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Fault Detection Based on Wavelets Transform. Application to a Roughing Mill
9 IFAC Fault Detection, Supervision and Safety of Technical Processes, Beijing 2006
ELSEVIER
IFAC
PUBLICATIONS FAULT DETECTION BASED ON WAVELETS TRANSFORM. APPLICATION TO A ROUGHING MILL. S. Lesecq, S. Gentil, S. Taleb
Laboratoire d'Automatique de Grenoble (UMR 5528 CNRS-INPG-UJF) ENSIEG- BP 46, F-38402 Saint Martin d'Hbres Cedex {samir.taleb, suzanne.lesecq, sylviane.gentil}@inpg.fr
Abstract: In this paper, the wavelet analysis is used to detect particular vibration faults. The proposed detection method is based on the stationary wavelet transform. The wavelet coefficients allow analysing the signal changes in different scales. They are fuzzified, leading to partial criteria, relative to the frequency contents in the different frequency bands. Fuzzy aggregation of these partial criteria gives the final decision. The procedure depends on different parameters that must be tuned. Several aggregation possibilities are applied to industrial data recorded on the main drive of a roughing mill. Copyright 9 2006 IFAC K e y w o r d s - Fault detection, Stationary wavelet transform, Fuzzy aggregation, Thresholding, Fuzzy decision making, Hot rolling mill processes. 1. INTRODUCTION In control theory, diagnostic methods are essentially based on models that are supposed to represent the behaviour of the process, either in the normal mode or in a particular faulty one. The model constitutes a prototype behaviour that is compared to the actual state, evaluated with data acquired online from the process. Usually, analytical equations are used to represent the normal process behaviour. From the measurements and an analytical model, quantities called residuals are computed (Isermann, 1997). Signal processing is another tool to deal with fault diagnosis. It can be used to analyse the residuals, or to analyse directly the signals measured online. This last approach avoids system modelling. Its difficulty is to ensure that a change in some quantity, e.g. the signal mean or the frequency contents, is a characteristic due to a particular fault. Signals may be studied either with time-domain methods (including correlation, mean or standard deviation changes (Basseville and Nikiforov, 1993)), with frequency methods (e.g. Fast Fourier Transform), or with more sophisticated ones including time-frequency or timescale methods. Wavelet analysis is a promising approach that has been investigated during the last decade for monitoring and diagnosis in various industrial areas (Lesecq and Barraud, 2003; Peng and Chu, 2004). Rolling is the most important processing method for metal produced by metallurgical industries by melting. A lot of strips, plates, and sheets are actually produced in hot and cold rolling mills. The accuracy of the thickness of sheets manufactured, for instance, for the car industry has to be within the range of ktm. Therefore, the slightest irregularity of texture or surface roughness may lead to useless strips. The challenge is thus to achieve the tolerance and quality standard required by rolling mill plants. Note that these processes operate usually with several Mega
Watt of power. They rotate roll masses greater than 20 Tons with a rolling speed up to 2000 m/mn (Seelinger, et al., 2002). The global pressure towards a rise of productivity induces an increase of the rolling speed. While higher speeds may achieve an increase of the productivity, they also increase the potential of the plant to react with vibrations. Actually, different vibration frequencies may appear in rolling mill plants. They are usually split in two ranges, namely: - "low" frequencies from 1 Hz to 20 Hz; - "high" ones, from 0.1 kHz to more than 0.5 kHz. Each range corresponds to different faults. Moreover, some faulty frequencies are proportional to the rolling speed. Usually, the detection of these vibrational faults is based on Frequency or Time-Frequency methods. Multiresolution analysis (MRA) is another tool to deal with such faults (Taleb et al., 2004). In this paper, the Stationary Wavelet Transform (SWT) is used instead of the MRA for the detection of "low" frequency vibrations. Actually, the SWT is time invariant: the coefficients of a delayed signal are just a time-shifted version of the coefficient of the original signal. The paper is organised as follows. In section 2, the process under study is described. Section 3 reminds some classical results about Wavelet transforms and their coefficients thresholding. Section 4 summarizes fuzzy decision making. Section 5 is devoted to the description of the proposed detection method. It is based on the fuzzification of the thresholded wavelet coefficients, leading to partial criteria. The aggregation of these criteria allows generating symptoms. Results achieved with the proposed method are discussed in section 6. The real data have been recorded by SMS-DEMAG during normal operation of the roughing mill. Thus, the procedure adapts to the load changes: this point is confidential and not presented in the present paper. 1115
2. PROCESS DESCRIPTION
THRESHOLDING
The ever higher load capacities within the rolling process lead to increased stresses and vibrations of the main drives in hot rolling mills (Seelinger, et aL, 2002). Further vibrations could also be excited through frequency converters or badly parameterised controllers. They are propagated via the motors to the drive chain. The highest excitations are caused by impulses like torque changes during biting of the slab being rolled. During normal rolling, vibrations can also rise through defects of machine elements like defect gears, antifriction beatings, eccentrically reels, high backlash, increased friction or electrical induced vibrations. The present application deals with the main drive of a roughing mill, which is a part of a hot rolling mill. A permanent telemetric torque measuring system has been installed. The drive is a so called Twin Drive, where two individual motors drive the upper and lower work rolls directly. Between the upper and lower drive, no permanent mechanical link exists: only during rolling both drive chains are coupled through the rolled slab. Fig. 1 shows a diagram of the twin drive. It consists of driving motors for upper and lower drive, intermediate spindles, couplings, spade end shafts and roll sets. Torque sensors are located on the drive shafts and allow permanent torque monitoring (Deckers, et al., 2003).
Traditional spectral analyses provide a good description of stationary signals. Unfortunately, they have severe shortcomings. The first one is that the Fourier Transform is unable to accurately analyse and represent a signal with non periodic features (for instance, a transient signal). Another difficulty is its incapability of providing information about the time dependency of the frequency contents. In this case, time-frequency methods have to be used instead of traditional spectral analysis (Flandrin, 1999; Boashash, 2003). The main difficulty of timefrequency methods is their constant time and frequency resolution, according to the HeisenbergGabor uncertainty principle. In order to obtain variable time and frequency resolution (their product been constant), the Wavelet Transform (WT) has been introduced. A quick overview of a particular WT, the stationary wavelet transform (SWT), and the thresholding of the wavelet coefficients is now given.
3.1 Wavelets Transform and Stationary Wavelets Transform The Continuous Wavelet Transform (CWT) projects a signal x(t) on a family of zero-mean functions ga,b (the wavelets) deduced from an elementary function ~" (the mother wavelet) by means of translations and dilatations:
=
(,)a,,
(')=
where * stands for "conjugate", a is the scaling parameter (taking lal > 1 dilates the function ~) and b is the translation parameter. Theoretical background may be consulted in Strang and Nguyen, (1996). The redundancy introduced in the CWT can be reduced by discretizing the parameters a and b :
DWTj,k(x)= Zx(t)gj,k (t)dt
(1)
where ~j,k ( t ) : 1 / ~ o ~ ( a j t - k b o ) .
Fig. 1" Roughing mill drive schema and torque sensor Direct information can be derived from the torque signals for pass-schedule optimisation, improvement of lubrication and cooling in the roll gap in addition to protection against unscheduled downtime, the assessment of the current condition and the estimation of the remnant lifetime. Note that signal processing methods have already been used for detection and diagnosis of vibrations in the metallurgical industry. Some integrated monitoring systems are described for instance in (Deckers, et al., 2003; Seelinger, et al., 2002). 3. STATIONARY WAVELET TRANSFORM AND
The choice a 0 = 2, b0 = 1 corresponds to the dyadic sampling of the time-frequency plane (i.e. one set of coefficients per octave). Thanks to this particular sampling, it is possible to obtain for the set ~.k an orthonormal basis with a wavelet ~ well localized both in time and frequency. At level j = l: J , the approximations a~ and details d~ (Fig. 2) are recursively computed using a 2-channel filter bank with particular filters, respectively a low pass filter and a high pass one ~ (Fig. 3). This recursive algorithm is initialized by x(k), i.e. a ~ = x(k) and:
=
h2k_,a, ,
d~=Ztg2k_,a ,
Low freq. Level 3
a_~.
Level 2
a2
Level 1
m
High freq.
d2 ~l
dl
Fig. 2" Frequency domain splitting
1116
Freq.
(2)
Since (2) is basically an inner product, it is very attractive for DSP implementation (fused addmultiply operation). The main advantage of the Stationary Wavelet Transform (Fig. 3) (Coifman and Donoho, 1995; Pesquet, et al., 1996) is its timeinvariance property: the SWT coefficients of a delayed signal are just a time-shifted version of the original ones. This property is fundamental for diagnosis in order to provide a symptom that is timeinvariant, i.e. its value does not depend on the time occurrence of the fault.
the present paper, the minimax threshold has been chosen. Its main property is that the Risk function:
R ( g , ~ ) = I E ( ~ "~" ( g , - ~ , ) 2) n
(5)
\ zak=l
is minimum, ~ being the reconstructed signal. For each level j, the threshold ~) is given by: ,~J
* = crJ 2.~j
(6)
where crJ is the standard deviation of d~, nj is the number of coefficients d~ used for the threshold computation and 2.~*j is tabulated. As an alternative to
r-m
the use of minimax thresholds, one could employ the universal
threshold
2 j = crj X/2 log(n j)
which
requires no look-up tables.
Fig. 3: SWT decomposition algorithm
4. FUZZY DECISION MAKING
3.2 Wavelet coefficient thresholding The singularity occurrence in the behaviour of a signal is revealed by the size of the wavelet coefficients (Strung and Nguyen, 1996; Mallat and Zhong, 1999). Thus, the objective is to find when the noisy data: x(ti)= g(t i)+ 8(t i), i= 1: n (3) changes its behaviour, without any hypothesis about the parametric form ofg which contains this change. In (3), e is usually supposed to be a zero mean and o~ variance independent normally distributed noise, which allows interesting theoretical results about the "optimality" of thresholding (Donoho and Johnstone, 1994). Donoho (1995) proposes to extract these significant coefficients by soft thresholding: 5~ =
0
0_< d~
(4)
where 8~ are the thresholded coefficients, d~ is given in (2) and ,r is the threshold value. Thus, the coefficients below their threshold are set to zero (they are assumed to represent the "normal behaviour"), while exceeding coefficients indicate the occurrence of a signal abnormal behaviour (Fig. 4), corresponding to the appearance of a fault in the process. Note that the signal behaviour change may affect the different decomposition level not in the same way. Therefore, 2 j is level dependent.
I , -~' I-"":"'"/ ..r- d~ ...............-"]0
2.j
Fig. 4: SWT coefficients soft thresholding The threshold choice is tricky. Several methods may be used, and a bibliographical study reveals many possibilities. The optimal choice requires knowledge (or at least hypotheses) about the analysed signal (Donoho and Johnstone, 1994; Taleb et al., 2004). In
Fuzzy decision making for diagnostic decision allows formal modelling of decision-making for imprecise and uncertain conditions, in order to select a solution characterized by partial points of view, called partial criteria (Dubois and Prude, 1985). In a known environment, each decision d ~ D (where D is the set of possible decisions) is evaluated by a series of values [Cl(d), c2(d),..., cp(d)] where c,{d) measures the decision d in the sense of criterion i. d is defined as a fuzzy subset obtained by aggregation of p partial criteria. Thus, the membership function Pd is such that:
fld = h(cl (d),cE (d) .... ,cp (d))
(7)
where h is a fuzzy set operator connective to be determined. Necessary conditions on h are: 9 h is a continuous function; 9 h(0,0 .... , 0 ) = 0 a n d h ( 1 , 1 , . . . , 1 ) = l ;
9 V(ui,vi)~[O,1] 2,ifui>vi then h(Ul,..., Up) > h(Vl,..., Vp). Three main decision-making attitudes can be modelled: 1) for an operator h expressing that all the criteria are met simultaneously, a natural axiom is: V(Ul,//2 .....
Up),
h(Ul,//2 .....
Up)<_n~in(ui) , i=l'p
(8)
which means that the overall evaluation of a decision cannot be better than the smallest (i.e. "worst") value of the partial evaluations. These operators correspond to conjunctions; 2) to express the redundancy of the objectives, h must meet the following condition: V ( U l , U 2 .....
Up), n~i (lli)<_h(Ul,U 2.... ,Up), i=l'p
(9)
which means that the overall evaluation is determined by the highest (i.e. "best") value of the partial ones. These operators are disjunctions; 3) h becomes a compromise when the following axiom is satisfied: .....
min(ui)<-h(Ul,U 2 ..... Up)<_maix(ui), i=l"p
1117
(10)
5. FAULT DETECTION 5.1 Proceduredescription
A fault occurrence is supposed to be revealed by a signal singularity as defined in Mallat and Zhong, (1999). The proposed detection method tries to detect these singularity appearances. It is composed of four steps (see Fig. 5).
Si'~'ai
SWT
~m]'i~--
~Aggregation H
i"......................... ..... .....i"-~ i
HThresholding ] Fuzzification ]
In order to obtain an instantaneous decision and specifying the moment of singularity occurrence (12) (13) and (14) are proposed. (12) allows a robust decision: it is significant only when the singularity propagates to all scales. (13) realises a compromise when the singularity is not present in all scales. (14) allows a sensitive decision: it takes a significant value when at least one threshold exceeding exists, at any level. To improve the robustness of indicators (12) to (14), (15) and (16) propose to first aggregate the fuzzified coefficients on a sliding window of length N. These operators allow enhancing the progressiveness of indicators (12) to (14). 6. DETECTION RESULTS
Fig. 5: Detection procedure
6.1 Data description
The first one applies the SWT to the signal. The second step corresponds to the discrimination between the coefficients that characterize the existence, or not, of a change in the signal behaviour. This step can be done thanks to a wavelet coefficient thresholding, where the thresholds are computed as explained in section 2. The third step is a fuzzification one. In diagnosis area, the classical use of the thresholded coefficients is a crisp one. Actually, a gradual detection is more interesting than a Boolean one (i.e. "there is/there is not a fault") because attention can be paid to a component before a fault is completely installed. Therefore, a fuzzification of the thresholded coefficients is implemented (Fig. 6): __
lail>
1 |
/a J (6~) =
!
2aJ~j
o
The benchmark data are provided by SMS-Demag (Deckers, et al., 2003). Several experiments, each one being made up of 5 records (so called passes of the strip) are used. Each record has six measurements (torque, current and speed of upper and low motors) sampled with frequency fe = 250 Hz. Due to fe, only "low" frequency vibrations can be detected (Seelinger, et al., 2002). The data set has been split in two, namely "normal" behaviour and "faulty" one. Note that an abnormal vibration induces extra frequency contents in signals that are conspicuous in the torque spectrum. For this reason, only the upper drive torque measurement ~ is considered in this paper. The vibrations are growing with the pass number. Thus, the study is restricted to the 5th pass. Fig. 7 shows ~ for one experiment. Note that the aim is to detect the increase of particular vibration modes that lead to a decrease in the surface roughness of the rolled strip. U p p e r t o r q u e , 5 p a s s e s , n o r m a l operation I I I I I L I I
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a/= o
2
where a J is a parameter that defines the membership function of the coefficients
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Fig. 6: Thresholded coefficients fuzzification In order to give a progressive, but also unique indicator, the fourth step considers the various fuzzy coefficients as partial criteria and ensures their aggregation in different ways: D1 k = m!n(p~); j = l ' J (12) J D2 k = mean(p~); j = l ' J
(13)
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(14)
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(16)
]
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Fig. 7" Upper torque (in Newton) of a "normal behaviour" experiment (5 passes) 6.2 Choice o f the Wavelet
When time-scale methods are used, the relationship between scale and frequency is expressed through the pseudo-frequencyf~ (in Hz) corresponding to a given scale a. fa is computed thanks to the normalised "centre frequency"fc of the wavelet (Abry, 1997): f~ = fc
J
D4k =m~. {mean(lu~-t);t=O'N-1};j=l'J
10I
axT e
where Te
=
1/fe
(17)
The underlying idea is to associate with a given wavelet a purely periodic signal of frequency f~ that maximizes the FFT of the wavelet modulus. Thus,
1118
the centre frequency-based approximation captures the main wavelet oscillations. Therefore, fc is a convenient and simple characterization of the leading dominant frequency of the wavelet. A preliminary frequency analysis has shown that the torsional vibration that must be detected mainly occurs around 11.6 Hz. Therefore, the wavelet should possess a pseudo-frequency fo around 11.6 Hz (17). The Symmlet6 (Daubechies, 1992) "oscillates" with this frequency at the 4 th level of decomposition. This choice is not unique (e.g., the Coiflet2 wavelet has the same property) and is also related to the sampling period. The maximum level of decomposition must fulfil J > 4 but it must be moderate because of the computational cost. Here, some experiments have shown that J = 5 is sufficient to ensure a good detection of the faulty oscillations. Moreover, the symptoms D4 (15) and D5 (16) are computed with a sliding window of length N = 32.
6.3 Results Fig. 8 exhibits the upper torque Fu of the 5th pass for an unfaulty record, its SWT decomposition (5 levels of decomposition) and the thresholds computed with equation (6) using an unfaulty signal. x 10 5
Wavelet
decomposition
appears are depicted: this constitutes the second experiment. The thresholds are identical to those used for the 1 st experiment. Fig. 11 shows the symptoms D1 to D5. The symptoms computed with D1 (12) and D2 (13) are not sensitive to this faulty situation. As expected, D3 (14) and D5 (16) are highly sensitive and D4 (15) has an intermediary (robust) behaviour. Symptoms
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x 10 s
Fig. 8: SWT (unfaulty signal), experiment 1 D1
Fig. 9 depicts the symptoms Dk, k = 1 to 5, generated with operators (12) to (16) by considering only the levels j = 3 : 5 ( j = l , 2 do not clearly exhibit information on the faults to be detected). First, a jump in the torque must be detected: it corresponds to the time the slab is starting to be rolled out. Then, a transient period is elapsed before any significant symptom is computed. Therefore, at the beginning of the record, Dk = 0.5 which mean that the symptom is not evaluated. Dk =0 stands for "no change has been detected" while D k = l means that "the coefficients have significantly exceeded the thresholds, therefore, a important change has occurred". As expected, for an unfaulty situation, the indicators are nearly zero. Some transient nonzero values can be seen on D3 and D5 that are more sensitive than the other ones. Note that the torque nominal value F~0 depends on the experiment, and is not always constant. The procedure adapts this situation but the results are confidential. In Fig. 10, the upper torque F~ and its SWT decomposition when a vibration
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1119
7. CONCLUSION AND PROSPECTS
REFERENCES
In this paper, the capability for the stationary wavelet transform to deal with torsional vibration fault detection in a rolling mill has been investigated. A detection procedure based upon a wavelet coefficient thresholding that is level-dependent has been investigated. The coefficients are fuzzified and aggregated in order to provide a symptom. The tuning parameters of this procedure are the wavelet itself, the number of decomposition levels, the thresholds and the thresholding method. The wavelet choice depends on the features that must be detected in the signal under analysis. This selection is sometimes not unique. For detection purpose, the final choice is made in order to maximize the symptom sensitivity. This method has been customized for data that have been recorded on the main drive of a roughing mill, which is a part of a hot rolling mill. The first results that have been obtained are really encouraging. However, new aggregation methods of the fuzzified wavelet coefficients and the choice of wavelet are under study in order to take into account the pseudofrequency property of the analyzing wavelet.
Abry P. (1997), Ondelettes et Turbulence. Multir~solutions, Algorithmes de D~composition, Invarianee d'~chelles", Diderot Ed., Paris, (in French). Basseville M., Nikiforov L. (1993), Detection of abrupt changes, Prentice Hall Ed. Boashash B. (edited by) (2003), Time frequency signal analysis and processing, a comprehensive reference, Elsevier Ed. Coifman R.R, Donoho D. (1995), Translation Invariant De-noising, series Lecture Notes in Statistics, New York Springer-Verlag, Vol. 1t)3, pp. 125-150. Daubechies I. (1992), Ten lectures on Wavelets, SIAM Ed., Philadelphia, Pennsylvania. Deckers J., Jepsen O., Latzel S., Metz H., Stuecher R. (2003), Condition monitoring and failure diagnosis in plants of the metal industry, IFAC Safeprocess, Washington D.C., USA. Donoho D. (1995), De-Noising by soft-Thresholding, IEEE Trans. on Information Theory, Vol. 41(3), pp. 613-627. Donoho D., Johnstone I. (1994), Ideal Spatial Adaptation by Wavelet Shrinkage, Biometrika, Vol. 81. Dubois D. Prade H. (1985), A review of fuzzy set aggregation connectives, Information Sciences, Vol. 36, pp. 85-121. Flandrin P. (1999), Time-frequency~time-scale analysis, Academic Press Ed., San Diego, California. Isermann R. (1997), Supervision, fault-detection and fault-diagnosis m e t h o d s - An introduction, Control Eng. Practice, Vol. 5(5), pp. 639-652. Lesecq S., Barraud A. (2003) Arcing fault detection using wavelet transform, IFAC Safeprocess 2003, Washington D.C., USA. Mallat S., Zhong S. (1992), Characterization of Signals from Multiscale Edges, IEEE Trans. on Pattern Anal. and Machine Intelligence, Vol. 14, pp. 710-732. Peng Z.K., Chu F.L. (2004), Application of the wavelet transform in machine condition monitoring and fault diagnostics: a review with bibliography, Mechanical Systems and Signal Processing, Vol. 18, pp. 199-221. Pesquet J.C., Krim H., Carfatan H. (1996), TimeInvariant orthonormal wavelet representations, IEEE Trans. on Signal Processing, Vol. 44, pp. 1964-1970. Seelinger A., Mackel J., Georges D. (2002), Measurement and diagnosis of preprocessdisturbing oscillations in high-speed rolling plants, IMEKO 2002, Tampere, Finland. Strang G., Nguyen T. (1996), Wavelets and Filter Banks, Wellesley-Cambridge Press. Taleb S., Lesecq S., Stuecher R. (2004), Torsional vibration detection using torque measurement of a main drive of a hot rolling mill with wavelet transform, IFAC MMM'04, Nancy, France.
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-- _-- Z - - ~
,
I
.
'
I. . . .
2
. ~
t
.lJ,
. L I 9 _1.i 9 ~ 9 9 . . . . . .
|''~
I'w["
IT"
_t.._lJ.
4
'"'"
I. . . . .
==.
I. . . . . . .
_h.
w,,
,
""
~
6 T i m e [s]
- - -
,1
ii.l,
w-,-lw-,'w-,v-;
....
.I
t "'1
,
'
. ..11
I
-q
"'l
_~
]
,
" I'
8
t
I~
I
v
I
10
Fig. 12: SWT (faulty case), experiment 3 Symptoms
x 105
'E .
D1
05
"0
.
.
.
.
. . . . .
----
--
-t"
-
--
i'
~
r
I "
f
~
"
i ll-
. . . . . . . . . .
4
. . . . .
t- . . . . .
I. . . . .
-4------t
|
oo
o.. ~. z. _. .~ .: ~ _' 0J0
. . . . .
p...a2
-
t_ ---
q_
'Ivt
__r_~
-1". . . . . . 4
1- . . . . . 6
Time [ s ]
t8
~
RVI - " - ~ 10
-
-
--1
Fig. 13: Symptom generation, experiment 3 ACKNOWLEDGEMENTS The authors would like to thank SMS-Demag which allows using the data provided for the European project EU-IST-2000-30009 MAGIC.
1120
ELSEVIER
IFAC
PUBLICATIONS FAULT DETECTION BASED ON WAVELETS TRANSFORM. APPLICATION TO A ROUGHING MILL. S. Lesecq, S. Gentil, S. Taleb
Laboratoire d'Automatique de Grenoble (UMR 5528 CNRS-INPG-UJF) ENSIEG- BP 46, F-38402 Saint Martin d'Hbres Cedex {samir.taleb, suzanne.lesecq, sylviane.gentil}@inpg.fr
Abstract: In this paper, the wavelet analysis is used to detect particular vibration faults. The proposed detection method is based on the stationary wavelet transform. The wavelet coefficients allow analysing the signal changes in different scales. They are fuzzified, leading to partial criteria, relative to the frequency contents in the different frequency bands. Fuzzy aggregation of these partial criteria gives the final decision. The procedure depends on different parameters that must be tuned. Several aggregation possibilities are applied to industrial data recorded on the main drive of a roughing mill. Copyright 9 2006 IFAC K e y w o r d s - Fault detection, Stationary wavelet transform, Fuzzy aggregation, Thresholding, Fuzzy decision making, Hot rolling mill processes. 1. INTRODUCTION In control theory, diagnostic methods are essentially based on models that are supposed to represent the behaviour of the process, either in the normal mode or in a particular faulty one. The model constitutes a prototype behaviour that is compared to the actual state, evaluated with data acquired online from the process. Usually, analytical equations are used to represent the normal process behaviour. From the measurements and an analytical model, quantities called residuals are computed (Isermann, 1997). Signal processing is another tool to deal with fault diagnosis. It can be used to analyse the residuals, or to analyse directly the signals measured online. This last approach avoids system modelling. Its difficulty is to ensure that a change in some quantity, e.g. the signal mean or the frequency contents, is a characteristic due to a particular fault. Signals may be studied either with time-domain methods (including correlation, mean or standard deviation changes (Basseville and Nikiforov, 1993)), with frequency methods (e.g. Fast Fourier Transform), or with more sophisticated ones including time-frequency or timescale methods. Wavelet analysis is a promising approach that has been investigated during the last decade for monitoring and diagnosis in various industrial areas (Lesecq and Barraud, 2003; Peng and Chu, 2004). Rolling is the most important processing method for metal produced by metallurgical industries by melting. A lot of strips, plates, and sheets are actually produced in hot and cold rolling mills. The accuracy of the thickness of sheets manufactured, for instance, for the car industry has to be within the range of ktm. Therefore, the slightest irregularity of texture or surface roughness may lead to useless strips. The challenge is thus to achieve the tolerance and quality standard required by rolling mill plants. Note that these processes operate usually with several Mega
Watt of power. They rotate roll masses greater than 20 Tons with a rolling speed up to 2000 m/mn (Seelinger, et al., 2002). The global pressure towards a rise of productivity induces an increase of the rolling speed. While higher speeds may achieve an increase of the productivity, they also increase the potential of the plant to react with vibrations. Actually, different vibration frequencies may appear in rolling mill plants. They are usually split in two ranges, namely: - "low" frequencies from 1 Hz to 20 Hz; - "high" ones, from 0.1 kHz to more than 0.5 kHz. Each range corresponds to different faults. Moreover, some faulty frequencies are proportional to the rolling speed. Usually, the detection of these vibrational faults is based on Frequency or Time-Frequency methods. Multiresolution analysis (MRA) is another tool to deal with such faults (Taleb et al., 2004). In this paper, the Stationary Wavelet Transform (SWT) is used instead of the MRA for the detection of "low" frequency vibrations. Actually, the SWT is time invariant: the coefficients of a delayed signal are just a time-shifted version of the coefficient of the original signal. The paper is organised as follows. In section 2, the process under study is described. Section 3 reminds some classical results about Wavelet transforms and their coefficients thresholding. Section 4 summarizes fuzzy decision making. Section 5 is devoted to the description of the proposed detection method. It is based on the fuzzification of the thresholded wavelet coefficients, leading to partial criteria. The aggregation of these criteria allows generating symptoms. Results achieved with the proposed method are discussed in section 6. The real data have been recorded by SMS-DEMAG during normal operation of the roughing mill. Thus, the procedure adapts to the load changes: this point is confidential and not presented in the present paper. 1115
2. PROCESS DESCRIPTION
THRESHOLDING
The ever higher load capacities within the rolling process lead to increased stresses and vibrations of the main drives in hot rolling mills (Seelinger, et aL, 2002). Further vibrations could also be excited through frequency converters or badly parameterised controllers. They are propagated via the motors to the drive chain. The highest excitations are caused by impulses like torque changes during biting of the slab being rolled. During normal rolling, vibrations can also rise through defects of machine elements like defect gears, antifriction beatings, eccentrically reels, high backlash, increased friction or electrical induced vibrations. The present application deals with the main drive of a roughing mill, which is a part of a hot rolling mill. A permanent telemetric torque measuring system has been installed. The drive is a so called Twin Drive, where two individual motors drive the upper and lower work rolls directly. Between the upper and lower drive, no permanent mechanical link exists: only during rolling both drive chains are coupled through the rolled slab. Fig. 1 shows a diagram of the twin drive. It consists of driving motors for upper and lower drive, intermediate spindles, couplings, spade end shafts and roll sets. Torque sensors are located on the drive shafts and allow permanent torque monitoring (Deckers, et al., 2003).
Traditional spectral analyses provide a good description of stationary signals. Unfortunately, they have severe shortcomings. The first one is that the Fourier Transform is unable to accurately analyse and represent a signal with non periodic features (for instance, a transient signal). Another difficulty is its incapability of providing information about the time dependency of the frequency contents. In this case, time-frequency methods have to be used instead of traditional spectral analysis (Flandrin, 1999; Boashash, 2003). The main difficulty of timefrequency methods is their constant time and frequency resolution, according to the HeisenbergGabor uncertainty principle. In order to obtain variable time and frequency resolution (their product been constant), the Wavelet Transform (WT) has been introduced. A quick overview of a particular WT, the stationary wavelet transform (SWT), and the thresholding of the wavelet coefficients is now given.
3.1 Wavelets Transform and Stationary Wavelets Transform The Continuous Wavelet Transform (CWT) projects a signal x(t) on a family of zero-mean functions ga,b (the wavelets) deduced from an elementary function ~" (the mother wavelet) by means of translations and dilatations:
=
(,)a,,
(')=
where * stands for "conjugate", a is the scaling parameter (taking lal > 1 dilates the function ~) and b is the translation parameter. Theoretical background may be consulted in Strang and Nguyen, (1996). The redundancy introduced in the CWT can be reduced by discretizing the parameters a and b :
DWTj,k(x)= Zx(t)gj,k (t)dt
(1)
where ~j,k ( t ) : 1 / ~ o ~ ( a j t - k b o ) .
Fig. 1" Roughing mill drive schema and torque sensor Direct information can be derived from the torque signals for pass-schedule optimisation, improvement of lubrication and cooling in the roll gap in addition to protection against unscheduled downtime, the assessment of the current condition and the estimation of the remnant lifetime. Note that signal processing methods have already been used for detection and diagnosis of vibrations in the metallurgical industry. Some integrated monitoring systems are described for instance in (Deckers, et al., 2003; Seelinger, et al., 2002). 3. STATIONARY WAVELET TRANSFORM AND
The choice a 0 = 2, b0 = 1 corresponds to the dyadic sampling of the time-frequency plane (i.e. one set of coefficients per octave). Thanks to this particular sampling, it is possible to obtain for the set ~.k an orthonormal basis with a wavelet ~ well localized both in time and frequency. At level j = l: J , the approximations a~ and details d~ (Fig. 2) are recursively computed using a 2-channel filter bank with particular filters, respectively a low pass filter and a high pass one ~ (Fig. 3). This recursive algorithm is initialized by x(k), i.e. a ~ = x(k) and:
=
h2k_,a, ,
d~=Ztg2k_,a ,
Low freq. Level 3
a_~.
Level 2
a2
Level 1
m
High freq.
d2 ~l
dl
Fig. 2" Frequency domain splitting
1116
Freq.
(2)
Since (2) is basically an inner product, it is very attractive for DSP implementation (fused addmultiply operation). The main advantage of the Stationary Wavelet Transform (Fig. 3) (Coifman and Donoho, 1995; Pesquet, et al., 1996) is its timeinvariance property: the SWT coefficients of a delayed signal are just a time-shifted version of the original ones. This property is fundamental for diagnosis in order to provide a symptom that is timeinvariant, i.e. its value does not depend on the time occurrence of the fault.
the present paper, the minimax threshold has been chosen. Its main property is that the Risk function:
R ( g , ~ ) = I E ( ~ "~" ( g , - ~ , ) 2) n
(5)
\ zak=l
is minimum, ~ being the reconstructed signal. For each level j, the threshold ~) is given by: ,~J
* = crJ 2.~j
(6)
where crJ is the standard deviation of d~, nj is the number of coefficients d~ used for the threshold computation and 2.~*j is tabulated. As an alternative to
r-m
the use of minimax thresholds, one could employ the universal
threshold
2 j = crj X/2 log(n j)
which
requires no look-up tables.
Fig. 3: SWT decomposition algorithm
4. FUZZY DECISION MAKING
3.2 Wavelet coefficient thresholding The singularity occurrence in the behaviour of a signal is revealed by the size of the wavelet coefficients (Strung and Nguyen, 1996; Mallat and Zhong, 1999). Thus, the objective is to find when the noisy data: x(ti)= g(t i)+ 8(t i), i= 1: n (3) changes its behaviour, without any hypothesis about the parametric form ofg which contains this change. In (3), e is usually supposed to be a zero mean and o~ variance independent normally distributed noise, which allows interesting theoretical results about the "optimality" of thresholding (Donoho and Johnstone, 1994). Donoho (1995) proposes to extract these significant coefficients by soft thresholding: 5~ =
0
0_< d~
(4)
where 8~ are the thresholded coefficients, d~ is given in (2) and ,r is the threshold value. Thus, the coefficients below their threshold are set to zero (they are assumed to represent the "normal behaviour"), while exceeding coefficients indicate the occurrence of a signal abnormal behaviour (Fig. 4), corresponding to the appearance of a fault in the process. Note that the signal behaviour change may affect the different decomposition level not in the same way. Therefore, 2 j is level dependent.
I , -~' I-"":"'"/ ..r- d~ ...............-"]0
2.j
Fig. 4: SWT coefficients soft thresholding The threshold choice is tricky. Several methods may be used, and a bibliographical study reveals many possibilities. The optimal choice requires knowledge (or at least hypotheses) about the analysed signal (Donoho and Johnstone, 1994; Taleb et al., 2004). In
Fuzzy decision making for diagnostic decision allows formal modelling of decision-making for imprecise and uncertain conditions, in order to select a solution characterized by partial points of view, called partial criteria (Dubois and Prude, 1985). In a known environment, each decision d ~ D (where D is the set of possible decisions) is evaluated by a series of values [Cl(d), c2(d),..., cp(d)] where c,{d) measures the decision d in the sense of criterion i. d is defined as a fuzzy subset obtained by aggregation of p partial criteria. Thus, the membership function Pd is such that:
fld = h(cl (d),cE (d) .... ,cp (d))
(7)
where h is a fuzzy set operator connective to be determined. Necessary conditions on h are: 9 h is a continuous function; 9 h(0,0 .... , 0 ) = 0 a n d h ( 1 , 1 , . . . , 1 ) = l ;
9 V(ui,vi)~[O,1] 2,ifui>vi then h(Ul,..., Up) > h(Vl,..., Vp). Three main decision-making attitudes can be modelled: 1) for an operator h expressing that all the criteria are met simultaneously, a natural axiom is: V(Ul,//2 .....
Up),
h(Ul,//2 .....
Up)<_n~in(ui) , i=l'p
(8)
which means that the overall evaluation of a decision cannot be better than the smallest (i.e. "worst") value of the partial evaluations. These operators correspond to conjunctions; 2) to express the redundancy of the objectives, h must meet the following condition: V ( U l , U 2 .....
Up), n~i (lli)<_h(Ul,U 2.... ,Up), i=l'p
(9)
which means that the overall evaluation is determined by the highest (i.e. "best") value of the partial ones. These operators are disjunctions; 3) h becomes a compromise when the following axiom is satisfied: .....
min(ui)<-h(Ul,U 2 ..... Up)<_maix(ui), i=l"p
1117
(10)
5. FAULT DETECTION 5.1 Proceduredescription
A fault occurrence is supposed to be revealed by a signal singularity as defined in Mallat and Zhong, (1999). The proposed detection method tries to detect these singularity appearances. It is composed of four steps (see Fig. 5).
Si'~'ai
SWT
~m]'i~--
~Aggregation H
i"......................... ..... .....i"-~ i
HThresholding ] Fuzzification ]
In order to obtain an instantaneous decision and specifying the moment of singularity occurrence (12) (13) and (14) are proposed. (12) allows a robust decision: it is significant only when the singularity propagates to all scales. (13) realises a compromise when the singularity is not present in all scales. (14) allows a sensitive decision: it takes a significant value when at least one threshold exceeding exists, at any level. To improve the robustness of indicators (12) to (14), (15) and (16) propose to first aggregate the fuzzified coefficients on a sliding window of length N. These operators allow enhancing the progressiveness of indicators (12) to (14). 6. DETECTION RESULTS
Fig. 5: Detection procedure
6.1 Data description
The first one applies the SWT to the signal. The second step corresponds to the discrimination between the coefficients that characterize the existence, or not, of a change in the signal behaviour. This step can be done thanks to a wavelet coefficient thresholding, where the thresholds are computed as explained in section 2. The third step is a fuzzification one. In diagnosis area, the classical use of the thresholded coefficients is a crisp one. Actually, a gradual detection is more interesting than a Boolean one (i.e. "there is/there is not a fault") because attention can be paid to a component before a fault is completely installed. Therefore, a fuzzification of the thresholded coefficients is implemented (Fig. 6): __
lail>
1 |
/a J (6~) =
!
2aJ~j
o
The benchmark data are provided by SMS-Demag (Deckers, et al., 2003). Several experiments, each one being made up of 5 records (so called passes of the strip) are used. Each record has six measurements (torque, current and speed of upper and low motors) sampled with frequency fe = 250 Hz. Due to fe, only "low" frequency vibrations can be detected (Seelinger, et al., 2002). The data set has been split in two, namely "normal" behaviour and "faulty" one. Note that an abnormal vibration induces extra frequency contents in signals that are conspicuous in the torque spectrum. For this reason, only the upper drive torque measurement ~ is considered in this paper. The vibrations are growing with the pass number. Thus, the study is restricted to the 5th pass. Fig. 7 shows ~ for one experiment. Note that the aim is to detect the increase of particular vibration modes that lead to a decrease in the surface roughness of the rolled strip. U p p e r t o r q u e , 5 p a s s e s , n o r m a l operation I I I I I L I I
2.5 x 106
a/= o
2
where a J is a parameter that defines the membership function of the coefficients
-
I I 1.
. . . .
.
.
I. . . . . . I I. . . .
.
|
1
/1/
.....................................
t L ~
1.5
~.0.5_
-~. . . .
.
.
.
.
.
.
.
.
. . . . .
.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -0.5.1
-
-
-
-1.5 I _2aJ2j
_ajgj
0
aJ,~
2a~gJ
"20
Fig. 6: Thresholded coefficients fuzzification In order to give a progressive, but also unique indicator, the fourth step considers the various fuzzy coefficients as partial criteria and ensures their aggregation in different ways: D1 k = m!n(p~); j = l ' J (12) J D2 k = mean(p~); j = l ' J
(13)
D3 k = m a x ( P ~ ) ; j = 1" a
(14)
]
(15)
D5k =max. {max(l-Jk_,);l=O'N-1};j=l'J
(16)
]
J
20
T i m e [s]
4~0
30
50
Fig. 7" Upper torque (in Newton) of a "normal behaviour" experiment (5 passes) 6.2 Choice o f the Wavelet
When time-scale methods are used, the relationship between scale and frequency is expressed through the pseudo-frequencyf~ (in Hz) corresponding to a given scale a. fa is computed thanks to the normalised "centre frequency"fc of the wavelet (Abry, 1997): f~ = fc
J
D4k =m~. {mean(lu~-t);t=O'N-1};j=l'J
10I
axT e
where Te
=
1/fe
(17)
The underlying idea is to associate with a given wavelet a purely periodic signal of frequency f~ that maximizes the FFT of the wavelet modulus. Thus,
1118
the centre frequency-based approximation captures the main wavelet oscillations. Therefore, fc is a convenient and simple characterization of the leading dominant frequency of the wavelet. A preliminary frequency analysis has shown that the torsional vibration that must be detected mainly occurs around 11.6 Hz. Therefore, the wavelet should possess a pseudo-frequency fo around 11.6 Hz (17). The Symmlet6 (Daubechies, 1992) "oscillates" with this frequency at the 4 th level of decomposition. This choice is not unique (e.g., the Coiflet2 wavelet has the same property) and is also related to the sampling period. The maximum level of decomposition must fulfil J > 4 but it must be moderate because of the computational cost. Here, some experiments have shown that J = 5 is sufficient to ensure a good detection of the faulty oscillations. Moreover, the symptoms D4 (15) and D5 (16) are computed with a sliding window of length N = 32.
6.3 Results Fig. 8 exhibits the upper torque Fu of the 5th pass for an unfaulty record, its SWT decomposition (5 levels of decomposition) and the thresholds computed with equation (6) using an unfaulty signal. x 10 5
Wavelet
decomposition
appears are depicted: this constitutes the second experiment. The thresholds are identical to those used for the 1 st experiment. Fig. 11 shows the symptoms D1 to D5. The symptoms computed with D1 (12) and D2 (13) are not sensitive to this faulty situation. As expected, D3 (14) and D5 (16) are highly sensitive and D4 (15) has an intermediary (robust) behaviour. Symptoms
x I 05
10
-
'
. . . .
1
,
r
,
lk D1
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. . . .
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lrll iii - .- iill
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q
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Wavelet decomposilion of the f a u l t y sigrlai
,-
1[ . . . . . . . 0 ~__-~-------
-21
I
-
-
1
T
I . . . .
,
4
Oo
d5
I
"1 I. . . .
-
--
T
. . . .
,
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-{
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~
1. . . .
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1
I I. . . . .
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20
:-- :-- :7'--
I
i
-r
. . . .
,
-,
-II-
-1~--
1. . . .
~
,
'
---I -
-~
-
--t
Fig. 9: Symptom generation, experiment 1
d1
i
+. . . .
'
~ ~Os~'-----7-P l-J~-ltlt-.-,I . . . . . ~I ~ - - - ~ . . . . I- . . . . ~ ''lLll-rul--I~n
of the unfaulty signal
--T
.
-r'-----
[-
.
.~/,,;:
d3
. '. . . . . .
i. . . . .
ok
D2
'
,
..
....
, ....
'7"
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,l
,
",
'-t'
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7
-
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-
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, ....
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E::-__----:_L----::27--'__ /:'--'.-----:
.
i . . . . .
'--
-
,...
S--:Z-----:-
,,,i........ Jr"
-
,
j
.
'
"l
T i m e [s]
8
. "
'
. "
i
. . . .
i"
"
il.i
I
"i'T
10
I
12
Fig. 10: SWT (faulty case), experiment 2 Symptoms
x 10 s
Fig. 8: SWT (unfaulty signal), experiment 1 D1
Fig. 9 depicts the symptoms Dk, k = 1 to 5, generated with operators (12) to (16) by considering only the levels j = 3 : 5 ( j = l , 2 do not clearly exhibit information on the faults to be detected). First, a jump in the torque must be detected: it corresponds to the time the slab is starting to be rolled out. Then, a transient period is elapsed before any significant symptom is computed. Therefore, at the beginning of the record, Dk = 0.5 which mean that the symptom is not evaluated. Dk =0 stands for "no change has been detected" while D k = l means that "the coefficients have significantly exceeded the thresholds, therefore, a important change has occurred". As expected, for an unfaulty situation, the indicators are nearly zero. Some transient nonzero values can be seen on D3 and D5 that are more sensitive than the other ones. Note that the torque nominal value F~0 depends on the experiment, and is not always constant. The procedure adapts this situation but the results are confidential. In Fig. 10, the upper torque F~ and its SWT decomposition when a vibration
O.
i. . . . . ti i
. . . .
11 . . . . .
.
.
.,
I-
-
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l?
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.
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+
.
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.
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.
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.
.
.
i
o+ oil .... . . 0
. 2
.
.
. 4
.
. 6
. T i m e Is]
. 8
10
12
Fig. 11: Symptom generation, experiment 2 In the third experiment, the vibrations become more important. Fig. 12 depicts this signal with its SWT decomposition. Fig. 13 shows the symptoms D1 to D5. It can be observed that D1 (12) is no more sensitive to the fault while D3 (14) is very sensitive. (15) and (16) correspond to an integral of the instantaneous indicators. Thus, they are less corrupted by local variations. The evolution of (15) and (16) reflects the periodicity of the fault; they take significant values every pseudo-period during the pass.
1119
7. CONCLUSION AND PROSPECTS
REFERENCES
In this paper, the capability for the stationary wavelet transform to deal with torsional vibration fault detection in a rolling mill has been investigated. A detection procedure based upon a wavelet coefficient thresholding that is level-dependent has been investigated. The coefficients are fuzzified and aggregated in order to provide a symptom. The tuning parameters of this procedure are the wavelet itself, the number of decomposition levels, the thresholds and the thresholding method. The wavelet choice depends on the features that must be detected in the signal under analysis. This selection is sometimes not unique. For detection purpose, the final choice is made in order to maximize the symptom sensitivity. This method has been customized for data that have been recorded on the main drive of a roughing mill, which is a part of a hot rolling mill. The first results that have been obtained are really encouraging. However, new aggregation methods of the fuzzified wavelet coefficients and the choice of wavelet are under study in order to take into account the pseudofrequency property of the analyzing wavelet.
Abry P. (1997), Ondelettes et Turbulence. Multir~solutions, Algorithmes de D~composition, Invarianee d'~chelles", Diderot Ed., Paris, (in French). Basseville M., Nikiforov L. (1993), Detection of abrupt changes, Prentice Hall Ed. Boashash B. (edited by) (2003), Time frequency signal analysis and processing, a comprehensive reference, Elsevier Ed. Coifman R.R, Donoho D. (1995), Translation Invariant De-noising, series Lecture Notes in Statistics, New York Springer-Verlag, Vol. 1t)3, pp. 125-150. Daubechies I. (1992), Ten lectures on Wavelets, SIAM Ed., Philadelphia, Pennsylvania. Deckers J., Jepsen O., Latzel S., Metz H., Stuecher R. (2003), Condition monitoring and failure diagnosis in plants of the metal industry, IFAC Safeprocess, Washington D.C., USA. Donoho D. (1995), De-Noising by soft-Thresholding, IEEE Trans. on Information Theory, Vol. 41(3), pp. 613-627. Donoho D., Johnstone I. (1994), Ideal Spatial Adaptation by Wavelet Shrinkage, Biometrika, Vol. 81. Dubois D. Prade H. (1985), A review of fuzzy set aggregation connectives, Information Sciences, Vol. 36, pp. 85-121. Flandrin P. (1999), Time-frequency~time-scale analysis, Academic Press Ed., San Diego, California. Isermann R. (1997), Supervision, fault-detection and fault-diagnosis m e t h o d s - An introduction, Control Eng. Practice, Vol. 5(5), pp. 639-652. Lesecq S., Barraud A. (2003) Arcing fault detection using wavelet transform, IFAC Safeprocess 2003, Washington D.C., USA. Mallat S., Zhong S. (1992), Characterization of Signals from Multiscale Edges, IEEE Trans. on Pattern Anal. and Machine Intelligence, Vol. 14, pp. 710-732. Peng Z.K., Chu F.L. (2004), Application of the wavelet transform in machine condition monitoring and fault diagnostics: a review with bibliography, Mechanical Systems and Signal Processing, Vol. 18, pp. 199-221. Pesquet J.C., Krim H., Carfatan H. (1996), TimeInvariant orthonormal wavelet representations, IEEE Trans. on Signal Processing, Vol. 44, pp. 1964-1970. Seelinger A., Mackel J., Georges D. (2002), Measurement and diagnosis of preprocessdisturbing oscillations in high-speed rolling plants, IMEKO 2002, Tampere, Finland. Strang G., Nguyen T. (1996), Wavelets and Filter Banks, Wellesley-Cambridge Press. Taleb S., Lesecq S., Stuecher R. (2004), Torsional vibration detection using torque measurement of a main drive of a hot rolling mill with wavelet transform, IFAC MMM'04, Nancy, France.
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Fig. 13: Symptom generation, experiment 3 ACKNOWLEDGEMENTS The authors would like to thank SMS-Demag which allows using the data provided for the European project EU-IST-2000-30009 MAGIC.
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