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PROCESS DESIGN AND CONTROL Model Gain Scheduling Control of an Ethyl tert-Butyl Ether Reactive Distillation Column Budi H. Bisowarno, Yu-Chu Tian, and Moses O. Tade´ * Department of Chemical Engineering, Curtin University of Technology, G.P.O. Box U1987, Perth, WA 6845, Australia
Reactive distillation (RD) is a favorable alternative to the conventional series of reactionseparation processes. However, the control of RD is challenging because of its complex dynamics resulting from the integrated functionality of reaction and separation. A linear proportionalintegral-derivative algorithm with fixed parameters has been shown not to be satisfactory to handle the high directionality of the process gain. It needs to be retuned adequately over a wide range of operating conditions. In this work, model gain scheduling is investigated for an ethyl tert-butyl ether reactive distillation column. It employs a set of simple local models, which cover relevant operating conditions and cope with the nonlinear characteristics of the process. The simple models are then integrated by using a proper switching scheme. Simulation results have shown that the proposed control strategy outperforms the standard proportional-integral control in both set-point tracking and disturbance rejection. Introduction The conventional proportional-integral-derivative (PID) algorithm is still widely used in process industries because of its simplicity and robustness. However, its performance is not sufficient for nonlinear characteristics, which is actually inherent in most chemical processes. The changing sign and directionality of the process gain complicates the design of a control system. In some cases, the changing sign of the process gain can be overcome by employing inferential control such that the relationship between the control variable and the manipulated variable is always monotonic. However, the directionality of the process gain, which may result from the inferential control system, cannot be handled well by using standard PID algorithms with fixed parameters. Two important approaches to design nonlinear control are based on differential geometric theory1 and gain scheduling control.2 One major problem for the differential geometric approach is that the requirement of the whole state of the nonlinear process for design of the control structure may not be met from the measured process variables.3 Besides, an exact nonlinear model is generally not available. These problems have limited the application of the differential geometric approach. On the other hand, gain scheduling can be used in the absence of complete plant models. A gain-scheduling controller is also able to respond quickly to changes in the operating conditions. Directionality of a chemical process means that a vector of inputs (e.g., manipulated variables) is differently amplified according to its direction. Directionality * To whom correspondence should be addressed. Tel.: +61 8 9266 7581. Fax: +61 8 9266 2681. E-mail: tadem@che. curtin.edu.au.
of the process gain has been known to create a considerably complex design of the control system for multivariable processes such as those in conventional distillation.4 Incorrect models of the process directionality may be quite useless for controller design.5 On the other hand, gain scheduling is based on linear time invariance of the process at a number of operating points, and then a linear controller is designed for each of these operating points.6 Therefore, a set of simple local models could be developed and then integrated to form a global multiple model. The number of the local models and the selection of the operating points are largely dependent on the directionality of the process gain. Reactive distillation is a favorable alternative to the conventional series of reaction-separation processes. Control of reactive distillation is challenging because of its complex dynamics resulting from its integrated functionality of reaction and separation. It has highly nonlinear behaviors such as directionality and changing sign of the process gain so that a standard proportionalintegral (PI) algorithm with fixed parameters is not satisfactory. It needs to be retuned adequately over a wide range of operating conditions. Although the dynamics and behaviors have been investigated extensively, control of reactive distillation is still an open research area. Previous studies on reactive distillation in the open literature mainly focus on steady-state design and openloop dynamics along with the existence of multiplicity phenomena. A limited number of reports have discussed the control aspects of reactive distillation. Among these reports are investigations on control strategies for batch reactive distillation7 and structure for optimizing batch reactive distillation8 and recent advances as detailed below.
10.1021/ie020763q CCC: $25.00 © 2003 American Chemical Society Published on Web 06/20/2003
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For continuous reactive distillation, a nonlinear inputoutput linearizing controller and a nonlinear controller have been designed for ethylene glycol production.9 A robust PI control configuration has been proposed for the same system.10 Selection of control configurations and the application of linear and nonlinear control strategies have been applied for an ethyl acetate reactive distillation.11 A variety of control structures have also been explored for a two-product reactive distillation column.12 It has been found that the inventory of one of the reactants needs to be detected so that a feedback trim can balance the reaction stoichiometry. Reactive distillation has been accepted as a preferred technology for ether productions such as ethyl tert-butyl ether (ETBE). ETBE is an oxygenate, which has better additive fuel properties and less water contamination than those of methyl tert-butyl ether (MTBE). Besides, ETBE is synthesized from isobutylene and ethanol, which can be produced from biomass. For ETBE reactive distillation, general control considerations have been presented.13 The advantages of combining composition and conversion have been discussed.14 Disturbance rejection properties of a variety of one-point control schemes have been explored as well.15 Pattern-based predictive control has recently been proposed for controlling the product purity.16 Standard PI algorithms, which were employed for all cases, indicated that a more advanced controller is needed to improve the control performance. In this paper, model gain scheduling (MGS) will be developed and implemented on one-point control (product purity) of an ETBE reactive distillation column. Several local models, which cover relevant operating conditions and cope with nonlinear characteristics (e.g., directionality of the process gain), will be derived and formulated in transfer functions. A switching scheme will be employed to integrate the local models. The performance of MGS will be evaluated and compared to that of a standard PI controller in both set-point tracking and disturbance rejection. Model Gain Scheduling MGS as a form of adaptive control requires online estimation of the process model or its process gain. A single highly nonlinear model may be used to model the process dynamics for the entire operating conditions. However, the development of such a nonlinear model is difficult in most cases, and the model will complicate the controller design. A promising approach is to decompose the entire operating conditions into a set of operating condition ranges, which cover the relevant operating regions.17 Simple multiple models, which cope with nonlinear and time-varying characteristics for each operating condition, are then developed, and a proper switching scheme is formed to integrate them. Therefore, wider operating conditions can be identified, and the requirement of a rapid control action can be satisfied. Powerful linear control algorithms can then be employed directly for MGS control of the nonlinear system. Multiple Models and Switching. For control system design, process dynamics should be first understood. If the operating conditions change and/or the disturbances occur, the input-output of the system will generally change as well. Therefore, multiple local models are developed to identify the different operating conditions as well as to control them rapidly.
The local models can be derived either from linearization of the nonlinear model or directly from inputoutput identification. State-space18 and polynomial methods19 have been employed to establish local models. In this work, the transfer function local models are derived to approximate the nonlinear stable system. In general, they may be formulated as single input single output (SISO) with time delay as shown in eq 1,20 where -θs y Kpe Gp ) ) u Ts + 1
(1)
y and u are output and input variables, respectively, and Gp is the transfer function of the process. T and θ are the process time constant and time delay, respectively, which may have to be determined by experiment. Therefore, the local models can be found in a straightforward manner. The parameters of the model can be either fixed or adjustable. The adjustable (adaptive) model is inefficient from the computational view especially when the operating conditions of the plant vary widely with time. This results from a large demand on the real-time identification of the model parameters. On the other hand, the fixed model can represent exactly only a finite number of operating conditions. However, use of a proper switching strategy can overcome this problem and yields a reasonable speed of response. The switching scheme consists of monitoring the performance index based on the identification error for each model and then switching to the model with the smallest index value. The need to switch to the smallest index comes from a small identification error that leads to a small tracking error.21 The performance index in eq 2 is adopted in this work, where R and
∫
I ) Re2 + β e2 dt
(2)
β are free design parameters, which determine the relative importance of the instantaneous and long-term measurement of the error and provide a smooth transition between different process models. Here, the design parameters of R g 0 and β > 0 chosen by the trial and error method can improve transient responses dramatically.21 The measurement of the error is needed to reliably estimate the accuracy of the local model. Two issues that arise from the application of multiple models are the number of local models and the selection of the operating points. The simplest way is to distribute the local models uniformly over the entire operating conditions. However, prior knowledge of the process can be used by employing more local models in the sensitive region so that these models can be distributed more efficiently.20 In other words, more local models should be established in the operating regions where the process directionality significantly varies. Controller Design. MGS employs powerful linear control algorithms directly for the control of nonlinear systems. It is based on linear time invariance of the process at a number of operating points, and then a linear controller is designed for each operating point.21 Therefore, the parameters of the controller should be interpolated or scheduled. Although all controller parameters can be scheduled, the controller gain is commonly scheduled because processes are usually characterized by process directionality with relatively constant dynamics.
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Figure 1. General structure of MGS with multiple models.
The conventional way of gain scheduling is switching between local linear controllers. The controller gain can also be interpolated as a function of the scheduling variable. A manipulated variable varies rapidly and is therefore not practical to be used as the scheduling variable. Scheduling based on either the set-point or measured output is more common.6 A measured output may be represented by an inferred variable, which can then be used as a scheduling variable. In this work, programmable or model gain scheduling (MGS) was employed, which is formulated in eq 3,22
Kc0Kp0 Kc ) Kp
(3)
where Kc0 and Kp0 are the reference values of the controller and process gains, respectively. The idea is to keep the overall gain of the closed-loop system constant. However, the overall gain should be kept at less than unity, particularly at the crossover frequency, to ensure system stability.23 The reference control parameters can be tuned by using the popular ZieglerNichols method, which attempts to achieve a quarter amplitude decay. However, the Abbas method24 can specify a desired closed-loop response25 and is thus adopted in this work. The tuning method relates the controller parameters to the characteristics of a firstorder plus time delay model as well as the desired overshoot of the closed-loop system. For a PI controller, the Abbas method specifies the controller gain (Kc) and integral time or reset time (Ti) respectively as
θ 2 Kc ) Kp(λ + θ) T+
Ti ) T +
θ 2
(4) (5)
where Kp, T, and θ are the open-loop process gain, time constant, and time delay, respectively, and λ is the desired closed-loop time constant. The controller gain is affected by the desired closed-loop time constant, which determines the desired closed-loop performance.24 Therefore, the local controller can be found in a straight-
forward manner as well. The time-varying process gain (Kp) is identified and computed online from the inferred variable and the manipulated variable. Thus, this method requires advanced programming of the process gain. The general structure of the proposed strategy, which employs MGS with multiple models, is shown in Figure 1. Case Studies ETBE Reactive Distillation Column. Reactive distillation has highly nonlinear behavior, which creates considerable complex control problems such as directionality and changing of the sign of the process gain. These problems have been identified through experimental work as well as simulation results. For simulation study, most mathematical models are based on the equilibrium-stage approach, which assumes that vapor and liquid leaving each stage are in thermodynamic equilibrium. A rate-based approach is an alternative to the previous approach, but its use is still limited because of its complexity. In this work, the equilibrium-stage model has been developed and implemented in SpeedUp for an ETBE reactive distillation and is rewritten in Aspen Custom Modeler. A full kinetic model of the reaction is employed so that a variety of conditions of the reactive system can be handled. The detailed model can be found in the previous publications from our group.13,26 Figure 2 shows the schematic diagram of the pilot reactive distillation column under consideration. This column was designed to synthesize ETBE from isobutylene and ethanol. It consists of 10 theoretical stages, including three reactive stages where the etherification reaction is carried out. The column pressure is controlled by adjusting the condenser duty. The inventory control for a reflux accumulator and reboiler holdup is maintained by adjusting the distillate flow rate and the bottom flow rate, respectively. The reboiler duty as the primary manipulated variable is manipulated to control the stage 7 temperature. Previous studies have found that controlling the stage 7 temperature is appropriate to infer the ETBE purity.27 The reflux rate as the second manipulated variable is kept constant to achieve high isobutylene conversion. Therefore, the LV configuration,
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Figure 3. Stage 7 temperature and ETBE purity vs reboiler duty.
Figure 2. ETBE reactive distillation column with the controllers (LV).
which outperforms other control configurations,15 is employed for one-point control (ETBE purity). The ETBE with a satisfactory purity is withdrawn from the bottom column. The feed and nominal operating conditions as well as process outputs are shown in Table 1. This typical set of operating conditions is chosen at the reboiler duty of 0.520 MJ/min at which the maximum value of the ETBE purity is obtained at a reflux rate of 2.2 L/min. It is considered as the nominal condition where the control system will be designed. The relationship between the ETBE purity and the reboiler duty reveals the nonlinearity of the process (e.g., input multiplicity), as shown in Figure 3. Therefore, inferential control is adopted to overcome the nonlinearity and to reduce the time delay. Figure 3 also shows the relationship between the stage 7 temperature as the inferred variable versus the reboiler duty. The process gain (∆T7/∆Qr), which can be determined by the derivatives of the curve, is high around the nominal operating condition of the reboiler duty and becomes small outside this range, as shown in Figure 4. Both figures indicate the directionality of the process gain in this reactive distillation process. Development of Multiple Models. The simplified input-output dynamic models of manipulation and disturbance paths are identified around the optimum reboiler duty at constant reflux rate. Referring to Figure 3, the relationship between the ETBE purity and the
Figure 4. Directionality of the process gain (L ) 2.2 L/min).
reboiler duty reveals the existence of input multiplicity, which creates a difficult control problem. Three fixed local models, which relate the stage 7 temperature (e.g., the measurable/inferred input) and the reboiler duty as the primary manipulated variable, are developed to capture the directionality of the process gain around the optimum reboiler duty for each reflux rate. The openloop response of the stage 7 temperature for a sequence of step changes in the reboiler duty at the nominal operating conditions is shown in Figure 5. The process gains are derived by using the appropriate function in the Aspen Custom Modeler package, and the time constants are determined through time-series analysis of the open-loop operation. Despite having 10 theoretical stages, the open-loop responses show that the time delay is much smaller than the time constant. Therefore, firstorder transfer functions are sufficient to describe these models, and the time delay inclusion was not pursued further. The models of the manipulation path resulting from linearization at several reboiler duties are shown in Table 2. The models of the disturbance path resulting from step changes in the feed rate and feed composition at the nominal operating conditions (Qr ) 0.520 MJ/min and L ) 2.2 L/min) are also shown in the same
Table 1. ETBE Reactive Distillation Column Characteristics and Inputs temperature (°C) rate (L/min) overall excess EtOH (mol %)
30 0.76 5.0
no. of rectification stages no. of reaction stages no. of stripping stages total no. of stages
2 3 5 10
isobutylene conversion (mol %) bottom ETBE purity (mol %) reboiler duty (MJ/min) bottom rate (L/min) distillate rate (L/min) reflux rate (L/min)
97.87 93.96 0.52 0.47 0.43 2.2
Feed Conditions composition
Column Specifications feed stage overhead pressure reflux rate Column Outputs condenser temperature (°C) bottom rectification section temperature (°C) middle reactive section temperature (°C) top stripping section temperature (°C) middle stripping section temperature (°C) reboiler temperature (°C)
29.1% ETBE 9.1% EtOH 7.3% iBut 54.5% nBut 6 950 kPa 2.2 L/min
69.5 69.7 72.4 101.2 127.7 157.1
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Figure 5. Open-loop response of the stage 7 temperature to step changes in the reboiler duty.
table. Recall that the PI controller, which is used for the basis comparison, was designed at these nominal operating conditions. In this work, the simplified input-output first-order models shown in Table 2 are employed or scheduled according to the performance index as formulated in eq 2. MGS is then implemented as formulated in eq 3, where the time-varying process gain (Kp) is identified and computed online from the measured stage 7 temperature as the inferred variable and the reboiler duty. Control Performance. The performance of the MGS control is evaluated and compared to that of a standard PI controller. The standard PI controller is tuned in the range of operating conditions by using the Abbas method. First-order models are used to represent the reactive distillation at different operating points. The desired closed-loop time constant is chosen to be 5 min, and the calculated control gain and time constant in the range of operating conditions are 0.00622 °C/(MJ/min) and Ti ) 84.53 min, respectively. For the MGS, the controller gain was computed online as a function of the scheduling variable. The time integral is kept constant at 84.53 min. The switching parameters, R ) 0.0001 and β ) 1, are chosen arbitrarily to guarantee smooth switching and stability of the system. The effectiveness of the control strategy is evaluated for changes in the set point (tracking problem) as well as in the feed rate and feed composition (regulation problem). The dynamic responses resulting from a (5 °C step change in the set point using the standard PI controller and MGS are shown in Figures 6 and 7, respectively. The step change was introduced instantly at 10 min, and the controlled variable (stage 7 temperature) and ETBE purity responses were recorded. The corresponding values of the integral of absolute error (IAE) and integral of time-weighted absolute error (ITAE) criteria of the stage 7 temperatures are shown in Table 3. The MGS clearly has better set-point tracking because of a far shorter settling time of the stage 7 temperature as well as the ETBE purity. The MGS improves the ITAE index by about 62% and 55% over the standard PI
Figure 6. Dynamic responses of a +5 °C step change in the set point.
Figure 7. Dynamic responses of a -5 °C step change in the set point.
controller for the (5 °C set-point step changes, respectively. The IAE index also shows the dramatic improve-
Table 2. Multiple Transfer Function Models Based on Open-Loop Step Responses reflux rate (L/min)
Qr < Qr optimum
Qr optimum
Qr > Qr optimum
2.0
4709.1/(238.4Tis + 1) at Qr ) 0.4825 MJ/min 6442.75/(197.4Tis + 1) at Qr ) 0.515 MJ/min 960.5/(122.4Tis + 1) at Qr ) 0.545 MJ/min
4679.5/(78.5Tis + 1) at Qr ) 0.4875 MJ/min 4675/(73.2Tis + 1) at Qr ) 0.520 MJ/min 9043.5/(126.1Tis + 1) at Qr ) 0.550 MJ/min
498/(21Tis + 1) at Qr ) 0.4925 MJ/min 493/(23.2Tis + 1) at Qr ) 0.525 MJ/min 1472/(54.9Tis + 1) at Qr ) 0.555 MJ/min
2.2 2.4
Disturbances (L ) 2.2 L/min) feed rate feed composition
0.412/(23.75Tis + 1) 0.163/6.75Tis + 1)
Ind. Eng. Chem. Res., Vol. 42, No. 15, 2003 3589 Table 3. Comparison of IAE and ITAE Indices of the Stage 7 Temperature IAE magnitude
PI
+5 °C T7 19.0 -5 °C T7 15.4 +10% Ff 160.1 -10% Ff 160.1 80-70% 58.1 prereacted sign change 308.9 in the reflux rate
ITAE
improve MGS (%) 4.1 4.3 8.3 8.3 2.2
78.4 72.1 94.8 94.8 96.2
4.2
98.6
PI
MGS
improve (%)
389.0 334.8 8831.7 8841.5 3358.3
288.2 293.6 145.7 148.1 147.7
25.9 12.3 98.3 98.3 95.5
20283.4 290.7
98.5
Figure 9. Dynamic responses of a -10% step change in the volumetric feed rate.
Figure 8. Dynamic responses of a +10% step change in the volumetric feed rate.
ment in the transient responses. The fast set-point tracking was also reflected in the faster reboiler duty responses of the MGS than those of the standard PI controller. For feed rate disturbance rejection, a (10% step change in the volumetric feed rate was introduced instantly at 10 min. The dynamic responses of the stage 7 temperature as well as the ETBE purity responses are shown in Figures 8 and 9 for the standard PI controller and MGS, respectively. The figures show that the MGS has better rejection of this disturbance than the standard PI controller. The stage 7 temperature was almost perfectly controlled at the set point of 127.68 °C by using the MGS. It results in far less overshoot and a shorter settling time. However, the ETBE purity slightly decreases (for a step increase in the feed rate) or increases (for a step decrease in the feed rate) to a new steady-state condition even though the stage 7 temperature was tightly controlled, as shown by the MGS. The corresponding values of the IAE and ITAE criteria are shown in Table 3. The control criteria confirm that the disturbance rejection of the feed rate is significantly improved by using the MGS. The reboiler duty moved smoothly to the new steady-state value for the standard PI controller. For the MGS, the transient responses of the reboiler duty were reflected on the fluctuation of the ETBE purity responses. The ETBE purity responses of
Figure 10. Dynamic responses of a 10% prereacted step change in the feed composition.
the standard PI controller are closer to the initial ETBE purity than those of the MGS. The dynamic changes resulting from a step change in the feed composition using the standard PI controller and MGS are shown in Figure 10. These changes, which were represented by changes in the prereacted feed from 80% to 70% of the isobutylene conversion, are introduced to test the feed composition rejection ability. The figure shows that the rejection ability of the MGS is again superior to that of the standard PI controller. The corresponding values of the IAE and ITAE criteria are shown in Table 3. However, the ETBE purity decreased significantly to about 96.58% and 96.71% for the MGS
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Figure 11. Dynamic responses of sinusoidal change in the reflux rate.
and PI controller, respectively. This result indicates that an estimator that relates the stage 7 temperature and the ETBE purity is required to estimate the cumulative effects of the simplified models. The reboiler duty response again shows the quick response of the MGS. Although the MGS is clearly superior, the performance of the standard PI controller designed at the nominal operating conditions (Table 1) is acceptable in the case studies as shown by the responses of the stage 7 temperature. For the regulatory problems, the PI controller even yields better rejection with respect to the ETBE purity responses. This is due to the reduced degree of the process nonlinearity. The reflux rate was kept constant at 2.2 L/min, and the reboiler duty was manipulated to control the stage 7 temperature as the inferred variable of the ETBE purity at the bottom product. A proper PI controller can also produce comparable performance to that of a pattern-based predictive control scheme for one-point control (ETBE purity control).16 For the pattern-based predictive control system, input-output experimental data can be utilized to provide process feature patterns qualitatively and quantitatively and are incorporated with a conventional controller. This could eliminate the requirement of exact process models, which could not be well described because of a large degree of uncertainties. Alternatively, input-output experimental data can be used to derive several simplified models for online process identification and tuning of the controller, which was conducted in this MGS control scheme. More local models could consequently complicate the control system design. However, the proposed MGS control scheme employs a switching strategy, which is therefore an important part of the control scheme. The capability of the proposed method to handle sinusoidal changes in the reflux rate, in which the MGS control scheme employs more local models, is discussed below. The reflux rate was varied in a sinusoidal pattern to simulate the uncertainties of the reflux rate, which may result from the condenser duty and/or distillate rate changes. In reality, the condenser duty may be directly affected by ambient temperature changes. Starting at a reflux rate of 2.2 L/min, the change was introduced at 10 min, reached the maximum of 2.4 L/min at 20 min, and then reached the minimum of 2.0 L/min at 40 min. It can be expected that the reboiler duty should be monitored and adjusted to maintain acceptable operating conditions. The dynamic changes resulting from a sinusoidal pattern over the period of 40 min in the reflux rate using the standard PI controller and MGS are shown in Figure 11. The figure and the corresponding values of the IAE and ITAE criteria shown in Table 3 clearly indicate the superiority of MGS over the stan-
Figure 12. Dynamic responses of a +10% step change in the volumetric feed rate (with noise).
Figure 13. Effects of switching parameters.
dard PI controller. However, both controllers result in fluctuation of the ETBE response in an acceptable range ((
