Chem. Eng. Comm., 193:782–800, 2006 Copyright # Taylor & Francis Group, LLC ISSN: 0098-6445 print/1563-5201 online DOI: 10.1080/00986440500267196
Control of a Continuous Polymerization Reactor by Wiener Input/Output Data-Based Predictive Controller with Direct Inverse Identification IN-HYOUP SONG, RO-JI OH, MYUNG-JUNE PARK, AND HYUN-KU RHEE School of Chemical and Biological Engineering and Institute of Chemical Processes, Seoul National University, Seoul, South Korea
KEE-YOUN YOO Department of Chemical Engineering, Seoul National University of Technology, Seoul, South Korea This article reports an experimental study for the identification and predictive control of a continuous methyl methacrylate (MMA) solution polymerization reactor. The Wiener model was introduced to identify the polymerization reactor in a more efficient manner than the conventional methods of Wiener model identification. In particular, the method of subspace identification was employed and the inverse of the nonlinear part was directly identified. The input variables in this work were the jacket inlet temperature and the feed flow rate, while the monomer conversion and the weight average molecular weight were selected as the output variables. On the basis of the identified model a Wiener-type input=output data-based predictive controller was designed and applied to the property control of polymer product in the continuous MMA polymerization reactor by conducting an on-line digital control experiment with online densitometer and viscometer. Despite the complex and nonlinear characteristics of the polymerization reactor, the proposed controller was found to perform satisfactorily for property control in the multiple-input multiple-output system with input constraints for both set-point tracking and disturbance rejection. This was also confirmed by simulation results. Keywords MMA; MPC; Polymerization reactor; Process control; Subspace identification
Introduction For the control of polymer properties, several advanced control algorithms have been adapted to the polymerization reactor. Among these, model predictive control (MPC) has been found the most successful. Although many researchers adopted the model predictive control algorithm for the control of polymerization reactors, there have been only a few published reports on the actual implementation of control schemes for continuous polymerization reactor systems through experiments. This Address correspondence to Hyun-Ku Rhee, School of Chemical and Biological Engineering and Institute of Chemical Processes, Seoul National University, Kwanak-ku, Seoul 151742, South Korea. E-mail:
[email protected]
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may have been because it is difficult to measure the physical properties on-line and to implement the advanced control algorithm in actual polymerization processes. Garcia (1984) implemented NLMPC for the control of block synthetic rubber production in a semibatch reactor and obtained excellent results. The author linearized the nonlinear model at each sampling period to optimize the objective function via quadratic programming. Hidalgo and Browsilow (1990) used a similar strategy to control the operation of an unstable styrene solution polymerization reactor. They calculated optimal inputs via nonlinear programming instead of successive linearization. A new nonlinear model-based predictive control algorithm was proposed by Mutha et al. (1997), who illustrated its performance by applying it to a semibatch acrylonitrile-butadiene (NBR) emulsion copolymerization process. Ahn et al. (1999) conducted control experiments for a methyl methacrylate (MMA) polymerization reactor on the basis of extended Kalman filter-based nonlinear model predictive control. Review articles for the control strategies for polymerization reactors have also been published (Embirucu et al., 1996; MacGregor et al., 1994). Despite their remarkable control performance, control schemes based on the first principles model are rather scarce in commercial polymerization processes since the first principles model, consisting of a large number of differential and difference equations, frequently exhibits deviation from the real polymerization reactor. The deviation is generally caused by the simplification of the first principles model, which is intended to make the controller design procedure simple. It seems that such a bias may be removed by using a more elaborate model. However, there exist indispensable features in modeling polymerization reactors that would inherently give rise to a bias such as unknown rate constants or model parameters for polymerization reactor systems. Consequently, using a more complex model cannot be a recommended solution because it would make the advanced controller design more complex. Furthermore, even a more complex model cannot remove a certain bias due to the causes mentioned above. Model predictive control based on identification has been selected as an alternative to nonlinear predictive control based on the first principles model. Since the dynamic behavior of a polymerization reactor is inherently nonlinear, various nonlinear input-output models such as the polynomial autoregressive moving average model (ARMA) or the fuzzy model have been tried by several researchers to describe the polymerization reactor (Na and Rhee, 2002; Altinten et al., 2003). Nonlinear model predictive controllers based on those models show good control performance, despite the severe nonlinearity of the polymerization reactor system. However, it is quite complicated to design nonlinear MPCs based on those models, and this prevents the user from adapting those nonlinear models. To avoid complication in designing a nonlinear controller, an appropriate choice of the nonlinear model would be the Wiener model. The Wiener model is a special kind of nonlinear model that consists of a static nonlinear block and a dynamic linear time invariant (LTI) part. Because of its relatively simple structure, this model has become increasingly popular as a ‘‘nextstep-beyond-linear modeling’’ of chemical processes (Pearson and Ogunnaike, 1997). In industrial practice, the steady-state behavior is often better understood than the dynamic behavior by virtue of knowledge about the physical and chemical phenomena, process experience, experimental knowledge, or previous modeling efforts. The Wiener approach, in which the static nonlinear part represents the steady-state description of the process, is of great interest because the extraction of a static
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model from the overall model structure would simplify the initialization and validation of the model. Wigren (1994) has analyzed the problem of the Wiener model by using the prediction error framework and the stochastic method. In recent years, model predictive control based on the Wiener model has been extensively studied (Norquay et al., 1998; Gerksic et al., 2000; Cervantes et al., 2003). More recently, several researchers have proposed the subspace approach for the identification of the state space model as the LTI part of the Wiener model (Westwick and Verhaegen, 1996; Jeong et al., 2001). Jeong et al. (2001) employed this scheme to design a model predictive controller and implemented it experimentally for the control of polymer quality in a continuous MMA polymerization reactor. However, the identification method of Wiener models is difficult and inefficient because of the complicated controller design procedure and the lack of a good representation of the output nonlinearity for identification purpose (Jeong et al., 2001). Unlike Jeong et al.’s work, the ‘‘model-free’’ approach will be employed for the identification of the linear part in this work. By doing so, it is expected that the controller design procedure can be made simpler and the accuracy of the identified model can be increased. Furthermore, the directly identified inverse static nonlinearity is implemented experimentally. In previous works the polynomial as static nonlinear function was required to be of low order for the guarantee of invertability. The commonly used polynomial representation for the output nonlinearities makes the identification in terms of unknown parameters very hard. The present study employs the linear input=output data-based prediction model for the identification of the LTI part and considers the direct identification of an inverse nonlinear function, which was first suggested by Kalafatis et al. (1997), from the angle of controller design. We then design a Wiener input=output data-based predictive controller and apply it to property control in a continuous solution polymerization reactor. It is worth noting that this is the first effort to incorporate the input=output data-based prediction model and the inverse identification of static nonlinear block into the Wiener model identification, apply the identified model to the design of a predictive controller, and implement the designed controller experimentally. On-line measurements of the monomer conversion and the weight average molecular weight (Mw) as controlled variables and direct sensing of feed flow rate and jacket inlet temperature as manipulating variables allow us to construct the input-output model of the continuous solution polymerization reactor and to realize the control scheme in an actual polymerization process. To close this section, we give an outline of the remainder of the article. Next rather detailed explanation for the identification and controller design is presented. Following that, the experimental system is described along with some remarks. Finally, the designed controller is implemented for a continuous polymerization reactor system, and its performance is evaluated by a series of simulation and experimental studies.
Control System Design Wiener Model Identification The Wiener model, as shown in Figure 1(a), consists of a dynamic linear part in series with a static nonlinear block. In the conventional Wiener model identification,
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Figure 1. Schematic representation of the Wiener model; (a) structure of Wiener model, (b) control scheme based on Wiener model.
the static nonlinear block is confined to those functions that are of one-to-one correspondence to give the inverse nonlinear function. This is essential to implement the nonlinear controller based on the Wiener model. In this study, we use a more convenient method to identify the Wiener model with the assumption that the measurement and process noises influence the LTI part only as shown in Figure 1(a). This assumption allows one to obtain the inverse function of static nonlinear block directly. In addition, the LTI part is identified by applying the method of subspace identification. This then leads to the input=output data-based prediction model, which has several advantages over other LTI models. LTI Part Identification Let us consider a linear system that can be described by the following state space equations: xkþ1 ¼ Axk þ Buk þ wk
ð1Þ
yk ¼ Cxk þ Duk þ vk
ð2Þ
where xk 2 Rn is the state variable, uk 2 Rm is the input vector, and yk 2 Rl is the output vector. Starting from the state space model and using a simple matrix calculation (Favoreel et al., 1999), one can derive the prediction model YLk ¼ CN Xf þ HN Uf þ HNS Zf þ Vf LW Wp þ Lu Uf
ð3Þ
where CN , HN, and HNS denote the extended observability matrix, the deterministic lower triangular block-Toeplitz matrix, and the stochastic lower triangular blockToeplitz matrix, respectively. YLk represents the future output, where the superscript L indicates the output of the LTI model. For the matrix input-output equation, Equation (3), one may refer to the article by Favoreel et al. (1999) or Van Overschee and De Moor (1996).
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Here, the ultimate goal of linear part identification is to calculate the gain matrices, Lw and Lu, of the prediction model by using the projection of data block Hankel matrices, Wp ; YLk , and Uf. A numerically efficient and robust way to implement this projection is the one using the following QR-factorization: 0 1 0 10 T 1 Uf R11 0 0 Q1 C B C B CB B ð4Þ 0 A@ QT2 C @ Wp A ¼ @ R21 R22 A L T Yk R31 R32 R33 Q3 The orthogonal projection may be expressed as YLk =
WP Uf
¼
YLk
WP
þ
Uf
Uf
¼ ð R31
R32 Þ
WP R11 R21
¼
YLk
WP
T
Uf þ 0 WP
R22
Uf
WP
WP
T !1
Uf Uf WP L Uf
WP
Uf ð5Þ
It then follows from Equations (3) and (5) that Lw ¼ Lð:; 1 : Nðm þ lÞÞ;
Lu ¼ Lð:; Nðm þ lÞ þ 1 : endÞ
ð6Þ
which are expressed in Matlab notation. One may consult the article by Favoreel et al. (1999) for the detail. The optimal predicted output sequence b yLk is expressed as L ^ yk ¼ Lw wp þ Lu uf
ð7Þ
in which wp is defined as ðyTp uTp ÞT with the last M known values of inputs up 2 RMm and outputs yp 2 RMm, respectively, where yp ¼ ðyIkMþ1 yIk1 yIk ÞT ;
up ¼ ðuTkMþ1 uTk1 uTk ÞT
ð8Þ
All the block Hankel matrices in Equation (3) are now replaced by their first column vectors, and ylk denotes the output of the inverse nonlinear function, which is yet to be identified. For the purpose of prediction and controller design, Equation (7) is employed. Static Nonlinear Block Identification and Iteration It is worth noting that there is no need to obtain the forward static nonlinear part because it is the inverse static nonlinear function that is needed for the design of a predictive controller in practical situations. The linear counterparts of the set-point and process output are obtained by using the inverse of the static nonlinear function as shown in Figure 1(b). Therefore, it would be better if one can determine the inverse nonlinear function directly to avoid the troublesome procedure of obtaining the inverse of the static nonlinear function. One can then reduce one step in calculating the optimal control inputs. To express the nonlinear function, we use an equation defined by a linear combination of the basis functions. Thus, the main structure may be expressed as yIl ¼
nb X i¼1
fi ðBi ðzk ÞÞ
ð9Þ
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where nb indicates the number of basis functions that should be determined in advance by trial and error. Here, zk indicates the vector consisting of the process outputs, which are the real outputs of process, over the prediction horizon. In practice, a high order usually causes a high sensitivity to noise and brings about model uncertainty in the identification. In this study, we select the Tchebychev polynomial as the basis function because of its minimax property, which enables the polynomial to approximate a monomial best by lower powers or alternatively to maximize the leading coefficient of a polynomial of fixed degree and size (Rivlin, 1974). This type of approximation is important because, when truncated, the error is spread smoothly over its normalized domain, i.e., [1, 1]. In addition, Tchebychev polynomials form a complete orthogonal set in the interval [1, 1]. By using this orthogonality, a piecewise continuous function in [1, 1] can be expressed in terms of Tchebychev polynomials. This polynomial crops up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods. Since the process output is taken as the output of the LTI part, the first identified LTI model is contaminated by the nonlinear output, i.e., the process output. To separate the nonlinearity from the LTI part, it is necessary to iterate the identification procedure of the LTI part and the static nonlinear part. More specifically, the linear part is first estimated while the nonlinear part is fixed. Then, these two sets are interchanged to evaluate the nonlinear part while the linear part is fixed (Kalafatis et al., 1997). It is possible to obtain the output of the LTI part by using the inverse function obtained in the previous step with the process output considered as the input of the inverse function. This procedure is depicted in detail in Figure 2. Although there is no guarantee for the global convergence, this iterative method is expected to work just like its counterpart in the Hammerstein or HammersteinWiener model, for which it has been demonstrated that the iterative method is usually very effective and converges quickly (Narendra and Gallman, 1966). Moreover, the case of divergence is rare (Stoica, 1981).
Figure 2. Iterative identification scheme of the LTI part and the inverse of the static nonlinear function.
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Iteration continues until the variance accounted for (VAF) index between the output b yLk from the LTI part and the output yIk from the inverse of the static nonlinear part obtains a desired value specified a priori. This iteration scheme results in the effect of the LTI part being separated from the static nonlinear part. The VAF index is defined as 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 PN L 2 I yi y i Þ C i¼1 ðb B VAF ¼ 100@1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ PN L 2ffi A ðb y Þ i¼1 i where N, b yLi , and yIi denote the number of data points, the outputs of the identified linear model, and the outputs of the identified inverse nonlinear function, respectively. Wiener-Type Input/Output Data-Based Predictive Controller Figure 3 presents the block diagram of the Wiener input=output data-based predictive control system. We use the identified Tchebychev polynomial function that approximates the inverse nonlinear function. This feature then allows the linear model based MPC algorithm to control nonlinear processes. The model equation, Equation (7), of the LTI part will be used in the MPC algorithm. The control parameters Lw and Lu are determined during the identification procedure, and the past input=output data wp are continuously updated so that the predicted output of the LTI part is also updated within the prediction horizon. On the basis of Equation (7), a sequence of optimal control moves can be computed, which minimizes the following objective function with the input magnitude constraints, i.e., min Jðuf Þ ¼ ðLu uf þ Lw wp rf ÞT QðLu uf þ Lw wp rf Þ þ uTf Ruf uf
ð11Þ
subject to the input constraints Cu uf Cf
ð12Þ
where rf denotes the set-point trajectory. Q and R are the weighting matrices for the output and input, respectively, and both are assumed diagonal. The optimization is solved via quadratic programming (QP), and only the first computed control move is implemented. The entire optimization procedure is repeated at the next sampling time. With the proposed control algorithm, there is
Figure 3. Block diagram of the Wiener-type input=output data-based predictive control system.
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no need to determine the state estimate or the state space model in order to calculate the optimal predicted output. Nowadays, instead of the computational load, the reduction of the number of tuning parameters can be a major concern in industrial applications. In this regard, the control scheme proposed in this study clearly has merit compared to the conventional model predictive control.
Experimental Section Experimental Apparatus Here, we consider a continuous MMA solution polymerization reactor system. The continuous polymerization reactor system consists of the reaction part and the automatic control part. The reaction part is composed of a polymerization reactor, heat-transfer apparatus, and a stirrer. The automatic control part includes a programmable logic controller for input and output data, a computer for control and data acquisition, and an actuator. In Figure 4, our lab-scale solution polymerization reactor system is depicted. The jacketed glass overflow reactor has a capacity of 1 L and is equipped with a stirrer for the mixing of the reactants. Heating or cooling of the reaction mixture is carried out with water flowing through the jacket. The valve stem positions of the hot and cold water lines are adjusted in a cascade control configuration, in such a way that the jacket inlet temperature is kept equal to the desired value specified by the master controller. A variable-speed remote set-point pump, which is a pistonoperated metering pump, is used for pumping the solution of monomer, solvent, and initiator into the reactor. The reaction product flows out of the reactor via an
Figure 4. Schematic diagram of the experimental setup.
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overflow line. To measure the density and viscosity of the reaction mixture, a circulation line is attached to the reactor. A small portion of the reaction mixture is circulated by a diaphragm metering pump through the circulation line, in which the online densitometer and viscometer are installed. Online Measurement of Physical Properties For the proper interpretation of process dynamics and estimation of process output, some properties such as density and viscosity must be evaluated. Especially, in the control scheme based on identification, the accurate estimation of polymer properties to be controlled is indispensable. The density of reaction mixture can be used effectively to follow the course of polymerization and to examine the variation of monomer conversion X. Ahn et al. (1999) proposed the correlation equation, which may be used to calculate the monomer conversion from the online density measurement as follows: 1 1 1 1 qd qm Wsf qs qm X¼ ð13Þ 1 1 q q p
m
in which the density qd of the reaction mixture is measured at the online densitometer and qm, qs, and qp denote the densities of monomer, solvent, and polymer, respectively, at the measuring temperature. Wsf represents the weight fraction of solvent in the reaction mixture, which may be assumed constant. For the on-line measurement of molecular weights, the correlation Equation (14) is employed to describe the relationship between the specific viscosity (gsp ) measured by the viscometer and the intrinsic viscosity ([g]), which is used to calculate the Mw from the Mark-Houwink equation: kH ½gC 0:55 2kH ½gC 0:55 2 0:55 2 bðC Þ ½g exp 1 bC 0:55 1 bC 0:55 0:55 3k ½gC H cðC 0:55 Þ3 ½g3 exp 1 bC 0:55
gsp ¼ aC 0:55 ½g exp
½g ¼ K½Mw a
ð14Þ ð15Þ
where C indicates the mass concentration (g=mL) of the polymer in the reactor that can be obtained by using on-line densitometer data, i.e., C ¼ Xqd, and the parameters, kH and b, are given as 3.0 g0.45mL0.55=cm3 and 0.9 mL0.55=g0.5, respectively. In Equation (15), the Mark-Houwink constants, K and a, are given as 3.99 105 cm3mol0.797g1.797 and 0.797, respectively. These parameters are determined by fitting the on-line measurement with the off-line measurements obtained by the gravimetric method for the conversion and by gel permeation chromatography (GPC) for the weight-average molecular weight. For the details, one may refer to the previous work (Ahn et al., 1999; Oh, 2003). Reactor Start-up Procedure Ethyl acetate (EA), benzoyl peroxide (BPO), and methyl methacrylate (MMA) are used as solvent, initiator, and monomer, respectively. The inhibitor contained in
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Table I. Reference conditions for the continuous MMA polymerization reactor system used in this work Items Initial charge Monomer Solvent Initiator Feed concentration Monomer Solvent Initiator Operating conditions Jacket inlet temperature Feed flow rate
Values 460 mL 340 mL 4.00 g 5.42 mol=L 4.35 mol=L 0.02 mol=L 75C (60 80C) 10 mL=min (5 25 mL=min)
the monomer is removed by using a 0.1 M NaOH solution. Nitrogen is bubbled through the reaction mixture that consists of solvent and monomer to purge oxygen in the reaction mixture. After 30 min, the reactor temperature is raised to the desired starting temperature by manipulating the jacket inlet temperature manually. Then the circulation pump is turned on to measure the density and intrinsic viscosity of the reaction mixture, the initiator dissolved in solvent is introduced, and the feed pump is also turned on simultaneously to start the polymerization reaction. After the polymerization reaction starts, the properties of polymer continuously change until the process reaches a steady state. The feed flow rate and reactor temperature are still manipulated manually. As soon as the process reaches a steady state, identification or on-line control is started with the control input manipulated automatically by computer. The reaction mixture is sampled at successive times with an interval of 20 min after the start-up procedure is over. The samples are used to measure the monomer conversion by the gravimetric method and Mw by gel permeation chromatography. The monomer conversion and Mw are also calculated from the on-line measurements of density and viscosity of the reaction mixture. These two kinds of measurement are to be compared with each other to validate the accuracy of the on-line measurement. The reference operating conditions for the experiments are given in Table I.
Results and Discussion Identification We conduct the identification experiment for the continuous MMA solution polymerization reactor to obtain an experimentally identified model and this will be used for the design of the predictive controller. After the start-up procedure, the input signal is introduced to the reactor system to obtain the on-line data of the plant output for a period of 340 min, and the product samples are taken from the reactor every 20 min to obtain the off-line data of the plant output. This procedure is useful for accurate identification and essential for property control of the continuous
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Figure 5. Experimentally obtained output data against the pseudo-random four-level input signals.
polymerization reactor system. The sample time for data acquisition is 1 min. Before the reactor reaches its steady state, the jacket inlet temperature Tjin is set at 75C and the feed flow rate qf is maintained at 10 mL=min for five hours. Then the jacket inlet temperature and the feed flow rate are changed to the four-level pseudo-random input signals whose lower and upper bounds are given in Table I. The experimental data set to be used for the identification are presented in Figure 5. Here, the two lower diagrams show the inputs used in the identification experiment, and the two upper diagrams present the outputs measured off-line (solid circle) and on-line (solid curve). From these diagrams, it is clearly seen that the on-line measurement is in good agreement with the off-line data. This indicates that the on-line measurements are quite reliable and can be used for the identification of the continuous MMA solution polymerization reactor. With these 340 sets of data, the identification is carried out following the procedure described above. Figure 6 shows the linear counterpart b yIk of the actual
Figure 6. Comparison between the linear counterparts from the identified inverse of the static nonlinear part (cross symbol) and the LTI part (solid curve) from the experimental work.
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reactor output (solid curve) and the output b yLk of identified linear block (cross symbol) for which the VAF indices are 99.2% and 99.1%, respectively. The three-dimensional plots for the static nonlinearity are illustrated for the MMA polymerization reactor over a wide operation range in Figure 7. These plots clearly demonstrate strong nonlinearity, especially in the region of low conversion as well as low and high Mw. Now, we are ready to design the Wiener-type input=output data-based predictive controller on the basis of the LTI model identified by experiment.
Figure 7. Nonlinear relation between the process output and the output of the LTI system.
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Control Simulation The proposed control scheme needs to be validated by the simulation study against various operation scenarios in advance. For this purpose, the first principles model of the MMA continuous polymerization reactor is used as the plant (Ahn et al., 1999). For the control purpose, we set the sampling time as 1 min and the prediction and control horizons are selected as 15 and 7 min, respectively. For the purpose of performance validation, the performance of the proposed controller is compared with that of the linear controller. Both controllers use the same LTI model and are best tuned by trial and error. The linear controller is designed using the method proposed by Kadali et al. (2003) in such a way that the controller includes the integral action. The weighting matrices of the Wiener controller for output and input are determined as diag(50, 30) and diag(1, 1), respectively, while those of linear controller are chosen as diag(50, 100) and diag(1, 1), respectively. Figures 8 and 9 show the multilevel set-point tracking performances of the Wiener input=output data-based predictive controller and the linear input=output data-based predictive controller, respectively. To examine the control performance in the presence of measurement noise (vk), 5% noises are added to both outputs of the identified LTI part. Here, 5% noise means that its standard deviation is 0.05. During the first 285 min, the controller proposed in this work keeps both the conversion and Mw at their respective set-points. With the linear control, however, Mw exhibits an offset.
Figure 8. Servo performance of the Wiener input=output data-based predictive controller for step changes in the set-points of conversion and=or weight-average molecular weight: simulation study.
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Figure 9. Servo performance of the linear input=output data-based predictive controller for step changes in the set-points of conversion and=or weight-average molecular weight: simulation study.
At 300 min, the set-point for Mw is changed from 150,000 to 170,000 g=mol while the set-point for the conversion is maintained. Here again, there exist offsets in both the conversion and Mw in the case of linear control whereas the nonlinear predictive controller accomplishes its goal satisfactorily. We assume that the decision on the set-point change is introduced at least one prediction horizon earlier than the actual change in the set-point for Mw. The predictive controller compares the predicted output with the set-point trajectory over one prediction horizon. Hence, if the set-point change occurs within the prediction horizon, the controller starts to act to manipulate the input variables one prediction horizon earlier than the setpoint change. In this study the prediction horizon is set to be 15 min and thus the controller is able to observe the set-point change 15 min before it really occurs. Therefore, the controller starts to manipulate the control input variables from 285 min and requires the jacket inlet temperature to decrease from 285 min in order to increase Mw. The feed flow rate is also made decrease at the same time in order to maintain the conversion at its set-point of 0.12. At 400 min, the set-point for conversion is increased from 0.12 to 0.17 while that for Mw is unchanged. It is clearly seen that the feed flow rate begins to decrease at 385 min, one prediction horizon earlier, to increase the conversion to 0.17, and the jacket inlet temperature also decreases to keep Mw at its set-point of 170,000 g=mol. In contrast to the excellent performance of the Wiener input=output data-based predictive controller, the linear counterpart cannot help exhibiting small offsets in both conversion and Mw. At 600 min, the set-points for conversion and Mw are changed from 0.17 to 0.15 and from 170,000 to 190,000 g=mol, respectively, at the same time. In this case the
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set-point changes in the conversion and Mw are in directions that can be handled by manipulating the jacket inlet temperature alone and thus the controllability of the system becomes better than in the previous cases. As expected, the decrease in the jacket inlet temperature is found good enough to drive both outputs to their respective set-points. It is to be noted that the feed flow rate plays a complementary role and its change is much smaller than in the previous cases, and yet the control performance is quite satisfactory, although the linear controller yields a small offset in the conversion. Figure 10 shows the control performance of the proposed controller when the feed pump stops for 50 min starting from 400 min. Without the control, the conversion increases beyond the level of 0.30 and Mw decreases to the level of 124,000 before they recover their respective steady-state values. When the controller works, the conversion does not reach the level of 0.30 and the deviation of Mw is much smaller than before. Above all, both outputs return to their set-points much more rapidly than in the case of no control action. Experiment An experiment was carried out to implement the Wiener-type input=output data-based predictive controller for the control of monomer conversion and Mw in a continuous MMA polymerization reactor. The jacket inlet temperature and feed flow rate are selected as the manipulated variables. The lower and upper bounds are set as 65C and 85C for the jacket inlet temperature and as 5 and 25 mL=min for the feed flow rate. The identified Wiener model is used in the predictive controller. The sampling time of the discrete-time control algorithm is 1 min. The set-point tracking control is performed under a prediction horizon of p ¼ 30 and a control horizon of
Figure 10. Disturbance rejection performance of the Wiener input=output data-based predictive controller against feed pump failure: simulation study.
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Figure 11. On-line digital control for set-point tracking in the continuous MMA polymerization reactor: experimental work.
m ¼ 10 with Q ¼ diag(500, 50) and R ¼ diag(1, 1). The weighting matrices are determined by trial and error. Figure 11 shows the profiles of the controlled outputs when there are step changes in the set-points for the conversion and the molecular weight. Here, the set-points were selected considering the range of outputs obtained by the identification experiment. At 120 min, the set-point for conversion is raised from 0.12 to 0.17, while the set-point for Mw is maintained at 120,000 g=mol. The set-point change was introduced at 50 min, well in advance of the actual set-point change. Because of the interactive dynamics of the reactor system, the controller increases the jacket inlet temperature from 90 min, one prediction horizon earlier, to increase the conversion and at the same time increases the feed flow rate to maintain Mw at 120,000 g=mol. One may note that without the increase in the feed flow rate Mw would decrease as the jacket inlet temperature increases. When the set-point for Mw is increased from 120,000 to 170,000 g=mol at 250 min, the controller increases the feed flow rate to increase Mw. However, the conversion tends to decrease, and thus the controller increases the jacket inlet temperature to regulate the conversion. With these experimental runs for set-point tracking control, it is demonstrated that the proposed controller shows an excellent tracking performance in real-time control of conversion and Mw under the presence of hard input constraints. The control experiment for disturbance rejection is also conducted to validate the performance of the proposed controller. The validation procedure for disturbance rejection is important because the feed concentration can change frequently in industrial polymerization processes. The disturbance rejection control is performed with p ¼ 30, m ¼ 10, Q ¼ diag(800, 80), and R ¼ diag(1, 1). In a practical sense, one may maintain the previously determined weighting factors simply because any disturbance would occur unexpectedly during normal operations. In this work, however, the weighting factors are best tuned by trial and error to obtain as good a control performance as possible. As a disturbance here, we consider a decrease in the monomer feed concentration from 5.42 to 3.35 mol=L and an increase in the solvent feed concentration
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Figure 12. On-line digital control for disturbance rejection in the continuous MMA polymerization reactor: experimental work.
from 4.35 to 6.03 mol=L between 100 and 200 min of normal operation. The monomer conversion and Mw are to be kept at their original values of 0.17 and 120,000 g=mol, respectively. Because the monomer concentration is decreased, the controller gradually decreases both the jacket inlet temperature and the feed flow rate to keep both the monomer conversion and Mw at their respective original values, as shown in Figure 12. It is also noted that when the disturbances in feed monomer and solvent feed concentrations are removed at 200 min, both the conversion and Mw recover their respective steady values rather rapidly. Here again, the Wiener-type input=output data-based predictive controller proves itself to have an excellent disturbance rejection property in real-time control of polymer quality.
Conclusions The method of subspace identification was introduced to identify a continuous MMA solution polymerization reactor by the Wiener model and the inverse of the nonlinear function was directly identified. With this approach the identification procedure of polymerization reactor systems can be made more efficient than the conventional method of Wiener model identification. On the basis of the identified Wiener model a Wiener input=output data-based predictive controller was designed and implemented for the property control of polymer product in the continuous MMA polymerization reactor. Simulation studies clearly demonstrated that the predictive controller designed in this work performs better than its linear counterpart not only for set-point tracking but also for disturbance rejection. Digital control experiments were also conducted with an actual polymerization reactor system by using an on-line densitometer and on-line viscometer to observe satisfactory performance of the designed controller in both set-point tracking and disturbance rejection. Consequently, we expect that the proposed identification method and control scheme may be successfully applied to nonlinear chemical processes.
Wiener Input=Output Data-Based Predictive Control
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Acknowledgment The financial support from the Brain Korea 21 project is greatly appreciated.
Nomenclature C HN HNS K Mw m N p Q R rf uf Wsf YLk yLi yIi CN gsp ½g qd qm qp qs
mass concentration deterministic lower triangular block-Toeplitz matrix stochastic lower triangular block-Toeplitz matrix as a subscript, k-th sampling time weight-average molecular weight control horizon number of data points prediction horizon weighting matrices of output weighting matrices of input set-point trajectory control input sequence over control horizon weight fraction of solvent in the reactor future optimal predicted output outputs of the identified linear model outputs of the identified inverse function extended observability matrix specific viscosity intrinsic viscosity measured density qd of the reaction mixture density of monomer density of polymer density of solvent
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