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 1. Stephanopoulos, G, and Han,C. 1996. Intelligent systems in process engineering: a review. Computers Chem. Engng 20(6/7), 743-791. 2. Cheung, J.T.Y, and Stephanopoulos, G 1990. Representation of process trends. Computers Chem. Engng 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. 9. Wu L., Xie Z, 1990. Scale theorms for zero-crossings. IEEE Trans.Pat.Anal.Mach. Intel., 12(1), 46-54. 10. Zhao Zhong, Jiang Weisun and Gu Xingsheng. 1998. Application of wavelet transform to process monitoring. Journal of Chemical Industry and Engineering (China), 49(6), 668-674. Received: 10 June 1999; Accept after revision: 12 May 2000.

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