Control of Dividing-Wall Columns via Fuzzy Logic - Industrial

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Control of Dividing-Wall Columns via Fuzzy Logic Salvador Tututi-Avila and Arturo Jiménez-Gutiérrez* Departamento de Ingeniería Química, Instituto Tecnológico de Celaya, Ave. Tecnológico y García Cubas S/N, 38010 Celaya, Gto., México ABSTRACT: Because of their potential energy savings, dividing-wall columns (DWCs) have been viewed as an interesting alternative to the use of sequences based on conventional distillation columns. The energy savings, however, must be supported by suitable controllability properties for a proper operation. In this study we use a design methodology to optimize the gains of the Takagi−Sugeno fuzzy controller and show its implementation for the control of a DWC. Comparison with optimal proprotional−integral−derivative (PID) controllers was carried out to further analyze the performance of the proposed controller under load disturbances in feed composition and set point changes. The case studies considered here are the separation of pentane−hexane−heptane and the industrially relevant separation of benzene−toluene−xylene. The results show that the fuzzy controller provides a suitable option for DWC control and that it produces improved control performance with respect to the conventional PID controller.

1. INTRODUCTION Distillation provides an effective way to separate many fluid mixtures and is the preferred separation process in the chemical and petrochemical industries. However, the operation of distillation columns requires significant amounts of energy; this factor, together with tighter environmental regulations, has provided an incentive toward the search for more energy-efficient distillation structures. For the separation of ternary mixtures, thermally coupled arrangements have shown significant energy savings with respect to the conventional direct (one by one as overheads) or indirect (one by one as bottom products) sequences. The fully thermally coupled structure, or Petlyuk column,1,2 has been considered with particular interest. Figure 1 shows how a sequence based on conventional columns (Figure 1a) can be integrated to provide the Petlyuk system (Figure 1b). A practical implementation of the Petlyuk column has been achieved through the use of a dividing-wall column (DWC, Figure 1c); such a structure splits the middle section of a single vessel into two sections by inserting a vertical wall within the vessel, thus implementing the Petlyuk configuration into a single shell.3−7 Dividing-wall columns provide a good example of process intensification since they can bring significant reductions in both capital and operational expenditures of up 30%.4,8,9 DWC for ternary separations can be considered as a proven technology, with over 100 columns reported in operation worldwide.10 The economic benefits provided by the DWC structure, however, can only be fully exploited if a proper control structure can be implemented for a suitable operation of the process.11 Previous works on the control of DWC have been mostly based on some type of proportional−integral (PI) or proportional−integral−derivative (PID) loops within a multiloop framework (e.g., DB/LSV, DV/LSB). Another option would be to develop a formal mathematical model and then apply some type of model-based control algorithm. An alternative approach, addressed in this work, is the application of a fuzzy logic control algorithm. Fuzzy logic control (FLC) has become one important field of fuzzy set theory. It is applied for designing nonlinear © XXXX American Chemical Society

controllers, mainly due to its simplicity, easy design, and implementation. Compared to classic control approaches, FLC uses more information from human heuristics and experience and relies less on mathematical models of the physical system,12 a characteristic particularly suitable for complex, nonlinear systems. FLC has been applied to several processes such as reactors,13 heat exchangers,14 conventional distillation,15,16 aircraft flight control,17 and nuclear reactors.18 The advantages of FLC over standard PI controllers have also been shown in several applications.12,19−23 The design of a fuzzy controller, however, remains a difficult task due to insufficient analytical design methods in contrast to the well-developed linear control techniques. In most cases fuzzy controller design is accomplished by trial and error methods. This work deals with the design of an optimal fuzzy logic control system using a systematic methodology to tune the fuzzy controller parameters and its implementation to a DWC to control the products purities. The analysis is supported by using rigorous simulations. The performance of the controller is investigated in terms of dynamic tests that include servo and regulatory tasks. The fuzzy control performance is compared to the dynamic responses provided by conventional feedback controllers.

2. FUZZY LOGIC CONTROLLER A fuzzy logic controller (FLC) can be viewed as a dynamic expert system in which the knowledge about the system is transferred into a fuzzy rule-base (FRB) using linguistic variables that do not have precise values. A fuzzy rule is a conditional statement, expressed in the form of IF−THEN clauses. The deduction of the rule (or inference) requires the definition of a membership function to characterize it. Information from the system leads to fuzzy decision-making via the firing of rules in the FRB through Received: October 16, 2012 Revised: February 14, 2013 Accepted: May 10, 2013

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dx.doi.org/10.1021/ie302836y | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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Figure 1. Separation of a ternary mixture: (a) Distributed sequence with conventional columns, (b) Petlyuk configuration, (c) DWC configuration.

functions and the type of membership function shape that should be used to design a fuzzy controller based on twelve considerations. Two membership function shapes, Gaussian and trapezoidal, are the most widely used. Some studies have been reported on the comparison of control performance using Gaussian and triangular membership functions.27−30 Doungri26 pointed out that control performance is the most important consideration to take into account, and he concludes that the best membership function shape is highly application-dependent. Once the membership functions and the rule-base of the FLC are set up, the next problem is the controller tuning, which remains a fairly difficult and sophisticated procedure since there is no general tuning method for FLC. Both cardinality (number of adjectives) and scaling factor parameters should be determined from an optimization procedure based on performance indexes, such as ISE, IAE, or ITAE. A small cardinality should be first employed (fast inference calculations), although one may be aware that large cardinality may provide a higher resolution in the control output. Fine tuning can also be obtained from the choice of scaling factors. For a two-input FLC, a cardinality of 3, 5, 7, 9, or 11 for each input is commonly used. In order to obtain optimal fuzzy controllers capable of stabilizing the DWC system, we follow the following systematic approach. The idea is to start developing a linear fuzzy PID controller.25 This kind of controller basically performs the same control action as the conventional one, but the control strategy is formulated as a set of fuzzy rules. Next, the fuzzy controller is considered as nonlinear. Finally, an optimization strategy for fine-tuning is used. From the solid foundation of linear control theory, it is safer to move to fuzzy control, rather than starting from scratch.25 The outlined procedure is applied to each individual control loop and involves the following steps: (1) Build and tune a conventional PID controller f irst. According to Janzen,25 one can start by tuning a conventional controller using any classical tuning technique such as Ziegler−Nichols; in our case, we optimize the PID parameters using the IAE criterion. (2) Replace it with an equivalent linear f uzzy controller. We transfer the optimal PID gains into the fuzzy controller domain; the fuzzy controller is set up using triangular premise membership functions. Furthermore, the error phase plane can be used to define the rule base structure. Janzen25 has presented a complete error phase plane study on a nine-rule base fuzzy controller. The relationship

the application of fuzzy set-theoretical calculations. The decisions of the fuzzy expert systems are then translated into control actions at the actuator level. A relevant part of the fuzzy theory, which focuses on principles of fuzzy control, is available in Driankov et al.24 The FLC used in this work is depicted in Figure 2, where F, RB, and D represent fuzzification block, a rule-base block, and a

Figure 2. Structure of the fuzzy−PID controller.

defuzzification block, respectively.25 This is basically the same architecture proposed by Han-Xiong and Gatland,19 called simplified fuzzy three term controller. In the fuzzification block, crisp variables are transformed into fuzzy sets. In the rule-base block, a control algorithm is coded using fuzzy statements, taking into account the control objectives and system behavior. The control actions are encoded by means of fuzzy inference rules, and the appropriate fuzzy sets are defined on the domains of the involved variables; then, fuzzy logic operators and inference methods are implemented in computational terms. Finally, in the defuzzification block the fuzzy calculations are changed to reflect real values. As reported in the literature,25 the scaling factors are the ones that affect most significantly the performance of the control system. Scaling factors play a similar role to that of the gain coefficients of a conventional controller. From the scaling factors, the controller input and output values are mapped into the universe of discourse of the fuzzy set definitions. 2.1. Optimal Tuning of Scaling Factors. According to Han-Xiong and Gatland19 fuzzy tuning is a difficult task due to the need of multiple simultaneous adjustments. There is not a well-defined methodology for the choice of membership functions and for tuning these types of controllers. In terms of real time requirements (minimal use of memory and fast calculations), the use of a triangular-shaped membership function is generally recommended. Dongrui26 has presented a comprehensive work regarding the shape of membership B



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Figure 3. Effect of liquid and vapor interconnecting streams on energy consumption for case study 1.

3. DESIGN OF THE DIVIDING-WALL COLUMN When conventional sequences are used for the separation of ternary mixtures, a remixing of the intermediate component affects the total energy required for the separation. Triantafyllou and Smith31 and Hernández and Jiménez32 have shown how Petlyuk/DWC columns can be designed and optimized for energy consumption by eliminating remixing effects. In this work we use the methodology proposed by Hernández and Jiménez32,33 for the design of divided wall columns with minimum energy consumption. Once design specifications are given, the procedure begins with a preliminary design for the DWC system, which is obtained from the use of shortcut methods for a nonintegrated counterpart based on conventional columns (a prefractionator with two binary separation columns, as in Figure 1a). Steady-state rigorous simulations of the thermally coupled structure are then conducted using Aspen Plus to test the preliminary design; in this step adjustments in the tray structure are made as required so that the design meets the specified product purities. Finally, a search procedure over the values of the interconnecting streams is carried out to detect the conditions under which the validated DWC design achieves minimum energy consumption. The search procedure provides the optimal values of the interconnecting vapor flow rate (VF) and the interconnecting liquid flow rate (LF) for the dividing-wall column.

between a fuzzy−PID controller and the conventional PID is given as ⎡ kek u − Kc ⎤ ⎢ ⎥ ⎢ kd /ke − Td ⎥ = 0 ⎢ ⎥ ⎣ k iTi /ke − 1⎦

(1)

where Kc, Ti, and Td are the proportional gain, integral time constant, and derivative time constant of the PID controller. (3) Make the f uzzy controller nonlinear. So far, we have a fuzzy controller that performs just like a conventional PID. The next step is to insert a nonlinear rule base. This is carried out for example by using smooth trapezoidal or Gaussian membership functions to generate a nonlinear fuzzy control surface. (4) Fine-tune it. From eqs 1 it can be noted that the fuzzy−PID controller has one degree of freedom, since it has one more gain factor than the conventional PID controller.25 The degree of freedom could be used to optimize the fuzzy controller. Also, eq 1 simplifies the tuning of the scaling factors of the fuzzy controller. Once the PID controller has been tuned, the parameters Kc, Ti, and Td are known, such that k u = Kc/ke ,

kd = Tdke ,

k i = ke/Ti

(2)

4. CASE STUDY 1 A mixture of n-pentane (nC5), n-hexane (nC6), and n-heptane (nC7) as used in reported works34,35 was taken as the first case for this study. The feed flow rate is 100 lbmol/h with a molar composition (A = nC5, B = nC6, C = nC7) equal to (0.40, 0.20, 0.40). Specified product purities of 98 mol % for all components were assumed. As in the reported references, thermodynamic properties were predicted with the Chao−Seader correlation. Energy efficiency is a major issue for the design of DWC systems, so the first aspect for the analysis is to detect a basic design that minimizes energy consumption. Figure 3 shows the results of the search for the optimum values of interconnecting

It can be noticed that eqs 2 are related to ke; therefore, an optimization problem could be formulated so as to optimize ke using the values of the tuned PID parameters, but this problem would be restricted by the selected tuning method. Instead, we use such equations only as initial guesses for the optimization phase, set a value of ke = 100 (as suggested by Janzen25) and the remaining three parameters (i.e., ku, kd, and ki) are then optimized. We use the IAE performance function (eq 3) as a basis for proper closed-loop behavior. J=

∫0

t

|e(t )| dt

(3) C



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streams. The details of the resulting design and operating conditions are given in Table 1. Liquid compositions profiles in Table 1. Final Design of the Dividing-Wall Column design parameter prefractionator part stages feed tray pressure (top, psi) pressure (bottom, psi) main column part number of trays vapor interconnecting tray liquid interconnecting tray sidestream tray condenser duty (MBTU/h) reflux ratio temperature (top, °F) pressure (top, psi) liquid to prefractionator, LF (lbmol/h) vapor to prefractionator, VF (lbmol/h) reboiler duty (MBTU/h)

value 17 9 22.4 26.1 36 11 27 17 1.5139 2.424 120 21.1 37.33 80 1.5462

both sections of the column are presented in Figure 4; one can notice how the production of the intermediate component is obtained in the tray with the maximum composition, consistent with the principles behind an optimal operation of the base design. The dynamic analysis was carried out with product compositions as the controlled variables. Three nominal controllers were used for the control of the three product streams, using as manipulated variables the reflux flow rate (L), the side stream flow rate (S), and the reboiler heat duty (QR); for inventory control, the liquid levels in the reflux tank and in the reboiler were controlled by manipulating the distillate (D) and bottoms (B) flow rates. An additional control loop for DWC systems has been proposed by Ling and Luyben36 and is included in this work. It consists of the manipulation of a liquid split ratio (βL) that aims for a better control of the heavy component composition (yP(X)) in the top of the prefractionator (Figure 5). The simulations were carried out using Aspen Plus Dynamics.

Figure 5. Control configurations of the dividing-wall column (four points control).

As for the particular type of controller, a fuzzy−PID controller was chosen because it is probably the preferred type among the fuzzy controllers. The input variables are the error and its derivatives, as in conventional PID algorithms. For the fuzzy− PID controller, the universes of discourse of error (e) and change of error (ce) were partitioned into three fuzzy sets (N negative, Z zero, and P positive), as shown in Figure 6; the output (u) was partitioned into five fuzzy sets (NB negative big; NS negative small; Z zero; PS positive small; and PB positive big). For the output linguistic variable (u), the singleton fuzzy sets of PB = 200, PS = 100, Z = 0, NS = −100, and NB = −200 were assigned. The rule-base of the FLC contains nine rules (Table 2).37 The intersection of the first row and the first column stands for the rule IF e = N AND ce = N THEN u = NB. In fuzzy logic, the

Figure 4. Composition profiles in the dividing-wall column. D



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Figure 6. Membership functions of error and change of error for the fuzzy controller.

of the prefractionator was tuned with the other three loops closed. 4.2. Dynamic Responses with a Control Scheme of Four Points. This section presents the results for the multiloop control of the DWC under input constraints with the four control loops closed, both for set point tracking and feed disturbance rejection. The servo problem was formulated by implementing a simultaneous change in the three set points (xA, xB, xC from 0.98 to 0.985 mol fraction), and the responses of the DWC system are shown in Figure 7. Fuzzy controllers yield a better dynamic response. It can be seen for instance how the composition of the light component reaches the new set point in 0.6 h under fuzzy control (Figure 7b), whereas the PID controller takes about 1 h to reach the desired value when the step change in the set point was imposed and also shows some overshooting in the composition of the intermediate component (Figure 7a). Overall, the structure with fuzzy control shows settling times for the three components lower than 0.6 h and provides smoother and faster responses for the three composition control loops. 4.3. Sensitivity Analysis. Additional tests were conducted to examine the fuzzy control performance under a series of process feed disturbances, and compare them to the system response under standard PID controllers. For the first disturbance, the feed flow rate was increased by 10% (from 100 to 110 lbmol/h). The second test consisted of a 10% increase in the feed composition of the light product. Finally, a simultaneous change with both disturbances was considered. Figure 8 shows the dynamic responses of the system under the feed flow rate disturbance. The mole fractions of components A in the top distillate (xA), B in the side stream (xB), and C in the bottom product (xC) return to their set point values within short settling times. Although settling times are quite comparable for both types of controllers, the dynamic responses show noticeable differences, particularly for the control of the light component. One can notice how the fuzzy controller provides a response with

Table 2. Rule Set for the Fuzzy−PID Controllers E CE

N

Z

P

N Z P

NB NS Z

NS Z PS

Z PS PB

AND operator is performed by a class of operators, being the min (minimum) operator the most widely used. The firing strength αk of rule k reflects the rule fulfillment degree. Therefore, the result α1 = min(μN(e), μN(ce)) represents the activation level for the rule, which implies that the NB membership function to the controller output variable is also set to this level. After fuzzy inference is applied to each rule, the activation level for each output variable membership function is obtained, and the defuzzification procedure takes place. The most commonly used procedure is based on the center of gravity of the profile described by the membership functions.38 Considering singleton conclusions and sum-accumulation, the resulting defuzzificated value in discrete terms is

u0 =

∑k αk*Sk ∑ αk*

(1)

k

where Sk is the position of the singleton in rule k of the discourse universe, and αk* is the weighted firing strength of rule k.25 The tuning procedure for PID and fuzzy controllers was done by minimizing the integral of the absolute value of the error (IAE) criteria. Table 3 shows the optimal parameters for both control structures. The four composition control loops were tuned with a sequential method as in Ling and Luyben.36 The loop of the reboiler heat input was tuned first. Then, this loop was closed and the loop of the reflux was tuned. After this step, the side stream loop was tuned with the other two loops set on automatic operation. Finally, the loop of the impurities in the top

Table 3. Parameters of the Controllers Obtained after the Minimization of IAE Values for Case Study 1 PID

Fuzzy−PID

controlled variable

manipulated variable

Kc

Ti (min)

Td (min)

kd

ki

ku

xD(nC5) xS(nC6) xB(nC7) xP(nC7)

L S QR βL

35 58 90 18.44

30 13.26 30 100

0.36 0.015 0.6 0.05

2 1 0.83 1

100 50 71.22 65

7.70 12.36 1.28 9.22

E



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Figure 7. Dynamic response under a process disturbance in the set point: (a) PID controller, (b) fuzzy controller.

Figure 8. Dynamic response under a process disturbance in the feed flow rate: (a) PID controller, (b) fuzzy controller.

Figure 9. Dynamic response under a disturbance in the feed composition (xA): (a) PID controller, (b) fuzzy controller.

disturbance. The composition of component A is the one more noticeably affected. Overall, the fuzzy controller provided a better control action that the PID controller. For a final comparison of the effectiveness of both controllers, Figure 11 gives a summary of the overall performance by evaluating the integral of the absolute error (IAE) for each of the servo and regulatory tasks. The superior performance of the fuzzy controller is supported by the lower IAE values.

smaller overshoot and undershoot, which is translated into lower IAE values; the associated lower control actions also favor the energy efficiency of the separation system. Overall, the fuzzy controller provided a fast attenuation of the load disturbance. Figure 9 shows the result for the second disturbance test. Both types of controllers are effective to handle the disturbance and return the product compositions to their steady state values. Nonetheless, it can be observed how the fuzzy controller provides a better response, with a smaller overshoot of the composition of the light component and a shorter settling time. Figure 10 shows the dynamic responses of the DWC when a simultaneous disturbance in the feed flow rate and feed composition of the light component was considered. The dynamic response of the system under each control strategy is shown in Figure 10, and the larger settling times and more sensitive transient response reflect the effect of a more aggressive

5. CASE STUDY 2 (BTX SYSTEM) The second case study is the industrially important ternary separation of benzene, toluene, and o-xylene (BTX), which has been the subject of several design and control studies.8,23,27,39−41 The feed flow rate was taken as 1 kmol/s with a molar composition (A, B, C) equal to (0.30, 0.30, 0.40). Specified product purities of 99 mol % for all components were assumed. F



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Figure 10. Dynamic response under a disturbance in the feed flow rate and the composition (xA): (a) PID controller, (b) fuzzy controller.

Figure 11. Comparison of (IAE) values for the responses under each type of controller.

Figure 12. Effect of liquid and vapor interconnecting streams on energy consumption for the BTX case study.

for the DWC system with minimum energy consumption. Liquid compositions profiles in both sections of the column are presented in Figure 13. The four composition control loops were tuned similarly as in the first case study, and Table 5 reports the resulting controller

All simulations were carried out using rigorous distillation column models with the use of Aspen Plus. The divided-wall column was designed similarly as in the first case study. Figure 12 shows the results of the search for the optimum values of interconnecting streams, and Table 4 gives the resulting design G



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showed a more oscillatory behavior than with the action of the PID controller. The column behavior in response to the combined flow rate and feed composition disturbance is displayed in Figure 17. As expected, the system takes longer to stabilize than the response obtained against the individual disturbances, but both controllers are effective to stabilize the column, with the PID controller taking longer settling times. The composition of the intermediate component (toluene) shows the highest transient deviations from the steady state conditions. To complete the analysis, IAE values for each dynamic response were evaluated. The results are shown in Figure 18; the lower IAE values obtained for both the servo and regulatory types of problems reflect the superior performance of the fuzzy controller.

Table 4. Final Design of the BTX Dividing-Wall Column design parameter prefractionator part stages feed tray pressure (top, atm) pressure (bottom, atm) main column part number of trays vapor interconnecting tray liquid interconnecting tray sidestream tray condenser duty (MW) reflux ratio temperature (top, K) pressure (top, atm) liquid to prefractionator, LF (kmol/s) vapor to prefractionator, VF (kmol/s) reboiler duty (MW)

value 24 12 0.41 0.61 46 34 10 20 34.54 2.49 322 0.34 0.21 0.67 37.59

6. ADDITIONAL TESTS FOR THE FUZZY CONTROLLER Fuzzy controllers have a more complex structure but provide more options for improving the expected performance of the DWC separation system. As mentioned above, Gaussian and triangular are the membership shapes most widely used to design fuzzy controllers. We implemented a triangular membership shape for both case studies to compare to the results obtained under a Gaussian shape, and a summary of results is provided in Figure 19, where the overall IAE values are compared. It can be noted that the controller performance for the first case study is fairly similar under both membership shapes, and that some differences are noted for the second case study, although there is not a clear trend as to which membership is better. It can be stated that in general both membership shapes provide a similar effectiveness as far as the overall control task and that in all of the tests conducted they outperformed the actions of the PID controller. A final test related to the design of the fuzzy controller was conducted. It has been reported that increasing the rules and number of membership functions beyond a certain limit is not convenient, since it only increases the complexity of FLC and does not yield an incentive as far as the system performance.42 The control system for the DWC in both case studies was further tested using 25 and 49 fuzzy rules with no significant improvement in the dynamic responses; in those cases, a fineretuning process was necessary because of the interactions among loops. It should also be mentioned that some authors

parameters. The control performance was examined under set point tracking and disturbance rejection tests. The servo problem was formulated by implementing a simultaneous change in the three set points (xA, xB, xC from 0.99 to 0.995 mol fraction), and the responses of the DWC system under each control action are shown in Figure 14. The composition of benzene is more effectively taken to the new set point by the fuzzy controller. An inverse response is observed in the composition profile of toluene, the intermediate component of the ternary mixture, as a result of mass balance effects and loops interactions; nonetheless, the fuzzy controller provides a faster stabilization toward the new steady state. The dynamic response of xylene is also aided by the use of the fuzzy controller with respect to the PID implementation. Similar disturbance tests as in the previous case study were considered. Figure 15 shows the system responses for the flow rate disturbance under each control option. Both controllers successfully rejected the disturbance to bring the product compositions back to their design values, although the settling times under the fuzzy controller action were lower. Figure 16 shows the responses for the disturbance test on the feed composition of the light component. The fuzzy controller yielded lower settling times, although the composition of toluene

Figure 13. Composition profiles for the BTX column. H



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Table 5. Parameters of the Controllers Obtained after the Minimization of IAE Values for Case Study 2 PID

fuzzy

controlled variable

manipulated variable

Kc

Ti (min)

Td (min)

kd

ki

ku

xD(B) xS(T) xB(X) xP(X)

L S QR βL

53.86 0.45 36.40 1.65

41.52 55 93.49 51

0.38 0.02 0.01 0.06

1.5 2.85 1.29 1

15 50 5 100

2.28 5.49 17.63 19.75

Figure 14. Dynamic responses for set point changes for the BTX column: (a) PID controller, (b) fuzzy controller.

Figure 15. Dynamic response under a disturbance in the feed flow rate for the BTX column: (a) PID controller, (b) fuzzy controller.

Figure 16. Dynamic response under a disturbance in the feed composition (xA) for the BTX column: (a) PID controller, (b) fuzzy controller.

recommend exploring a reduction on the number of rules for fuzzy controllers to reduce the complexity of the design task.43 This observation is consistent with the applications shown in this

work, since the design of the fuzzy controller was based on a small number of rules and provided an effective control action for the DWC. I



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Figure 17. Dynamic response under a disturbance in the feed flow rate and the composition (xA) for the BTX column: (a) PID controller, (b) fuzzy controller.

Figure 18. Comparison of controller performance in terms of IAE values.

Figure 19. Overall performance of PID controller and fuzzy controller with Gaussian and triangular membership shapes for the case studies in terms of their IAE values.

7. CONCLUSIONS A systematic approach for the design of fuzzy controllers has been presented and applied to the composition control of a DWC system. Three control loops, one for the composition control of each product, was used. In addition, another loop was included to control the impurities at the top of the prefractionator. Set point tracking and responses to feed load disturbances in composition were carried out, and the dynamic

behavior of the fuzzy controller was compared to the DWC performance under a proportional−integral−derivative controller. After both the fuzzy−PID and the PID controllers were tuned by minimizing IAE values, the closed loop responses showed that the fuzzy−PID controller was effective in controlling the DWC system and that its implementation can significantly improve the set point tracking and the disturbance rejection provided by the use of the PID controller. The J



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effectiveness of the control strategy for the DWC systems enhances the energy savings that have been shown under steady state conditions. The fuzzy control does not require complex computation and it can be easily implemented. This results from this work show that fuzzy controllers provide a suitable option to control a complex distillation system such as the dividing-wall column. Future work includes exploring model-based control approaches and their comparison to fuzzy control techniques using rigorous DWC models.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +52-461-611-7575 Ext 139. E-mail: [email protected]. mx. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge financial support from Conacyt, Mexico, through project CB-84493.

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REFERENCES

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Article

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Control of Dividing-Wall Columns via Fuzzy Logic - Industrial

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