Fuzzy Model-Based Predictive Hybrid Control of Polymerization

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Ind. Eng. Chem. Res. 2009, 48, 8542–8550

Fuzzy Model-Based Predictive Hybrid Control of Polymerization Processes Na´dson M. Nascimento Lima,*,† Flavio Manenti,‡ Rubens Maciel Filho,† Marcelo Embiruçu,§ and Maria R. Wolf Maciel† Department of Chemical Processes, Faculty of Chemical Engineering, State UniVersity of Campinas (UNICAMP), P.O. Box 6066, 13081-970, Campinas, Saõ Paulo, Brazil, CMIC Department “Giulio Natta”, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy, and Department of Chemical Engineering, Polytechnique Institute, Federal UniVersity of Bahia (UFBA), Federaçaõ, 40210-630, SalVador, Bahia, Brazil

This paper presents a fuzzy model-based predictive hybrid controller (FMPHC) for polymerization processes based on Takagi-Sugeno models and moving horizon methodology. Such processes are characterized by strongly nonlinear behaviors, which may either require significant effort to tune model-based controllers or render them ineffective. The proposed FMPHC is a promising integrated approach to handle nonlinearities and control issues. An industrial copolymerization process of ethylene and 1-butene is adopted to validate the proposed approach and to compare it to the most widespread advanced multivariable control. 1. Introduction Polymerization processes enable the production of many commodities of relevant industrial interest and high added value. Given their flexibility, such processes exhibit complex nonlinear behaviors and strong interactions among variables which can easily lead to instabilities if not adequately controlled. In addition, their nonlinearities make the development of a satisfactory deterministic model to accurately reproduce main phenomena that characterize the system particularly challenging. Difficulties are especially related to the resulting complexity and/or large scale of the differential-algebraic equations (DAE) system to be integrated numerically. It is also worth mentioning that the on-line feasibility of predictive control and optimization procedures unavoidably depends on the iterative (and faster than the real-time) integration of DAE systems, by requiring high performing numerical solvers. An approach to face this problem is to develop reduced models, which require less computational time even though they cannot adequately represent the system in the whole operating domain. Such a limitation affects even the design of a reliable and robust control system as already discussed in the literature. Basically, two of the most widespread approaches in chemical process modeling and control are the linear model predictive control1 and the artificial intelligence in terms of fuzzy systems. Rather than using these techniques separately a combination can be used with fuzzy models as a support in the controller design and as an internal model in a moving horizon predictive control structure as well, by obtaining a nonlinear predictive control. The resulting approach allows opportunely representing the process on the overall operating domain and with different types of data (e.g., even operators’ information can be included). Compared to existing methods, this approach is more effective to handle modeling and control issues that involve processes with nonlinear behaviors and complex dynamics (i.e., polymer production plants). Moreover, it allows developing specific * 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.

models that account for both uncertainty concepts and probabilistic logic by giving the approach some more importance. According to Sala et al.,2 the current research on novel modeling and control methods is based on the application of fuzzy systems, and many authors have discussed the benefits and importance of these systems in process control. Alexandridis et al.3 introduced a systematic methodology based on fuzzy systems to face the problem of nonlinear system identification. Habbi et al.4 proposed a nonlinear, dynamic, fuzzy model for describing the natural circulation in a drum-boiler-turbine system. Abdelazin and Malik5 used the fuzzy models to approximate real continuous functions with a selected accuracy by using Takagi-Sugeno’s fuzzy models for the real-time identification of nonlinear systems and to predict the system output, to mention a few. It is also important to highlight that fuzzy logic may significantly simplify integration and implementation of specific algorithms as well as reduce the computational time required to model and simulate complex systems. At the same time, it is worthwhile emphasizing that many conventional control algorithms may be inadequate to meet highspec qualities that are required by market for an increasing number of industrial processes. This is common even for polymerization processes, which involve evaluation of specific physico-chemical properties (i.e., molecular weight distribution and average molecular weight) to characterize macromolecular dynamics and processability. An efficient approach is the model-based predictive control (MPC), where the dynamic model is directly implemented in the control system. According to Campello et al.,6 most of the main advantages of the MPC are in its ability and easiness to introduce lower and upper bounds on process and control variables. In particular, for this reason, many literature works have focused attention on such an optimal control technique. To mention a few, Schnelle and Rollins7 adopted the MPC to control a continuous polymerization process prototype, Santos et al.8 implemented an online nonlinear MPC to regulate liquid level and temperature in a CSTR pilot plant, Park and Rhee9 applied an extended Kalman filter based on a nonlinear MPC to run a semibatch copolymerization reactor, Ramaswamy et al.10 used the MPC methodology to make stable a CST bioreactor, and Manenti and Rovaglio11 implemented a nonlinear

10.1021/ie900352d CCC: $40.75  2009 American Chemical Society Published on Web 08/17/2009

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

MPC for optimal grade transitions of an overall polyethylene terephthalate plant. Most of the MPC applications are based on the dynamic matrix control (DMC) strategy, which is particularly simple to design and implement. Dougherty and Cooper12 asserted that this family of controllers is nowadays the reference structure for advanced multivariable control in chemical process industry, and they introduce a multiple model adaptive control strategy for multivariable DMC. Guiamba and Mulholland13 developed and implemented an adaptive linear dynamic matrix control (ALDMC) algorithm to control an integrated pump-tank system consisting of two input and two output variables. Haeri and Beik14 proposed an extended nonlinear DMC algorithm able to handle constrained multivariable systems. An interesting approach is to integrate features of DMC structure to facilitate the characterization of nonlinearities in designing controllers by using fuzzy logic. On this subject, Mollov et al.15 discussed a predictive controller for nonlinear processes based on the Takagi-Sugeno fuzzy model. In addition, the same authors described the feasibility of this approach and tested its closed-loop stability on a laboratory plant. Mendonça et al.16 successfully introduced a generalized framework of fuzzy predictive filters for multivariable processes, making the nonconvex optimization problems encountered in nonlinear MPC applications feasible. This paper proposes a fuzzy model-based predictive hybrid controller (FMPHC) in which a fuzzy model is introduced into the MPC structure as internal model so as to take into account both process restrictions and nonlinearities. The copolymerization 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 little information, and they are imbued with a high degree of uncertainty. In 1965, Zadeh17 introduced his seminal idea in a continuous-valued logic named fuzzy set theory to deal with these concepts.18 Zadeh’s work had a radical influence on the way to approach the uncertainty issue, since it challenged not only probability theory as the sole representation for uncertainty but also the foundations of probability theory (see also classical binary or two-valued logic19). What Zadeh did was introduce flexibility into the processing of doubtful numerical data by not requiring exact and rigid answers as in probability theory and included qualitative information within the analysis methodology. A fuzzy set contains elements with different degrees of membership in this set (that is, degrees of membership ) (0: 1] ) {x ∈ R/0 e x e 1}), and a crisp (classical) set contains elements that would not be members unless their membership was complete in the same set (that is, degrees of membership ) 1). Thus, crisp sets are special cases of fuzzy sets when the membership is complete, without any ambiguity in their membership. Elements of a fuzzy set can be even members of other fuzzy sets on the same universe since their membership does not need to be complete for any set (possibility of overlapping among fuzzy regions). All the pieces of information of a fuzzy set are described by membership functions. The algorithm developed in this work incorporate Gaussian membership functions µ(x) for the inputs x20,21 given by eq 1

[ ( )]

µ(xi) ) exp -

1 xi - c i 2 σi

2

(1)

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Figure 1. Typical Gaussian membership function. Table 1. Main t-norms and t-conorms type

Zadeh

probabilistic

Lukasiewicz

t-norm

min (x1, x2)

x1 · x2

max (x1 + x2 -1, 0)

t-conorm

max (x1, x2)

x1 + x2 - x1 · x2

min (x1 + x2, 1)

Weber

{ {

x1, ifx2 ) 1 x2, ifx1 ) 1 0, ifnot

x1, if x2 ) 0 x2, if x1 ) 0 1, if not

where xi is the ith input variable, ci is the ith center of the membership function, and σi is the standard deviation of the ith membership function. Figure 1 illustrates a typical Gaussian membership function and their parameters. 2.1. Fuzzy Set Operations. Operations of union, intersection, and complement are standard fuzzy operations. They are defined as for crisp sets, when the range of membership values is limited to the unit interval. For each standard operation, there is a broad class of functions whose members can be considered fuzzy generalizations of the standard operations. In such a case, fuzzy intersections and fuzzy unions are usually denoted by t-norms and t-conorms (or s-norms), respectively. t-norms and t-conorms are so named because they were first introduced as triangular norms and triangular conorms, respectively, in study of statistical metric spaces.18 Common t-norms and t-conorms for two fuzzy sets X1 and X2 with elements x1 and x2, respectively, are shown in Table 1. Probabilistic t-norm and t-conorm shall be applied in the calculation of the inferred exit condition of the fuzzy rule base for the process considered in this work.20,21 2.2. Fuzzy Modeling. Fuzzy models are based on a set of rules with the aim to represent at the best the process understanding. Fuzzy modeling consists of the sequential implementation of three essential steps: fuzzification, inference, and defuzzification. Using membership functions, the fuzzification process converts numerical inputs into fuzzy sets, ready to be used by the fuzzy system.22 By doing so, some problematic issues may arise, especially because the fuzzy variable is characterized by inaccuracy, ambiguity, and/or vagueness. In accordance with Lima et al.,21 these concepts allow incorporating both operator and process information in the model. Moreover, different process operating conditions can be even accounted for in the fuzzy model by opportunely modify kinetic parameters, heat and mass transfer properties, transport factors, and so on. The nature of the expression governing the inference mechanism is

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IF premise(antecedent) THEN conclusion(consequent) (2) Such an IF-THEN rule type is generally referred to as the deductive form where each rule represents a cause-effect relationship with a specific corresponding action, according to the 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 the fuzzy sets. 3. Takagi-Sugeno Fuzzy Model The Takagi-Sugeno fuzzy model is a special case of functional fuzzy models. Its structure was proposed by Takagi and Sugeno.23 In this approach, the fuzzy model replaces the consequent fuzzy sets in a fuzzy rule by a linear equation of the input variables. Thus, a fuzzy model can be regarded as a collection of several linear models locally applied in the fuzzy regions, which are defined by the rule premises and where the overall model of the system consists of the interpolation of these linear models. Therefore, it has a suitable dynamic structure and some well-established theories on linear systems can be considered and easily applied for the theoretical analysis and design of the closed-loop system.24 The Takagi-Sugeno model for fuzzy rules generation from a given input-output data set (specifically, two inputs x1 and x2 and one output y) can assume the following general form _ 1 and x2 is X _ 2 THEN y is y ) f(x1, x2) IF x1 is X

(3)

where X1 and X2 are fuzzy sets (membership functions) of x1 and x2, respectively, and y ) f(x1,x2) is a crisp consequent function. Generalizing expression 3 to the n-input case, the following form of the Takagi-Sugeno model takes place _ i,1) and (x2 is X _ i,2) and ... and IF (x1 is X (xj is X _ i,j) and ... and (xn is X _ i,n) THEN yi ) ai1 · x1 + ai2 · x2 + ... + aij · xj + ... + ain · xn (4) where i ) 1, ..., rules number; j ) 1, ..., n; and aij are parameters of the consequent function of the fuzzy model. Given the input, the output of the fuzzy model is inferred through the ith weighed mean output of each rule R

∑ f · µ (x) i

y)

i

i)1 R

(5)

∑ µ (x) i

i)1

where µi(x) are membership functions, fi is a consequent function to each rule i, and R is the total amount of rules of the fuzzy model. 4. Identification of Functional Fuzzy Models Before starting the development of a hybrid controller through the integration of fuzzy logic concepts into the predictive structure of DMC, it is of primary importance to generate the functional fuzzy model. Then, the fuzzy model shall be used as an internal model of the DMC by replacing its typical convolution (prevision) model. As a consequence, the basic

structure of the DMC controller shall be deeply modified by obtaining a novel FMPHC technique. An interesting characteristic of the proposed controller is the ability to meet nonlinearities. This is possible because the fuzzy model starts representing the process linearly through membership functions defined for the initial operating region; nevertheless, as the process operating conditions change, the same operating point may move from the original region to the other regions governed by other linear membership functions. By doing so, in the overlapped regions, membership functions governing the process behavior are the combination of the rules of adjacent operating regions, making possible the handling of the nonlinear nature of the system. 4.1. Developing Functional Fuzzy Models. Important decisions on the quality of models must be taken in the initial phases of 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, and their interconnection form. Numbers and types of the selected variables should be in accordance with the problem requirements. For the sake of simplicity, let us consider as a goal the development of a fuzzy model for a SISO (single-input, singleoutput) control system in order to represent only the relationship between a controlled variable and a manipulated variable. The fuzzy model shall be able to represent the dynamic process behavior on a predefined time horizon. Generally, fuzzy models use the last information in their structure. Thus, the number of data for each variable used to develop the model is an important parameter of performance of the prediction model. Really, it can be seen as an optimization topic that must be considered in the same model development. The next step is the data generation for the model identification. Maximum and minimum variation limits of the variable have to be defined for determining the operating range of the model. First, the training data set is generated and the model parameters are evaluated. This model is then validated through the test data. Data generation is carried out through a random disturbance on the input variable.20,21 It must be observed that data generation of training and test is carried out by using disturbances with different frequency and width. Another important point in the development of dynamic fuzzy models is the determination of the sampling rate, in accordance with the characteristic time of the process; alternatively, when the model is adopted for controlling the process, as per this work, the value of the sampling rate is related to the controller action interval. 4.2. Generation of Functional Fuzzy Models. As already described, the functional fuzzy model developed here is a Takagi-Sugeno type with the structure of eq 4; the initial stage in developing the model deals with process identification, given an adequate data set, then, the fuzzification stage (membership functions), the inferentiation (through t-norms and t-conorms), and the evaluation of the inferred numerical output by means of the weighed mean (eq 5). However, the dimension of the model is unknown a priori: number of rules, number and parameter values of membership functions associated to each variable (expected values and standard deviations, since Gaussian membership functions are adopted), and parameters of the consequent functions of the rules. In order to minimize model dimensions, the following methods can be adopted according to the problem as follows. (1) Subtractive clustering method: it determines the number of rules and the parameters of the membership functions. (2) Gradient method: the quality of the fuzzy model can be


improved by modifying input parameters. The gradient method operates on the input data. (3) Learning from example (LFE): it develops the rules only. It trusts the complete specification of the membership functions. (4) Least-squares algorithm: it needs the number of rules and the membership functions to calculate the parameters of the consequent functions. (5) Modified learning from example (MLFE): contrary to LFE, MLFE calculates rules and parameters of the membership functions. In accordance with the aim of this work, the functional fuzzy model shall be generated through the combination of the subtractive clustering method with the least-squares algorithm. Details on subtractive clustering and least-squares methods are given by Chiu25,26 and Passino and Yurkovich,22 respectively. 4.3. Validation of Functional Fuzzy Models. The mean square error is adopted for validating the model

error )

m

∑ (y¯k - y )

2

k

k)1

(6)

m

where k is the time interval, m is the number of discrete time intervals, jyk is the output predicted by the fuzzy model at the kth time interval, and yk is the output of the process at the kth time interval (deterministic model).

The dynamic matrix control, DMC, was developed at Shell Oil Company in 1979. The basic idea is to use a time-domain step-response model (called the convolution model) of the process to calculate the future changes in the manipulated variable that minimize some performance indexes. In the DMC approach, one would have the future output responses matching some optimal trajectories in the PH (prediction horizon) by finding the best set of values of manipulated variables in the CH (control horizon). This is exactly the concept of a leastsquares problem of fitting PH data points with CH coefficients. The aim of predictive control is to drive future outputs close to the reference trajectory. The computation sequence is to calculate at first the reference trajectory and estimate the output predictions by using the convolution model. Then, the errors between predicted and reference trajectories are calculated.27 The next step is to estimate the sequence of the future controls by minimizing an appropriate quadratic objective function J. However, only the first control action is implemented. Such a procedure is iterated by means of a moving horizon methodology. Objective function J is defined as follows PH

J)

∑ (y i)1

CH

d i

2 2 - ypred CL,i) + f

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

actual is the vector of the current measured value of the where yi-1 set is the vector of controlled variable at sampling time i - 1, yi-1 set-point of the controlled variable at sampling time i - 1, and R is the reference trajectory parameter with 0 e R e 1. pred in eq 7 can be directly obtained from a Predicted values yCL,i process model. However, since the model is approximated, the controller is not robust enough and the following correction is applied

actual ypred CL,i ) yCL,i + (yi-1 - yCL,i-1)

5. Dynamic Matrix Control

∑ [(∆u ) ]

future 2

k

(7)

k)1

where i and k are the time interval, y is the output variable (controlled variable), u is the input variable (manipulated variable), ∆uk ) uk - uk-1, and f 2 is the suppression factor for changes of the manipulated variable. In the original DMC strategy the term ydi is the set point; to prevent severe control actions, a term based on the model algorithmic control strategy28 is introduced here. The desired output is calculated through an optimal trajectory defined by a first-class filter actual set + (1 - R) · yi-1 ydi ) R · yi-1

(8)

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(9)

where yCL,i is defined by the convolution model. It is assumed that the difference between predicted and actual values of the previous sampling time is still valid for the current time interval. Thus, the system reaches the desired value for successive corrections of the shunting line. Luyben29 discussed details on DMC and convolution models. 6. Fuzzy Model-Based Predictive Hybrid Controller In the FMPHC, the functional fuzzy model replaces the convolution model of the original configuration of DMC. The fuzzy model operates in a predictive way in the moving horizon control structure, and the control action is obtained by minimizing an objective function corresponding to eq 7, except for the term yCL,i of eq 9, which is calculated through the Takagi-Sugeno fuzzy model. In general, the fuzzy model makes predictions for the output variable in correspondence with the last and current signals of inputs and with the last signals of output. The hybrid control scheme is schematically shown in Figure 2. In Figure 2, it is observed that the DMC model-based structure receives the prediction data from the fuzzy model as well as the information on the reference trajectory yd. As a result, the hybrid controller minimizes the objective function (eq 7) and provides the control signal u to the process, as well as to the fuzzy model for a new prediction. This loop is realized until the output variable to achieve the desired value. The controller tuning is performed through the integral of the absolute value of the error (IAE) IAE )

∫

tf

t0

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

(10)

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

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Figure 3. Basic process configuration.30 Table 2. Nominal Operating Conditions normalized values main feed, PFR1 total flow % mass ethylene ethylene/1-butene mass ratio hydrogen concentration (CH2) temperature

9.306 2.48 1.8 1.250 0.17

secondary feed, zone 4 of NONIDEAL CSTR (near the top of the CSTR) total flow % mass ethylene ethylene/1-butene mass ratio hydrogen concentration temperature

1.034 0.28 0.2 0 0.33

catalyst feed, zone 1 of NONIDEAL CSTR (bottom of the CSTR) total flow catalyst concentration cocatalyst/catalyst mass ratio catalyst impurities temperature

0.001 1.42 1.36 0 1.50

system parameters input pressure stirrer rotation

1.25 0.80

outputs conversion polymer production rate weight-average molecular weight number-average molecular weight (Mn) reactor temperature

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

19.2576 2.196 6.5012 2.5746 1.68

7. Case Study: Copolymerization of Ethylene with 1-Butene The selected case study is the copolymerization of ethylene with 1-butene in Ziegler-Natta catalyst solution. Reactions take place in a series of continuous stirred tank and tubular reactors.30,31 The feed flow rate is assumed to be a mixture of ethylene, 1-butene, cyclohexane (as solvent), hydrogen, and a mixture of Ziegler-Natta catalysts and cocatalysts. Catalysts and cocatalysts are both preactivated before feeding reactor vessels, and hence, it may be assumed that catalysts are fed in their active form. Feed flow rates with different compositions may be inserted into different process locations by making flexible the feed policies. 7.1. Process Configuration. The system consists of two tubular reactors (PFR1 and PFR2) and a nonideal continuous stirred tank reactor (NONIDEAL CSTR), and no cooling devices are used (adiabatic operations). The basic system configuration is shown in Figure 3. There are different operational configurations that may be adopted since every reactor vessel is equipped with injection lines for all the chemical species, even though all the components are usually fed into the first reactor of the

Figure 5. Dynamic trend of the average molecular weight (Mn) by perturbing hydrogen concentration (CH2) within the main feed ( 40%.

series (which may be either PFR1 or NONIDEAL CSTR). On the other hand, hydrogen is even injected along the process in order to modify the polymer grade. PFR2 is used as a trimmer for increasing the polymer yield by reducing the amounts of monomer residual at the output stream. Moreover, according to the process requirements, the stirred reactor may become unstirred in order to convert it into a tubular reactor with large diameter. By doing so, the whole process may consist of either a series of tubular reactors or a continuous stirred tank reactor or some other mixed configurations. Switching the process operational configuration, significant changes in the molecular weight distribution (MWD) of the final polymer may be obtained, 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 one catalyst feed point are used. Lateral feed points are used to improve the mixing degree in the stirred tank reactor, which is controlled by the stirrer rotational speed and the lateral feed flow rate as well. The process consists of a series of one nonideal stirred tank and one tubular reactor, and it is used to produce polymer grades with narrower MWDs. Configuration 2: tubular mode. Monomers and catalysts are injected into PFR1, and hydrogen is injected along the reactor series to control the MWD. The CSTR is not operating as usual; in fact, the stirrer is turned off and the process is converted


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Figure 6. Identification data. Table 4. Controller Parameters for the Regulatory Problem

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 2 3 4 5 6 7 8

48.50 73.71 91.04 25.65 10.84 60.20 39.86 39.86

32.32 19.52 9.68 45.02 77.96 25.76 17.31 42.99

20.53 20.53 20.53 20.53 20.53 20.53 20.53 20.53

19.63 19.63 19.63 19.63 19.63 19.63 19.63 19.63

consequent part rule i

ai1 × 102 u(k), u(k - 1)

ai2 × 102 w(k)

bi1 × 102 u(k), u(k - 1)

i)1 i)2 i)3 i)4 i)5 i)6 i)7 i)8

–54.22 –2.98 –0.77 1.38 –12.54 32.54 –0.57 163.01

6.46 7.29 –0.41 21.08 –15.68 –10.09 –5.53 78.22

–22.74 44.07 83.97 150.41 100.59 126.22 126.72 –66.59

w(k)

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

Figure 7. Validation of the fuzzy model.

parameters

DMC

FMPHC

convolution horizon prediction horizon (PH) control horizon (CH) suppression factor (f) reference trajectory parameter (R) sampling time (normalized value) IAE (normalized value)

25 4 1 1.0 0.5 0.5 0.041

7 1 1.0 0.5 0.5 0.008

The stirred mode only (configuration 1) is analyzed hereinafter. The CSTR is divided into 5 zones, each one considered as an ideal stirred tank. There are two feed streams: one for PFR1, and the other one at an intermediate point of the CSTR. Catalyst and cocatalyst are added at the bottom of the CSTR. Relevant process variables for monitoring the product quality are the conversion, polymer production rate, average molecular weights, and reactor temperature. Relevant inputs are the system feed rate, the lateral feed stream fraction of the monomer and hydrogen concentrations, the catalyst and cocatalyst concentrations, the feed temperature, the stirrer speed rotation, and the inlet pressure. The catalysts may contain some impurities, which are typical process disturbances. A nonlinear mathematical model is considered as plant for data generation and identification of the fuzzy model. Details on the deterministic mathematical model and kinetic mechanism are given elsewhere.30,31 Nominal steady-state operating conditions adopted to simulate the polymer plant and to design the control loops are illustrated in Table 2 (the numerical values are in a normalized form as the process real data cannot be published for industrial confidentiality reasons; the normalized value for each variable was defined from standard values). 7.2. Process Simulation. The mathematical model of the process comprises a relatively large set of partial-differential algebraic equations, which must be solved simultaneously. The partial-differential equations that constitute the tubular reactor balance equations were discretized along the flow direction by using the standard method of characteristics. By doing so, the resulting set of differential and algebraic equations was solved. The integration of the discretized model was numerically performed by means of a DASSL (Differential Algebraic System Solver) routine, which uses backward differentiation formula for discretizing and integrating the model. DASSL updates the integration step automatically, depending on the stiffness of the local integration properties of the set of equations. The resulting

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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 prediction horizon (PH) control horizon (CH) suppression factor (f) reference trajectory parameter (R) sampling time (normalized value) IAE (normalized value)

25 9 1 1.0 0.5 0.5 6.672

7 1 1.0 0.5 0.5 7.681

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 the average molecular weight (Mn), and the manipulated variable is the hydrogen concentration in the main feed (CH2). Figure 4 shows the dynamic evolution of the average molecular weight in an open-loop response for a feed temperature disturbance equal to -33% occurring at the normalized time value of 30. The system response begins approximately in correspondence with time 32, as a consequence of the inherent process dynamics delay. Figure 5 presents the dynamic evolution of the average molecular weight for step disturbances of (40% in the hydrogen concentration within the main feed in order to highlight process nonlinearities. 7.3. Dynamic Fuzzy Modeling. An algorithm for functional dynamic fuzzy modeling, which considers SISO systems, is developed by using subtractive clustering and least-squares methods, and it is introduced into the open-loop scenario. A sampling rate equal to 0.5 and a simulation interval equal to

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

120 are adopted. Three inputs are considered for the model: the manipulated variable at the kth sampling times, the manipulated 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 and the servo mechanism controls. Figure 6 shows the training (model generation) and test (validation) data for input and output variables. Table 3 shows parameters for the fuzzy model. In Table 3, k refers to time instant, u refers to CH2, w and yi refer to Mn, where i ) 1, ..., R, with R being the amount of rules of the fuzzy model, and ain and bih are consequent functions parameters of the fuzzy model 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 (membership functions) for input and output variables is 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) 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 trajectory equal to 2.5746. It is worth noting that a very small error is obtained. 7.4. Performance of the FMPHC. The FMPHC algorithm developed in Fortran 90 is introduced in the simulation program


in order to check performances and benefits of the hybrid controller by comparing it to the DMC algorithm on both the regulatory and servo mechanism problems. The regulatory problem is set up by implementing a feed temperature step disturbance equal to -33%. Table 4 shows the parameters used for DMC and FMPHC controllers. In addition, it reports the control errors. Figure 8 presents a graphical analysis of the dynamic trend of the average molecular weight Mn and the manipulated variable CH2. A comparison between the controllers and the open-loop system behavior is also given. Table 4 and Figure 8 clearly show improvements deriving from the FMPHC approach against DMC, having a smaller IAE value, a quicker time response, and a smaller overshoot. Also, Figure 8 shows that FMPHC has a more stable dynamic behavior for the manipulated variable CH2. A servo problem is given by implementing a step-wise setpoint trajectory on the controlled variable, and Figure 9 shows the dynamic behavior of average molecular weight Mn and manipulated variable CH2. According to these results and the values of IAE of Table 5, both controllers present a similar and satisfactory performance for 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 model for simulating process open-loop responses was formulated and validated on a real industrial data set. The proposed controller was also compared to DMC for both regulatory and servo problems. In general, FMPHC presents a satisfactory performance by showing more stable responses against DMC, especially in the regulatory case. It is even important to know that the main advantage of the FMPHC is that it does not need any deterministic mathematical model but only input-output process information. Moreover, FMPHC can easily handle process nonlinearities with a relatively small computational effort for its algebraic fuzzy rules which it is based on. Thus, the use of internal fuzzy dynamic models in the moving horizon predictive control structure seems to be a promising and appealing approach, and it presents interesting potentialities in the development of advanced control strategies, especially for processes with strongly nonlinear behaviors such as polymerization plants. Acknowledgment This work was supported by FAPESP (Fundaçaõ de Amparo a` Pesquisa do Estado de Saõ Paulo) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico). Notation a, b ) parameters of the consequent function of the fuzzy model c ) center of the Gaussian membership function C ) normalized concentration CH ) control horizon CSTR ) continuous stirred tank reactor DMC ) dynamic matrix controller f ) suppression factor FMPHC ) fuzzy model-based predictive hybrid controller IAE ) integral of the absolute value of the error k ) discretized time interval (normalized value) m ) number of discrete time intervals

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M ) normalized molecular weight MPC ) model predictive control MWD ) molecular weight distribution PFR ) plug flow reactor PH ) prediction horizon R ) total amount of fuzzy model rules u ) process inputs, fuzzy model inputs (manipulated variables) w ) fuzzy model inputs (controlled variables) W ) fuzzy sets of process outputs x ) fuzzy model inputs X ) fuzzy sets of process inputs y ) process outputs jy ) predicted response Greek Letters R ) reference trajectory parameter µ ) Gaussian membership function σ ) standard deviation of Gaussian membership function Subscripts CL ) closed-loop f ) final time H2 ) hydrogen within the main feed i ) fuzzy model rule index j ) fuzzy model input index k ) discretized time interval index n ) average polymer property index 0 ) initial time Superscripts actual ) actual value d ) desired output value future ) future value pred ) predicted value set ) set-point

Literature Cited (1) Qin, S. J.; Badgwell, T. A. A Survey of Industrial Model Predictive Control Technology. Control Eng. Pract. 2008, 11, 733. (2) Sala, A.; Guerra, T. M.; Babuska, R. Perspectives of Fuzzy Systems and Control. Fuzzy Sets Syst. 2005, 156, 432. (3) Alexandridis, A. P.; Siettos, C. I.; Sarimveis, H. K.; Boudouvis, A. G.; Bafas, G. V. Modelling of Nonlinear Process Dynamics Using Kohonen’s Neural Networks, Fuzzy Systems and Chebyshev Series. Comput. Chem. Eng. 2002, 26, 479. (4) Habbi, H.; Zelmat, M.; Bouamama, B. O. A Dynamic Fuzzy Model for a Drum-Boiler-Turbine System. Automatica 2003, 39, 1213. (5) Abdelazim, T.; Malik, O. P. Identification of Nonlinear Systems by Takagi-Sugeno Logic Grey Box Modeling for Real-Time Control. Control Eng. Pract. 2005, 13, 1489. (6) Campello, R. J. G. B.; Von Zuben, F. J.; Amaral, W. C.; Meleiro, L. A. C.; Maciel Filho, R. Hierarchical Fuzzy Models within the Framework of Orthonormal Basis Functions and their Application to Bioprocess Control. Chem. Eng. Sci. 2003, 58, 4259. (7) Schnelle, P. D.; Rollins, D. L. Industrial Model Predictive Control Technology as Applied to Continuous Polymerization Processes. ISA Trans. 1998, 36, 281. (8) Santos, L. O.; Afonso, P. A. F. N. A.; Castro, J. A. A. M.; Oliveira, N. M. C.; Biegler, L. T. On-line Implementation of Nonlinear MPC: an Experimental Case Study. Control Eng. Pract. 2001, 9, 847. (9) Park, M.; Rhee, H. Property Evaluation and Control in a Semibatch MMA/MA Solution Copolymerization Reactor. Chem. Eng. Sci. 2003, 58, 603. (10) Ramaswamy, S.; Cutright, T. J.; Qammar, H. K. Control of a Continuous Bioreactor Using Model Predictive Control. Process Biochem. 2005, 40, 2763. (11) Manenti, F.; Rovaglio, M. Integrated Multilevel Optimization in Large-scale Poly(ethylene Terephthalate) Plants. Ind. Eng. Chem. Res. 2008, 47, 92.

8550


(12) Dougherty, D.; Cooper, D. A Practical Multiple Model Adaptive Strategy for Multivariable Model Predictive Control. Control Eng. Pract. 2003, 11, 649. (13) Guiamba, I. R. F.; Mulholland, M. Adaptive Linear Dynamic Matrix Control Applied to an Integrating Process. Comput. Chem. Eng. 2004, 28, 2621. (14) Haeri, M.; Beik, H. Z. Application of Extended DMC for Nonlinear MIMO Systems. Comput. Chem. Eng. 2005, 29, 1867. (15) Mollov, S.; Boom, T.; Cuesta, F.; Ollero, A.; Babuska, R. Robust Stability Constraints for Fuzzy Model Predictive Control. IEEE Trans. Fuzzy Sys. 2002, 10, 50. (16) Mendonça, L. F.; Sousa, J. M.; Sa´ da Costa, J. M. G. Optimization Problems in Multivariable Fuzzy Predictive Control. Int. J. Approximate Reasoning 2004, 36, 199. (17) Zadeh, L. Fuzzy Sets. Inf. Control 1965, 8, 338. (18) Ross, T. J. Fuzzy Logic with Engineering Applications; John Wiley & Sons Ltd.: Chichester, 2004. (19) Klir, G.; Yuan, B. Fuzzy Sets and Logic Fuzzy: Theory and Applications; Prentice Hall: Upper Saddle River, 1995. (20) Lima, N. M. N.; Zun˜iga Linãn, L.; Maciel Filho, R.; Wolf Maciel, M. R.; Medina, L. C.; Embiruçu, M. Development of a Cognitive Approach to a Molecular Distillation Process of Heavy Liquid Petroleum Residue. In European Symposium on Computer Aided Process Engineering, ComputerAided Chemical Engineering 26; Proceedings of the 19th ESCAPE, Cracow, June 14-17, 2009; Jezowski, J., Thullie, J., Eds.; Elsevier Science: Amsterdam, The Netherlands, 2009; pp 1435-1444. (21) Lima, N. M. N.; Maciel Filho, R.; Embiruçu, M.; Wolf Maciel, M. R. A Cognitive Approach to Develop Dynamic Models: Application to Polymerization Systems. J. Appl. Polym. Sci. 2007, 106, 981. (22) Passino, K. M.; Yurkovich, S. Fuzzy Control; Addison-WesleyLongman: Menlo Park, 1998.

(23) Takagi, T.; Sugeno, M. Fuzzy Identification of Systems and its Applications to Modeling and Control. IEEE Trans. Syst. Man. Cybern. 1985, 15, 116. (24) Chen, B.; Liu, X. Reliable Control Design of Fuzzy Dynamic Systems with Time-Varying Delay. Fuzzy Sets Syst. 2003, 1. (25) Chiu, S. A Cluster Estimation Method with Extension to Fuzzy Model Identification. IEEE 1994, 1240. (26) Chiu, S. Method and Software for Extracting Fuzzy Classification Rules by Subtractive Clustering. IEEE 1996, 461. (27) Rodrigues, J. A. D.; Maciel Filho, R. Production Optimization with Operating Constraints for a Fed-Batch Reactor with DMC Predictive Control. Chem. Eng. Sci. 1999, 54, 2745. (28) Vasco de Toledo, E. C.; Rodrigues, J. A.; Maciel Filho, R. A Tuned Approach of the Predictive-Adaptive GPC Controller Applied to a FedBatch Bioreactor Using Complete Factorial Design. Comput. Chem. Eng. 2002, 26, 1493. (29) Luyben, W. L. Process Modeling, Simulation, and Control for Chemical Engineers; McGraw-Hill: New York, 1989. (30) Embiruçu, M.; Lima, E. L.; Pinto, J. C. Continuous Soluble ZieglerNatta Ethylene Polymerizations in Reactor Trains. I. Mathematical Modeling. J. Appl. Polym. Sci. 2000, 77, 1574. (31) Embiruçu, M.; Pontes, K. V.; Lima, E. L.; Pinto, J. C. Continuous Soluble Ziegler-Natta Ethylene Polymerizations in Reactor Trains. III. Influence of Operation Conditions upon Process Performance Macromolecular Reaction Engineering. Macromol. React. Eng. 2008, 2, 161.

ReceiVed for reView March 3, 2009 ReVised manuscript receiVed July 17, 2009 Accepted July 31, 2009 IE900352D

Fuzzy Model-Based Predictive Hybrid Control of Polymerization

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