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Combining Kernel Partial Least-Squares Modeling and Iterative Learning Control for the Batch-to-Batch Optimization of Constrained Nonlinear Processes Yingwei Zhang,* Yunpeng Fan, and Pengchao Zhang Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern UniVersity, Shenyang, Liaoning 110004, P. R. China
A new approach to the optimal control with constraints is proposed to achieve a desired end product quality, and a modified kernel partial least-squares (KPLS) is used to build the combining model of nonlinear processes. The particle swarm optimization algorithm is used to solve the optimal problem. The contributions of the article are as follows: The modified KPLS is proposed for the optimal control purpose, and the optimal manipulated variables are computed for the next batch run based on modified KPLS. The proposed approach is applied to a bulk polymerization of styrene batch process and fused magnesium furnace. Simulation results show the proposed approach is effective for predicting the control profile of next batch run. 1. Introduction Batch processes are widely used because of their flexibility in handling many products.1,2 In batch-to-batch processes, the control problems are not easily solved because of the high dimensionalities of the industrial processes, the presence of changing conditions, and disturbances.3-5 The demand for effective end quality control and optimal operation in the industry has propelled research into iterative learning control methods over the past few decades. Hence, many approaches based on iterative learning control (ILC) have been proposed to solve the problems.6-10,24-27 These approaches exploit the repetitive nature of batch processes and use process knowledge obtained from previous batches to accommodate the next input strategy so that the output trajectory tracks asymptotically to the desired objective. Clarke-Pringle and MacGregor10 introduced the batch-to-batch adjustments to optimize the molecularweight distribution. Lee et al.11,12 proposed the ILC-based quadratic criterion approaches for quality control of batch processes. Doyle et al.13 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 end quality prediction immediately instead of performing an off-line analysis in the lab to determine the following batch control trajectory.14,15 The optimum steady-state operation of an industrial plant and online process steady-state optimization under uncertainty are studied.33,34 The terminal-cost optimization of a control of an affine nonlinear system is investigated in a piecewise manner, and the input spaces are separated into constraint-seeking and sensitivity-seeking directions.35 An iterative optimization strategy is proposed and applied to the set-point optimization of batch chromatography in the presence of a plant-model mismatch.36 The problem of minimizing the batch time of the copolymerization of acrylamide is considered.37 Flores-Cerillo and MacGregor16,17 utilized the PLS model to reject persistent process disturbances and achieve new end-product quality targets. The above-mentioned PLS methods assume that the deviations from the mean trajectories at each one of the time instances along the batch are linear. Since in PLS the mean trajectories are subtracted and then the PLS gives a different weight to each variable at every different time interval, then PLS can provides a time varying nonlinear model (or more * To whom correspondence should be addressed. E-mail:
[email protected];
[email protected].
correctly a locally linearized model at each time point). Linear approaches are not viable for describing nonlinear processes. There are some approaches to solve the nonlinear issue. The five layer neural network method was used to solve the nonlinear issue by Krammer.18 The principle component curve-neural network algorithm was proposed by Dong et al.19 The basic idea of the kernel method is to first map the input space into a feature space via a nonlinear map and then extract the dominant components in the feature space.20-23,40 Sequential quadratic programming coupling with multiway kernel partial leastsquares (MKPLS) model is used to solve the optimization problem without constraints.38 The kernel partial least-squares regression issue is investigated in reproducing the kernel Hilbert space.39 With the simple computing, the kernel methods are available for the optimal problems. Combining of a nonlinear model with iterative learning control for batch process optimization with constraints has not been previously proposed. In this article, the training data from several normal batches are used to build the model base on the modified KPLS. The PSO is used to solve the optimal problems. In this article the operating variables space is divided into three parts: output variables, manipulated variables, and state variables subspaces and the process constraints are considered. The contributions of the article are as follows: The modified KPLS is proposed for the optimal control purpose, and the optimal manipulated variables are computed for next batch run based on modified KPLS. The remainder of this article is organized as follows. In section 2, the details of KPLS are described and the batch-tobatch iterative learning control approach is proposed based on the KPLS model. The PSO algorithm is introduced in section 3. The proposed monitoring approach is applied to a bulk polymerization of styrene batch process and fused magnesium furnace in section 4. Conclusions are summarized in section 5. 2. Iterative Learning Control Using Modified KPLS Model 2.1. Modified KPLS Model. KPLS is an extension of PLS.31 We assume a data point xi ∈ Rd. The data point is mapped into a higher-dimensional feature space F. xi f Φ(xi)
(1)
In this space, standard PLS is performed. The trick herein is that the PLS can be computed such that the vectors appear
10.1021/ie1004702 2010 American Chemical Society Published on Web 07/16/2010
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Table 1. NIPALS for KPLS for comprehension
for computation
Initialize ωstate,i
Initialize ωstate,i
1 wstate,i )
2
T Kiωstate,i tstate,i ) Kiωstate,i / √ωstate,i
ΦT(xstate,i)ωstate,i /|ΦT(xstate,i)ωstate,i | 3
tstate,i ) ΦT(xstate,i)wstate,i
4
T qstate,i ) Yitstate,i /|tstate,i tstate,i |
ωstate,i )
T qstate,i ) Yitstate,i /|tstate,i tstate,i |
Yiqstate,i
ωstate,i )
T qstate,i qstate,i
loop until ωstate,i converges
5
loop until ωstate,i converges
Φstate,i+1)
6
Yiqstate,i T qstate,i qstate,i
T T T T Ki+1 ) (I - tstate,itstate,i /tstate,i tstate,i)Ki(I - tstate,itstate,i /tstate,i tstate,i)
T T (I - tstate,itstate,i /tstate,i tstate,i)Φstate,i
T T /tstate,i tstate,i)Yi Yi+1 ) (I - tstate,itstate,i
T T Yi+1 ) (I - tstate,itstate,i /tstate,i tstate,i)Yi
go to step 2
go to step 2
only within scalar products. Thus, mapping eq 1 can be omitted. Instead, we only work with a kernel function k(x,y), which replaces the scalar products (Φ(x),Φ(y)). Nonlinear iterative partial least squares (NIPALS) algorithm is used for the computation of KPLS. This algorithm is listed in Table 1. The training data consist of the normal operating state data and manipulated input data, which are used to build the model. Each batch consists of a response matrix Y and a regression X. Each row vector of X, denoted as xT, is T composed of the operating state information xstate and manipulated input information uT. T xT ) (xstate , u T)
and 1N )
[ ]
)
tTu,kPTu,k
+
fTk
T T T yTk ) tstate,k Qstate,k + hstate,k + tTu,kQTu,k + hTu,k
2
T T wu,i ) uu,i ωu,i /|uu,i ωu,i |
3
T tu,i ) uu,i wu,i
4
T qu,i ) Yitu,i /|tu,i tu,i |
ωu,i )
Yiqu,i T qu,i qu,i
5
loop until ωu,i converges
6
T T /tu,itu,i)ui ui+1 ) (I - tu,itu,i
T T Yi+1 ) (I - tu,itu,i /tu,itu,i)Yi
go to step 2
vectors. tTu,k is the score vector, which can be obtained by implementing PLS in Table 2.
2.2. ILC in the Score Space. The modified KPLS model is built as follows for a batch k:
uTk
Initialize ωu,i
(3)
1 ··· 1 1 l ··· l N 1 ··· 1
T T T Φ(xstate,k) ) tstate,k Pstate,k + fstate,k
for comprehension 1
(2)
Before applying KPLS, mean centering in the high-dimensional space should be performed. This can be done by j , where substituting the kernel matrix K with K ¯ ) K - 1NK - K1N + 1NK1N K
Table 2. NIPALS for PLS
For batch k, the following can be obtained by using the ILC algorithm:7
(4) eTk ) yTd - yTk
(7)
T T ek-1 ) yTd - yk-1
(8)
(5) (6)
T T T For Φ(xstate,k ) and yTk , Pstate,k and Qstate,k are the loading vectors. T tstate,k is the score vector, which can be obtained by implementing KPLS in Table 1. For uTk and yTk , PTu,k and QTu,k are the loading
T where ek-1 is the error vector obtained from batch k - 1, yTd is the target product grade,
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T T ) ekT - ek-1 ) yTd - ykT - (yTd - yk-1 T ) ) -(ykT - yk-1 T T T T T T + )Qstate,k + [-(tu,k - tu,k-1 )]Qu,k - tstate,k-1 ) -(tstate,k T T T T ) - hstate,k-1 ) + (hu,k - hu,k-1 (hstate,k
(9)
For the case of no errors: T T T eTk ) ek-1 - ∆tstate,k Qstate,k - ∆tTu,kQTu,k
(10)
T T T T where ∆tstate,k ) tstate,k - tstate,k-1 and ∆tTu,k ) tTu,k - tu,k-1 . T T To obtain ∆tstate,k and ∆tu,k, the following quadratic optimization is proposed:
T min J ) eTk Wek + ∆tstate,k Rstate∆tstate,k + ∆tTu,kRu∆tu,k
∆tstate,k,∆tu,k
(11) subject to u_ e |uTk | e uj
(12)
_y e |yTk | e jy
(13)
In this article, PSO is used to solve the optimization problem with constraints, which will be introduced in the next section. 3. Particle Swarm Optimization (PSO) Algorithm The particle swarm optimization (PSO) was originally designed by Kennedy and Eberhart for efficiently finding optimal or near-optimal solutions in large search spaces.28 Each particle is defined as a potential solution to a problem in D-dimensional space. Regarding the minimum problem, suppose f(X) is the objection function, Xi ) (Xi1, Xi2, ..., XiD) is the current position of particle, Vi ) (Vi1,Vi2, ..., ViD) is the current speed of particle, Pi ) (Pi1, Pi2, ..., PiD) is the best position, then the best position of particle i can be computed according to the following formulation: Pi(t + 1) )
{
Pi(t) if f(Xi(t + 1) g f(Pi(t))) (14) Xi(t + 1) if f(Xi(t + 1) < f(Pi(t)))
Table 3. Parameter Values
If the population is s, and Pg(t) is the global best position, then Pg(t) ) {P0(t), P1(t), ..., Ps(t)|f(Pg(t)) ) min{f(P0(t)), ..., f(Ps(t))}
(15)
According to the theory of particle swarm optimization, the following equation presents the process of evolution: Vi(t + 1) ) WVi(t) + c1r1(t)(Pij(t) - Xi(t)) + c2r2(t)(Pg(t) - Xi(t)) (16) Xij(t + 1) ) Xij(t) + Vij(t + 1)
Figure 1. Steps of the particle swarm optimization algorithm.
(17)
where r1 and r2 are two independently uniformly distributed random variables with range [0, 1]; c1 and c2 are positive constant parameters called acceleration coefficients which control the maximum step size; W is the inertia weight that controls the impact of previous velocity of particle on its current one. Generally, the value of each component in Vi can be clamped to the range [-Vmax, Vmax] to control excessive roaming of particles outside the search space. Then the particle flies toward a new position according to eqs 16 and 17. This process is repeated until a user-defined stopping criterion is reached.
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
The flowchart given in Figure 1 shows the steps of the particle swarm optimization algorithm. 4. Case Studies 4.1. Bulk Polymerization of Styrene. In this section, the proposed optimal method is applied to the bulk polymerization of styrene in a reactor29,30 and is compared with PLS plus ILC optimizing results. The state equation is shown as follows.
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( )
F0 F Em (1 - x1)2 exp(2x1 + 2Mnx12)Am exp Mm T 2
x˙1 )
(
x˙1x2 1400x2 x˙2 ) 11 + x1 Aw exp(B/T)
(
)
)
x˙1 Aw exp(B/T) - x3 x˙3 ) 1 + x1 1500 F)
1 - x1 x1 + r1 + r2Tc r3 + r4Tc F0 ) r1 + r2Tc Tc ) T - 273.15
where x1 is the conversion, x2 and x3 are the dimensionless number-average and weight-average chain lengths, respectively. u ) T/Tref is the dimensionless reactor temperature as control variable; Tc is the temperature in degrees Celsius; Aw and B are coefficients in the relation between the weight-average chain lengths and the temperature obtained from experiments; Am and Em are the frequency factor and energy, respectively, of the overall monomer reaction; the constants r1-r4 are density-temperature corrections; Mm and Mn 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. The process parameter values are given in Table 3. The number of latent variables is 12 for both the PLS and KPLS. The objective of the batch is to archive a conversion of 80% with values of dimensionless number-average and weight-average chain lengths equal to 1.0. The objective is to derive an optimal temperature profile so that the conversion is maximized and values of dimensionless NACL and WCAL are close to 1.0 at the end of a batch; that is,
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Table 4. Summary of Modeling and Test Results batch
RMSE training
RMSE test
2 3 5
0.1359 0.1184 0.1258
0.3698 0.2689 0.3589
ˆ i is the predicted value, and where Yi is the reference value, Y n is the total number of samples in a batch. It can be noted that the model is accurate but the model errors still exist. To compare with the proposed method, the control strategy with the PLS model is also used for simulation. Two cases are tested to investigate the performance of the proposed method. Case 1. The normal case without any disturbance is first considered. The initial reactor temperature is set to 400 K. The PLS model with 12 latent variables explains 76.3% of the X block and 82.4% of the Y block. The modified KPLS model with 12 latent variables explains 81.6% of the X block and 94.1% of the Y block. The number of latent variables needed to build the model was determined by the cross validation procedure. Figures 2-4 show that the errors in different models, ekT(tf) ) |ydT - yT(tf)|, converge from batch to batch. The performance of using KPLS under iterative control overcomes the control strategies based on PLS models due to its capabilities of capturing the nonlinearities. Figures 2-4 show the end product quality variable variations with KPLS model from batch
T min J ) eTk Wek + ∆tstate,k Rstate∆tstate,k + ∆tTu,kRu∆tu,k
∆tstate,k,∆tu,k
subject to 0.85 e |uTk | e 1.15 0.6 e |yT1,k | e 0.8 0.8 e
|yT2,k |
Figure 2. Convergence performance of conversion.
e 1.2
0.8 e |yT3,k | e 1.2 In this section, the final time tf is fixed to 400 min and the desired end quality is yTd ) [0.8 1 1]. The control variable is divided into 20 equal intervals and is constrained within the range 0.85 e |uTk | e 1.15. The output variable is yT ) [x1 x2 x3]T. During each interval, Xstate is the end dimensionless reactor temperature as the measurement variable. These data (Xstate(20 × 20), Y(20 × 3)) are unfolded and scaled to zero mean and unit variance, and then used to train the model. The weight matrices are selected as Q ) 0.8I and R ) I to ensure the convergence by trial and error. The prediction results using KPLS are shown in Table 4. The performance index was defined as n
RMSE ) (
∑ (Yˆ i)1
i
- Yi)2 /n)1/2 Figure 3. Convergence performance of number-average chain lengths.
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Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010 Table 5. Sum of Errors of Final Quality Variables Using Different Models in Case 1 and Case 2 in Polymerization Case 1
Case 2
sum of errors
Figure 4. Convergence performance of weight-average chain lengths.
sum of errors
batch
PLS
KPLS
PLS
KPLS
5th 10th 15th 18th
0.0773 0.0458 0.0335 0.0227
0.0189 0.0156 0.0154 0.0098
0.0846 0.0472 0.0312 0.0297
0.0485 0.0296 0.0251 0.0143
73.7% of the X block and 81.6% of the Y block. The modified KPLS model with 12 latent variables explains 82.8% of the X block and 93.7% of the Y block. The number of latent variables needed to build the model was determined by the cross validation procedure. A batch is divided into 20 stages with equal length. Also 20 normal batch runs are generated from the mechanism model as the training data sets. Table 5 shows the results of the 5th, 10th, 15th, and 18th batch runs by using different models. In all cases, the sum of errors calculated by the control method with the modified KPLS model reduces and converges much more quickly than the PLS model-based method. This demonstrates that by modifying the input trajectory for the next batch run, the tracking strategy is improved. Figure 6 shows optimal control trajectories of first and fourth batches. These control trajectories show the batch-to-batch correction. 4.2. Fused Magnesium Furnace. The fused magnesium furnace (FMF) used to produce fused magnesia belongs to a kind of mine hot electric arc furnace. With the development of technology of melting, FMF has already gotten extensive application in the industry.32 FMF refining technology can
Figure 5. Optimal control trajectories in Case 1 (solid, first batch; solid, third batch).
to batch, and from the figure, it can be seen that the final quality converges to the desired target in the third batch. Figure 5 shows optimal control trajectories of the first batch and the third batch. These control trajectories show the manipulated variable movements, which need to be performed. Case 2. To generate the training data sets for building the modified KPLS model, the random changes with uniform distribution and magnitude of (10% are added to the nominal trajectory. The PLS model with 12 latent variables explains
Figure 6. Optimal control trajectories in Case 2 (solid, first batch; solid, fourth batch).
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Figure 7. Diagram of electrical smelting magnesium furnace: (1) transformer, (2) short circuit network, (3) electrode holder, (4) electrode, (5) furnace shell, (6) trolley, (7) electric arc, (8) burden controller.
Figure 9. Optimal control trajectories in FMF (solid, first batch; solid, third batch).
Figure 8. Tracking performance of end quality variables. Table 6. Sum of Errors of Final Quality Variables Using Different Models in FMF Case 1
Case 2
sum of errors
sum of errors
batch
PLS
KPLS
PLS
KPLS
5th 7th 9th 10th
0.0996 0.0846 0.0810 0.0734
0.0687 0.0623 0.0576 0.0483
0.1370 0.0951 0.0924 0.0869
0.0874 0.0735 0.0721 0.0539
enhance the quality and increase the production variety. The FMF smelting process is shown in Figure 7. Voltage flicker is the types of power quality problems that is introduced to the power system as a result of the furnace’s very large power input ratings. The FMF in this article takes the light-burned magnesia as the raw material. It makes use of the heat generated by both the burden resistance when the current flows through the burden and the arc between the electrodes and the burden that melts the burden and then obtains the fused magnesia crystals with higher purity. A characteristic of the fused magnesium furnace is that it requires a large electrical power. The capacity of a three-phase furnace transformer is up to several thousand or tens of thousands volt-amperes. The required values of the
power fluctuate drastically and dramatically when the furnace is working, and the effect of the electrode regulator just adjusts the power though adjusting the location of the electrodes. In this example, the reactor temperature T is the input variable, divided into 20 equal stages and is constrained within the range 2000 e T e 2800. Xstate contains A, B, and C phase currents, which varies from 900 to 1200. The objective is to derive an optimal temperature profile so that the A, B, and C phase currents are close to 1000 at the end of a batch; that is, T min J ) eTk Wek + ∆tstate,k Rstate∆tstate,k + ∆tTu,kRu∆tu,k
∆tstate,k,∆tu,k
subject to 2000 e |uTk | e 2800 900 e |yTl,k | e 1200,
l ) 1, 2, 3
In total, 20 normal batch runs under various operation conditions are generated. These data (Xstate(20 × 20), Y(20 × 3)) are unfolded and scaled to zero mean and unit variance and then used to train the model. The weight matrices are selected as Q ) 0.5I and R1 ) I for ensuring the convergence. The number of latent variables is 12 for both the PLS and KPLS. The PLS model with 12 latent variables explains 71.5% of the X block and 81.2% of the Y block. The modified KPLS model with 12 latent variables explains 82.3% of the X block and 91.5% of the Y block. The number of latent variables needed to build the model was determined by the cross validation procedure. To compare with the proposed method, the control
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strategy with PLS model is also used for simulation. The initial reactor temperature is set to 2500. Figure 8 shows that actual controlled variables approach to the target. From Table 6, the performance using modified KPLS under iterative control overcomes that using PLS due to its capabilities of capturing the nonlinearities. Table 6 also shows that by modifying the input trajectory for the next batch run, the tracking strategy is improved, and in the figure the end quality approach to the target in the sixth batch can be seen. Figure 9 shows optimal control trajectories of first batch and third batch. These control trajectories show the manipulated variable movements, which need to be performed. 5. Conclusions A new batch-to-batch iterative learning control approach based on modified KPLS is proposed in this article. The ILC is developed to solve the problems of modifying the input trajectory for the next batch run, which can accommodate the next input trajectory for online implementation. The particle swarm optimization algorithm is used to solve the optimal problem. The proposed approach is applied to a bulk polymerization of a styrene batch process and fused magnesium furnace to demonstrate its effectiveness. Acknowledgment The authors appreciate the valuable comments and suggestions of the anonymous reviewers. The work is supported by China’s National 973 program (2009CB320600) and NSF in China (Grant 60974057). Nomenclature F ) feature space xi ) data point xi ∈ Rd X ) input data matrix X ∈ Rm×d Yi ) output data matrix after i - 1 deflation Φ(X) ) N s-dimension data in feature space Φi ) data after i - 1 deflation K ) kernel matrix K ) Φ(X)Φ(X)T Ki ) kernel matrix after i - 1 deflation corresponding to Φi T xstate ) the state data uT ) the input data ωi ) the scores of output qi ) the loadings of output 1N ) N × N matrix with all elements equal to 1/N pi ) the ith loadings vector wi ) the weight matrix ti ) the projection of Φ on pi I ) identity matrix yTd ) the desired output T Pstate ) the loadings of xstate T Pu ) the loadings of u T Qstate ) the loadings of output according to xstate T Qu ) the loadings of output according to u T tstate,k ) eigenvectors of the dominant eigenvalue of KYkYkT J ) the objective function u_ ) the constraint of input, u_ e |ukT| e uj uj ) the constraint of input, u_ e |ukT| e uj y ) the constraint of output, y e |ykT| e jy jy ) the constraint of output, _y e |ykT| e jy c1 ) acceleration coefficients in the PSO c2 ) acceleration coefficients in the PSO W ) inertia weight in the PSO Xi ) the current position of particle in the PSO
Vi ) the current speed of particle in the PSO Pi ) the best position of particle in the PSO T ) the reactor temperature Tc ) the temperature in degrees Celsius Tref ) the dimensionless reactor temperature perameter Aw, B ) coefficients in the relation between the weight-average chain lengths and the temperature obtained from experiments Am ) the frequency factor Em ) energy Mm ) the monomer molecular weight Mn ) the polymer-monomer interaction parameter tf ) the final time of the batch
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ReceiVed for reView December 3, 2009 ReVised manuscript receiVed June 24, 2010 Accepted July 4, 2010 IE1004702
