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Application of Wavelet Transform to Process Operating Region Recognition.

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Dev. Chem. Eng. Mineral Process., 9(IL2). pp.89-100,2001.
Application of Wavelet Transform to Process
Operating Region Recognition
Zhong Zhao' and Yihui Jin
Department of Automation, TSinghua University, Beijing 100084,
I? R. China
Based on the analysis ofwavelet transform,we studied how to extract thefiequency
band and multi-scale qualitative features of process measurement signals. It was
proved that the extractedfeatures could describe the process peflormance region
in a stable and complete w q .Combining the pattern inductive algorithm, a method
of process region recognition based on wavelet transform has been proposed.
Results of an actual application have verijied itsfeasibiliq and effectiveness.
Introduction
Chemical processes are large-scale, complex systems operated under close monitoring
and control by trained operators using computer-based control systems. During the
c o m e of normal, steady-state operations, simple observation of scores of displays is
sufficient to confirm the process' status. But, when the process is transient or crises
occur, the dynamic evolution of displayed data can confound even the best of the
operators. It is often very difficult for human operators to cany out the following
process monitoring tasks [ 11: (1) Distinguish normal fiom abnormal operating regions;
(2) Identify the causes of process trends; (3) Evaluate current process trends and
anticipate hture operating states; (4) Plan and schedule sequences of operatingsteps to
bring the plant to the desired operatingregion. The key cognitive skill, required to cany
out the above tasks, is the formation of a "mental" model of the process operation that
fits the current facts about the process and enables the operators to correctly recognize
the process operating region [2]. The "mental" model, which describes the relationship
of the process representation space and the process operating region, can be built based
on the extraction of the process features fiom the measured process data [3]. To
establish such a "mental" model, we need to resolve the interrelated questions: (1) How
to extract the prominent, stable and complete process features which denote the process
operating regions fiom noisy measured data? (2) How to determine the relationship of
the extracted process features and process operating regions? In the following sections,
we present a systematic approach to resolve the above questions using the wavelet
transform.
*Authorfor correspondence (e-mail:zhaoz@roc. au.tsinghua.edu.cn).
89
Z. UUU, and
Y.Jin
Description of Process Operating Region Recognition Based on
Pattern Recognition
Process operating region recognition via the analysis of the process measured data can
be regarded as the problem of pattern recognition. Define P as a pattern of process
representation space Sx, then P = [ x ,,x2,...,x M] where x i ,i = 1,...,
are the
process measured signals, Mis the number of the measured signals. With the extraction
of the process features, the pattern P of the process representation space S, can be
classified into k mutually exclusive feature subspaces corresponding to different
I
satisfies the following
process operating region C,. The feature subspace
conditions,
sx
...(1)
S , ' n S X J =O,I#J,I,J=1,2
k
...(2)
,..., k
.
us,' =sx
...(3)
I =1
wheref* is the mapping function of process features. The classification rule of process
representation space Sx can be described by the inductive discriminate function as
d ( ~E )R(s,) . The pattern P of process representation space implicates the
operating region C, if d, (P) > dJ(P), J
+ I , I , J = 1,..., k .
Process Operating Region Recognition via Wavelet Transform
The real process measured data are nonstationary signals, which include the process
feature signals in different regions on the time-fiequency plane. The recent
development of the wavelet transform provides a formal fiamework for representing a
signal in the time-frequency domain, therefore we can apply the wavelet transform to
the process feature extraction from process measured data.
According to the principle of multi-resolution analysis [4], when choose the basis
scale
function
&)
and
wavelet
function
~ ( r ) properly,
( ~ ~ ( t ) = 2 - / ~ 2 - l ? - k ) j = ...j,,
i o + kl € Z ) and {@, k =2-Jm&2-Jmf-k),kEz ) Can form
I))'
the Riesz basis in L2(R) space, wherej is called scale factor, k is called location factor.
Then, any r(t)E L2(R) can be expanded as,
...(4)
...(5 )
...(6 )
The wavelet coefficients dx(j,k),j = j o+ 1,...,j,,, and the scale coefficient cx(j,,,,k)
90
Application of Wavelet TrMsfonn to Process Operating Region Recognition
are called the wavelet transform of x(t). They can be calculated with the fast wavelet
transform algorithm of Mallat [5].
According to the Parseval Theorem [4],the wavelet transform defined in (5) and
(6) can be rewritten as:
h
A
where x(w), v / ~ , ~ ( and
w ) ,4 , m k ( ~are
) the Fourier transforms of x(t) , ty,&(t), and
41m/c(t)respectively.
Definition 1: The energy distribution feature of the wavelet transform of process
measurement data is defined as :
where L is the width of the chosen data window.
Definition 2: The fluctuant feature of the wavelet transform of the process measured
data is defined as,
where d A j m+l,k,m+l)= cx(jm,k,m1.
The energy distribution feature defined as Definition 1 reflects the macro energy
distribution feature of a process measurement signal in the different wavelet subspaces.
Since v / , , ~ ( wis) not the ideal band-pass function, then there are cross-sections in
A
different frequency bands of v, , especially between the adjacent fiequency bands.
Therefore, the frequency band distribution features are reflected both in the absolute
and relative values of energy distribution in the different wavelet subspaces. The
fluctuant feature defined in Definition 2 just describes the micro wave shapes of
process measurement signal in different wavelet subspaces.
Lemma 1 [6]: Define the dyadic wavelet transform of x(t) with discrete scale and
continuous time as,
W,(2J,-T)
=1 jx(t)w(-M,
t --T
2J
6
when
w(t)
j = 0,l)....
...(1 1 )
is a real function, for the stationary white noise n(t) with variance
02,
Wr,(2’,7) is a stationary process and the energy distribution of Wr,(2’,7) is
91
Z. Zhao and Y.Jin
independent of scalej ,
j = O,l,...
..
Iklr
where
is the integration of v2(t)in time domain.
Theorem 1: When w ( t ) is a real function, for the stationary white noise n(t) with
variance c 2 ,d,,(j,k) using the wavelet transform defined in equation (2) is a
stationary process and its energy distribution is independent of scalej,
Eld,(j,k)l2 = o2Ily112,
j = 0,I,...
...(13)
Proof: At every scalej, wT,(2’,1) is a stationary stochastic signal, then its statistic
characters are independent of the time variable 7 . Since d,,(j,k) is the sample value
of WT,(2-’,r)at the instant 7 = 2’k, (k = 1,2,3,...) ,then, according to the Lemma, we
can get equation (1 3). This completes the proof.
From Theorem 1, Definition 1, and Definition 2, we can see that the energy
distribution feature and fluctuant feature of white noise are independent of scale factor
j . As the wavelet transform can extract the features of fiequency structure of process
measured data and it has sparse expansion character in scale space, then the extracted
energy distribution feature and fluctuant feature with wavelet transform which reflect
the changes of frequency structure of process measured data are very sensitive of scale
factorj. Therefore, using the pattern inductive algorithm, we can determine the main
resolution range of scale factorj , in which the extracted energy distribution feature and
fluctuant feature of process measured data are affected by the noise only slightly.
The closed operating regions can be described prominently and stably with the
memorable fiequency band feature, but it is usually very difficult to describe the
unclosed and semi-closed operating regions, which are usually unknown operating
regions, with quantitative analysis method [7].
Definition 3 [2]: A qualitative pattern of process measured data is a time interval where
the sign of the second derivative is a qualitative constant (+,O,-). It is bounded on both
sides by inflection points, the geometric prominent point is the local extreme, then a
qualitative pattern can be described as,
where ( t l l , x l l ) , ( t 1 3 , x 1 3 are
)
the inflection points, (t12, x 1 2 ) is the local extreme in
(Il,,tl, ) , comt denotes the qualitative constant.
According to Definition 3, the process trend can be qualitative described by the
nonoverlapping sequence of qualitative patterns. Therefore, we can explain the
extracted process qualitative trend with the real operating knowledge to aid the operator
to deduce the unknown operating regions.
Definition 4: If the two quditative pattern sequences of process measured data
p = ( p I , p 2...,
, prl and q = (q1,q2,
...,qsl satisfy r = s,P, = qI = { ~ l l 1 2 1 3 ( ~ ) } =y ~1,2,..r,
92
Application of Wavelet Transform to Process Operating Region Recognition
we call p is qualitativly equal to q (p<QE>q),
which means they are in the same
operating region.
From Definition 3, we can see that the qualitative patterns are defined by the local
extrema and inflexion points. Choose the smooth function 4 ( t ) as scale function, the
first order derivative of 4 ( t ) as wavelet function, then ~ ( t= )@ ( t ) / d t . According to
the definition in (5), (6) and the property of smooth function [8], we can get the wavelet
transform with scalej, as follows,
d,(/,k)=
X(Q*V,,k(O
=
w *d( diA
(1) =
- d( x ( o * 4 , . k )
df
...(14)
...(1 5 )
c , ( j , k ) = x(t)*4,,k(t)
where * denotes the convolution. From equations (14) and (1 5), we can see that the
local extrema of process measured data at scalej appear as the local extrema of scale
coefficients at scale (j+1) and the inflexion points of process measured data at scale j
appear as the local extrema of wavelet coefficients at scale (j+1). Therefore, according
to the local extrema of scale and wavelet coefficients at every scalej ( j o+ 1I j 5 j,) ,
we can construct the multi-resolution qualitative trend of process measured data.
Lemma 2 [9]:According to the principle of multi-scale analysis, with the increment of
scalej, the local extrema of scale and wavelet coefficients disappear gradually, but new
extrema of scale and wavelet coefficients will not appear in the time interval where the
local extrema disappear.
According to Lemma 2, the local extrema of scale and wavelet coefficients are
stable of scale j . Furthermore, we can reconstruct the original signal from the
qualitative trend with the projection algorithm [ 6 ] . But it has been proved that the
original signal can not be uniquely reconstructed just with the sequence of the local
extrema of wavelet transform [8]. Since the real operating trajectories are allowed to
change in limited ranges, if we can prove the deviation of reconstructed signal and the
original is bounded, then we can say the extracted qualitative trend completely describe
the operating regions.
Theorem 2: Let x = {x(n)>f==,denote the process measured data in the data window,
(W,X);:=, denote the wavelet transform of x , vx ={max(lW,xl)~~,}denote the
sequence of the local extrema of (W,x)$, , h = {/~(n)}f=~denote the reconstructed
signal with the projection approximation algorithm [6] from vx, (W,h);:,
wavelet transform of h, vh = {max(lW,hl)::,
of (W,h)::,
denote the
denote the sequence of the local extrema
,then
1p.l - x[j5 kll=II, k '0
where 1 -1 denote the Euclidean norm of linear space.
...(1 6 )
Proof: According to the theory of wavelet frame [4], there exist k,,k, > 0, which
satisfy:
93
Z UUK, and Y.Jin
J=1
According to the projection approximation algorithm [6], we can get:
vh = vx
...(19)
Iwjx(n)l I maxlwjx(n)l
I (IvxII
n
(wJh(n)(
I rnaxl~~h(n)(=m a x l ~ ~ x ( n111~x1
)(
n
n
...(20)
...(21)
...(22)
...(23)
IIWjxllI fiIIvxII
According to the inequalities (17), (1 8) and inequalities (22), (23), it is easy to see that
there are k3,k4 > 0 , which satisfy:
...(24)
...(25)
...(26)
In order to determine the relationship of the extracted process features and process
operating regions, we can use the pattern inductive learning algorithm to leam the
relationship, the learning procedure is as follows:
(1) According to the operating regions, choose the average values of energy
distribution and fluctuant features at different scales as the classifying values. Choose
10% as the permitted deviation to construct the classifying sample set.
(2) Choose a sample from the sample set. Consider a classifying feature (energy
distribution or fluctuant feature at different scalej), if all the samples that include the
feature are in the same operating regions, then the feature forms a rule in the classifying
rule set, otherwise it forms a rule condition in the partial rule set. The classifying
information of a condition is defined as:
1
l-~(I-E,cIE)(l-EIE)
...(27)
where EC is the number of samples which are covered by the rule condition, E is total
number of samples, E,' is the number of samples which are correctly divided into the
ith operating regions by the rule condition, E, is the number of samples which belong
to the ith operating region.
(3) According to H, add new rule conditions sequentially until complete
classification or the number of rule conditions reaches the chosenthreshold. According
94
Application of Wavelet Transform to Process Operating Region Recognition
to the final rule conditions, form the new Classifying rule.
(4) For a new sample, if there is only one rule covers the new sample, then the new
sample is regarded as the correct classification sample. If there are more than one rule
covering the new sample, then choose the rule with the biggest Has the classifying rule.
If there is no rule covering the new sample or the sample is classified incorrectly, then
the new sample is regarded as the classifying sample, repeat the classifying step (2) and
step (3).
( 5 ) According to the classifying rule set, determine the main resolution range J , of
scale factorj and the nominal values of frequency band features for classifying the
operating regions.
The procedure of process operating region recognition based on wavelet transform
can be summarized as follows:
(i) Choose time instant t o ,calculate kJo+l,kJo+2,...,kJm
that satisfy:
...(28)
(ii) At off-line, calculate fiequency band distribution features of different process
operating regions according to Definition 1 and Definition 2. Classify the extracted
fiequency band features with the proposed pattern inductive learning algorithm and
determine the main resolution range j&, ...j,,
jo
,,
~,j l o , j l 5
, ,j,,,)
,
and the nominal
values of frequency band features to form the rule set of classifying the operating
regions. According to Definition 3, construct the multi-resolution qualitative trend of
different process operating regions.
(iii) At on-line, with fixed center time of wavelet transform mode as equation (281,
calculate the frequency band distribution features in the main resolution range j , .
According to the rule set of classifying the operating regions, classify the operating
regions. The operating regions, which can not be explained by the classifying rule set,
are regarded as the unclosed operating regions. Construct their qualitative trends
according to Definition 3. According to Definition 4, compare the unknown
qualitative trends with the known qualitative trends. Explain the unmatched qualitative
patterns with operating knowledge.
Case Study
The research process is the FS-4OG copolymerization process [ 101. According to the
real operation, the normal operating regions can be divided into: (i) early reaction stage;
(ii) middle reaction stage; (iii) final reaction stage. If the process was operated
improperly, there would be two fault operating regions: (i) irregular chain propagation;
(ii) gel effect. But since the process is very complex, it is very difficult for the operators
to recognize the above operating regions directly. The character trajectories of different
operating regions are shown in Figure 1, where the sample interval T, = 10s .
95
Z Zhao and Y. Jin
sru,
Lregulschmnpropasdon
I
(I1
r
no
1
\t
I
k
h
wo
I
h
s
Figure 1. Character trajectories at dgerent regions.
The distribution and fluctuant features of the character trajectories of different
operating regions are shown in Figures 2 and 3. We can see that the extracted
hquency band features by wavelet transform of different operating regions appear
obviously change. Apply wavelet transform to the real operating samples and classify
them with the proposed pattern inductive learning algorithm to construct the
classifying rule set of operating regions, where the numbers of the learning samples at
five operating regions are 10, 19, 14, 16, 12 respectively, while the numbers of the
verifying samples are 8, 19, 14, 16, 10 respectively. After the learning, the following
operating regions recognition rule set are obtained.
96
Application of Wavelet Transformto Process Operating Region Recognition
Figure 2. Energy dktributionfeatures.
Figure 3. Fluctuantfeatures.
97
RULE1 :
IF
THEN
RULE2:
IF
THEN
RULE3:
IF
THEN
RULE4:
[0,--0.37291
AND [S2=2.7620]
[CiIrregular chain propagation] H-0.2655
[03=0.3518]A N D [04=0.3412]A N D [5',=4.9901]
[C,= Gel effect] H4.3876
[04=0.3619]A N D [03=0.2339]AND [&=3.6604]AND [S3=1.5472]
[C,= Early reaction stage] H4.2315
IF
[043.311 11 AND [4=0.2442]AND [0,=0.2362]AND [S2=1.3948)
THEN [C,= Middle reaction stage] H4.2741
RULES:
[D4=0.3942]AND (0,=0.24]AND [D3=0.2341]AND [S2=3.1592]
AND
IF
[S3=l.68391
THEN [C,= Final reaction stage] H4.3934.
The comparison of the recognition results with the recognition rule set, the neural
networks with sigmoid as activation functions (SBFN),the neural networks with radial
basis as activation functions (RBFN) and the wavelet based neural networks with
nonlinear PLS structure [lo] are shown in Table 1. From Table 1 , we can see that the
proposed operating region recognition method has strong classifying ability and the
classifying knowledge can be represented and explained with the classifying rule set.
Therefore, the classifyingknowledge can be used for fiuther reasoning analysis.
Table I . Comparison of the operating region recognition results.
Accuracy (%)
Wavelet transform
SBFN
RBFN
WBFN with nonlinear
PLS structure
Learning
samples
100.0
92.72
96.34
100.0
Verifying samples
97.5
84.57
91.25
98.50
To verify the recognition ability of unclosed operating regions, add the ramp
function to the middle reaction stage character trajectory to simulate the sensor drift
fault, corresponding trajectories are shown in Figure 4.
98
Application of Wavelet Transform to Process Operating Region Recognition
eloh
I!
k
k
Figure 4. Character trajectories of middle stage and unknown operating region.
Because of the affects of process noise, it is difficult to recognize the change of
process operating region and also we can not recognize the unknown operating
regions with the classifying rule set. Therefore, we can use the sequence of qualitative
patterns to infer the change of the operating regions. The qualitative patterns are
shown in Figure 5 and we can see that there is an abnormal rising pattern in the
process measured data. Therefore, we can infer that there may be a sensor drift fault.
7o
690
10
1
20 30 40
50
m
70
k
j
a
m miwiioi~
j
0
-
10 20 30
a
5O 60 70
en
I
j
90 1w110120
k
Figure 5. Qualitative trenh of middle reaction stage and unknown operating region.
Conclusions
In this paper, we have proposed a systematic method of recognizing the process
operating regions with wavelet transform. From the above analysis and case study, we
make the following conclusions. (1) The frequency band distribution features of
nonstationary process measured data can be extracted with wavelet transform. The
extracted fiequency band distribution features have prominent and stable characters.
(2) The multi-resolution qualitative trend of process measured data can be constructed
with wavelet transform. The constructed qualitative trend can describe the operating
regions in a stable and complete manner. (3) The relationship of process frequency
99
Z Urao and Y.Jin
distribution features and process operating regions can be learned by the proposed
inductive learning algorithm and represented with classifying rule set. (4) The real
application results have verified the feasibility and effectiveness of the proposed
method.
Acknowledgement
This work was supported by the National Post-Doctoral Science Foundation of China.
References
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14(4/5), 495-510.
3. Bakshi, B. R., and Locher G 1994. Analysis of operating data for evaluation, diagnosis and control. 1.
Proc. Contr. 4(4), 179-194.
4. Chui C. K. 199O.An introduction to wavelet. Academic Press: New York, 25-32.
5. Mallat S. 1989. A theory for multiresolution signal decomposition: the wavelet representation. IEEE
Trans.Pat.Anal. Mach. Intel. 11(7), 674-691.
6. Mallat S. 1992. Characterizationof signals 60m multiscale edges. IEEE Trans.Pat.Anal. Mach.Intel.
14(7),710-732.
7. Kuipers B.J. 1989. Qualitative reasoning: modeling and simulation with incomplete knowledge.
Automatica, 25(2), 571-585.
8. Zhao Zhong,Jiang Weisun and Gu Xingsheng,. 1999. Process monitoring based on wavelet transform,
Journal of Control and Decision (China), 14(1),19-24.
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Received: 10 June 1999; Accept after revision: 12 May 2000.
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