Nonlinear Control of Polymerization Reactor by Wiener Predictive

Ind. Eng. Chem. Res. 2004, 43, 7261-7274

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Nonlinear Control of Polymerization Reactor by Wiener Predictive Controller Based on One-Step Subspace Identification In-Hyoup Song and Hyun-Ku Rhee* School of Chemical Engineering and Institute of Chemical Processes, Seoul National University, Kwanak-ku, Seoul 151-742, Korea

A subspace identification algorithm for the Wiener model is reformulated from a control point of view. The linear time-invariant part and the inverse static nonlinear function are simultaneously identified without an iterative procedure, and then the Wiener input/output data-based predictive controller is designed on the basis of the identified model. Through simulation and experimental studies for the multivariate control of polymer properties in a continuous methyl methacrylate (MMA) polymerization reactor, the proposed identification method is validated, and the performance of the designed predictive controller is examined. The Wiener model identified without any iterative procedure is found to predict accurately the output of the polymerization reactor, and the predictive controller designed in this work performs quite satisfactorily for polymer property control in the polymerization reactor. 1. Introduction Many researchers have adopted advanced control algorithms for polymer property control in polymerization reactors. These reactors often exhibit highly interactive nonlinear dynamic behavior. Schmidt et al.,1 for instance, demonstrated the existence of steady-state multiplicities, parametric sensitivity, and limit cycles for free-radical polymerization in a continuous stirredtank reactor. It is apparent that what is in demand is not only a more effective control algorithm such as predictive control but also nonlinear control and its actual implementation in continuous polymerization reactor systems. Recently, various researchers have reported experimental works for the nonlinear model predictive control of continuous solution polymerization reactors.2-9 The term “predictive control” indicates not a specific control strategy but a wide spectrum of control algorithms that make an explicit use of a process model in cost function minimization to obtain the control signal.10,11 When such a model is not available, the design of predictive control thus requires a stage of model construction. A prediction model needed to calculate the predicted output can be obtained from the process data, e.g., the dynamic matrix constructed using step response coefficients in dynamic matrix control (DMC). It has been found that this prediction model can be obtained from the input/output data by using the subspace matrices without going through the intermediate steps of model identification and design of a stochastic observer. Because no parametric model is required for the controller design, this controller is referred to as the “model-free controller”.12,13 Because no parametric model is explicitly calculated and no a priori information on the model such as the structure and order of model is required, this approach requires fewer user-set parameters than conventional approaches such as generalized predictive control (GPC). Furthermore, the calculation of prediction model without model reduction gives alleviated bias * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: +(82) 2-880-7405. Fax: (+82) 2-8887295.

errors. A comparison between the parametric approach and the model-free approach has been provided by Rossiter and Kouvaritakis.14 It is often argued that the requirement of massive amount of data might be a drawback of the model-free approach compared to the parametric model approach based on the prediction error method. Reducing the amount of data required still remains an open problem. However, it is evident that there are several attractive advantages that compensate for the drawback as mentioned in the above paragraph and as also reported by Rossiter and Kouvaritakis.14 Although it was first adopted in the design of a linear-quadratic-Gaussian (LQG) controller, this approach is in harmony with the design of predictive controllers because the subspace identification technique involves the minimization of the summation of multiple-step-ahead prediction errors. Several researchers in the field of predictive controller design have been attracted by these merits of the so-called model-free approach. Favoreel et al.15 proposed a model-free predictive control for bilinear systems. In our previous work,16 we designed and evaluated a linear input/output (I/O) data-based predictive controller with integral action, whereas Yoo and Rhee proposed a linear-fractional-linear-parameter-varying (LF-LPV) I/O databased predictive controller.17 More recently, Kadali et al.18 proposed a data-driven subspace approach for the design of a predictive controller, in which a linear feedforward controller as well as a feedback model-free subspace predictive controller were designed. Although the names of these controllers are different, the basic idea of controller design stems from the same root, i.e., the subspace identification algorithm. Despite the remarkable performances of nonlinear model-free predictive controllers reported in the abovementioned works, these controllers suffer from the huge dimensions of the block Hankel matrices involved.15,17 Moreover, they require some sort of iterative procedure to obtain the control input during online control. In case of the model-free subspace-based bilinear predictive control, the optimal control criterion is a nonlinear and nonconvex function for the control input. An iterative

10.1021/ie034279e CCC: $27.50 © 2004 American Chemical Society Published on Web 06/05/2004

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Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004

Gauss-Newton optimization method, which is only guaranteed to converge to a local minimum, was proposed to solve this problem. Although the optimal control criterion is a square and convex function for the control input in the case of an LF-LPV I/O data-based predictive controller, this controller is not free from the iterative calculation because it is necessary to update the parameter trajectories at every sampling time. The Wiener model attracts our interest as an alternative to the bilinear and LF-LPV models. Because of its relatively simple structure, this model has become increasingly popular as the “next-step-beyond-linear modeling” of chemical processes.19 The Wiener model is a special kind of nonlinear model that consists of a static nonlinear block and a dynamic linear part. This model is widely encountered in various fields of engineering and science.20,21 A well-known Wiener model encountered in the process industries is the pH neutralization reactor.22 Kalafatis et al. clarified the conditions under which the pH process can be treated by Wiener model.23 The Wiener model was also well described by Wigren.24 A general treatment of Wiener and Volterra models was presented by Rugh,25 and the direct estimation of the kernels was treated by Korenberg.20 Tools for semiphysical modeling were further discussed by Lindskog and Ljung.26 The aim of this work is the design of a Wiener I/O data-based predictive controller as a nonlinear modelfree predictive controller. Indeed, the prediction model was already used as the linear time-invariant (LTI) part of the Wiener model in our previous work.9 In the present study, unlike in Song et al.,9 we extend the batch least-squares algorithm to the design of the predictive controller. The novel ideas lie in the choice of nonparametric I/O data-based prediction model to describe the dynamic LTI part and the direct identification of the inverse nonlinear function. The matrix input/ output equation is reformulated in such a way that the output of the nonlinear process can be directly inserted into the matrix input/output equation. From the angle of Wiener model identification, the present scheme does not require any iteration to obtain the final LTI and nonlinear parts in the case of a stochastic system with insignificant output noise or a deterministic system, in contrast to the previous Wiener model identification methods. In the presence of output noise including both white and colored noises, iteration of the whole identification scheme can be applied to eliminate the biased estimation of model parameter caused by the noise. In addition, this scheme is simple and robust because actual calculation of the respective identified models is carried out via QR decomposition of the modified block Hankel matrices of the past input and output data. For a stochastic system, an iterative identification of the LTI part and the inverse nonlinear function is applied to remove the bias in the estimates as proposed by Kalafatis et al.23 From the control point of view, by virtue of adopting the Wiener model, the predictive controller can be designed efficiently without a tremendous increase of the dimensions of the data block Hankel matrices and without any iterative calculations during online control. To close this section, we provide an outline of the present article. In section 2, the problem is formulated, and some preliminaries are presented. With regard to the identification of Wiener model, the main results are given in section 3 along with some remarks, whereas

Figure 1. Wiener system with additive noise at the output.

the design of predictive controller is discussed in section 4. Finally, in section 5, the designed controller is implemented on a continuous polymerization reactor system, and its performance is evaluated by a series of simulation and experimental studies. 2. Problem Statement and Preliminaries Consider the Wiener system shown in Figure 1, which consists of an LTI part and a static nonlinearity. We assume that the measurement and process noises influence only the LTI part of the Wiener model as depicted in Figure 1. The unknown LTI part contains a prediction representation of the state space model, the so-called I/O data-based prediction model

yf ) Lwwp + Luuf

(1)

in which wp is defined as (ypTupT)T with the known values of inputs up ∈ Rpnu and outputs yp ∈ Rpny, respectively, collected during the last p time intervals, where

yp ) (yk-p ‚‚‚ yk-2 yk-1)T up ) (uk-p ‚‚‚ uk-2 uk-1)T yf ) (yk yk+1 ‚‚‚ yk+p-1)T and

uf ) (uk uk+1 ‚‚‚ uk+p-1)T with uk ∈ Rnu and yk ∈ Rny as the input vector and the output vector, respectively, of the LTI part. The output nonlinearity is given by

zk ) f(yk + ν′k)

(2)

where zk ∈ Rny is the output vector of the static nonlinearity and ν′k denotes the measurement noise that is assumed to be zero-mean white noise. It is assumed that f(‚) is one-to-one and continuous so that its inverse exists and admits a polynomial representation m

yk )

m

cizki - ν′k ) zk + ∑cizki + vk ∑ i)1 i)2

(3)

For brief notation, the power of vector zk indicates the powers of its elements. To obtain a unique parametrization of the Wiener model, it can be assumed that c1 ) [1 ‚‚‚ 1]T without loss of generality. The remaining coefficients ci (i ) 2, 3, ..., m) and the prediction gain matrices Lw and Lu are parameters to be identified.

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Because ν′k is white noise, vk ) - ν′k is also zero-mean white noise. In the next section, the I/O data-based prediction model will be reformulated in such a way that the output of a nonlinear process can be directly inserted into the matrix input/output equation.

where

vp and vf are defined in the same way, and

zk-p2 l zk-pm zk-p+12 l wzp ) zk-p+1m l zk-12 l zk-1m

3. Identification of Wiener Model The I/O data-based prediction model, eq 1, can be rewritten as

( )

y yf ) Lwwp + Luuf ) (LwyLwu) up + Luuf ) p Lwyyp + Lwuup + Luuf (4)

() ( ) ([ ) ( ) ( )]

Substituting the polynomial representation of the inverse function, eq 3 into eq 4, i.e., into each element of vectors yf and yp, we obtain

( )( )

m

cizki ∑ i)2

yk zk m yk+1 zk+1 cz i ) + i)2 i k+1 yf ) l l l yk+p-1 zk+p-1 m

( )

cizk+p-1i ∑ i)2

m

cizk+1i ∑ i)2

cizk-p ∑ i)2

zk-p vk-p m zk-p+1 v cizk-p+1i + i)2 + k-p+1 l l l zk-1 vk-1

∑

)

l

m

cizk+p-1i ∑ i)2

+

m

cizk-1 ∑ i)2

i

Lwuup + Luuf (5)

Transposition of the second and third terms on left-hand side of eq 5 to the right-hand side gives

( ) ( ) ( ) ( ) ( )( ) zk-p zk-p+1 + Lwu l zk-1

uk-p uk-p+1 + l uk-1

( )( ) m

∑ i)2

)

cizki ∑ i)2

i

zk z ) Lwy zf ) k+1 l zk+p-1

0 l 0 cm

m

m

) Lwy

(

zk2 l zkm zk+12 l wzf ) zk+1m l zk+p-12 l zk+p-1m

c2 c3 ‚‚‚ cm 0 ‚‚‚ c2 c3 ‚‚‚ cm 0 Lzf ) l ‚‚‚ 0 0 c2 c3 ‚‚‚ Lzp ) LwyLzf

vk v + k+1 l vk+p-1

∑

() ()

zp ) (zk-pT zk-p+1T ‚‚‚ zk-1T)T

(

c2 c3 ‚‚‚ cm 0 ‚‚‚ c2 c3 ‚‚‚ cm 0 l l 0 ‚‚‚ 0 c2 c3 ‚‚‚

0 l 0 cm

m

cizki ∑ i)2

cizk-pi

uk m m i uk+1 cz cz i + Lwy i)2 i k-p+1 - i)2 i k+1 Lu l l l uk+p-1

∑

∑

m

m

cizk-1i ∑ i)2

+

cizk+p-1i ∑ i)2

Lwy

vk-p vk vk-p+1 v - k+1 l l vk-1 vk+p-1

) Lwyzp + Lwuup + Luuf + Lzpwzp - Lzfwzf + Lwyvp - vf (6)

()

)

zk2 l zkm zk+12 l zk+1m l zk+p-12 l zk+p-1m

The vector equation is easily extended to the matrix equation by augmenting the vectors columnwise in sequence of increasing sampling time. In matrix form, eq 6 is equivalent to

Zf ) LwyZp + LwuUp + LuUf + LzpWzp - LzfWzf + Ef (7) in which the data block Hankel matrices for zk and uk, represented as Zf and Uf, respectively, with p-block rows and N-block columns are defined as

(

zk zk+1 ‚‚‚ zk+N zk+1 zk+2 zk+N+1 Zf ) l l zk+p-1 zk+p ‚‚‚ zk+N+p-1 and

)

7264

(

)


uk uk+1 ‚‚‚ uk+N uk+1 uk+2 uk+N+1 Uf ) l l uk+p-1 uk+p ‚‚‚ uk+N+p-1

Ll ) L(:, 1: nyp + 2ny(m - 1)p) Ln ) L(:, nyp + 2ny(m - 1)p + 1:end) and thus

and Wzp and Wzf are defined for wzp and wzf in the same way. Here, Ef represents LwyVp - Vf. Because vk is zeromean white noise, Ef consists of zero-mean noise signals. To calculate Lwy, Lwu, Lu, and Lzf, the block Hankel matrices are rearranged as follows

Zf ) (Lwy Lzp

( )

( )

Zp U Lzf) Wzp + (Lwu Lu) Up + Ef f -Wzf

) LlWl + LnWn + Ef

(8)

where Ll and Ln are given as (Lwy Lzp Lzf) and (Lwu Lu), respectively. If (i) the measurements of the input ut and the process output yt for t ∈ (k - p, k + N + p - 1) are available, (ii) the deterministic input is persistently exciting of order 2p, and (iii) the number of measurements go to infinity as N f ∞, the open-loop models can be consistently identified, and the identification involves finding the prediction of the future outputs Zf. The leastsquares prediction of Zf can be found by solving the least-squares problem

( )||

||

W min Zf - (LlLn) W1 Lw,Lu n

2

(9)

F

The solution can be easily obtained by calculating the orthogonal projection of the row space of Zf onto the row space spanned by Wl and Wn, i.e.

( )

W Z ˆ f ) Zf/ Wl ) LlWl + LnWn n

(10)

Lw ) [Ll(:, 1:nyp)Ln(:, 1:nup)] Lu ) Ln(:, nup + 1:end) Lzf ) Ln(:, nyp + ny(m - 1)p + 1:end)

Zf/WlWn ) LnWn

and

(11)

A numerically efficient and robust way to realize this projection is the following QR factorization

()(

)( )

Q1T Zf R11 0 0 Wl ) R21 R22 0 Q2T Wn R31 R32 R33 Q T 3

( ) ( )( ) +

(12)

( ) [( )( ) ] ( ) )( ) ( )

(

R 0 ) (R31R32) R11 R 21 22

+

T

Wl Wl Wn Wn

(16)

and

yfI ) zf + Lzfwzf,

rfI ) rf + Lzfrzf

(17)

where

in which rk denotes the reference input of the controller at the kth sampling time. 4. Wiener-type Predictive Control

Wl Wn )

W Zf Wl n

yfL ) Lwwp + Luuf

rf ) (rkT rk+1T ‚‚‚ rk+p-1T)T

The orthogonal projection is expressed as

W W Zf/ Wl ) Zf Wl n n

(15)

where nu and ny denote the numbers of manipulated variables and controlled variables, respectively. The first colon in L(:, j:k) refers to all the elements in a column of matrix L, and the second colon means to take all the columns from jth column to the kth column. The keyword “end” refers to the last column. [Ll Ln] denotes the columnwise augmentation of Ll and Ln. In the deterministic system, Lw, Lu, and Lzf can be directly calculated without iteration because this method is built without the assumption that the output of the LTI part is known, i.e., the output of the LTI part is equivalent to the process output. For the stochastic system, the iterative least-squares algorithm proposed by Kalafatis et al.23 can be useful in obtaining the unbiased matrices Lw, Lu, and Lzf. The identified set is divided into two parts, a linear part and a nonlinear part. The predicted output of the identified LTI part is calculated by eq 16, and the reference input and process output are converted to their respective linear counterparts by eq 17. As intended, the output prediction model is the I/O databased prediction model, and the inverse nonlinear function is represented by the identified matrix Lzf consisting of polynomial coefficients of the inverse nonlinear function. It is worth noting that the polynomial can be represented by LzfWzf. These two parts can be obtained from

In terms of the oblique projection, LlWl and LnWn can be obtained, respectively, by

Zf/WnWl ) LlWl

(14)

T -1

Wl Wl Wn ≡ L Wn

Wl Wn

(13)

where the symbol + denotes the pseudo-inverse. Using the Matlab notation, we obtain from eqs 10, 12, and 13 that

Figure 2 shows the conventional schematic block diagram of the nonlinear controller based on the Wiener model. If the performance index is chosen to be quadratic and the constraints are linear, then the optimization can be reduced to a series of quadratic programming (QP) problems. The future reference trajectory rf and the future input sequence uf are defined as

()

rt+1 r rf ) t+2 l rt+p

and

()

ut+1 u uf ) t+2 l ut+p

(18)


eq 17 can convert the reference input and process output to their respective linear counterparts simultaneously. Substitution of eqs 16 and 17 into eq 22 gives

J ) (rf + Lzfrzf - Lwwp - LuΛu j f)TQ ˆ (rf + Figure 2. Schematic diagram of a Wiener-type control system.

For the purpose of reducing the computational complexity and producing a less-aggressive controller, it is a common practice to introduce constraints on the future control inputs. An often-used approach is to assume that control increments are equal to zero after the control horizon Nc, i.e.

∆ut+k ) ut+k - ut+k-1 ) 0

for k > Nc

(19)

Taking into account the fact that the control horizon is typically shorter than the prediction horizon p and that eq 19 is required, one can express uf as follows

()

ut+1 ut+2 l jf ) Λ u uf ) Λu t+Nc

l ut+Nc

( )

where

I nu 0 0 Inu l l Λ) 0 0 l l 0 0

‚‚‚ ‚‚‚ l ‚‚‚

l I ‚‚‚ nu

∆u j f ) Du j f - ut

D)

(

0

l l ‚‚‚ -Inu

0 l 0 I nu

)

and

where rzf is constructed for rk in the same manner as wzf is constructed for zk. It has been reported that the objective function of a nonlinear process can be optimized with the control input obtained by solving its linear counterpart.27,28 For the constrained case, the constraint should also be transformed into its linear counterpart via eq 17, and then the optimization is solved by quadratic programming (QP). In conventional nonlinear predictive control, sequential quadratic programming (SQP) is used to optimize the objective function. With the nonlinear predictive controller designed in this work, however, a simpler QP approach can be applied to calculate the control input. Here, it should be emphasized that the computational burden is reduced in comparison to that required for SQP. For the unconstrained case, the minimizing control input is obtained explicitly by means of the ordinary least-squares theory as follows

ˆ (rf + Lzfrzf - Lwwp)] (24) ΛTLuTQ

The control input increments can also be expressed in vector form

Inu 0 ‚‚‚ -Inu Inu

ˆ ∆(Du j f - ut) (23) (Du j f - ut)TR

ˆ + DT R ˆ ∆D + ΛTLuTQ ˆ LuΛ)-1[DTR ˆ ∆ ut + uf ) (R

0 0 l I nu

by introducing the auxiliary quantities

j f) + u j fTR û jf + Lzfrzf - Lwwp - LuΛu

(20)

()

ut 0 ut ) l 0

Then, the optimal control criterion can be written as

J ) (rf - zf)TQ ˆ (rf - zf) + u j fTR û jf + ˆ ∆(Du j f - ut) (21) (Du j f - ut)TR where Q ˆ, R ˆ , and R ˆ ∆ are diagonal weighting matrices having appropriate dimensions for the output, input, and input increment, respectively. If the forward nonlinear function is one-to-one and continuous, the optimal criterion can be minimized by the control input that minimizes the following objective function27,28

ˆ (rfI - yfL) + u j fTR û jf + J ) (rfI - yfL)TQ ˆ ∆(Du j f - ut) (22) (Du j f - ut)TR in which rfI represents the linear counterpart of rf calculated by eq 17 and yfL denotes the linear prediction value of yf calculated by eq 16. It is worth noting that

With the proposed control algorithm, there is no need to calculate the state estimate or the state space model to calculate the optimal predicted output.15 Currently, instead of the reduction of the computational load, the reduction of the number of tuning parameters can be a major concern in industrial applications. In these regards, the control scheme proposed in this study clearly has merit compared to conventional model predictive control. The following list summarizes the main contributions of this work: (1) The linear I/O data-based predictive controller is extended to a nonlinear I/O data-based predictive controller using the Wiener model. (2) Wiener model identification can be performed in one step by adopting the inverse polynomial representation, and the proposed scheme is designed to be free from the convergence problem in the case of systems without noise or with insignificant noise. (3) The proposed controller requires fewer tuning parameters than the parametric model-based predictive controller by virtue of the modelfree approach. (4) Nonlinear I/O data-based predictive control that guarantees the global minimum of control objective function is first designed. 5. Application to Polymer Quality Control in a Continuous Polymerization Reactor 5.1. Validation of Control Scheme by Simulation Study. The proposed control scheme needs to be validated by a simulation study against various scenarios before application to an actual polymerization reactor. For this purpose, we will consider polymer quality control in a continuous methyl methacrylate (MMA) solution polymerization reactor system. For a detailed description, one can refer to the previous work.6 The first-principles model to be used as the plant can be

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Table 1. Reference Conditions for the Continuous MMA Polymerization Reaction System Used in the Simulation Study and Experimental Work for Polymer Property Control parameter initial charge monomer solvent initiator feed concentration monomer solvent initiator operating conditions jacket inlet temperature feed flow rate

value 460 mL 340 mL 4.00 g 5.42 mol/L 4.35 mol/L 0.02 mol/L 75 °C (62.5-82.5 °C) 7 mL/min (3-23 mL/min)

found in the same reference. Table 1 shows the reference operating conditions and the content of reaction and feed mixtures. (1) Wiener Model Identification. For the identification, we use the input and output data obtained from numerical simulation of the continuous polymerization reactor. Here, the inlet temperature of the reactor jacket Tjin (from 62.5 to 82.5 °C) and the feed flow rate qf (from 3 to 23 mL/min) are taken as the input variables, whereas the monomer conversion and the weightaverage molecular weight (Mw) are considered as the outputs. Pseudo-random multilevel input signals are employed because these are closer to the white autocorrelation function and adequately excite nonlinear modes of multivariate systems.29 Figures 3 and 4 present the input data and the corresponding output data without noises (vk ) 0) and with 5% measurement noises to both outputs, respectively, for which the sampling time is 1 min. Here, 5% noise means that the

standard deviation of the noise is 0.05, which is the same as in Wigren24 and Kalafatis et al.23 The numbers of past input and output data points used to predict the future output are both 15, and the order of polynomial is determined as 3. Figure 5 shows the results of the identification involving the LTI part and the inverse nonlinear function for the noise-free case, whereas the results of onestep identification for 5% measurement noises to both outputs are illustrated in Figure 6. The predicted output of the LTI part is compared with the output of the inverse nonlinear function. The variance-accounted-for (VAF) indices for the conversion and Mw are 99.97 and 99.99%, respectively, in the noise-free case and 98.89 and 98.97%, respectively, with 5% noises. These results are in good agreement with the observation that the batch least-squares approach is comparable to the method reported by Wigren.24 The VAF index is defined as

( ) x∑ x∑ Nv

(yLk - yIk)2

VAF ) 100 1 -

k)1

(25)

Nv

(yLk )2

k)1

where Nv, yLk , and yIk denote the number of data points used in validation, the outputs of the identified linear model, and the outputs of the identified inverse nonlinear function, respectively. It is clearly demonstrated that the Wiener model identified in one step without an iterative procedure can provide an excellent prediction of the outputs.

Figure 3. Output data without measurement noises and the pseudo-random four-level input signal obtained by simulation.


Figure 4. Output data with 5% measurement noises and the pseudo-random four-level input signal obtained by simulation.

Figure 5. Validation of the identified Wiener model. The solid curve (s) denotes the output of the LTI part, and the symbols (+) represent the output of the inverse nonlinear function.

(2) Wiener Input/Output Data-Based Predictive Control. The objective here is to examine the perfor-

mance of the Wiener I/O data-based predictive controller before implementing the controller for the control of

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Figure 6. Validation of the identified Wiener model in the presence of 5% measurement noises. The solid curve (s) denotes the output of the LTI part, and the symbols (+) represent the output of the inverse nonlinear function.

Figure 7. Servo performance of the Wiener input/output data-based predictive controller for step changes in the set points of conversion and weight-average molecular weight.


Figure 8. Servo performance of the Wiener predictive controller based on iterative identification for step changes in the set points of conversion and weight-average molecular weight.

Figure 9. Servo performance of the Wiener input/output data-based predictive controller for step changes in the set points of conversion and weight-average molecular weight in the presence of 5% measurement noises.

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Figure 10. Schematic diagram of the experimental apparatus.

Figure 11. Four-level pseudo-random input and output data generated by experiment.

polymer properties in a continuous polymerization reactor. For control purposes, we set the sampling time as 1 min and select the prediction and control horizons as 15 and 7 min, respectively. The input constraints are taken into account in the calculation of the predictive control input. The jacket inlet temperature ranges between 62.5 and 82.5 °C, and the feed flow rate is bounded between 3 and 23 mL/min. The weighting matrices are determined by trial and error.

Figures 7 and 8 show the multilevel set-point-tracking performances of the controller proposed in this work and the Wiener predictive controller based on iterative identification, respectively. Both controllers use the same LTI model and are best tuned by trial and error. To examine the control performance in the presence of measurement noise (vk), 1, 5, and 10% noises are added to both outputs of the identified LTI part. In Figure 9, we observe that the demand on the set-point changes


Figure 12. Validation of the experimentally identified Wiener model.

is satisfactorily accomplished, although both the input and output start to oscillate as the variance of the noise increases. The upper two diagrams present the outputs, whereas the lower two diagrams show the control inputs. To compare the input variations to each other more clearly in one diagram, we simply added the values 20 and 40, respectively, to the values of both inputs when 5 and 10% noises were introduced to the system. Here, it is clearly seen that the proposed controller performs quite satisfactorily and its performance appears to be as good as that of the Wiener predictive controller based on the conventional identification scheme. The sampling rate of 1 min seems demanding for the control of chemical processes. Such a high sampling rate was used to guarantee a good control performance. However, improvements in computer technology can make the high sampling rate a less serious problem. Continuous-time identification of the Wiener model and the controller design can be a breakthrough for this drawback. A continuous-time multi-input, multi-output Wiener modeling method was proposed by Bhandari and Rollings,30 who demonstrate its excellent prediction performances. A continuous version of the controller pro-

posed in this study is now under investigation, and it is expected that the slow sampling rate can be treated thereby. 5.2. Experimental Implementation of Control Scheme. (1) Experimental System. The continuous polymerization reactor system consists of the reaction part and the automatic control part. The former is composed of a glass polymerization reactor, a heattransfer apparatus, and a mechanical stirrer, whereas the latter includes a programmable logic controller for input and output data, a computer for control and data acquisition, and an actuator. In Figure 10, a schematic diagram of the polymerization reactor system is presented. The jacketed glass overflow reactor has a capacity of 1 L and is equipped with a stirrer for mixing of the reactants. A circulation line is attached to the reactor to measure the density and viscosity of the reaction mixture online. A small portion of the reaction mixture is circulated by a diaphragm metering pump through the circulation line, in which the online densitometer (model DMA401 YHEW from Anton Paar) and viscometer (model MIVIADF from SOFRASER) are installed. The detailed

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Figure 13. Servo performance of the Wiener input/output data-based predictive controller: Experimental results.

experimental system and procedure are described elsewhere.6,7 The online measurements are essential for the property control in a polymerization reactor. In our previous works,6-8 the online measurement techniques were described and experimentally validated. By comparing the online measurements with the offline data, it was found that the monomer conversion and molecular weight could be determined from the online measured density and viscosity data by using appropriate correlations. (2) Identification and Control Experiments. The identification experiment was conducted following the same procedure as in the simulation study. Figure 11 shows the four-level pseudo-random input signals and the corresponding outputs, which are fully excited. The solid curves in the upper two diagrams denote the online measurements calculated from the online density and viscosity data, while the solid circles represent offline measurements of conversion and Mw, which are determined by gravimetry and gel permeation chromatography, respectively. These results indicate that the online measurements can be used for identification purposes. The bounds of the inputs are set on the basis of the capabilities of the experimental system. The Wiener model is obtained by using the input and output data and validated as shown in Figure 12. The outputs from the linear part and the inverse nonlinear function are compared to each other to obtain the VAF indices of 99.0% for conversion and 98.2% for Mw. Using the identified Wiener model, we design the Wiener I/O data-based predictive controller and implement it to the control of monomer conversion and Mw

in the continuous MMA polymerization reactor. The reference operating conditions and the constraints are the same as for the simulation study. The sampling time of the control algorithm is 1 min. The set-point-tracking control is conducted with a prediction horizon of p ) 30, a control horizon of Nc ) 10, Q ) diag(125, 10), and R ) diag(1, 1). The weighting matrices, Q and R, for the output and input, respectively, are determined by trial and error. Figure 13 shows the profiles of the controlled outputs when there are demands for changes in the set points for both the conversion and the molecular weight. Here, the new set points are selected considering the range of outputs covered by the identification experiment. At 130 min, the set point for Mw is raised from 100 000 to 110 000 g/mol, whereas the set point for conversion is maintained at 0.12. Predictably, the controller increases the feed flow rate to raise Mw. Because the reactor system is dynamically interactive, the conversion tends to decrease, and this is compensated by making the jacket inlet temperature increase. When the set point for conversion increases from 0.12 to 0.17 at 230 min, the controller mainly increases the jacket inlet temperature. However, Mw tends to decrease because of the increased reactor temperature, and thus the controller slightly increases the feed flow rate to regulate Mw. Finally, the set points for conversion and Mw are changed simultaneously from 0.17 to 0.15 and from 170 000 to 190 000 g/mol, respectively. Here, we note that a decrease in the jacket inlet temperature or an increase in the feed flow rate can bring about a decrease in the conversion and at the same time an increase in


Mw. As expected, the controller exercises a predictive action on both inputs to drive both outputs to their respective set points. As a whole, the results of the control experiment demonstrate that the nonlinear predictive control scheme proposed in this study can be applied to polymer property control in actual polymerization processes. 6. Conclusions Here, the Wiener model is chosen to identify a continuous polymerization reactor system by the method of subspace identification. It is shown that the linear time-invariant part and the inverse static nonlinear function can be identified simultaneously without any iteration procedure by reformulating the subspace identification algorithm. This identification scheme is then validated by a simulation study as well as by experimental work on a continuous MMA polymerization reactor. Then, the Wiener input/output data-based predictive controller is designed on the basis of the identified model and applied to polymer property control in a polymerization reactor. By conducting a series of simulation and experimental studies it is demonstrated that the predictive controller designed in this work performs quite satisfactorily for the control of polymer properties in a continuous polymerization reactor, which gives rise to a multivariate control system with input constraints. Acknowledgment The financial support from the Brain Korea 21 Project is greatly appreciated. Nomenclature ci ) polynomial coefficients J ) quadratic performance index Lu ) gain matrix for the future input in the linear I/O databased prediction model Lw ) gain matrix for the past input and output in the linear I/O data-based prediction model Lzf ) gain matrix for the inverse nonlinearity consisting of polynomial coefficients Mw ) weight-average molecular weight m ) order of polynomial N ) number of data points Nc ) control horizon Nv ) number of output data points used in validation p ) prediction horizon Q ) weighting matrix for output R ) weighting matrix for input R∆ ) weighting matrix for input movement rf ) set-point trajectory uf ) control input sequence over the control horizon uk ) control input at the kth sampling time step up ) past input sequence within the latest p interval wp ) past data sequence consisting of yp and up yf ) predicted output yk ) outputs of the identified linear part in Wiener model at the kth sampling time step yp ) past output sequence within the latest p interval zk ) process output at the kth sampling time step

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Received for review November 28, 2003 Revised manuscript received April 2, 2004 Accepted May 3, 2004 IE034279E

Nonlinear Control of Polymerization Reactor by Wiener Predictive

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