Model Predictive Statistical Process Control of Chemical Plants

May 30, 2002 - The approach is model based, and it uses principal component analysis to compress the selected process measurements into scores, which ...
1 downloads 9 Views 253KB Size
Download PDF
Ind. Eng. Chem. Res. 2002, 41, 6337-6344

6337

Model Predictive Statistical Process Control of Chemical Plants Thomas McAvoy Department of Chemical Engineering, Institute for Systems Research, University of Maryland, College Park, Maryland 20742

Statistical process control aims at improving process operation by distinguishing abnormal process conditions from common cause variations. Improved process operation then results from correcting the abnormal conditions. In this paper an alternative, feedback-based approach to process quality improvement is discussed. The goal of the approach is to use existing process measurements to help reduce the variability of product quality when its online measurement is not feasible. The approach is model based, and it uses principal component analysis to compress the selected process measurements into scores, which are controlled. One or more manipulated setpoints are chosen and varied to counteract the effect of stochastic process disturbances on product quality. The approach discussed assumes that the selected process measurements correlate with product quality and that the stochastic disturbances that cause product variability are stationary. The methodology is illustrated on the Tennessee-Eastman process, where a 44% reduction in product variation is achieved. When nonstationary upsets such as steps occur, the score control setpoints need to be adjusted. A steady-state model predictive controller is discussed which overcomes the problem caused by step upsets. The application presented in the paper involves one manipulated variable, but the methodology is multivariable in nature. Introduction Statistical process control (SPC) is aimed at improving processes and products. Shewart1 laid out the basic concepts for SPC in 1931, and the Shewart chart has been widely used in industry. In addition to Shewhart charts, cumulative sum (CUMSUM) and exponentially weighted moving average (EWMA) charts2,3 have also been used to compare current operation to normal operation. A major drawback of these approaches in terms of their use for process systems is that they were developed for univariate systems. Process plants have correlated variables, and it is relatively easy to construct cases in which the correlation causes the univariate methods to be in error. The proper approach to SPC for process systems involves the use of multivariate methods such as principal component analysis (PCA)4 and partial least squares (PLS).5 With both PCA and PLS, linear combinations of raw variables, called scores, are calculated and used for monitoring a process. Kresta et al.6 and Wise and Gallagher7 have discussed the application of PCA and PLS in the process industries. SPC approaches seek to distinguish between common cause variations in a process, which involve normal process variations, and special cause events, e.g., a faulty sensor or component. Once an abnormal situation is detected, it is up to a process engineer to find its root cause. Thus, the word control in SPC has a somewhat different meaning than when it is used in feedback process control. Control in SPC is carried out by determining and correcting an abnormal situation, and as such it is typically not carried out in real time. This paper discusses a model predictive control approach to SPC in which continuous feedback of information is used. The control objective is to reduce the variability of product quality by changing the setpoints of one or more control loops in a plant. The measured variables used for feedback are a linear combination of the process scores calculated from a PCA model.

This paper addresses the issue of reducing the variation in product quality when stochastic upsets enter a process that is operating normally. Under such an operation three broad cases can be considered for product quality control. First, one can use an online process analyzer and close control loops around it. This first case can be further subdivided depending on the speed of the analyzer that is used. If the analyzer responds in real time, then it can be used in a straightforward manner. When the analyzer is much slower than other measurements, then a multirate approach can be employed. Li et al.8 have recently discussed an inferential multirate approach applied to an industrial octane quality control problem, and they have given a number of references to multirate systems. Second, if an online analyzer is not available, one may be able to build a soft sensor to predict product quality from readily measured process variables.9 In building such a sensor, one would need either to install an analyzer on a temporary basis or to time stamp process measurements so that they could be correlated with laboratory assays. Third, one may not have either an online analyzer or a soft sensor and may have to rely on taking samples to a laboratory and waiting for them to be assayed, typically on the order of 8 h or longer. It is this third case that is addressed in this paper. An approach is presented that makes use of both real-time process measurements and laboratory assays. This paper makes use of data-based multivariate statistical process models within a feedback control system. It is well-known that one has to be very careful when such models are incorporated into a feedback system.10 If normal process operating data are used for model development, then the data have a correlation structure. The resulting model can be extrapolated by a feedback system to a region in which the underlying correlation structure is no longer valid, and therefore the model should not be used. To avoid this problem, either some plant testing must be conducted or the

10.1021/ie020067q CCC: $22.00 © 2002 American Chemical Society Published on Web 05/30/2002

6338

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002

database used for model development must be rich enough that it can be used in feedback control. Yabuki and Macgregor11,12 and Flores-Cerrillo and Macgregor13 have used multiway PLS models of batch polymer processes to calculate mid-course corrections for batches projected to have product quality targets outside of acceptable limits. The database used included midcourse changes that operators had made in the past, and so it was rich enough that accurate models for trajectory adjustment could be developed from it. Shah et al.14 present results for an industrial batch reactor in which a PCA model is inverted to force the reactor to track a desired score trajectory. Chen et al.15 discuss the use of PCA models in what Piovoso and Kosanovich16 termed score control. In ref 15, a lagged PCA model is developed and model predictive control is used to force the PCA scores toward the origin. Application of the approach to a binary distillation tower and the Tennessee-Eastman process17 was discussed. It was found that the score controller resulted in a significant reduction in the variability of product quality in both examples. This paper also presents a model predictive score control approach. In the earlier paper,15 the score control is used in addition to a feedback quality controller that employs a composition analyzer. Here it is assumed that an analyzer is not available. Also, the score control approach is further developed by using orthogonal PCA18-20 to focus on that part of the variability that can be altered using the chosen manipulated variables. In this paper control in a reduced dimension is discussed, and several other papers have treated this subject.19,22 Clark-Pringle and Macgregor21,22 discuss batch to batch control in which a subset of quality variables is controlled in order to indirectly control the entire quality space. For example, rather than controlling the full molecular weight distribution in a polymer system, an average of the distribution is controlled. In ref 21 these authors generalize their approach and apply it to a Kamyr digester. They formulate a control problem that involves minimizing the predicted error in certain output variables. Featherstone and Braatz19 discuss modeling and control of sheet and film processes, and they use a pseudo singular value model to determine which parts of a dynamic model can be identified accurately enough that they can be used in a feedback controller. Although 45 outputs could be controlled, their analysis lead to only 10 output directions being controlled because the identification data did not allow for accurate determination of many process gains. Featherstone and Braatz20 have extended the ideas in ref 19 to general processes. The approach presented in this paper differs from other reduced dimension control approaches in that it is a two-time-scale approach in which both real-time measurements and slow laboratory measurements are employed. Also, for the real-time calculations, it is assumed that stationary disturbances occur and, to handle these disturbances, a constraint on the control moves is employed. The structure of this paper is as follows. After an introduction to the problem, model predictive score control is presented. The basic assumptions in score control are that process operation is stationary and that the scores of the selected measurements correlate with product quality. Thus, reducing the variation in the process scores produces a corresponding reduction in the variation of the product quality. Next the methodology is applied to the Tennessee testbed process,17 on which

Figure 1. Typical process plant.

Figure 2. Plant database.

its effectiveness is demonstrated. A discussion of how to handle nonstationary disturbances is given, and finally conclusions are drawn. Model Predictive Control in the Score Space In the approach used in this paper, it is assumed that the plant is running under an existing control system and that it is subjected to stationary stochastic disturbances. The plant is shown schematically in Figure 1. Only a single controller (XC) and a single measurement (XM) are shown for illustration, but obviously the plant has a number of control loops and measurements. Typically the setpoints of the controllers are set to a constant value. As a result, the entering disturbances cause stochastic fluctuations in the product quality. It is assumed that this quality is measured infrequently via a laboratory assay. To reduce the fluctuations in product quality, it is proposed to add a model predictive controller on top of the existing plantwide control system. The model predictive controller manipulates the setpoints of a small number of selected loops, as compared to keeping these setpoints constant, to counteract the effect of the disturbances. Because the process operation is assumed to be stationary, no net steadystate adjustment is made to the manipulated setpoints. In the methodology described below, the disturbances do not have to be measured directly. Rather, the effect of the disturbance is inferred through the use of existing measurements. To reduce the product variations, a group of real-time measurements, xi(t), and a group of setpoints to be manipulated, sj(t), are selected. Pseudo random binary signal (PRBS) forcing is used on the setpoints to develop a database for modeling the process. The resulting data are shown schematically in Figure 2, where the area corresponding to the setpoints is shaded. The total time for the PRBS forcing is Tf, and the values of x1 to xn and the setpoints s1 to sm are measured from t ) 0 to Tf. A sampling interval of ∆t is used for the measurements. In a typical case the number of setpoints will be much smaller than the number of process measurements. In addition, the variation in the measurements will be caused both by the unmeasured stochastic disturbances and by the PRBS forcing that is used to generate the database. In using a dynamic PCA model

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002 6339

for control, it is useful to focus on the space of measurements that is affected by the setpoints themselves. The database in Figure 2 can be written as [X, S], where X designates the measurements and S the setpoints. First, X is compressed to a set of dynamic scores using PCA as

in the setpoints, ∆s, are to be calculated and that the effect of these changes on z is given by

[

a1 0 0 a2 a1 0 a1 z ) a3 a2 l l l aP aP-1 aP-2

r

X=

tipiT ∑ i)1

(1)

]

... 0 ... 0 ... 0 ∆s ≡ AD∆s l l ... aP-M

(5)

where ti are the scores, pi are the loading vectors, and r is the number of PCA components used.4 The scores can be arranged into a matrix, T ) [t1, t2, ..., tr]. To focus on the part of the score space that can be affected by S, orthogonal PCA, as discussed in ref 18, is used. A matrix W is defined as

where AD is the dynamic matrix that contains the step response coefficients. At time k the future values of z, zf, can be predicted as

W ≡ TTS

where ∆z is the difference between the measured value of z at t ) k and the predicted value and zp are the contributions to z that are the result of past moves in s. Feedback is incorporated into the control system through ∆z. The error between zf and the z setpoint, zsp, if no control action is taken is defined as

(2)

If W is used to transform T, its null space defines a linear combination of the scores that are not affected by the S variables. Similarly, the range of W defines a linear combination of the scores that S can affect. It is this latter combination of scores that is used for control. Clark-Pringle and Macgregor21,22 have also used the same approach in their work on reduced dimension control. The control objective is to shrink this score region by changing the chosen setpoints. The major assumption that is made is that a reduction in this combination of real-time scores translates into a reduction in the variability of product quality. One way of viewing this assumption is that it implies that principal component regression applies to the product quality; i.e., the scores in the range of W correlate with the product quality. If the singular value decomposition (SVD) of W is calculated as U∑VT, the first m columns of U define the range of W, while the remaining columns define the null space of W. If the first m columns of U are designated U1, then the scores can be transformed as

Z ) TU1

(4)

In the MPC formulation discussed below for the Tennessee-Eastman process, a step response model developed from eq 4 is employed. Any MPC approach can be employed, and the particular MPC formulation used for illustration is discussed next. Assume that M changes

(6)

E ≡ zsp - ∆z - zp

(7)

A model predictive control approach to SPC can be defined as P

min

∑||Γlz([El - AD,l∆s])||2 +

∆s(t)...∆s(t+M-1) l)1

M

||Γls([∆s(t+l-1)])||2 ∑ l)1 subject to P

[∆s(t+l-1)] + ∑ ∆s ) 0 ∑ l)1 past

(3)

The elements of Z are linear combinations of the raw measurements, X, that are affected by the setpoints, S. In addition, the unmeasured stochastic disturbances also affect Z. The dimension of Z is the same as the dimension of S, assuming that each of the setpoints used is independent and that dim T g dim S. Because the measurements are affected both by unmeasured disturbances and by the setpoints, this inequality will hold in general. In the development of a model predictive SPC approach to the reduction of product quality variability, a discrete-time linear dynamic model is used. The model, which can be any type, is identified from the data matrix [Z, S]. In the example below involving the TennesseeEastman process, a single-input single-output ARX model is identified as

z(t+1) ) Az(t) + Bs(t-nk) + e(t)

zf ) ∆z + AD∆s + zp

(8)

where M is the control horizon, P is the prediction horizon, and Γlz and Γls are diagonal weighting matrices. The objective function involves a tradeoff between the errors in the score and the amount of control action employed. The constraint on the changes in s forces the sum of control moves to be zero. The logic behind this constraint is as follows. The model predictive SPC approach is designed to counteract the effect of stationary stochastic upsets around steady state. To keep the process at steady state, the changes in s are forced to sum to 0. Equation 8 can be solved by using a Lagrange multiplier and adding the constraint to the objective function as P

∑

||Γlz([El - AD,l∆s])||2 + ∆s(t)...∆s(t+M-1),λ l)1 M M ||Γls([∆s(t+l-1)])||2 + λ( [∆s(t+l-1)] l)1 l)1 min

∑

∑

+

∑ ∆s)

past

(9) For the case where Γlz ) w1 and Γls ) w2 for all l, the solution of eq 9 can be calculated in a straightforward

6340

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002

Figure 3. Schematic of the Tennessee-Eastman process. Table 1. PI Controller Constants controlled

manipulated

Kc

A flow C flow D flow E flow purge separator exit product flow steam condenser cooling water temperature reactor cooling water temperature

Inner Cascade Controllers A valve C valve D valve E valve purge valve separator exit valve product valve steam valve condenser cooling water valve reactor cooling water valve

150%/kscmh 14%/kscmh 0.026%/kg‚h-1 0.017%/kg‚h-1 200%/kscmh 2.0%/m3‚h-1 3.0%/m3‚h-1 2.5%/kg‚h-1 -8.0%/°C -10%/°C

reactor level separator level stripper level reactor pressure product flow A/C reactor feed B in purge

Outer Cascade Controllers E feed setpoint separator exit setpoint product setpoint reactor cooling water temperature setpoint C feed setpoint A feed setpoint purge setpoint

500 kg‚h-1/% -0.12 m3‚h-1/% -0.07 m3‚h-1/% -0.02 °C/kPa 0.6 kscmh/m3‚h-1 1.0 kscmh -0.0366 kscmh/%

manner as

∆s ) (2w1ADTAD + 2w2I)-1(2w1ADTE - λR) (10) λ ) [rT(2w1ADTAD + 2w2I)-1(2w1ADTE) +

∆s]/[rT(2w1ADTAD + 2w2I)-1r] ∑ past

(11)

where r is an M × 1 vector of ones, r ) [1; 1; ...; 1]. At each time, k, E, and ∑past∆s are known from the past calculations and measurements. From eq 11 the Lagrange multiplier, λ, can be calculated, and then eq 10 can be solved for ∆s. The first row of ∆s is implemented, and the entire calculation is repeated at the next time step. The application of eqs 10 and 11 to the TennesseeEastman process is discussed next. Results on the Tennessee-Eastman Process A detailed description of the Tennessee-Eastman process, including typical disturbances and baseline operating conditions, is given in ref 17. The process involves the production of two products, G and H, from four reactants, A, C, D, and E. In addition, there are two side reactions that occur and an inert, B, essentially

TR (min) 0.075 0.13 0.10 0.10 0.06 0.12 0.12 1.5 1.8 0.70 200 200 200 30 60 164 250

all of which enter with one of the feed streams. A process diagram for the Tennessee-Eastman process is shown in Figure 3. Included in the figure is the control system that is being used to run the plant. The tuning parameters for the various loops shown are given in Table 1. In the original problem statement, an analyzer was used to measure the composition of the product stream. Here it is assumed that this composition is not available online but rather that it is measured offline in a laboratory. The product composition control objective is to maintain the G/H ratio in the product stream constant. As discussed by McAvoy and Ye,23 the ratio of the D and E feeds can be used to control the G/H ratio in the product. If the small side reactions are neglected, then basically component D is used to produce G and component E is used to produce H. As a result, the molar G/H ratio in the product is approximately equal to the molar D/E ratio in the feed to the process. Because an online analyzer is not being used, a fixed value for the D/E ratio of 0.8126 is employed. The disturbance that is considered here is a random change in the composition of the C feed, labeled IDV(8) in ref 17. The response of the G/H ratio to IDV(8) resulting from the existing control system is shown in Figure 4 as the dashed line. A 3-day period is simulated, and for the purposes of

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002 6341

Figure 4. Comparison of constant D/E policy and model predictive score control. Table 2. Scale Factors and Loads variable

mean

A feed (kscm/h) 0.3166 reactor pressure (kPa) 2704 reactor temperature (°C) 120.6 separator level (%) 50.0 separator pressure (kPa) 2632.8 D feed setpoint (kg/h) 3655.7

standard deviation 0.1484 23.35 1.307 1.000 22.83 75.73

load 1

load 2

0.3727 -0.5718 0.5371 0.2681 0.4124 -0.5616 -0.3380 -0.4616 0.5369 0.2697

clarity, only the responses for the first 24 h are shown in Figure 4. The remainder of the transients shows the same trends. Over the 3-day period, the G/H ratio oscillates around a steady-state value of 1.226, and it ranges from 1.167 to 1.280. The standard deviation of the G/H response is a measure of the product variability, and its value over the 3 days is 0.0229. The model predictive control approach discussed here is used to reduce this standard deviation either without using a product analyzer or without measuring the disturbance. The model predictive results are also shown in Figure 4, where it can be seen that a significant reduction of the product variability is produced. How these results are achieved is discussed next. To illustrate score control, five process measurements are selected for feedback and the D feed is manipulated for control. It should be noted that it is straightforward to use a multivariable SPC approach and employ additional manipulated setpoints in the algorithm, e.g., the reactor level and/or pressure setpoints. The measurements are feed flow, reactor pressure, reactor temperature, separator level, and separator pressure. These measurements have three key characteristics in common. First, they are affected by the random upset in the C feed. Second, their response tends to lead the response of the G/H ratio in the product. Last, these variables are affected by the manipulated D feed setpoint. To develop a dynamic score model, a PRBS signal of (1% ((36.56 kg/h) is added to the D feed setpoint for a period of 10 h. During this period, the random disturbance in the C feed also occurs. It can be noted that the variance of the change in the D feed is larger than that resulting from the PRBS forcing alone because the D/E ratio contributes to changing the D feed setpoint as well. The E feed responds to the random upset because it is used for reactor level control. The E feed then affects the D feed setpoint through the constant D/E ratio. A sampling time of 5 min is used. The five measurements as well as the D feed setpoint measurement are scaled to zero mean and unit variance, and

Figure 5. Comparison of five-step-ahead prediction of identified score model and data.

the various scale factors are given in Table 2. The five measurements are reduced to two scores using PCA, i.e., eq 1. The two scores explain 88.3% of the variance in the five measurements, and the PCA loading vectors are also given in Table 2. The orthogonal transformation matrix, U1 in eq 3, can be calculated from SVD as

U1 )

[

-0.3360 -0.9419

]

(12)

If the scores are multiplied by U1, a one-dimensional score vector, z, results. Because the mean of z is 0, the score setpoint, zsp, is set to 0 for model predictive control. The MATLAB Identification Toolbox24 is used to develop a dynamic model between the D feed setpoint and z. A third-order ARX model was identified using the instrumental variable option in the Identification Toolbox. The resulting A and B matrixes in eq 4 are

A ) [0.172 75 0.409 61 -0.326 13]

(13)

B ) [0.426 10 0.497 01 0.123 14]

(14)

and the value of nk is 0. Figure 5 shows the five-stepahead predictions of z calculated using the ARX model together with the measured values. As can be seen, excellent agreement is achieved. Similar accuracy is achieved for a 10-step-ahead prediction. Next, model predictive control is implemented by calculating step response coefficients from the ARX model. A prediction horizon, P, is selected as 1 h (12 time steps), and a control horizon, M, is chosen as 15 min (four step changes). The score errors all are weighted with a value of w1 ) 1, and the changes in the scaled D feed setpoint are all weighted with w2 ) 10. The results achieved by this controller are shown in Figure 4. As can be seen, the G/H ratio oscillates around a steady-state value of 1.226. The maximum G/H value is 1.26, and the minimum is 1.195 over the 3 days simulated. The standard deviation of the G/H response is 0.0128 over the 3 days, which is a reduction in the product quality variability of 44% compared to the constant D/E policy. This result is excellent, and it is achieved by using measurements that are already available. Figure 6 can be used to explain how the model predictive SPC approach achieves this result. The response of the G/H ratio and the score variable, z, are

6342

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002 Table 3. Comparison of Constant D/E and Score Controllers for IDV(1) + IDV(8) time period (min) 0-500 501-1000 1001-1500 1501-2000 2001-2500

Figure 6. Plot of the G/H ratio and z for model predictive control.

Handling Step Upsets A key assumption that is made in formulating the model predictive score control is that the stochastic disturbances encountered are stationary in nature. If a step disturbance enters the process, then continued use of the model predictive SPC controller can cause problems. Figure 7 shows the responses produced by the SPC controller compared to a controller that uses a fixed D/E ratio when a step disturbance occurs in the composition of the C feed stream in addition to the random IDV(8) disturbance. The step disturbance is labeled IDV(1) in

score controller

1.211 1.231 1.224 1.223 1.230

1.188 1.173 1.193 1.193 1.193

ref 17. The step upset results in the plant being brought to a new, nonzero steady-state operating point in terms of the z score variable. When the model predictive SPC controller tries to bring the process back to zsp ) 0, its action results in a very slow transient in G/H compared to the constant D/E controller. As can be seen in Figure 7, the dynamic score controller does reduce the variability of the G/H ratio significantly, but the desired G/H setpoint of 1.226 is not achieved. Table 3 gives G/H results averaged over 500 min intervals. The score controller produces an average G/H ratio of 1.193 even during the period from 1000 to 2500 min. By contrast, the constant D/E policy produces a G/H value that is much closer to setpoint, 1.226, during the same period. To bring G/H back to setpoint, it is necessary to change the D/E ratio setpoint when the score controller is used, and a steady-state approach to this calculation is discussed next. To overcome the problem, a steady-state MPC controller is used on top of the dynamic score controller to change the D/E ratio every time a laboratory assay is obtained. The steady-state controller is calculated by solving the following optimization problem:

min [(gˆ /h)2 + w∆(d/e)k2] [∆(d/e)]k

Figure 7. Plot of the G/H ratio in the product for IDV(1) + IDV(8).

plotted in this figure. As can be seen, the shape of the z response is similar to the shape of the G/H response. When a maximum (or minimum) occurs in the z response, it is followed in time by a maximum (or minimum) in the G/H response. However, the z variable leads the G/H response by about 15 min. This point is illustrated in the figure by using arrows to highlight the first maximum and minimum in each plot. Because the D feed responds to changes in z, it begins to change before a peak in the G/H ratio occurs, and as a result, better control of the product quality is achieved through the inferential z variable. It can be noted that the MPC tuning parameters of M ) 4 and P ) 12 were simply chosen and not optimized. Thus, an even better performance might be achievable through optimization of these tuning parameters. When step upsets occur, the MPC approach discussed above has a problem because it tries to drive the plant to the original setpoint. This problem and how to overcome it are discussed next.

constant D/E policy

subject to gˆ /h ) m∆(d/e)k + ∆(g/h)k

(15)

where gˆ /h is a deviation between the predicted value of G/H and 1.226, ∆(g/h)k is the deviation between the measured G/H value at time k and 1.226, ∆(d/e)k is the difference between the deviation variable d/e ≡ (D/E 0.813) at time k and the same variable at time k - 1, and w is a scalar weight. For the Tennessee-Eastman process, the stoichiometry is such that 1 mol of G is produced by 1 mol of D and 1 mol of H is produced by 1 mol of E. Thus, the deviation in G/H, g/h, can be modeled as directly proportional to the deviation in D/E, d/e, and the proportionality constant, m, can be calculated from the steady-state values of G/H and D/E as 1.50. Equation 15 can be solved analytically to give

∆(d/e)k ) -

[m∆(g/h)k)] [m2 + w]

(16)

Figure 8 shows the responses produced by the addition of the steady-state G/H controller on top of the SPC controller. For Figure 8 the disturbances used are IDV(1) plus IDV(8), and w is 2. It is assumed that G/H is measured every 8 h and the sample used is a composite 8 h sample, which results in the average value of G/H over the time period being determined. As can be seen, the addition of the steady-state controller improves the response of the SPC controller, which in turn reduces the variability of product quality. Table 4 gives G/H results averaged over 500 min intervals for the constant

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002 6343

Tennessee-Eastman example, which is used here for illustration, involves only a single manipulated variable. It is straightforward to use a multivariable approach in which several setpoints are manipulated. If a steadystate model is available, it is also straightforward to extend eq 15 so that optimum values can be calculated for all of the manipulated setpoints so that step upsets can be handled effectively. Conclusions

Figure 8. Plot of the G/H ratio in the product for IDV(1) + IDV(8).

Figure 9. Plot of the D/E ratio for score + steady-state model predictive control. Table 4. Comparison of Constant D/E and Score + Steady-State MPC Controllers for IDV(1) + IDV(8) time period (min) 0-500 501-1000 1001-1500 1501-2000 2001-2500

constant D/E policy

score controller with steady-state update

1.211 1.231 1.224 1.223 1.230

1.188 1.190 1.232 1.235 1.228

D/E and steady-state D/E plus score controllers. The addition of the steady-state D/E controller produces an average G/H ratio of ∼1.232 during the period from 1000 to 2500 min, which is a significant improvement compared to using just the score controller alone. A comparison of Figures 7 and 8 shows that the score controller with the steady-state adjustment greatly reduces the variability in the product G/H ratio compared to the constant D/E approach. It takes the steadystate controller approximately 16 h to eliminate the effect of the IDV(1) step upset. From 1000 to 2500 min the standard deviation of G/H produced by the constant D/E controller is 0.022, while the proposed approach produces a standard deviation of 0.0124, which is a 43% reduction in variability. Figure 9 shows how the D/E setpoint is manipulated by the steady-state controller to achieve the results shown in Figure 8. It is straightforward to show that using d/e given by eq 16 results in no steady-state offset in g/h. Equation 16 shows that ∆(d/e)k only equals 0 when ∆(g/h)k is 0, which happens when the measured G/H value is at setpoint. The

A model predictive control approach to SPC in chemical plants is discussed. The methodology holds promise as a means of reducing product quality variability in plants where online quality measurement is not feasible. Rather than holding setpoints in a plant control system constant, a small number are varied in response to the effect of process disturbances on selected measurements. The approach uses PCA to compress these measurements into a set of scores that can be controlled. The selected measurements used should correlate with the product quality, and their dynamic response should lead that of the product quality. By controlling the process scores, one indirectly controls the product quality. A key assumption made in the approach is that the process is experiencing stationary stochastic disturbances. The methodology is illustrated on the Tennessee-Eastman process where a 44% reduction in the variability of the product quality is achieved. When nonstationary upsets such as steps occur, the score control setpoints need to be adjusted. It has been shown that the addition of the steady-state MPC controller overcomes the problem caused by step upsets. The application discussed in the paper involves one manipulated variable, but the methodology is multivariable in nature. Literature Cited (1) Shewart, W. Economic Control of Quality of Manufactured Products; van Nostrand: New York, 1931. (2) Woodward, R.; Goldsmith, P. Cumulative Sum Techniques; Oliver and Boyd: London, 1964. (3) Hunter, J. Exponentially Weighted Moving Average. J. Qual. Technol. 1986, 18, 203. (4) Wold, S.; Geladi, P.; Esbensen, K. Principal Component Analysis. Chemom. Intell. Lab. Syst. 1987, 2, 37. (5) Geladi, P.; Kowalski, B. Partial Least squares Regression: A Tutorial. Anal. Chim. Acta 1986, 185, 1. (6) Kresta, J.; Macgregor, J.; Marlin, T. Multivariate Statistical monitoring of Process Operating Performance. Can. J. Chem. Eng. 1991, 69, 35. (7) Wise, B.; Gallagher, N. The Process Chemometrics Approach to Process Monitoring and Fault Detection. J. Process Control 1996, 6, 329. (8) Li, D.; Shah, S.; Chen, T.; Qi, K. Application of Dual-Rate Modeling to CCR Octane Quality Inferential Control. Proceedings of the DYCOPS Symposium, Cheju, Korea, June 2001. (9) Dong, D.; McAvoy, T. Pollution Monitoring Using Multivariate Soft Sensors. Proceedings of 1995 ACC, Seattle, WA, June 1995; American Automatic Control Council: Seattle, WA, p 1857. (10) Macgregor, J.; Marlin, T.; Kresta, J. Some Comments on Neural Networks and Other Empirical Modeling Methods. Proceedings of Chemical Process Control IV; CACHE Corp.: Austin, TX, 1991; p 665. (11) Yabuki, Y.; Macgregor, J. Product Quality Control in Semibatch Reactors Using Midcourse Correction Policies. Ind. Eng. Chem. Res. 1997, 36, 1268. (12) Yabuki, Y.; Nagasawa, T.; Macgregor, J. An Industrial Experience with Product Quality in Semi-batch Processes. Comput. Chem. Eng. 2000, 24, 585. (13) Flores-Cerrillo, J.; Macgregor, J. Control of Particle Size Distribution in Emulsion Semi-batch Polymerization Using Mid-

6344

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002

course Correction Policies. Ind. Eng. Chem. Res. 2002, in press. (14) Shah, S.; Miller, R.; Takada, H.; Satou, T. Modeling and Control of a Tubular Reactor: A PCA-based Approach. Proceedings of the DYCOPS Symposium, Corfu, Greece, June 1998. (15) Chen, G.; McAvoy, T.; Piovoso, M. A Multivariate Statistical Controller for On-line Quality Improvement. J. Process Control 1998, 8, 139. (16) Piovoso, M.; Kosanovich, K. Applications of Multivariate Statistical Methods to Process Monitoring and Controller Design. Int. J. Control 1994, 59, 743. (17) Downs, J.; Vogel, E. A Plantwide Industrial Process Control Problem. Comput. Chem. Eng. 1993, 17, 245. (18) Zheng, L.; McAvoy, T.; Huang, Y.; Chen, G. Application of Multivariate Statistical Analysis in Batch Processes. Ind. Eng. Chem. Res. 2001, 40, 1641. (19) Featherstone, A.; Braatz, R. Control-oriented Modeling of Sheet and Film Processes. AIChE J. 1997, 43, 1989.

(20) Featherstone, A.; Braatz, R. Integrated Robust Identification and Control of Large Scale Processes. Ind. Eng. Chem. Res. 1998, 37, 97-106. (21) Clarke-Pringle, T.; Macgregor, J. Product Quality Control in Reduced Dimension Spaces. Ind. Eng. Chem. Res. 1998, 37, 3992. (22) Clarke-Pringle, T.; Macgregor, J. Reduced Dimension Control of Dynamic Systems. Ind. Eng. Chem. Res. 2000, 39, 2970. (23) McAvoy, T.; Ye, N. Base Control of the Tennessee Eastman Problem. Comput. Chem. Eng. 1994, 18, 383. (24) Ljung, L. System Identification Toolbox; Math Works: Natick, MA, 1995.

Received for review January 22, 2002 Revised manuscript received April 23, 2002 Accepted April 26, 2002 IE020067Q