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Application of adaptive model-predictive control to a batch MMA polymerization reactor
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Chemical Engineering Science, Vol. 53, No. 21, pp. 3729—3739, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009–2509(98)00166–3 0009—2509/98/$—See front matter
Application of adaptive model-predictive control to a batch MMA polymerization reactor Hyung-Jun Rho, Yun-Jun Huh and Hyun-Ku Rhee* Department of Chemical Engineering, Seoul National University, Kwanak-ku, Seoul 151-742, Korea (Received 30 May 1997; accepted 6 March 1998) Abstract—An adaptive model-predictive control (AMPC) algorithm was applied to the temperature control of a batch polymerization reactor for polymethylmethacrylate (PMMA) and the tracking performance for set point changes was investigated. An experimental control system for the polymerization reactor was constructed with cascade structure. The process parameters of auto-regressive moving average exogenous (ARMAX) model were estimated by the recursive least squares (RLS) method with self-varying forgetting factor. To make the estimation robust, we introduced a dead-zone and normalization algorithm in the RLS estimator. The modelpredictive controller of unified approach was then implemented to control the reactor temperature on the basis of this process model. This indirect adaptive model-predictive controller showed better performance than the conventional PID controller for tracking set point changes, especially in the latter part of the reaction course when the gel effect became significant. Also, the control experiments were successfully conducted for tracking the optimal temperature trajectory which would yield polymer product having desired properties. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive model-predictive; batch reactor; control experiment; ARMAX; robust RLS; MMA polymerization. 1. INTRODUCTION
Model-predictive control is a control scheme in which the controller determines a control input profile that optimizes a given objective function on a time interval extending from the current time to the prediction horizon. Since the concept of model-predictive control was introduced to the industry by Richalet et al. (1978), various kinds of predictive controllers have been proposed in the literature whereas Soeterboek (1992) proposed the predictive controller of unified approach including the features of several well-known model-predictive controllers. Due to the complex reaction mechanism, the strong inherent nonlinearities, and the absence of steady state, the control of a batch polymerization process presents many problems. Furthermore, the model uncertainties are unavoidable as in most other chemical processes, and these features make the application of model-based controllers difficult. Adaptive control scheme is a potential solution and some studies have been performed in the domain of polymerization.
* Corresponding author. Tel.: #82-2-880-7405; fax: #822-888-7295; e-mail: hkrhee @plaza.snu.ac.kr.
Several kinds of adaptive, nonlinear and modelpredictive control algorithms have been evaluated for the control of a batch polymerization reactor. Kiparissides and Shah (1983) applied adaptive control algorithms and a fixed gain PID controller to a batch polyvinyl chloride reactor, and carried out simulation studies. An adaptive pole-assignment controller was developed and corroborated experimentally by Tzouanas and Shah (1989). The performance of the generalized minimum variance controller was compared to that of the fixed gain PID controller. However, these adaptive pole-assignment controllers are difficult to be tuned in comparison to the modelpredictive controller though it may be reinterpreted as the model predictive controller with adaptive algorithm (Soeterboek, 1992). More recently, Soroush and Kravaris (1992) applied the globally linearizing control method, one of the nonlinear control algorithms, to a batch polymerization reactor and experimentally compared its performance with the conventional PID controller. Nevertheless, the nonlinear control algorithms are hard to be applied to the industrial processes because they still have critical problems such as lack of robustness and stability. Houston and Schork (1987) applied the adaptive model-predictive control algorithm to a semibatch polymerization reactor with a linear approximation
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whereas Kiparissides et al. (1990) used the long-range predictive control algorithm for the control of molecular weights in a batch polymerization reactor. These authors, however, confined their studies to the numerical simulation of the control system using mathematical model. Mendoza-Bustos et al. (1990) applied the robust adaptive control algorithm to the control of monomer conversion in a continuousstirred tank reactor (CSTR) in the presence of reactive impurities. In their study, the forgetting factor was one of the tuning parameters, so it should be specified a priori. In the present work, a control system for a batch MMA polymerization reactor was constructed by applying the robust recursive least squares (RLS) method with time-varying forgetting factor and the adaptive model-predictive controller of unified approach. The controller performance was tested and corroborated by experiments. 2. REACTOR MODEL
The system considered in this study is the batch solution polymerization process of methylmethacrylate (MMA) using benzoylperoxide (BPO) as the initiator. The physical system consists of jacketed reactor, reaction mixture, heating-cooling water, and other utilities as shown in Fig. 1. 2.1. Kinetic mechanism The kinetic mechanism for the free radical solution polymerization of MMA is described in Table 1. In the table, I, M and S denote initiator, monomer and solvent, respectively, while /) represents the primary free radical. Also, R and P denote living and dead j j polymer with chain length j, respectively.
Fig. 1. Schematic diagram of the batch MMA polymerization reactor system. Table 1. Kinetic mechanism for the MMA solution polymerization kd I —P2/
Initiation
kj / )#M —PR 1
2.2. Mass balance equations From the kinetic mechanism of free radical solution polymerization, it is straightforward to formulate the following differential equations which describe the dynamics of polymerization reactor; i.e.
Propagation
kp R #M —PR i`1 i
Termination by combination
kic R #R —PR i j i`j
Initiator:
Termination by disproportionation
td #P R #R ) —kPP i j i j
Chain transfer to monomer
53. R #M —kPP #R i i 1
Chain transfer to solvent
534 R #S —kPP #S i i
1 d(I») "!k I d » dt
(1)
Monomer: 1 d(M») "!2f k I!(k #k )MG d p trm 0 » dt
(2) 1 d(G ») 1 "2f k I #k MG !k G G d p 0 t 0 1 » dt
Solvent: 1 d(S») " !k SG trs 0 » dt
(3)
(5)
1 d(G ») 2 "2f k I #k M(G #2G )!k G G d p 0 1 t 0 2 » dt
Moment of living polymer concentration: 1 d(G ») 0 "2f k I!k G2 d t 0 » dt
#(k M#k S)(G !G ) trm trs 0 1
(4)
#(k M#k S)(G !G ) trm trs 0 2
(6)
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Moment of dead polymer concentration: 1 d(F ») 1 1 " (k #k )G2#(k M#k S)G td 0 trm trs 0 » dt 2 t
(7)
1 d(F ») 1 "k G G #(k M#k S)G t 0 1 trm trs 1 » dt
(8)
1 d(F ») 2 "k G G #k G2#(k M#k S)G t 0 2 5# 1 trm trs 2 » dt (9) where I, M and S are the concentrations of initiator, monomer and solvent, respectively, and » denotes the volume of reaction mixture. Here, G and F are the k k kth moments of living and dead polymer concentrations, respectively, defined by = = G " + jkR and F " + jkP , k"0, 1, 2 k j k j j/1 j/1
(10)
from which the number- and weight-average molecular weights can be determined as G #F G #F 2]MW . 1]MW and Mw" 2 Mn" 1 m m G #F G #F 1 1 0 0 (11) The gel effect and glass effect were taken into account by applying the free volume correlation proposed by Schmidt and Ray (1981) whereas the change in volume » was considered by introducing the overall mass balance equation. The gel and glass effect correlations used in this study are given in Table 2, in which » , » , » represent the mean free volume, f fpc ftc the critical free volume for propagation and the critical free volume for termination, respectively. While the physical properties were taken from the literature (Brandrup and Immergut, 1989), the kinetic parameters were determined by applying the parameter estimation technique to experimental data (Chang, 1995). Presented in Fig. 2 are the time evolutions of the monomer conversion and the number-
Table 2. Correlations for the gel and glass effects used in this study For the propagation reactions: k "k g p p0 p 1.0 for » '» f fpc g" p 7.1]10~5 exp(171.53 » for » )» f f fpc » "0.05 fpc For the termination reactions:
G
k "k g t t0 t 0.10575 exp(17.5 » !0.01715 ¹) for » '» f f ftc g" t 2.3]10~6 exp(75.0 » ) for » )» f f ftc » "0.18564!2.965]10~4 ¹. ftc
G
Fig. 2. Comparison between the experimental and simulation results under various isothermal conditions.
and weight-average molecular weights obtained by experiments under five different isothermal conditions and these are compared with the results of model analysis. It is clearly seen that the simulation results are in good agreement with the experimental data at all temperatures. This indicates that the mathematical model together with the kinetic parameter values can be employed effectively for the simulation and openloop optimal design of batch MMA polymerization reactors. 2.3. Energy balance equations The energy balances for reactor and jacket can be formulated as follows: d(o C »¹ ) r pr r "Q »!ºA(¹ !¹ ) g r j dt
(12)
d(o C » ¹ ) c pc j j "o C q (¹*/!¹065)#ºA(¹ !¹ ) c pc c j j r j dt !º A (¹ !¹ ) (13) = = j = where Q , º and A denote the heat generation rate g per unit volume, the overall heat transfer coefficient and the heat exchange area, respectively. Further, the subscripts r, j, c and R represent the reaction mixture, jacket, heating-cooling water and ambient air, respectively. As the monomer conversion increases, the heat transfer efficiency becomes poorer and the heat generation rate becomes undetectable. Therefore, it is important to predict both the heat generation rate
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and the heat transfer coefficient as functions of the monomer conversion. In this study, the heat generation rate and the heat transfer coefficient were determined by employing the following equations: Q "(!*H )k MG g p p 0 º(X)"º(0)][0.2#0.8 exp (!5X5)]
(14) (15)
where X denotes the monomer conversion. The correlation for the latter was suggested by Soroush and Kravaris (1992), but here the parameter values were determined from the experimental data of this study. 2.4. Open-loop optimal reaction temperature trajectory The molecular weight distribution of the polymer is the most important element to the properties of the product. However, it is difficult to measure the molecular weight in real time. Therefore, the control of the molecular weight is often carried out by manipulating the reaction temperature which influences the monomer conversion and the molecular weight significantly. In this work, the optimal control theory was applied to calculate the optimal temperature trajectory for a batch MMA polymerization reactor (Ponnuswamy et al., 1987; Secchi et al., 1990; Kim et al., 1996). Application of the Pontryagin’s minimum principle to the mathematical model developed for the polymerization reactor system led to a two-point boundary value problem. The discretization control method was adopted to solve the two-point boundary value problem while the first-order gradient method was used to minimize the Hamiltonian. 3. CONTROL SYSTEM DESIGN
Cascade control strategy was introduced in order to improve the dynamic response to load changes while the recursive least-squares method was used for real-time system identification with the parametric process model. Based on the estimated model, the adaptive model-predictive controller calculates the set point of the jacket inlet temperature as the master controller, and then the slave PI controller manipulates the control valves for the heating and cooling water streams. Figure 3 shows the structure of the cascade control system. 3.1. Process model The parametric process model is represented by q~dB(q~1) C y(k)" u(k)# e(k) D A(q~1)
(16)
where q~1 is the shift operator, d is the time-delay of the process and the polynomials A and B are given by A(q~1)"1#a q~1#a q~2 1 2
(17) B(q~1)"b #b q~1. 1 2 The second term of eq. (16) represents the disturbance appearing in the output of the process. Here, C and D represent the disturbance characteristics while e(k) denotes a discrete white noise sequence.
Fig. 3. Configuration of the cascade control system.
3.2. System identification In adaptive systems, two features must be taken into account. First, the excitation condition may not be satisfied because the system consists of a closed loop. Second, the issue of robustness of adaptive systems has become of crucial interest. The excitation condition can be satisfied by requiring that the controller order be higher than the excitation order (As stro¨m and Wittenmark 1995). Another approach is to use the cascade structure. According to the constant-trace normalized RLS method with dead-zone (Mosca, 1995), the RLS parameter estimation algorithm with self-varying forgetting factor j is realized by the equations summarized in Table 3. Since the forgetting factor changes with the property of regression vector, it is not necessary to specify j. In Table 3, the symbol tilde denotes the normalized increment of a variable; e.g. yJ "dy(t)/m(t!1)
(18)
dy(t)"y(t)!y(t!1)
(19)
and the normalization factor m(t!1) is defined as m(t!1)"maxMm , Eu(t!1)EN, m ' 0. (20) 0 0 Data normalization procedure causes the parameter estimation law to see the effect of unmodeled dynamics as a bounded disturbance or to transform a possibly unbounded dynamic component into a bounded disturbance. This ensures that the normalized perturbation sequence is bounded. This RLS method is superior to the conventional RLS method. In particular, the dead zone facility is
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Table 3. Constant-trace normalized RLS method with dead-zone Gain matrix: i(t)P(t!1)uJ (t!1) i(t)" 1#uJ T(t!1)P(t!1)uJ (t!1) Parameter update: h(t)"h(t!1)#K(t)eJ (t) eJ (t)"yJ (t)!uJ (t!1)h(t!1) Exponential forgetting: uJ T(t!1)P(t!1)K(t) j(t)"1! Trace[P(0)]
Fig. 4. Block diagram of the adaptive model-predictive control system.
Covariance update: 1 P(t)" [1!K(t)uJ T(t!)]P(t!1) j(t)
Hp J" + [PyL (k#i)!P(1)w(k#i)]2 i/Hm Hp~dª Q 2 nu(k#i!1) #o + (23) Q d i/1 and this is minimized under following constraint:
Dead zone setting:
G
i(t)"
A
i3 0, 0,
C
B
k , if De8 (t)D*(1#k)1@2) 1#k otherwise
with k'0
used to switch off the estimate update whenever the absolute value of the prediction error is small compared to the expected size of a disturbance upper bound. The tuned parameters are d"2, m "0.01, 0 k"3.0 and )"0.01. Because the measured jacket inlet temperature was used as the input signals instead of the controller output, ‘the lack of excitation’ was not observed. 3.3. Predictive control algorithm of unified approach The i-step-ahead predictor used in adaptive model-predictive control (AMPC) is expressed as
D
/Pu(k#i!1)"0, 1)H (i)H !dK (24) c p where o is the input weighting factor and H , H and p m H represent the prediction horizon, the minimumc cost horizon and the control horizon, respectively. Both Q and Q denote polynomials introduced for n d loop-shaping of the controller output. The polynomial Q does not exist in other model-predictive d control algorithms such as dynamic matrix control. When the criterion function is quadratic and there are no constraints on the controller output, the criterion function can be minimized analytically. If the criterion function J is minimized with respect to the vector u, then the optimum u can be calculated by applying the condition LJ/Lu"0.
(25)
(22)
The resulting structure of the AMPC is depicted in Fig. 4. The detailed calculation procedure is not repeated here and can be found in Soeterboek (1992). It is well known that the closed-loop stability may not be guaranteed for indirect adaptive control algorithms. This happens when the controller and the estimator are not worked independently. This kind of excitation problems can be solved by order analysis between the controller and the estimator. Through the control experiments as well as the order analysis, the model-predictive controller used in this study was proven to be able to excite the input signals of the estimator persistently while the conventional PI or PID controller was not (As stro¨m and Wittenmark, 1995; Mosca, 1995).
The unified criterion function including weighting and structuring for the controller output is given by
The computer control algorithm was realized by the IBM 486 PC for the control of the reactor temperature. A cascade digital control system with PID
H PyL (k#i)"G u(k#i!dK !1)# i u(k!1) i AK F # i [y(k)!yL (k)] ¹
(21)
where ‘'’ denotes the estimated variable and P is a polynomial which is introduced to improve the servo behavior of the control system whereas G , H , i i and F are obtained from the following Diophantine i equations: CK P F "E #q~i i i DK DK BK P H "G #q~i`dK i . i DK AK
4. EXPERIMENTAL
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controller was set up as shown in Fig. 1. The jacketed stainless steel reactor had a capacity of two liters and was provided with a turbine-type stirrer for mixing. The speed of the stirrer was kept constant by a linked inverter throughout the course of the reaction and the average flow rate into the jacket was maintained at 3.0 l/min. Heating and cooling of the reaction mixture was carried out by heating—cooling water through the jacket. The valve stem positions of the hot and cold water lines were adjusted, in a cascade control configuration, in such a way that the jacket inlet temperature was kept equal to the desired value specified by the master controller. The conventional PID controller supported by control software FIX-DMACSTM was used as the slave controller. The temperatures of the reaction mixture, jacket inlet and outlet flows, and hot and cold water streams were measured by resistance-temperature detector (RTD) and the resulting 4—20 mA analog signals through transmitters were used for control and data acquisition by the computer. Texas Instrument 505 programmable logic controller (PLC) was used to convert analog signals to digital ones, and vice versa. The program bodies for the calculation of AMPC output and system parameters were written in Microsoft C language. For the purpose of tuning the AMPC parameters, we first investigated the characteristics of disturbance. From the result of control experiments, we noticed that disturbance of almost constant size was introduced to the control system from the start till the end of the reaction course. Therefore, the control parameters were tuned based on the zeroth-order disturbance model. In practice, the parameters of the AMPC were selected by the rule of thumb method (Soeterboek, 1992) and fine tuning was performed by trial-and-error method. In this study, the important parameters for control stability were found to be o and Q whereas those for control robustness were d H and C. The control parameters for the AMPC and c the slave PI controller are listed in Tables 4 and 5, respectively. When the PID controller was used as the master controller for comparison, the control parameters were tuned by trial-and-error method to obtain proportional band (PB) ("110%), q ("3.5 min) and I q ("0.001 min). We also tried with the Cohen and D Coon PID tuning method without success because a large overshoot could not be avoided. Ethylacetate (EA), benzoylperoxide (BPO), and methylmethacrylate (MMA) were used as solvent, initiator, and monomer, respectively. The inhibitor included in the monomer was eliminated by extraction before the operation (Collins et al., 1973). After the monomer and the solvent were charged into the reactor, dissolved oxygen was purged out with nitrogen gas. The reactor temperature was raised to the desired starting temperature and then the purified initiator dissolved in solvent was fed to start the polymerization. The set of reference conditions for the experiment is given in Table 6.
Table 4. Settings of the AMPC parameters used in the experiment Parameter
Value
H p H m d P u Q n Q d o C ¹ (s) s
36 7 6 1!0.85q~1 1!q~1 1!q~1 1—0.97q~1 0.02 1 5
Table 5. Parameter settings for the slave PI controller Parameter
Value
Proportional band (PB) q I ¹ s
55.0% 4.00 min 1s
Table 6. Reference conditions for the batch MMA solution polymerization Item
Value
Monomer charge Solvent charge Initiator charge Hot water temperature Cold water temperature
800 ml 800 ml 8.0 g 92°C 25°C
The reaction mixture was sampled with a syringe at successive times and the conversion was measured by the gravimetric method. The average molecular weights were measured by the gel permeation chromatography. 5. RESULTS AND DISCUSSION
5.1. ¹racking of set point changes Presented in Fig. 5 are the experimental results for tracking the stepwise changes in set point by the conventional PID controller while Fig. 6 shows the experimental results by the adaptive model-predictive controller. In each figure, Part (a) presents the reaction temperature tracking the set point and Part (b) depicts the slave controller output (valve stem position) and the master controller output which is the slave controller set point. Part (c) shows the simulation results of the monomer conversion and the heat generation rate divided by the thermal conductance, which may be regarded to represent the characteristics of nonlinearities. The reaction time was set equal to zero when the initiator solution was injected. The reaction
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Fig. 5. Experimental results for tracking the stepwise changes in set point for the reaction temperature by the conventional PID controller.
Fig. 6. Experimental results for tracking the stepwise changes in set point for the reaction temperature by the adaptive model-predictive controller.
temperature was lower than the set point at the start of reaction. In the earlier part of the reaction course, both controllers show similarly good performances. However, in the latter part when the monomer conversion is high and the heat transfer efficiency becomes low, the AMPC obviously performs better than the PID controller. It can be seen that the plant response with the AMPC is more active and faster than that with the PID controller at the instant of set point changes with only one overshoot. It is also noticed that the plant output is better regulated and the controller output is less sensitive to disturbances when the AMPC was applied. Furthermore, the PID controller shows a little delay in action, which causes the deviation from the set point, especially in the latter part. This is due to the fact that, while the AMPC uses the prior information about the set point changes and the time delay, this is not the case with the PID controller. In addition, the AMPC takes into account the process change caused by the decrease in the heat transfer efficiency and the change in the heat generation rate, and thus it does manipulate the jacket inlet temperature more satisfactorily in the latter part of the reaction course. It was indeed impossible to tune the parameters of PID controller to obtain the same rate of response. When the PID controller was used as the master controller as in Fig. 5, the output signal of the slave PI controller becomes irregular and spike-like though the slave controller parameters are the same as those in the case of AMPC. The spike-like signal is observed especially when abrupt changes occur in the set point.
The monomer conversion experimentally measured is found to be in good agreement with the predicted monomer conversion in Part (c) of both Figs 5 and 6. It is also noticed that the ratio Q /ºA is strongly g dependent on the reaction temperature, so small changes in the reaction temperature may induce large deviation of the reactor temperature. The transient behavior of monomer conversion turned out to be almost the same between the two experiments [cf. Figs 5(c) and 6(c)]. In addition, the variation of the heat generation rate divided by the thermal conductance was found very similar. These indicate that the two control experiments were conducted under almost the same thermal conditions. Nevertheless, different molecular weights were obtained because they are very sensitive to the reaction temperature unlike the monomer conversion as shown in Fig. 7, in which the average molecular weights were calculated by model simulation based on the reaction temperature measured experimentally. It is clearly seen from Fig. 7 that the deviation between the estimated values of the average molecular weights and the corresponding set point values is much larger with the PID controller than with the AMPC, especially when the reaction temperature is high. To make quantitative comparison of AMPC and PID controller, three different criteria for good control were applied. These are the integral of the square of the error (ISE), the integral of the absolute value of the error (IAE) and the integral of time-weighted absolute error (ITAE). The results are presented in Table 7. All the three criteria tested clearly shows that the AMPC is superior to the PID controller. It is
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evident that the quality of polymer produced can be controlled more effectively by using the AMPC. 5.2. Estimator performance with AMPC Proper parameter estimation is essential for the acceptable performance of an adaptive controller. Figure 8 illustrates the performance of the RLS parameter estimator used in the AMPC. Part (a) shows the measured reaction temperature and the predicted reaction temperature calculated by using the parametric plant model. It is clearly seen that the estimated reactor temperature is almost identical to the measured one. This indicates that the plant parameters are reasonably well estimated throughout the reaction course.
In Part (b), the variations in the jacket inlet temperature and the forgetting factor are illustrated. The reactor temperature and the jacket inlet temperature were used as the input signals to the RLS estimator. Since the forgetting factor varies with the input signal property, good estimation performance can be expected. That is, the trace of the covariance matrix is maintained constant during the estimation. The estimated values of the plant model parameters by the RLS method are presented in Part (c). The parameters kept changing during the operation. It is evident that the plant model was updated properly as the monomer conversion and the temperature set point changed.
Fig. 7. Simulation results for the average molecular weights and the polydispersity index under the conditions of Figs 5 and 6.
5.3. ¹racking of the optimal temperature trajectory with AMPC In general, the ultimate goal of the polymerization reactor temperature control is to obtain polymer product having desired properties. Therefore, the batch operation is often carried out for tracking a given temperature trajectory. Figure 9 shows the experimental results for tracking the optimal temperature trajectory which was calculated in an open-loop manner (off-line). For this the AMPC was employed as the master controller. In Part (a) of Fig. 9, it is shown that the reactor temperature traced the optimal trajectory sharply throughout the operation. The initial reactor temperature was different from the set point but this difference disappeared within the initial period of 15 min. Part (b) shows variations of the master and slave controller outputs. Although slave controller shows oscillatory behavior, its period falls in the range of 4—5 min. The estimated on-line parameters are presented in Part (c). Here it is noticed that the parameters a and a obtain larger values between 1 2 90 and 180 min. This type of change indicates that there occurred a significant change in the plant such as gel effect during this period. Figure 10 shows the time evolution of the monomer conversion, the number- and the weight-average molecular weights of polymer and the polydispersity index. The curves and the filled keys represent the simulation results and the experimental data,
Table 7. Comparison of various control criteria between the AMPC and the PID controller Control algorithm
ISE
IAE
IATE
Reaction temperature
AMPC PID control
58.50 379.46
60.12 134.80
98.71 12500.05
Monomer conversion
AMPC PID control
0.004 0.006
0.4 0.6
44.5 46.3
Number average molecular weight
AMPC PID control
6.14]107 5.42]108
6.73]104 1.68]105
3.55]106 7.72]106
Weight average molecular weight
AMPC PID control
1.35]108 9.67]108
9.43]104 2.55]105
4.57]106 1.45]107
Criteria
Batch MMA polymerization reactor
Fig. 8. RLS estimator performance for tracking the stepwise changes in set point by the adaptive model-predictive controller.
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Fig. 10. Time evolution of monomer conversion, average molecular weights and polydispersity index obtained by the tracking experiment of Fig. 9.
experimental data and simulation results. The polydispersity index increases gradually as the reaction proceeds. Clearly, the number- and weight-average molecular weights approach the desired values set a priori. This implies that the polymer product having desired properties can be obtained by controlling the reactor temperature in such a way to track the optimal trajectory. 6. CONCLUSIONS
Fig. 9. Experimental results for tracking the optimal temperature trajectory (target:conversion"0.7, Mn"100,000 g/mol, Mw"170,000 g/mol).
respectively. The initial temperature was 63.7°C and the total reaction time was 241 min. The desired number- and weight-average molecular weights were 100,000 g/mol and 170,000 g/mol, respectively. Not only the monomer conversion but also the average molecular weights show good agreement between the
The model-predictive controller of unified approach was combined with the robust RLS estimator. This adaptive control structure was then realized experimentally by the temperature control of the batch MMA polymerization process. Based on the experimental results, it is concluded that the adaptive modelpredictive controller (AMPC) performs better than the conventional PID controller in the sense of fast response, better regulation and less sensitive controller output. The RLS estimator with self-varying forgetting factor was also corroborated by experiment. The estimated plant model parameters tended to take larger values when significant changes occur in the plant. The AMPC showed a satisfactory performance in the control experiment for tracking the optimal temperature trajectory determined by the off-line optimization of the process model. Consequently, the AMPC is proven effective for the temperature control of batch polymerization processes to obtain polymer product having desired properties. It seems evident that the adaptive model-predictive controller developed using robust RLS method can be applied to other polymerization processes without additional efforts.
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Acknowledgement This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the Automation Research Center (ARC) at Pohang University of Science and Technology.
» » f » fpc
NOTATION
X y(k) w(k)
A B C C p d D e(k) f F k g p g t G k H p H m H c I J k i k p0 k i0 K m m 0 M MW m Mn Mw P P j q q~1 Q g Q ,Q n d R j S ¹ ¹*/ j ¹065 j ¹ s º u(k)
polynomial in the plant model, eq. (16) or heat transfer area, cm2 polynomial in the plant model, eq. (16) polynomial in the plant model, eq. (16) heat capacity, cal/g °C time delay of the process, eq. (16) polynomial in the plant model, eq. (16) discrete white noise with zero mean at t"k initiator efficiency, eq. (2) kth moment of dead polymer concentration, k"0, 1, 2 correction factor of propagation reaction rate correction factor of termination reaction rate kth moment of living polymer concentration, k"0, 1, 2 prediction horizon, eq. (23) minimum-cost horizon, eq. (23) control horizon, eq. (23) initiator or its concentration, mol/l criterion function, eq. (25) reaction rate constant of reaction i, 1/s or cm3/mol s initial rate constant for propagation reaction, cm3/mol s initial rate constant for termination reaction, cm3/mol s Kalman gain matrix, Table 3 normalization factor, eq. (20) minimum normalization factor, eq. (20) monomer or its concentration, mol/l molecular weight of monomer, g/mol number-average molecular weight of polymer, g/mol weight-average molecular weight of polymer, g/mol error covariance matrix, Table 3 dead polymer of chain length j or its concentration, mol/l flow rate, l/sec backward shift operator, q~1x(k)"x(k—1) heat generation rate per unit volume, cal/sec l polynomials in the unified criterion function living polymer of chain length j or its concentration, mol/l solvent or its concentration, mol/l temperature, °C jacket inlet temperature, °C jacket outlet temperature, °C sampling period, s overall heat transfer coefficient, cal/cm2 s °C controller output and process input at t"k
» ftc
volume of reaction mixture, l mean free volume, As /molecules critical free volume for propagation, As /molecules critical free volume for termination, As /molecules monomer conversion plant output at t"k set point at t"k
Greek letters d increment of a variable * difference operator, *"1!q~1 e prediction error h parameter vector i dead zone switch j forgetting factor k constant used in Table 3 o input weighting factor or density, g/l q derivative time of PID controller, min D q integral time of PID controller, min I u regression variable / characterizing polynomial of a disturbance primary radical created by initiator de/. composition ) expected size of the disturbance upper bound Subscripts c heating—cooling mixture d initiator decomposition i initiation j jacket p propagation reaction r reactor or reaction mixture t termination tc termination by combination td termination by disproportionation trm chain transfer to monomer trs chain transfer to solvent REFERENCES
As stro¨m, K. J. and Wittenmark, B. (1995) Adaptive Control. Addison-Wesley, Reading, MA, U.S.A. Brandrup, J. and Immergut, E. H. (1989) Polymer Handbook, 3rd edn. Wiley, New York, U.S.A. Chang, S. C. (1995) Modeling and control of a batch PMMA polymerization reactor. Ph.D. thesis, Seoul National University, Seoul, Korea. Collins, E. A., Bares, J. and Billmeyer, F. W. (1973) Experiments in Polymer Science. Wiley, New York, U.S.A. Houston, W. E. and Schork, F. J. (1987) Adaptive predictive control of a semibatch polymerization reactor. Polym. Process Engng 5, 119—147. Kim, I. S., Ahn, S. M. and Rhee H.-K. (1996) Physical property control for a batch polymerization reactor. Proceedings of 11th KACC in Seoul, pp. 263—266. Kiparissides, C. and Shah, S. L. (1983) Self-tuning and stable adaptive control of a batch polymerization reactor. Automatica 19, 225—235.
Batch MMA polymerization reactor
Kiparissides, C., Sidiropoulou, E., Voutetakis, S. and Frousakis, C. (1990) Control of molecular weight in a batch polymerization reactor using long-range predictive control methods. Chem. Engng Comm. 92, 1—22. Mendoza-Bustos, S. A., Penlides, A. and Cluett, W. R. (1990) Robust adaptive process control of a polymerization reactor. Comput. Chem. Engng 14, 251—258. Mosca, E. (1995) Optimal Predictive and Adaptive Control. Prentice-Hall, Englewood Cliffs, NJ. Ponnuswamy, S. R., Shah, S. L. and Kiparissides, C. A. (1987) Computer optimal control of batch polymerization reactor. Ind. Engng Chem. Res. 26, 2229—2236. Richalet, J., Rault, A., Testud, J. L. and Papon, J. (1978) Model predictive heuristic control: ap-
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plications to industrial process. Automatica 14, 413—428. Schmidt, A. D. and Ray, W. H. (1981) The dynamic behavior of continuous polymerization reactors—I. Isothermal solution polymerization in a CSTR. Chem. Engng Sci. 36, 1401—1410. Secchi, A. R., Lima, E. L. and Pinto, J. C. (1990) Constrained optimal batch polymerization reactor control. Polym. Engng Sci. 30, 1209—1219. Soeterboek, A. R. M. (1992) Predictive Control—A ºnified Approach. Prentice-Hall International, UK. Soroush, M. and Kravaris, C. (1992) Nonlinear control of a batch polymerization reactor: an experimental study. A.I.Ch.E. J. 38(9), 1429—1448. Tzouanas, V. K. and Shah, S. L. (1989) Adaptive pole-assignment control of a batch polymerization reactor. Chem. Engng Sci. 44(5), 1183—1193.
Chemical Engineering Science, Vol. 53, No. 21, pp. 3729—3739, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009–2509(98)00166–3 0009—2509/98/$—See front matter
Application of adaptive model-predictive control to a batch MMA polymerization reactor Hyung-Jun Rho, Yun-Jun Huh and Hyun-Ku Rhee* Department of Chemical Engineering, Seoul National University, Kwanak-ku, Seoul 151-742, Korea (Received 30 May 1997; accepted 6 March 1998) Abstract—An adaptive model-predictive control (AMPC) algorithm was applied to the temperature control of a batch polymerization reactor for polymethylmethacrylate (PMMA) and the tracking performance for set point changes was investigated. An experimental control system for the polymerization reactor was constructed with cascade structure. The process parameters of auto-regressive moving average exogenous (ARMAX) model were estimated by the recursive least squares (RLS) method with self-varying forgetting factor. To make the estimation robust, we introduced a dead-zone and normalization algorithm in the RLS estimator. The modelpredictive controller of unified approach was then implemented to control the reactor temperature on the basis of this process model. This indirect adaptive model-predictive controller showed better performance than the conventional PID controller for tracking set point changes, especially in the latter part of the reaction course when the gel effect became significant. Also, the control experiments were successfully conducted for tracking the optimal temperature trajectory which would yield polymer product having desired properties. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive model-predictive; batch reactor; control experiment; ARMAX; robust RLS; MMA polymerization. 1. INTRODUCTION
Model-predictive control is a control scheme in which the controller determines a control input profile that optimizes a given objective function on a time interval extending from the current time to the prediction horizon. Since the concept of model-predictive control was introduced to the industry by Richalet et al. (1978), various kinds of predictive controllers have been proposed in the literature whereas Soeterboek (1992) proposed the predictive controller of unified approach including the features of several well-known model-predictive controllers. Due to the complex reaction mechanism, the strong inherent nonlinearities, and the absence of steady state, the control of a batch polymerization process presents many problems. Furthermore, the model uncertainties are unavoidable as in most other chemical processes, and these features make the application of model-based controllers difficult. Adaptive control scheme is a potential solution and some studies have been performed in the domain of polymerization.
* Corresponding author. Tel.: #82-2-880-7405; fax: #822-888-7295; e-mail: hkrhee @plaza.snu.ac.kr.
Several kinds of adaptive, nonlinear and modelpredictive control algorithms have been evaluated for the control of a batch polymerization reactor. Kiparissides and Shah (1983) applied adaptive control algorithms and a fixed gain PID controller to a batch polyvinyl chloride reactor, and carried out simulation studies. An adaptive pole-assignment controller was developed and corroborated experimentally by Tzouanas and Shah (1989). The performance of the generalized minimum variance controller was compared to that of the fixed gain PID controller. However, these adaptive pole-assignment controllers are difficult to be tuned in comparison to the modelpredictive controller though it may be reinterpreted as the model predictive controller with adaptive algorithm (Soeterboek, 1992). More recently, Soroush and Kravaris (1992) applied the globally linearizing control method, one of the nonlinear control algorithms, to a batch polymerization reactor and experimentally compared its performance with the conventional PID controller. Nevertheless, the nonlinear control algorithms are hard to be applied to the industrial processes because they still have critical problems such as lack of robustness and stability. Houston and Schork (1987) applied the adaptive model-predictive control algorithm to a semibatch polymerization reactor with a linear approximation
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whereas Kiparissides et al. (1990) used the long-range predictive control algorithm for the control of molecular weights in a batch polymerization reactor. These authors, however, confined their studies to the numerical simulation of the control system using mathematical model. Mendoza-Bustos et al. (1990) applied the robust adaptive control algorithm to the control of monomer conversion in a continuousstirred tank reactor (CSTR) in the presence of reactive impurities. In their study, the forgetting factor was one of the tuning parameters, so it should be specified a priori. In the present work, a control system for a batch MMA polymerization reactor was constructed by applying the robust recursive least squares (RLS) method with time-varying forgetting factor and the adaptive model-predictive controller of unified approach. The controller performance was tested and corroborated by experiments. 2. REACTOR MODEL
The system considered in this study is the batch solution polymerization process of methylmethacrylate (MMA) using benzoylperoxide (BPO) as the initiator. The physical system consists of jacketed reactor, reaction mixture, heating-cooling water, and other utilities as shown in Fig. 1. 2.1. Kinetic mechanism The kinetic mechanism for the free radical solution polymerization of MMA is described in Table 1. In the table, I, M and S denote initiator, monomer and solvent, respectively, while /) represents the primary free radical. Also, R and P denote living and dead j j polymer with chain length j, respectively.
Fig. 1. Schematic diagram of the batch MMA polymerization reactor system. Table 1. Kinetic mechanism for the MMA solution polymerization kd I —P2/
Initiation
kj / )#M —PR 1
2.2. Mass balance equations From the kinetic mechanism of free radical solution polymerization, it is straightforward to formulate the following differential equations which describe the dynamics of polymerization reactor; i.e.
Propagation
kp R #M —PR i`1 i
Termination by combination
kic R #R —PR i j i`j
Initiator:
Termination by disproportionation
td #P R #R ) —kPP i j i j
Chain transfer to monomer
53. R #M —kPP #R i i 1
Chain transfer to solvent
534 R #S —kPP #S i i
1 d(I») "!k I d » dt
(1)
Monomer: 1 d(M») "!2f k I!(k #k )MG d p trm 0 » dt
(2) 1 d(G ») 1 "2f k I #k MG !k G G d p 0 t 0 1 » dt
Solvent: 1 d(S») " !k SG trs 0 » dt
(3)
(5)
1 d(G ») 2 "2f k I #k M(G #2G )!k G G d p 0 1 t 0 2 » dt
Moment of living polymer concentration: 1 d(G ») 0 "2f k I!k G2 d t 0 » dt
#(k M#k S)(G !G ) trm trs 0 1
(4)
#(k M#k S)(G !G ) trm trs 0 2
(6)
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Moment of dead polymer concentration: 1 d(F ») 1 1 " (k #k )G2#(k M#k S)G td 0 trm trs 0 » dt 2 t
(7)
1 d(F ») 1 "k G G #(k M#k S)G t 0 1 trm trs 1 » dt
(8)
1 d(F ») 2 "k G G #k G2#(k M#k S)G t 0 2 5# 1 trm trs 2 » dt (9) where I, M and S are the concentrations of initiator, monomer and solvent, respectively, and » denotes the volume of reaction mixture. Here, G and F are the k k kth moments of living and dead polymer concentrations, respectively, defined by = = G " + jkR and F " + jkP , k"0, 1, 2 k j k j j/1 j/1
(10)
from which the number- and weight-average molecular weights can be determined as G #F G #F 2]MW . 1]MW and Mw" 2 Mn" 1 m m G #F G #F 1 1 0 0 (11) The gel effect and glass effect were taken into account by applying the free volume correlation proposed by Schmidt and Ray (1981) whereas the change in volume » was considered by introducing the overall mass balance equation. The gel and glass effect correlations used in this study are given in Table 2, in which » , » , » represent the mean free volume, f fpc ftc the critical free volume for propagation and the critical free volume for termination, respectively. While the physical properties were taken from the literature (Brandrup and Immergut, 1989), the kinetic parameters were determined by applying the parameter estimation technique to experimental data (Chang, 1995). Presented in Fig. 2 are the time evolutions of the monomer conversion and the number-
Table 2. Correlations for the gel and glass effects used in this study For the propagation reactions: k "k g p p0 p 1.0 for » '» f fpc g" p 7.1]10~5 exp(171.53 » for » )» f f fpc » "0.05 fpc For the termination reactions:
G
k "k g t t0 t 0.10575 exp(17.5 » !0.01715 ¹) for » '» f f ftc g" t 2.3]10~6 exp(75.0 » ) for » )» f f ftc » "0.18564!2.965]10~4 ¹. ftc
G
Fig. 2. Comparison between the experimental and simulation results under various isothermal conditions.
and weight-average molecular weights obtained by experiments under five different isothermal conditions and these are compared with the results of model analysis. It is clearly seen that the simulation results are in good agreement with the experimental data at all temperatures. This indicates that the mathematical model together with the kinetic parameter values can be employed effectively for the simulation and openloop optimal design of batch MMA polymerization reactors. 2.3. Energy balance equations The energy balances for reactor and jacket can be formulated as follows: d(o C »¹ ) r pr r "Q »!ºA(¹ !¹ ) g r j dt
(12)
d(o C » ¹ ) c pc j j "o C q (¹*/!¹065)#ºA(¹ !¹ ) c pc c j j r j dt !º A (¹ !¹ ) (13) = = j = where Q , º and A denote the heat generation rate g per unit volume, the overall heat transfer coefficient and the heat exchange area, respectively. Further, the subscripts r, j, c and R represent the reaction mixture, jacket, heating-cooling water and ambient air, respectively. As the monomer conversion increases, the heat transfer efficiency becomes poorer and the heat generation rate becomes undetectable. Therefore, it is important to predict both the heat generation rate
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and the heat transfer coefficient as functions of the monomer conversion. In this study, the heat generation rate and the heat transfer coefficient were determined by employing the following equations: Q "(!*H )k MG g p p 0 º(X)"º(0)][0.2#0.8 exp (!5X5)]
(14) (15)
where X denotes the monomer conversion. The correlation for the latter was suggested by Soroush and Kravaris (1992), but here the parameter values were determined from the experimental data of this study. 2.4. Open-loop optimal reaction temperature trajectory The molecular weight distribution of the polymer is the most important element to the properties of the product. However, it is difficult to measure the molecular weight in real time. Therefore, the control of the molecular weight is often carried out by manipulating the reaction temperature which influences the monomer conversion and the molecular weight significantly. In this work, the optimal control theory was applied to calculate the optimal temperature trajectory for a batch MMA polymerization reactor (Ponnuswamy et al., 1987; Secchi et al., 1990; Kim et al., 1996). Application of the Pontryagin’s minimum principle to the mathematical model developed for the polymerization reactor system led to a two-point boundary value problem. The discretization control method was adopted to solve the two-point boundary value problem while the first-order gradient method was used to minimize the Hamiltonian. 3. CONTROL SYSTEM DESIGN
Cascade control strategy was introduced in order to improve the dynamic response to load changes while the recursive least-squares method was used for real-time system identification with the parametric process model. Based on the estimated model, the adaptive model-predictive controller calculates the set point of the jacket inlet temperature as the master controller, and then the slave PI controller manipulates the control valves for the heating and cooling water streams. Figure 3 shows the structure of the cascade control system. 3.1. Process model The parametric process model is represented by q~dB(q~1) C y(k)" u(k)# e(k) D A(q~1)
(16)
where q~1 is the shift operator, d is the time-delay of the process and the polynomials A and B are given by A(q~1)"1#a q~1#a q~2 1 2
(17) B(q~1)"b #b q~1. 1 2 The second term of eq. (16) represents the disturbance appearing in the output of the process. Here, C and D represent the disturbance characteristics while e(k) denotes a discrete white noise sequence.
Fig. 3. Configuration of the cascade control system.
3.2. System identification In adaptive systems, two features must be taken into account. First, the excitation condition may not be satisfied because the system consists of a closed loop. Second, the issue of robustness of adaptive systems has become of crucial interest. The excitation condition can be satisfied by requiring that the controller order be higher than the excitation order (As stro¨m and Wittenmark 1995). Another approach is to use the cascade structure. According to the constant-trace normalized RLS method with dead-zone (Mosca, 1995), the RLS parameter estimation algorithm with self-varying forgetting factor j is realized by the equations summarized in Table 3. Since the forgetting factor changes with the property of regression vector, it is not necessary to specify j. In Table 3, the symbol tilde denotes the normalized increment of a variable; e.g. yJ "dy(t)/m(t!1)
(18)
dy(t)"y(t)!y(t!1)
(19)
and the normalization factor m(t!1) is defined as m(t!1)"maxMm , Eu(t!1)EN, m ' 0. (20) 0 0 Data normalization procedure causes the parameter estimation law to see the effect of unmodeled dynamics as a bounded disturbance or to transform a possibly unbounded dynamic component into a bounded disturbance. This ensures that the normalized perturbation sequence is bounded. This RLS method is superior to the conventional RLS method. In particular, the dead zone facility is
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Table 3. Constant-trace normalized RLS method with dead-zone Gain matrix: i(t)P(t!1)uJ (t!1) i(t)" 1#uJ T(t!1)P(t!1)uJ (t!1) Parameter update: h(t)"h(t!1)#K(t)eJ (t) eJ (t)"yJ (t)!uJ (t!1)h(t!1) Exponential forgetting: uJ T(t!1)P(t!1)K(t) j(t)"1! Trace[P(0)]
Fig. 4. Block diagram of the adaptive model-predictive control system.
Covariance update: 1 P(t)" [1!K(t)uJ T(t!)]P(t!1) j(t)
Hp J" + [PyL (k#i)!P(1)w(k#i)]2 i/Hm Hp~dª Q 2 nu(k#i!1) #o + (23) Q d i/1 and this is minimized under following constraint:
Dead zone setting:
G
i(t)"
A
i3 0, 0,
C
B
k , if De8 (t)D*(1#k)1@2) 1#k otherwise
with k'0
used to switch off the estimate update whenever the absolute value of the prediction error is small compared to the expected size of a disturbance upper bound. The tuned parameters are d"2, m "0.01, 0 k"3.0 and )"0.01. Because the measured jacket inlet temperature was used as the input signals instead of the controller output, ‘the lack of excitation’ was not observed. 3.3. Predictive control algorithm of unified approach The i-step-ahead predictor used in adaptive model-predictive control (AMPC) is expressed as
D
/Pu(k#i!1)"0, 1)H (i)H !dK (24) c p where o is the input weighting factor and H , H and p m H represent the prediction horizon, the minimumc cost horizon and the control horizon, respectively. Both Q and Q denote polynomials introduced for n d loop-shaping of the controller output. The polynomial Q does not exist in other model-predictive d control algorithms such as dynamic matrix control. When the criterion function is quadratic and there are no constraints on the controller output, the criterion function can be minimized analytically. If the criterion function J is minimized with respect to the vector u, then the optimum u can be calculated by applying the condition LJ/Lu"0.
(25)
(22)
The resulting structure of the AMPC is depicted in Fig. 4. The detailed calculation procedure is not repeated here and can be found in Soeterboek (1992). It is well known that the closed-loop stability may not be guaranteed for indirect adaptive control algorithms. This happens when the controller and the estimator are not worked independently. This kind of excitation problems can be solved by order analysis between the controller and the estimator. Through the control experiments as well as the order analysis, the model-predictive controller used in this study was proven to be able to excite the input signals of the estimator persistently while the conventional PI or PID controller was not (As stro¨m and Wittenmark, 1995; Mosca, 1995).
The unified criterion function including weighting and structuring for the controller output is given by
The computer control algorithm was realized by the IBM 486 PC for the control of the reactor temperature. A cascade digital control system with PID
H PyL (k#i)"G u(k#i!dK !1)# i u(k!1) i AK F # i [y(k)!yL (k)] ¹
(21)
where ‘'’ denotes the estimated variable and P is a polynomial which is introduced to improve the servo behavior of the control system whereas G , H , i i and F are obtained from the following Diophantine i equations: CK P F "E #q~i i i DK DK BK P H "G #q~i`dK i . i DK AK
4. EXPERIMENTAL
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controller was set up as shown in Fig. 1. The jacketed stainless steel reactor had a capacity of two liters and was provided with a turbine-type stirrer for mixing. The speed of the stirrer was kept constant by a linked inverter throughout the course of the reaction and the average flow rate into the jacket was maintained at 3.0 l/min. Heating and cooling of the reaction mixture was carried out by heating—cooling water through the jacket. The valve stem positions of the hot and cold water lines were adjusted, in a cascade control configuration, in such a way that the jacket inlet temperature was kept equal to the desired value specified by the master controller. The conventional PID controller supported by control software FIX-DMACSTM was used as the slave controller. The temperatures of the reaction mixture, jacket inlet and outlet flows, and hot and cold water streams were measured by resistance-temperature detector (RTD) and the resulting 4—20 mA analog signals through transmitters were used for control and data acquisition by the computer. Texas Instrument 505 programmable logic controller (PLC) was used to convert analog signals to digital ones, and vice versa. The program bodies for the calculation of AMPC output and system parameters were written in Microsoft C language. For the purpose of tuning the AMPC parameters, we first investigated the characteristics of disturbance. From the result of control experiments, we noticed that disturbance of almost constant size was introduced to the control system from the start till the end of the reaction course. Therefore, the control parameters were tuned based on the zeroth-order disturbance model. In practice, the parameters of the AMPC were selected by the rule of thumb method (Soeterboek, 1992) and fine tuning was performed by trial-and-error method. In this study, the important parameters for control stability were found to be o and Q whereas those for control robustness were d H and C. The control parameters for the AMPC and c the slave PI controller are listed in Tables 4 and 5, respectively. When the PID controller was used as the master controller for comparison, the control parameters were tuned by trial-and-error method to obtain proportional band (PB) ("110%), q ("3.5 min) and I q ("0.001 min). We also tried with the Cohen and D Coon PID tuning method without success because a large overshoot could not be avoided. Ethylacetate (EA), benzoylperoxide (BPO), and methylmethacrylate (MMA) were used as solvent, initiator, and monomer, respectively. The inhibitor included in the monomer was eliminated by extraction before the operation (Collins et al., 1973). After the monomer and the solvent were charged into the reactor, dissolved oxygen was purged out with nitrogen gas. The reactor temperature was raised to the desired starting temperature and then the purified initiator dissolved in solvent was fed to start the polymerization. The set of reference conditions for the experiment is given in Table 6.
Table 4. Settings of the AMPC parameters used in the experiment Parameter
Value
H p H m d P u Q n Q d o C ¹ (s) s
36 7 6 1!0.85q~1 1!q~1 1!q~1 1—0.97q~1 0.02 1 5
Table 5. Parameter settings for the slave PI controller Parameter
Value
Proportional band (PB) q I ¹ s
55.0% 4.00 min 1s
Table 6. Reference conditions for the batch MMA solution polymerization Item
Value
Monomer charge Solvent charge Initiator charge Hot water temperature Cold water temperature
800 ml 800 ml 8.0 g 92°C 25°C
The reaction mixture was sampled with a syringe at successive times and the conversion was measured by the gravimetric method. The average molecular weights were measured by the gel permeation chromatography. 5. RESULTS AND DISCUSSION
5.1. ¹racking of set point changes Presented in Fig. 5 are the experimental results for tracking the stepwise changes in set point by the conventional PID controller while Fig. 6 shows the experimental results by the adaptive model-predictive controller. In each figure, Part (a) presents the reaction temperature tracking the set point and Part (b) depicts the slave controller output (valve stem position) and the master controller output which is the slave controller set point. Part (c) shows the simulation results of the monomer conversion and the heat generation rate divided by the thermal conductance, which may be regarded to represent the characteristics of nonlinearities. The reaction time was set equal to zero when the initiator solution was injected. The reaction
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Fig. 5. Experimental results for tracking the stepwise changes in set point for the reaction temperature by the conventional PID controller.
Fig. 6. Experimental results for tracking the stepwise changes in set point for the reaction temperature by the adaptive model-predictive controller.
temperature was lower than the set point at the start of reaction. In the earlier part of the reaction course, both controllers show similarly good performances. However, in the latter part when the monomer conversion is high and the heat transfer efficiency becomes low, the AMPC obviously performs better than the PID controller. It can be seen that the plant response with the AMPC is more active and faster than that with the PID controller at the instant of set point changes with only one overshoot. It is also noticed that the plant output is better regulated and the controller output is less sensitive to disturbances when the AMPC was applied. Furthermore, the PID controller shows a little delay in action, which causes the deviation from the set point, especially in the latter part. This is due to the fact that, while the AMPC uses the prior information about the set point changes and the time delay, this is not the case with the PID controller. In addition, the AMPC takes into account the process change caused by the decrease in the heat transfer efficiency and the change in the heat generation rate, and thus it does manipulate the jacket inlet temperature more satisfactorily in the latter part of the reaction course. It was indeed impossible to tune the parameters of PID controller to obtain the same rate of response. When the PID controller was used as the master controller as in Fig. 5, the output signal of the slave PI controller becomes irregular and spike-like though the slave controller parameters are the same as those in the case of AMPC. The spike-like signal is observed especially when abrupt changes occur in the set point.
The monomer conversion experimentally measured is found to be in good agreement with the predicted monomer conversion in Part (c) of both Figs 5 and 6. It is also noticed that the ratio Q /ºA is strongly g dependent on the reaction temperature, so small changes in the reaction temperature may induce large deviation of the reactor temperature. The transient behavior of monomer conversion turned out to be almost the same between the two experiments [cf. Figs 5(c) and 6(c)]. In addition, the variation of the heat generation rate divided by the thermal conductance was found very similar. These indicate that the two control experiments were conducted under almost the same thermal conditions. Nevertheless, different molecular weights were obtained because they are very sensitive to the reaction temperature unlike the monomer conversion as shown in Fig. 7, in which the average molecular weights were calculated by model simulation based on the reaction temperature measured experimentally. It is clearly seen from Fig. 7 that the deviation between the estimated values of the average molecular weights and the corresponding set point values is much larger with the PID controller than with the AMPC, especially when the reaction temperature is high. To make quantitative comparison of AMPC and PID controller, three different criteria for good control were applied. These are the integral of the square of the error (ISE), the integral of the absolute value of the error (IAE) and the integral of time-weighted absolute error (ITAE). The results are presented in Table 7. All the three criteria tested clearly shows that the AMPC is superior to the PID controller. It is
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evident that the quality of polymer produced can be controlled more effectively by using the AMPC. 5.2. Estimator performance with AMPC Proper parameter estimation is essential for the acceptable performance of an adaptive controller. Figure 8 illustrates the performance of the RLS parameter estimator used in the AMPC. Part (a) shows the measured reaction temperature and the predicted reaction temperature calculated by using the parametric plant model. It is clearly seen that the estimated reactor temperature is almost identical to the measured one. This indicates that the plant parameters are reasonably well estimated throughout the reaction course.
In Part (b), the variations in the jacket inlet temperature and the forgetting factor are illustrated. The reactor temperature and the jacket inlet temperature were used as the input signals to the RLS estimator. Since the forgetting factor varies with the input signal property, good estimation performance can be expected. That is, the trace of the covariance matrix is maintained constant during the estimation. The estimated values of the plant model parameters by the RLS method are presented in Part (c). The parameters kept changing during the operation. It is evident that the plant model was updated properly as the monomer conversion and the temperature set point changed.
Fig. 7. Simulation results for the average molecular weights and the polydispersity index under the conditions of Figs 5 and 6.
5.3. ¹racking of the optimal temperature trajectory with AMPC In general, the ultimate goal of the polymerization reactor temperature control is to obtain polymer product having desired properties. Therefore, the batch operation is often carried out for tracking a given temperature trajectory. Figure 9 shows the experimental results for tracking the optimal temperature trajectory which was calculated in an open-loop manner (off-line). For this the AMPC was employed as the master controller. In Part (a) of Fig. 9, it is shown that the reactor temperature traced the optimal trajectory sharply throughout the operation. The initial reactor temperature was different from the set point but this difference disappeared within the initial period of 15 min. Part (b) shows variations of the master and slave controller outputs. Although slave controller shows oscillatory behavior, its period falls in the range of 4—5 min. The estimated on-line parameters are presented in Part (c). Here it is noticed that the parameters a and a obtain larger values between 1 2 90 and 180 min. This type of change indicates that there occurred a significant change in the plant such as gel effect during this period. Figure 10 shows the time evolution of the monomer conversion, the number- and the weight-average molecular weights of polymer and the polydispersity index. The curves and the filled keys represent the simulation results and the experimental data,
Table 7. Comparison of various control criteria between the AMPC and the PID controller Control algorithm
ISE
IAE
IATE
Reaction temperature
AMPC PID control
58.50 379.46
60.12 134.80
98.71 12500.05
Monomer conversion
AMPC PID control
0.004 0.006
0.4 0.6
44.5 46.3
Number average molecular weight
AMPC PID control
6.14]107 5.42]108
6.73]104 1.68]105
3.55]106 7.72]106
Weight average molecular weight
AMPC PID control
1.35]108 9.67]108
9.43]104 2.55]105
4.57]106 1.45]107
Criteria
Batch MMA polymerization reactor
Fig. 8. RLS estimator performance for tracking the stepwise changes in set point by the adaptive model-predictive controller.
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Fig. 10. Time evolution of monomer conversion, average molecular weights and polydispersity index obtained by the tracking experiment of Fig. 9.
experimental data and simulation results. The polydispersity index increases gradually as the reaction proceeds. Clearly, the number- and weight-average molecular weights approach the desired values set a priori. This implies that the polymer product having desired properties can be obtained by controlling the reactor temperature in such a way to track the optimal trajectory. 6. CONCLUSIONS
Fig. 9. Experimental results for tracking the optimal temperature trajectory (target:conversion"0.7, Mn"100,000 g/mol, Mw"170,000 g/mol).
respectively. The initial temperature was 63.7°C and the total reaction time was 241 min. The desired number- and weight-average molecular weights were 100,000 g/mol and 170,000 g/mol, respectively. Not only the monomer conversion but also the average molecular weights show good agreement between the
The model-predictive controller of unified approach was combined with the robust RLS estimator. This adaptive control structure was then realized experimentally by the temperature control of the batch MMA polymerization process. Based on the experimental results, it is concluded that the adaptive modelpredictive controller (AMPC) performs better than the conventional PID controller in the sense of fast response, better regulation and less sensitive controller output. The RLS estimator with self-varying forgetting factor was also corroborated by experiment. The estimated plant model parameters tended to take larger values when significant changes occur in the plant. The AMPC showed a satisfactory performance in the control experiment for tracking the optimal temperature trajectory determined by the off-line optimization of the process model. Consequently, the AMPC is proven effective for the temperature control of batch polymerization processes to obtain polymer product having desired properties. It seems evident that the adaptive model-predictive controller developed using robust RLS method can be applied to other polymerization processes without additional efforts.
H.-J. Rho et al.
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Acknowledgement This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the Automation Research Center (ARC) at Pohang University of Science and Technology.
» » f » fpc
NOTATION
X y(k) w(k)
A B C C p d D e(k) f F k g p g t G k H p H m H c I J k i k p0 k i0 K m m 0 M MW m Mn Mw P P j q q~1 Q g Q ,Q n d R j S ¹ ¹*/ j ¹065 j ¹ s º u(k)
polynomial in the plant model, eq. (16) or heat transfer area, cm2 polynomial in the plant model, eq. (16) polynomial in the plant model, eq. (16) heat capacity, cal/g °C time delay of the process, eq. (16) polynomial in the plant model, eq. (16) discrete white noise with zero mean at t"k initiator efficiency, eq. (2) kth moment of dead polymer concentration, k"0, 1, 2 correction factor of propagation reaction rate correction factor of termination reaction rate kth moment of living polymer concentration, k"0, 1, 2 prediction horizon, eq. (23) minimum-cost horizon, eq. (23) control horizon, eq. (23) initiator or its concentration, mol/l criterion function, eq. (25) reaction rate constant of reaction i, 1/s or cm3/mol s initial rate constant for propagation reaction, cm3/mol s initial rate constant for termination reaction, cm3/mol s Kalman gain matrix, Table 3 normalization factor, eq. (20) minimum normalization factor, eq. (20) monomer or its concentration, mol/l molecular weight of monomer, g/mol number-average molecular weight of polymer, g/mol weight-average molecular weight of polymer, g/mol error covariance matrix, Table 3 dead polymer of chain length j or its concentration, mol/l flow rate, l/sec backward shift operator, q~1x(k)"x(k—1) heat generation rate per unit volume, cal/sec l polynomials in the unified criterion function living polymer of chain length j or its concentration, mol/l solvent or its concentration, mol/l temperature, °C jacket inlet temperature, °C jacket outlet temperature, °C sampling period, s overall heat transfer coefficient, cal/cm2 s °C controller output and process input at t"k
» ftc
volume of reaction mixture, l mean free volume, As /molecules critical free volume for propagation, As /molecules critical free volume for termination, As /molecules monomer conversion plant output at t"k set point at t"k
Greek letters d increment of a variable * difference operator, *"1!q~1 e prediction error h parameter vector i dead zone switch j forgetting factor k constant used in Table 3 o input weighting factor or density, g/l q derivative time of PID controller, min D q integral time of PID controller, min I u regression variable / characterizing polynomial of a disturbance primary radical created by initiator de/. composition ) expected size of the disturbance upper bound Subscripts c heating—cooling mixture d initiator decomposition i initiation j jacket p propagation reaction r reactor or reaction mixture t termination tc termination by combination td termination by disproportionation trm chain transfer to monomer trs chain transfer to solvent REFERENCES
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