Fuzzy Control of a Nylon Polymerization Semi-batch Reactor

8/11/2019 Fuzzy Control of a Nylon Polymerization Semi-batch Reactor

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Fuzzy Sets and Systems 160 (2009) 537 553

www.elsevier.com/locate/fss

Fuzzy control of a nylon polymerization semi-batch reactor

C. Wakabayashi, M. Embiruu, C. Fontes, R. Kalid

Escola Politcnica, Universidade Federal da Bahia, Rua Aristides Novis, 2 Federao, CEP-40210-630 Salvador, BA, Brazil

Received 26 April 2007; received in revised form 26 August 2008; accepted 27 August 2008

Available online 5 September 2008

Abstract

Batch and semi-batch polymerization reactors with specified trajectories for certain process variables present challenging control

problems. This work reports, results and procedures related to the application of PI (proportional and integral) fuzzy control in a

semi-batch reactor for the production of nylon 6. Closed loop simulation results were based on a phenomenological model adjusted

for a commercial reactor and they attest to the potential benefits and versatility of the use of PI fuzzy control in polymerization

systems.

2008 Elsevier B.V. All rights reserved.

Keywords: Nylon polymerization; Fuzzy control; Optimal tuning

1. Introduction

Fuzzy logic has become one of the most successful technologies for developing sophisticated process control strate-

gies. The most important feature of fuzzy control is that it does not require an internal mathematical model. Control

is achieved with expertise from human reasoning which can be expressed through a set of heuristic rules which are

quantified according to fuzzy set theory. In this sense fuzzy control algorithms present many advantages[1] such as

simplicity, robustness, no need to find transfer functions, nonlinear behavior and adaptability.

Despite the fact that fuzzy control has been used in several production processes since the 1970s[13,12]and that the

fuzzy systems are universal function approximations[14],few applications have been found in polymerization systems

[1,2,6]and none of these have been related to the production of nylon, considering the control challenges and intrinsic

features existing in this polymerization system.

Batch and semi-batch polymerization are typical of some processes such as PVC (poly vinyl chloride) and nylon

production. The difficulty of measuring controlled variables, the existence of interactions, dead time and constraints,

added to the nonlinear and multivariable nature of the system, pose challenging problems for the control of polymeriza-

tion reactors[15].For the batch and semi-batch reactors in particular, additional difficulties arise concerning process

variables, such as reactor temperature and pressure, which have to follow set-point trajectories to assure the quality of

the final product[2].In some cases, control is achieved using an open loop control approach[19].

Corresponding author. Tel.: +55 71 32839505; fax: +55 71 32839801.

E-mail addresses:[email protected](C. Wakabayashi),[email protected](M. Embiruu),[email protected](C. Fontes),

[email protected](R. Kalid).

0165-0114/$ - see front matter 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.fss.2008.08.009
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538 C. Wakabayashi et al. / Fuzzy Sets and Systems 160 (2009) 537 553

Nomenclature

A jacket area (m2)

[Ci] concentration of -caprolactam (1) and cyclic dimmer (2) in liquid phase (molkg-1 mixture)

C(i) controller output signal

Cvp,mi x specific heat of liquid reaction mixture (kJ kg1 K1)Cvp,w specific heat of pure water in vapor phase (kJ kg

1 K1)

c control action variationE error error error variationF mass of liquid in reactor at timet(kg)

Hi enthalpy change forith reaction (J mol1)

Ki equilibrium reaction constants forith reaction

K gain

ki rate constant forith reaction (kgmol1 h1)

k time instant

[Mv] concentration of caprolactam in vapor phase (molm

3)[Nv] concentration of nitrogen in vapor phase (molm3)

P total pressure (kPa)

Rvm rate of evaporation of caprolactam (molh1)

Rvw rate of evaporation of water (molh1)

RvT total rate of evaporation (mol h1)

Si concentration of linear oligomers in liquid (molkg1)

S1 amino-caproic acid concentration in liquid (molkg1)

T temperature (K)

Tj jacket output temperature (K)

Tr reference temperature ( = 473.15K)

Tset set-point temperature (K)Vg volume of vapor space (m3)

VT Rate of vapor escape from reactor (molh1)

U overall he at transfer coefficient (kJ h1 m2 K1)

W water concentration in liquid (mol h1)

[Wv] concentration of water in vapor (molm3)

Subscript

C proportional action

I integral action

Superscript

reverse reactions

Greek letters

i moments of theSndistribution

m latent heat of vaporization of monomer (kJ mol1)

w latent heat of vaporization of water (kJ mol1)

dimensionless time constant

Antunes et al.[2]present the design and analysis of a fuzzy control algorithm for temperature control in a Methyl

MethAcrylate (MMA) batchpolymerization reactor. The authors used a PID (proportional, integral and derivative) fuzzy

controller taking only the reaction temperature error and the error change as input linguistic variables. The tuning was

carried out altering the output scaling factors and the results show that the PID fuzzy was suitable and outperformed the


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C. Wakabayashi et al. / Fuzzy Sets and Systems 160 (2009) 537 553 539

PID velocity algorithm. In this system the major source of nonlinearity is due to its autocatalytic nature and additional

challenges for the control problem include the increase in viscosity together with reactor wall fouling.

Hanai et al.[10], working with another batch polymerization process (polybutadiene production), propose an iden-

tification procedure using a fuzzy neural network structure to estimate the physicochemical characteristics and the

conversion through the initial conditions of the batch.

This work shows the results and procedures associated with the application of a fuzzy control strategy in a semi-batch reactor for the production of nylon 6, including variable set-points for pressure and temperature. A complete

phenomenological model[8,9,16,18]represents the process. The operational conditions and the design variables were

selected according to an industrial unit installed in Brazil and some model parameters were adjusted to resemble the

operational conditions.

This paper is organized as follows. Section 2 presents the process description, the case studies analyzed and the

control objectives. Section 3 and Appendix A present aspects related to the phenomenological model including the

kinetic mechanism, the reaction rate equations of the caprolactam (monomer) and amino-caproic acid together with

the moment balances and the macro-scale equations (mass and energy balances). The control problem description, the

results and all aspects involved in the control design are detailed in Section 4.

2. Process description

The process comprises a semi-batch reactor for the production of nylon 6 (2.7 m3 volume), equipped with stirrer and

jacket, where hot oil or cold oil flow (not simultaneously) promotes heating or cooling of the reaction mass (Fig. 1).

One complete batch takes about 6h to complete. At the first stage -caprolactam and water are fed into the reactor.

The reaction is endothermic initially and the reaction mass has to be heated for 3h from the initial temperature (90 C)

up to 260 C. At this point, the-caprolactam rings open, resulting in amino-caproic acid formation. Over the following

3 h, the reactor temperature is maintained at 260 C and the polyaddition reaction (exothermic reaction) of the amino-

caproic acid takes place. As a result the reaction mass becomes overheated causing liquid desorption from the free

surface and desorption through bubble formation resulting in water vaporization and also vaporization of a small amount

of caprolactam. Subsequently the initial pressure of 2atm is increased for 3h to reach the required 5atm pressure and

this is maintained for a further hour. After this a pressure reduction procedure starts up to reach atmospheric pressure.

A vacuum pump is then started and depressurization begins for 1h, reaching 420mmHg. The polymer is immediatelydischarged from the reactor, solidified with cool water and set out for drying and palletizing processes.

Vacuum pump

Condenser

Scrubber

Motor

Hot oil

Cold oil

Vapor output

Oil output

Fig. 1. Semi-batch reactor for the production of nylon 6 (heating and cooling systems).
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0 1 2 3 4 5 6

0

50

100

150

200

250

Temperature controller setpoint

Time (hours)

Temperature(C)

0 1 2 3 4 5 6

0

50

100

150

200

250

300

350

400

450

500

550

Pressure controller setpoint

Time (hours)

Pressure(kPa)

Fig. 2. Set-point trajectories for temperature and pressure.

The control problem dealt with in this work concerns temperature and pressure set-point tracking, as shown in

Fig. 2. In this sense, the control problem is typically a set-point tracking or servo problem. The reaching of thesespecified requirements is of great importance to assure the desired quality of the final product.

3. Kinetic and mathematic modeling

The equations of the phenomenological model adopted in this work are presented by Wajge et al.[18],Gupta et al.

[8], Gupta and Kumar[9]and Tai and Tagawa[16]. This model uses the following assumptions:

Heat transfer occurs from the hot oil inside the jacket to the reaction mass in the heating step and from the reaction

mass to the cold oil in the cooling step. Water and-caprolactam vaporization from the reaction mass.

Pressure build-up due to the heating process. Variable viscosity of the reaction medium during the batch and its effects on the mass and energy transfer coeffici-

ents. Mass and energy balances of the system.

The following kinetic model was considered:

Ring opening reaction of the -caprolactam (C1) monomer, with the amino-caproic acid (S1) formation (Wrefers to

water):

C1

+Wk1

k1=k1/K1

S1

(1)


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Polycondensation reaction (Sm,Snand Sm+nare polymer chains):

Sm +Snk2

k2=k2/K2

Sm+n +W m, n = 1, 2, 3,. . . (2)

Polyaddition reaction:

Sn +C1 k3k3=k3/K3

Sn+1 (3)

Oligomer ring opening:

Cm +Wk4,m

k4,m =k4,m/K4,m

Sm m =2,. . . , 5 (4)

Oligomer polyaddition:

Sn +Cmk5.m

k5,m =k5,m/K5,m

Sn+m (5)

The reaction rates, equilibrium constants, mass and energy balances are described in Appendix A.

4. The control problem

The difficulty of obtaining on-line measurements of the variables associated with the macromolecular structure

means that secondary variables, such as temperature and pressure, must be monitored and controlled during the whole

semi-batch processes to assure the quality of the final product. Considering the intrinsic transient nature of the process

analyzed, the control problem consists of tracking the temperature and pressure set-point trajectories established for

these variables. These trajectories are directly associated with each step of the process and their objectives. Hot oil

and cold oil are both available throughout the process and used separately for reactor heating or cooling procedures,

while temperature increases or decreases from its set-point value. This configuration offers better control, minimizing

the occurrence of overshoot at the beginning of the constant temperature step. The control system is performed using

two loops, one for temperature and another for pressure control. In the first case, a split range configuration with twocontrol valves is used (Fig. 3).

For each simulation, two control strategies were analyzed and compared: conventional PID and PI fuzzy control.

In the first, the tuning parameters were selected through an optimal tuning method that comprises the solution of an

0

100

0 100

control signal (%)

valvestemp

osit

ion(%)

0

100

cold oil

hot oil

reference

point

open

closed

heatingcooling

open

closed

Fig. 3. Split range configuration.


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optimization problem to minimize the deviations between the controlled variables and their set-points and the following

procedure was adopted:

Open loop simulation. Identification of a first order model plus dead time. Application of CohenCoon and ITAE (integral of time-weighted absolute error) tuning methods. Selection of the best results and use of these values as an initial estimate for the optimal tuning method.

Altinten et al. [1] also adopted an optimal tuning procedure for tuning a PID fuzzy controller using a genetic algorithm

technique in order to obtain the membership functions which give maximum performance. The system analyzed was

the temperature control of a batch jacketed reactor for the solution polymerization of styrene. Jantzen[11]proposed

a design strategy making use of the PID tuning methods to design the rules and membership functions of the fuzzy

controller.

4.1. PID fuzzy controller

While conventional control is based on the knowledge of differential equations that represent the dynamic behavior

of a system, fuzzy control uses heuristic rules to control the process and can be regarded as a nonmathematical controlalgorithm[21]. In some cases, heuristic rules can be established based on previous experience, practical rules and

strategies frequently used. The linguistic rules are processed simultaneously and each consequent rule contributes to

the final control action. The main feature of this approach is the need for good heuristic knowledge of the process or the

relationship among the manipulated and controlled variables so as to be able to establish a set of rules to achieve the

control objectives. An in-depth discussion of the techniques of fuzzy model design from measurable data is presented

by Babuka and Verbruggen[3]and highlights the possibility of obtaining control rules that can be validated by experts

and may be combined with their prior knowledge. In this sense, fuzzy systems can provide a good guess for the initial

system model[14].

Another fuzzy control method comprises modeling the system using a TakagiSugeno fuzzy model where the local

dynamics in different state-space regions are represented by linear models and the overall model is depicted as a fuzzy

interpolation of these linear models. The linear matrix inequality (LMI) method [5,17]based on a TakagiSugeno

fuzzy model together with the use of local linear models in the rule consequents can be adopted for the control design

in this case. It can be applied to cases where the output variables involved are not measurable[7],for systems with

state delay[20]or for systems with time-varying delay[4]and where stability conditions are also established. Despite

this, this LMI fuzzy control method is particularly difficult to implement in polymerization reactors because of the

complexity of modeling such systems, the high number of state equations involved, the nonlinearity and the existence

of non measurable states, among others.

PIDfuzzy control is a fuzzified version of the conventional PID controller and can be an efficientmethodto the control

of complex systems such as polymerization reactors. Despite its intrinsic relationship with conventional PID, the fuzzy

approach provides a nonlinear control law, a high number of tuning parameters, robustness features, the possibility of

a multivariable approach and, in general, requires less effort to implement than other fuzzy control methods.

In PIDfuzzycontrol the rulesthat define thecontrollers model aredirectly associated with thePID velocity algorithm.

This approach expresses the intrinsic relationships, established by the conventional PID, between the error and/or theerror variation with the corresponding control action. The existence of a pattern of rules is an advantage of this method.

From the PID digital controller equations, the following options for rule construction may be considered:

Proportional fuzzy controller. PI fuzzy controller. PID fuzzy controller.

The PI-fuzzy controller considers error (Ei) and error variation Eias inputs and the output is the variation in control

actionCi. This controller is directly associated with the conventional PI controller:

C(k) C(k 1)= KC[E(k) E(k 1)]+T KC

I E(k) (6)
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error-

++ Fuzzification DefuzzificationFuzzy inference z/(z-1) Control valve

1/z- error

Actual temperature

Setpoint Fuzzy Rule Base

Fig. 4. PI fuzzy controller diagram.

unfavourable favourable

unfavourable favourableTsp

T

T Negative

error area

Positive

error area

Negative

error variation

Positive

error variation

Negative

error variationPositive

error variation

Fig. 5. Configurations for the error and the error variation (temperature control).

or

C(k)= KC

E(k)+T KC

IE(k) (7)

PI fuzzy control hasthe following generic rule:If error=Ei and error variation = Ei, then control actionvariation =

Ci; whereEi,Ei and Ci are fuzzy sets. The PI-fuzzy control diagram for the reactor temperature is presented in

Fig. 4.

4.2. Fuzzy control design

The fuzzy inference system adopted in this work is based on the Mamdani model. The universe of discourse of

each variable is defined through simulations with the phenomenological model. Based on these universes of discourse,

membership functions are initially established, following a symmetric disposal (positive error and negative error).

Considering the temperature,Fig. 5presents all the possibilities that can appear for error and error variation accordingto the measurement of this controlled variable.

Table 1shows the rules for temperature fuzzy control, where the following membership functions are presented:

Zero error. Positive error. Negative error. Zero error variation. Positive error variation. Negative error variation. Great control action variation. Small control action variation.

Zero control action variation.


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Table 1

Rules for the fuzzy control of temperature

Rule Error Oil flow change Condition Error variation Consequent Control action variation

1 Iferroris zero cis zero

2 Iferroris positive Hot oil and erroris zero then cis small opening

3 Iferroris positive Hot oil and erroris positive then cis large opening4 Iferroris positive Hot oil and erroris negative then cis small opening

5 Iferroris negative Cold oil and erroris zero then cis small opening

6 Iferroris negative Cold oil and erroris positive then cis small opening

7 Iferroris negative Cold oil and erroris negative then cis large opening

Table 2

Rules for the fuzzy control of pressure



2 Iferroris positive Hot oil and erroris zero then cis large closing

3 Iferroris positive Hot oil and erroris positive then cis large closing

4 Iferroris positive Hot oil and erroris negative then cis large closing

5 Iferroris negative Cold oil and erroris zero then cis small opening

6 Iferroris negative Cold oil and erroris positive then cis small closing

7 Iferroris negative Cold oil and erroris negative then cis large opening

Considering the non-simultaneous availability of hot and cold oils a negative error situation implies opening the

cold oil valve (cooling), and a positive error implies opening the hot oil valve (heating). In each case the valve is

changed from its previous position. Both valves have the same size coefficient. Table 2shows similar rules for the

reactor pressure control.AnalyzingTable 1the following statements can be made concerning these rules:

Rule 1 implies that if error and error variation are zero, there is no control action for both hot and cold oil streams.

The fuzzy controller will behave as a purely proportional controller. The integral action does not exist because in

this situation the error is zero. The derivative action does not exist because the variation of the error is zero. In rule 2 the fuzzy controller will behave like a PI controller. This rule guarantees the existence of an action similar

to a conventional integral action as the existence of error will provide the variation in the control action. As well as proportional and integral actions, rule 3 introduces a derivative (anticipatory) action through the direction

of the error variation. Rule 4 is similar to rule 3. In this case negative error variation implies a favorable situation (Fig. 5) and a less

aggressive variation in the control action may be performed. Rules 57 (negative error) have analog interpretations relative to rules 24, respectively. The inclusion of additional rules considering the cases of zero error together with the negative or positive change

of the error would be equivalent to inserting an anticipatory action in the rule base which would actuate when the

controlled variable leaves its set-point. The impact of the inclusion of this feature in the control performance is not

meaningful for the system studied. Furthermore, the inclusion of these new rules would require the definition of two

new membership functions associated with the control action variation, namely, very slight opening and very

slight closing, in this case to provide a smoother action than in the case of negative or positive errors. The inclusion

of these new membership functions would mean the inclusion of new tuning parameters that would need adjusting.

Rule 3 imposes a great variation in control action and rule 2 imposes a slight one. This suggests that the fuzzy

controller has nonlinear gain. Although this fuzzy control strategy is based on the PI digital controller, the rules

presented inTables 1 and 2also provide a derivative feature for the PI fuzzy controller.
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closedBig closing Small openingSmall closing Big opening

-15 -4.5 -0.75 -0.375 0 0.375 0.75 4.5 15 (% opening)

-1 0.3 -0.05 -0.025 0 0.025 0.05 0.3 1 (normalized)

Negative error Zero error Positive error

-40 -4 0-0.4 0.4 4 40 (C)

-1 0.1 -0.01 0 0.01 0.1 1

Zero error variationPositive error variation

-0.01

-1 -0.5 -0.2 0 0.2 0.5 1 (normalized)

Negative error variation

(normalized)

-0.005 -0.002 0 0.002 0.005 0.01 (C)

Fig. 6. Membership functions of error, error variation and control action variation (temperature fuzzy control).

Fig. 6presents the initial membership functions considered for error, error variation and control action variation

together with the universes of discourse which have been normalized in the interval [1; +1]. In order to provide a

simple control structure with the desired performance the rule base established for the controller considered a minimum

number of membership functions for the error and error variation. Other functions such as small negative error or

small positive error variation could have been included, but a meaningful improvement was not confirmed in this

case. With regards to the control action variation, a minimum choice of membership functions was also applied. The

intermediary actions (slight closing and slight opening) were included due to their anticipatory effects. These

actions were taken or not according to the favorable or unfavorable situations presented in Fig. 5. The same scalefactors were considered for both heating and cooling actions. Despite the centers or positions of the membership

functions, the initial choice presented inFig. 6was heuristic and based on the universe of discourse of the variables.

In the case of the control action variation, a maximum variation of 15% was assumed for each valve (cold and hot

oil), and a maximum variation of 5% of this limit was initially assumed for the small opening and small closing

actions. Even given the intuitive feature associated with the initial selection of positions and formats of the member-

ship functions; this was effectively useful only as an initial estimate for the optimal tuning procedure that was done

afterwards.

Adopting the same procedure used in PID tuning, some membership function parameters were adjusted through the

solution of the following optimization problem:

min

[T(t) Tsp(t)]2

+[ P(t) Psp(t)]2

s.t. min< < max(8)

where T(t) and P(t) are the measured values of temperature and pressure along the batch (simulated through the

phenomenological model),Tsp(t) andPsp(t) are the respective set-point trajectories,ais the set of parameters adjusted

andaminand amaxare their lower and upper bounds, respectively.

Table 3 presents 12 parameters associated with temperature control considered in the optimization, their initial values

according toFig. 6and the optimal values obtained for each. The selection of these parameters (vertex of membership

functions) was based onTable 1which establishes the membership functions directly related to temperature control.

The feature of the control problem, given two input flows (hot and cold oil) selected according to the error sign,

also suggested the possibility of different optimal values associated with the same membership function in different

situations (cooling or heating).

Although not all the possible membership functions(Fig. 6)are used inTable 1(rule base for temperature control),

the inclusion of any other functions in the rule base have to consider a symmetric disposal. Therefore, if the small


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Table 3

Tuning parameters and optimal values

Parameter Initial (normalized,Fig. 6) Optimal (normalized)

Second vertex of positive error (heating) 0.1 0.0211

Second vertex of negative error (cooling) 0.1 0.1

Second vertex of positive error variation (heating) 0.5 0.7294Second vertex of positive error variation (cooling) 0.5 0.1747

Third vertex of small opening (cooling, cold oil flow) 0.05 0.299

Third vertex of small opening (heating, hot oil flow) 0.05 0.05

Second vertex of big opening (heating, hot oil flow) 0.3 0.5755

Second vertex of big opening (cooling, cold oil flow) 0.3 0.9

Third vertex of zero error variation (heating) 0.2 0.2935

Third vertex of zero error variation (cooling) 0.2 0.01

First vertex of big opening (heating, hot oil flow) 0.025 0.2

First vertex of big opening (cooling, cold oil flow) 0.025 0.1127

0 1 2 3 4 5 6

0

50

100

150

200

250

300

Time (hours)

Temperature(C)Valveopening(%)

Temperature

Setpoint

Hot oil valve opening

Cold oil valve opening

Fig. 7. Temperature and manipulated variables (conventional PID control).

closing function is inserted in the consequent of some rules, its parameters should have the same absolute values of

the small opening function, given a cooling or heating situation.

All the triangular membership functions were also maintained symmetrical after the optimal tuning

procedure.

4.3. Results

In conventional PID control, four tuning parameters are optimized: proportional gain, integral time and derivative

time for the temperature controller and proportional gain for the pressure controller.Fig. 7presents simulation results

of the temperature with conventional PID control. Figs. 8a and b present simulation results (controlled and manipulated

variables) of the temperature with two different PI-fuzzy controllers: without anticipatory action and with anticipatory

action. In the first case, the fuzzy inference system follows the rules presented inTable 1.In the second case, only

rule 4 was changed by inserting an anticipatory action during the reactor heating step according toTable 4.This slight

change provides better control performance with the elimination of overshoot.

In all simulation tests a sampling time of 1min was adopted for the control action of the whole batch. The total time

for the simulation of one batch with defined tuning parameters was about 1 hour (less than the batch time). The fuzzy

control block is extremely fast and the total computational time is almost entirely associated with the model (plant)

simulation. The tuning procedure therefore, given the closed loop simulation of the whole batch at each iteration, took

considerable computation. A Pentium 4 with 3.2 GHz was used.


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0 1 2 3 4 5 6

0

50

100

150

200

250

300

Time (hours)

Temperature(C)

Hot&ColdOilflow(m3/h)

Temperature

Setpoint

Hot oil flowCold oil flow

0 1 2 3 4 5 6

0

50

100

150

200

250

300

Time (hours)

Te

mperature(C)


Temperature

Setpoint

Hot oil flow

Cold oil flow

Fig. 8. (a) Temperature and manipulated variables (PI-fuzzy control) (without anticipatory action) and (b) Temperature and manipulated variables

(PI-fuzzy control) (with anticipatory action).

Table 4

Change in rule 4 (temperature control) (anticipatory action)

Rule Error Condition Error variation Consequent Control action variation

4 If erroris positive and erroris negative then cis small closing

Both the conventional PID and PI-fuzzy controllers performed well, as shown in Table 5.In all cases, ISE (integral

of square error) values for temperature control were high due to the fact that heating oil temperature was not sufficient

to supply energy to the reaction mass at the beginning of the process, based on the set-point established.

Overshoot was eliminated with PI-fuzzy controller with anticipatory action (Table 4)but an increase in the sum of

squared errors was also observed in this case. It is important to highlight that according to the polymerization system

features the existence of high overshoot can cause a drop in product quality with off spec polymer production.


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Table 5

ISE results of PID and fuzzy controllers for temperature (Figs. 7 and 8)

Control strategies Overshoot (C) Offset (C) ISE

PID 9.05 1.45 5.276109

Fuzzy without anticipatory action 7.77 0.03 3.65109

Fuzzy with anticipatory action 0.52 0.03 5.8109

Table 6

New rules for the fuzzy control of temperature



2 Iferroris positive Hot oil and erroris zero then cis zero

3 Iferroris positive Hot oil and error is positive then cis small opening

4 Iferroris positive Hot oil and erroris negative then cis small closure

5 Iferroris negative Cold oil and erroris zero then cis zero

6 Iferroris negative Cold oil and erroris positive then cis small closure7 Iferroris negative Cold oil and erroris negative then cis small opening

0 1 2 3 4 5 6

0

50

100

150

200

250

300

Time (hours)

Tem

perature(C)


TemperatureSetpointCold oil flowHot oil flow

Fig. 9. Temperature and manipulated variables (PI-fuzzy control) (rules according to the Table 6).

The offset values presented in this paper were computed by calculating the difference between the controlled variable

and its set-point at the end of the batch.Although the PI-fuzzy control for the temperature (rules described in Table 1) performs well for the controlled

variable, the same was not observed for the hot and cold oil flows (manipulated variables) where control actions exhibit

high frequency oscillations hampering control. In this case the fuzzy control does not behave well and is similar to

a relay-type control. One way to overcome this problem is the inclusion of explicit constraints for the control action

variations in the fine tuning of the process. In this case the optimal membership function parameters would not be

able to provide control action variations during the whole semi-batch process above a prescribed limit. In this work,

however, additional adjustments were only made to the original rules (Table 1)in an attempt to better tune this control

problem.Table 6presents the new rule-set adopted for the PI-fuzzy control where other membership functions (fuzzy

sets) associated with the output of rules 27 were changed. New rules 2 and 4 (positive error) and 5 and 6 insert an

anticipatory action in the fuzzy control. New rules 3 (positive error) and 7 (negative error) reduce the control action in

relation to the original case (Table 1). In the latter better results obtained with control action reduction may be related

to overdesign of the valve size coefficients (Cv) associated with both cold and hot oil flows. In all simulation tests the
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Table 7

ISE results of PID and fuzzy temperature controllers (Figs. 79)

Control strategies Overshoot (C) Offset (C) ISE

PID 9.05 1.45 5.276109

Fuzzy without anticipatory action 7.77 0.03 3.65109

Fuzzy with anticipatory action 0.52 0.03 5.8109Fuzzy (rules ofTable 6) 0.52 0.03 6.3109

0 1 2 3 4 5 6

0

100

200

300

400

500

600

Time (hours)

P

ressure(kPa)

Vapourflow(mol/h)/30

Pressure

Setpoint

Vapour Flow

Fig. 10. Pressure and manipulated variable (conventional PID control).

values of the valve size coefficient were established according to those of the actual industrial reactor analyzed in this

work.

Fig. 9 presents simulation results of the temperature (controlled and manipulated variables) using the PI-fuzzy controlwith the rules described inTable 6.A notable improvement was achieved in the profile of the manipulated variables

without a drop in performance associated to temperature. This can be verified inTable 7which presents the ISE result

related toFig. 9together with the results inTable 5.

Fig. 10presents simulation results of the pressure (controlled and manipulated variables) with conventional PID

control. Figs. 11a and b show simulation results of the pressure with PI-fuzzy control. The pressure controller performed

well and the anticipatory action included in the PI-fuzzy temperature controller did not cause any significant impact,

as illustrated inFig. 11a. Like temperature control, pressure did not follow the set-point in the whole initial ramp since

the vaporization rate was not sufficiently high during this heating step.

5. Conclusions

The application of a PI fuzzy controller in a nylon polymerization system using a phenomenological model with

parameters that resemble the operational conditions of a commercial reactor was presented. Not only the simulation

results but also the methodology and the definition and interpretation of the rules based on the features of a real system

were examined. Some parameters of the membership functions, inserted in the rule base, were adjusted based on an

optimization procedure that minimizes the differences between the real trajectories of process variables (temperature

and pressure) and the desired set-points. In this tuning procedure, the behavior of manipulated variables was also

considered and consequent rule adjustments proved to be a powerful tool for control tuning, providing not only good

results of the controlled variable but also smooth behavior of the manipulated variable. Moreover, rule adjustment

was also able to detect the existence of possible faults in the system design and correct their effects on the control

performance.

The PI-fuzzy control strategy showed better results, considering the nonlinearities, the multivariability and the in-

teractive behavior of the system. The easy understanding of the heuristic rules and adaptability of the fuzzy inference


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0 1 2 3 4 5 6

0

100

200

300

400

500

600

Time (hours)

Pressure(kPa)Vapour

flow(mol/h)10

Temperature

Setpoint

Vapour flow

0 1 2 3 4 5 6

0

100

200

300

400

500

600

Time (hours)

Pressure(kPa)Vapourflow(mol/h)

10

Pressure

Setpoint

Vapour flow

Fig. 11. (a) Pressure and manipulated variable (PI-fuzzy control) (without anticipatory action) and (b) pressure and manipulated variable (PI-fuzzy

control) (with anticipatory action,Table 4).

system, even with process changes, suggest that the PI-fuzzy strategy is a potential control alternative for the poly-

merization system studied. Moreover, this strategy also provides an efficient way to reduce overshoot in temperature

control considering the importance of this performance parameter for batch and semi-batch polymerization systems

with heating ramp in this case.

Acknowledgment

The authors acknowledge FAPESB (Fundao de Amparo Pesquisa do Estado da Bahia) and CNPq (Conselho

Nacional de Desenvolvimento Cientfico e Tecnolgico) for their financial support.

Appendix A. Model description (Mass/energy balances)

The reaction rate for the-caprolactam monomer, [C1], and amino-caproic acid, [S1], in the liquid phase are given

by

rc1 = k1[C1][W]+ k

1[S1] k3[C1]0 +k

3(0 [S1]) (A.1)

rS1 =k1[C1][W] k

1[S1] k2[S1](0 +[S1])+ k

2[W](0 [S1]+ [S2])

k3[S1][C1]+ k3[S2] k5[S1][C2]+ k5[S3] (A.2)


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The three first moment balances can be expressed by

d0dt

= k3[C1]0 k5[C2]0 +0

0.113 Rvm +0.018 Rvw

F

+k1[C1][W] k

1[S1] +(k4[W][C2] k

4[S2])

k

2[W]

41

0

2

+k3(1 [S1])+ k5(0 [S1] [S2])

k2 0 0 +k

2[W]1 0 +

02

k3(0 [S1])

k5(0 [S1][S2])+ k3[C1]0 +k5[C2]0 +k2 202

(A.3)

d1dt

= k3[C1]1 k5[C2]1 +1

0.113 Rvm +0.018 Rvw

F

+k1[C1][W] k

1[S1]+ 2(k4[W][C2] k

4[S2])

k2[W]

4

2

1

2

+k3(1 0)+ k

5(1 2 0 +[S1])

k2 0 1 + k2[W]2 1

2 + 1

4

k3(1 [S1])

k5(1 [S1] 2[S2])+ k3[C1](1 +0)+ k5[C2](1 + 2 0)+ k2 0 1 (A.4)

d2dt

= k3[C1]2 k5[C2]2 + 2

0.113 Rvm +0.018 Rvw

F

+k1[C1][W] k

1[S1]+ 4(k4[W][C2] k

4[S2])

k2[W]

4

3

2

2

+k3(2 2 1 +0)+ k

5(2 4 1 +4 0 [S1])

k2 0 2 +k

2[W]

1

3

3

3 22

+12

+

28

k3(2 [S1])

k5(2 [S1]4[S2])+ k3[C1](2 +2 1 + 0)

+k5[C2](2 +4 0 +4 1)+ k2(0 2 +21) (A.5)

The material balances for cyclic dimerous, [C2], and water, [W], are given by

d[C2]

dt= k4[C2][W] + k

4[S2] k5[C2]0 + k

5(0 [S1] [S2])

+[C2]

0.113 Rvm +0.018 Rvw

F

(A.6)

d[W]

dt= k1[C1][W]+ k

1[S1]+k2

20

2 k2[W](0 [S1]) k4[C2][W]

+k4[S2]

RvwF

+[W]

0.113 Rvm +0.018 Rvw

F

(A.7)

A liquid phase global balance furnishes:

dFT

dt= RvT (A.8)

The vaporization rate is due to monomer and water, and given by

RvT =0.113 Rvm +0.018 Rvw (A.9)

For one component C, the masse balance gives:

d[C]

dt = rC Rvm

FT +[C]

RvT

FT (A.10)


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A global mass balance in the gas phase gives:

VT = Rvm + Rvw Vg

R T

d P

dt+

Vg

T

N Ci =1

[Ci ]dT

dt(A.11)

The monomer, water and nitrogen balances in the vapor phase give:d[Mv]

dt=

Rvm

Vg

VT[Mv]

Vg([Mv]+[Wv]+ [Nv]) (A.12)

d[Wv]

dt=

Rvw

Vg

VT[Wv]

Vg([Mv]+[Wv]+ [Nv]) (A.13)

d[Nv]

dt=

VT[Nv]

Vg([Mv]+ [Wv]+[Nv]) (A.14)

The energy balance for the reactor results in:

U A(Tj T)+F

1000

5i =1

ri (Hi ) (Rvm m (Tr)+ Rvw w(Tr))

(0.113 Rvm Cvp,m +0.018 Rvw C

vp,w)(T T)

+dT

dt=

+Clp,mix(0.113 Rvm +0.018 Rvw)(T Tr)

(Clp,mix +2.0925 103(T Tr))F

(A.15)

Kinetic constants, correlations for the vaporization rates, phase equilibria, heat exchanges and physical properties

can be found in[18].

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Fuzzy Control of a Nylon Polymerization Semi-batch Reactor

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