EconomicOptTaguchi AK LD JA 2013

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Optim Eng (2013) 14:547563DOI 10.1007/s11081-013-9237-3

Economic oriented stochastic optimization in processcontrol using Taguchis method

Andrs Kirly Lszl Dobos Jnos Abonyi

Received: 5 March 2011 / Accepted: 6 September 2013 / Published online: 24 October 2013 Springer Science+Business Media New York 2013

Abstract The optimal operating region of complex production systems is situatedclose to process constraints related to quality or safety requirements. Higher protcan be realized only by assuring a relatively low frequency of violation of these con-straints. We dened a Taguchi-type loss function to aggregate these constraints, tar-get values, and desired ranges of product quality. We evaluate this loss function byMonte-Carlo simulation to handle the stochastic nature of the process and apply the

gradient-free Mesh Adaptive Direct Search algorithm to optimize the resulted robustcost function. This optimization scheme is applied to determine the optimal set-pointvalues of control loops with respect to pre-determined risk levels, uncertainties andcosts of violation of process constraints. The concept is illustrated by a well-knownbenchmark problem related to the control of a linear dynamical system and the modelpredictive control of a more complex nonlinear polymerization process. The applica-tion examples illustrate that the loss function of Taguchi is an ideal tool to representperformance requirements of control loops and the proposed Monte-Carlo simula-tion based optimization scheme is effective to nd the optimal operating regions of controlled processes.

Keywords Monte-Carlo simulation Model predictive control Economicassessment Stochastic optimization

1 Introduction

Due to the dynamic and signicant changes of the economic environment, perfor-mance assessment of process control is a highlighted area of chemical engineer-ing (HoLee et al. 2010). Our aim is to develop an optimization framework based

A. Kirly L. Dobos J. Abonyi (B )Department of Process Engineering, University of Pannonia, Egyetem street 10, Veszprm, 8200,Hungarye-mail: [email protected]
mailto:[email protected]:[email protected]


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548 A. Kirly et al.

on Taguchis method (Taguchi et al. 1989) designed to determine optimal operatingregimes of chemical processes by taking process constraints, desired maximum num-ber (frequency) of constraint violations and process uncertainties into consideration.

Variance in the closed control loop caused by unmeasured disturbances and badly

designed controllers might cause variations in the product quality. In case of increas-ing variance of process variables the probability and frequency of violation of qualityrequirements is increasing that might lead to increase the amount of less valuableoffset products. Typical examples when reduced number of violations of the pre-determined process constraints is acceptable can be found in the eld of statisticalprocess control (SPC ; Oakland 2007). In SPC statistical tools are applied to monitorthe performance of the production process and detect signicant deviations that maylater result in offset products.

We follow a more sophisticated model based approach during this work. Mod-

ern process analysis, monitoring, control and optimization tools are mainly based onsome kind of process model. It is obvious to utilize these process models also in theeconomic assessment and optimization. Usually the output of cost-benet analysis iscost reduction or prot increment expressed by a cost function. These functions in-corporate the costs of the operation, raw materials, current prices of products (Bauerand Craig 2008), and risks of malfunctions. In our economic oriented optimizationstrategy the aim is to nd steady state operation points (controller set points) whereprot is realized. This task is fullled at the supervisory control level (Chen et al.2012).

The general approach for economic performance evaluation comprises the fol-lowing steps: reduce the variance of the controlled variable and shift the set points(process mean) closer to the operation limits (Lee et al. 2008) without increasing thefrequency of the violation. This operation is referred to the improved control (Zhaoet al. 2009). The variance reduction might mean to re-tune the existing controllers,or, in more radical cases, change the whole control strategy.

To handle uncertainty and effects of measurement noise, a novel Monte-Carlo(MC) simulation based approach is proposed. MC simulation is frequently appliedin various areas (Rubinstein and Kroese 2008). This tool has also been proven itsefciency in risk related optimization of chemical processes, e.g. it is applied in opti-mizing maintenance strategies of operating processes (Borgonovo et al. 2000). Thereis a common characteristic in these solutions: the stochastic nature of the studied sys-tem has to be modeled. In the applied methodology this simulation is related to themodeling of the unmeasured disturbances of the control loops. To handle this ran-dom effect, MC simulation is applied with the characterized noise. An economic costfunction is calculated in each case to measure the economic efciency of the process.

We use Taguchis quality loss function to adjust the economic benet, which isalso based on MC simulation. Using this function, soft constraints can be applied forthe optimization. Taguchis loss function is a widely used method for quality control(Taguchi et al. 1989). Di Mascio and Barton proposed a novel technique for measur-ing dynamic control quality within the Taguchi framework to compare several con-trollers based on performance and stability (Mascio and Barton 2001). Maghsoodlooet al. (2004) give a comprehensive analysis of Taguchis contribution to the eld of quality from a statistical (presenting Statistical Quality Control) and an engineering


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Economic oriented stochastic optimization in process control 549

viewpoint. Using Taguchis loss function in robust stochastic optimization togetherwith MC simulation can be often found in the literature. Zang et al. (2005) give areview of robust design based on these concepts in dynamics, while Yang and Choupropose a hybrid Taguchi method to solve a multi-response simulation-optimization

problem (Yang and Chou 2005). Rezaie et al. present a robust tool to handle un-certainty permutations by MC simulation (Rezaie et al. 2007). Recent methods canbe found in Subbaraj et al. (2011), where Taguchis function is used to develop aself-adaptive real-coded genetic algorithm, while Yang and Peng (2012) propose animprovement of Taguchi method by the integration of particle swarm optimization.

Integrating this benet analysis tool into the mesh adaptive direct search optimiza-tion algorithmwhere the task is to nd the most benecial steady state operationpointresulted the proposed economic oriented optimization framework. In the pro-posed multilayer optimization framework, the application of gradient based method-

ologies for maximizing the economic throughput is not possible, due to the stochasticcharacteristics caused by the closed loop variance. Therefore, the utilization of directsearch methods are necessary. Mesh Adaptive Direct Search (MADS) (Audet andDennis 2006) class of algorithms is a relatively new set of direct search methods fornonlinear optimization, i.e. these algorithms are capable of calculate the extremumsof a non-smooth functions, like our economic objective function. Since the steadystate operation points are mainly determined by the variance of the controlled vari-ables, incorporating this effect into the model is inevitable. The created optimizationframework functions as an industrial Advanced Process Control system (Ray 1981;

Bauer and Craig 2008).The paper is organized as follows: in Sect. 2 the economic cost function basedmultilayer optimization framework is introduced. In Sect. 3 the applied methodologyis explained in detail. In Sect. 4, the efciency of the proposed methodology is illus-trated throughout a linear benchmark control problem. The economic performancemeasure has been formalized as a basis for optimizing the set point signal. In thissimple case of the benchmark example the process is controlled with a PI controller.To determine optimal set point of the controller, Taguchis quality loss function isapplied to maximize the economic benet. As a second example a non-linear processcontrolled by a linear model based predictive controller MPC ( MPC ; Garca et al.1989) is considered, since this combination highly applicable for variance-reductionpurposes is widely applied in chemical process industry. In this case study the processvariance is caused by an unmeasured disturbance, model mismatch, and noise addedto the controlled variable. In this example the effect of the unmeasured disturbancewith different amplitude is examined in detail. These examples show the realisticbenets of the proposed methodology.

2 Taguchi loss function based multilayer optimization

The proposed optimization framework could be considered as a multilayer optimiza-tion problem, as depicted in Fig. 1.

This framework is rather similar to the Advanced Process Control (APC) systemsapplied in process industry for online prot optimization (Bauer and Craig 2008).


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550 A. Kirly et al.

Fig. 1 The layers of an economic optimization of an operating technology

In the following sections the main aspects and tasks are introduced, which have tobe taken into consideration in different optimization levels.

2.1 Applying Taguchis loss function at the supervisory control levelThe main task at the supervisor level is to maximize the economic throughput byvarying the steady state set point signal. In general the economically optimal setpoint is close to the operation limits of the process. For that reason the reductionof the closed loop variance is necessary. Due to the process variancecaused bydisturbances, noise, etc.there is a risk of process constraint violation which hasto be taken when the new set point is determined. The essence of this economicoptimization approach is depicted in Fig. 2.

The aim of the economic oriented process optimization framework could be for-mulated as minimizing a quality cost function. As we discussed earlier, the economicloss can be expressed by the Taguchi loss function which is formulated mathemati-cally in basic case as follows:

L(y) =(y t)2 LSL y USLC r otherwise

(1)

where L(y) is the loss associated with the value of quality characteristic y, t is thetarget value. Cr is the cost of rejection while LSL and USL are the lower and up-

per specication limits respectively. Based on the equation above, Kapur and Cho(1996) developed a multivariate loss function for the multivariate quality characteris-tic y1, y 2, . . . , y n :

L(y 1, y 2, . . . , y n ) =pi = 1

ij = 1 K ij (y i t i )(y j t j ) LSLi y i USLi

C r (y 1, y 2, . . . , y n ) otherwise (2)


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Fig. 2 Approach to economic benet estimation with variance reduction

Fig. 3 Traditional rectangularloss quality function

where t j is the target, K ij is the loss coefcient of the j th quality characteristic.In many situations, target value is dened as a range instead of a number, and

within this range the same loss or prot is realized. The traditional loss function inthis case is depicted in Fig. 3.

In our optimization problem target is dened as a range, however, using classicalfunction is not appropriate because of its sharp boundaries. Therefore we have denedsteps in quality function as sigmoid (see Fig. 7 and Fig. 9) and the loss function isformed as follows.

L(y) =

a

1+ ey b

c LSL y USL

a

1+ e (y b)

c LSL < y USL +

C r otherwise(3)

where a , b, c are the parameters of the sigmoid functions and is a necessary smallnumber, e.g. USL- LSL100 .

In our case the optimization can be expressed as a minimization problem accordingto the set points of local controllers of operative control level w = [ w1, . . . , w p ]T .p is the number of the controlled variables, denoted by yi . The task at the operativecontrol level can be summarized as yi should be as close to w i as possible:

minw

L(y 1, y 2, . . . , y n ) (4)


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simulation is applied. The Monte Carlo method is applied frequently in the solution of stochastic optimization problems, in stochastic linear programming (Prkopa 1995;Marti et al. 2004). Kjellstrm (1969) was the rst using Monte Carlo estimators forthe iterative improvement of convergence behavior in nonlinear stochastic optimiza-

tion.In the proposed multilayer optimization framework, the application of gradientbased methodologies for maximizing the economic throughput is not possible, due tothe stochastic characteristics caused by the closed loop variance. Integrating the sim-ulation based economic performance assessment methodology into a direct searchoptimization algorithm an effective optimization framework is obtained. Mesh Adap-tiveDirect Search ( MADS ) (Audetand Dennis2006) class of algorithms is a relativelynew set of direct search methods for nonlinear optimization, i.e. these algorithms arecapable of calculate the extremums of a non-smooth objective functions, like our

economic objective function.Our methodology is the following:

Economic performance assessment of the considered steady state operation point.It means applying a set point (w) and calculating the value of the economic costfunction, Eq. (4), with respect to the process constraints. Because of consideringthe process variance as random phenomena, Monte Carlo simulation with multipleruns of augmented process simulator is applied to aggregate the effect of randomvariances in a nal economic cost function.

Integrate the economic performance evaluation tool into the MADS optimizationalgorithm to nd the economically optimal steady state operation point. The pre-viously applied economic cost function (the quality loss function) has to be min-imized with respect to the proposed constraints with varying set point signal (w).This algorithm can handle constraint limits of process variables at certain con-dence levels.

In the following section way of application of Monte Carlo simulation and MADSoptimization algorithm is introduced briey.

3.1 Monte Carlo simulation

Monte Carlo Simulation ( MCS ) methods are widely applied in mathematical mod-eling problems, where some kind of random phenomena must be handled. In theproposed multilayer optimization framework, process variance caused by noise andunmeasured disturbances is considered as random phenomena. The Monte Carlomethod consists of the following steps:

Dene the domain of possible inputs. Generate inputs from this domain randomly using a specied probability distribu-

tion. Execute deterministic computation using the inputs. Aggregate the results of the computations into the nal result.

In engineering practice, the normal distribution is considered as an adequate as-sumption for characterizing many uncertainties. In the modeling of the considered


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process, the next steps are followed: at rst the mathematical model of the processis created. Then noise signals with different variance have been added to input andoutput variables to approximate the real process variance. By setting the added noisesignals randomly using a specied probability distribution, different result can be ob-

tained in each individual simulation. In case of economic oriented optimization, theresult is obviously the value of the economic objective function, Eq. (4). Aggregat-ing the economic performances of the individual runs into a nal value the statisticaleconomic performance can be obtained with respect to the examined steady stateoperation point. Since the complex production processes are mostly characterizedwith non-linear process models, the economic assessment and optimization need anoptimization algorithm which is able to handle the non-linear optimization problemwith respect to the process constraints. In addition the economic cost function is non-smooth, caused by the closed loop variance.

3.2 The mesh adaptive direct search methodology

Since the evaluation of gradient of the economic objective function respect to thesteady state operation points is highly CPU time intensive, there is a need to uti-lize any gradient free optimization method. Mesh Adaptive Direct Search (MADS)(Audet and Dennis 2006) class of algorithms is a relatively new set of direct searchmethods for nonlinear optimization, i.e. these algorithms are capable of minimize anon-smooth function, like our economic cost function (Eq. (4)) under the proposed

constraints. According to Audet and Dennis (2006), Abramson et al. (2009), MADScan be interpreted as a generalization of Generalized Pattern Search (GPS) (Torczon1997) algorithms, with the restriction to nitely many pool direction removed.

MADS is an iterative algorithm, where at each iteration a nite number of testpoints are generated. At a beginning of an iteration, the infeasible test points areltered (discarded), i.e. innite objective value is assigned to them (f(x) = + ).Thereafter the feasible test points are evaluated by the objective function, and com-pared with the current best objective function value found so far. Each of these testpoint lies on the current mesh, which is constructed from a nite set of nD directionsD R n and scaled by the mesh size parameter m

k R n . If we nd a point with

lower objective value than the current best one, this test point is a so-called improved mesh point and the iteration is a successful iteration .

Each iteration consist of two steps, the so-called SEARCH step and POLL step.SEARCH step can return any point of the underlying mesh, it is trying to nd an un-ltered point. If it fails to generate an improved mesh point, then the second step, thePOLL is invoked. POLL step consists of a local exploration around the current bestsolution, and the test points are generated in some directions scaled by the mesh sizeparameter. MADS are novel in the number of usable directions, since in GPS, POLLdirections belong to a nite set, while POLL direction in MADS belongs to a muchlarger set, in fact if the iteration number k goes to innity, the union of the normalizedPOLL directions over all k become dense in the unit sphere. According to Audet andDennis (2006), this algorithmic construction allows faster convergence. Another im-portant difference between MADS and GPS is the so called poll size parameter , pk .This parameter determines the size of the frame where the POLL step can operate.


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Fig. 5 Example of GPS frames (above ) and MADS frames (bottom ) P k = { xk + mk d : d D k } ={p 1, p 2, p 3} for different values of mk =

pk . In all six gures, the mesh M k is the intersection of all

lines (Audet and Dennis 2006)

In case of GPS, mesh size and poll size are equal ( mk = pk ), while in MADS thesetwo parameters can differ. This difference is depicted in Fig. 5. Additional pieces of information like convergence analysis or practical implementations can be found inAbramson et al. (2009).

In the economic oriented multilayer optimization framework (see Fig. 1) MADSis applied in the supervisory level to maximize the economic performance formalizedas Eq. (4). The optimization problem is solved with respect to the process constraints,with varying set point signal (w). Since MADS needs a reduced number of runs of the augmented process simulator, the optimal value of the set point signals can be

quickly obtained. The low number of iteration during optimization is necessary, sinceMonte Carlo simulation of the operative control level (augmented process simulator)is applied, which is highly computation demanding process.

In the following section the effective application of the proposed framework isgoing to be examined throughout the case studies of a benchmark, linear process anda MPC controlled highly non-linear technology.

4 Application examples

In this section, two application examples are presented to demonstrate the applicabil-ity of the proposed framework for enhancing the economic benet of the operatingtechnologies. The calculations for both examples are based on closed loop-data, gen-erated using MATLAB-Simulink. To ensure reproducibility the MATLAB les aredownloadable from the website of the authors at www.abonyilab.com.
http://www.abonyilab.com/http://www.abonyilab.com/


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Fig. 6 Block diagram of the SISO closed loop system

4.1 Optimization of a constrained single-input process

Consider a single-input single-output (SISO) process, characterized by G p shown inFig. 6 subject to disturbance dynamics G d described by:

yk = G p u k + G d k =0.6299z 1

1 0.8899z 1 uk 2 +

1 0.8z 11 0.8899z 1 k k

= 1, . . . , q (6)

where t is a normally distributed white noise sequence of mean 0 and variance 1. qdenotes the last time step of the considered simulation. The objective in the supervi-sory control level is to minimize the loss function, L(y) mean of the Taguchis lossfunction value on the considered time horizon, respect to the process constraints. Theoptimization problem can be formalized as:

minw

2L(y) where L(y) =

501+ ey 0.5 LSL y USL

501+ e (y 0.5) LSL < y USL +

50 otherwise

(7)

where the loss function of the outputs is depicted in Fig. 7 with = 2. The appliedTaguchi functions have a larger is better shape. The specication limits for inputsand outputs are the following:

LSL = 10 yk 10 = USL; 5 uk 5; where k = 1, . . . , q (8)

In the base case a PI controller has been chosen, and the controller parametersare K c = 1.926, T I = 0.6. As previously remarked, the probability of not crossingthe limits of the output means a non-linear constraint for the optimization problem.During the examinations this condence level is assigned as 95 % and 90 %. In theliterature (Zhao et al. 2009) the same SISO process is applied as an example, withthe same probabilities. The means of the output are y = 1.4844 and y = 2.4531 atcondence level of 95 % and 90 %, respectively. The output data in 95 % condencelevel is depicted in Fig. 8.


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Fig. 7 Taguchi loss function foroutputs in the SISO process

Fig. 8 Base case operation withprobability constraint level of 95 %

As we mentioned earlier, the number of individual economic performance evalu-ations in the Monte Carlo simulation has been set to 100. There has been an attemptto apply quadratic programming as optimization algorithm (utilizing MATLAB, Op-timization Toolbox), but the computational demand was extremely high, almost onehour even in this simple example, but with the same result. By applying MADS,the computation demand has been signicantly decreased into 2 minutes. It can bepossible, since the computation of the gradient of the economic cost function is notnecessary. The initial set point for the optimizer was set equal to the upper constraintof the output variable, w0 = 10.

Thanks to the application of Taguchis loss function, soft constraints and qualityevaluation can be easily realized. Utilizing the previously introduced Monte Carlosimulation based optimization methodology new steady state operation points havebeen determined with multiplied economic performance respect to the dened con-dence level.

4.2 The polymerization process

The process under consideration is a polymerization process controlled by a linearMPC, the previously mentioned DMC (Ricker 1988). The controlled system pos-sesses all those difculties which exist in an operating polymerization process.


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The reactor what has been studied is a CSTR where a free radical polymerizationreaction of methyl-methacrylate is considered using azobisisobutironitil (AIBN) asinitiator, and toluene as solvent. The aim of the process is to produce different kindsof product grades. The number-average molecular weight is used for qualifying the

product and process state. The polymerization process can be described by the fol-lowing model equations (Silva-Beard and Flores-Tlacuahuac 1999):

dC mdt

= (kp + kf m )C m P 0 +F (C min Cm )

V (9)

dC I dt

= kI C I +F I C I in F C I

V (10)

dD 0dt

= (0.5ktc + ktd )P 20 + kf m Cm P 0 F D 0

V (11)

dD 1dt

= M m (kp + kf m )C m P 0 F D 1

V (12)

where

P 0 = 2f C I kI ktd + ktc (13)The actual values of parameters used in the equations can be seen in Table 1.

The number average molecular weight ( NAMW ) is dened by the ratio of D1/D 0.By assuming an isotherm operation model the process model consists of four states,represented by four differential equations Eq. (9)Eq. (12) (Maner and Doyle 1997).During the simulations T s = 0.03 h is applied as sample time.

The qualication of the product and process operation is based on the numberaverage molecular weight. Due to the non-linear model equations, the developmenteconomic performance turns into a highly non-linear optimization problem.

The control objective on the supervisory control level is to maximize the economicperformance of the process. The objective function is formalized as

minw

2L(y) where L(y) =

1001+ e y 25000100 LSL y USL

1001+ e

(y 25000)100

LSL < y USL +

100 otherwise

(14)

where the loss functions of the inputs and outputs are depicted in Fig. 9 with = 20.The applied Taguchi functions have a larger is better shape. The specication limitsare the following:

LSL = 24000 yk 26000 = USL (15)

An important characteristic of the process is the increasing of product quantitywhen shifting the steady state operation closer to the lower limit, hence the optimalsteady state operation point is expected near to the lower limit. The maximum prob-ability of violating the process constraints is 1 %, so the mentioned condence levelis 99 %. The number of Monte Carlo simulations is 100, similarly to the previous


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560 A. Kirly et al.

Fig. 9 Taguchi loss function foroutputs in the polymerizationprocess

Fig. 10 Improved case, optimalsteady state operation point of PMMA reactor

case. The closed loop variance of the process is caused by the noise added to the inletmonomer owrate (F ) with mean of 0 and = 0.014. Other source of the closed loopvariance is the noise added to the controlled variable with the mean of 0 and = 143.

On the operative control level a linear MPC, a Dynamic Matrix Controller (DMC)is installed (Ricker 1988). The DMC applies the linear convolution model of theprocess for predicting the effects of the considered manipulated variable sequence.The manipulated variable in the control strategy of the reactor is the initiator inletow rate. The tuning parameters of the applied DMC are H m = 30, H p = 3, H c = 3.The value of is chosen as 4 1012.

As base case the safest steady state operation point has been chosen which is in themiddle of the specied operation range (w = 25000). As the result of the economicperformance optimization the optimal steady state set point is w = 25632. Thanks tothis set point modication the quantity of the produced polymer has been increasedwith 5 % and throughout this the economic performance also increased with 5 %. Theresult of the closed loop simulation with the optimal setpoint is depicted in Fig. 10.


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As it is mentioned the number of individual economic performance evaluations inthe Monte Carlo simulation has been set to 100. In this case study there has been anattempt to apply quadratic programming as optimization algorithm. Using this tool,the same result has been obtained but the computational demand was extremely high,

almost 10 hours. By applying MADS, the computation demand has been decreasedinto half an hour. As it can be seen, the quadratic programming might be applicablebut its computation demand is exaggerated. The initial set point for the optimizerwas set equal to the lower constraint of the output variable, w0 = 24000.

As Fig. 10 shows the frequency of constraint violation is conspicuously low, theprocess constraint has been violated only once. It means 99.85 % probability of notviolating the limits, in contrast to the previously determined condence level, whichwas 99 %. It may happen since the off-specication product (products which does notfulll the requirements) means extra outgoings in the economic objective function.

Accordingly it is not worthwhile to produce even just 1 % off-specication product,however this amount can be accepted technologically.The results conrmed the assumption that the economically optimal operation is

close to the process constraints. However 99 % was set as condence level of limitviolation, the way of formulating the economic cost function does not allow sucha low quantity of off-specication product, since it causes extra outgoings duringoperation.

5 Conclusion

We introduced an economic oriented optimization framework for the optimization of operating regimes of controlled processes. We applied Taguchis quality loss functionto aggregate process constraints, target values, and desired ranges of product quality.Due to unmeasured disturbances and noise caused process variance, the determina-tion of the optimal operating region (setpoints of the local controllers) is rather dif-cult, since shifting the operation point closer to the process limits means a risk of violation of constraints and reducing product quality. Economic performance and risk

level of the operation can be measured as quality loss. Monte Carlo simulation is ap-plied with multiple runs of process model (augmented with the model of the controlsystem) with economic performance assessment to handle the random phenomena of process variance and to provide robust estimate of the loss function. We showed thestochastic nature of the cost function is effectively handled by utilized gradient-freeMesh Adaptive Direct Search algorithm. The application of MADS reduced the timedemand of optimization by one order of magnitude compared to applying quadraticprogramming.

The resulted optimization scheme is applied to determine the optimal set-pointvalues of control loops with respect to pre-determined risk levels, uncertainties andcosts of violation of process constraints. The efciency of the proposed frameworkis demonstrated throughout an example of a benchmark, linear system and a non-linear process. In the rst benchmark problem, signicant economic benet could berealized, the prot could be increased substantially by nding the optimal set pointsignal after variance reduction.


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Table 1 Design parameters forMMA polymerization reactor F 1.0 m

3h

Cmin 6.4678 kmolm3C I in 8 kmolm3

V 0.1 m3M m 100.12 kgkmolf 0.58

R 8.314 kJkmolKkp 2.4952 106 m

3kmolh

kI 1.0224 10 1 1hkf m 2.4522 103 m

3kmolh

ktc 1.3281 1010 m3

kmolh

ktd 1.0930 1011 m3

kmolh

These application examples illustrate that Taguchis loss function is an ideal tool torepresent performance requirements of control loops. This is important result sincethe proposed methodology helps to develop and improve control performance as-sessment tools and integrate hard to difcult requirements of process engineers intooptimization algorithms. With this tool risk management and optimal set point cal-culation can be handled together. In this paper we demonstrated the application of our framework in a optimal control problem, however it can also be applied in otheroptimization and risk management problems where uncertainties should be handledand the user can express the requirements by tuning the shapes of the loss function.

Acknowledgements This publication/research has been supported by the European Union and the Hun-garian Republic through the projects GOP-1.1.1-11-2011-0045 and TMOP-4.2.2.C-11/1/KONV-2012-0004National Research Center for Development and Market Introduction of Advanced Information andCommunication Technologies. The research of Janos Abonyi was realized in the frames of TMOP 4.2.4.A/2-11-1-2012-0001 National Excellence ProgramElaborating and operating an inland student and re-searcher personal support system. The project was subsidized by the European Union and co-nanced bythe European Social Fund.

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