Predictive Dynamic Modeling of Key Process ... - ACS Publications

Department of Chemical Engineering, University of California at Santa Barbara, Santa Barbara, California 93106-5080, United States, Procter & Gamble T...
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Ind. Eng. Chem. Res. 2011, 50, 1419–1426

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Predictive Dynamic Modeling of Key Process Variables in Granulation Processes Using Partial Least Squares Approach Doron Ronen,† Constantijn F. W. Sanders,† Hong Sing Tan,‡ Paul R. Mort,§ and Francis J. Doyle III*,† Department of Chemical Engineering, UniVersity of California at Santa Barbara, Santa Barbara, California 93106-5080, United States, Procter & Gamble Technical Centre, Ltd. Whitley Road, Longbenton, Newcastle Upon Tyne, NE12 9TS U.K, and Procter & Gamble Co. 5299 Spring GroVe AVenue, Cincinnati, Ohio 45217, United States

Granulation is a complex multivariable process with significant industrial importance. In this paper, a dynamic partial least-squares (PLS) approach is used to develop empirical predictive models of key process variables. PLS is tested on a detailed process simulator as well as on an industrial mixer-granulator process. The applicability of the nonlinear dynamic kernel-PLS (KPLS) for granulation process is also demonstrated. Accurate predictions obtained by these methods motivate the development of model predictive controllers for these processes. 1. Introduction Granulation is a complex multivariable process in which many input variables influence multiple product properties. Granulation processes are characterized by several physical attributes that are essential for product performance, such as granule size and size distribution. These processes are represented by a set of mass transfer, solid transport, as well as population balance equations. The population balance is mathematically described by partial differential-integral equations. The overall granulation model is highly nonlinear and large-scale with multiple coordinates and uncertain parameters.1 An optimally operated granulation process will yield, in a reproducible manner, product with tightly controlled performance attributes. In particular, it is desired to keep the product attributes regardless of unavoidable natural changes in the powder feed. Applying closed-loop control to such processes requires knowledge of the dynamic nature of the process, as well as accurate and continuous measurement of the key process properties, some of which are complex in nature such as the product size distribution. Although, as Iveson et al.2 described in a review paper, the understanding of the fundamental processes that control granulation behavior and product properties have increased in recent years, applying such theoretical tools to model the dynamics of a real industrial process is still a difficult task. In the search for a more practical solution for the industries that utilize granulation, this work addresses the development of an empirical predictive modeling methodology, that will produce accurate models for model-based control.3 A suitable strategy would be model predictive control (MPC), which is an effective method to control such multiple input, multiple output processes.4 The majority of MPC applications in the chemical process industries utilize empirical models that are constructed from plant data. In this work, we explore the use of dynamic partial least-squares (PLS) as it is currently the most promising dynamic multivariable linear modeling technique that has proven industrial applications. In addition we explore the applicability of a recently introduced nonlinear variant of PLS, the kernel partial * To whom correspondence should be addressed. E-mail: [email protected]. † University of California at Santa Barbara. ‡ Procter & Gamble Technical Centre, Ltd. § Procter & Gamble Co.

least squares (KPLS), which can be easily implemented for the problem and has the potential of capturing complex nonlinear process attributes. 2. Methods 2.1. Dynamic Empirical Modeling. Process modeling and model-based process control rely on the availability of appropriate mathematical models to represent the system of interest.5 Empirical modeling techniques require the collection of experimental data on those variables believed to be representative of important process behaviors, and of the key properties of the product or system output. In practice, some of the key properties measurements that are relevant to product quality, may not be available on a regular and frequent enough time basis or may require expansive and time demanding laboratory analysis, leading to unacceptable time delays.6 A large number of methodologies are available for dynamic empirical modeling, such as state space models, time series models, and time transfer function models. In this work we use the autoregressive with exogenous inputs (ARX) discrete regression model defined as nu

-1

A(q )yi(k) )

∑ B (q

-1

j

)uj(k - nkj)

(1)

j)1

A(q) ) 1 + a1q-1 + a2q-2 + ... + anyq-ny

(2)

Bj(q) ) bj,1q-1 + bj,2q-2 + ... + bj,nuq-nn

(3)

where u(k) and y(k) are the process input and output data vectors, respectively. The terms A and B contain the autoregressive and exogenous terms of the model, respectively. The autoregressive term captures dynamics through time lagged terms of the output, and the exogenous term captures dynamics through time lagged terms of the input. In this way the output values at time k are a consequence of previous process conditions. The advantage of this approach is that steady state regression techniques can be used directly for the modeling of dynamic processes, as is clearly expressed in the matrix from

10.1021/ie100836w  2011 American Chemical Society Published on Web 01/04/2011

Y(k) ) CX(k)

(4)

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X(k) ) [y(k - 1), y(k - 2), ..., y(k - ny), u(k - 1), u(k - 2), ..., u(k - nn)]

Table 1. PLS and KPLS NIPALS Algorithms

(5)

step

(6)

1 2 3 4 5 6 7

The ordinary least-squares solution for C is given by C ) (XTX)-1XTY

However, in an industrial environment, it is more often the case that many of the predictor variables (X) are highly correlated with one another and their covariance matrix is nearly singular, which renders classical regression methods intractable. Autoregressive models introduce even more colinearity into the predictor matrix along the different lagged values of the variables used. Reduced space methods such as PLS and PCA can overcome this problem.6,7 Keeping in mind that the ultimate target of the granulation process modeling is its use in an MPC controller design, it is important to test the resulting ARX model’s ability to accurately predict multistep-ahead moves of the process output. Although ARX models provide for one-step-ahead predictions with the input and output data available at time k being used to predict the output at time (k + 1), multistep-ahead predictions can be calculated using the predicted output values instead of the measured output values in eq 5, that is, the regressor vector x is defined as X(k) ) [yˆ(k - 1), yˆ(k - 2), ..., yˆ(k - ny), u(k - 1), u(k - 2), ..., u(k - nn)]

PLS w ) XTu t ) Xw,trt/|t|

KPLS randomly initialize u t ) ΦΦTu ) Ku,trt/|t|

c ) YTt u ) Yc,uru/|u| repeat steps 2-5 until residual value of t converge XrX - ttTX Kr(I - ttT)K(I - ttT) YrY - ttTY YrY - ttTY

and the weight vectors w,c (as detailed in Table 1) from the X and Y matrices in decreasing order of their corresponding singular values. The PLS regression model can be expressed with regression coefficient B and residual matrix R as follows: Y ) XB + R

(10)

B ) W(PTW)-1CT

(11)

Following the equalities derived by Ranner et al.,13 the matrix B can be written as eq 12, which will be used to make predictions in PLS regressions: B ) XTU(TTXXTU)-1TTY

(12)

(7)

Where yˆ(k) denotes the predicted output at time k. 2.2. Partial Least Squares. Partial least squares (PLS) methods have been demonstrated as a useful tool for analysis of noisy and highly colinear data and modeling of the systems from which the data are collected.8 It represents an appealing alternative to the classical multiple linear regression (MLR) approach since it has been shown to be more robust to noise, colinearity, and high dimensionality in the data than the conventional least-squares methodology. PLS has established itself as an important analytical tool in chemometrics. Various examples of the implementation of PLS analysis to industrial process modeling and control can be found in the literature (e.g., Dayal et al.,9 MacGregor and Kourti,7 and others). Applications to dynamic process modeling, yielding an ARX type of model, are also described in recent literature.6,8-10 Unlike related methods, such as principal component analysis (PCA), which finds factors that capture the greatest amount of variance in the predictor (X) only, the PLS method attempts to find factors which both capture variance and achieve correlation. PLS handles this by projecting the information in high dimensional spaces (X,Y) down to low dimensional spaces defined by a small number of latent vectors. These new latent vectors summarize all the important linear information contained in the original data sets, by representing the scaled and mean centered values of X (n × N) and Y (n × M) matrices as9,11 X ) TPT + E

(8)

Y ) UQT + F

(9)

where n is the number of input/output samples, N is the number of predicting variables, M is the number of output variables, T and U are (n × k) matrices of the extracted k latent (score) vectors, P (N × k) and Q (M × k) are matrices of loadings, and E (n × N) and F (n × M) represent matrices of residuals. The PLS model is easily calculated using the NIPALS algorithm,11,12 which sequentially extracts the latent vectors t,u

Typically, the majority of the covariance of X and Y can be accounted for by the first few latent variables, while the higher order latent variables are typically associated with the random noise (or nonlinearities) in the data. The optimal number of latent vectors retained in the model is often determined by cross validation.6,9 2.3. Kernel Partial Least Squares. When applying linear PLS to nonlinear problems, the minor latent variables cannot always be discarded since they may not only describe noise or negligible variance-covariance structures in the data. For a nonlinear problem, the minor latent variables may capture significant information about the nonlinear nature of the problem. In fact, nonlinear structures may be modeled using a combination of higher-order and lower-order latent variables calculated from linear PLS.6 However, a common consequence of this approach is an overfitted model that is too sensitive to the noise in the modeling data. Different approaches that incorporate nonlinear functions into the linear PLS framework were offered in the literature. Recently, a Kernel-based PLS approach was formulated12 which does not involve any complex nonlinear optimizations. According to Cover’s theorem,14 the nonlinear data structure in the input space is more likely to be linear after highdimensional nonlinear mapping. Kernel-based learning algorithms proceed by mapping original data X into a (possibly highdimensional) Hilbert space F, usually called featured space, and then by using linear pattern analysis to detect relations in the feature space. The KPLS regression was developed by Rosipal and Trajero,12 where the objective is to construct a linear PLS regression in F or equivalently a nonlinear PLS regression model in X. Consider the nonlinear transformation of the input variable xi, i ) 1,...,n into the feature space F: xi ∈ RN f Φ(xi) ∈ F

(13)

KPLS, like similar Kernel techniques, uses implicit mapping based on the “kernel trick”, where mapping is performed by

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011 -1 T

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Yˆ ) ΦB ) KU(T KU) T Y

(18)

Yˆt ) ΦtB ) KtU(TTKU)-1TTY

(19)

T

Here, Φt is the matrix of the mapped test points and Kt is the (nt × n) test Gram matrix whose elements are Kij) K(xi,xj), where xi and xj are the testing and training points, respectively. In order for the PLS to be able to work in the feature space, the mapped data must be centralized in the feature space. This can be done as follows:12 Figure 1. Simulator structure. Five inputs are included in the simulator: solid feed rate, binder spray rate, fine powder feed-rate, drum rotationrate,and the drum inclination angle. The model is divided into three well mixed drum compartments, each described by an individual set of ODEs, a retention time model, and a set of global parameters that influence the model behavior (taken from Glaser et al.16).

specifying the inner product between each pair of data rather than by explicitly calculating their coordinates: Φ(xi)TΦ(xj) ) K(xi, xj)

(14)

- xj | 2

K(xi, xj) ) (axixTj + r)d

)(

)

(20)

where 1n represents a vector whose elements are ones with length n. Kt should also be centralized in the feature space by applying 1 1 Kt ) Kt - 1nt1Tn K I - 1n1Tn n n

(

)(

)

(21)

(15) (16)

3. Computational Study

KPLS can be easily calculated using a modified NIPALS algorithm as shown in Table 1. Unlike other nonlinear PLS techniques, the KPLS requires no nonlinear optimizations and is directly derived from the PLS algorithm by modifying steps 2 and 3 of the PLS procedure so as to use the matrix Φ of mapped input data instead of using the data itself. As in eq 12 the matrix of the regression coefficient B in KPLS will have the form B ) ΦTU(TTKU)-1TTY

(

Recent works have shown the applicability of KPLS to (static) industrial applications: Woo et al.15 established excellent prediction for 3 key process variables of an industrial cokes wastewater treatment plant’s effluent based on 27 measured process variables. Kim et al.11 successfully applied KPLS to two different industrial test cases: a slurry-fed ceramic melter, using 20 temperature readings to predict the molten glass level, and a polymer test plant correlating 10 controlled variables to predict one controlled variable.

According to Mercer’s theorem, any symmetric and positive definite kernel can be used,12 where the most common are the Gaussian (eq 15) and the Polynomial (eq 16) kernels. Notice that ΦΦT represents the (n × n) kernel Gram matrix K of the cross dot products between all mapped input data points. K(xi, xj) ) e-γ|xi

1 1 K ) I - 1n1Tn K I - 1n1Tn n n

(17)

and predictions on training data and test data having nt points can be made as follows:

To evaluate the applicability of dynamic PLS modeling for the granulation process, and to estimate the type and amount of data needed for industrial process modeling, a computational study was performed based on a granulation process simulator (Figure 1) that was developed in a previous study.16 This simulator uses a nonlinear one-dimensional population balance model (1D-PBM) that has been successfully calibrated to model a laboratory continuous drum granulation process with fine particle recycle. Simulated data were generated using this simulator by randomly manipulating four input variables: solid feed flow rate, binder feed flow rate, drum rotation speed, and a recycle rate, which recycles rejected powder to the first compartment. These variables were randomly perturbed around their nominal values

Figure 2. Perturbed input variables and the resulting d50 calculated with the granulation process simulator.

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Figure 3. RMSEP versus length of training data set based on simulation data, at different excitation rates.

at steady state sequentially, that is, one after the other in fixed time gaps (Figure 2). The predictor matrix (X) included five variables: the four manipulated variables together with the simulator’s computed recycle flow. The simulators output of particle median size (d50) was used to construct the output vector (Y). Delay times of the various predicting variables relative to the output variable were estimated using a crosscorrelation function, and the time lag of the output variable was estimated using an autocorrelation function resulting in one lagged output variable added to the predictor matrix. For the predicting variables no time lags were used. The resulting PLS-based ARX model’s short horizon predictive power was estimated using a root-mean-square error in prediction (RMSEP) (eq 22) that provides an estimate of the average deviation of the model from cross validation data. The cross validation data were produced with a set of separately calculated simulation sequences with different excitation regimes.

RMSEP )



∑ (yˆ

i

- yi)2

n

(22)

where yi is the reference value, yˆi is the predicted value, and n is the total number of samples. To estimate the required size of the data set needed for reliable granulation process modeling, a sensitivity test was performed. The convergence of the RMSEP related to the length of data set used for the PLS model training is shown in Figure 3. This plot is based on averaging 100 multiple simulations and modeling runs for each training length. The sample rate was set to 2 min, and the simulated process step response time (τ) was set to 4.5 min. The RMSEP was calculated for a short prediction horizon of eight samples (i.e., 16 min) using a set of separately calculated simulation sequences with different excitation regimes. The short horizon start point was then moved along the time axis of the data on time step after another thus creating

Figure 5. PLS-based dynamic model validation for different training set lengths (A, 120; B, 220; C, 520 min). Circles represents simulation results, lines represent 8-point horizon prediction.

a set of RMSEP measurements out of which an average RMSEP could be calculated. Minimum RMSEP values were obtained using two latent variables in the PLS model. From this figure one can note that most of the dynamic features are captured by the model in the first 200 min of training data, as the mean RMSEP converges to low values. However using training data of up to 600 min would improve the model predictions. Notice that these results are not so sensitive to the excitations rate used in the modeling data set (i.e., time between two successive input variable perturbations), as long as this time is in the order of magnitude of the expected variations in process variables. Figure 4 depicts the response of the process model obtained with 520 min of training data to the process simulation for a step response in one of the input variables. Selected PLS models obtained using three different lengths of data sets (120, 220, and 520 min long) from a single simulation run with excitations every 15 min are shown in Figure 5. In this example, the predictions of these models are cross-validated using data from a separate simulation run with randomly timed excitations of the inputs. It is noticeable that the longer the training set used, the more fine details of the process dynamics are captured by the models, confirming Figure 3 results. Considering that PLS models capture covariance in Table 2. Percent Variance Captured by PLS Models Based on Different Lengths of Training Data Sets X block training set length 120 min 220 min 520 min

Figure 4. Step response to a 1.2% step change in binder feed flow-PLS model based on 520 min training data vs process simulation.

520 min (with 5% white noise)

Y block

LVs

LVi

total

LVi

total

1 2 1 2 1 2 1 2

76.04 7.66 59.45 14.21 41.35 17.29 33.93 33.10

76.04 83.70 59.45 73.66 41.35 58.64 33.93 67.03

82.44 11.42 93.05 2.80 97.88 1.04 82.75 1.61

82.44 93.85 93.05 95.85 97.88 98.92 82.75 84.37

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Figure 6. PLS-based dynamic model validation, simulated process with 5% white measurement noise. Circles represents simulation results, lines represent 8-point horizon prediction.

Figure 7. Representative P&G process flow diagram for mixer-granulator (Mort et al.17). For simplicity, this diagram omits the usual operations for classification and recycle.

X and Y, it is possible to calculate the percentage of variance captured by each of their latent variables by dividing the variance predicted by the latent variables to the total variance in the original data. The percentage of variance captured by the above-mentioned 3 PLS models (from the training data) is detailed in Table 2. Notice that high values of explained variance do not guarantee good prediction of validation data by these models. The robustness of the PLS-based models to measurement noise is demonstrated in Figure 6 and Table 2, where the simulated process was subject to 5% white noise on the output and input variable measurements. 4. Dynamic Modeling of an Industrial Process Plant 4.1. Process Plant Description and Test Methodology. Following the computational study described above, a set of tests were performed on a Procter & Gamble (P&G) industrial granulation process. It is important to notice that there are significant distinctions between the P&G study and the one reported in section 3. The P&G process study is not meant to be used as a direct comparison (or validation) for the process simulation studies in section 3. Rather, we present both as separate case studies to demonstrate the feasibility of using PLS methodology as an empirical modeling tool for granulation process control. A complex industrial granulation process, such as the flowsheet shown in Figure 7, was subjected to a series of (designed) random perturbations in a number of input parameters. This plant is equipped with an online granule size

measurement system that measures particle size on the basis of image analysis of 2-D camera images. The analysis constructed size distributions on the basis of the measured cross sectional area of the 2-D images. Granule size data along with all other plant variables were then sent to the UCSB team for modeling. All plant data given hereafter is in normalized units. The low-shear drum-granulation pilot plant that was used to design the simulator in section 3 produced particles with d50 values of several millimeters; on the other hand, the mediumhigh shear mixer-granulation process shown in Figure 7 typically produced particles with d50 values less than 1 mm. While the underlying physical mechanisms of growth and consolidation may be similar, the flow and shear fields are very different for the two processes (the drum granulator is relatively low shear compared to the medium-high shear mixer-granulation process), the process layouts and control handles are different, as are the material properties. As such, the choices of process variables (manipulated and measured) are unique for each process. Multiple trials were run on the industrial plant on different dates. In each trial, up to five manipulated variables were subjected to random perturbations around their nominal values at steady state in a similar way to the simulation work described earlier (adjusted to the process τ), while other adjustments were continuously made to other plant variables (i.e., normal plant operations). A total of 81 process variables were continuously measured (in addition to the granules size distribution captured by the online camera) out of which predictor variables were chosen on the basis of either engineering judgment, genetic algorithm based variable selection (PLS Toolbox 5.0 by eigenvector research incorporated), and trial and error. Lag times and process delay times were evaluated using the auto- and cross-correlation functions, respectively, as described in section 3. 4.2. Modeling an Industrial Process PlantsCase I. In this first case, the data of a single plant run, containing a total of 147 sampling points, were collected. The sampling time was 0.4 times the process characteristic time τ. Perturbations were introduced to four of the manipulated variables. A total of nine variables, including the manipulated variables, were selected for the predictor matrix (Figure 8). The granules mean size (cross-section area), as calculated from the camera’s data, was used as the process output variable. The PLS model for the training data, using two latent variables, is shown in Figure 9. For an independent cross validation of the PLS model, a less accurate PLS model based only on half the available data for training was calculated, yielding a RMSEP value of 0.26 (Figure 10 and Table 3). Notice that these results capture most of the variance in the data and are in good agreement with the simulation results for the same training length to τ ratio (Figure 3 and Figure 5A).

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Figure 8. Values of the nine predictor variables used in Case I PLS modelsfour manipulated variables (top) and five additional process variables (bottom).

Figure 9. Dynamic PLS model fitting to plants mean size data.

The same X and Y matrices used to construct the PLS model based on half of the available measurements were used to evaluate the feasibility of constructing a predictive dynamic KPLS model for this process. Two KPLS models were calculated using both Gaussian and linear kernels. While using the Gaussian kernel resulted in very poor prediction, using the linear kernel (xixTj ) resulted in significantly better predictions compared to the PLS model, having a RMSEP of only 0.19 (Figure 10). The long-term goal of this work is to implement a closedloop control on an industrial granulation process. As a critical validation of the model at this stage, an advisory MPC was demonstrated using both of the above-mentioned PLS and KPLS models: The two models were transformed to ARX models, then an MPC controller was designed based on the PLS-ARX model, while the KPLS-ARX model was used as the plant model

Figure 10. Cross-validation of the dynamic PLS and dynamic KPLS models for mean size data. Table 3. Percent Variance Captured by PLS ModelssIndustrial Case I X block

Y block

training set length

LVs

LVi

total

LVi

total

60τ

1 2 1 2

24.97 25.09 39.46 23.31

24.97 50.06 39.46 62.77

79.34 9.04 72.13 16.8

79.34 88.38 72.13 88.92

36τ

(Figure 11). Two out of the four originally perturbed variables were chosen as the control variables, while the rest of the measured variables were treated as disturbance variables. Figure 12 demonstrates simulated step response of this system. Figure 13 simulates disturbance rejection while subjecting the simulated plant to real disturbance data measured in the process plant.

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Figure 11. Block diagram of closed loop simulation.

Figure 14. PLS model predictions for first plant test data set (Case I) using a model trained on second plant test data set (Case II). Solid line is the model prediction; dashed line is the same prediction with a constant bias.

Figure 12. Simulated step response of MPC controlled granulation process using two manipulated variables.

Figure 15. PLS model prediction for third plant data set (Case III) based on training data from second data set (Case II) using mean granule area as the output variable.

Figure 13. Simulated disturbance rejection of MPC controlled granulation process using two manipulated variables.

Both demonstrate stable and efficient control of the process, as is desired by the industry. 4.3. Modeling an Industrial Process PlantsCase II. A greater challenge for the dynamic PLS models is the prediction of a plant’s output based on training data obtained on a different day, using different lots of raw materials. This kind of test gets us closer to the ultimate goal of designing a robust industrial process controller for this type of processes. In this case, a similar test run performed on a different day used five manipulated variables for excitations. On the basis of this run’s data, a PLS model using only six predictor variables was trained

and used to predict the process output of the first test run described earlier in section 4.2. This model used four LVs and five lagged samples of the output variable in the predictor matrix. Results, shown in Figure 14, demonstrate very good prediction for the most part of the test, but with a constant bias. This bias may have been caused by unaccounted material property differences between the two runs or by differences in the particle size distribution (PSD) pattern, which are reflected as a bias in the calculated mean granules size. It should be noted that for application in a closed loop model predictive control scheme, such a bias is not significant since the closed loop mechanism can effectively eliminate it. 4.4. Modeling an Industrial Process PlantsCase III. Results from a third test run emphasize the complexity of predicting the size attributes of a granulation process product. In this test, a similar run was made in the same process plant. The PLS model developed in section 4.3 failed to accurately predict the product mean size (Figure 15). However, for this case, using the area-based median (instead of mean) granule size for both model training and prediction resulted in a much better prediction (Figure 16). Analyzing the full PSD data

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of granulation processes may require multiple measurements to accurately and consistently capture the required complexity of the product PSD. Although not demonstrated here, both PLS and MPC are capable of dealing with multiple output variables, thus making this challenge feasible. KPLS was also demonstrated to be an effective nonlinear modeling technique for this kind of process, yielding a significant improvement in model prediction relative to the classical PLS. KPLS simplicity makes it an attractive alternative to the PLS in cases where higher accuracy is required or more complex and nonlinear processes are involved. The analysis of both modeling techniques in this work was limited by the amount of data at hand. Producing more reliable models should be done by using longer plant runs as indicated by the numerical study section. Acknowledgment Funding for this work from IFPRI (International Fine Particle Research Institute) is gratefully acknowledged. Figure 16. PLS model prediction for third plant data set (Case III) based on training data from second data set (Case II) using median granule area as the output variable.

Figure 17. PLS modeling for granules PSD (area standard deviation): Cross validation using different test days for training and validation data.

gathered by the online camera reveals that indeed this run resulted in different PSD shape relative to the earlier tests. In addition, the granule sizes obtained in this run were small compared to the resolution of the online camera, which may explain the inaccuracy of the obtained model. 4.5. Modeling an Industrial Process PlantsCase IV. In this test the PSD width (calculated as the granules area standard deviation) of the product was modeled. A PLS model with two LVs was trained with data from a single plant run using three input variables and one lagged output variable. The excitation scheme in this example was limited to one input variable. Figure 17 demonstrates the excellent prediction obtained by this model when tested against data of a different plant run. 5. Conclusions Dynamic PLS modeling was demonstrated to be an effective tool in modeling key process variables for two types of continuous granulation processes. These models are applicable to the design of industrially relevant close-loop control schemes. When modeling complex attributes such as particle size distribution, it is very important to ensure the use of appropriate measures (and measuring devices). Robust modeling and control

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ReceiVed for reView April 7, 2010 ReVised manuscript receiVed September 27, 2010 Accepted December 1, 2010 IE100836W