

1

83 66 50 42

20 00 00 20

00 006 00 003

00 00 005 00

92 91 53 47

21 60 52 59

00 0 056 00 0 029

002 0042 0051 0043

1 1 2 1

2 3 4

8-

47

6.

3-

,

1

2-

4. :: d

x 2-

l.

O-

0

-1 .

-21 0

40

80 Ud.

120

I 160

(a)

-2’ 0

40

80 ffmn

120

160

(b)

Fig 4 (a) Simulated ( ) and estimated bias No 4 (b) Hypothesesfor fault No 3 - 1, no fault, 0, no dewon, 1, bias, 2, sf 3, bad data

-30’ -20

1

0

20

40

60

80

100 120

-30’ -20

&la (a)

0

20

40 60 bin

80

100

120

@I

Fig 5 (a) Error signals wth dynamic model faults (b) Error signals wth stationary model faults

mean #O are correlated with the sensor falures, so that the decorrelatlon method does not work here, since the latter cannot be detected The problem arises from the fixed parameter set of the process model It 1s

proposed to track the parameters by using an on-lme ldentlfication algonthm [9] and more a pnorr knowledge about the measured process It 1snow exammed whether the statlonary accuracy of the process model can

49

c

n

-10

\

-10. -20

-20 .

I -3OL -20

’ 0

20

40

60

80

!

100

-?20

l20

0

20

40

60

80

!

100

120

Gill

Gi9

(a) (b) Fig 6 Error signals before (dotted curve) and after decorrelatlon (a) Dynamic model faults only, (b) with stationary model faults

t 3

15

IS-

10

10-

5

s-

.-

t 0

1

-5 -10 -20

o-5

0

20

40 60 umia

80

100

120

(a)

-10 -20

0

20

40 60 11,.

80

100

120

@I

Fig 7 Simulated (dotted curve) and estimated bias (a) Dynamic model faults only, (b) wth stationary model faults

be maintained by these methods without reducing the ablhty to identify the sensor fallures of interest

5. Conclusions A new approach for the detection, locahzatlon and ldentlficatlon of ‘small’ sensor fallures using analytic redundancy methods has been mvestlgated The method includes the estlmatlon of the srze and type of the malfunctlon A mathematical process model IS required to check different measurements for consistency Unavoidable model faults are suppressed by decorrelatlon The advantage of the proposed method IS the use of one non-linear model of high order Instead of

many hneamed or otherwise simplified models The test with real data from a steam generator has shown that the method works very well if dynamic model faults are considered, but has to be combmed with an ldentificatlon algorithm to ensure stationary accuracy Acknowledgement The authors wish to acknowledge Slemens AG for financing the project and for the contnbutlon of the measurement data References 1 R Patton, P M Frank and R Clark (eds), Fault Dlagnosls VI Dynamic Systems Theory and Apphcatlon, Prentice-Hall International, Englewood Clfis,

NJ, 1989, pp 2f, pp 21f

50 2 P M Frank and J Wunnenberg, Robust fault dlag-

noses usmg unknown input observer schemes, m R Patton, P M Frank and R Clark (eds), Fault Dlagnosrs m Dynamzc Systems

Theory and Appbca-

Iron, Prentice-Hall International, Englewood Chffs,

NJ, 1989, pp 47-98 3 W Ptacek and U Appel, Mehrkanahge KlemsteQuadrate-Schatzung autoregresslver Parameter zur Sensorausfallerkennung, Proc 6 Aachener Symp fur Signaitheorre ASST, Sept 1987, pp 88-92 4 U Appel and A v Brandt, Adaptive sequential segmentation of pIecewIse stationary time series, Inf Scr , 29 (1983) 27-56 5 E G Gay and R E Curry, A model of the human

observer m fadure detection tasks, IEEE Tram Syst Man Cybern , MC-6

(1976) 85-94

6 A

Wemz, Dokumentatton

analytcsche Sensoruberwachung,

des

IAR-Tedprojektes

Instltut fur Automabon und Robohk, Umversltat Karlsruhe, F R G , 1990 7 P Strobach, Schnelle adaptive Algonthmen zur ordnungsrekurslven Kbnste-Quadrate-Schatzung autoregresslver Parameter, Dr Jng Dlssertatlon, Umversltat der Bundeswehr, Neublberg, F R G , 1985 8 W Thomann, Em mchthneares mathematlsches Model1 hoher Genamgkett fur emen Dampferzeuger, Dr -1ng hsertatzon, Umversltat Karlsruhe, F R G , 1990 9 M Lang, Identlfikatlon physlkahscher Parameter am Dampferzeuger, Dr -1ng Dlssertatlon, Umversltat Karlsruhe, F R G , 1990