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A statistical procedure based on wavelets for faultdetection applied on the three tank systemAbdelmalek Kouadri a & Mimoun Zelmat aa Applied Control Laboratory, University of Boumerdes , Av.del’independance,Boumerdes , 35000 , AlgeriaPublished online: 14 Jun 2013.

To cite this article: Abdelmalek Kouadri & Mimoun Zelmat (2010) A statistical procedure based on wavelets for faultdetection applied on the three tank system, Journal of Statistics and Management Systems, 13:5, 949-960, DOI:10.1080/09720510.2010.10701513

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A statistical procedure based on wavelets for fault detection applied onthe three tank system2

Abdelmalek Kouadri ∗

Mimoun Zelmat4

Applied Control LaboratoryAv.del’independance6

University of Boumerdes35000, Boumerdes8

Algeria

Abstract10

In this paper, the use of discrete wavelet transform (DWT) and a statistical techniqueanalysis for fault detection are presented. The detection procedure is based on a statistical12

analysis of the components of approximation and details from the measurement datacollected in healthy state. Thus, the statistical characteristics obtained are then used to14

formulate an appropriate index of fault detection. The fault detection index sensitivity isevaluated in relation to confidential intervals established in healthy mode. This strategy is16

validated experimentally on a system of three reservoirs type DTS-200.

Keywords and phrases : Fault detection, discrete wavelet transform, statistical analysis, standard18

deviation, confidential interval.

1. Introduction20

In recent years, several studies have focused on the problem of faultdetection and diagnosis [1, 2, 3]. In its fine description of the monitoring22

system [4], Isermann gave special attention to the problem of faultdetection which is a fundamental step in a monitoring procedure. More24

precisely, it is to emphasize any-anomalies that may occur during theprocess operating to allow workers to take or assist in taking appropriate26

corrective actions. Monitoring approaches based on model and withoutmodel have been established and applied in various fields. In the area28

∗E-mail: a kouadri@hotmail.com

——————————–Journal of Statistics & Management SystemsVol. 13 (2010), No. 5, pp. 949–960c© Taru Publications

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950 A. KOUADRI AND M. ZELMAT

of signal processing, various approaches depending on the time andfrequency analysis have been developed. Hold up as examples spectral2

analysis by the Hilbert and Fourier transforms, the analysis by filteringor again the cepstral analysis. In the context of the analysis by short term4

Fourier transform, an alternative based on the use of sliding windows wasconsidered [5].6

The disadvantage of these techniques is to achieve the compromisebetween the frequency and temporal resolutions. Indeed, a good time8

resolution requires a short period of observation, while a good frequencyresolution is achieved not only through a fairly long observation.10

Given these conclusions and in order to allow a diagnosis in timeand on the whole spectral frequency signal analysis, techniques based12

on wavelet transform [6, 7] have been studied and applied in manydifferent fields. In [8], the fault detection and isolation based on the discrete14

wavelet transform (DWT) was presented and implemented on a systemof ball bearings. The faults in such a system are detected by analysing16

the statistical properties of the DWT coefficients. In addition, Yang etal. presented in [9] a study on the implementation of the DWT to the18

protection of electrical distribution lines, based on an approach that helpsto show up the faults caused by high impedance. The method is based20

on the determination of tolerance called “confidential intervals” for theDWT coefficients to decide on the presence or absence of anomalies in the22

electrical distribution networks.The joint consideration of the concept of confidential intervals and24

statistical properties of DWT coefficients of a problem in fault detection isin fact the main purpose of this contribution. Indeed, this article presents26

a formal fault detection based on an appropriate statistical analysis ofthe discrete wavelet transform coefficients. More specifically, the fault28

detection algorithm uses the repeated measures which aim to extractthe statistical characteristics of the analysed signal in order to define the30

best random phenomena associated with it. The proposed algorithm isvalidated on a benchmark three tanks type DTS-200 for the detection of32

faults abrupt sensors and actuators.

2. Wavelet transform34

The wavelet transform decomposes the signal on the basis of anal-ysed functions construct from a function called “mother wavelet” by36

dilation and translation. However, the most used in practice is to filter

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WAVELETS FOR FAULT DETECTION 951

a signal in cascade using low and high pass complementary filters. Thecontinuous wavelet transform of a signal f (t) ∈ L2(<) is given by [10]:2

W f (a, b) =1√a

∫ +∞

−∞f (t)ψ

(t − b

a

)dt , (1)

where ψ is a function called wavelet mother subdued to the particular4

conditions of eligibility and orthogonality [10]. The wavelet coefficientsquantify the resemblance of the analysed signal f (t) with the analysed6

wavelet ψ(t) . In this case, the notion of scale replaces the concept offrequency. The constant a representing the inverse of the frequency is8

known as scale factor, its variation describes the behaviour of expansionand concentration of wavelet. The constant b is called the translation of10

the wavelet; it is a location parameter related to the time.In practice, the discrete version of the continuous wavelet transform12

(1) is often used. The discrete wavelet transform DWT is used to undergothe signal f (t) succession of low and high pass filters. Each elementary14

operation of filtering corresponds to a resolution. A signal which isrelatively easier to interpret can be obtained from a complex signal. The16

discreet version of the continuous wavelet transform (I) is obtained bysampling according to 2 j values, j ∈ Z . It is given by:18

W f ( j, b) =∫ +∞

−∞f (t)2− j/2ψ

(t − b

2 j

)dt , (2)

where j is a relative integer.20

The transformed W f ( j, b) is characterized by its approximation anddetails components. These components of the function f (t) are given at j22

by the scalar products:

a jn = 〈 f ,ϕ j,n〉 , (3)24

d jn = 〈 f ,ψ j,n〉 , (4)

where n is an integer and ϕ represents the adjustable scale function in26

dilation and translation of the wavelet mother.

3. Proposed methodology28

Any dynamic system is defined by a number of parameters listedas principal and they affect its dynamics. These parameters may be30

not sufficient and have not enough consideration on a problem of faultdiagnosis through a single experiment. Indeed, several experiments are32

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952 A. KOUADRI AND M. ZELMAT

necessary to understand and analyse the variability of such variables.Therefore, the use of more efficient strategies for a robust condition2

monitoring is required. In this context, the decomposition of signals bydiscrete wavelet transform and statistical evaluation of their properties4

represent one of the most considered strategies. Fault diagnosis issuerequires ensuring the accurate detection of the faults in the presence of6

random factors such as parameters uncertainties, measurement errors,and the disturbances affecting the system. The wavelet transform, by8

the local property of its analysis, decomposes the measured data intothe frequency components according to an adapted scale. In the first10

step, the multi-scale analysis through the components from the wavelettransform provides a signal. Other signals are then extracted to complete12

the analysis. Thus, as well as, in the presence of several random factorsand the un-determination of failures situations, the statistics assessment14

methods can be used. In general, the results of such evaluation lead todecide the existence or neither of faults.16

In a random situation, the proposed fault detection approach isbased on three principle steps: repeated data sets collection (acquisition),18

wavelet decomposition of measured signals, and statistical analysis of thesignal.20

3.1 Adapted fault detection index

The fault detection index is based on the variability measure of22

the signal by the standard deviation. The objective of the variabilitymeasure of a signal is to characterise the dispersion in the presence of24

faults, random factors, and disturbances on the system. Based on the factthat the standard deviation changes from experiment to another in the26

same conditions, it becomes possible to classify the standard deviationas a random variable representing the studied phenomenon. Thus, for28

each experiment, a standard deviation σi , i = 1, . . . , N , is evaluated.Under several assumptions (see section 3.2), the sequence {σ1,σ2, . . . ,σN}30

defines a random variable Σ of mean mσ and variance σ2σ , so:

σ2σ =

∑Ni=1(σi − mσ )2

N. (5)32

To increase the sensitivity of the fault detection index σ , an as-sessment of its accuracy is considered. This is obtained by an estimate34

from a single sample. Specifically, it should be constructed, from a givendeduction, a random interval containing σi with an acceptable probability.36

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WAVELETS FOR FAULT DETECTION 953

This probability will define a confidence level. In this context, a confidencelevel of 95% is retained. Each sample corresponds to the observed crt2

from the used estimate at. Considering the centred interval in σi , i.e.]σi −ε,σi +ε[,ε is determined as:4

P(σi −ε ≺ σi ≺ σi +ε) ≥ 0.95 . (6)

Bienayme-Chebyshev inequality [11] allows calculating a solution to this6

problem, but it is inefficient. To get an acceptable result, it is necessary toinvolve the distribution law fora,. To do this, first one defines the random8

interval ]σi − ε, σi + ε [, known as the confidential interval at significantlevel (1 −α) , where α represents the error risk incurred by assert that σi10

is located in the considered interval.For a number relatively large of experiment (N ≥ 50) , the law of12

the reduced variable can be approximated by the normal distribution lawN (0, 1) . The confidential interval for σi at a significant level α for which14

the confidential condition of the equation 6 verifies, so:

CIα =]mσ − 19.6σσ , mσ + 1.96σσ [ (7)16

with

ε = 1.96σσ . (8)18

For the purposes of faults detection, it is required to evaluate the confi-dential interval for each DWT component using the equation (7). A fault20

is detected if the standard deviation of one component of wavelet analysisof signal does not belong to the corresponding confidential interval.22

3.2 Considerations on the proposed strategy start-up

In the previous section, the probabilistic computing of the fault24

detection index σ and its associated confidential interval is based onthe fact that the random variable Σ follows a normal distribution law26

of mean mσ and variance σ2σ . This hypothesis should apply to all DWT

components of the analysed signal. However, such a condition is not28

met and requires the use of an appropriate statistical test to examine thevalidity of results.30

To do this, we opted for the Kolmogorov-Smirnov test [12]. Thischoice is closely linked to the measures, frequencies distribution of the32

random variable Σ , and the number of available samples. Specifically, itcreates on a given sample, a vector of cumulative frequency of the random34

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954 A. KOUADRI AND M. ZELMAT

variable Σ obeying a normal distribution law N (mσ→η,σσ→η) , whereη represents a DWT component (details or approximation component).2

These cumulative frequencies, which represent the distribution function,obtained from the probabilities pk assessed for any σk less than σ and are4

given by:

Fk = ∑σk<σ

pk . (9)6

Thus, we define an empirical integral law F of mean mσ→η and standarddeviation σσ→η in order to verify for all σ ∈ < the below condition:8

supσ∈<

|Fk −F| → 0 . (10)

4. Test rig10

The experimental system consists of three tanks connected togetherby three valves to control on the transfer fluid in the system, as well12

as, they are three valves leak as shown in Figure 1 [13]. Two pumps areused to supply the water to tank 1 and 2 respectively. The differential14

pressure sensor is mounted on each tank and two flow sensors are fixedon the output pumps. The parameters L1, L2 and L3 represent the levels16

in tanks 1, 2 and 3 respectively and Q1, Q2 indicate the output flow fromthe pumps 1 and 2 respectively.18

Figure 1diagram of three tank system DTS-20020

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WAVELETS FOR FAULT DETECTION 955

This system is regulated by two PI decentralised controllers. TheNIDAQ 6024E acquisition card is used to collect the data from the2

benchmark by MATLAB software. The collected data of 1310 samples isrepeated for N = 50 experiments.4

5. Results and discussion

The discrete wavelet transform type Daubechies 5 at level 5 of6

decomposition is calculated for each measured signal, in order to extractthe statistical characteristics (mean and standard deviation) of the compo-8

nents approximation and details in healthy condition. Table 1 contains themean and standard deviation of signals.10

Table 1Mean and standard deviation of DWT approximation anddetails signals associated with input and output variables ofthe experimental system in healthy state.

L1 L2 L3 Q1 Q2

mσ→a5 0.0009 0.0075 0.0017 0.0101 0.0987

σa→a5 0.0002 0.0012 0.0004 0.0028 0.0206

mσ→d5 0.0003 0.0018 0.0003 0.0020 0.0169

σa→d5 4.82e-5 0.0002 4.40e-5 0.0003 0.0022

mσ→d4 0.0003 0.0012 0.0003 0.0019 0.0122

σa→d4 3.72e-5 0.0001 3.03e-5 0.0002 0.0011

mσ→d3 0.0003 0.0007 0.0003 0.0021 0.0109

σa→d3 2.18e-5 4.69e-5 2.79e-5 16.10e-5 99.53e-5

mσ→d2 0.0004 0.0005 0.0004 0.0029 0.0136

σa→d2 2.69e-5 1.87e-5 2.41e-5 0.0001 0.0010

mσ→d1 0.0007 0.0006 0.0006 0.0044 0.0191

σa→d1 3.47e-5 1.98e-5 3.13e-5 0.0002 0.001412

The confidential intervals are calculated for all components of ap-proximation and details at a significant level equal to 0.95 as shown in14

Table 2.In this work, three cases of failure mode are presented:16

Case 1: 10% of failure on the sensor level 1,

Case 2: 10% of failure on pump output 1,18

Case 3: 10% of failure on the sensor level 2.

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956 A. KOUADRI AND M. ZELMAT

Table 2Confidential interval established for DWT approximation anddetails signals for the experimental system input and outputvariables mode

L1 L2 L3 Q1 Q2

CIσ→a5 [0.0005 0.0013] [0.0052 0.0097] [0.0009 0.0025] [0.0047 0.0155] [0.0583 0.1391]

CIσ→d5 [0.0002 0.0004] [0.0014 0.0022] [0.0002 0.0004] [0.0014 0.0026] [0.0125 0.0213]

CIσ→d4 [0.0002 0.0003] [0.0010 0.0013] [0.0003 0.0004] [0.0015 0.0023] [0.0100 0.0144]

CIσ→d3 [0.0002 0.0003] [0.0006 0.0008] [0.0003 0.0004] [0.0018 0.0024] [0.0089 0.0128]

CIσ→d2 [0.0004 0.0005] [0.0004 0.0005] [0.0004 0.0005] [0.0026 0.0032] [0.0116 0.0156]

CIσ→d1 [0.0006 0.0008] [0.0005 0.0006] [0.0005 0.0007] (0.0041 0.0048] (0.0163 0.0218]2

Figure 2 illustrates the change in water level in tank 1 and itscomponents approximation and details obtained by the discrete wavelet4

transform when a fault of 10% occurs on the sensor level 1.

6

Figure 2Water level in reservoir 1 and its wavelet decomposition obtainedbefore and after the occurrence of fault (case 1)

Upon occurrence of the fault, all the components of DWT signals8

which were affected can be seen in Figure 2. This effect was not moresignificant on the three first details signals (d1, d2, d3) . The latest details10

components (d4, d5) diagnose clearly the fault in the system. However,when the fault on the system is quite small or the system is controlled by12

adequate controllers; the changing in curve of each signal is not obviousso we can not detect the fault from the relevant figure. Table 3 contains the14

standard deviations of each DWT component of each measured signal.The fault is detected when the standard deviation of DWT signals does16

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WAVELETS FOR FAULT DETECTION 957

not belong to the corresponding confidential interval. Each bold valueindicates the occurring fault. This change in the standard deviation is2

mainly due to the incorrect information provided by the sensor levelmeasurement in the first tank to the controller, which is acting on the first4

feed water pump in order to reduce the difference between the set pointand this output measure. This affects the statistical characteristics of the6

signal L1 and the signal Q1 . The standard deviations of signals L3 andQ2 are also affected because of the interactions in this process.8

Table 3Standard deviation of DWT approximation and details signalsof the experimental system variables in the faulty state case 1.

L1 L2 L3 Q1 Q2

σa5 0.0330 0.0090 0.0481 0.4948 0.1485σd5 0.0051 0.0017 0.0003 0.0544 0.0133σd4 0.0047 0.0011 0.0003 0.0348 0.0123σd3 0.0053 0.0006 0.0004 0.0236 0.0100σd2 0.0020 0.0005 0.0004 0.0195 0.0125σd1 0.0029 0.0006 0.0005 0.0128 0.0182

10

The results of case 2 clearly illustrate the fault effects on all thecomponents of signal Q1 as shown in Figure 3. When pump 1 output is12

reduced, the water level in tank I starts to gradually decrease for which thecommand loop begins to recover the difference between the desired level14

in tank 1 and the measurement level (see Table 4).

16

Figure 3Pump 1 output and its wavelet decomposition obtained before andafter the occurrence of fault (case 2)

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958 A. KOUADRI AND M. ZELMAT

Table 4Standard deviation of DWT approximation and details signalsof the experimental system variables in the faulty state case 2

L1 L2 L3 Q1 Q2

σa5 0.0039 0.0061 0.0025 0.0305 0.0808

σd5 0.0003 0.0020 0.0003 0.0107 0.0182

σd4 0.0002 0.0010 0.0003 0.0059 0.0102

σd3 0.0003 0.0007 0.0004 0.0078 0.0105

σd2 0.0004 0.0005 0.0005 0.0109 0.0135

σd1 0.0007 0.0006 0.0007 0.0144 0.01802

Figure 4 describes the level L2 changes and its DWT componentsin which the fault is detected. In this case pump 2 begins to increase4

the feed flow in tank 2 to reduce the difference between the measuredand the requested level L2 . Therefore, the statistical characteristics are6

quite changed in the system signals which are the subject of the fault (seeTable 5).8

Figure 4Water level in reservoir 2 and its wavelet decomposition obtainedbefore and after the occurrence of fault (case 3)10

6. Conclusion

The proposed methodology proves its effectiveness that the sensitiv-12

ity of the fault index to any changes in approximation and details signals isbased on the confidential intervals. The fault index based on the standard14

deviation of the DWT components has detected the fault and at the sametime identified the fault source.16

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WAVELETS FOR FAULT DETECTION 959

Table 5Standard deviation of DWT approximation and details signalsof the experimental system variables in the faulty state case 3

L1 L2 L3 Q1 Q2

σa5 0.0009 0.0135 0.0191 0.0247 0.2238

σd5 0.0003 0.0039 0.0003 0.0034 0.0217

σd4 0.0003 0.0021 0.0003 0.0037 0.0198

σd3 0.0003 0.0023 0.0004 0.0053 0.0104

σd2 0.0005 0.0011 0.0004 0.0066 0.0118

σd1 0.0007 0.0010 0.0005 0.0109 0.01572

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