US20170061305A1 - Fuzzy curve analysis based soft sensor modeling method using time difference Gaussian process regression - Google Patents
Fuzzy curve analysis based soft sensor modeling method using time difference Gaussian process regression Download PDFInfo
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- the invention relates to a fuzzy curve analysis based time difference Gaussian process regression soft sensor modeling method (FCA-TDGPR), and belongs to the field of complex industrial process modeling and soft sensing.
- FCA-TDGPR fuzzy curve analysis based time difference Gaussian process regression soft sensor modeling method
- the traditional soft sensor modeling methods mostly consider the characteristics of zero delay, that is, considering the input and output with the same sampling interval and input variable collected at time t corresponds to the t-th dominant variable sample in the database.
- the existing time delay estimation methods are trying to find the auxiliary variables which are most closely related to the dominant variables for modeling. When applying such methods to practical applications, it is needed to get a tradeoff between the complexity and the accuracy of the algorithm.
- the existing methods include mutual information (MI) method, correlation coefficient method, etc.
- MI mutual information
- the invention adopts fuzzy curve analysis method to introduce variable time delay information into the soft sensor model, and the characteristic of this method is low computational complexity while being easy to understand. It is possible to visually and effectively determine the importance degree of the input variables.
- the performance of soft sensor model needs to be maintained by periodic reconstruction in order to dynamically track and effectively take control of the process.
- the main methods include Moving Window (MW), iterative approach and Just-in-time Learning (JITL), but these methods often need to update the model frequently.
- Time difference model not only can deal with the problem of model performance degradation due to time lapse and achieve the effect of tracking process dynamics exactly as model updating it also is able to minimize the likelihood of model reconstruction.
- the soft sensor model without delay can't be modeled by the sequence of auxiliary variables which are the most relevant to the dominant variable.
- the estimation accuracy of the dominant variable will be greatly affected.
- this invention provides a soft sensor modeling method based on FCA-TDGPR.
- Soft sensor modeling method based on FCA-TDGPR which comprises the following steps: aiming at the time delay process, first of all, collect enough samples of data and constitute the historical database with the historical values of auxiliary variables and dominant variable.
- the FCA method is used to determine the time delay parameters of each auxiliary variable with respect to the dominant variable sequence, which is used to reconstruct the soft sensor modeling data;
- TDGPR time difference Gaussian process regression
- FIG. 1 is a flow chart of online soft sensor method based on FCA-TDGPR;
- FIG. 2 is a schematic diagram of debutanizer process
- FIG. 3 is a schematic diagram of TDGPR modeling approach
- FIG. 4 contains fuzzy curve distribution diagrams of the original variables and optimal time delay variables
- FIG. 5 contains scatterplots of butane concentration predictions with different j values.
- debutanizer is an important part of naphtha desulfurization and separation device of oil refining production process, and one of the dominant variables needed to be controlled for this process is the concentration of the bottom butane (C4).
- C4 concentration of the bottom butane
- FIG. 2 The schematic diagram of the process is shown in FIG. 2 , due to the value of C4 cannot be directly measured, therefore, there is a delay issue in analyzing and obtaining C4 concentration values.
- different auxiliary variables show different degrees of time delay.
- Experimental data is derived from the actual industrial process which contains 2394 samples, a total of 7 auxiliary variables. As shown in FIG.
- x 1 is the top temperature
- x 2 is the top pressure
- x 3 is the reflux flow
- x 4 is the top product outflow
- x 5 is the 6th tray temperature
- x 6 is the bottom temperature 1
- x 7 is the bottom temperature 2.
- the 1 dominant variable is the bottom butane concentration, which in this invention is predicted as the key variable of the process.
- the maximum time delay T max of 6 variables is set to 19.
- Step3 Determine the importance of each variable in the time delay input variable set by FCA, for (x i (t),y(t)), fuzzy membership function of variable x i is defined as:
- fuzzy curve C i, ⁇ with the condition that ⁇ is the i-th variable delay value can be obtained by making centroid defuzzification of each new expanded variable using formula (3); as shown in the formula (4), d i is the ⁇ which can make the maximum coverage of fuzzy curve C i, ⁇ ; C i, ⁇ ( ⁇ ) max is the maximum value of the fuzzy curve point range, while C i, ⁇ ( ⁇ ) min is the minimum value of the fuzzy curve point range;
- the input variable x i (t ⁇ ) is more important.
- the importance degree of each variable can be determined by sorting the coverage of C i, ⁇ ( ⁇ ).
- the optimal delay parameter d i as well as time delay variable x i (t ⁇ d i ) can thus be obtained by FCA method, which later on can be used for soft sensor modeling data reconstruction.
- Step4 Based on the previous step, the time delay parameters d 1 , d 2 , . . . , d m are used to reconstruct the training input sample set for on-line modeling, the reconstructed input dataset is denoted as X d (t), X d (t)[x 1 (t ⁇ d 1 ), x 2 (t ⁇ d 2 ), x 3 (t ⁇ d 3 ), x 4 (t ⁇ d 4 ), x 5 (t ⁇ d 5 ), x 6 (t ⁇ d 6 )]. If there is a new input sample X(t+1), then the delay input set could be restructured based on historical database samples with the same parameters, then go to step 5, otherwise, wait for the arrival of new data.
- Step 5 After the reorganization procedure, the training set and the new data are processed by j order time difference treatment (the value of j can be determined according to the sampling period and property of dominant variable):
- the GPR method can obtain the mapping relationship through the given training input and output samples. In this way, the corresponding predictive value and the uncertainty degree can be obtained given the new input data, which means the result will be probabilistic.
- the GPR algorithm is shown as below.
- the mean function and covariance function are determined, then the distribution of the Gaussian process is well-determined.
- the mean function is usually preprocessed into 0.
- Covariance function can transform the correlation of output data into the function of input data. As similar inputs produce similar outputs, the covariance function can be selected according to the characteristics of the sample distribution. One condition which must be satisfied is that the closer the distance of samples is, the more correlated the two samples are, and vise versa.
- the covariance function form of this invention is shown in formula (7):
- x p , x q ⁇ R D , v controls the magnitude of the covariance function.
- ⁇ d describes the relative importance of each input attribute x d .
- the optimization of the parameters can be realized by using the conjugate gradient method.
- test data x * Based on test sample and training data, the posterior distribution of test data x * can be calculated, and its predictive value obey the joint Gaussian distribution described in formula (9), where K(X,X) is n-dimensional covariance matrix of training samples; k(x * ,X) is the covariance vector of test sample and training samples; k(x * ,x * ) is the autocovariance of test sample, and f gp is a predictive value of GPR.
- the first 1519 samples are selected in 2394 samples to reconstruct 1500 training samples.
- the final 875 samples are then used as test samples, and a soft sensor is established for on-line prediction of butane concentration.
- FIG. 5 contains scatterplots of butane concentration prediction results with two methods respectively denoted as the FCA-TDGPR method (present method) which involves delay estimation, and t-TDGPR method which is without time delay estimation.
- the predicted results of the present invention can be better close to the true value of the butane concentration when the time difference increases. This suggests that extracted delay information is in line with the actual causal relationship of the process, and the soft sensor model with variable time delay estimation is more accurate.
- FIGS. 4 and 5 have jointly validated that fuzzy curve analysis based time difference Gaussian process regression soft sensor modeling method has good accuracy for on-line prediction of bottom butane concentration.
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Abstract
Description
- This application claims the benefit of priority to Chinese Application No. 201510541727.5, entitled “A fuzzy curve analysis based soft sensor modeling method using time difference Gaussian process regression”, filed on Aug. 28, 2015, which is herein incorporated by reference in its entirety.
- Field of the Invention
- The invention relates to a fuzzy curve analysis based time difference Gaussian process regression soft sensor modeling method (FCA-TDGPR), and belongs to the field of complex industrial process modeling and soft sensing.
- Description of the Related Art
- The traditional soft sensor modeling methods mostly consider the characteristics of zero delay, that is, considering the input and output with the same sampling interval and input variable collected at time t corresponds to the t-th dominant variable sample in the database. However, there is a significant time lag between the input data collected by each sensor and the output data obtained through the laboratory analysis or online instrumentation. If we continue using steady-state modeling approaches, the established model will be unable to fully explain the characteristics of the process, and it does not meet the causality of actual process. In order to ensure that the soft sensor model can achieve accurate predictions of the key variables in a long time, it is necessary to take measures to introduce the time delay and dynamic information.
- Essentially speaking, most of the existing time delay estimation methods are trying to find the auxiliary variables which are most closely related to the dominant variables for modeling. When applying such methods to practical applications, it is needed to get a tradeoff between the complexity and the accuracy of the algorithm. In view of the process time delay estimation problem, at present, the existing methods include mutual information (MI) method, correlation coefficient method, etc. The invention adopts fuzzy curve analysis method to introduce variable time delay information into the soft sensor model, and the characteristic of this method is low computational complexity while being easy to understand. It is possible to visually and effectively determine the importance degree of the input variables.
- The performance of soft sensor model needs to be maintained by periodic reconstruction in order to dynamically track and effectively take control of the process. The main methods include Moving Window (MW), iterative approach and Just-in-time Learning (JITL), but these methods often need to update the model frequently. Time difference model not only can deal with the problem of model performance degradation due to time lapse and achieve the effect of tracking process dynamics exactly as model updating it also is able to minimize the likelihood of model reconstruction.
- Up to now, data-driven based soft sensor modeling methods have received more and more attention. Some commonly used methods like partial least squares (PLS) and principal component analysis (PCA). They can describe well the linear relationship between input variables and output variables. And artificial neural networks (ANN), support vector machine (SVM) and least squares support vector machine (LS-SVM) can effectively deal with the nonlinear characteristics of the process. In recent years, as a non-parametric probabilistic model, GPR not only can give the predictive value, but also can get the uncertain degree of predictive value. Therefore, the GPR model is selected as the basic soft sensor modeling algorithm in the present invention, and combined with the TD modeling strategy to effectively deal with the drift between input and output data in the process.
- In summary, considering the establishment of a time delay online soft sensor model for the strict control of key variables in the process is of great significance.
- For time delay process, the soft sensor model without delay can't be modeled by the sequence of auxiliary variables which are the most relevant to the dominant variable. When such a model is employed to do estimation, the estimation accuracy of the dominant variable will be greatly affected. In order to effectively extract the process delay information and set up an online soft sensor model without the case of frequent model updating, it is necessary to provide a more effective online strategy for the prediction of key variable. Therefore, this invention provides a soft sensor modeling method based on FCA-TDGPR.
- The invention is realized by the following technical scheme:
- Soft sensor modeling method based on FCA-TDGPR which comprises the following steps: aiming at the time delay process, first of all, collect enough samples of data and constitute the historical database with the historical values of auxiliary variables and dominant variable.
- The FCA method is used to determine the time delay parameters of each auxiliary variable with respect to the dominant variable sequence, which is used to reconstruct the soft sensor modeling data;
- Then, When new input samples are available, a time difference Gaussian process regression (TDGPR) model is employed for current time online prediction based on historical variable values collected certain moments ago, thus making it possible to realize real-time estimation and control of the key variable, obtain more accurate results, increase the yield and reduce production costs.
-
FIG. 1 is a flow chart of online soft sensor method based on FCA-TDGPR; -
FIG. 2 is a schematic diagram of debutanizer process; -
FIG. 3 is a schematic diagram of TDGPR modeling approach; -
FIG. 4 contains fuzzy curve distribution diagrams of the original variables and optimal time delay variables; -
FIG. 5 contains scatterplots of butane concentration predictions with different j values. - The modeling flow chart, which is shown in
FIG. 1 below, is further detailed in the present invention: - Take the actual chemical process as an example, debutanizer is an important part of naphtha desulfurization and separation device of oil refining production process, and one of the dominant variables needed to be controlled for this process is the concentration of the bottom butane (C4). The schematic diagram of the process is shown in
FIG. 2 , due to the value of C4 cannot be directly measured, therefore, there is a delay issue in analyzing and obtaining C4 concentration values. At the same time, different auxiliary variables show different degrees of time delay. Experimental data is derived from the actual industrial process which contains 2394 samples, a total of 7 auxiliary variables. As shown inFIG. 2 , x1 is the top temperature; x2 is the top pressure; x3 is the reflux flow; x4 is the top product outflow; x5 is the 6th tray temperature; x6 is thebottom temperature 1; x7 is thebottom temperature 2. The 1 dominant variable is the bottom butane concentration, which in this invention is predicted as the key variable of the process. - Step1: Collect historical input and output data to form a training database which contains N continuous samples. Assuming that the data is expressed as {X(t),y(t)},
t - Step2: For each of the original variables xi, i∈{1, 2, . . . , 6}, they are extended to the input variables with time delay {xi(t−λ), λ=0, 1, . . . , Tmax} by formula (1), and a set of 120 dimensional delay variables will be obtained for subsequent analysis
- Step3: Determine the importance of each variable in the time delay input variable set by FCA, for (xi(t),y(t)), fuzzy membership function of variable xi is defined as:
-
- For each xi, {Φit, y(t)} provides a fuzzy rule which is described as {if xi is Φit(xi), then y is y(t)}, and Φit is a fuzzy membership function of input variable xi at t-th data point; In formula (2) a Gaussian fuzzy membership function is selected; b is determined as 20% the range of variable xi. As a result, For N training samples, each sample corresponding to each variable has N fuzzy rules. In the fuzzy membership function, Φit=1 holds true at each point {xi(t),y(t)}.
- For time delay process, by introducing time delay information the original variable xi becomes (Tmax+1)-dimensional, which can be expressed as xi(t−λ), λ=0, 1, . . . , Tmax, λ is a variable delay value to be introduced; fuzzy curve Ci,λ with the condition that λ is the i-th variable delay value can be obtained by making centroid defuzzification of each new expanded variable using formula (3); as shown in the formula (4), di is the λ which can make the maximum coverage of fuzzy curve Ci,λ; Ci,λ(λ)max is the maximum value of the fuzzy curve point range, while Ci,λ(λ)min is the minimum value of the fuzzy curve point range;
-
- If the scope of the Ci,λ(λ) range is closer to that of y, then the input variable xi(t−λ) is more important. In view of this point, the importance degree of each variable can be determined by sorting the coverage of Ci,λ(λ). Finally, the optimal delay parameter di as well as time delay variable xi(t−di) can thus be obtained by FCA method, which later on can be used for soft sensor modeling data reconstruction.
- Step4: Based on the previous step, the time delay parameters d1, d2, . . . , dm are used to reconstruct the training input sample set for on-line modeling, the reconstructed input dataset is denoted as Xd(t), Xd(t)[x1(t−d1), x2(t−d2), x3(t−d3), x4(t−d4), x5(t−d5), x6(t−d6)]. If there is a new input sample X(t+1), then the delay input set could be restructured based on historical database samples with the same parameters, then go to
step 5, otherwise, wait for the arrival of new data. - Step 5: After the reorganization procedure, the training set and the new data are processed by j order time difference treatment (the value of j can be determined according to the sampling period and property of dominant variable):
-
ΔX d,j(t)X d(t)−X d(t−j) -
Δy j(t)=y(t)−y(t−j) (5) - Next, make a regression of the relationship between ΔXd,j(t) and Δyj(t) by GPR, which satisfies Δ(t)=f(ΔXj(t))+e(t). The GPR method can obtain the mapping relationship through the given training input and output samples. In this way, the corresponding predictive value and the uncertainty degree can be obtained given the new input data, which means the result will be probabilistic. The GPR algorithm is shown as below.
- In general, the relationship between the observed output value y and noise e satisfies:
-
y i =f(x i)+ε -
ε˜N(0,σn 2) (6) - If the mean function and covariance function are determined, then the distribution of the Gaussian process is well-determined. For simplicity, the mean function is usually preprocessed into 0. Covariance function can transform the correlation of output data into the function of input data. As similar inputs produce similar outputs, the covariance function can be selected according to the characteristics of the sample distribution. One condition which must be satisfied is that the closer the distance of samples is, the more correlated the two samples are, and vise versa. The covariance function form of this invention is shown in formula (7):
-
- In the formula, xp, xq∈RD, v controls the magnitude of the covariance function. πd describes the relative importance of each input attribute xd. The determination of the hyper-parameter Θgp=(v, π1, . . . , πD, σR 2) in the Gaussian process is generally estimated by the MLE method. The optimization of the parameters can be realized by using the conjugate gradient method. Based on test sample and training data, the posterior distribution of test data x* can be calculated, and its predictive value obey the joint Gaussian distribution described in formula (9), where K(X,X) is n-dimensional covariance matrix of training samples; k(x*,X) is the covariance vector of test sample and training samples; k(x*,x*) is the autocovariance of test sample, and fgp is a predictive value of GPR.
-
- when the new input data arrives at
time t+ 1, the calculating formula of predictive value yj,pred(t+1) with TDGPR method is: -
ΔX(t+1)=X(t+1)−X(t+1−j) -
Δy j,pred(t+1)=f GPR(ΔX j(t+1)) -
y j,pred(t+1)=y j(t+1−j)+Δy j,pred(t+1) (10) - In the actual industrial process, there will be the case of instrument damage or laboratory analysis with delay, and the circumstance that the time interval of obtaining dominant variable is large and the quantity is small or there is a lack of dominant analysis value in the database. Thus, shown in
FIG. 3 , for new incoming test data X(t+1), based on yi(t+1−j) stored in the database j moment ago, the predictive value of the dominant variable at time t+1 can be obtained. The predicted output yj,pred(t+1) of the online model is calculated by formula (10), and the predicted result of bottom butane concentration can be obtained. - As shown in
FIG. 4 , compared the original variables without time delay, 6 reconstructed variables contribute more to the dominant variable, which introduce more relevant modeling data for online modeling. At the same time, in order to verify the effectiveness of this invention for on-line estimation, the first 1519 samples are selected in 2394 samples to reconstruct 1500 training samples. The final 875 samples are then used as test samples, and a soft sensor is established for on-line prediction of butane concentration. -
FIG. 5 contains scatterplots of butane concentration prediction results with two methods respectively denoted as the FCA-TDGPR method (present method) which involves delay estimation, and t-TDGPR method which is without time delay estimation. - From
FIG. 5 , when the time difference order j increases from 1 to 10, the time interval of prediction based on historical database is gradually increasing, and the prediction accuracy is declining. This is because the more recent the analysis value is, the better tracking ability the model has for current process dynamics. - Although the accuracy of the two methods are in decline, compared with the TDGPR method without considering the time delay, the predicted results of the present invention can be better close to the true value of the butane concentration when the time difference increases. This suggests that extracted delay information is in line with the actual causal relationship of the process, and the soft sensor model with variable time delay estimation is more accurate.
- After fuzzy curve analysis method is taken to determine the optimal parameters, reconstructed data is proved to be capable of enhancing the accuracy of online model significantly by introducing more contributing auxiliary variables to dominant variable sequence. At the same time, it reflects that the GPR method can explain the dynamic change of the process well. The online soft sensor model based on TDGPR method can adaptively estimate real-time butane concentration with historical variables collected j time ago.
-
FIGS. 4 and 5 have jointly validated that fuzzy curve analysis based time difference Gaussian process regression soft sensor modeling method has good accuracy for on-line prediction of bottom butane concentration. - While the present invention has been described in some detail for purposes of clarity and understanding, one skilled in the art will appreciate that various changes in form and detail can be made without departing from the true scope of the invention. All figures, tables, appendices, patents, patent applications and publications, referred to above, are hereby incorporated by reference.
Claims (2)
ΔX d,j(t)=X d(t)−X d(t−j)
Δy j(t)=y(t)−y(t−j) (5)
y i =f(x i)+ε
ε˜N(0,σn 2) (6)
f gp |X,y,x * ˜N(
s.i.
cov(f gp)=k(x * ,x *)−k(X,x *)T [K(X,X)+σn
ΔX d,j(t+1)=X d(t+1)−X d(t+1−j)
Δy j,pred(t+1)=f GPR(ΔX d,j(t+1))
y j,pred(t+1)=y j(t+1−j)+Δy j,pred(t+1) (10)
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US20200118015A1 (en) | 2020-04-16 |
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US11164095B2 (en) | 2021-11-02 |
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