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Partial least square-based model predictive control for large-scale manufacturing processes

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The Model Predictive Control (MPC) method has been widely adopted as a useful tool to keep quality on target in manufacturing processes. However, the conventional MPC methods are inadequate for large-scale manufacturing processes particularly in the
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  Partial least square-based model predictive controlfor large-scale manufacturing processes KYUCHUL SONG, PYOUNG YOL JANG, HYUNBO CHO* and CHI-HYUCK JUN Department of Industrial Engineering, Pohang University of Science and Technology, San 31 Hyoja, Pohang 790-784, KoreaE-mail: hcho@postech.ac.kr Received February 2000 and accepted July 2001The Model Predictive Control (MPC) method has been widely adopted as a useful tool to keep quality on target in manufacturingprocesses. However, the conventional MPC methods are inadequate for large-scale manufacturing processes particularly in thepresence of disturbances. The goal of this paper is to propose a Partial Least Square (PLS)-based MPC methodology to accom-modate the characteristics of a large-scale manufacturing process. The detailed objectives are: (i) to identify a reliable predictionmodel that handles the large-scale ‘‘short and fat’’ data; (ii) to design an effective control model that both maximizes the requiredquality and minimizes the labor costs associated with changing the process parameters; and (iii) to develop an efficient optimizationalgorithm that reduces the computational burden of the large-scale optimization. The case study and experimental results dem-onstrate that the presented MPC methodology provides the set of optimal process parameters for quality improvement. In par-ticular,thequalitydeviationsarereducedby99.4%,thelaborcostsby84.2%,andthecomputationaltimeby98.8%.Asaresult,theproposed MPC method will save on both costs and time in achieving the desired quality for a large-scale manufacturing process. 1. Introduction For over a decade, quality has been considered a keycompetitive criterion in manufacturing processes. Relatedacademic research efforts have been stimulated in manydifferent topics, such as Statistical Process Control (SPC),Quality Control (QC), Total Quality Control (TQC),Taguchi method, Model Predictive Control (MPC), etc(Escalante, 1999). Among those techniques, MPC is aclass of algorithms that optimize a set of controllableprocess variables with the objective of keeping the qualityon target as is discussed in Camacho (1995).According to Hidalgo and Brosilow (1990) MPC canbe defined in terms of the following three tasks: (i)to identify a functional relationship, prediction model,which maps process variables into quality variables; (ii) todesign an optimization problem, control model, consist-ing of the objective function and constraints; and (iii) toobtain the optimal settings of process variables throughon-line optimization. The main features of MPC include:(i) explicit prediction of the future quality; (ii) explicithandling of the constraints; and (iii) on-line optimization(Camacho, 1995). Due to these advantages, MPC hasbeen widely used as a convenient and effective tool forachieving the desired level of quality in manufacturingprocesses.However, traditional MPC methods cannot be effec-tively applied to process industries with respect to someaspects. Its application has been solely focused on small-scale manufacturing processes that requires the moni-toring and optimization of a small number of process andquality variables. A large-scale manufacturing process,such as a chemical process, a food process or a steel-making process, is not easy to control, since it isassociated with a large number of controllable and un-controllable variables. For instance, a typical photo-etching process to produce the shadow masks for a colormonitor from a coil of raw metal consists of four sub-processes: (i) coating with photo resist; (ii) imaging withultra-violet light; (iii) developing of the unexposed por-tion; and (iv) etching. The quality of the products pro-duced in the process is often affected by a few hundreds tothousands of variables. The process types have somesignificant properties. First, it is difficult to construct areliable prediction model from the large-scale ‘‘short andfat’’ data consisting of many variables (columns) and asmall number of observations (rows). Second, the costsnecessary to change the settings of the process variablesare burdensome for operators. Therefore, the number of process variables that change must be minimized. Third,both the uncontrollable variables (for example, initial *Corresponding author 0740-817X    2002 ‘‘IIE’’ IIE Transactions  (2002)  34 , 881–890  conditions of raw materials, external temperature andhumidity) and the controllable variables that the opera-tors want to remain unchanged must be incorporated intothe constraints in the control model. Hereafter thesevariables are denoted by ‘‘  fixed variables ’’. Fourth, thelarge-scale optimization problem cannot be appropriatelysolved in real-time.The goal of the paper is to suggest a Partial LeastSquare (PLS)-based MPC methodology that accommo-dates the characteristics of a large-scale manufacturingprocess. The proposed methodology can be applied tooptimize the quality of products produced in process in-dustries with a large number of process and qualityvariables. The detailed objectives are: (i) to identify areliable prediction model that handles the large-scale‘‘short and fat’’ data; (ii) to design an effective controlmodel that minimizes both the discrepancy in quality andthe labor costs associated with changing the settings of process variables; (iii) to incorporate the fixed variablesinto the constraints in the control model; and (iv) to de-velop an efficient optimization algorithm that reduces thecomputational burden.Some assumptions are made in the paper:1. The unidentified disturbances are negligible or non-existent. When disturbances occur then robust MPCcan be applied, in which the difference between thecurrent measured output and the current predictedoutput is added to the prediction model for use insubsequent predictions. This is beyond the scope of this paper.2. The srcinal data on the process and quality variablesare refined through statistical methods such as outlierand noise removal techniques.The remainder of this paper is organized as follows:Section 2 provides a review of related work. In Section 3,the conceptual framework of the MPC method is pre-sented. The PLS-based prediction and control models areaddressed in Sections 4 and 5, respectively. Section 6provides the optimization algorithm. Section 7 shows acase study and experimental results. Section 8 providesthe conclusion of the paper. 2. Related work From the end of the late 1970s, an increasing demand formore reliable process control methods and advances inmicroprocessor technology have driven the developmentof MPC methods. Dynamic Matrix Control (DMC), us-ing a linear prediction model, is one of the earliest MPCmethods. DMC formulates an unconstrained controlproblem for Multi-Input Multi-Output (MIMO) systems,after which the constraints on the process variables andquality variables are explicitly included in the problemformulation (Camacho, 1995). Finally, the control modelis formulated as an optimization problem formed with anobjective function and constraints. The objective functionis usually to minimize the deviations of quality variablesfrom the desired levels as well as the magnitude of thechanges in the process variables from their current set-tings. The resulting optimization problem is solved byQuadratic Programming (QP) techniques.While linear MPC methods are easy to implement,methods using a nonlinear model may be necessary inpractice since many manufacturing systems are inherentlynonlinear. The prediction model in the nonlinear MPCmethod is formulated as the nonlinear relationship be-tween the process variables and the quality variables as isdiscussed in Shaw and Doyle (1997) and Patwardhan et al.  (1998). Whilst nonlinear programming techniquesare adopted for approximating the control model, feed-back approaches are necessary in process control becauseof the disturbances and model inaccuracies inherent in allreal processes. The objective of the robust control meth-ods discussed in Bemporad and Morari (1999) is to designcontrollers that preserve stability and performance inspite of the model inaccuracies and disturbances. In otherwords, the robust control methods consider the discrep-ancy between the model and the real process explicitly.Although MPC methods have been widely accepted asuseful tools for obtaining the optimal parameters inmanufacturing processes, the conventional MPC methodsare inadequate for large-scale manufacturing processes. Adetailed investigation on previous studies in the area re-veals a number of defects that are discussed below.Neural network techniques have been extensively usedto build nonlinear prediction models since it is oftensuitable to approximate an arbitrary nonlinear function(Hidalgo and Brosilow, 1990; Kim  et al. , 1997; Botto andCosta, 1998; Bemporad and Morari, 1999). However,neural network models for a large-scale manufacturingprocess have several problems. First, they cannot be ac-curately constructed with the ‘‘short and fat’’ practicaldata as pointed out in Erriksson  et al.  (1995). Second,the large-scale MIMO network is too sensitive to over-parameterization and local minima (Lakshminaraynan et al. , 1997; Patwardhan  et al. , 1998).As far as control models are concerned, the costs as-sociated with changing the settings of process variablesdepends on the number of changes rather than on themagnitude of the changes since the setting of a largenumber of process variables incurs a tremendous laborcost. Further, both the controllable and controllablevariables that the operators want to remain unchangedmust be incorporated into the constraints in the controlmodel. Since the control model is formulated as a non-linear optimization problem in real manufacturing pro-cesses, analytical approaches cannot be appropriatelyemployed. Hence, it is usually solved by a means of it-erative optimization algorithms, such as the Gauss– Newton method, the Levenberg–Marquardt method, on 882  Song  et al.  Successive Quadratic Programming (SQP) as in Biegler(1998), etc. Although interior-point methods have beensuggested by Rao  et al.  (1998) for fast optimization insmall-scale manufacturing processes the previous meth-ods have rapidly increasing computational burdens as thenumber of process variables expands to a few thousands.Consequently, the previous MPC approaches experienceda large computational burden in simultaneously opti-mizing numerous process variables in real-time andtherefore on-line optimization has been nearly impossible(Pottmann and Sebord, 1997). 3. Conceptual framework The conceptual framework of the proposed MPC pro-cedure is illustrated in Fig. 1. It consists of two phases:off-line and on-line. The off-line phase is performed on aset of historical data that has been stored previouslycollected, whereas the on-line phase is based on a data setthat is being currently collected. The central motivationunderlying the adoption of the two-phase approach isthat it becomes possible to efficiently and effectivelygenerate the optimal settings of the manufacturing pro-cess from the accumulated process and quality data.In the off-line phase, the prediction model mapping of process variables into quality variables is constructedusing the historical data set. In the on-line phase, thecontrol model is formulated as a nonlinear programmingmodel incorporated with the constructed predictionmodel. The control model is then optimized to obtain theoptimal settings.In order to describe the prediction and control models,some notations need to be defined. Let  X   j  denote the  j thprocess variable (  j  ¼  1 ;  . . .  ; m ) and let  Y  k   be the  k  thquality variable ( k   ¼  1 ;  . . .  ; r  ). The required data set isdepicted in Fig. 2. Matrices  X  ¼ ð  x ij Þ  and  Y  ¼ ð  y  ik  Þ  arethe historical data on the process variables and the cor-responding quality variables, respectively, where  x ij  is the i th observation of   X   j  and  y  ik   is the  i th observation of  Y  k  ð i  ¼  1 ;  . . .  ; n Þ .  v 0 are the current settings of controllablevariables, and  d 0 are the current measurements of fixedvariables.  y d is the desired level of the quality variable.The number of observations is usually smaller than thatof process variables (‘‘short and fat’’) in practice. 4. Identification of prediction model using PLS The traditional model building methods can handle nei-ther the practical ‘‘short and fat’’ data nor the large-scaleMIMO data. To overcome such limitations, multivariateprojection approaches that reduce the srcinal large-scaledata to lower dimensional data have been developed. Inparticular, the Partial Least Square (PLS) method ispopular for dealing with highly nonlinear correlated databetween process variables and quality variables (Geldadiand Kowalski, 1986; Lakshminaraynan  et al. , 1997;Malthouse  et al. , 1997). Its application to MPC methodslooks promising but is yet to appear. Fig. 1.  Conceptual framework of the proposed MPC procedure. Partial least square-based model predictive control   883  In the PLS method,  X  and  Y  are decomposed into asum of series of lower dimensional matrices as follows: X  ¼  t 1 p T1  þ t 2 p T2  þþ t  A p T  A  þ E  ¼  TP T þ E ; Y  ¼  u 1 q T1  þ u 2 q T2  þþ u  A q T  A  þ F  ¼  UQ T þ F : ð 1 Þ T ð n   A Þ  and  U ð n   A Þ  represent the score matrix, while P ð m   A Þ  and  Q ð r    A Þ  represent the loading matrix for X ð n  m Þ  and  Y ð n  r  Þ , respectively. To determine thedominant directions in which to project data, a maximaldescription of the covariance within  X  and  Y  is used as acriterion. The first set of loading matrices (the directioncosines of the dominant directions in the  X  and  Y ), p 1 ð m  1 Þ  and  q 1 ð r   1 Þ , is obtained by maximizing thecovariance between  X  and  Y . The respective projection of the  X  and  Y  onto  p 1  and  q 1  gives the first set of scorematrix,  t 1 ð n  1 Þ  and  u 1 ð n  1 Þ . This procedure is calledan ‘‘outer relation’’. X  and  Y  are indirectly related through their scores byan ‘‘inner relation’’, which is a functional mapping modelfrom  t 1  to  u 1 , that is,  u 1  ¼  h 1 ð t 1 Þ . Denoting  E 1  ¼  X  and F 1  ¼  Y , the residuals at this stage are computed by thedeflation process: E 2  ¼  X  t 1 p T1  ¼  E 1   t 1 p T1  ; F 2  ¼  Y  u 1 q T1  ¼  Y  h 1 ð t 1 Þ q T1  ¼  F 1   h 1 ð t 1 Þ q T1  : ð 2 Þ The procedure of determining the scores and loadingmatrices of the inner relation is continued (with the re-siduals obtained at each stage) until the required numberof PLS dimensions ( A ) is extracted. In practice, thenumber of PLS dimensions is determined based on eitherthe percentage of variance explained or the use of sta-tistically sound approaches such as cross-validation. Thedirections irrelevant in the data set (such as noise andredundancies) are confined to the error matrices, that is, E  and  F .Once  P ,  Q  and the function  H  ¼ ð h 1 ;  . . .  ; h  A Þ  are esti-mated from the historical data, the quality data corre-sponding to a new process data set  X 0 ¼ ½ v 0 ; d 0   can bepredicted by the following model: Y  ¼  UQ T þ F  ¼  H ð T Þ Q T þ F  ¼  H ð X 0 P Þ Q T þ F :  ð 3 Þ Conceptually, the PLS algorithm is based on decompos-ing the original variables of   X  and  Y  into the scorevariables of   T  and  U  that summarize a great deal of correlated and redundant information. Here, the columndimension of   T  and  U , which is  A , is much less than thatof   X ð m Þ  or that of   Y ð r  Þ . This means that the higher di-mension of the srcinal variables is reduced to the lowerdimension of the score variables. This is called ‘‘dimen-sion reduction’’.In the nonlinear PLS algorithm, the nonlinear func-tional relationship between  T  and  U ,  U  ¼  H ð T Þ , can beconstructed by neural networks due to their nonlinearapproximation properties (Qin and McAvoy, 1992;Malthouse  et al. , 1997). A direct benefit of such a strategyis that only one SISO (Single-Input-Single-Output) net-work is trained at a time. A centered sigmoid neuralnetwork is used to construct the nonlinear mapping be-tween the input and output score variables, every time alatent variable is extracted. According to Hornik  et al. (1989), a three-layered neural network with a back-propagation structure is sufficient to approximate thenonlinear mapping with the arbitrarily desired accuracy.This mapping process is repeated until the concernednumber of latent variables is reached, as is shown inFig. 3. This concept is characterized by the integration of PLS and nonlinear mapping among score variables. As a Fig. 2.  Detailed descriptions of the data set. 884  Song  et al.  result, nonlinear PLS builds the prediction model fast andeffectively between  X  and  Y  variables. Further, this ap-proach circumvents the over-parameterization and con-vergence to local minima that are known to be thedisadvantages of neural networks. 5. Design of control model To effectively accommodate the large-scale manufactur-ing process, two modifications are required in the tradi-tional control model. First, the magnitude of the changesin the process variables in the objective function shouldbe replaced by the number of changes. This modificationreduces the number of changes in the process variables.Second, fixed variables, that is, both the uncontrollableand controllable variables that the operators want to re-main unchanged should be included in the constraints.Those variables should remain constant (at their currentmeasurements) during the optimization. This modifica-tion reduces the number of variables to be optimized, andtherefore it may exclude impractical solutions.Let  I   j  be a binary variable whose value is one or zerodepending on whether or not the  j th controllable variable v  j  changes from the current settings (  j  ¼  1 ;  . . .  ; c ). That is,  I   j ð v  j Þ ¼  1 ;  if   v  j  6¼  v 0  j ,0 ;  otherwise.  Then, the control model can be formulated as follows:min v a  y d   ^yy  2 þ b X c j ¼ 1  I   j ð v  j Þ ;  ð 4 Þ subject to v min   v    v max ;  ð 5 Þ d  d 0 ¼  0 :  ð 6 Þ The first term in the objective function is the penalty onthe deviations of predicted quality variables  ^yy  from thedesired levels  y d , whereas the second term is the penaltyon the number of changes in the controllable variablesfrom the current settings.  a  and  b  are the weight factorson each term. Constraint (5) restricts the controllableranges of the controllable variables and Constraint (6)makes the fixed variables remain unchanged at theircurrent measurements.  v min and  v max are the feasibleranges of the controllable variables. 6. Optimization procedure The designed control model, nonlinear optimizationproblem, is usually solved using iterative optimizationalgorithms such as the Gauss–Newton method, theLevenberg–Marquardt method or Successive QuadraticProgramming (SQP), etc. The principal idea of the iter-ative algorithms is to transform the srcinal problem intoan easier sub-problem that can be easily solved and thenused as a basis of an iterative procedure.Of various iterative optimization algorithms, the mostfavored is the SQP approach. It is known from Schit-towski (1985) that the SQP outperforms other methods interms of efficiency, accuracy, and percentage of successfulsolutions, over a large number of problems. Furthermore,the SQP approach can be applied to problems whose Fig. 3.  An overview of nonlinear PLS algorithm. Partial least square-based model predictive control   885
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