Fuzzy Model-Based Predictive Hybrid Control of Polymerization Processes

Nadson M. Nascimento Lima,*,† Flavio Manenti,‡ Rubens Maciel Filho,† Marcelo Embirucu,§ andMaria R. Wolf Maciel†

Department of Chemical Processes, Faculty of Chemical Engineering, State UniVersity of Campinas(UNICAMP), P.O. Box 6066, 13081-970, Campinas, Sao Paulo, Brazil, CMIC Department “Giulio Natta”,Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy, and Department of ChemicalEngineering, Polytechnique Institute, Federal UniVersity of Bahia (UFBA), Federacao,40210-630, SalVador, Bahia, Brazil

This paper presents a fuzzy model-based predictive hybrid controller (FMPHC) for polymerization processesbased on Takagi-Sugeno models and moving horizon methodology. Such processes are characterized bystrongly nonlinear behaviors, which may either require significant effort to tune model-based controllers orrender them ineffective. The proposed FMPHC is a promising integrated approach to handle nonlinearitiesand control issues. An industrial copolymerization process of ethylene and 1-butene is adopted to validatethe proposed approach and to compare it to the most widespread advanced multivariable control.

1. Introduction

Polymerization processes enable the production of manycommodities of relevant industrial interest and high added value.Given their flexibility, such processes exhibit complex nonlinearbehaviors and strong interactions among variables which caneasily lead to instabilities if not adequately controlled. Inaddition, their nonlinearities make the development of asatisfactory deterministic model to accurately reproduce mainphenomena that characterize the system particularly challenging.Difficulties are especially related to the resulting complexityand/or large scale of the differential-algebraic equations (DAE)system to be integrated numerically. It is also worth mentioningthat the on-line feasibility of predictive control and optimizationprocedures unavoidably depends on the iterative (and faster thanthe real-time) integration of DAE systems, by requiring highperforming numerical solvers.

An approach to face this problem is to develop reducedmodels, which require less computational time even though theycannot adequately represent the system in the whole operatingdomain. Such a limitation affects even the design of a reliableand robust control system as already discussed in the literature.

Basically, two of the most widespread approaches in chemicalprocess modeling and control are the linear model predictivecontrol1 and the artificial intelligence in terms of fuzzy systems.Rather than using these techniques separately a combinationcan be used with fuzzy models as a support in the controllerdesign and as an internal model in a moving horizon predictivecontrol structure as well, by obtaining a nonlinear predictivecontrol.

The resulting approach allows opportunely representing theprocess on the overall operating domain and with different typesof data (e.g., even operators’ information can be included).Compared to existing methods, this approach is more effectiveto handle modeling and control issues that involve processeswith nonlinear behaviors and complex dynamics (i.e., polymerproduction plants). Moreover, it allows developing specific

models that account for both uncertainty concepts and proba-bilistic logic by giving the approach some more importance.

According to Sala et al.,2 the current research on novelmodeling and control methods is based on the application offuzzy systems, and many authors have discussed the benefitsand importance of these systems in process control. Alexandridiset al.3 introduced a systematic methodology based on fuzzysystems to face the problem of nonlinear system identification.Habbi et al.4 proposed a nonlinear, dynamic, fuzzy model fordescribing the natural circulation in a drum-boiler-turbinesystem. Abdelazin and Malik5 used the fuzzy models toapproximate real continuous functions with a selected accuracyby using Takagi-Sugeno’s fuzzy models for the real-timeidentification of nonlinear systems and to predict the systemoutput, to mention a few.

It is also important to highlight that fuzzy logic maysignificantly simplify integration and implementation of specificalgorithms as well as reduce the computational time requiredto model and simulate complex systems.

At the same time, it is worthwhile emphasizing that manyconventional control algorithms may be inadequate to meet high-spec qualities that are required by market for an increasingnumber of industrial processes. This is common even forpolymerization processes, which involve evaluation of specificphysico-chemical properties (i.e., molecular weight distributionand average molecular weight) to characterize macromoleculardynamics and processability.

An efficient approach is the model-based predictive control(MPC), where the dynamic model is directly implemented inthe control system. According to Campello et al.,6 most of themain advantages of the MPC are in its ability and easiness tointroduce lower and upper bounds on process and controlvariables. In particular, for this reason, many literature workshave focused attention on such an optimal control technique.To mention a few, Schnelle and Rollins7 adopted the MPC tocontrol a continuous polymerization process prototype, Santoset al.8 implemented an online nonlinear MPC to regulate liquidlevel and temperature in a CSTR pilot plant, Park and Rhee9

applied an extended Kalman filter based on a nonlinear MPCto run a semibatch copolymerization reactor, Ramaswamy etal.10 used the MPC methodology to make stable a CSTbioreactor, and Manenti and Rovaglio11 implemented a nonlinear

* To whom correspondence should be addressed. Tel.: 55 +19 + 3521-3971. Fax: 55 +19 + 3521-3965. E-mail: nadson@feq.unicamp.br.

† State University of Campinas.‡ Politecnico di Milano.§ Federal University of Bahia.

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10.1021/ie900352d CCC: $40.75 2009 American Chemical SocietyPublished on Web 08/17/2009

MPC for optimal grade transitions of an overall polyethyleneterephthalate plant.

Most of the MPC applications are based on the dynamicmatrix control (DMC) strategy, which is particularly simple todesign and implement. Dougherty and Cooper12 asserted thatthis family of controllers is nowadays the reference structurefor advanced multivariable control in chemical process industry,and they introduce a multiple model adaptive control strategyfor multivariable DMC. Guiamba and Mulholland13 developedand implemented an adaptive linear dynamic matrix control(ALDMC) algorithm to control an integrated pump-tank systemconsisting of two input and two output variables. Haeri andBeik14 proposed an extended nonlinear DMC algorithm able tohandle constrained multivariable systems.

An interesting approach is to integrate features of DMCstructure to facilitate the characterization of nonlinearities indesigning controllers by using fuzzy logic. On this subject,Mollov et al.15 discussed a predictive controller for nonlinearprocesses based on the Takagi-Sugeno fuzzy model. Inaddition, the same authors described the feasibility of thisapproach and tested its closed-loop stability on a laboratoryplant. Mendonca et al.16 successfully introduced a generalizedframework of fuzzy predictive filters for multivariable processes,making the nonconvex optimization problems encountered innonlinear MPC applications feasible.

This paper proposes a fuzzy model-based predictive hybridcontroller (FMPHC) in which a fuzzy model is introduced intothe MPC structure as internal model so as to take into accountboth process restrictions and nonlinearities. The copolymeriza-tion of ethylene with 1-butene is considered as a case study.

2. Fuzzy Systems To Model Complex Processes

Many engineering problems are characterized by very littleinformation, and they are imbued with a high degree ofuncertainty. In 1965, Zadeh17 introduced his seminal idea ina continuous-valued logic named fuzzy set theory to dealwith these concepts.18 Zadeh’s work had a radical influenceon the way to approach the uncertainty issue, since itchallenged not only probability theory as the sole representa-tion for uncertainty but also the foundations of probabilitytheory (see also classical binary or two-valued logic19). WhatZadeh did was introduce flexibility into the processing ofdoubtful numerical data by not requiring exact and rigidanswers as in probability theory and included qualitativeinformation within the analysis methodology.

A fuzzy set contains elements with different degrees ofmembership in this set (that is, degrees of membership ) (0:1] ) {x ∈ R/0 e x e 1}), and a crisp (classical) set containselements that would not be members unless their membershipwas complete in the same set (that is, degrees of membership) 1). Thus, crisp sets are special cases of fuzzy sets when themembership is complete, without any ambiguity in theirmembership. Elements of a fuzzy set can be even members ofother fuzzy sets on the same universe since their membershipdoes not need to be complete for any set (possibility ofoverlapping among fuzzy regions).

All the pieces of information of a fuzzy set are described bymembership functions. The algorithm developed in this workincorporate Gaussian membership functions µ(x) for the inputsx20,21 given by eq 1

where xi is the ith input variable, ci is the ith center of themembership function, and σi is the standard deviation of theith membership function. Figure 1 illustrates a typical Gaussianmembership function and their parameters.

2.1. Fuzzy Set Operations. Operations of union, intersection,and complement are standard fuzzy operations. They are definedas for crisp sets, when the range of membership values is limitedto the unit interval. For each standard operation, there is a broadclass of functions whose members can be considered fuzzygeneralizations of the standard operations. In such a case, fuzzyintersections and fuzzy unions are usually denoted by t-normsand t-conorms (or s-norms), respectively. t-norms and t-conormsare so named because they were first introduced as triangularnorms and triangular conorms, respectively, in study of statisticalmetric spaces.18 Common t-norms and t-conorms for two fuzzysets X1 and X2 with elements x1 and x2, respectively, are shownin Table 1. Probabilistic t-norm and t-conorm shall be appliedin the calculation of the inferred exit condition of the fuzzyrule base for the process considered in this work.20,21

2.2. Fuzzy Modeling. Fuzzy models are based on a set ofrules with the aim to represent at the best the processunderstanding. Fuzzy modeling consists of the sequentialimplementation of three essential steps: fuzzification, inference,and defuzzification.

Using membership functions, the fuzzification process con-verts numerical inputs into fuzzy sets, ready to be used by thefuzzy system.22 By doing so, some problematic issues may arise,especially because the fuzzy variable is characterized byinaccuracy, ambiguity, and/or vagueness. In accordance withLima et al.,21 these concepts allow incorporating both operatorand process information in the model. Moreover, differentprocess operating conditions can be even accounted for in thefuzzy model by opportunely modify kinetic parameters, heatand mass transfer properties, transport factors, and so on.

The nature of the expression governing the inference mech-anism is

µ(xi) ) exp[-12(xi - ci

σi)2] (1)

Figure 1. Typical Gaussian membership function.

Table 1. Main t-norms and t-conorms

type Zadeh probabilistic Lukasiewicz Weber

t-norm min(x1, x2)

x1 · x2 max(x1 + x2 -1, 0) {x1, ifx2 ) 1

x2, ifx1 ) 1

0, ifnot

t-conorm max(x1, x2)

x1 + x2 - x1 · x2 min(x1 + x2, 1) {x1, if x2 ) 0

x2, if x1 ) 0

1, if not

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Such an IF-THEN rule type is generally referred to as thedeductive form where each rule represents a cause-effectrelationship with a specific corresponding action, according tothe assigned operating condition.

Lastly, defuzzification converts fuzzy results into crisp results.This procedure allows selecting a crisp single-valued quantity(or a crisp set) among all the possible alternatives within thefuzzy sets.

3. Takagi-Sugeno Fuzzy Model

The Takagi-Sugeno fuzzy model is a special case offunctional fuzzy models. Its structure was proposed by Takagiand Sugeno.23 In this approach, the fuzzy model replaces theconsequent fuzzy sets in a fuzzy rule by a linear equation ofthe input variables. Thus, a fuzzy model can be regarded as acollection of several linear models locally applied in the fuzzyregions, which are defined by the rule premises and where theoverall model of the system consists of the interpolation of theselinear models. Therefore, it has a suitable dynamic structureand some well-established theories on linear systems canbe considered and easily applied for the theoretical analysis anddesign of the closed-loop system.24

The Takagi-Sugeno model for fuzzy rules generation froma given input-output data set (specifically, two inputs x1 andx2 and one output y) can assume the following general form

where X1 and X2 are fuzzy sets (membership functions) of x1

and x2, respectively, and y ) f(x1,x2) is a crisp consequentfunction. Generalizing expression 3 to the n-input case, thefollowing form of the Takagi-Sugeno model takes place

where i ) 1, ..., rules number; j ) 1, ..., n; and aij are parametersof the consequent function of the fuzzy model.

Given the input, the output of the fuzzy model is inferredthrough the ith weighed mean output of each rule

where µi(x) are membership functions, fi is a consequent functionto each rule i, and R is the total amount of rules of the fuzzymodel.

4. Identification of Functional Fuzzy Models

Before starting the development of a hybrid controller throughthe integration of fuzzy logic concepts into the predictivestructure of DMC, it is of primary importance to generate thefunctional fuzzy model. Then, the fuzzy model shall be usedas an internal model of the DMC by replacing its typicalconvolution (prevision) model. As a consequence, the basic

structure of the DMC controller shall be deeply modified byobtaining a novel FMPHC technique.

An interesting characteristic of the proposed controller is theability to meet nonlinearities. This is possible because the fuzzymodel starts representing the process linearly through member-ship functions defined for the initial operating region; neverthe-less, as the process operating conditions change, the sameoperating point may move from the original region to the otherregions governed by other linear membership functions. Bydoing so, in the overlapped regions, membership functionsgoverning the process behavior are the combination of the rulesof adjacent operating regions, making possible the handling ofthe nonlinear nature of the system.

4.1. Developing Functional Fuzzy Models. Important deci-sions on the quality of models must be taken in the initial phasesof the modeling procedure.20,21

First, it is necessary to define the structure of the fuzzy model,which includes the set of rules, the variables to be used, andtheir interconnection form. Numbers and types of the selectedvariables should be in accordance with the problem require-ments. For the sake of simplicity, let us consider as a goal thedevelopment of a fuzzy model for a SISO (single-input, single-output) control system in order to represent only the relationshipbetween a controlled variable and a manipulated variable. Thefuzzy model shall be able to represent the dynamic processbehavior on a predefined time horizon.

Generally, fuzzy models use the last information in theirstructure. Thus, the number of data for each variable used todevelop the model is an important parameter of performanceof the prediction model. Really, it can be seen as an optimizationtopic that must be considered in the same model development.

The next step is the data generation for the model identifica-tion. Maximum and minimum variation limits of the variablehave to be defined for determining the operating range of themodel. First, the training data set is generated and the modelparameters are evaluated. This model is then validated throughthe test data. Data generation is carried out through a randomdisturbance on the input variable.20,21 It must be observed thatdata generation of training and test is carried out by usingdisturbances with different frequency and width.

Another important point in the development of dynamic fuzzymodels is the determination of the sampling rate, in accordancewith the characteristic time of the process; alternatively, whenthe model is adopted for controlling the process, as per thiswork, the value of the sampling rate is related to the controlleraction interval.

4.2. Generation of Functional Fuzzy Models. As alreadydescribed, the functional fuzzy model developed here is aTakagi-Sugeno type with the structure of eq 4; the initial stagein developing the model deals with process identification, givenan adequate data set, then, the fuzzification stage (membershipfunctions), the inferentiation (through t-norms and t-conorms),and the evaluation of the inferred numerical output by meansof the weighed mean (eq 5).

However, the dimension of the model is unknown a priori:number of rules, number and parameter values of membershipfunctions associated to each variable (expected values andstandard deviations, since Gaussian membership functions areadopted), and parameters of the consequent functions of therules. In order to minimize model dimensions, the followingmethods can be adopted according to the problem as follows.(1) Subtractive clustering method: it determines the number ofrules and the parameters of the membership functions. (2)Gradient method: the quality of the fuzzy model can be

IF premise(antecedent) THEN conclusion(consequent)(2)

IF x1 is X_1 and x2 is X_2 THEN y is y ) f(x1, x2) (3)

IF (x1 is X_i,1) and (x2 is X_i,2) and ... and

(xj is X_i,j) and ... and (xn is X_i,n)THEN yi ) ai1 · x1 + ai2 · x2 + ... + aij · xj + ... + ain · xn

(4)

y )∑i)1

R

fi · µi(x)

∑i)1

R

µi(x)

(5)

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improved by modifying input parameters. The gradient methodoperates on the input data. (3) Learning from example (LFE):it develops the rules only. It trusts the complete specificationof the membership functions. (4) Least-squares algorithm: itneeds the number of rules and the membership functions tocalculate the parameters of the consequent functions. (5)Modified learning from example (MLFE): contrary to LFE,MLFE calculates rules and parameters of the membershipfunctions.

In accordance with the aim of this work, the functional fuzzymodel shall be generated through the combination of thesubtractive clustering method with the least-squares algorithm.Details on subtractive clustering and least-squares methods aregiven by Chiu25,26 and Passino and Yurkovich,22 respectively.

4.3. Validation of Functional Fuzzy Models. The meansquare error is adopted for validating the model

where k is the time interval, m is the number of discrete timeintervals, yjk is the output predicted by the fuzzy model at thekth time interval, and yk is the output of the process at the kthtime interval (deterministic model).

5. Dynamic Matrix Control

The dynamic matrix control, DMC, was developed at ShellOil Company in 1979. The basic idea is to use a time-domainstep-response model (called the convolution model) of theprocess to calculate the future changes in the manipulatedvariable that minimize some performance indexes. In the DMCapproach, one would have the future output responses matchingsome optimal trajectories in the PH (prediction horizon) byfinding the best set of values of manipulated variables in theCH (control horizon). This is exactly the concept of a least-squares problem of fitting PH data points with CH coefficients.

The aim of predictive control is to drive future outputs closeto the reference trajectory. The computation sequence is tocalculate at first the reference trajectory and estimate the outputpredictions by using the convolution model. Then, the errorsbetween predicted and reference trajectories are calculated.27

The next step is to estimate the sequence of the future controlsby minimizing an appropriate quadratic objective function J.However, only the first control action is implemented. Such aprocedure is iterated by means of a moving horizon methodol-ogy. Objective function J is defined as follows

where i and k are the time interval, y is the output variable(controlled variable), u is the input variable (manipulatedvariable), ∆uk ) uk - uk-1, and f 2 is the suppression factor forchanges of the manipulated variable.

In the original DMC strategy the term yid is the set point; to

prevent severe control actions, a term based on the modelalgorithmic control strategy28 is introduced here. The desiredoutput is calculated through an optimal trajectory defined by afirst-class filter

where yi-1actual is the vector of the current measured value of the

controlled variable at sampling time i - 1, yi-1set is the vector of

set-point of the controlled variable at sampling time i - 1, andR is the reference trajectory parameter with 0 e R e 1.

Predicted values yCL,ipred in eq 7 can be directly obtained from a

process model. However, since the model is approximated, thecontroller is not robust enough and the following correction isapplied

where yCL,i is defined by the convolution model. It is assumed thatthe difference between predicted and actual values of the previoussampling time is still valid for the current time interval. Thus, thesystem reaches the desired value for successive corrections of theshunting line. Luyben29 discussed details on DMC and convolutionmodels.

6. Fuzzy Model-Based Predictive Hybrid Controller

In the FMPHC, the functional fuzzy model replaces theconvolution model of the original configuration of DMC. Thefuzzy model operates in a predictive way in the moving horizoncontrol structure, and the control action is obtained by minimiz-ing an objective function corresponding to eq 7, except for theterm yCL,i of eq 9, which is calculated through the Takagi-Sugenofuzzy model.

In general, the fuzzy model makes predictions for the outputvariable in correspondence with the last and current signals ofinputs and with the last signals of output. The hybrid controlscheme is schematically shown in Figure 2. In Figure 2, it isobserved that the DMC model-based structure receives theprediction data from the fuzzy model as well as the informationon the reference trajectory yd. As a result, the hybrid controllerminimizes the objective function (eq 7) and provides the controlsignal u to the process, as well as to the fuzzy model for a newprediction. This loop is realized until the output variable toachieve the desired value.

The controller tuning is performed through the integral ofthe absolute value of the error (IAE)

where t0 and tf are the starting and the arrival times of theevaluation period.

error ) �∑k)1

m

(yk¯ - yk)

2

m(6)

J ) ∑i)1

PH

(yid - yCL,i

pred)2 + f 2 ∑k)1

CH

[(∆uk)future]2 (7)

yid ) R · yi-1

actual + (1 - R) · yi-1set (8)

Figure 2. Integrated structure of the predictive control and fuzzy model.

yCL,ipred ) yCL,i + (yi-1

actual - yCL,i-1) (9)

IAE ) ∫t0

tf|yset(t) - yactual(t)|dt (10)

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7. Case Study: Copolymerization of Ethylene with1-Butene

The selected case study is the copolymerization of ethylenewith 1-butene in Ziegler-Natta catalyst solution. Reactions takeplace in a series of continuous stirred tank and tubularreactors.30,31 The feed flow rate is assumed to be a mixture ofethylene, 1-butene, cyclohexane (as solvent), hydrogen, and amixture of Ziegler-Natta catalysts and cocatalysts. Catalystsand cocatalysts are both preactivated before feeding reactorvessels, and hence, it may be assumed that catalysts are fed intheir active form. Feed flow rates with different compositionsmay be inserted into different process locations by makingflexible the feed policies.

7.1. Process Configuration. The system consists of twotubular reactors (PFR1 and PFR2) and a nonideal continuousstirred tank reactor (NONIDEAL CSTR), and no cooling devicesare used (adiabatic operations). The basic system configurationis shown in Figure 3. There are different operational configura-tions that may be adopted since every reactor vessel is equippedwith injection lines for all the chemical species, even thoughall the components are usually fed into the first reactor of the

series (which may be either PFR1 or NONIDEAL CSTR). Onthe other hand, hydrogen is even injected along the process inorder to modify the polymer grade. PFR2 is used as a trimmerfor increasing the polymer yield by reducing the amounts ofmonomer residual at the output stream. Moreover, accordingto the process requirements, the stirred reactor may becomeunstirred in order to convert it into a tubular reactor with largediameter. By doing so, the whole process may consist of eithera series of tubular reactors or a continuous stirred tank reactoror some other mixed configurations. Switching the processoperational configuration, significant changes in the molecularweight distribution (MWD) of the final polymer may beobtained, allowing production of many resin grades.

The most common operational configurations are as follows.Configuration 1: stirred mode. CSTR is normally operating,

and PFR1 is not used. Two monomers feed points and onecatalyst feed point are used. Lateral feed points are used toimprove the mixing degree in the stirred tank reactor, which iscontrolled by the stirrer rotational speed and the lateral feedflow rate as well. The process consists of a series of one nonidealstirred tank and one tubular reactor, and it is used to producepolymer grades with narrower MWDs.

Configuration 2: tubular mode. Monomers and catalysts areinjected into PFR1, and hydrogen is injected along the reactorseries to control the MWD. The CSTR is not operating as usual;in fact, the stirrer is turned off and the process is converted

Figure 3. Basic process configuration.30

Table 2. Nominal Operating Conditions

normalized values

main feed, PFR1total flow 9.306% mass ethylene 2.48ethylene/1-butene mass ratio 1.8hydrogen concentration (CH2

) 1.250temperature 0.17

secondary feed, zone 4 of NONIDEALCSTR (near the top of the CSTR)

total flow 1.034% mass ethylene 0.28ethylene/1-butene mass ratio 0.2hydrogen concentration 0temperature 0.33

catalyst feed, zone 1 of NONIDEALCSTR (bottom of the CSTR)

total flow 0.001catalyst concentration 1.42cocatalyst/catalyst mass ratio 1.36catalyst impurities 0temperature 1.50

system parametersinput pressure 1.25stirrer rotation 0.80

outputsconversion 19.2576polymer production rate 2.196weight-average molecular weight 6.5012number-average molecular weight (Mn) 2.5746reactor temperature 1.68

Figure 4. Open-loop simulation response to temperature disturbance.

Figure 5. Dynamic trend of the average molecular weight (Mn) by perturbinghydrogen concentration (CH2

) within the main feed ( 40%.

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into a series of three tubular reactors. The proper control of thefeed temperature is crucial in this configuration in order to avoidpolymer precipitation inside the reactor. This operation modeis used to produce polymer grades with broader MWDs.

The stirred mode only (configuration 1) is analyzedhereinafter. The CSTR is divided into 5 zones, each oneconsidered as an ideal stirred tank. There are two feedstreams: one for PFR1, and the other one at an intermediatepoint of the CSTR. Catalyst and cocatalyst are added at thebottom of the CSTR.

Relevant process variables for monitoring the product qualityare the conversion, polymer production rate, average molecularweights, and reactor temperature. Relevant inputs are the systemfeed rate, the lateral feed stream fraction of the monomer andhydrogen concentrations, the catalyst and cocatalyst concentra-tions, the feed temperature, the stirrer speed rotation, and theinlet pressure. The catalysts may contain some impurities, whichare typical process disturbances.

A nonlinear mathematical model is considered as plant fordata generation and identification of the fuzzy model. Detailson the deterministic mathematical model and kinetic mechanismare given elsewhere.30,31

Nominal steady-state operating conditions adopted to simulatethe polymer plant and to design the control loops are illustratedin Table 2 (the numerical values are in a normalized form asthe process real data cannot be published for industrialconfidentiality reasons; the normalized value for each variablewas defined from standard values).

7.2. Process Simulation. The mathematical model of theprocess comprises a relatively large set of partial-differentialalgebraic equations, which must be solved simultaneously.The partial-differential equations that constitute the tubularreactor balance equations were discretized along the flowdirection by using the standard method of characteristics. Bydoing so, the resulting set of differential and algebraicequations was solved. The integration of the discretized modelwas numerically performed by means of a DASSL (Dif-ferential Algebraic System Solver) routine, which usesbackward differentiation formula for discretizing and inte-grating the model. DASSL updates the integration stepautomatically, depending on the stiffness of the localintegration properties of the set of equations. The resulting

Figure 6. Identification data.

Table 3. Parameters of the Fuzzy Model

rules number ) 8

antecedent part

ci × 102 σi × 102

rule i u(k), u(k - 1) w(k) u(k), u(k - 1) w(k)

1 48.50 32.32 20.53 19.632 73.71 19.52 20.53 19.633 91.04 9.68 20.53 19.634 25.65 45.02 20.53 19.635 10.84 77.96 20.53 19.636 60.20 25.76 20.53 19.637 39.86 17.31 20.53 19.638 39.86 42.99 20.53 19.63

consequent part

ai1 × 102 ai2 × 102 bi1 × 102

rule i u(k), u(k - 1) w(k) u(k), u(k - 1) w(k)

i ) 1 –54.22 6.46 –22.74i ) 2 –2.98 7.29 44.07i ) 3 –0.77 –0.41 83.97i ) 4 1.38 21.08 150.41i ) 5 –12.54 –15.68 100.59i ) 6 32.54 –10.09 126.22i ) 7 –0.57 –5.53 126.72i ) 8 163.01 78.22 –66.59

Figure 7. Validation of the fuzzy model.

Table 4. Controller Parameters for the Regulatory Problem

parameters DMC FMPHC

convolution horizon 25prediction horizon (PH) 4 7control horizon (CH) 1 1suppression factor (f) 1.0 1.0reference trajectory parameter (R) 0.5 0.5sampling time (normalized value) 0.5 0.5IAE (normalized value) 0.041 0.008

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software (open-loop software) is fully written in Fortran 90,and it is tested and validated on a real industrial database.31

The output variable analyzed as the control objective is theaverage molecular weight (Mn), and the manipulated variableis the hydrogen concentration in the main feed (CH2

). Figure 4shows the dynamic evolution of the average molecular weightin an open-loop response for a feed temperature disturbanceequal to -33% occurring at the normalized time value of 30.The system response begins approximately in correspondencewith time 32, as a consequence of the inherent process dynamicsdelay.

Figure 5 presents the dynamic evolution of the averagemolecular weight for step disturbances of (40% in the hydrogenconcentration within the main feed in order to highlight processnonlinearities.

7.3. Dynamic Fuzzy Modeling. An algorithm for functionaldynamic fuzzy modeling, which considers SISO systems, isdeveloped by using subtractive clustering and least-squaresmethods, and it is introduced into the open-loop scenario. Asampling rate equal to 0.5 and a simulation interval equal to

120 are adopted. Three inputs are considered for the model:the manipulated variable at the kth sampling times, themanipulated variable at the (k - 1)th sampling time (nu ) 2),and the controlled variable at the (k - 1)th sampling time (ny) 1). Such a model is then adopted for both the regulatory andthe servo mechanism controls. Figure 6 shows the training(model generation) and test (validation) data for input and outputvariables.

Table 3 shows parameters for the fuzzy model. In Table 3, krefers to time instant, u refers to CH2

, w and yi refer to Mn, wherei ) 1, ..., R, with R being the amount of rules of the fuzzy model,and ain and bih are consequent functions parameters of the fuzzymodel for the variables CH2

and Mn, respectively, where n )1, ..., nu and h ) 1, ..., ny).

The ith rule where Xin and Wih are fuzzy sets (membershipfunctions) for input and output variables is

Results of fuzzy model validation are illustrated in Figure 7,which shows a very good prediction for the output variable,since fuzzy and deterministic models are practically overlapped.The mean square error (calculated by eq 6) is equal to 0.005,which corresponds to 0.18% of a reference set-point trajectoryequal to 2.5746. It is worth noting that a very small error isobtained.

7.4. Performance of the FMPHC. The FMPHC algorithmdeveloped in Fortran 90 is introduced in the simulation program

Figure 8. Closed-loop and open-loop simulations for a feed temperature step disturbance equal to -33%.

Table 5. Controller Parameters for the Servo Problem

parameters DMC FMPHC

convolution horizon 25prediction horizon (PH) 9 7control horizon (CH) 1 1suppression factor (f) 1.0 1.0reference trajectory parameter (R) 0.5 0.5sampling time (normalized value) 0.5 0.5IAE (normalized value) 6.672 7.681

Figure 9. Closed-loop response to the piece-wise set-point trajectory.

IF (u(k) is X_i,1) and (u(k - 1) is X_i,2) and (w(k - 1) is W_ i,1)THEN yi(k + 1) ) ai1 · u(k) + ai2 · u(k - 1) + bi1 · w(k - 1)

(11)

8548 Ind. Eng. Chem. Res., Vol. 48, No. 18, 2009

in order to check performances and benefits of the hybridcontroller by comparing it to the DMC algorithm on both theregulatory and servo mechanism problems.

The regulatory problem is set up by implementing a feedtemperature step disturbance equal to -33%. Table 4 showsthe parameters used for DMC and FMPHC controllers. Inaddition, it reports the control errors. Figure 8 presents agraphical analysis of the dynamic trend of the average molecularweight Mn and the manipulated variable CH2

. A comparisonbetween the controllers and the open-loop system behavior isalso given.

Table 4 and Figure 8 clearly show improvements derivingfrom the FMPHC approach against DMC, having a smaller IAEvalue, a quicker time response, and a smaller overshoot. Also,Figure 8 shows that FMPHC has a more stable dynamicbehavior for the manipulated variable CH2

.A servo problem is given by implementing a step-wise set-

point trajectory on the controlled variable, and Figure 9 showsthe dynamic behavior of average molecular weight Mn andmanipulated variable CH2

.According to these results and the values of IAE of Table 5,

both controllers present a similar and satisfactory performancefor set-point changes in the average molecular weight Mn.

8. Conclusions

A fuzzy model-based predictive hybrid controller (FMPHC)was developed and applied to a copolymerization process,which is intrinsically strongly nonlinear. A specific modelfor simulating process open-loop responses was formulatedand validated on a real industrial data set. The proposedcontroller was also compared to DMC for both regulatoryand servo problems. In general, FMPHC presents a satisfac-tory performance by showing more stable responses againstDMC, especially in the regulatory case. It is even importantto know that the main advantage of the FMPHC is that itdoes not need any deterministic mathematical model but onlyinput-output process information. Moreover, FMPHC caneasily handle process nonlinearities with a relatively smallcomputational effort for its algebraic fuzzy rules which it isbased on. Thus, the use of internal fuzzy dynamic models inthe moving horizon predictive control structure seems to bea promising and appealing approach, and it presents interest-ing potentialities in the development of advanced controlstrategies, especially for processes with strongly nonlinearbehaviors such as polymerization plants.

Acknowledgment

This work was supported by FAPESP (Fundacao de Amparoa Pesquisa do Estado de Sao Paulo) and CNPq (ConselhoNacional de Desenvolvimento Científico e Tecnológico).

Notation

a, b ) parameters of the consequent function of the fuzzy modelc ) center of the Gaussian membership functionC ) normalized concentrationCH ) control horizonCSTR ) continuous stirred tank reactorDMC ) dynamic matrix controllerf ) suppression factorFMPHC ) fuzzy model-based predictive hybrid controllerIAE ) integral of the absolute value of the errork ) discretized time interval (normalized value)m ) number of discrete time intervals

M ) normalized molecular weightMPC ) model predictive controlMWD ) molecular weight distributionPFR ) plug flow reactorPH ) prediction horizonR ) total amount of fuzzy model rulesu ) process inputs, fuzzy model inputs (manipulated variables)w ) fuzzy model inputs (controlled variables)W ) fuzzy sets of process outputsx ) fuzzy model inputsX ) fuzzy sets of process inputsy ) process outputsyj ) predicted response

Greek Letters

R ) reference trajectory parameterµ ) Gaussian membership functionσ ) standard deviation of Gaussian membership function

Subscripts

CL ) closed-loopf ) final timeH2 ) hydrogen within the main feedi ) fuzzy model rule indexj ) fuzzy model input indexk ) discretized time interval indexn ) average polymer property index0 ) initial time

Superscripts

actual ) actual valued ) desired output valuefuture ) future valuepred ) predicted valueset ) set-point

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ReceiVed for reView March 3, 2009ReVised manuscript receiVed July 17, 2009

Accepted July 31, 2009

IE900352D

8550 Ind. Eng. Chem. Res., Vol. 48, No. 18, 2009