A Technique for Integrated Quality Control, Profile Control, and

Ind. Eng. Chem. Res. 2000, 39, 693-705

693

PROCESS DESIGN AND CONTROL A Technique for Integrated Quality Control, Profile Control, and Constraint Handling for Batch Processes In S. Chin,† Kwang S. Lee,*,† and Jay H. Lee*,‡ Department of Chemical Engineering, Sogang University, Shinsoo-1, Mapogu,Seoul 121-742, Korea, and School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283

We propose a novel control technique for batch and semibatch processes. The main novelty of the technique lies in that it can accommodate several different aspects of industrial batch process control, such as the following of prespecified reference trajectories, the satisfaction of process constraints, and the meeting of product quality targets, altogether in a single framework. This simplifies the design and enables the user to make a systematic trade-off among the different objectives. Another novelty is the feature of iterative learning that allows the controller to use the information collected in the previous batches. This is done by establishing a batch-to-batch correlation model giving the integral control action with respect to the batch index. This feature allows the performance to be improved gradually as the batch run is repeated and makes it possible to attain precise control despite significant model errors and disturbances. The performance of the proposed technique is illustrated using a simple nonlinear semibatch reactor model. I. Introduction An important objective in most industrial batch process operations is to achieve consistent end-product quality. A major obstacle to achieving this objective is that on-line sensors for quality measurement are very often unavailable. The current industrial practice is to control the directly measured variables such as temperatures and pressures at various locations so that they track some preassigned trajectories, while eliminating the disturbances at the source. The product quality is analyzed in the laboratory after each batch run and the information is relayed back to the operator for adjusting the condition of upcoming batch runs based on some established guidelines (e.g., SQC charts) or, simply, experience. Motivated by this industrial approach, the research on batch process control has thus far centered around the problem of tracking a given reference trajectory for nonlinear systems. Various nonlinear control methods have been suggested1 while most industrial problems have been solved by “gain-scheduling” PID controllers.2 Besides the research on the conventional feedback control, a stream of recent research has focused on the notion of iterative learning control (ILC),3-5 which seeks gradual run-to-run improvements by feeding back the error signals from previous batch runs. The tracking controls are there not only to assure a stable and economic operation but also to provide some * To whom all correspondence should be addressed. Phone: (82-2)705-8477. Fax: (82-2)3272-0319. E-mail: [email protected]. ac.kr. Phone: (765)494-4088. Fax: (765)494-0805. E-mail: [email protected]. † Sogang University. ‡ Purdue University.

protection against disturbances that would otherwise affect the final product quality. This strategy would work well if all the disturbances affecting the quality variables do so by first affecting the controlled process variables (such as the heat-transfer disturbances affecting the temperature and thereupon the product quality through kinetics). On the other hand, practical evidence suggests that because of often significant runto-run changes in batch ingredients and process behavior, maintaining consistent temperature and/or pressure trajectories alone does not render consistent product quality. Feeding back the results of laboratory analysis helps to combat sustained changes but not those occurring on a run-to-run basis. As a way to improve the quality control aspect of a batch process operation, one may consider the option of inferential control, a strategy of predicting the final product quality using the correlation with on-line process measurements. The keystone in this approach is the correlation model (called “soft sensor”), which must reliably predict the final product quality. Suggested methods for developing the correlation model range from simple static linear regression, such as the least squares and its variants (e.g., partial least squares), to more elaborate optimal dynamic estimation methods like the Kalman filtering6,7 and methods based on nonlinear regression tools like the neural networks.8 This paper focuses on how the two approaches, tracking of reference trajectories and inferential (as well as laboratory-analysis-based) control of quality variables, can be merged. An obvious and straightforward way to implement the two together is to have the quality controller determine the reference trajectories for the tracking controllers. On the other hand, by separation of the two designs, the trade-off between the two

10.1021/ie990305q CCC: $19.00 © 2000 American Chemical Society Published on Web 03/06/2000

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controls cannot be made systematically. For instance, if the process variables should be kept within certain bounds, it is not clear how to reflect this requirement in the quality controller design because the optimal design would depend on, among other things, the performance of the tracking controller. In addition, on the basis of the economic considerations, one may wish to systematically trade off the quality control performance for a reduced deviation of relevant process variables from their economic target values (or trajectories). Motivated by this, we propose in this paper an integrated framework to design the two controllers as a single unit. The starting point of our development is our previous work on model predictive control for batch processes (BMPC),9 which was constructed by incorporating the feature of ILC into the regular model predictive control (MPC) algorithm (formulated on the basis of a timevarying linear model description). This algorithm, however, handles only tracking control of measured process variables. Hence, the focus will be given here on how we can embed quality correlation models (established through a data regression) into the BMPC algorithm so that the objective of inferential quality control can be incorporated into the optimal control calculation. We call the generalized method QBMPC (quality control combined with BMPC). Because QBMPC has the same mathematical structure of BMPC, the advantages of BMPC are inherited as they stand. The most outstanding feature is that both the quality and tracking offsets can be eliminated as the run number is increased, even in the presence of model error and batchwise repeating disturbances when there are enough degrees of input freedom and no random disturbance. Given the objective, we draw heavily from the earlier framework of BMPC and emphasize the differences. We also offer some guidelines on model development and choice of design/tuning parameters. The paper is organized as follows: In the second section, we consider a general semibatch reactor of which the operation pattern has a complexity typical of industrial batch reactors and develop a linear timevarying model structure capable of describing such an operation. In the modeling framework, models for the controlled outputs, secondary outputs, and quality variables are to be derived separately and then combined into one. We conclude the section with comments on the model structure and relevant identification techniques. In the third section, the QBMPC algorithm is derived based on the model developed in the second section. Because the algorithm is similar to that of BMPC, we focus on describing the necessary tailoring and other practical aspects such as tuning techniques. This section is followed by the fourth section, a numerical example involving a simulated nonlinear semibatch reactor with exothermic reaction. Control of end-product quality is demonstrated together with reactor temperature tracking under various scenarios. Finally, conclusions are drawn with a brief discussion on the relevance and limitations of the QBMPC technique in real process applications. II. Model Development for Batch Processes The batch process has many unique features not found in continuous processes, as has been indicated by Bonvin10 for the case of batch and semibatch reactors.

Figure 1. A typical operation pattern of a semibatch reactor.

The model developed through this study is one that can accommodate most such features, though not completely. In this study, we confine ourselves to a discrete-time batch process of which the total run length is fixed with N sampling steps. Unlike in continuous processes, operational modes of a batch process can vary with time. For example, some input variables may be varied continuously over the entire batch run while others can be manipulated only at a specific time or during a specific time interval. The situation can be further complicated by the fact that some output variables are measured at every sampling instant while others only once every several sampling instances. Some measurements may even be gathered in an aperiodic manner. In addition, just as inputs, an output may be measured over the entire batch run or only during a limited interval. End-product quality can be measured only upon the completion of the batch. Finally, some of the output variables are to be controlled along prespecified time-varying trajectories while others are measured only for monitoring purposes. Although some of the above features, especially the multirate output measurements, are also present in continuous processes, complication by all the above features is usually observed only in batch processes. Figure 1 shows a hypothetical but conceivable pattern of a semibatch reactor operation and demonstrates the potential complexity involved in representing a batch process operation. To represent such complex operational patterns with a single-model structure, it is convenient to allow the process variables to have time-varying dimensions. The process variables associated with a batch operation can be classified into the following four groups: (1) manipulated input variable, u(t) ∈ Rnu(t); (2) controlled output variable, y(t) ∈ Rny(t), which requires tracking control; (3) secondary output variable, s(t) ∈ Rns(t), used only for monitoring or product quality estimation; (4) quality variable, q ∈ Rnq, measured only after the completion of a batch. For future uses, we define the following: t-1

Nu(t) }

∑ l)0

t

nu(l), Ny(t) }

∑ l)1

t

ny(l), Ns(t) }

ns(l) ∑ l)1

Nys(t) } Ny(t) + Ns(t), NT(t) } Ny(t) + Ns(t) + nq (1) Although the time evolution of the variables within a

Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 695

batch are subject to the dynamics of an underlying process, the relationship between the input and output sequences for the whole batch run can be represented as an algebraic map. Such a representation proves to be very convenient in developing and analyzing a control system. Model for the Controlled Output. The model structure for the controlled output described below is the same one as that used to develop the BMPC algorithm. Hence, details can be found in Lee et al.9 For the sake of completeness, we summarize the model development process and the logic behind it. Define the input, output, and disturbance sequences over the entire batch interval as

u } [uT(0) uT(1) ‚‚‚ uT(N - 1)]T ∈ RNu(N) y } [yT(1) yT(2) ‚‚‚ yT(N)]T ∈ RNy(N)

(2)

d } [dT(1) dT(2) ‚‚‚ dT(N)]T

these vectors can be correlated, to reflect the temporal/ spatial correlations that exist among the signals within the batch. Note that, in the above model, xyk represents the part of ryk that repeats itself in the subsequent batches, while vyk represents the part that disappears after the kth batch. More precisely, in the absence of xy, ry of one batch would be completely independent from those of other batches. On the other hand, in the absence of the noise terms wy and vy, there will be complete correlation among all the batches. The above reflects the deterministic nature of ry as well as the random nature. Strictlyspeaking, it is not correct to represent the neglected higher order terms as an additive noise term independent of the input, but this is a well-accepted approach for linear controller design. Now, we let e j yk } [xyk ˆ d)]. Writing the expression for e j y for two Gy(uk - u consecutive batch indices and taking the difference yields the following transition model for the tracking error trajectory with respect to the batch index:

These variables are assumed to be related through the following general nonlinear algebraic map:

y e j k+1 )e j yk - Gy∆uk+1 + wyk

y ) N (u, d)

eyk ) e j yk + vyk

(3)

Let yd and ud represent the specified output reference trajectory and the corresponding nominal input trajectory (i.e., yd ) N (ud, 0)). We define the output error trajectory as

ey } yd - y ) yd - N (u, d)

(4)

In general, precise ud is not known a priori because the nonlinear map N is not known exactly. We let u ˆ d be an estimate of ud. Now we linearize the above error trajectory equation with respect to u and d around u ˆ d and 0, respectively. Then, we obtain

(5)

(7)

where ∆uk+1 } uk+1 - uk. The model is in the standard form for state-space optimal controller design. Model for the Secondary Output. The procedure employed in the previous subsection can also be applied to the modeling between secondary output and manipulated input variables. Secondary measurements are often obtained through laboratory tests and therefore are subject to time delays. For the model development, however, we can neglect these delays (because they can be added back in forming the measurement equation later) and define the sequence of the secondary output s similarly to y in (2). If we denote the nominal trajectory of the secondary output as sd and es } sd - s and follow the same procedure as that above, the error transition model for the secondary output is obtained as s )e j sk - Gs∆uk+1 + wsk e j k+1

where Gy } (∂N /∂u)|(u)uˆ d,d)0), Hy } (∂N /∂d)|(u)uˆ d,d)0), and ry represents the residual. If we look at the terms comprising ry, we can see that it has components repeated through several batches (e.g., the effect of the error in the nominal input trajectory and the effect of disturbances of persisting type, etc.) as well as those changing from batch to batch (the effect of batch-tobatch random disturbances and that of higher order terms, etc.). The noises that enter the state and the measurements can also be included in ry and these would belong to the latter type. For design purposes, we conveniently represent ry as an output of a linear stochastic system given. For the sake of simplicity, we restrict ourselves to the following model description:

Model for Quality Variables. The measurement of end-product quality is available only after the completion of a batch run. If the quality does not deviate much from its target value (denoted hereafter as qd), q can be reasonably expressed as a linear combination of y, s, and u plus a bias term. More preferably, the linear relation can be written in terms of the deviation variables. If we define eq } qd - q, the relationship can be written as

y xk+1 ) xyk + wyk

where rq denotes the residual term. Once again, we represent rq using the stochastic model,

(6)

ryk ) xyk + vyk where the subscript k denotes the batch index. In the above, both vyk and wyk are assumed zero-mean i.i.d. (independent and identically distributed) sequences in terms of k. For a particular k, however, the elements of

esk ) e j sk + vsk

ˆ d - u) + rq eq ) A1ey + A2es + A3(u

(8)

(9)

q ) xqk + w j qk xk+1

rqk ) xqk + v j qk

(10)

j qk are again zero-mean batchwise i.i.d. where w j qk and v

696


sequences. The integrator reflects the fact that, even with all the variables set at their nominal values, one may still get errors in the quality because of the errors in the determination of the nominal trajectories, errors in the correlation model, and so forth. The noise terms are added to model the random run-to-run changes. Now, writing (9) at two consecutive batch indices, taking the difference, and then substituting (7) and (8)into the resulting equation yield y s q (ek+1 - eqk) ) A1(ek+1 - eyk) + A2(ek+1 - esk) q - rqk) A3∆uk+1 + (rk+1

) -(A1Gy + A2Gs + A3)∆uk+1 + y - vyk) + A1(wyk + vk+1

A2(wsk

+

s vk+1

-

vsk)

+

q (rk+1

-

rqk)

(11)

j qk, the above If we let e j qk } eqk - A1vyk - A2vsk - v equation together with (10) can be rearranged to q )e j qk - Gq∆uk+1 + wqk e j k+1

eqk ) e j qk + vqk

(12)

where

Gq ) A1Gy + A2Gs + A3 wqk ) A1wyk + A2wsk + w j qk

(13)

j qk vqk ) A1vyk + A2vsk + v Combined Model. Equations 7, 8, and 12 have the same structure and can be combined into a single model. For this, we define the augmented variables as follows:

[] [] [] []

eyk wyk vyk Gy ek } esk , G } Gs , wk } wsk , vk } vsk eqk wqk vqk Gq

(14)

[

G1,0

0

· · ·

0

]

G2,0 G2,1 · · · 0 Gi,j ∈ Rny(t)×nu(t) , · · · 0 or Rns(t)×nu(t) · · · · · · GN,0 · · · · · · GN,N-1

(16)

where Gi,j represents the impulse response coefficient matrix composed of output responses at time i to independently applied unit pulse inputs at time j. Identification of Gy. Because of the large dimensionality, it is generally impractical to independently estimate all the parameters in Gy using input-output data. For example, when ny(t) ) nu(t) ) 1 and N ) 200, the number of nonzero elements in Gy is 20100 while the number of data points from a single-batch run is only 200. This means that even in the most ideal case, one would need at least 101 batch data sets to determine all the parameters in Gy. For an initial estimate of Gy, one may identify a simple time-varying linear model such as a linear combination of several time-invariant linear models (e.g., ARX models) with time-varying weights. More specifically, if we let ∆yk(t) } yk(t) - yk-1(t) and ∆uk(t) } uk(t) - uk-1(t), we can introduce

∆yk(t) ) [µ1(t,η1)g1(q-1, θ) + ‚‚‚ + µn(t,ηn)gn(q-1, θ)]∆uk(t) + ∆r(t) (17)

The resulting model is

e j k+1 ) e j k - G∆uk+1 + wk

on the model structure and identification. Identification of time-varying linear models for nonlinear batch systems is a subject that remains largely unexplored at the current time. In fact, it is logical to develop a wide array of tools because different problems are likely to call for different techniques. Here, we discuss a simple technique that worked well for the problems we tried. We note that because of the ILC feature embedded in the model structure and the (subsequently presented) control algorithm, a precise model is often not needed to achieve precise control on an asymptotic basis. Model Structure. Given the causality, the structure of Gy and Gs is restricted to the following lower-blocktriangular form:

(15)

ek ) e j k + vk The last two relations in (13) dictate how their covariance matrices should be structured (see the section Parameter Tuning). The error transition model of (15) can be viewed as a predictor of some sort. It has a structure to infer e j k+1 (comprising the error trajectories of the controlled output, secondary output, and end-product quality) for an upcoming batch based on the information from a completed batch (e j k), for a given change in the input j k can be trajectory (∆uk+1). An optimal estimate of e easily constructed using the Kalman filter. On the other hand, we would like to perform the prediction in real time, that is, as the measurements become available throughout the batch. This will be discussed in the next section. Comments on Model Structure and Identification Technique. Before moving on to the control algorithm derivation, we make some relevant comments

where g1, g2, and so forth are appropriate transfer functions and µi’s represent the time-varying weights (whose functional forms can be specified as trapezoids, Gaussian curves, etc.). ηi’s and θ denote the parameters. They can be easily estimated using the standard leastsquares method or its variations. The reason for using the incremental changes of the variables (in terms of the run index) for the identification is to remove the effect of bias that can change from one batch run to another. Identification of the above model can be done using a relatively small number of batch data sets. Gy can be constructed by computing the impulse responses using the model. Another practical option is to fit hinging-hyperplanes to obtain a piecewise-linear model. This method performs a regime decomposition based on data and fits a separate linear model for each regime. This naturally yields a time-varying linear model (without linearization) for the local behavior around a fixed trajectory. For details, see Chikkula et al.11 Once the initial estimate of Gy is obtained, it can be refined using recursive least squares as more batch data become available from closed-loop runs.


Identification of Gs can be done in the same way as that above. In the case that the measurements of the secondary outputs are sparse, one can interpolate the data to fill in the missing measurements. Another option is to develop the model in terms of the larger sampling interval (i.e., the sampling interval of the secondary output) and treat all the input movements for each interval as a vector variable. Quality Regression Model. Regression of ey, es, and u ˆ d - u to the quality error eq as in (9) can be done using the standard linear regression techniques. Given the fact that the mean of rq changes with k according to (10), it is recommended to recast (9) to

∆qk ) A1∆yk+ A2∆sk + A3∆uk + ∆rqk

(18)

before regression by taking the difference between the relations at two batch indices. In this model, ∆y, ∆s, and ∆u usually have dimensions that are by orders-ofmagnitude larger than ∆q. Given the large dimensionality, it is conceivable that the regression matrix be highly ill-conditioned. This problem can be alleviated to an extent through a careful design of input perturbation signals, but the problem may persist because the temporal and spatial patterns of the output variables are dependent on the process dynamics. Theoretically, the accurate process model is most important to modelbased control techniques. To reduce the adverse effect of ill-conditioning on the parameter estimation, it is desirable to reduce the dimension of the regressor beforehand, as in PCR (principal component regression) or PLS (partial least squares). Detailed procedures for PCR and PLS can be found in the vast literature of chemometrics, for example, Geladi and Kowalski12 and Jolliffe.13 Indeed, one need not strive to be very accurate in the aforementioned identification. It is because the control technique proposed in this paper still assures asymptotically minimum offsets for both tracking and quality variables, even in the presence of relatively large model error. Of course, this does not exclude the necessity of model accuracy. An accurate model can enhance the convergence rate as well as real-time control performance.

[ ] [] e j k(t) e j } ek ek(t) k

[

with ∆uk(t) ) ‚‚‚ ) ∆uk(N - 1) ) 0

e j k-1 - G(0)∆uk(0) - ‚‚‚ G(t - 1)∆uk(t - 1) + wk-1 ) e j k-1 - G(0)∆uk(0) - ‚‚‚ G(t - 1)∆uk(t - 1) + wk-1 + vk

]

(20)

On the basis of this definition, (15), and the fact that j k-1(N) ) e j k-1, we can obtain the ek-1(N) ) ek-1 and e following periodically time-varying state-space system (with the period of N):

[ ] [ ][ ] [ ] [ ] [ ] [ ][ ] [ ] [ ]

e j k(0) j k-1(N) I 0 e I 0 ) + w + v ek(0) I 0 ek-1(N) I k-1 I k

(21)

e j k(t + 1) j k(t) G(t) I 0 e ) ∆uk(t) ek(t + 1) 0 I ek(t) G(t)

ek(t) ) [0 H(t)]

e j k(t) ek(t)

t ) 0, ‚‚‚, N - 1 (22)

The output measurement vector ek(t) and the related measurement matrix H(t) change with time. At the time only the controlled output is measured,

e(t) ) ey(t),

H(t) ) [Hy(t) 0 0]

(23)

where

When the secondary output is measured together with the controlled output,

e(t) )

[ ]

ey(t) , es(t)

H(t) )

[

Hy(t) 0 0 0 Hs(t) 0

]

(25)

Delays in the secondary measurements can be easily incorporated because Hs(t) can be chosen to pick whichever elements of es corresponding to the measurements at time t. For example, if all the secondary measurements has delays of τ time units, Hs(t) can be represented by

III. Derivation of Quality Control-Combined Batch MPC (QBMPC) Finally, at the end of a batch run, we have Developments here are similar to those given in Lee et al.9 for BMPC. Hence, we keep the discussion brief and focused on those aspects unique to QBMPC. Formulation of a State-Space Model along the Time Index. In order to design a controller that performs output feedback control in real time, we convert the batch transition model of (15) into a timetransition model. For this, we partition G as a block column matrix according to the time index.

G } [G(0) G(1) ‚‚‚ G(N - 1)], G(j) ∈ RNT(N)×nu(t) (19) Define the state as

e(N) )

[ ] ey(N) e

q

,

H(N) )

[

Hy(N) 0 0 0 0 I

]

(27)

Note that this measurement is not needed until the start of the next run, so some delays in the laboratory analysis of the product quality can be tolerated. If the delays take longer than the intermittent period between batches, delayed eq can be added to the state accordingly. Predictor Construction. Let ek(t + m|t) represent the optimal prediction of the error sequence for the kth batch on the basis of the measurements up to time t and the m future control moves. Given the definition of the state, ek(t + m|t) can be represented by

698


ek(t + m|t) ) ek(t|t) - Gm(t)∆um k (t)

(28)

dictates the weighting matrix Q to have the shape of

[

Qy(t) 0 0 Q(t) ) 0 0 0 0 0 Qq(t)

where ek(t|t) is the optimal estimate of ek(t) at time t during the kth batch andThe state estimate can be

Gm(t) ) [G(t), ‚‚‚, G(t + m - 1)] and ∆um k (t)

[

∆uk(t) l ) ∆uk(t + m - 1)

(29)

]

]

] [

]

P h k(1) P ˆ k(1) ) P ˆ k(1) Pk(1) P h k-1(N + 1) + Rw P h k-1(N + 1) + Rw P h k-1(N + 1) + Rw P h k-1(N + 1) + Rw + Rv

[

(30)

]

(31)

and

[

] [

]

e j k(t|t - 1) e j (t - 1|t - 1) ) k ek(t|t - 1) ek(t - 1|t - 1) G(t - 1) ∆uk(t - 1) (32) G(t - 1)

[ ] [

]

[

]

e j k(t|t) e j (t|t - 1) ) k + ek(t|t) ek(t|t - 1) Kk(t)(ek(t) - H(t)ek(t|t - 1)) (33) Kk(t) )

[

[ ] ] [

An analytical solution to (36) can be obtained and is given by ∆um k (t) is applied to the process starting at

(GmT(t)Q(t)Gm(t) + R)-1GmT(t)Q(t)ek(t|t) (38)

e j k(0|0) e j (N|N) ) k-1 ek(0|0) e j k-1(N|N)

[

(37)

∆um k (t) )

obtained using a non-steady-state Kalman filter designed for (21) and (22). With the covariance matrices of wk and vk given as Rw and Rv, the Kalman filter equation looks like

[

]

P ˆ k(t) T H (t)(H(t)Pk(t)HT(t))-1 Pk(t)

(34)

]

P h k(t + 1) P ˆ k(t + 1) P h (t) P ˆ k(t) ) k P ˆ k(t + 1) Pk(t + 1) P ˆ k(t) Pk(t) ˆ k(t) Pk(t)] (35) Kk(t)H(t)[P for t ) 1, ‚‚‚, N

In the above, ek(t|t1) represents the optimal estimate of ek(t) on the basis of the measurements at time t1 of the kth batch and Pk(t) is the covariance matrix of the h k(t) is the covariance estimate ek(t|t - 1). Likewise, P ˆ k(t) represents the crossmatrix of e j k(t|t - 1) and P j k(t|t - 1), covariance matrix between ek(t|t - 1) and e respectively. Input Calculation. The input movements can be determined by solving an optimization at each sample time. When the conventional quadratic objective function is adopted, the optimization is cast as

1 min {eTk (t + m|t)Q(t)ek(t + m|t) + m 2

∆uk (t)

m ∆umT k (t)R(t)∆uk (t)} (36)

subject to (28) The fact that the secondary output is not controlled

time t until the next sample time t + 1 when a new optimization is solved with the updated prediction equation. Hence, only the first element of ∆um k (t) is actually implemented. To the above minimization, constraints can be added to reflect the input limits and allowable output ranges. With the constraints formulated as linear inequalities of ∆um k (t), the minimization becomes a standard QP (quadratic programming) problem. For further details on the constrained case, readers may refer to Lee et al.9 In the above, Qy(t) and Qq(t) are weighting matrices that are used to scale the variables and to assign relative importance to them. One can also reflect the confidence in the quality inference by choosing Qq(t) to be proportional to the inverse of the covariance matrix of eq(t + m|t). This way, less control action will be applied when there is a higher level of uncertainty in the prediction. Another possible formulation is to include only the end quality error and the input penalty term in the objective function (i.e., to set Qy(t) ) 0) and express the tracking requirements as constraints. This is a natural formulation in many cases as there is no intrinsic benefit gained from precisely tracking a fixed trajectory run after run. Constraining the variables within some distances from the reference trajectory would provide the needed stability of the operation and also assure the validity of the linear model. The QBMPC algorithm as formulated has the integral action with respect to the batch index. This is because the batch transition model on which our design is based contains integrators for the entire output trajectory error. Drawing an analogy to the role of the integral action in continuous control, it is reasonable to expect that the controller will eventually get rid of the errors in the outputs completely that would otherwise persist (as runs are repeated). For the BMPC algorithm, this idea is formalized and the convergence is rigorously proved in Lee and Lee.14,15 The proof should extend to the QBMPC algorithm after some modification. Summary of the Algorithm. The procedure for QBMPC implementation is summarized as follows: Step 1. Obtain the system matrix G in (14) using an appropriate method. Step 2. Initialize the state estimator ((30) and (31)). For (30), one can choose e j 0(N|N) and e0(N|N)) as the error trajectory obtained from some previous batch run. u0 is then the input trajectory from the same run. If there is no previous run, one can use the zero vector for e j 0(N|N) and the best available estimate for u0. Also, for (31) let

[

] [ ]

P h 0(N + 1) P ˆ 0(N + 1) I I )γ P ˆ 0(N + 1) P0(N + 1) I I

(39)


where γ > 0 is chosen to be “large” to reflect the fact that the error for the initial run may be large. Finally, set k ) 1. Step 3. Before the start of each fresh batch, initialize the state and the covariance matrix using (30) and (31). Set t ) 1. Step 4. At time t, obtain the state estimate using (32)-(34). Also, update the covariance matrix using (35). Step 5. Calculate ∆uk(t) using (38) or, in the presence of constraints, by solving the constrained optimization. Implement uk(t) ) uk-1(t) + ∆uk(t) as the input until the next sample time. If t < N, set t r t + 1 and go back to step 4. If t ) N, set k r k + 1, update the model matrices with the gathered data if necessary, and go to step 3. Parameter Tuning. The QBMPC algorithm has six tuning factors: Qy(t), Qq(t), R, Rw, Rv, and the control horizon, m. In an earlier section, we already talked about how to choose Qy(t) and Qq(t). By default, the output prediction is made up of the terminal time of the batch for all t. The best choice for the control horizon (m) in view of performance is the maximum possible value, that is, m ) N - t at time t. Such a choice can be computationally expensive, however, especially when the sample time is short in relation to the total batch time. In this case, m may be set as a small number. Too small an m, however, can cause sluggish performance, especially in the beginning. For Rw and Rv, we first note that (13) dictates the covariance matrices to have a specific structure. More q yT specifically, if we define Rqy w } E{wkwk } and so forth, we obtain

[

Ryy 0 Ryq w w sq R Rw ) E{wkwTk } ) 0 Rss w w qs qq Rqy w Rw Rw

]

(1) The ratio between Rw and Rv determines the speed of batch-to-batch learning. In this sense, it has a similar effect as the input weighting matrix. As Rw is increased in relation to Rv, the learning from batch to batch becomes faster (i.e., more aggressive), increasing the convergence speed but at the expense of reduced robustness. In other words, as Rw gets larger, the past batch measurements are trusted increasingly more for the prediction. ss yy ss (2) The relative sizes of Ryy w , Rw (and also Rv , Rv ) q jq j versus Rw affect the trust ratio between the inferential estimate of the end quality from the ongoing (and past) batch measurements and the lab measurements of the end quality of the previous batch. As Ryy w and/or q jq j Rss w become larger in relation to Rw , increasingly more trust is put on the inferential estimate (on the basis of the measurements of y and s) in relation to the lab measurements. Increasing Rqvj qj in relation to Rqwj qj should have a similar effect because batch-to-batch independent disturbances cannot be predicted using the lab measurements of previous batches. In summary, to make the lab measurement of the end quality count more in the prediction, one should increase Rqwj qj in relation to all the other matrices.

IV. Numerical Illustration: Control of a Nonlinear Semibatch Reactor Model Process Description. We consider a jacketed semibatch reactor where an exothermic series-parallel firstorder reaction (with respect to each reactant) takes place.

(40)

k1

A + B 98 C k2

B + C 98 D

where yy T qy yy Ryq w ) Rw A1 , Rw ) A1Rw ss T qs ss Rsq w ) Rw A2 , Rw ) A2Rw

Rqq w

)

T A1Ryy w A1

+

T A2Rss w A2

+

(42)

The following equations describe the reactor system:

(41)

Rqwj qj

with Rqwj qj } E{w j qkw j qT k }. In the above, we assumed that the basic driving noises for the three residuals, wyk, wsk, and w j qk (see (10)), are uncorrelated with one another. Rv has the same structure. ss Now, the problem is reduced to choosing (Ryy w , Rw , q j qj yy ss q jq j Rw ) and (Rv , Rv , Rv ). These can be estimated using data, in principle. However, an accurate estimation of the covariance matrices usually requires exorbitant amounts of data. Hence, it is more practical to parameterize them in some reasonable way and use the parameters as tuning knobs. For example, one may model the time sequence of these noise terms as outputs to a discrete linear system driven by white noise. For some useful parameterizations derived from this approach and tuning guidelines, see Lee et al.9 The relative sizes of the covariance matrices affect the performances of tracking control and quality control. Here are some useful guidelines in adjusting the covariance matrices in relation to one another:

d(VT) UA (T - Tj) + QBTB )dt FCp V∆H2 V∆H1 k10e-E1/RTCACB k e-E2/RTCBCC, FCp FCp 20 T(0) ) TI d(VCA) ) -Vk10e-E1/RTCACB, CA(0) ) CAI dt d(VCB) ) CBFQB - Vk10e-E1/RTCACB dt Vk20e-E2/RTCBCC, CB(0) ) 0 (43) d(VCC) ) Vk10e-E1/RTCACB - Vk20e-E2/RTCBCC, dt CC(0) ) 0

{

0 for t < 31 dV ) QB, QB(t) ) , V(0) ) VI QB(t) for t g 31 dt with

700


UA/FCp ) 0.375 (L/min); TI ) 25 (°C); TB ) 35 (°C) CAI ) 1 (mol/L); CBF ) 0.90, 0.95 (mol/L); VI ) 50 (L)

∆H1/FCp ) -28.50 (K‚L/mol); ∆H2/FCp ) -20.5 (K‚L/mol) k10 ) 5.0969 × 1016 (L/mol‚min); E1/R ) 12 035 (K) k20 ) 2.2391 × 1017 (L/mol‚min); E2/R ) 13 450 (K) The reactor operation is displayed in Figure 2. A is charged initially and the heat-up is followed until B starts to be fed at t ) 30 min. The reaction commences at this point and continues until the batch terminal time of tf ) 100 min. During this period, the concentration of A is sampled every 10 min and measured with a 5-min delay for analysis. The desired product is C and maintaining the final yield of C (which is V(tf)Cc(tf)) at a target value (42 mol) is the main objective of the operation. We considered two manipulated variables: jacket temperature (denoted by Tj(t)) and flow rate of B (denoted by QB(t)). For these variables, the following constraints were imposed:

20 e Tj(t) e 45 °C

(44)

Figure 2. An overview of the operation of the reactor model.

Figure 3. Time-varying weighting functions for the model construction.

where

0.5 e QB(t) e 1.5 (L/min)

∆y(t) ) y2(t) - y1(t), ∆u(t) ) u2(t) - u1(t)

The sample time for control was chosen to be 1 min. Correspondence between the reactor variables and those in the QBMPC algorithm is as follows:

A(q-1) ) 1 + a1q-1 + ‚‚‚ + a3q-3

y ) T′, s ) C′A, q ) (V(tf)C(tf)′, u ) [T′j, Q′B]T (45) where ′ represents the deviation variable. The target value for V(tf)C(tf) was chosen as 42 mol. We considered three different scenarios. In the first scenario, only the tracking control of T is carried out under BMPC by manipulating Tj(t) while keeping QB(t) at a constant value. In the second scenario, quality control is conducted together with tracking control under QBMPC but again manipulating only Tj(t). Under this situation, the input does not have sufficient degrees of freedom to control both the quality and the reactor temperature without offset. Hence, a trade-off for control error between the two outputs needs to be made. Finally, in the third scenario, both Tj(t) and QB(t) are manipulated for combined quality and tracking control. In all three cases, CBF is assumed to be 0.95 (mol/L) for the first 19 batches and then drops to 0.90 (mol/L) starting at the 20th batch. Consequently, CBF acts as a batchwise persisting disturbance. To be realistic, some errors were intentionally introduced to the nominal trajectories for {T, CA} and {Tj, QB(t)} such that the resulting quality does not coincide with the target value, even without the disturbance. Identification. Various parts of the process model for QBMPC design were identified through separate experimental runs. For estimation of Gy, we used data for two batches only. The input-output relationship was represented by the following simple time-varying linear model:

A(q-1, θ)∆y(t) ) [µ1(t, η1)B1(q-1, θ) + µ2(t, η2)B2(q-1, θ)] ∆u(t) (46)

(47)

Bi(q-1) ) b0,iq-1 + b1,iq-2 + b2,iq-3 The shapes of µ1(t, η1) and µ2(t, η2) are shown in Figure 3, which reflect the fact that the reactor behaves like a heating tank during the initial heat-up stage and also as the reaction fades out approaching the batch termination time. The same procedure as that in Lee et al.9 was used for the estimation of the parameters. Gy was constructed using the impulse response coefficients generated from the above model. The same approach was employed for the estimation of Gs. The only major issue was that the measurements were collected every 10 min instead of every minute. Linear interpolation was made to fill in the missing points. The quality regression model of (18) was obtained by applying principal component regression (PCR) on a data set consisting of data from 30 batches, which were generated from various open-loop and closed-loop experiments. In PCR, three principal components were retained for each of ∆yk, ∆sk, and ∆uk by applying PCA independently to each measurement vector. Indeed, three principal directions were not enough to capture the data set, especially for ∆uk. We, however, chose not to include more modes as we were interested in examining the effect of various model errors on the performance of QBMPC. In the same manner, we obtained an alternative quality regression model using only ∆yk and ∆uk for the purpose of demonstrating the effect of the secondary variable on quality control. As the foregoing discussion indicates, the model used for QBMPC design was a rather coarse one. We could have spent more time refining it, but we also wanted to leave in enough model errors, to really test the zero tracking/quality-offset performance and robustness of QBMPC.


Choice of Tuning Parameters. Among the six tuning parameters, the control horizon m was chosen to be N - t throughout the simulation study. The input weight R was designed as a diagonal matrix with diagonal elements fixed at 0.1 for Tj and 7.5 for QB, respectively, considering the variations of each variable. Nominal values of other tuning factors were chosen as below but the specific values were changed depending on simulations to demonstrate their effects on control performance. Qy(t) was chosen as a constant diagonal matrix with a nominal value of 0.02I. Qq(t) were designed as

Qq ) 100R[cov{eqk(t + m|t)}]-1, R ) 1 nominally (48) so that less weight is given to the quality control when the uncertainty in quality prediction is high. In the above, “100” is introduced for a scaling purpose; there is only one squared term in the cost function for eq while there are 100 terms for ey. cov{eqk(t + m|t)}} can be estimated as a function of Pk(t) in a straightforward manner. Rw and Rv were constructed as shown in (40) and (41). The covariance matrices for the construction of Rw and Rv were chosen nominally as follows: ss q jq j Ryy w ) 0.2J + 2.0I, Rw ) 0.2J + 3.0I, Rw ) 1.0 (49) ss q jq j Ryy v ) 0.1J + 1.0I, Rv ) 0.1J + 1.5I, Rv ) 0.5

where

[ ] 1 1 1 · · · 1

1 2 2 · · · 2

J} 1 · · · 1

2 · · · 2

3 · · · · · · · · · 3 · · ·

3 · · · N

(50)

In the above, J and I represent covariance matrices for an integrated white noise process and a white noise process, respectively. Hence, these noises are modeled as a mixture of white noise and integrated white noise, which is reasonable in most cases. More detailed rationale behind the above formula can be found in Lee et al.9 Results and Discussion Tracking Only Control. First, with Figure 4, we summarize the results obtained when only the tracking control of the reactor temperature was conducted. This can be done by setting Qq(t) ) 0 and e(N) ) ey(N) (see (27)), which means no end-quality measurements were used. In this simulation, Tj(t) was manipulated for the control with QB(t) fixed at 1 L/min. As can be observed from Figure 4b, the reactor temperature shows an obvious tendency to converge to the reference trajectory. In the first batch, T(t) shows a modest oscillation around the reference trajectory, mainly because of errors in the initialized nominal input trajectory, but the discrepancy almost completely disappears after 19 consecutive batch runs. Similarly, as CBF drops from 0.95 to 0.90 (mol/L) in the 20th batch, some additional tracking error is induced. After 30 successive runs, however, T(t) again converges to the reference trajectory so that the actual

Figure 4. Simulated performances of QBMPC for the first scenario (tracking control only): (a) responses of the end-quality variable and (b) reactor temperature profiles.

profile cannot be distinguished from the reference profile. On the other hand, the quality variable, being unattended, deviates significantly from its target value. The offset gets even bigger after the disturbance in CBF enters at the 20th batch. Combined Tracking and Quality Control Using Tj(t). Tradeoff between the Quality and Tracking Offsets. Figure 5 shows the performance of QBMPC when only Tj(t) is used as a manipulated variable and QB is kept constant at 1 L/min. Because Tj(t) alone does not provide enough degrees of freedom to control both the reactor temperature and end-quality variable, offsets result. However, offsets can be distributed between them by adjusting the output weights in the quadratic cost. In Figure 5a, we compare the responses of the quality variable for three different cases: the nominal case (R ) 1, Qy ) 0.02I), less weight given to the quality error (R ) 1, Qy ) 1.0I), and more weight given to the quality error (R ) 100, Qy ) 0.02I). The other tuning factors were fixed at the nominal values. From this figure, we show that the quality offset can be adjusted simply by selecting the output weights. For example, in the case of R ) 100, the quality offset can be removed almost completely. On the other hand, the tracking performance deteriorates significantly in this case, as can be seen in Figure 5b. In the case of Qy ) 1.0I, the quality error becomes significant but the tracking offset, though not shown in the figure, is diminished almost to the level of previous tracking-only control.

702


Figure 5. The effect of the output weight ratio for the second scenario (combined tracking/quality control with a single manipulated input): (a) responses of the end-quality variable for three different output weights and (b) the reactor temperature profiles for the case of R ) 100.

This simulation also demonstrates that to achieve tight quality control with tracking control only, the reference trajectory for tracking variables need to be modified according to the situation. The temperature profiles obtained at the 19th and 50th batches in Figure 5b can be said to represent the desired trajectories in terms of achieving the target quality for CBF ) 0.95 mol/L and CBF ) 0.90 mol/L, respectively. Combined Tracking and Quality Control Using Both Tj(t) and QB(t). This time, both Tj(t) and QB(t) are manipulated. In this case, the control input has sufficient degrees of freedom to steer both the tracking variable and the quality variable to the zero offset state, as can be seen from Figures 6 and 7. In this part, we discuss various aspects of QBMPC including the effect of tuning factors on the control performance. Effects of Q/R and Rw/Rv on the Learning Rate and Robustness. First, in Figure 6a, we show the quality variation for four choices of Q/R. Here, both Qy and Qq are decreased/increased together by factors of 0.5, 2, and 5 from their nominal values. From Figure 6a, one can observe that the 0.5-times-the-nominal setting results in a sluggish response compared to those of the nominal case. The 2-times-the-nominal setting gives a shorter rising time but induces a modest overshoot. As the ratio is further increased to 5-times-the-nominal setting, the quality variable diverges. These results demonstrate that as one might expect, the Q/R ratio is a critical

Figure 6. The effect of the Q/R ratio for the third scenario (combined tracking/quality control with two manipulated inputs): (a) responses of the quality variable for four Q/R ratios and (b) the reactor temperature profiles for the setting of Q/R ) 2× (the nominal case).

factor that determines the learning rate as well as robustness of QBMPC. A larger Q/R means a faster learning rate but reduced stability margin. In Figure 6b, the resulting temperature profiles for the 2-times-the-nominal case are shown for selected batches. No tracking offsets can be observed before and after the feed disturbance. Though not compared here, temperature responses for other choices of Q/R (except the 5-times-the-nominal case) exhibited a tendency similar to the quality variable in terms of learning rate and robustness. As asserted in the Parameter Tuning section, similar effects can be obtained for the Rw/Rv ratio, which is evidenced in Figure 7. To show the effect of the Rw/Rv ratio more clearly, the Q/R ratio is increased by letting R be very small (0.1 times-the-nominal setting, 0.01I). As Rw is decreased (0.025-times-the-nominal setting) in relation to Rv, measurements from the previous batches are filtered more and thus reflected less on the state estimate for the present batch. As a consequence, the learning rate is decreased but the robustness improves. On the other hand, a larger Rw (10-times-the-nominal setting) yields faster learning than that in the nominal case. Unlike the results from the Q/R ratio change, however, increasing Rw further did not induce instability but only gave results similar to those in the 10-timesthe-nominal setting. Indeed, it can be seen from (34)


Figure 8. Responses of the quality variable under no quality control, inferential control only, inferential control with the feedback of previous batch’s quality measurement, and the same as the third except the secondary variable.

Figure 7. The effect of the Rw/Rv ratio for the third scenario (combined tracking/quality control with two manipulated inputs): (a) responses of the quality variable for three different Rw/ Rv ratios and (b) the reactor temperature profiles for Rw/Rv ) 10× (nominal setting).

and (35) that the Kalman gain, Kk(t), converges to the state of full measurement feedback as Rw/Rv f ∞. Hence, virtually no change in response can be observed beyond a certain Rw/Rv ratio. In addition, it can be said that the uncertainty in the present regression model is not large enough to induce instability for any Rw/Rv ratio with other tuning parameters fixed as stated above. Effectiveness of Inferential Control of the Quality Variable. QBMPC is capable of utilizing the on-line measurements of y and s for prediction and control of the end quality. It can also use the quality measurements from previously finished batches for the same purpose. To perceive the effectiveness of the former (i.e., “inferential control”), four simulations were carried out and the results are summarized in Figure 8. The four simulated cases correspond to those of no quality control, inferential quality control without any feedback of previous batch’s quality measurement, inferential control with feedback of quality measurement, and the same as the third but with alternative quality model constructed without the secondary variable, all under the nominal tuning factors. Results for the first and third cases are from Figures 4 and 6 (or 7), respectively. The inferential-only control was implemented by restricting e(N) to include only ey(N) (of (27)) in QBMPC. From Figure 8, it is obvious that the inferential-only

control can improve the quality but cannot remove the offset completely because of some errors in the quality prediction model. On the other hand, quality control without the secondary variable steers the quality variable to the target value but with a lower convergence rate than the nominal case. This demonstrates that the reliable quality inference by a more accurate quality model is effective in improving the transition but not crucial in convergence. Considering the fact that we had constructed the quality regression model with the secondary variable in a rather simplistic manner, additional refinement of the quality model (achieved, for example, by taking extra secondary measurements such as CB and/or CD, if available, and/or including more principal modes in PCR) should give better results than those in the previous cases. Effects of Rqwj qj in Relation to Other Noise Covariance Matrices. A large value of Rqwj qj implies that the previous batches’ quality measurements are reflected more to the quality control in relation to the estimate built from online measurements of other process variables. When the quality measurements contain large uncertainties, more weight needs to be given to the on-line estimate and vice versa. To investigate the effect of Rqwj qj in relation to other noise covariance matrices, we conducted simulations for two different values of Rqwj qj with the other covariance matrices set at their nominal values. The results are shown in Figure 9. Note that we did not include any quality measurement error in this simulation. First, we can see that, at the 20th batch, the quality variable with the large Rqwj qj shows a larger error than the one with the small Rqwj qj . This is what we expect because the real-time estimate is counted less for larger Rqwj qj . As the batch runs are repeated, however, quality measurements are fed back more strongly and, as a result, the quality converges to the target value faster than that in the small Rqwj qj case. Benefit of Real-Time Feedback against Batchwise Random Disturbances. To demonstrate the benefit of real-time control of QBMPC, we assume this time that CBF changes in a completely random manner from the 11th batch on, as shown in Figure 10a. Under this disturbance, we implemented iterative learning control (ILC) and QBMPC and compared the quality control results in Figure 10b. The ILC was implemented by applying the input sequence ∆uN k (0) calculated with

704


here performs some (batchwise) filtering of measurements and therefore should be less susceptible to this weakness than the conventional ILC. Nevertheless, Figure 10b indicates that the QBMPC which uses online measurements to perform the control in real time can yield a far less quality error variance compared to the ILC. Conclusions

Figure 9. Responses of the quality variable for two Rqwj qj s.

A novel multivariable control technique for batch and semibatch processes, named QBMPC, has been proposed. The QBMPC is based on a simple time-varying linear model that can describe the dynamics of fairly general batch and semibatch processes (of which operating modes can change during a batch run). The technique is capable of handling several critical control tasks for industrial batch operations, such as the end-product quality control, profile control, and constraint handling, all in a single framework. Another strength of the technique is that it makes maximum use of available measurements, including the measurements of previous batches, laboratory analysis results for end-product quality, and real-time measurements. Feeding back measurements from the previous batch enables offsetfree control of end-quality variables and batch profiles, while the use of on-line measurements endows the technique with the capability of rejecting disturbances as they occur as well as providing control of the end quality for the on-going batch albeit in an inferential manner. The proposed QBMPC has six major tuning factors (matrices). Choosing appropriate values for them is important but a blind search can lead astray given the large number of parameters one must decide. We analyzed their effects on the control performance and suggested some guidelines. The properties of QBMPC including the effect of tuning factors are demonstrated through numerical simulations performed with a semibatch reactor model. The biggest contribution here, we believe, lies in integrating many different facets of batch process optimization and control into a single technique. The unifying framework the QBMPC provides should facilitate the design in many cases and enable engineers to make systematic trade-offs among different objectives that best suit the particular situation. Acknowledgment

Figure 10. QBMPC vs ILC in the presence of batchwise completely random disturbances: (a) the pattern of CBF simulated; (b) responses of the quality variable under QBMPC and ILC.

QBMPC at the start of the kth batch, in an open-loop manner. Hence, the control decisions are made prior to each batch operation rather than in real time. Nominal tuning factors were used for both controls. Fundamentally, ILC is capable of rejecting only batchwise repeating (or correlated) disturbances because it must rely exclusively on previous batches’ measurements. Hence, when the disturbance changes in a completely random manner as in this simulation, the ILC may actually amplify the error significantly. The ILC we practiced

The second author would like to acknowledge the financial support from the Korea Science and Engineering Foundation through the Automation Research Center at Pohang University of Science and Technology. The third author gratefully acknowledges the financial support from the National Science Foundation’s Young Investigator Program (USA) under Grant CTS #9357827. Literature Cited (1) Berber, R. Control of Batch Reactors: A Review. Trans. Inst. Chem. Eng. 1996, 74, 3-20. (2) Tan, S. H.; Hang, C. C.; Chai, J. S. Gain SchedulingsFrom Conventional to Neuro-fuzzy. Automatica 1997, 33, 411. (3) Bien, Z., Xu, J., Eds. Iterative Learning Control; Kluwer Academic Publisher: Boston, 1998.

Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 705 (4) Chen, Y. A Bibliographical Library for ILC Research. http:// www.ee.nus.sg/∼yangquan/ILC/ilcref.html, 1998. (5) Moore, K. Iterative Learning Control-An Expository Overview. Appl. Comput. Controls Signal Process. Circuits 1998, 1. (6) Lee, J. H.; Datta, A. K. Nonlinear Inferential Control of Pulp Digesters. AIChE J. 1994, 40, 50. (7) Russell, S.; Kesavan, P.; Lee, J. H.; Ogunnaike, B. Recursive Data-Based Prediction and Control of Product Quality. AIChE J. 1998, 44, 2442. (8) Qin, S. J. Neural Networks for Intelligent Sensors and ControlsPractical Issues and Some Solutions. In Neural Systems for Control; Omidvar, O., Elliott, D. L., Eds.; Academic Press: New York, 1997; Chapter 8, p 213. (9) Lee, K. S.; Lee, J. H.; Chin, I. S.; Lee, H. J. A Model Predictive Control Technique Combined with Iterative Learning for Batch Processes. AIChE J. 1999, 45 (10), 2175. (10) Bonvin, D. Optimal Operation of Batch Reactors: A Personal View. J. Process Control 1998, 8, 355-368. (11) Chikkula, Y.; Lee, J. H.; Ogunnaike, B. A. Dynamically Scheduled MPC of Nonlinear Processes Using Hinging Hyperplane Models. AIChE J. 1997, 44, 2658.

(12) Geladi, P.; Kowalski, B. R. Partial Least-Squares Regression: A Tutorial. Anal. Chim. Acta 1985, 185, 1. (13) Jolliffe, I. T. Principal Component Analysis; SpringerVerlag: New York, 1986. (14) Lee, K. S.; Lee, J. H. Model-Based Predictive Control Combined with Iterative Learning for Batch or Repetitive Processes. In Iterative Learning Control; Bien, Z., Xu, J., Eds.; Kluwer Academic Publisher: Boston, 1998; Chapter 16, p 313. (15) Lee, K. S.; Lee, J. H. Convergence of Constrained ModelBased Predictive Control for Batch Processes. IEEE Trans. Automat. Control. To be published in July 2000.

Received for review April 30, 1999 Revised manuscript received November 5, 1999 Accepted November 9, 1999 IE990305Q

A Technique for Integrated Quality Control, Profile Control, and

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