Design and Application of Model-on-Demand Predictive Controller to

Ind. Eng. Chem. Res. 2003, 42, 847-859

847

Design and Application of Model-on-Demand Predictive Controller to a Semibatch Copolymerization Reactor Su-Mi Hur, Myung-June Park, and Hyun-Ku Rhee* School of Chemical Engineering and Institute of Chemical Processes, Seoul National University, Kwanak-ku, Seoul 151-742, Korea

This article considers the design of a multivariable model-on-demand predictive controller (MoDPC) and its application to polymer quality control in a semibatch MMA/MA copolymerization reactor. The MoD-PC is designed by combining the model-on-demand (MoD) framework with the conventional model predictive controller. For this purpose, a local autoregressive exogenous input model is constructed with a small portion of data located in the region of interest at every sample time when a model equation is needed for output prediction. This model equation is then used to calculate the optimal control input sequence. Through the open-loop test, the reaction temperature and free volume are shown to be an adequate choice for the elements of the regressor state vector in the data searching step. The validity of the identified model is corroborated by the prediction method. The results of simulation studies for the regulation and disturbance rejection problems with and without noise demonstrate that, despite the nonlinearity of the system, the MoD-PC is an effective strategy with a low computational load for the production of copolymers with desired properties. 1. Introduction One of the control objectives in the field of polymerization reaction engineering is to produce polymers with desired properties. For this purpose, the control of polymerization reactors has been studied by various researchers.1,2 However, such control still remains as a difficult task because of the complex reaction mechanisms involved and the highly nonlinear nature of polymerization systems. Copolymers have a variety of better properties than homopolymers because two (or more) monomers with different characteristics are used to compose the polymer chains. The use of two monomers, however, brings about complexity of a higher degree and requires the consideration of several features for the production of copolymers with desired end properties. Hence, a great deal of research has been directed toward the development of operating strategies for the control of copolymerization reactors. Semibatch copolymerization is often performed in an attempt to maintain copolymer composition reasonably constant when the comonomers have different reactivities. Quantitative strategies for semibatching have been developed through empirical experimentation or classical trajectory optimization techniques.3 In addition to open-loop/feed-forward trajectories,4 on-line optimization has been proposed as a means of dealing with nonideal operating conditions caused by model mismatch and disturbances.5 Model predictive control (MPC) has established its reputation as a powerful and broadly applicable tool, especially in chemical plants.6 One of the most important factors to be considered in the design and realization of MPC is the acquisition of an accurate model, which should be able not only to capture the system dynamics but also to be easily incorporated into the * To whom all correspondence should be addressed. E-mail: [email protected]. Tel.: (+82) 2-880-7405. Fax: (+82) 2-888-7295.

controller design procedure. The first-principles model of a polymerization reactor usually contains a large number of kinetic parameters that are neither readily found in the literature nor easily determined by experiment. To overcome the difficulties imposed by the use of first-principles models, various types of empirical models are applied for the identification of polymerization reactors and used in the design of MPC. Because linear models fail to predict the nonlinear behavior of polymerization reactors, it is recommended that nonlinear models be used in the control of polymer properties, and a number of studies on the nonlinear control of polymerization reactors have been conducted.7,8 Nonlinear models, however, bring about difficulties in the identification and optimization of the controller. The optimization problem requires a sophisticated, timeconsuming procedure and is numerically very difficult to solve.9 Moreover, in some cases such as semibatch reactors, inherent properties of the system make the identification of an adequate model a highly effortdemanding task and do not guarantee the consistent estimate of the model. We adopt the concept of model-on-demand (MoD) and an MoD-based predictive controller (MoD-PC)10 to overcome the shortcomings of existing identification methods while maintaining their merits. Its application is then illustrated by using it for polymer-property control in a semibatch reactor for the solution copolymerization of methyl methacrlyrate (MMA) and vinyl acetate (VAc). MoD modeling is a novel paradigm proposed by Cybenko11 and Stenman,12 who called it “just-in-time modeling”. Since then, it has been further developed and modified by Stenman10,13 and other researchers14,15 with analysis and illustrative applications. However, all applications are restricted to single-input-single-output (SISO) systems. The basic concept of MoD is to identify a model with the data that belong to a small neighborhood around the current operating point rather than to estimate a complex global model covering the entire

10.1021/ie020291f CCC: $25.00 © 2003 American Chemical Society Published on Web 01/21/2003

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Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003

input-output domain. This algorithm provides a good fit by using a very simple model in a prespecified domain. When the MoD approach is used, the MoD-PC is expected to have increased flexibility and to show improved control performance compared to global modelbased controllers. 2. Model-on-Demand Predictive Controller (MoD-PC) MoD is a so-called data-mining technology that was proposed as an alternative to traditional modeling approaches. The concept of MoD is that a model is estimated only when it is needed. The idea is to store all observations from a process in a database and then to estimate a local model by selecting portions of data that belong to a small neighborhood around the current operating point. The detailed procedure is as follows: Given a database including the observed regressor state vector φ(k) and lagged input-output pairs, a local prediction is formed by retrieving relevant data from the database (data searching) and applying them to a suitable modeling operation (local modeling). Because of lack of space, the main points of MoD are briefly introduced. For details on MoD, one is referred to the article by Stenman.10 The essential objective of data searching is to define the relevant data. These data are used for model parameter estimation and might affect the accuracy of the prediction. Among the state variables that have been stored in the database, one must compose the regressor space by selecting the main variables that can reflect the dynamic characteristics of the system. The regressor states (state variables in the regressor space) are investigated to determine whether they exist within a domain of a prespecified distance from the current state, and the regressor states within the domain are selected. Also, the state variables that correspond to the selected regressor states are retrieved from the database. In general, there are many ways to define the distance function on which the shape of the neighborhood depends. The most common choice is the weighted Euclidian norm

d(φ(k),φ(‚)) ) x[φ(k) - φ(‚)]M[φ(k) - φ(‚)]T (1) where φ(k) denotes the current regressor state vector at time k and φ(‚) represents all regressor state vectors to be tested. M is the weighting matrix, which determines the shape and the orientation of the neighborhood of the current operating point. The idea of local modeling is not new. Any identification approach that has been studied in the past can be utilized in the MoD framework. The main difference of the MoD framework from the traditional approaches is that it considers the local data belonging to the neighborhood instead of the entire set of data. The system dynamics can be modeled as

y(k) ) m(ψ(k),θ) + He(k)

(2)

where y(k) and ψ(k) denote the output vector and the regression vector observed and stored in the database, respectively, and θ is a model parameter vector. Then, a local model is identified via the weighted regression problem

θˆ ) arg min θ

∑i l(H-1(q,θ)[y(i) - m(ψ(i),θ)]‚wi(ψ(k))

(3)

Here, wi(‚) is the weight that controls the locality, and l(‚) is a scalar-valued, positive-norm function. As it is possible to use any model structure as a local model, the technique of local linear model is appealing for the modeling of complex systems. If a quadratic norm [l(e) ) 0.5e2] is used and the model is linear in the parameters, the estimate can be explicitly computed using the ordinary weighted least-squares method. One of the practical uses of the MoD technique is to merge it with model-based control theory. Among several candidate controllers by which the MoD technique is accommodated, we decided to use MPC because of the merits stated in the previous section. The most obvious way of incorporating the MoD approach into the MPC formulation is to obtain a local model by conducting the MoD method whenever a model is needed, that is, at every sample time, and to minimize the objective function J that is constructed on the basis of this local model. If the MoD method uses a linear model as the local model, the design of MoD-PC converges to that of conventional linear MPC between sample times. 3. Semibatch Copolymerization Reactor The multivariable MoD-PC algorithm is applied to the quality control of a semibatch copolymerization reactor in which the free-radical polymerization of methyl methacrylate (MMA) and vinyl acetate (VAc) takes place. This particular system shows a strong nonlinearity, with highly correlated states and time-variant output behavior. The physical system consists of a semibatch reactor with an inlet flow of initiator, solvent, and the more reactive monomer (MMA). The initiator used in this work is azobis(isobutyronitrile) (AIBN), which is dissolved in the initially charged reactants as well as in the feed solution. Note that VAc monomer is initially charged in the reactor, but it is not contained in the feed solution. The reaction temperature (Tr) and the feed flow rate (qf) are taken as the manipulated variables, while the weight-average molecular weight (Mw) and the instantaneous copolymer composition (F1) are the controlled outputs. The instantaneous composition F1 refers to the composition of the comonomers in the copolymer chains produced at any given instant, which depends on the relative rates of the addition of monomers 1 and 2 into the polymer and is expressed as the ratio of the respective rates of propagation16

F1 )

Rp1 Rp1 ) Rp Rp1 + Rp2

(4)

where Rpi denotes the propagation reaction rate of monomer i and subscript 1 represents the more reactive monomer, MMA. In the simulation study, the first-principles model for the semibatch reactor is considered as the plant. The reaction kinetics is assumed to follow a free-radical polymerization mechanism including chain-transfer reactions to solvent and two monomers. For the model equations, one can refer to the article by Park et al.17 There are 11 state equations describing the mass balances for the two monomers, initiator, and solvent

Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 849 Table 1. Physcal and Kinetic Parameters and Gel Effect Correlation Used in the First-Principles Model Fm1 (g/L) Fm2 (g/L) Fs (g/L) Fp1 (g/L) Fp2 (g/L)

Physical Properties 965.4 - 1.090(T - 273.15) - 9.7 × 10-4(T - 273.15) 958.4 - 1.3276(T - 273.15) 925.0 - 1.239(T - 273.15) Fm1/[0.724 - 9.0 × 10-4(T - 273.15 - 70)] 1211 - 8.496 × 10-1(T - 273.15)

kd (s-1) kp11 (L/mol/s) kp22 (L/mol/s) kt11 (L/mol/s) kt22 (L/mol/s) ktrm11 (L/mol/s) ktrm11 (L/mol/s) r1 r2 ktd/ktc ktrs1 (L/mol/s) ktrs2 (L/mol/s) ktrm12 (L/mol/s) ktrm21 (L/mol/s)

Rate Constants 1.2525 × 1014 exp(-29 350/RT) 4.92 × 105 exp(-4353/RT) 3.20 × 107 exp(-6300/RT) 9.8 × 107 exp(-701/RT) 3.70 × 109 exp(-3200/RT) 0.17 × 10-4(kp11) 2.38 × 10-4(kp22) 20.3 0.07 2.483 × 103exp(-4353/R/T) 1.0 × 10-5(kp11) 1.07 × 10-4(kp22) 2.38 × 10-4(ktrm11) 0.17 × 10-4(ktrm22)

vf g vfpc 1 gp11 ) 7.1 × 10-5 exp(171.53vf) vf e vfpc gt11 )

{

0.105 75 exp[17.15vf - 0.017 15(T - 273.15)] vf g vftc vf e vftc 2.3 × 10-6 exp(75vf)

vfpc ) 0.05 vftc ) 0.1856 - 2.965 × 10-4(T - 273.15) gp22 ) 1 gt22 ) exp (-0.4407X - 6.7530X2 - 0.3495X3)

monomer1 solvent initiator

180 mL 120 mL 300 mL 4.8 g

Feed Concentration 4.6826 mol/L 5.0739 mol/L 0.0812 mol/L

Operating Conditions reaction temperature 55-90 °C feed flow rate 0-8 mL/min

and the first three moments of living and dead polymer concentrations and an equation for the total volume of the reaction mixture. To reduce the complexity of the rate expressions, the pseudo-kinetic rate constant method (PKRCM)18 is applied to the copolymerization system. The physical properties and kinetic parameters are taken from Zhang and Ray,19 and the correlations for the gel effect are taken from Pinto and Ray.20 Actual values are listed in Table 1. It is worth noting that the gel effect is partially responsible for the autocatalytic nature of the polymerization and brings about the strong nonlinearity. The reference conditions for the simulation study are given in Table 2. To reduce the differences among the orders of magnitude, the inputs and the outputs are normalized between 0 and 1 by defining the scaled variables as follows

u1 )

Tr - Tr,min Tr,max - Tr,min

u2 )

reaction temperature (Tr, u1) (°C) feed flow rate (qf, u2) (mL/min) free volume (vf) (Å3/molecule) rate of reaction temperature change (°C) rate of feed flow rate change (mL/min)

Mw - Mw,min Mw,max - Mw,min

y2 )

Mw,min Mw,max F1,min F1,max Tr,min Tr,max qf,min qf,max vf,max vf,min ∆Tr,min ∆Tr,max ∆qf,min ∆qf,max

40 000 80 000 0.8 1 55 90 0 8 0.23 0.12 -0.5 0.5 -4 4

F1 - F1,min F1,max - F1,min

The scaling factors and the constraints are listed in Table 3. The requirements for the reaction rate, heattransfer limitations, and safe operating conditions determine the upper and lower bounds on the reaction temperature, which are defined as constraints and scaling factors. Similarly, the upper and lower bounds of the feed flow rate are defined as constraints to avoid negative flow rates and to limit the maximum flow rate that can be handled by the pump. 4. Design of the MoD-PC

Table 2. Reference Conditions for the Simulation Study Initial Charge monomer 1 (MMA) monomer 2 (VAc) solvent (EA) initiator (AIBN)

weight-average molecular weight (Mw, y1) (g/mol) copolymer composition (F1, y2)

y1 )

Gel Effect Correlation

{

Table 3. Scaling Factors and Constraints for the Input and Output Variables

qf - qf,min qf,max - qf,min

(5)

In copolymerization, the more reactive monomer is usually added to the reactor to produce copolymer with a uniform composition. If the copolymer composition is measured or estimated on-line, the more reactive monomer can be added in a closed-loop manner. Determination of the trajectory for the reaction temperature or for the addition of feed to a batch or semibatch polymerization reactor, respectively, is mostly done off-line. These trajectories can be obtained on the basis of operating experience, or they can be calculated by achieving certain goals subject to constraints. The rigorous calculation of optimal trajectories requires a reasonably accurate model of the polymerization process. Moreover, in most industrial applications, one has to take into account the disturbances that can prevent the optimal trajectory from being realized in the reactor under open-loop conditions. Although on-line optimization is suggested as a solution to this problem, it can suffer from practical limitations caused by the lack of a detailed mathematical model and the computational time, if any, required to recalculate an optimal trajectory. All of these shortcomings certainly motivated us to seek a highly efficient model-on-demand predictive controller. Construction of the Regressor State Vector and the Regressor Space. To select the variables to be used in the regressor space, we first examine the effects of parameters and operating conditions on the states of the reactor system during the entire course of the polymerization reaction. Figure 1 presents the output responses for the constant inputs at various reaction temperatures. It is observed that the process of interest here presents some problems and challenges when nonlinear model predictive control is applied. Because of the nature of the semibatch process, there is no steady state, and the system shows no regular variation even when there is no change in the inputs. Moreover, the

850


Figure 1. Output responses for the constant inputs at the three different reaction temperatures 60, 75, and 85 °C.

gel effect makes the nonlinear feature of the system more pronounced, and different reaction temperatures produce considerably different output responses. Therefore, use of a global model might not provide a good fit for control purposes. Because it assumes an important role in the system dynamics, as observed in Figure 1, the reaction temperature is chosen as one of the elements of the regressor state vector. The free volume of the reaction mixture is selected as another element of the regressor state vector. From a physical point of view, the irregular variations of the output responses might be caused by the gel effect.21 The gel effect describes the decreasing tendency of two long chains to come together and become dead polymers. In this study, it is assumed that the gel effect occurs in the reactor when the free volume is decreased below a critical value, as proposed by Hamer et al.22 Figure 2 shows the profile of the free volume during the reaction under isothermal condition at 75 °C. In Figure 1, it is noticed that the output trajectories undergo changes in their slopes at the time around 170 min when the free volume crosses its critical value. Although the free volume is easily obtained in the simulation, it has to be

replaced by a measurable variable such as viscosity, or a suitable correlation must be introduced in the actual situation. To select relevant data, the weighted Euclidian norm is used as the distance function, d(x,y), in the regressor space. The weighting matrix M is assumed to be diagonal for convenience. Larger values are assigned to the elements for the free volume because the free volume reflects the time-varying feature of the semibatch reactor. We also add an extra weight to the recent elements of the regressor state vector. vj f(k) is the free volume scaled in the same manner as in eq 5 with the scaling parameters listed in Table 3.

φ(k) ) [vj f(k), vj f(k-1), vj f(k-2), u1(k-1), u1(k-2), u1(k-3)] (6) M ) diag(3, 1.9, 1.7, 1, 1, 1),

b ) 0.15

In the data retrieval step, the data within the distance b from the current state in the regressor space are selected. In computing θˆ , it should be noted that a small

Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 851

Figure 2. Time evolution of the free volume of the reaction mixture at 75 °C.

Figure 3. Output data for the identification generated by eight-level input signals for four different batch runs.

neighborhood would strengthen the influence of the measurement noise on the resulting estimate. On the other hand, a large neighborhood would make the Taylor expansion inappropriate and introduce a bias. The distance criterion thus involves a trade off between bias and variance, which causes a deterioration of the prediction accuracy. This is why the determination of the distance criterion requires much effort. Several classical methods are available for determining the neighborhood such as the false nearest neighborhood (FNN) method, cross-validation, the Akaike information criteria (AIC), and related methods. In this work,

however, we determine the neighborhood by trial-anderror, and the data that are located within a distance of 0.15 in the regressor space are retrieved. For the design of MoD-PC, a database that is composed of regressor states and input-output data is generated by imposing a pseudo-random eight-level input signal onto the first-principles model and saving the responses for the entire reaction time. Because the performance of MoD-PC is largely dependent on the quality and amount of data, it is important that the data contain much information about the reactor dynamics. It is reasonable to use more frequently changed input

852


Figure 4. Comparison of the output predicted by the local ARX model with the plant output.

signals with many levels and to perform independent four-batch runs for a sufficient supply of data. The generated output data are shown in Figure 3. Model Structure and Controller Design. In this study, we use the ARX model of third order for both inputs and outputs, which is one of the simplest linear model structures. Because the system has two inputs and two outputs, the model equation is defined as follows

y(k) ) [y1(k) y2(k)]T

u(k) ) [u1(k) u2(k)]T (7)

y(k) ) A(k) y(k-1) + B(k) y(k-2) + C(k) y(k-3) + D(k) u(k-1) + E(k) u(k-2) + F(k) u(k-3) The parameters for the model, that is, the elements of the 2 × 2 matrices A(k), B(k), C(k), D(k), E(k), and F(k), are determined by using the least-squares error method. For the sake of simplicity in calculation, y1(k) and y2(k) are assumed to be decoupled in the model equation. Hence, A(k), B(k), and C(k) are used in diagonal form. The model parameters are obtained by using pseudo-

inverse, which makes the parameter estimation procedure simple

[ ]

yi(k+3) yi(k+4) Yi ) l , yi(k+N-1) yi(k+N)

[]

aii bii cii dii θi ) dij , eii eij fii fij

[

yi(k+2) yi(k+1) y (k) ψi ) i u(k+2) u(k+1) u(k)

Yi ) ψiTθi

yi(k+3) yi(k+2) yi(k+1) u(k+3) u(k+2) u(k+1)

‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚

yi(k+N-1) yi(k+N-2) yi(k+N-3) u(k+N-1) u(k+N-2) u(k+N-3)

θˆ i ) (ψi ψiT)-1ψi Yi

]

(8)


Figure 5. Regulatory performance of the MoD-PC.

Because the ARX MIMO structure includes too many terms in the prediction, the model equation is reconstructed without loss of generality as a state-space model so that the model predictive controller can be designed effectively.

xk+1 ) Akxk + Bkuk,

y(k) ) Ckxk

[

uk ) [u(k)T u(k-1)T u(k-2)T]T

]

[][ ]

Ck Bk CkAk CkAkBk + CkBk CkAk2 ) xk + l uk + p-1 l i p CkAk Bk CkAk

(9)

xk ) [y(k)T y(k-1)T y(k-2)T]T, A(k) B(k) C(k) Ak ) I , 0 0 0 I 0

[ ]

yˆ(k+1/k) yˆ(k+2/k) Yk+1|k ) l yˆ(k+p/k)

[

]

D(k) E(k) F(k) Bk ) 0 , 0 0 0 0 0 Ck ) [I 0 0]

The p-step prediction equation is written as follows

[

∑ i)0

‚‚‚ 0 ‚‚‚ 0 l l

C kB k 0 CkAkBk + CkBk CkBk l l p-1

∑C A B

p-2

i

k

i)0

k

∑C A B

p-m+1

i

k

k

k

i)0

)Y ˜ k + Sk∆Uk

0 0 l

k

‚‚‚

∑ C A B ∑C A B i

k

i)0

k

i

k

k

i)0

k

]

‚∆Uk

p-m

k

(10)

854


Figure 6. Regulatory performance of the MoD-PC when the MMA concentration in the feed is reduced by 90% between 200 and 300 min.

where ∆Uk ) [∆u(k)T, ‚‚‚, ∆u(k+m-1)T]T denotes input changes within the control horizon. The cost function is defined in the form

min{|Λy[Yk+1|k - Rk+1|k]||22 + |Λ∆U∆Uk||22} (11) ∆Uk

subject to

u(k+l)min e u(k+l) e u(k+l)max, 0 e l e m - 1 -∆u(k+l)max e ∆u(k+l) e ∆u(k+l)max,

0elem-1

y(k+l)min e y(k+l/k) e y(k+l)max, 1 e l e p where Λy and Λ∆U represent the penalties on the control errors and the rate of control input changes, respectively. As usual, p and m denote the prediction and control horizons, respectively. Among the m future control actions that minimize the cost function, only the first one is applied to the control process. When new

measurements become available, a new optimization problem is formulated, and the solution provides the next control action. Model Validation. Prior to control application, the model has to be validated. For this purpose, we generate another eight-level pseudo-random input signal as shown in the lower two diagrams of Figure 4. At every step, a local ARX model is obtained by the MoD method, and the p-step-ahead prediction is carried out. Both the predicted output yˆ (k+p/k) and the real (plant) output y(k+p) with p ) 5 are plotted in the upper two diagrams of Figure 4. The outputs predicted by the identified local ARX model are in good agreement with those from the first-principles model. Because large input changes can result in a significant deviation in the output prediction, we introduce constraints on the input as well as on the rate of input changes ∆u(k) in the design of MoD-PC as given in Table 3. To evaluate the reliability of the identified model, the percentage variance accounted for (VAF) index is used; that is


( )

Figure 7. Regulatory performance of the MoD-PC when the feed pump becomes inactive between 200 and 300 min.

x∑ x∑

diagonal form for convenience, and their elements are determined by the trial-and-error method as follows

N

(yi - yˆi)

VAF ≡ 100 1 -

i)1

2

%

(12)

Λy ) diag(100,130), Λ∆U ) diag(10,10)

N

yi2

i)1

where yi and yˆ i denote the plant output and the predicted output, respectively. The VAF values for the results of Mw and F1 are 97.7 and 98.8%, respectively. 5. Application of MoD-PC to Semibatch Copolymerization Reactor In what follows, we investigate the performance of the MoD-PC based on the local ARX model by applying it to the semibatch copolymerization reactor that was described in section 3. Regulatory Performance. The sample time and the prediction and control horizons are specified as 1 min, 5, and 3, respectively. Weighting matrices are used in

Figure 5 presents the simulation results for the performance of the MoD-PC when reactor operation is started with the setpoints for Mw and F1 specified as 62 000 and 0.95, respectively. The initial deviations of the outputs from the desired values are quite large because of the intrinsic nature of free-radical polymerization. Therefore, the reaction temperature undergoes a sudden change, and the feed rate drops to its lower bound. These control actions make the outputs Mw and F1 decrease simultaneously. In thermally initiated freeradical polymerization, the higher the reaction temperature, the lower the average molecular weight. Also, the lower feed rate causes the concentration of more reactive monomer to decrease. Although Mw shows somewhat oscillatory behavior, the controller drives both of the controlled variables to their respective setpoints within 100 min. After 200 min, the control inputs undergo

856


Figure 8. Regulatory performance of the EKF-based NLMPC when the feed pump ceases working between 200 and 300 min.

sluggish changes to maintain the outputs at the setpoints. Because MMA is more reactive, additional feeding of MMA is required to maintain the constant instantaneous composition until 500 min. After 500 min, however, the VAc concentration in the reactor becomes low, and thus, the feed flow rate decreases steadily. As the low concentrations of two monomers decrease the rate of monomer addtion to copolymer chains, the controller makes the reaction temperature decrease to maintain Mw at the desired value. Disturbance Rejection Problems. As another illustration, a simulation study is conducted for disturbance rejection. For this purpose, consider two cases in which two different significant disturbances are introduced in the feed stream. All of the parameters used in the model identification and control such as the prediction and the control horizons, the distance criterion, and the weighting matrices remain the same as in the previous case. In the first case, it is assumed that the feed concentration of MMA is reduced to 10% of the normal concentration between 200 and 300 min. Figure 6 shows the regulatory performance of the MoD-PC under such circumstances. The abrupt decrease in the MMA feed concentration causes both Mw and F1 to decrease, but the effect on Mw does not seem significant because Mw is related to the accumulated mass in the reactor. Although the system inputs and outputs are highly correlated, it appears that the reaction temperature decreases to increase Mw and that the feed flow rate increases rapidly to compensate for the loss of MMA. After the recovery of the MMA concentration at 300 min, the controller changes the direction of change in each input, and finally, both Mw and F1 are driven quickly to their setpoints after somewhat oscillatory behavior of about 150 min. This case study illustrates that the

MoD-PC performs quite satisfactorily and proves itself effective for the purpose of disturbance rejection. In the second case, the pump is assumed to stop working between 200 and 300 min. As shown in Figure 7, the malfunction of the feed pump causes Mw to increase and F1 to decrease from their respective setpoints. As the feed stream is composed of initiator, solvent, and MMA, the cessation of the feed supply leads to a more complicated situation than in the previous case. Without additional supply, the amount of initiator in the reactor is continuously decreased, and this causes Mw to increase. On the other hand, the interruption of the MMA supply brings about a rather rapid decrease in F1. It is also worth noting that, because there is no feed, the weight fraction of polymer increases, and the gel effect becomes more pronounced to make Mw increase. With one of the control inputs, qf, having lost its function, the temperature alone does not seem capable of providing sufficient control action. Only after the feed pump recovers its function at 300 min does the feed flow rate jump up almost instantly to raise the level of F1 and the temperature gradually increase to bring Mw down to its setpoint. As a result, the outputs return to their respective setpoints after mild oscillation for about 100 min. For the purpose of comparison, a simulation study is performed with the extended Kalman filter (EKF) based nonlinear MPC23,24 for the same case as shown in Figure 7, and the result is presented in Figure 8. Compared to the MoD-PC, the EKF-based nonlinear MPC performs somewhat better most probably because the firstprinciples model of the reactor system is applied to the EKF-based MPC and also used as the plant model. Hence, there is no model mismatch in this case. In realistic situations, however, the plant would not be so perfectly described by the first-principles model that the


Figure 9. Regulatory performance of the MoD-PC when measurement noise is present.

control performance must be deteriorated as a result of the model mismatch. Although it is a linear controller and has its own limitations in comparison to the nonlinear MPC, the MoD-PC shows satisfactory control performance that is comparable to the performance of the nonlinear MPC and yet has merits in that the MoDPC does not require a mathematical model and imposes much lower computational load. Regulatory Performance in the Presence of Measurement Noise. Because the recursive prediction adopted in this study is based on on-line measurements and the database, the quality of the measurements and the database certainly affects the prediction capabilities and accordingly the control performance. Several levels of measurement noise are added to both the generated database and the on-line measurements of output variables so as to test the sensitivity. The database is generated with a white noise of zero mean and 0.001 standard deviation. Because noise with a standard deviation of 0.01 contaminates the data so severely that the quality of the database becomes inadequate for identification, the standard deviation of the noise added

to the database is taken as 0.001. For the control, white noise with zero mean and a standard deviation of 0.01 is added to all measured variables. As shown in Figure 9, unexpected oscillations are observed during the period between 200 and 400 min. Apparently, the onset of the gel effect in the presence of noise makes control of the system more difficult. After 400 min, however, the controller exhibits its good control performance in the sense that the controlled outputs follow the setpoints satisfactorily. Thus, this result demonstrates the robustness of the MoD-PC against the measurement noise. 6. Conclusions A multivariable model-on-demand-based predictive controller (MoD-PC) is designed to overcome the shortcomings associated with global model identification and on-line optimization that nonlinear MPC can encounter while maintaining the advantages of a local linear model structure and a linear MPC algorithm. The performance of the MoD-PC is corroborated by conducting simulation studies for the control of the weight-average molecular

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weight and the copolymer composition in a semibatch MMA/VAc copolymerization reactor. The regressor state vector used in the data searching process consists of the reaction temperature and the free volume of the reaction mixture. These variables are chosen by investigating the responses of all of the variables under open-loop operation. The prediction ability of the local ARX model is demonstrated through the model validation procedure, and the predictive controller is designed on the basis of this model. The MoD-PC proposed in this study exhibits excellent regulatory performances for both outputs even when interruption by a severe disturbance occurs. Also, satisfactory control performance in overcoming measurement noise demonstrates the robustness of the controller. Considering that the MoD-PC has merits in terms of the computational load, optimality, and flexibility as demonstrated by the illustrative case studies, one can suggest MoD-PC as a potential control strategy for the production of copolymers with desired properties. Acknowledgment Financial support from the Brain Korea 21 Program sponsored by the Ministry of Education is gratefully acknowledged. Nomenclature A(k) ) polynomial matrix in the ARX model Ak ) system matrix in the modified state-space model b ) distance criterion B(k) ) polynomial matrix in the ARX model Bk ) system matrix in the modified state-space model C(k) ) polynomial matrix in the ARX model Ck ) system matrix in the modified state-space model d(x,y) ) distance between x and y D(k) ) polynomial matrix in the ARX model e(k) ) discrete white noise with zero mean at time k E(k) ) polynomial matrix in the ARX model Fi ) instantaneous copolymer composition for monomer i F(k) ) polynomial matrix in the ARX model l(‚) ) positive norm function m ) control horizon m1 ) monomer 1 m(ψ(k),θ) ) model equation M ) weighting matrix Mw ) weight-average molecular weight (g/mol) p ) prediction horizon q ) shift operator qf ) feed flow rate (L/min) Rk+1|k ) setpoint trajectory vector Rpi ) propagation reaction rate of monomer i Tr ) reaction temperature (°C) u(k) ) control input vector at time k uk ) input vector at time k in the modified state-space model ∆Uk ) control input sequence vf(k) ) free volume of reactants vj f(k) ) normalized free volume of reactants wi(‚) ) weight used in the weighted regression problem xk+1 ) state vector at time k in the modified state-space model y(k) ) plant output vector at time k yˆ (k+p/k) ) p-step-ahead output prediction at time k Yk+1|k ) predicted output trajectory vector

Greek Letters θ ) model parameter θˆ ) estimated model parameter Λ∆U ) weighting matrix for the rate of input changes Λy ) weighting matrix for the control errors φ(k) ) regressor state vector at time k ψ(k) ) regression vector at time k Ωk(x) ) neighborhood of x at time k

Literature Cited (1) Schork, F. J.; Deshpande, P. B.; Leffew, K. W. Control of Polymerization Reactors; Marcel Dekker: New York, 1993. (2) Embirucu, M.; Lima, E. L.; Pinto, J. C. A Survey of Advanced Control of Polymerization Reactors. Polym. Eng. Sci. 1996, 36, 433. (3) Choi, K. Y.; Butala, D. N. An Experimental Study of Multiobjective Dynamic Optimization of a Semibatch Copolymerization Process. Polym. Eng. Sci. 1991, 31, 353. (4) Derrick, J. K.; MacGregor, J. F. Feedback Control of Polymer Quality in Semi-Batch Copolymerization Reactors. Chem. Eng. Sci. 1992, 47, 929. (5) Helbig, A.; Abei, O.; Marquardt, M. Model Predictive Control for On-line Optimization of Semibatch Reactors. In Proceedings of the American Control Conference; IEEE Press: Piscataway, NJ, 1998, Vol. 3, p 1695. (6) Garcia, C. E.; Prett, D. M.; Morari, M. Model Predictive Control: Theory and PracticesA Survey. Automatica 1989, 25, 335. (7) Perterson, T.; Hernandez, E.; Arkun, Y.; Schork, F. J. A Nonlinear DMC Algorithm and Its Application to a Semibatch Polymerization Reactor. Chem. Eng. Sci. 1992, 47, 737. (8) Pan, Y.; Lee, J. H. Recursive Data-based Prediction and Control of Product Quality for a Batch PMMA Reactor. In Proceedings of the American Control Conference; IEEE Press: Piscataway, NJ, 2000; Vol. 3, p 1747. (9) Henson, M. A. Nonlinear Model Predictive Control: Current Status and Future Directions. Comput. Chem. Eng. 1998, 23, 187. (10) Stenman, A. Model on Demand: Algorithms, Analysis and Applications. Ph.D. Thesis, Linko¨ping University, Linko¨ping, Sweden, 1999. (11) Cybenko, G. Just-in-Time Learning and Estimation. In Identification, Adaptation, Learning: The Science of Learning Models from Data; Bittani, S., Picci, G., Eds.; Springer: New York, 1996. (12) Stenman, A.; Gustafsson, F.; Ljung, L. Just in Time Models for Dynamical Systems. In Proceedings of the 35th IEEE Conference on Decision and Control; IEEE Press: Piscataway, NJ, 1996; Vol. 1, p 1115. (13) Stenman, A. Model-free Predictive Control. In Proceedings of the 38th IEEE Conference on Decision and Control; IEEE Press: Piscataway, NJ, 1999; Vol. 4, p 3712. (14) Braun, M. W.; Rivera, D. E.; Stenman, A.; Foslien, W. Comparison of Global Nonlinear Models and “Model-on-Demand” Estimation Appied to Identification of a RTP Water Reactor. In Proceedings of the 38th IEEE Conference on Decision and Control; IEEE Press: Piscataway, NJ, 1999; Vol. 4, p 3950. (15) Braun, M. W.; McNamara, B. A.; Rivera, D. E.; Stenman, A. Model-on-Demand Identification for Control: An Experimental Study and Feasibility Analysis for MoD-based Predictive Control. IFAC Symposium on System Identification; Santa Barbara, CA, June 21-23, 2000. (16) Hammouri, H.; McKenna, T. F.; Othman, S. Application of Nonlinear Observers and Control: Improving Productivity and Control of Free Radical Solution Copolymerization. Ind. Eng. Chem. Res. 1999, 38, 4815. (17) Park, M.-J.; Ahn S.-M.; Rhee, H.-K. Kinetic Parameter Estimation for the MMA/MA Copolymerization System. J. Appl. Polym. Sci. 2000, 78, 2554. (18) Hamielec, A. E.; Xie, T. Modelling Free-Radical Copolymerization Kinetics-Evaluation of the Pseudo-Kinetic Rate Constant Method 1. Macromol. Theory Simul. 1993, 2, 455. (19) Zhang, S. X.; Ray, W. H. Modeling of Imperfect Mixing and Its Effects on Polymer Properties. AIChE J. 1997, 43, 1265. (20) Pinto, J. C.; Ray, W. H. The Dynamic Behavior of Continuous Solution Reactors-VII. Experimental Study of a Copolymerization Reactor. Chem. Eng. Sci. 1995, 50, 715.

Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 859 (21) Soots, C.; Stanford, T. Modeling the Polymerization Processs The Gel Effect Function. 23rd Southeastern Symposium on Theory; Columbia, South Carolina, March 1991. (22) Hamer, J. W.; Akarmov, T. A.; Ray, W. H. The Dynamic Behavior of Continuous Polymerization Reactors II. Nonisothermal Solution Homopolymerization and Copolymerization in a CSTR. Chem. Eng. Sci. 1981, 36, 1987. (23) Lee, J. H.; Ricker, N. L. Extended Kalman Filter Based Nonlinear Model Predictive Control. Ind. Eng. Chem. Res. 1994, 33, 1530.

(24) Park, M.-J.; Hur, S.-M.; Rhee, H.-K. Online Estimation and Control of Polymer Quality in a Copolymerization Reactor. AIChE J. 2002, 48, 1013.

Received for review April 16, 2002 Revised manuscript received August 16, 2002 Accepted December 12, 2002 IE020291F

Design and Application of Model-on-Demand Predictive Controller to

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