- Email: [email protected]
PLS Based Monitoring and Control of Batch Digesters
Copyright © IFAC Dynamics and Control of Process Systems, Corfu, Greece, 1998
PLS BASED MONITORING AND CONTROL OF BATCH DIGESTERS Parthasarathy Kesavan Jay H. Lee 1 Victor Saucedo Gopal A. Krishnagopalan
Department of Chemical Engineering, A uburn University, Auburn, AL 36849
Abstract: In this paper, a data-based control method for reducing product quality variations in batch pulp digesters is presented. Compared to the previously proposed techniques, the new technique uses more on-line liquor measurements in predicting the final pulp quality. The liquor measurements obtained at different time instances are related to the final Kappa number through the PLS regression. In using the PLS regression model for control, two approaches are proposed: The first approach computes the optimal control moves directly from the PLS model while the second approach uses the prediction from the PLS model to adapt key parameters of a Hfactor model. The effectiveness of the control algorithms is examined first through a simulation study and then through an experimental study performed on a lab-scale batch digester. Copyright © 1998 lFAC Keywords: Partial Least Square, Digester, Quality prediction and control.
difficulty in maintaining uniform pulp quality stems from the fact that on-line Kappa number measurements are unavailable. This is because online pulp sampling is infeasible. The problem is further complicated by the absence of feed-stock quality measurements.
1. INTRODUCTION
Over the last few decades, increased global competition has stepped up the need to produce paper of uniform quality at competitive prices. This demands a reduction in the variability seen in both paper machines and pulping processes. Pulp can be produced by using either batch or continuous digesters. In this paper, we will address the issue of controlling pulp quality variations in batch digesters.
The current practice consists of statistical quality control (SQC) and model based control (MBC) approaches. In the SQC approach, historical data are used to form a statistical base for the final Kappa number. If the Kappa number deviations of the recent cooks fall outside the statistical bounds, the cooking conditions are altered by the operator to offset the deviation. This is essentially an off-line method that cannot deal with uncorrelated, i.e., batch-to-batch, changes. Because of this, what is considered to be "normal variation" by SQC may still be quite significant.
Variations in pulp quality can be attributed largely to the variations in the characteristics of wood chips used for pulping. Moisture content of chips, chip size and chip composition can all vary significantly. The most important quality variable is the Kappa number, which is a measure of the residual lignin content in the pulp. The main 1 corresponding author:phone (334)844-2060, (334)844-2063, e-mail: [email protected]
In the MBC approach, a model establishing the relation between the feed-stock disturbances, measured outputs, manipulated inputs and the final
fax
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2. PROBLEM DESCRIPTION
Kappa number is first developed. The model is then utilized in predicting the final Kappa number on-line and modifying the cooking conditions appropriately to achieve the target Kappa number. The model can be either fundamental (Williams et al., 1982; Datta and Lee, 1994) or emphical (Hatton, 1973; Chari, 1973). Due to the complexity of fundamental models, current industrial practices rely almost exclusively on simple empirical or semi-empirical models. Some of these models (Chari, 1973; Venkateswarlu and Gangiah, 1992) utilize on-line Effective Alkali (EA) measurements available at a particular time instant (for example, at the beginning of the bulk phase), to compensate for the uncertainty in the initial conditions. However, single EA measurement alone cannot account for all the uncertainties in the feed-stock condition, and use of more on-line measurements is highly desired. Recently, methods for measuring various concentrations of the cooking liquor in real time have been developed (Vanchinathan and Krishnagopalan, 1995). The utility of these additional measurements in fundamental model based estimation and control has already been demonstrated by Datta and Lee (Datta and Lee, 1994).
Fig. (1) shows the batch digester set-up at Auburn University with the facility to take on-line liquor measurements. Wood chips and white liquor (consisting mainly of sodium hydroxide and sodium sulfide) are fed to the digester. The temperature of the digester is raised to 1700 C and maintained at that value by recirculating the liquor through a heater controlled by a temperature controller. The chemicals react with the wood to remove lignin, the binding material present in the wood, to generate pulp containing cellulose fibers . The amount of residual lignin present in the pulp is a measure of pulp quality and is represented by Kappa number. The control objective is to minimize the deviation of the Kappa number from the specified target value. Main disturbances affecting the pulp quality are the feed-stock disturbances such as changes in the wood composition, chip size and initial moisture content. Since on-line pulp sampling is infeasible, direct feedback control design is not possible. Kappa number measurements available at the end of the batch may be useful for future batches, but not for the current batch. Recently, techniques have been developed (Vanchinathan and Krishnagopalan, 1995) to obtain additional on-line secondary liquor measurements such as the concentrations of effective alkali (EA), lignin (Li), solids (So) and sulfide (Su). Details of the on-line liquor measurement techniques can be found in (Vanchinathan and Krishnagopalan, 1995). Online liquor samples are taken at 5 min sample time intervals. These liquor measurements contain information about feed-stock disturbances that affect the final Kappa number. The objective of this paper is to develop algorithms that use these online measurements in predicting and controlling the final Kappa number. The available manipulated variables are the temperature and batch end time.
The main contribution of this paper is in how the on-line liquor measurements are utilized for better Kappa number control. We propose ways to use these measurements for control without resorting to a complex fundamental model. We discuss the issues that arise in developing a statistical model between the liquor measurements and the final Kappa number through regression. We also present two different ways to use this statistical model for control purposes. In the first approach, the manipulated variables (the temperature of the cook and the batch end time) are included in the statistical model and the control law is developed directly on the basis of this model. In the second approach, the statistical model is used to adapt the key parameters of an empirical model (for example Chari 's model (Chari, 1973» to compensate for feed disturbances. The empirical model is then used to adjust the manipulated variables.
3. STATISTICAL MODELING In this section, we will develop a model relating the final quality variable to the measurements available during a batch and the manipulated variables. This model will then be used to predict the final quality variable and to manipulate the inputs to achieve a target quality.
This paper is organized as follows: In Section 2, the batch digester control problem is formulated. Section 3 deals with the development of the regression model between the on-line measurements, manipulated variables and the final quality variables. Two different approaches are proposed in Section 4 to use the regression model for controller design. The effectiveness of the prediction model and the control algorithms are examined through simulation and experimental studies in Section 5 and 6, respectively. Finally, some concluding remarks are made in Section 7.
3.1 Model Structure Let us assume the following underlying model for quality variables: q=
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h (0, UQ, • . . , uT-d
(1)
the fact that all the information available at the time k is utilized. We simply rewrite the above to obtain
r - - - - - Wood Chq,. & Wh . . Liquor
(6) Batch ~r
where Vk = [uk ... U~_I ( . Note that at time k, Vk is the set of manipulated variable moves to
J
be implemented in the future. For monitoring, we assume Vk to be at its reference value; for control, V k is determined by the controller. We can linearize equation (6) with respect to the nominal trajectory y[ , . . . , Yk and corresponding u(j, .. . , ur_I to arrive at
qjk = AY~ 1. Lignin cone. 2. Solid cone. 3. Sulfldlty
1. Temperatu..
2.Dene/ty
3.E_..........
+ BU~_I + CU~
(7)
where q' = q - qr, qr being the target quality. We represent the above with the notation (8)
Fig. 1. Schematic diagram of the batch digester set-up and its instrumentation
where M = [A,B,C] and Xk = [Yk'T ,Uk'T ,Uk'T ] T . For convenience, we will drop the superscript from here on.
In the above, q is a vector containing the quality variables at the end of a batch, () contains unknown parameters that change from one batch to another (including feed-stock conditions, process parameters, etc.), and u represents known process input variables whose paths can potentially be manipulated (most often through tracking control systems). The subscript [']i denotes the ith sample time. T is the total number of samples within a single batch. Here, It is an unknown, possibly nonlinear function representing the effect of feedstock disturbance and manipulated variables on the final quality variables.
{f
3.2 Regression Problem The problem is to identify matrix M appearing in the predictor of (8), given historical data for Xk and q. Let us form the following matrices with data available from N batches:
x = [xk(I) .. . Xk(j) ... xk(N)] Q = [q(l) . . . q(j) ... q(N)]
We assume that we have additional process measurements that are determined through the following relation:
(9)
In the above, the index (j) is used denotes the data vector corresponding to the ith batch. The least squares solution that minimizes IIQ-M XIIF. where 11 ·IIF is the Frobenious norm, is given by
MLS
where y represents measurements of process variables at any given sample time k, Uk-I = [u6 ... Uk_I ] T, and v is the measurement error. Let
= QXT(XXT)-l
(10)
Then,
In practice, since X contains the measurements and manipulated input moves made at various sample time instants during a batch, the regressor vector Xk is of very high dimension and its elements exhibit strong correlation. Hence, the matrix (X XT) tends to be highly ill-conditioned and the least squares estimator is generally a poor choice (in terms of variance, robustness, etc.). Multivariate statistical modeling techniques such as the Partial Least Squares (PLS) and Principal Component Regression (PCR) should alleviate this problem (Geladi and Kowalski, 1986).
The "hat" notation ( is used to denote an estimate of a variable. The notation Ik represents
The multivariable statistical modeling methods utilize the correlation among the different measurements and manipulated variables, as exhibited by the model-building data, in reducing the
Yk
= [ YIT...T Vk ]T
(3)
Since Yk contains information about (), it can be used to estimate (). Let us denote the estimate of () based on Yk as
(4)
A
)
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4. CONTROL
number of regression variables. The reduction in the dimensionality is achieved by projecting the original variables onto a lower dimensional subspace called the principal component space. The projection of the original data to the principal component space can be represented as
s=PX
In contrast to the monitoring, batch quality control on the basis of a statistical model is still at its infancy. Most of the earlier works were oriented towards controlling continuous processes using static PCR/PLS models (Kaspar and Ray, 1992; Pivoso and Kosanovich, 1994; Chen and McAvoy, 1996). For semi-batch processes, Yabuki and MacGregor (Yabuki and MacGregor, 1997) suggested using off-line and on-line measurements to make mid-course correction based on PLS models. More recently, Russell et. al. (Russell et al., 1998) presented a similar, but more general approach to batch reactor control, equipped with a recursive method for quality prediction based on Kalman filtering.
(11)
where P is the projection matrix (which is a "fat" matrix) and S is a matrix containing the transformed variables for each batch called the principal components (P.C.). Now the regression problem is solved in the principal component space, i.e., using data matrix S. The predictor takes the form of qlk
= gST (SST)-l P, Xk ...
In this section we discuss two ways to use the PLSbased prediction model for control. In the first approach, we use the prediction directly in calculating the manipulated inputs. This approach is similar to those proposed in (Yabuki and MacGregor, 1997) and (Russell et al., 1998). In the second approach, we use the prediction indirectly, as a basis for adapting the key parameters of a prior available nonlinear model.
(12)
M
Note that the number of rows in S, i.e., the number of principal components, would be, in general, significantly smaller than that in X and SST be well-conditioned due to the nature of the projection. Ideally, the number of principal components would be equal to the number of modes of variations present in the process. However, in practice, due to the inherent nonlinearity of the process and measurement noise, a few more P.C.s may be needed to retain the relevant information present in the data matrix. The P.C.s can be chosen to maximize the variance of the X data explained, as in PCR, or to maximize the covariance between X and Q, as in PLS. Since our ultimate interest here lies in the output quality prediction, it should be to one's advantage to consider the correlation with the output quality, as is done in the PLS method(Geladi and Kowalski , 1986) . Several different techniques are available in the literature on how to decide on the number of P.C.s to be retained in building a regression model (see for example (Kresta et al., 1991».
4.1 Direct Method
For control, we want to manipulate the input trajectory u to control the quality. For this, let us divide [0, T) into m intervals: [0, . . . , kd,· .. , [km - 1 ,· . . , J Let us parameterize the trajectory Ui for the ith interval [k i - 1 , ··· , ki ) with parameter vector ()f. Based on the input parameterization, we can write the quality prediction equation at t = k i as
An appropriate resolution of the input parameterization for each interval depends on the length of the interval and sensitivity of the quality variables to high frequency changes during the interval. In the simplest case, the trajectory for each interval can be chosen constant, yielding only a single parameter. The other extreme is to pararneterize it with the input values at all sample times present within the interval. The more the parameters, the more data one would need for model development. Often, parameterization of very fine resolution is not needed. It only makes the identification problem more difficult.
P.C .s themselves can be used for monitoring batch processes, as suggested by MacGregor and coworkers in several related papers «MacGregor et al., 1994; Nomikos and MacGregor, 1994) and the references therein). Monitoring the P.C.s is useful for detecting the occurrence of disturbances. In this paper, however, we are concerned with how the disturbances affect the final quality. Hence, instead of monitoring the P.C.s, we monitor the final quality predictions derived from the regression model. If the quality prediction is significantly different from the desired quality, the control algorithms discussed in the next section can be utilized to minimize the final quality deviations.
Now, Or+l>·· · 'O~ can be determined at each k i so that the quality constraints are met. This can be formulated as an optimization of some sort, for instance,
740
n
min
8j,j~i+1
q~.Aqqlk. +
L
moves). More generally, one can use multiple data points, that is, choose 1]lk according to
(14)
j=i+1 k
subject to the prediction equation constraint (13) and other appropriate constraints. Aq and A8 are weighting matrices, the elements of which are chosen according to the importance of the variables.
min 'I
L Ilqli - 91(7J,Ui-1,Urom )112
(17)
i=1
The control law to compute the future input moves can now be obtained by using the quality model (15) with the new set of model parameters in the following quadratic minimization problem:
4.2 Indirect Method
When the relationship between the manipulated variables and the control variables (for example between the cooking temperature and Kappa number) is strongly nonlinear, the previously developed linear model may not provide a satisfactory control performance. This is especially true if the batch end-time is to be used as an input. In such circumstances, it would be advantageous to use a simple nonlinear model (either fundamental or empirical) and adapt a few key parameters to compensate for the unknown disturbances. This can be thought of as a form of adaptive control. In this study, we propose to use the quality predictions from the statistical model for model parameter adaptation.
subject to
Note that Uk may be parameterized with only a few parameters as before. 5. SIMULATION STUDIES 5.1 Data Generation
A fundamental model for the batch digester described in Section 2 was developed based on the information given in Datta and Lee (Datta and Lee, 1994). This model was used as if it were the true plant, in generating the batch digester data for the simulation study. On-line liquor measurements of EA, Li, So and Su were assumed to be taken every 5 min. Softwood was chosen as the main feed-stock. The nominal cooking temperature was 170°C and batch end time was 180 min. It was assumed that the batch temperature can be altered at the half way mark (Le., 90 minutes into the cook) to a new constant value (Tmid)' Batch end time (tb) and T mid were taken as the manipulated variables «(}U = [tb T mid)). For the model development, 50 batches were simulated while varying the feed-stock quality, T mid and tb in a random manner. Varying contents of hardwood were assumed to be present in the softwood feedstock, introducing large composition changes from one batch to next. The range of the feed-stock variation is shown in Table (1). Tmid was varied between 165 and 175°C and tb was varied between 160 and 220 min. The measurements, manipulated variables, and quality variable were all scaled to have zero mean and unit variance. The data from the 50 batches were then used to develop a PLS model relating the measurements, manipulated variables (Tmid,tb) and final pulp quality (Ka).
Assume that a nonlinear model of the following form is available: (15) In the above, 1] is a vector of model parameters that depends on () (see Section 3.1). The relationship between the model parameters 7J and the process parameters () is generally unknown. However, from (1) we see that the quality measurements contain information about the unknown parameter (). In the traditional adaptive control algorithms, the quality measurements and the manipulated variable measurements are used in modifying the parameters 1] to account for the unknown disturbances. In the batch quality control framework, however, the quality measurements are available only at the end of a batch. Hence, it is not possible to account for short-term (e.g., batch-to-batch) variations. However, if the quality predictions from the PLS model were used to adapt the key model parameters, batch-to-batch variations would be accounted for. For instance, assuming the invertibility of (15), an estimate of the model parameters 7} can be obtained as follows:
nom )
• = 92 (Aqlk, Uk-1, uk 7Jlk
(16)
5.2 Modeling and Control
where qlk is the quality prediction obtained from the statistical model (with U/e = u;:om and u;:om contains the nominal values for the future input
Initial simulation studies with different on-line measurements (EA, Li, So and Su) indicated
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Variable Chip moisture Chip size Hard wood content
Range 45 - 55% 3 - 4 mm 0 - 20%
Variable Chip moisture Chlp~re
Hard wood content
Table 1. Range of normal feed-stock variations used for building PLS model
Range 40 - 60% 3-6mm 0 - 45%
Table 2. Range of large feed-stock variations used for testing PLS model
tb that EA and Li had the maximum correlaH= fe43.2-~dt tion with final kappa number. Hence only these (21) measurements were incorporated into the ino put data matrix (X). The model was developed tbl tb at the half way mark. A total of 19 samples = f e43 .2-+Wi3dt + f e 43 .2 T~?1i'!173d(22) had been taken at the half way mark. Hence, o tu for the jth batch (j = 1,2, ... ,50) , Xk(j) = 43 .2 !O!!~ [EAI (j), .. . ,EA I9 (j), Lil (j), .. . ,Li I9 (j), tb(j), T mid(j)J . =HI + e T_id+ 73 (tb - tbt) (23) A PLS model as discussed in Section 3 was deIn the above equations simplification is achieved veloped using the data. A total of 6 P.C.s were by using the fact that T mid is held constant enough to explain 99% variance in X and 98% after the change at time tbl (= 90). HI is the variance in q. Thus the number of variables were H factor up to the time tbl when the change reduced from 40 (in X) to 6. A plot of the P.C.s in temperature is made and can be calculated versus the Kappa number indicated a linear trend. from past batches. The parameters of the Chari 's Hence a linear regression model was developed model are estimated by using data that was used between the P.C.s and the Kappa number. for the PLS model development. For parameter Additional models developed at later time inestimation, the logarithmic form of (20) as given stances during the batch did not yield any signifbelow was used since this results in a linear icant improvement in the final quality prediction. parameter estimation problem. Hence the control calculation was performed only once (at the 90 min mark.) . The control moves log(Ka) = log(Ac) + el * log(Vf~o) were calculated by the following two methods.
-e2 * log( Ca~o) - e3
Direct Method: For fresh batches, the PLS model was used to predict the final Kappa number at the nominal batch end time. If the predicted Kappa was different from the target Kappa, the control law as in (14) was used to calculate the values for the manipulated variables tb and T mid in order to achieve the target Kappa. Aq and AB were chosen as 1 and [10 IJ, respectively. Note that a higher weight is given to the batch end time, in order to minimize the batch scheduling and inventory problems.
* log(H) (24)
Ac was used as an adjustable parameter for each batch; it was estimated on-line at time tbl on the basis of the Kappa prediction from the PLS model (with tb and Tmid kept at their nominal values), the known values ofVfro and Caro and the nominal value of H. The logarithmic form of Chari's model with the adjusted Ac is then used in computing Tmid and tb according to the control law given by equation (18) . M and AU were chosen the same as in the direct algorithm.
Indirect Method: In this approach, in order to handle the nonlinearity between the temperature, batch end time and the final Kappa number, a semi-empirical model developed by Chari (Chari, 1973) was used. Chari's model is given by
5.3 Results and Discussion For testing the PLS model, 20 new batch data were generated with both normal and large random variations in the feed-stock in the range shown in Table (2). We included some large variations in the test data in order to test the prediction capability of the model outside the range for which it was developed. Fig (2) shows the predicted final Kappa number at the nominal batch end time by using the PLS model versus the true value, for different batches . From the figure we see that the model predictions of the final Kappa number are close to the actual values.
(20) where A c , el, e2, e3 are model parameters, Ka is the final Kappa number, Vf~o is the ratio of initial white liquor volume to dry wood weight, Ca~ is the ratio of initial active alkali to dry wood weight and H denotes the H-factor , which is defined as the area under the plot of relative delignification rate vs. time (K.E., 1957):
In order to test the control strategy, a Monte Carlo simulation was performed, with random
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6. EXPERIMENTAL STUDIES 6.1 Data Generation
.--
The batch digester as shown in Fig. (I) was used for cooking wood chips. 250g. of oven dry wood chips was used for cooking in each batch. Liquor to wood ratio was 5.5. Softwood was used for cooking. Chip size varied between 3-5 mm thickness, chip composition varied between 2628% lignin and chip moisture varied between 0.450.55. The temperature was ramped up to 170°C and maintained at that value. Batch end time was taken as 180 min. For the time being, data were gathered to build and test the prediction capability of a PLS model. Hence T mid and tb were kept at their nominal values for all batches. In the future, additional data will be generated to develop control relevant models by varying T mid and tb around their nominal values.
..... '
Fig. 2. Kappa number predictions obtained with the PLS model versus the actual Kappa number variations in the initial conditions. 20 batch runs were simulated. Fig. (3(a» shows the variations in the final Kappa number without any control, with the control based on the PLS model and with the control based on the combined PLS / modified Chari's model. The standard deviation (square root of variance) was reduced from 6 (without control) to 3 with the PLS model based control and to 1.24 with the combined PLS / modified Hfactor model based control. Figs. (3(b» and (3(c}) show the variations in the manipulated variables (Tmid and tb) from their nominal values (170 and 180). The better performance of the combined PLS/modified Chari's model based control can be attributed to better handling of the nonlinearity between the manipulated variables and the final pulp quality.
6.2 Modeling
Data for 30 batch runs were obtained. After an analysis of the off-line and on-line data, only 21 batches were found to be suitable for model building and testing. Since 21 batches were not sufficient for us to divide into separate modelbuilding and testing data sets, we decided to build the model using 20 batches, leaving only a single batch for testing. However, we tested all the possible choices for the model-building set, producing 21 data points for model testing.
Kappa It variations
On-line data consisting of EA, lignin, solids and sulfide available at 5 min. interval were used in the PLS model. A linear PLS model with 7 principal components was built relating the measurements collected up to the 90 min mark and the final Kappa number. Fig. (4) shows the predictions of final Kappa number by PLS model vs. the actual Kappa number. For comparison, the predictions from traditional Chari 's H-factor model is also shown (the word traditional is used since we used the Chari's model without the parameter adaptation described earlier) . From the figure we see that the PLS model is able to give better predictions because of its ability to use more on-line liquor measurements. The use of the PLS model results in a significant reduction in the mean square error (MSE) for the quality prediction when compared to the Chari's model. The prediction of the PLS model is slightly off in the Kappa range of 30 because the data sets used for modeling were mostly in the range 18-25. Thus the PLS model was forced to extrapolate beyond the range of model-building data. Use of more data around the 30 Kappa range would probably have resulted in a reduction in the prediction error.
Temp. variations
175..-----.--....----,.--.---'-.----,--....----,.--.---,
Batch end time variations
•
10
12
Batch index
.
,. ,
20
Fig. 3. (a) shows the Kappa number variations obtained without control, with PLS based control strategy and with combined PLS / modified-H-factor based control strategy. Target Kappa number is 40. (b) shows the variation in the mid batch temperature (Tmid) with respect to the nominal value (170) and (c) shows the variation in the batch end time (tb) with respect to the nominal value (180).
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Kaspar, M.H. and W.H. Ray (1992). Chemometric methods for process monitoring and high performance controller design. AIChE J. 38(10),1593- 1608 . K.E., Vroom (1957). The " h" factor: A means of expressing cooking times and temperatures as a single variable. Pulp and Paper Magazine of Canada Convention Issue, 228-233 . Kresta, J.V., J.F. MacGregor and T .E. Marlin (1991). Multivariate statistical monitoring of process operating performance. The Canadian Journal of Chem. Eng. 69, 35-47. MacGregor, J .F., C. Jaeckle, C. Kiparissides and M. Koutoudi (1994) . Process monitoring and diagnosis by multi block pis methods. AIChE J. 40(5), 35-47. Nomikos, P. and J . F. MacGregor (1994) . 'monitoring batch processes using multiway principal component analysis . AIChE J. 40(8), 1361-1375. Pivoso, M.J . and K.A. Kosanovich (1994). Applications of multivariate statistical methods to process monitoring and controller design. Int. Journal of Control 59(3), 743- 765. Russell, S.A., P. Kesavan, J.H. Lee and B.A. Ogunnaike (1998) . Recursive data based prediction and control of product quality for batch and semi-batch processes applied to nylon 6,6 autoclave. submitted to AIChE J. Vanchinathan, S. and G.A. Krishnagopalan (1995). Kraft delignification kinetics based on liquor analysis. TAPPI Journal 18(3), 127132. Venkateswarlu, C. and K. Gangiah (1992). Dynamic modeling and optimal state estimation using extended kalman filter for a kraft pulping digester. 1& EC Res. 31,848-855. Williams, T.J ., T. Christiansen and L.F. Albright (1982). A mathematical model of the kraft pulping process, voU and 2. Report 129. Purdue laboratory for applied industrial contro!. vo!. 1,2. Yabuki, Y. and J .F. MacGregor (1997). Product quality control in semi-batch reactors using mid-course correction policies. In : IFACADCHEM Preprints. pp. 189-193.
28
. 22
.
o •
0
,.
,.
20
22
,.
28
Actual
Fig. 4. Comparison of the prediction performance of PLS model (*) with that obtained from H-factor model (0) by using data from experimental studies. 7. CONCLUSION
In this paper, a data driven approach has been developed to model and control the final pulp quality in batch digester. The approach utilized new on-line liquor measurements available during the course of a batch in the PLS framework. Manipulated variables such as the batch end time and mid-batch temperature were incorporated into the model to develop a control algorithm. The potential of the technique for prediction and control of the final quality was demonstrated by means of simulation studies. The prediction capability of the PLS model was also demonstrated using experimental data. Currently experimental data are being obtained to test the control algorithms in real time. Acknowledgement : The authors gratefully acknowledge the financial and technical support of the U.S. Department of Energy, Pulp and Paper Consortium and Auburn Pulp and Paper Research and Education Center.
8. REFERENCES
Chari, N.C.S. (1973). Integrated control system approach for batch digester control. Tappi journal 56(7), 65-68. Chen, G. and T.J. McAvoy (1996) . Process control using data based multivariate statistical models. The Canadian Journal of Chem . Eng. 14, 1010-1024. Datta, A.K. and J.H . Lee (1994). On-line monitoring and control of pulp digester. Froc. of ACC pp. 500-504. Geladi, P. and B. R. Kowalski (1986). Partial least squares regression: A tutorial . Anal. Chim . Acta 185, 1- 17. Hatton, J .V. (1973) . Application of empirical equations to process contro!' Tappi Journal 56(8) , 108-111 .
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PLS BASED MONITORING AND CONTROL OF BATCH DIGESTERS Parthasarathy Kesavan Jay H. Lee 1 Victor Saucedo Gopal A. Krishnagopalan
Department of Chemical Engineering, A uburn University, Auburn, AL 36849
Abstract: In this paper, a data-based control method for reducing product quality variations in batch pulp digesters is presented. Compared to the previously proposed techniques, the new technique uses more on-line liquor measurements in predicting the final pulp quality. The liquor measurements obtained at different time instances are related to the final Kappa number through the PLS regression. In using the PLS regression model for control, two approaches are proposed: The first approach computes the optimal control moves directly from the PLS model while the second approach uses the prediction from the PLS model to adapt key parameters of a Hfactor model. The effectiveness of the control algorithms is examined first through a simulation study and then through an experimental study performed on a lab-scale batch digester. Copyright © 1998 lFAC Keywords: Partial Least Square, Digester, Quality prediction and control.
difficulty in maintaining uniform pulp quality stems from the fact that on-line Kappa number measurements are unavailable. This is because online pulp sampling is infeasible. The problem is further complicated by the absence of feed-stock quality measurements.
1. INTRODUCTION
Over the last few decades, increased global competition has stepped up the need to produce paper of uniform quality at competitive prices. This demands a reduction in the variability seen in both paper machines and pulping processes. Pulp can be produced by using either batch or continuous digesters. In this paper, we will address the issue of controlling pulp quality variations in batch digesters.
The current practice consists of statistical quality control (SQC) and model based control (MBC) approaches. In the SQC approach, historical data are used to form a statistical base for the final Kappa number. If the Kappa number deviations of the recent cooks fall outside the statistical bounds, the cooking conditions are altered by the operator to offset the deviation. This is essentially an off-line method that cannot deal with uncorrelated, i.e., batch-to-batch, changes. Because of this, what is considered to be "normal variation" by SQC may still be quite significant.
Variations in pulp quality can be attributed largely to the variations in the characteristics of wood chips used for pulping. Moisture content of chips, chip size and chip composition can all vary significantly. The most important quality variable is the Kappa number, which is a measure of the residual lignin content in the pulp. The main 1 corresponding author:phone (334)844-2060, (334)844-2063, e-mail: [email protected]
In the MBC approach, a model establishing the relation between the feed-stock disturbances, measured outputs, manipulated inputs and the final
fax
737
2. PROBLEM DESCRIPTION
Kappa number is first developed. The model is then utilized in predicting the final Kappa number on-line and modifying the cooking conditions appropriately to achieve the target Kappa number. The model can be either fundamental (Williams et al., 1982; Datta and Lee, 1994) or emphical (Hatton, 1973; Chari, 1973). Due to the complexity of fundamental models, current industrial practices rely almost exclusively on simple empirical or semi-empirical models. Some of these models (Chari, 1973; Venkateswarlu and Gangiah, 1992) utilize on-line Effective Alkali (EA) measurements available at a particular time instant (for example, at the beginning of the bulk phase), to compensate for the uncertainty in the initial conditions. However, single EA measurement alone cannot account for all the uncertainties in the feed-stock condition, and use of more on-line measurements is highly desired. Recently, methods for measuring various concentrations of the cooking liquor in real time have been developed (Vanchinathan and Krishnagopalan, 1995). The utility of these additional measurements in fundamental model based estimation and control has already been demonstrated by Datta and Lee (Datta and Lee, 1994).
Fig. (1) shows the batch digester set-up at Auburn University with the facility to take on-line liquor measurements. Wood chips and white liquor (consisting mainly of sodium hydroxide and sodium sulfide) are fed to the digester. The temperature of the digester is raised to 1700 C and maintained at that value by recirculating the liquor through a heater controlled by a temperature controller. The chemicals react with the wood to remove lignin, the binding material present in the wood, to generate pulp containing cellulose fibers . The amount of residual lignin present in the pulp is a measure of pulp quality and is represented by Kappa number. The control objective is to minimize the deviation of the Kappa number from the specified target value. Main disturbances affecting the pulp quality are the feed-stock disturbances such as changes in the wood composition, chip size and initial moisture content. Since on-line pulp sampling is infeasible, direct feedback control design is not possible. Kappa number measurements available at the end of the batch may be useful for future batches, but not for the current batch. Recently, techniques have been developed (Vanchinathan and Krishnagopalan, 1995) to obtain additional on-line secondary liquor measurements such as the concentrations of effective alkali (EA), lignin (Li), solids (So) and sulfide (Su). Details of the on-line liquor measurement techniques can be found in (Vanchinathan and Krishnagopalan, 1995). Online liquor samples are taken at 5 min sample time intervals. These liquor measurements contain information about feed-stock disturbances that affect the final Kappa number. The objective of this paper is to develop algorithms that use these online measurements in predicting and controlling the final Kappa number. The available manipulated variables are the temperature and batch end time.
The main contribution of this paper is in how the on-line liquor measurements are utilized for better Kappa number control. We propose ways to use these measurements for control without resorting to a complex fundamental model. We discuss the issues that arise in developing a statistical model between the liquor measurements and the final Kappa number through regression. We also present two different ways to use this statistical model for control purposes. In the first approach, the manipulated variables (the temperature of the cook and the batch end time) are included in the statistical model and the control law is developed directly on the basis of this model. In the second approach, the statistical model is used to adapt the key parameters of an empirical model (for example Chari 's model (Chari, 1973» to compensate for feed disturbances. The empirical model is then used to adjust the manipulated variables.
3. STATISTICAL MODELING In this section, we will develop a model relating the final quality variable to the measurements available during a batch and the manipulated variables. This model will then be used to predict the final quality variable and to manipulate the inputs to achieve a target quality.
This paper is organized as follows: In Section 2, the batch digester control problem is formulated. Section 3 deals with the development of the regression model between the on-line measurements, manipulated variables and the final quality variables. Two different approaches are proposed in Section 4 to use the regression model for controller design. The effectiveness of the prediction model and the control algorithms are examined through simulation and experimental studies in Section 5 and 6, respectively. Finally, some concluding remarks are made in Section 7.
3.1 Model Structure Let us assume the following underlying model for quality variables: q=
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h (0, UQ, • . . , uT-d
(1)
the fact that all the information available at the time k is utilized. We simply rewrite the above to obtain
r - - - - - Wood Chq,. & Wh . . Liquor
(6) Batch ~r
where Vk = [uk ... U~_I ( . Note that at time k, Vk is the set of manipulated variable moves to
J
be implemented in the future. For monitoring, we assume Vk to be at its reference value; for control, V k is determined by the controller. We can linearize equation (6) with respect to the nominal trajectory y[ , . . . , Yk and corresponding u(j, .. . , ur_I to arrive at
qjk = AY~ 1. Lignin cone. 2. Solid cone. 3. Sulfldlty
1. Temperatu..
2.Dene/ty
3.E_..........
+ BU~_I + CU~
(7)
where q' = q - qr, qr being the target quality. We represent the above with the notation (8)
Fig. 1. Schematic diagram of the batch digester set-up and its instrumentation
where M = [A,B,C] and Xk = [Yk'T ,Uk'T ,Uk'T ] T . For convenience, we will drop the superscript from here on.
In the above, q is a vector containing the quality variables at the end of a batch, () contains unknown parameters that change from one batch to another (including feed-stock conditions, process parameters, etc.), and u represents known process input variables whose paths can potentially be manipulated (most often through tracking control systems). The subscript [']i denotes the ith sample time. T is the total number of samples within a single batch. Here, It is an unknown, possibly nonlinear function representing the effect of feedstock disturbance and manipulated variables on the final quality variables.
{f
3.2 Regression Problem The problem is to identify matrix M appearing in the predictor of (8), given historical data for Xk and q. Let us form the following matrices with data available from N batches:
x = [xk(I) .. . Xk(j) ... xk(N)] Q = [q(l) . . . q(j) ... q(N)]
We assume that we have additional process measurements that are determined through the following relation:
(9)
In the above, the index (j) is used denotes the data vector corresponding to the ith batch. The least squares solution that minimizes IIQ-M XIIF. where 11 ·IIF is the Frobenious norm, is given by
MLS
where y represents measurements of process variables at any given sample time k, Uk-I = [u6 ... Uk_I ] T, and v is the measurement error. Let
= QXT(XXT)-l
(10)
Then,
In practice, since X contains the measurements and manipulated input moves made at various sample time instants during a batch, the regressor vector Xk is of very high dimension and its elements exhibit strong correlation. Hence, the matrix (X XT) tends to be highly ill-conditioned and the least squares estimator is generally a poor choice (in terms of variance, robustness, etc.). Multivariate statistical modeling techniques such as the Partial Least Squares (PLS) and Principal Component Regression (PCR) should alleviate this problem (Geladi and Kowalski, 1986).
The "hat" notation ( is used to denote an estimate of a variable. The notation Ik represents
The multivariable statistical modeling methods utilize the correlation among the different measurements and manipulated variables, as exhibited by the model-building data, in reducing the
Yk
= [ YIT...T Vk ]T
(3)
Since Yk contains information about (), it can be used to estimate (). Let us denote the estimate of () based on Yk as
(4)
A
)
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4. CONTROL
number of regression variables. The reduction in the dimensionality is achieved by projecting the original variables onto a lower dimensional subspace called the principal component space. The projection of the original data to the principal component space can be represented as
s=PX
In contrast to the monitoring, batch quality control on the basis of a statistical model is still at its infancy. Most of the earlier works were oriented towards controlling continuous processes using static PCR/PLS models (Kaspar and Ray, 1992; Pivoso and Kosanovich, 1994; Chen and McAvoy, 1996). For semi-batch processes, Yabuki and MacGregor (Yabuki and MacGregor, 1997) suggested using off-line and on-line measurements to make mid-course correction based on PLS models. More recently, Russell et. al. (Russell et al., 1998) presented a similar, but more general approach to batch reactor control, equipped with a recursive method for quality prediction based on Kalman filtering.
(11)
where P is the projection matrix (which is a "fat" matrix) and S is a matrix containing the transformed variables for each batch called the principal components (P.C.). Now the regression problem is solved in the principal component space, i.e., using data matrix S. The predictor takes the form of qlk
= gST (SST)-l P, Xk ...
In this section we discuss two ways to use the PLSbased prediction model for control. In the first approach, we use the prediction directly in calculating the manipulated inputs. This approach is similar to those proposed in (Yabuki and MacGregor, 1997) and (Russell et al., 1998). In the second approach, we use the prediction indirectly, as a basis for adapting the key parameters of a prior available nonlinear model.
(12)
M
Note that the number of rows in S, i.e., the number of principal components, would be, in general, significantly smaller than that in X and SST be well-conditioned due to the nature of the projection. Ideally, the number of principal components would be equal to the number of modes of variations present in the process. However, in practice, due to the inherent nonlinearity of the process and measurement noise, a few more P.C.s may be needed to retain the relevant information present in the data matrix. The P.C.s can be chosen to maximize the variance of the X data explained, as in PCR, or to maximize the covariance between X and Q, as in PLS. Since our ultimate interest here lies in the output quality prediction, it should be to one's advantage to consider the correlation with the output quality, as is done in the PLS method(Geladi and Kowalski , 1986) . Several different techniques are available in the literature on how to decide on the number of P.C.s to be retained in building a regression model (see for example (Kresta et al., 1991».
4.1 Direct Method
For control, we want to manipulate the input trajectory u to control the quality. For this, let us divide [0, T) into m intervals: [0, . . . , kd,· .. , [km - 1 ,· . . , J Let us parameterize the trajectory Ui for the ith interval [k i - 1 , ··· , ki ) with parameter vector ()f. Based on the input parameterization, we can write the quality prediction equation at t = k i as
An appropriate resolution of the input parameterization for each interval depends on the length of the interval and sensitivity of the quality variables to high frequency changes during the interval. In the simplest case, the trajectory for each interval can be chosen constant, yielding only a single parameter. The other extreme is to pararneterize it with the input values at all sample times present within the interval. The more the parameters, the more data one would need for model development. Often, parameterization of very fine resolution is not needed. It only makes the identification problem more difficult.
P.C .s themselves can be used for monitoring batch processes, as suggested by MacGregor and coworkers in several related papers «MacGregor et al., 1994; Nomikos and MacGregor, 1994) and the references therein). Monitoring the P.C.s is useful for detecting the occurrence of disturbances. In this paper, however, we are concerned with how the disturbances affect the final quality. Hence, instead of monitoring the P.C.s, we monitor the final quality predictions derived from the regression model. If the quality prediction is significantly different from the desired quality, the control algorithms discussed in the next section can be utilized to minimize the final quality deviations.
Now, Or+l>·· · 'O~ can be determined at each k i so that the quality constraints are met. This can be formulated as an optimization of some sort, for instance,
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n
min
8j,j~i+1
q~.Aqqlk. +
L
moves). More generally, one can use multiple data points, that is, choose 1]lk according to
(14)
j=i+1 k
subject to the prediction equation constraint (13) and other appropriate constraints. Aq and A8 are weighting matrices, the elements of which are chosen according to the importance of the variables.
min 'I
L Ilqli - 91(7J,Ui-1,Urom )112
(17)
i=1
The control law to compute the future input moves can now be obtained by using the quality model (15) with the new set of model parameters in the following quadratic minimization problem:
4.2 Indirect Method
When the relationship between the manipulated variables and the control variables (for example between the cooking temperature and Kappa number) is strongly nonlinear, the previously developed linear model may not provide a satisfactory control performance. This is especially true if the batch end-time is to be used as an input. In such circumstances, it would be advantageous to use a simple nonlinear model (either fundamental or empirical) and adapt a few key parameters to compensate for the unknown disturbances. This can be thought of as a form of adaptive control. In this study, we propose to use the quality predictions from the statistical model for model parameter adaptation.
subject to
Note that Uk may be parameterized with only a few parameters as before. 5. SIMULATION STUDIES 5.1 Data Generation
A fundamental model for the batch digester described in Section 2 was developed based on the information given in Datta and Lee (Datta and Lee, 1994). This model was used as if it were the true plant, in generating the batch digester data for the simulation study. On-line liquor measurements of EA, Li, So and Su were assumed to be taken every 5 min. Softwood was chosen as the main feed-stock. The nominal cooking temperature was 170°C and batch end time was 180 min. It was assumed that the batch temperature can be altered at the half way mark (Le., 90 minutes into the cook) to a new constant value (Tmid)' Batch end time (tb) and T mid were taken as the manipulated variables «(}U = [tb T mid)). For the model development, 50 batches were simulated while varying the feed-stock quality, T mid and tb in a random manner. Varying contents of hardwood were assumed to be present in the softwood feedstock, introducing large composition changes from one batch to next. The range of the feed-stock variation is shown in Table (1). Tmid was varied between 165 and 175°C and tb was varied between 160 and 220 min. The measurements, manipulated variables, and quality variable were all scaled to have zero mean and unit variance. The data from the 50 batches were then used to develop a PLS model relating the measurements, manipulated variables (Tmid,tb) and final pulp quality (Ka).
Assume that a nonlinear model of the following form is available: (15) In the above, 1] is a vector of model parameters that depends on () (see Section 3.1). The relationship between the model parameters 7J and the process parameters () is generally unknown. However, from (1) we see that the quality measurements contain information about the unknown parameter (). In the traditional adaptive control algorithms, the quality measurements and the manipulated variable measurements are used in modifying the parameters 1] to account for the unknown disturbances. In the batch quality control framework, however, the quality measurements are available only at the end of a batch. Hence, it is not possible to account for short-term (e.g., batch-to-batch) variations. However, if the quality predictions from the PLS model were used to adapt the key model parameters, batch-to-batch variations would be accounted for. For instance, assuming the invertibility of (15), an estimate of the model parameters 7} can be obtained as follows:
nom )
• = 92 (Aqlk, Uk-1, uk 7Jlk
(16)
5.2 Modeling and Control
where qlk is the quality prediction obtained from the statistical model (with U/e = u;:om and u;:om contains the nominal values for the future input
Initial simulation studies with different on-line measurements (EA, Li, So and Su) indicated
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Variable Chip moisture Chip size Hard wood content
Range 45 - 55% 3 - 4 mm 0 - 20%
Variable Chip moisture Chlp~re
Hard wood content
Table 1. Range of normal feed-stock variations used for building PLS model
Range 40 - 60% 3-6mm 0 - 45%
Table 2. Range of large feed-stock variations used for testing PLS model
tb that EA and Li had the maximum correlaH= fe43.2-~dt tion with final kappa number. Hence only these (21) measurements were incorporated into the ino put data matrix (X). The model was developed tbl tb at the half way mark. A total of 19 samples = f e43 .2-+Wi3dt + f e 43 .2 T~?1i'!173d(22) had been taken at the half way mark. Hence, o tu for the jth batch (j = 1,2, ... ,50) , Xk(j) = 43 .2 !O!!~ [EAI (j), .. . ,EA I9 (j), Lil (j), .. . ,Li I9 (j), tb(j), T mid(j)J . =HI + e T_id+ 73 (tb - tbt) (23) A PLS model as discussed in Section 3 was deIn the above equations simplification is achieved veloped using the data. A total of 6 P.C.s were by using the fact that T mid is held constant enough to explain 99% variance in X and 98% after the change at time tbl (= 90). HI is the variance in q. Thus the number of variables were H factor up to the time tbl when the change reduced from 40 (in X) to 6. A plot of the P.C.s in temperature is made and can be calculated versus the Kappa number indicated a linear trend. from past batches. The parameters of the Chari 's Hence a linear regression model was developed model are estimated by using data that was used between the P.C.s and the Kappa number. for the PLS model development. For parameter Additional models developed at later time inestimation, the logarithmic form of (20) as given stances during the batch did not yield any signifbelow was used since this results in a linear icant improvement in the final quality prediction. parameter estimation problem. Hence the control calculation was performed only once (at the 90 min mark.) . The control moves log(Ka) = log(Ac) + el * log(Vf~o) were calculated by the following two methods.
-e2 * log( Ca~o) - e3
Direct Method: For fresh batches, the PLS model was used to predict the final Kappa number at the nominal batch end time. If the predicted Kappa was different from the target Kappa, the control law as in (14) was used to calculate the values for the manipulated variables tb and T mid in order to achieve the target Kappa. Aq and AB were chosen as 1 and [10 IJ, respectively. Note that a higher weight is given to the batch end time, in order to minimize the batch scheduling and inventory problems.
* log(H) (24)
Ac was used as an adjustable parameter for each batch; it was estimated on-line at time tbl on the basis of the Kappa prediction from the PLS model (with tb and Tmid kept at their nominal values), the known values ofVfro and Caro and the nominal value of H. The logarithmic form of Chari's model with the adjusted Ac is then used in computing Tmid and tb according to the control law given by equation (18) . M and AU were chosen the same as in the direct algorithm.
Indirect Method: In this approach, in order to handle the nonlinearity between the temperature, batch end time and the final Kappa number, a semi-empirical model developed by Chari (Chari, 1973) was used. Chari's model is given by
5.3 Results and Discussion For testing the PLS model, 20 new batch data were generated with both normal and large random variations in the feed-stock in the range shown in Table (2). We included some large variations in the test data in order to test the prediction capability of the model outside the range for which it was developed. Fig (2) shows the predicted final Kappa number at the nominal batch end time by using the PLS model versus the true value, for different batches . From the figure we see that the model predictions of the final Kappa number are close to the actual values.
(20) where A c , el, e2, e3 are model parameters, Ka is the final Kappa number, Vf~o is the ratio of initial white liquor volume to dry wood weight, Ca~ is the ratio of initial active alkali to dry wood weight and H denotes the H-factor , which is defined as the area under the plot of relative delignification rate vs. time (K.E., 1957):
In order to test the control strategy, a Monte Carlo simulation was performed, with random
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6. EXPERIMENTAL STUDIES 6.1 Data Generation
.--
The batch digester as shown in Fig. (I) was used for cooking wood chips. 250g. of oven dry wood chips was used for cooking in each batch. Liquor to wood ratio was 5.5. Softwood was used for cooking. Chip size varied between 3-5 mm thickness, chip composition varied between 2628% lignin and chip moisture varied between 0.450.55. The temperature was ramped up to 170°C and maintained at that value. Batch end time was taken as 180 min. For the time being, data were gathered to build and test the prediction capability of a PLS model. Hence T mid and tb were kept at their nominal values for all batches. In the future, additional data will be generated to develop control relevant models by varying T mid and tb around their nominal values.
..... '
Fig. 2. Kappa number predictions obtained with the PLS model versus the actual Kappa number variations in the initial conditions. 20 batch runs were simulated. Fig. (3(a» shows the variations in the final Kappa number without any control, with the control based on the PLS model and with the control based on the combined PLS / modified Chari's model. The standard deviation (square root of variance) was reduced from 6 (without control) to 3 with the PLS model based control and to 1.24 with the combined PLS / modified Hfactor model based control. Figs. (3(b» and (3(c}) show the variations in the manipulated variables (Tmid and tb) from their nominal values (170 and 180). The better performance of the combined PLS/modified Chari's model based control can be attributed to better handling of the nonlinearity between the manipulated variables and the final pulp quality.
6.2 Modeling
Data for 30 batch runs were obtained. After an analysis of the off-line and on-line data, only 21 batches were found to be suitable for model building and testing. Since 21 batches were not sufficient for us to divide into separate modelbuilding and testing data sets, we decided to build the model using 20 batches, leaving only a single batch for testing. However, we tested all the possible choices for the model-building set, producing 21 data points for model testing.
Kappa It variations
On-line data consisting of EA, lignin, solids and sulfide available at 5 min. interval were used in the PLS model. A linear PLS model with 7 principal components was built relating the measurements collected up to the 90 min mark and the final Kappa number. Fig. (4) shows the predictions of final Kappa number by PLS model vs. the actual Kappa number. For comparison, the predictions from traditional Chari 's H-factor model is also shown (the word traditional is used since we used the Chari's model without the parameter adaptation described earlier) . From the figure we see that the PLS model is able to give better predictions because of its ability to use more on-line liquor measurements. The use of the PLS model results in a significant reduction in the mean square error (MSE) for the quality prediction when compared to the Chari's model. The prediction of the PLS model is slightly off in the Kappa range of 30 because the data sets used for modeling were mostly in the range 18-25. Thus the PLS model was forced to extrapolate beyond the range of model-building data. Use of more data around the 30 Kappa range would probably have resulted in a reduction in the prediction error.
Temp. variations
175..-----.--....----,.--.---'-.----,--....----,.--.---,
Batch end time variations
•
10
12
Batch index
.
,. ,
20
Fig. 3. (a) shows the Kappa number variations obtained without control, with PLS based control strategy and with combined PLS / modified-H-factor based control strategy. Target Kappa number is 40. (b) shows the variation in the mid batch temperature (Tmid) with respect to the nominal value (170) and (c) shows the variation in the batch end time (tb) with respect to the nominal value (180).
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Kaspar, M.H. and W.H. Ray (1992). Chemometric methods for process monitoring and high performance controller design. AIChE J. 38(10),1593- 1608 . K.E., Vroom (1957). The " h" factor: A means of expressing cooking times and temperatures as a single variable. Pulp and Paper Magazine of Canada Convention Issue, 228-233 . Kresta, J.V., J.F. MacGregor and T .E. Marlin (1991). Multivariate statistical monitoring of process operating performance. The Canadian Journal of Chem. Eng. 69, 35-47. MacGregor, J .F., C. Jaeckle, C. Kiparissides and M. Koutoudi (1994) . Process monitoring and diagnosis by multi block pis methods. AIChE J. 40(5), 35-47. Nomikos, P. and J . F. MacGregor (1994) . 'monitoring batch processes using multiway principal component analysis . AIChE J. 40(8), 1361-1375. Pivoso, M.J . and K.A. Kosanovich (1994). Applications of multivariate statistical methods to process monitoring and controller design. Int. Journal of Control 59(3), 743- 765. Russell, S.A., P. Kesavan, J.H. Lee and B.A. Ogunnaike (1998) . Recursive data based prediction and control of product quality for batch and semi-batch processes applied to nylon 6,6 autoclave. submitted to AIChE J. Vanchinathan, S. and G.A. Krishnagopalan (1995). Kraft delignification kinetics based on liquor analysis. TAPPI Journal 18(3), 127132. Venkateswarlu, C. and K. Gangiah (1992). Dynamic modeling and optimal state estimation using extended kalman filter for a kraft pulping digester. 1& EC Res. 31,848-855. Williams, T.J ., T. Christiansen and L.F. Albright (1982). A mathematical model of the kraft pulping process, voU and 2. Report 129. Purdue laboratory for applied industrial contro!. vo!. 1,2. Yabuki, Y. and J .F. MacGregor (1997). Product quality control in semi-batch reactors using mid-course correction policies. In : IFACADCHEM Preprints. pp. 189-193.
28
. 22
.
o •
0
,.
,.
20
22
,.
28
Actual
Fig. 4. Comparison of the prediction performance of PLS model (*) with that obtained from H-factor model (0) by using data from experimental studies. 7. CONCLUSION
In this paper, a data driven approach has been developed to model and control the final pulp quality in batch digester. The approach utilized new on-line liquor measurements available during the course of a batch in the PLS framework. Manipulated variables such as the batch end time and mid-batch temperature were incorporated into the model to develop a control algorithm. The potential of the technique for prediction and control of the final quality was demonstrated by means of simulation studies. The prediction capability of the PLS model was also demonstrated using experimental data. Currently experimental data are being obtained to test the control algorithms in real time. Acknowledgement : The authors gratefully acknowledge the financial and technical support of the U.S. Department of Energy, Pulp and Paper Consortium and Auburn Pulp and Paper Research and Education Center.
8. REFERENCES
Chari, N.C.S. (1973). Integrated control system approach for batch digester control. Tappi journal 56(7), 65-68. Chen, G. and T.J. McAvoy (1996) . Process control using data based multivariate statistical models. The Canadian Journal of Chem . Eng. 14, 1010-1024. Datta, A.K. and J.H . Lee (1994). On-line monitoring and control of pulp digester. Froc. of ACC pp. 500-504. Geladi, P. and B. R. Kowalski (1986). Partial least squares regression: A tutorial . Anal. Chim . Acta 185, 1- 17. Hatton, J .V. (1973) . Application of empirical equations to process contro!' Tappi Journal 56(8) , 108-111 .
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