Batch-to-Batch Iterative Learning Control of a Batch Polymerization

Clarke-Pringle and MacGregor(10) introduced the batch-to-batch ... The details of ELM and OS-ELM are described in section 2. ... For sequential learni...
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Batch-to-Batch Iterative Learning Control of a Batch Polymerization Process Based on Online Sequential Extreme Learning Machine Tang Ao,* Xiao Dong, and Mao Zhizhong College of Information Science and Engineering, Northeastern UniVersity, Shenyang, Liaoning ProVince, People’s Republic of China

This paper develops a batch-to-batch iterative learning control (ILC) strategy based on online sequential extreme learning machine (OS-ELM) for batch optimal control. On the basis of extreme learning machine (ELM), a data-based nonlinear model is first adopted to capture the batch process characteristics aiming to obtain superior predictive accuracy. Subsequently, due to the model-plant mismatch in real batch processes, an ILC algorithm with adjusting input trajectory by means of error feedback is employed focusing on the improvement of the final product quality. In order to cope with the problems of the unknown disturbances and process variations from batch to batch, when a batch run is completed, OS-ELM is utilized to update the model weights so as to guarantee the precision of the model for optimal control, which corresponds to a nonlinear updating procedure. The feasibility and effectiveness of the proposed method are demonstrated via the application to a simulated bulk polymerization of the styrene batch process, and the simulation results show superior performance. 1. Introduction Batch and semibatch processes play important roles in most industries such as special chemical, semiconductor, and biology for high-value products to meet high speed development programs. They can be characterized as systems which generate a desired product in a batch reactor through the execution of a planned schedule during finite time. Thus, the final quality of the product at the end of the batch takes an important role in our interests. Due to model-plant mismatch and batch-to-batch variation, however, the quality of a product at the end of the batch may deviate from the desired quality leading to bad product. Hence, much research based on iterative learning control (ILC) has been proposed to solve this issue,1-9 which exploits the repetitive nature of batch processes and uses process knowledge obtained from previous batches to revise the operating strategy of the current batch so that the output trajectory converges asymptotically to the desired trajectory. There are some previous reports related to this topic provided by a considerable number of researchers. Clarke-Pringle and MacGregor10 introduced the batch-to-batch adjustments to optimize the molecular-weight distribution. Lee and co-workers in several papers5,11-13 proposed ILC-based quadratic criterion approaches for quality control of batch processes. Moreover, Dong et al.14 developed batch-to-batch control based on a hybrid model to realize the particle size distribution control. The purpose of online operation is to obtain the final quality prediction immediately instead of off-line analyzing in lab to determine the following batch control trajectory. Nevertheless, the batch process has the nature of dynamic, time-varying, and strongly nonlinear behavior, which means that the mechanism modeling is complicated, inaccurate, and time-consuming. Hence, it is improper to adapt the mechanism model to predict the final product quality in some situations. Data-based modeling can be a useful alternative in this case. Campbell et al.15 outlined a brief survey of run-to-run control algorithms based on linear models for batch processes. Xiong et al.8 proposed a linear timevarying perturbation model for product quality trajectory * Corresponding author. Tel.: +86-024-83689525. Fax: +86-02483689525. E-mail: [email protected].

tracking. Furthermore, Flores-Cerillo and MacGregor7,16,17 utilized the partial least-squares model to reject persistent process disturbances and achieve new final product quality targets. However, the application of linear modeling to inherently nonlinear processes may lead to inefficient and unreliable process modeling, it is for these reasons that most linear models are usually inaccurate to describe the complex nonlinearity within the process variables. To cope with this issue, a number of strategies have been proposed using neural networks, which can approximate any complex nonlinear functions. Dong at el.14 used neural network models for batch-to-batch optimization. Zhang6 developed a stacked neural network model to overcome the difficulties in developing a mechanistic model and show the nonlinear characteristics. In the above literature, a neural network is capable of a building nonlinear model by means of approximating any nonlinear functions properly whereas it does not take the process variations from batch to batch and unknown disturbances into account, which will give rise to inferior performance of the model. Therefore, the model should be updated to capture the changes over batches. Xiong and Zhang8 developed an updated perturbation model to handle the dynamics, which is essentially considered as a linear model. Thus, exploring a nonlinear modeling method that can be updated online deserves more attention. Kernel-based algorithms are often used for nonlinear modeling and monitoring.18-23 An adaptive updating strategy for these algorithms, however, is unsuitable for some fast manufacturing systems due to the complexity and time consumption of the online calculation. Moreover, there is still no theoretical literature in this field. Another choice is to employ a kind of neural network which is able to recursively update while modeling online. Recently Huang et al.24 proposed a single hidden layer feedforward neural network (SLFN) with additive or radial basis function (RBF) nodes referred to as extreme learning machine (ELM), which tends to provide superior generalization performance and extremely fast learning speed, as well as less parameters to be identified by randomly choosing the input weights and analytically determining the output weights of SLFNs. They also proved the universal approximation capability of ELM in the literature.25-27 Comparing with

10.1021/ie9007979 CCC: $40.75  2009 American Chemical Society Published on Web 10/26/2009

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other modeling methods, ELM can describe complex nonlinearity within process variables better than the linear models as well as cope with many problems of conventional neural networks such as overfitting, local minima, and slow learning speed. Therefore, it is very suitable for data-based modeling in complicated chemical processes. According to its advantages, Liang and Huang28 presented the online sequential ELM algorithm for a batch process, which can update the network with newly arriving data one-by-one or chunk-by-chunk. To our knowledge, combining a nonlinear update model with iterative learning control for batch processes has not previously been reported. Motivated by the online sequential extreme learning machine (OS-ELM) and inspired by batch-to-batch iterative learning control, this paper aims to develop a recursive nonlinear model to handle the complex nonlinearity and update the model online with fast speed for batch-to-batch optimal control. First of all, we will collect training data from several normal batches and build the ELM model for the final quality prediction, then determine the next input trajectory according to iterative learning control, and finally add the current data into the data set and update the model using OS-ELM algorithm for the next batch. The remainder of this paper is organized as follows. The details of ELM and OS-ELM are described in section 2. In section 3, we introduce a batch-to-batch iterative learning control strategy on the basis of the OS-ELM model and give the outline of the proposed strategy. Section 4 shows the simulated results by applying it to the bulk polymerization of styrene in a batch reactor. Finally, conclusions are drawn in section 5. 2. Brief of ELM and OS-ELM 2.1. ELM. In supervised batch learning, the learning algorithms use a finite number of input-output samples for training. For N arbitrary distinct samples (xi,ti) ∈ Rn × Rm, where xi is a n × 1 input vector and ti is a m × 1 target vector, if an SLFN ˜ hidden (single-hidden layer feedforward neural network) with N nodes can approximate these N samples with zero error, it then implies that there exist βi, ai, and bi such that ˜ N

fN˜(xj) )

∑ β G(a , b , x ) + ε i

i

i

j

j

) tj,

j ) 1, ..., N

(1)

i)1

Where, ai and bi are the learning parameters of hidden nodes (weight vector connecting the input node to the hidden node and threshold of the hidden node) which are randomly selected according to the proof given by Huang et al.24,29 and βi is the weight connecting the ith hidden node to the output node. To avoid overfitting the noise in the data, an error term εj is added. G(ai,bi,x) is the output of the ith hidden node with respect to the input x and N˜ is the number of hidden nodes which can be determined by trial and error or prior expertise. Then, eq 1 can be written compactly as Hβ ) T

[

and

β1 β) l βN˜T

]

[]

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T

t1 and T ) l tNT

N˜×m

(4)

N×m

where H is called the hidden layer output matrix of the network; the ith column of H is the ith hidden node’s output vector with respect to inputs x1, x2, ..., xN and the jth row of H is the output vector of the hidden layer with respect to input xj. According to theorems in ref 24, the hidden node parameters ai and bi need not be tuned during training and may simply be assigned with random values. Equation 2 then becomes a linear system, and the output weights β are estimated as β˜ ) H+T

(5)

where H+ is the Moore-Penrose generalized inverse30 of the hidden layer output matrix H. More details are provided in ref 24. 2.2. OS-ELM. In real applications, the training data may arrive chunk-by-chunk or one-by-one; hence, the batch ELM algorithm has to be modified for this case so as to make it online sequential. The output weight matrix βˆ (βˆ ) H+T) given in (5) is a leastsquares solution of (2). Here, we consider the case where rank(H) ) N˜ is the number of hidden nodes. Under this condition, H+ of (5) is given by H+ ) (HTH)-1HT

(6)

If HTH tends to become singular, one can make it nonsingular by choosing smaller network size N˜ or increasing data number N in the initialization phase of OS-ELM. Substituting (6) into (5), βˆ becomes βˆ ) (HTH)-1HTT

(7)

Equation 7 is called the least-squares solution to Hβ ) T. Sequential implementation of the least-squares solution of (7) results in the OS-ELM. N0 Given a chunk of initial training set N0 ){(xi,ti)}i)1 and N0 ˜ , if one considers using the batch ELM algorithm, one needs gN to consider the solution of minimizing |H0β - T|, which is given by β0 ) K0-1H0TT0 where K0 ) H0TH0. Suppose that we have another chunk of data N1 N0+N1 ){(xi,ti)}i)N +1, where N1 is the number of samples in this 0 chunk. The problem becomes minimizing

|[ ] [ ]| H0 T0 βH1 T1

(8)

Considering both N0 and N1, the output weight β becomes

[ ][ ]

β1 ) K1-1

(2)

where H(a1, ..., aN˜, b1, ..., bN˜, x1, ..., xN) ) G(a1, b1, x1) · · · G(aN˜, bN˜, x1) l ··· l G(a1, b1, xN) · · · G(aN˜, bN˜, xN)

[] T

H0 H1

T

T0 T1

where K1 )

[ ][ ] H0 H1

T

H0 H1

(9)

For sequential learning, we have to express β1 as a function of β0, K1, H1, and T1 and not as a function of the data set N0. Now K1 can be written as

[ ]

(3)

K1 ) [H0T H1T ]

N×N˜

and

H0 ) K0 + H1TH1 H1

(10)

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[ ][ ] H0 H1

T

T0 T1

)H0TT0 + H1TH1 ) K0K0-1H0TT0 + H1TT1 )K0β0 + H1TT1 ) (K1 - H1TH1)β0 + H1TT1

where tf is the batch end time. For the kth batch, the actual product quality can be written as the sum of the model prediction and its error yk(tf) ) yˆk(tf) + εk

)K1β0 - H1TH1β0 + H1TT1 (11) Combining 9 and 11, β1 is given by

[ ][ ]

β1 )K1-1

H0 H1

T

T0 ) K1-1(K1β0 - H1TH1β0 + H1TT1) T1

The prediction of the (k + 1)th batch can be approximated expressed as follows yˆk+1(tf) ) yˆk(tf) +

)β0 + K1-1H1T(T1 - H1β0)

(19)

∂f ∂f (u k+1 - u1k) + · · · + (u k+1 - uNk) ) ∂u1 1 ∂uN N

yˆk(tf) + GT∆Uk+1

(20)

(12)

and

where K1 ) K0 + H1TH1. When the (k + 1)th chunk of data set

∆Uk+1 ) [∆u1k+1∆u2k+1 · · · ∆uNk+1]T

k+1

∑

Nk+1 ) {(xi, ti)}

j)0

i)(

Nj

(13)

k

∑

∂yi G) ) ∂uj

˜ N

∑ l)1

∂yi ∂Hl ) ∂Hl ∂uj

˜ N

∑β

l,i k Hk(1

- Hk)Rj,l

k)1

(21)

Nj)+1

j)0

is received, where k g 0 and Nk+1 denotes the number of samples in (k + 1)th chunk, we have Kk+1 ) Kk + Hk+1THk+1 βk+1 ) βk + Kk+1-1Hk+1T(Tk+1 - Hk+1βk)

(14)

Kk+1-1 rather than Kk+1 is used to compute βk+1 from βk in (14). The update formula for Kk+1-1 is derived using the Woodbury formula Kk+1-1 )(Kk + Hk+1THk+1)-1 )Kk-1 - Kk-1Hk+1T(I + Hk+1Kk-1Hk+1T)-1Hk+1Kk-1 (15)

where G is the gradient of model output with respect to the input of the OS-ELM. Rj,k k is the input layer weight from the jth input to the lth hidden node, βk,i is the output layer weight from the lth hidden node to the ith output layer node, Hk is the output ˜ is the number of the hidden nodes. of the lth hidden node, and N The subscript k represents the kth batch. Let e and eˆ respectively express the actual error and the predicting error corresponding to the desired target. Thus, for the kth batch, the following can be obtained eˆk ) yd - yˆk ek ) yd - yk ) yd - yˆk - εk

(22)

The same as above, the errors for the (k + 1)th batch are given by

Let Pk+1 ) Kk+1-1, then the equation for updating βk+1 can be written as

eˆk+1 ) yd - yˆk+1 ek+1 ) yd - yk+1 ) yd - yˆk+1 - εk+1

Pk+1 ) Pk - PkHk+1T(I + Hk+1PkHk+1T)-1Hk+1Pk

assuming that the model prediction errors for the kth batch and the (k + 1)th batch are the same. Combining (22) and (23) gives

βk+1 ) βk + Pk+1Hk+1T(Tk+1 - Hk+1βk)

ek+1 ) ek - GT∆Uk+1

(16) Equation 16 gives the recursive formula for βk+1. 3. Outline of Batch-to-Batch Iterative Learning Control Based on OS-ELM Due to the model-plant mismatch and unknown disturbances from batch to batch, the final quality does not always meet the desired product quality in real industry. Batch-to-batch iterative learning control can be used to solve this problem by using the information of previous batch and currant batch to revise the next batch input trajectory. The following model is used to express the input-output relationship6 Y ) f(U)

(17)

where Y represents the product quality variables, U ) [u1, u2, ..., uN]T is a vector of input variables. The nonlinear function f( · ) is expressed by the online sequential ELM. From ref , the first-order Taylor series expansion of (17) can be given by ∂f ∂f ∂f yˆ(tf) ) f0 + ∆u1 + ∆u2 + · · · + ∆un ∂u1 ∂u2 ∂un (18)

(23)

(24)

The objective of the ILC is to control the input trajectory in order to make the final product quality achieved the desired target. By solving the following optimal quadratic objective function,5 one can get the revised input trajectory for the (k + 1)th batch min J ) ek+1

∆Uk+1

2

+ ∆Uk+1

| | | | Q

2 R

(25)

)ek+1TQek+1 + ∆Uk+1TR∆Uk+1 where Q and R are positive definitive weight matrices. By setting ∂J/∂∆Uk+1 ) 0, the partial derivative of the quadratic objective function with respect to the input change ∆Uk+1 can be obtained ∆Uk+1 ) (GQGT + R)-1GQek Uk+1 ) Uk + ∆Uk+1

(26)

It is to be noted that the change of control trajectory ∆Uk+1 is directly updated by the actual tracking error of the process. In many chemical batch processes, however, the actual final quality is impossible to obtain immediately. Thus, the model

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prediction value could be an alternative. Thus, the tracking performance will depend on the accuracy of the model. The convergence of the ILC can directly be derived from the convergence theorems in the literature.5 It is shown that ek will converge as k f ∞ if

|

I - GK < 1

|

(27)

where K ) (GQGT + R)-1GQ. The hidden layer output matrices for the kth batch and (k 1)th batch, Hk and Hk-1, can be obtained based on eqs 3 and 4 with the input trajectories Uk andUk-1. Let Pk-1 ) (Hk-1THk-1)-1, Pk is given by Pk ) Pk-1 - Pk-1HkT(I + HkPk-1HkT)-1HkPk-1

(28)

With Hk-1 and Yk-1 from the previous batch, βk-1 is obtained + βk-1 ) Hk-1 Yk-1

(29)

G)

x˙1 )f1

x˙3

The outline of the proposed strategy is summarized as follows: Step 1. The three-way training data collected from normal batches are batchwise unfolded into two-way matrix and normalized to be zero mean and unit variance. Then determine the number of hidden nodes and choose the activation function which can be any bounded nonconstant piecewise continuous functions for ELM. Step 2. Randomly generate hidden nodes parameters (such as input weights and hidden biases, etc), then analytically determine the output weights β based on the training data. Finally, calculate matrix G as follows

F

(31)

l)1

Step 3. Set k ) 0, let βk ) β and Hk ) H. Select the weight matrices Q and R for optimal control strategy as well as the desired product quality, and set the initial input Uk+1. Step 4. Apply the input Uk+1 to the batch process and obtain the output Yk+1 at the end of the batch. Then the input for next batch, Uk+2, is calculated in the light of ∆U ) (GQGT + R)-1GQek+1 Uk+2 ) Uk+1 + ∆U

(32)

Step 5. According to Uk+1, the output of the hidden nodes, Hk+1, can be obtained. With Hk, Hk+1, and βk, the OS-ELM is carried out from eq 33 to get βk+1 for next batch Pk

)(HkTHk)-1

Pk+1 )Pk - PkHk+1T(I + Hk+1PkHk+1T)-1Hk+1Pk βk+1 )βk + Pk+1Hk+1T(Yk+1 - Hk+1βk) (33) Then recalculate the matrix G based on eq 34 for the next batch.

F0 Tc

(

Em F02F (1 - x1)2 exp(2x1 + 2χx12)Am exp Mm x4Tref f1x2 1400x2 )f2 ) 11 + x1 Aw exp(B/uTref) f1 Aw exp(B/uTref) - x3 )f3 ) 1 + x1 1500 x1 1 - x1 + ) r1 + r2Tc r3 + r 4 Tc )r1 + r2Tc )uTref - 273.15 )

(30)

∑ βH(1 - H)R

- Hk+1)R

(34)

In this section, the proposed method is applied to the thermally initiated bulk polymerization of styrene in a batch reactor which shows the typical nonlinear characteristics and is compared with a control strategy based on the recursive partial least square (PLS)31 model. The details on the process description are explained well by Kwon and Evans,32 and Gattu and Zafiriou33 reported the parameter values of the first-principles model. The differential equations are shown as

βk ) βk-1 + PkHkT(Yk - Hkβk-1)

˜ N

k+1Hk+1(1

l)1

4. Simulation and Results

x˙2

∂yi ) ∂uj

∑β

Step 6. Set k ) k+1 and return to step 4. The batch-to-batch iterative learning control scheme of the online sequential ELM based method is illustrated in Scheme 1.

Where H+k-1 is the Moore-Penrose generalized inverse of Hk-1. From both (28) and (29), the equation for updating βk can be written as

G)

∂yi ) ∂uj

11111

˜ N

( (

)

)

)

(35) where x1 is the conversion, x2 and x3 are the dimensionless number-average and weight-average chain lengths (NACL and WACL), respectively; T is the reactor temperature; Tc is the temperature in degrees celsius; Aw and B are coefficients in the relation between the WACL and the temperature obtained from experiments; Am and Em are the frequency factor and activation energy, respectively, of the overall monomer reaction; the constants r1-r4 are density-temperature corrections; Mm and χ are the monomer molecular weight and polymer-monomer interaction parameter, respectively; and tf is the final time of the batch. The initial values of the states are x1(0) ) 0, x2(0) ) 1, and x3(0) ) 1. Table 1 gives several reference values used to obtain the dimensionless variables as well as the values of process parameters. The objective of the batch is to achieve a conversion of 0.8 (80%) with values of dimensionless NACL and WACL equal to 1.0 in the minimum amount of time. In this study, the final time, tf, is fixed to 400 min and the desired product final quality is yd ) [0.8, 1, 1], respectively. The reactor temperature, T, is the control variable constrained within the range 370 e T e 470, which is divided into N ) 20 equal stages as the input variable. The output variables are the three final product qualities known as conversion, NACL, and WACL. The above mechanistic model is assumed to be unavailable, and the ELM model is utilized to build the relationship between T and y ) [x1, x2, x3]. Perturbations uniformly distributed over [-5, 5] are performed on the reactor temperature. In total, 30 normal batch runs under various operation conditions by the design of experiment are generated from the mechanism model. These data (X(30 × 20), Y(30 × 3)) are unfolded and scaled to zero mean and unit variance and, then, used to train the model. The hidden node number is

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Scheme 1. Scheme of the Batch-to-Batch ILC Based on OS-ELM

Figure 1. Comparison of error convergence with different models in case 1.

Table 1. Parameter Values parameter

value

Mm Aw Am Em B r1 r2 r3 r4 χ tf xnf xwf

104 kg/kmol 0.033454 4.26 × 105 m3/(kmol s) 10103.5 K 4364 K 0.9328 × 103 kg/m3 -0.87902 kg/(m3 °C) 1.0902 × 103 kg/m3 -0.59 kg/(m3 °C) 0.33 400 min 700 1500

selected as 20 in order to avoid the nonsingular problem in OSELM algorithm, and the sigmoid function is chosen for the hidden node. The weight matrices are selected as Q ) I and R ) 0.2I for ensuring the convergence by experiments. To compare with the proposed method, the control strategy with recursive PLS (RPLS) model is also used for simulation. Nine latent variables are determined for the PLS model by leaveone-out cross-validation. The RPLS updating procedure is achieved by performing PLS regression on a new data block in which the new data is added and the old one is deleted. Four cases are tested to investigate the performance of the proposed method. Case 1: Normal Case. The normal case without any disturbance is first considered. The initial reactor temperature is set to 400 K. The initial modeling time using the ELM model is 0.0246 s, while 0.2175 and 3.4116 s are needed for PLS and back propagation (BP) neural network, respectively. It can be seen that ELM shows the advantage of extremely fast learning speed which is very suitable for online fast modeling, especially rapid batch processes. Figure 1 shows that the errors with different models, ek(tf) ) |yd - y(tf)|, converge from batch to

Figure 2. Tracking performance of final quality variables with OS-ELM model in case 1.

batch. The performance using OS-ELM under iterative control overcomes the control strategies based on PLS and RPLS models due to its capabilities of both capturing the nonlinear and updating the model. Figure 2 shows the final product quality variable variations with the OS-ELM model from batch to batch; and from the figure, it can be seen that the final quality converges to the desired target in the third batch. Case 2: Set Point Change. To test the performance of the proposed control strategy, the desired qualities are changed from [0.8, 1, 1] to [0.9, 0.9, 0.9] in normal operation. The results are shown in Figures 3 and 4. Both figures indicate that the proposed method can achieve the new target precisely after three batch runs by means of updating the nonlinear model. Meanwhile, the trajectories corresponding to the other two models also converge whereas more errors and slow convergence speed are shown due to the model-plant mismatch. Figure 5 presents the input variable trajectories in the first, third, and fifth batch runs with the proposed method. It also compares them with the trajectory (Td) collected from the batch in which the final qualities achieve the desired target. The results prove that the ideal trajectory can be obtained after three batch runs as the same results are shown in Figures 3 and 4. Case 3: Fixed Parameter Disturbance. Assume that the parameter Aw is changed to 1.2Aw at the beginning of the third batch run in this case. Figure 6 depicts the results of the tracking

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Figure 3. Comparison of error convergence with different models in case 2.

Figure 4. Tracking performance of final quality variables with the OSELM model in case 2.

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Figure 6. Tracking performance of final quality variables with the OSELM model in case 3.

Figure 7. Comparison of error convergence with different models in case 3.

NACL and WACL can be ensured to converge and the conversion has small deviation from the desired target. Figure 7 shows the comparison of the error variations using different models. It is demonstrated that the tracking errors converge more quickly by using the OS-ELM algorithm to update the nonlinear model.

Figure 5. Trajectories of the reactor temperature for case 2.

performance using the proposed method. It is worth noting that the conversion is not changed in the third batch run and cannot converge from the fourth batch run. This is because the parameter Aw only affects the values of NACL and WACL directly and both values have an effect on calculating the conversion as well. Therefore, according to the analysis of the mechanism model, it is illustrated that only the qualities of

Case 4: Different Sampling Intervals. To investigate the performance of the proposed strategy under different numbers of batch stages, a different sampling time of 40 min for each batch has been applied. Thus, the batch is divided into 10 stages (N ) 10) with equal length. Here, four latent variables are retained in the PLS model in terms of leave-one-out crossvalidation. Also 30 normal batch runs are generated from the mechanism model as the training data sets. Cases 1-3 are simulated based on the new data. Table 2 shows the results of the third, fifth, seventh, and ninth batch runs in the case when N ) 10 by using different models. In all three cases, the sum of errors calculated by control strategy with OS-ELM model reduces and converges much more quickly than RPLS modelbased method. This demonstrates that by utilizing the nonlinear updated model, more meaningful information can be obtained and the tracking strategy is improved even though the number of the sampling point within a batch is reduced from 20 to 10.

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Table 2. Sum of Errors of Final Quality Variables Using Different Models when N ) 10 in Cases 1-3 Sum of Error case 1

3rd 5th 7th 9th

case 2

case 3

RPLS

OS-ELM

RPLS

OS-ELM

RPLS

OS-ELM

0.0843 0.0639 0.0351 0.0294

0.0254 0.0238 0.0238 0.0238

0.0927 0.0554 0.0512 0.0492

0.0385 0.0369 0.0368 0.0368

0.1708 0.1081 0.0968 0.0790

0.1638 0.0618 0.0602 0.0602

5. Conclusions A novel batch-to-batch iterative learning control method on the basis of OS-ELM is proposed in this paper. The ELM model is first constructed, which can better capture the complex nonlinearity than linear models. Subsequently, an ILC strategy is employed to address the problems of model-plant mismatching as well as modify the input trajectory for the next batch run. To overcome the disadvantages of the fixed model such as unknown disturbance and batch-to-batch variation, a recursive algorithm based on ELM named OS-ELM is exploited by adjusting the output weights to update the old model, which not only can accommodate the process information but also give fast speed for online implementation. The successful simulation of the proposed method to a bulk polymerization of styrene batch process has demonstrated its feasibility and effectiveness and shown its superior performance. Acknowledgment The authors wish to thank Yuou Situ and Associate Professor Zhang Yingwei for indispensable help when revising the paper. We would also like to thank the anonymous reviewers for their comments and suggestions that helped us to improve the paper. This project was supported in part by the Natural Science Foundation of China (No.60674063) and Hi-Tech Research and Development Program of China (No.2007AA04Z194). Literature Cited (1) Chen, J. H.; Lin, K. C. Batch-to-batch iterative learning control and within-batch on-line control for end-point qualities using MPLS-based dEWMA. Chem. Eng. Sci. 2008, 63, 977–990. (2) Lee, K. S.; Chin, I. S.; Lee, H. J.; Lee, J. H. Model predictive control technique combined with iterative learning control for batch processes. AIChE J. 1999, 45, 2175–2187. (3) Lee, K. S.; Lee, J. H. Iterative learning control-based process control technique for integrated control of end product properties and transient profiles of process variables. J. Process Control 2003, 13, 607–621. (4) Lee, J. H.; Lee, K. S. Iterative learning control applied to batch processes: an overview. Control Eng. Pract. 2007, 15, 1306–1318. (5) Lee, J. H.; Lee, K. S.; Kim, W. C. Model-based iterative learning control with a quadratic criterion for time-varying linear systems. Automatica 2000, 36, 641–657. (6) Zhang, J. Batch-to-batch optimal control of a batch polymerization process based on stacked neural network models. Chem. Eng. Sci. 2008, 63, 1273–1281. (7) Flores-Cerrillo, J.; MacGregor, J. F. Iterative learning control for final batch product quality using partial least squares models. Ind. Eng. Chem. Res. 2005, 44, 9146–9155. (8) Xiong, Z. H.; Zhang, J. Product quality trajectory tracking in batch processes using iterative learning control based on time-varying perturbation models. Ind. Eng. Chem. Res. 2003, 42, 6802–6814. (9) Xiong, Z. H.; Zhang, J.; Wang, X.; Xu, Y. M. Tracking control for batch processes through integrating batch-to-batch iterative learning control and within-batch on-line control. Ind. Eng. Chem. Res. 2005, 44, 3983– 3992.

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ReceiVed for reView May 15, 2009 ReVised manuscript receiVed September 7, 2009 Accepted October 14, 2009 IE9007979