Batch-to-batch control of particle size distribution in cobalt oxalate synthesis process based on hybrid model

Powder Technology 224 (2012) 253–259

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Batch-to-batch control of particle size distribution in cobalt oxalate synthesis process based on hybrid model Shuning Zhang a,⁎, Fuli Wang a, b, Dakuo He a, b, Runda Jia a a b

College of Information Science and Engineering, Northeastern University, Shenyang 110004, Liaoning Province, PR China State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, PR China

a r t i c l e

i n f o

Article history: Received 14 December 2011 Received in revised form 8 February 2012 Accepted 2 March 2012 Available online 9 March 2012 Keywords: Cobalt oxalate synthesis process Crystallization Population balance Particle size distribution Batch-to-batch control Hybrid model

a b s t r a c t A hybrid model based batch-to-batch control strategy is proposed for control of particle size distribution (PSD) in cobalt oxalate synthesis process. In order to enhance the model prediction accuracy and generalization capability, a hybrid modeling approach for cobalt oxalate synthesis process in cobalt hydrometallurgy is developed by combining simplified first principle model with PLS model. The simplified first principle model that captures the dominant characteristics of the synthesis process is built to describe PSD evolution. The PLS model is utilized to compensate the unmodeled characteristic of the simplified first principle model and to enhance model generalization capability. Due to the repetitive nature of the process, a batch-to-batch control strategy based on hybrid model is then presented to design the operating policy that drives the process to a target PSD. Applications to a simulated cobalt oxalate synthesis process demonstrate that the proposed approach can improve process performance from batch to batch in the presence of unknown disturbances. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Cobalt oxalate synthesis process which is an important composition unit of cobalt hydrometallurgy is a batch crystallization process in fact. The particle size distribution (PSD) of cobalt oxalate is an important factor in the production of a high quality production and determines the efficiency of downstream operations (e.g. filtration, drying). Therefore many problems in downstream processes can be attributed to poor particle characteristics established in crystallization step [1,2]. In this light, high-performance control on cobalt oxalate synthesis process to achieve PSD with desired characteristics is of great importance. Due to the repetitive nature of batch processes, it is possible to determine the optimal operating policy of the next batch using results from previous batches. Various batch-to-batch control strategies for the final product quality have been proposed in the literatures. Clarke-Pringle and MacGregor [3] used batch-to-batch manipulated variable trajectories adjustments to control MW distribution in linear polymers. They exploit fundamental knowledge by controlling two key process parameters that determine the overall distribution shape. However, the approach may not always produce on-spec products at the end of the batch, even with perfect tracking control of the process variables. This is because the regulatory model is hard to capture disturbances' subtle effects on final product quality sufficiently.

⁎ Corresponding author. Tel.: + 86 24 23911005; fax: + 86 24 23890912. E-mail address: [email protected] (S. Zhang). 0032-5910/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2012.03.001

Lee and co-workers [4] propose the quadratic criterion based ILC approach for tracking control for temperature of batch processes based on a linear time-varying error transition model. Park et al. [5] used a PLS model-based predictive controller to control the PSD at the final time in an emulsion copolymerization reactor. Flores-Cerrillo and MacGregor [6] proposed an iterative learning control strategy based on empirical PLS models to control the full PSD in a simulated condensation polymerization process. Zhang [7] presented a neural network based batch-to-batch optimal control strategy for a batch polymerization process. In these aforementioned approaches, datadriven models have been suggested as an alternative to overcome the computational limitations of mechanistic model. Though modelbased empirical control has the advantage of ease in model building, control actions and performance predictions calculated from such a model may not be reliable. Since an empirical model is valid only in the region of the model where identification data were available, one must exercise a caution in using it for model-based control. Therefore, Doyle and co-workers [8,9] controlled the endpoint PSD of an emulsion polymerization system with batch-to-batch control strategy. A hybrid model, consisting of a mechanistic emulsion polymerization model and a MPLS correlation, is used as predictor in the method. However, a comparison of a data-driven model of emulsion polymerization with the hybrid model is not studied in the paper. In summary, essential difference of the aforementioned batch-tobatch control strategy is model structure employed. Therefore the core of PSD control is the development of a model that can predict the evolution of PSD accurately. The most popular modeling methods are first principle model [10,11], data-driven model [5] and hybrid

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model [8,12]. First principle model using a population balance modeling framework can reflect the process characteristic and is easy to be explained, but it requires significant amount of time and effort, and may not be suitable for agile responsive manufacturing. Data-driven model can be a very useful alternative in this case. However, the data-driven model cannot be explained and is easily prone to over fitting. By combining the simplified model and data-driven model, the hybrid model can complement both methods to obtain good performance because the simplified first principle model can improve the extrapolation capability and the data-driven model can increase the prediction accuracy. Therefore the purpose of this paper is to present a batch-to-batch control strategy for control of particle size distribution in cobalt oxalate synthesis process based on hybrid model. The hybrid modeling approach for cobalt oxalate synthesis process combines simplified first principle model with PLS model. A major benefit of the hybrid model is the ability to exploit the improved extrapolative capability of the first principle model while the PLS model is developed to compensate the unmodeled characteristic and to enhance model generalization capability. The hybrid model is then used within a nonlinear optimization program to calculate manipulated variable trajectories (MVT) that drive the PSD to the target. The proposed batch-to-batch control strategy is effective in updating MVT to regulate the PSD of cobalt oxalate to a desired target in the presence of strong disturbances. The rest of this paper is organized as follows. A hybrid model for cobalt oxalate synthesis process is presented in Section 2. This is followed by the development of the batch-to-batch control strategy based on the aforementioned hybrid model in Section 3. Then simulation results are presented for the implementation of the control strategy to cobalt oxalate process in cobalt hydrometallurgy in Section 4. Finally, Section 5 draws some concluding remarks of this paper. 2. Hybrid model This section illustrates the development of a hybrid model for cobalt oxalate synthesis process. First, the cobalt oxalate synthesis process, its simplified first principle model, and the PLS model are explained. Subsequently, the hybrid model is developed.

with continuously stirring. In order to keep the constant temperature of reaction, a heating jacket is mounted in the crystallizer. Due to the complexity of the reactions and the evolution of cobalt oxalate crystal, a pure batch operation cannot be used as it leads to a risk of reaction runaway. Therefore, a semi-continuous fed-batch mode is used. A fixed volume of cobalt chloride is first charged to the crystallizer, after which, ammonium oxalate is fed. During the operation, the setpoints for temperature and agitation speed are constant while the setpoint for feed flow-rate of ammonium oxalate is changed. The final batch time is fixed. The process is very complex which includes a large number of interacting operating variables. The evolution of crystal size is dependent upon the reactor temperature, feed flowrate of ammonium oxalate, concentrations of cobalt chloride and ammonium oxalate, and agitation speed. The objective for each batch is to make cobalt oxalate crystal with a specified particle size distribution. Although measurement of PSD is available only at the end of the batch, other online measurements are available for monitoring and control purpose. These online measurements including reactor temperature (Tr), flow-rate of ammonium oxalate (Fao), and agitation speed (Na) are collected every 15 s. The initial concentration of cobalt chloride and the concentration of ammonium oxalate can be measured before the start of a batch. The manipulated variable is selected as the trajectory of the ammonium oxalate flow-rate (Fao). 2.2. Simplified first principle model for synthesis process Cobalt oxalate synthesis process is a batch crystallization process which is an ancient unit operation and is widely used in industries. The dynamic behavior of a crystallization process can be captured by a population balance equation (PBE), along with conservation equations and kinetic relations. According to Puel et al. [13], the PBE is firmly established as a basic theoretical framework for all crystallization processes. Furthermore, nucleation and crystal growth dominate the crystallization kinetics. Hence, based on the following basic assumptions [14]: the suspension is perfectly mixed; crystal agglomeration or breakage phenomena are neglected; and growth of crystals is size-independent, the PBE for cobalt oxalate synthesis process is described as:

2.1. Description of cobalt oxalate synthesis process A cobalt oxalate synthesis process in cobalt hydrometallurgy is a liquid phase reaction of cobalt chloride and ammonium oxalate, leading to the desired cobalt oxalate crystals. Cobalt oxalate and ammonia chloride are produced: CoCl2 þ ðNH4 Þ2 C2 O4 → CoC2 O4 ↓ þ 2NH4 Cl : A

B

P

D

The process flowsheet is shown in Fig. 1. The process consists of one ammonium oxalate dissolver and one crystallizer. The evolution of cobalt oxalate crystal is carried out in the crystallizer operated

1 ∂ðNðL; t ÞV s ðt ÞÞ ∂NðL; t Þ þG ¼ Rn V s ðt Þ ∂t ∂L

ð1Þ

where Vs represents the suspension volume, N(L, t) represents the number density at a characteristic length, L and time, t, G represents the crystal growth rate, and Rn represents the crystal birth rate. For the sake of simplicity, the expressions for the growth rate and crystal birth rate can be written respectively as:
 K γ α Rn ¼ K n exp − a Na ðΔC Þ Tr

ð2Þ


 K β G ¼ K g exp − b ðΔC Þ Tr

ð3Þ

where Kn is birth rate coefficient, Kg is growth rate coefficient, Ka and Kb are constants, Tr is the reactor temperature, Na is agitation speed, α and γ are birth rate exponent, β is growth rate exponent, and ΔC ¼ C p −C  is the supersaturation. The solubility (C∗) is a function of temperature (Tr). For cobalt oxalate synthesis process, the experimentally determined solubility curve (with Tr in K) is

Fig. 1. Schematic diagram of the cobalt oxalate synthesis process.



2

C ¼ 0:0001ðT r −273:15Þ þ 0:001ðT r −273:15Þ þ 0:1:

ð4Þ

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255

The solute concentration (Cp) which can be calculated from the material balance is given by dC P F C V ¼ ao B A −3ρp K v Gμ 2 dt ðV A þ F ao t Þ

ð5Þ

where VA is the volume of cobalt chloride, CB is the concentration of ammonium oxalate, ρP is the density of crystal, Kv is the shape factor, and μ2 is the second moment of the PSD. As is evident from Eq. (1), the population balance equation is a hyperbolic partial differential equation which does not possess an analytical solution in most cases. Therefore, numerical methods such like method of moments, discretization techniques, and finite elements method [13] have been established for PBE. Among these techniques, the discretization technique, where the partial differential equations are sectioned along the size domains into finite classes, appears to be robust. Marchal et al. [15] developed the method of classes, a method that recasts the PBE into a set of computationally affordable reduced-order ordinary differential equations (ODEs). Hence, under the assumption of constant number density function at each granulometric class, the PBE can be transformed into a set of ODEs using the method of classes, as represented in Eq. (6). 8 dN1 1 dV s G > > ¼ Rn − N − N > > V s dt 1 2ΔL1 1 dt > > > > ⋮ > < dNi 1 dV s G G ¼− Ni − Ni þ N i−1 > V 2ΔL 2ΔL dt dt > s i i > > > ⋮ > > > dN 1 dV s G G > : M ¼− N − N þ N V s dt M 2ΔLM M 2ΔLM−1 M−1 dt

Fig. 2. Architecture of hybrid model of cobalt oxalate synthesis process.

where B are the regression weight matrix and G are the residual matrix. The NIPALS algorithm are used to calculate P and Q in such a manner that the inner relationship between their scores is linear [17]. 2.4. The architecture of hybrid model The hybrid model for process exploits the improved extrapolative ability of first principle model while the inevitable modeling error is corrected through PLS model. Therefore the PSD prediction for the kth batch can be decomposed into two factors: k k k y^ hyb ¼ yfpm þ yPLS :

ð6Þ

ð10Þ

The architecture of hybrid model is shown in Fig. 2. Where e is output of the PLS model to compensate the difference between the measurement and the output of the simplified first principle model. In this paper, the inputs to train PLS is x ¼ ½C A ; C B ; T r ; F ao ; Na . The main steps of hybrid model calibration are summarized as follows:

where Ni is the number of crystals per unit volume in the ith class (Li) at time t, Li is the width of the ith class, and M is the number of granulometric classes. Until recent now, the particle technology, especially the particle synthesis from a solution, has not been well developed. Some semiexperiential models are difficult to establish in practical situation, and also make the detailed first principle model difficult to be solved online. Therefore, the simplified first principal model is developed to describe the process in general, then the PLS model is used to account for the phenomena such as particle aggregation which are not included in the first principle model and the noise that may occur during PSD measurement.

1. Determine the parameters, Kn and Kg for the simplified first principle model. The parameters needed to perform the calculation of nucleation rate and crystal growth rate are obtained from the available experimental data. 2. Predict the PSD of cobalt oxalate crystals yfpm using simplified first principle model, and save the prediction results and correspondent inputs x in database. 3. Calculate the prediction error e0 ¼ ylab −yfpm , and use the inputs x building the PLS model. 4. Calculate the PLS model output e, and predict the PSD of cobalt oxalate crystals yhyb .

2.3. PLS model

3. Batch-to-batch control strategy

In this study, PLS [16] is used to correct PSD residuals, the difference between the PSD calculated by the simplified first principle model and the PSD obtained from laboratory. Since the modeled PSD is discretized into a number of granulometric classes (the number of granulometric classes is 20 in this study), residuals are computed at the characteristic size of each class. Generally, the PLS method reduces the dimension of the predictor variables X and response variables mathbfY by projection to directions that maximize the covariance between input and output variables. The decompositions of X and Y into their score and loading matrices can be described as:

The PSD control problem is complicated by several disturbances including variations in initial raw material concentrations, human error in setpoint of ammonium oxalate flow-rate, as well as other conditions change during operation. Therefore, in order to limit the deterioration of control performance due to model plant mismatches

T

X ¼ TP þ E T

Y ¼ UQ þ F

ð7Þ ð8Þ

where T and U are the score matrices, P and Q are the loading matrices, and E and F are the residual matrices of X and Y, respectively. The score matrices T and U are linearly related as: U ¼ TB þ G

ð9Þ

Fig. 3. Batch-to-batch control based on model prediction modification.

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However, hybrid model only approximate the batch process because of the lack of data for building PLS model, prediction offsets can occur due to model plant mismatches. After the completion of the kth batch, the model prediction errors ek can be calculated as: k

k

k

e ¼ y −y^ hyb

ð11Þ

where yk and y^ khyb are the measured and hybrid model prediction of end-point quality of the kth batch. As batch processes are intended to be run repeatedly, it is reasonable to correct the model prediction y^ kþ1 of the (k +1)th batch by adding the hyb prediction errors ek of the kth batch [18]. However, if significant noise is contained in the product quality measurement and/or significant disturbance exist and last only one batch, an appropriate filter should be applied to the measurements to extract only the long term trend of the model prediction error [19]. Hence the average model errors [19] of all previous batches are used to correct the model prediction for the next batch.  k , of all previous batches can be calcuThe average model error, e lated as: Fig. 4. PSD measurements created by simulating virtual process with different input trajectories from the training set.

and unknown disturbance, a batch-to-batch control strategy is used at the end of the current batch. It utilizes the information of the current and previous batch run to enhance the operation of the next batch. In the approach, modeling of the process is very important since the prediction and control action calculation are based on it. A hybrid model proposed in Section 2 is built in our paper to model the relationship between the process variables and end-point quality.

k ¼ e

k 1X i e: k i¼1

ð12Þ

^ kþ1 Hence, the modified prediction y cor can be defined as: k ^ kþ1 ^ kþ1 y cor ¼ y hyb þ α e

ð13Þ

where α (0 b α ≤ 1) is a bias correction parameter which is the tradeoff between the convergence rate of iterative optimization and the accuracy of the modified model prediction.

Fig. 5. Comparison of the simulated PSD with prediction results from the three methods. (a) Hybrid model, (b) PLS model, and (c) simplified first principle model (dotted linesmeasurements, solid lines-predictions).

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Therefore, considering the constraints on the input trajectory, the optimal control of the (k + 1)th batch can be described as the following equation: h i h i kþ1 T kþ1 kþ1 T kþ1 kþ1 min J ¼ ysp −y^ cor W ysp −y^ cor þ Δxc RΔ xc s:t:y^ cor x kþ1 c

kþ1

¼ y^ hyb þ α e

k

kþ1

Δ xc

kþ1

¼ xc

k

−xc

kþ1

xc; min ≤xc

≤xc; max

ð14Þ

^ kþ1 where ysp represents the desired quality, y cor represents the prediction of product quality for the (k + 1)th batch after corrections of flow-rate of ammonium oxalate, xkþ1 is the flow-rate of ammonium c for the (k + 1)th batch, Δxkþ1 is the deviation of flow-rate of ammonic um between the (k + 1)th and the kth batch, W; R are the relative weighting matrixes for setpoint tracking and control penalty respectively, and xc;min , xc;max are the lower and upper bounds to be respected by the flow-rate of ammonium. The above quadratic programming problem considers constraints of the manipulated variable in the form of linear inequalities. These constraints are very useful to define the feasible region for the optimization problem using the sequential quadratic programming (SQP) implemented by the MATLAB Optimization Toolbox function, “fmincon”, in this study. The constraints in the form of nonlinear inequalities can also be considered. However, the computation cost of solving the optimization problem could increase significantly when using the SQP approach. As shown in Fig. 3, the procedure of hybrid model-based batch-tobatch control strategy for cobalt oxalate synthesis process to update the input trajectory based on off-line end-point quality measurements is outlined as follows:

257

Therefore, the simulated cobalt oxalate synthesis process can be used as a plant to generate a batch history. Sixty batches of simulated process operational data were generated with the measurements of the process variables which were corrupted by normally distributed random noise. Of the 60 batches of data, 40 batches were used to develop hybrid model and the remaining 20 batches were used as unseen testing data. Fig. 4 illustrates a set of training data that is used to build the PLS model. The target distribution explored in this batch-to-batch optimization study is also depicted in the figure.

4.2. Prediction results In order to demonstrate the efficiency of the developed hybrid model, the prediction performances are compared with the simplified first principle model and the PLS model. The number of latent variables retained determines the performance of the PLS model. In order to select the number of latent variables for PLS model, we apply a leave-one-out cross-

1. At the current kth batch, the input trajectory xck is implemented into the batch process and the outputs yk are measured after the completion of the current batch. 2. Calculate the model error ek using Eq. (10) and store them. Using  k−1 and ek to the average model error of the previous batches e k  compute the average model error e . 3. Modify the hybrid model predictions for the (k+1)th batch using Eq. ^ kþ1 (12) and obtain modified prediction y cor . Then the quadratic optimization problem described by Eq. (13) is solved and an updated input trajectory xkþ1 for the next batch (i.e. the (k+1)th batch) can be c obtained. 4. Update the batch k = k + 1 and go to Step 1. 4. Application to cobalt oxalate synthesis process In this section, we present a case study for optimal control of particle size distribution in a semi-batch cobalt oxalate synthesis process. First, the virtual process and the generation of simulated measurement data are explained. Then, the performance of hybrid model is compared with the simplified first principle model and PLS model. Finally, the performance of batch-to-batch control strategy based on hybrid model is demonstrated. 4.1. Virtual process and batch history In order to create a batch history of PSD measurements and associated input trajectories, a simulated cobalt oxalate synthesis process is constructed by making several changes to the model described in the previous section. The changes made on the model to obtain the virtual process are classified as changes of nucleation and growth phenomena, addition of agglomeration, and consideration of key process disturbances. With regard to the nucleation phenomena, the nucleation rate constant, Kn, is changed by incorporating a multiplicative factor of 1.05. With respect to the growth aspect, the change consist of -15% perturbation of the growth rate constant, Kg, i.e. Kg is changed into 0.85Kg. Furthermore, the agglomeration phenomena is added to the PBE, and random variation is added to the concentrations of cobalt chloride in the initial charge to the reactor.

Fig. 6. Convergence of PSD using the proposed control strategy for scenario 1. (a) Comparison of PSD obtained from initial, 1st and 15th batches with the target, (b) variation of 2-norm error percentage with batch number. (The (✯) represents the initial 2-norm error percentage (no batch-to-batch control), and (○), the 2-norm error percentage reduction using batch-to-batch control.).

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model using data from the batches history provides a means for enhancing the prediction accuracy. Given the good model performance of hybrid model, it is used for the PSD control. The batch-to-batch optimal control will be considered in the next section.

4.3. Batch-to-batch control In cobalt oxalate synthesis process, two control problems frequently arise: (1) the rejection of a persistent batch-to-batch disturbance to achieve consistent PSD at the end of the batch and (2) the designing of new desired PSDs to satisfy customer demands. Therefore, two different scenarios were studied in Section 4.3.1 and Section 4.3.2 respectively to investigate the performance of the proposed hybrid model based batch-to-batch control strategy. The weighting parameters W and R affect the contributions of the tracking errors and control changes. In this study, W and R are selected as W ¼ I20 and R ¼ 2  104 I3 respectively, where I is an identity matrix. In all two cases, the manipulated variable that will be considered is the feed rate of ammonium oxalate. This variable is selected due to its profound effect on the nucleation and growth Fig. 7. Ammonium oxalate input profiles of the initial, 1st, 5th, 10th, and 15th batches for scenario 1.

validation with the root mean squared error (RMSECV) as measure of performance. The RMSECV with a latent variables is defined as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u1 X ^ −i ∥2 ∥y −y RMSECV a ¼ t n i¼1 i

ð15Þ

where n represents the number of samples in the training set, y−i represents the predicted value of yi obtained from the PLS model with a latent variables when we have eliminated the ith sample. In our study, the best number of the latent variables for the PLS model is 8. Then the comparisons of the PSD obtained by the virtual process and predicted from the three methods are shown in Fig. 5. It can be observed from Fig. 5 that, compared with that of PLS and simplified first principal model, both the stability and the accuracy of the hybrid model are much better than that of these two methods. Therefore, a major benefit of the hybrid model is the ability to harness the extrapolative capability of the first principal model, while the PLS

Fig. 8. Comparison of target 2 to target 1.

Fig. 9. Convergence of PSD using the proposed control strategy for scenario 2. (a) Comparison of PSD obtained from initial, 1st and 15th batches with the target, (b) variation of 2-norm error percentage with batch number. (The (✯) represents the initial 2-norm error percentage (no batch-to-batch control), and (○), the 2-norm error percentage reduction using batch-to-batch control.).

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and a PLS correction has been built to predict the endpoint PSD of cobalt oxalate crystals. Then the hybrid model predictions are modified using model errors of previous batches. Finally, based on the modified model predictions, optimal control policy is calculated using the batch-to-batch control strategy. Applications to a simulated cobalt oxalate synthesis process demonstrate that the proposed approach can improve process performance from batch to batch in the presence of model plant mismatches and unknown disturbances. Further studies on the model update approaches will be carried out in the future. Acknowledgements This work was supported by the National High Technology Research and Development Program of China (No. 2011AA060204), National Natural Science Foundation of China (Nos. 61074074 and 61004083), Project 973 of China (No. 2009CB320601) and the Fundamental Research Funds for the Central Universities (No. N100604008). References

Fig. 10. Ammonium oxalate input profiles of the initial, 1st, 5th, 10th, and 15th batches for scenario 2.

phenomena. To be realistic, 0.5% measurement noise is added to the nominal trajectory for flow-rate of ammonium oxalate (Fao). 4.3.1. Scenario 1: changes in raw material purity Batch processes always exhibit batch-to-batch variations due to unknown disturbances such as variations in raw material properties, reactive impurities, and so on. In this case, a disturbance is introduced to the concentration of cobalt chloride in order to simulate the effect of change in raw material purity. The concentration of cobalt chloride is assumed to be 60 (g/L) which is less than the nominal value(65 (g/L)). Fig. 6a shows the progressive achievement of the desired target using the proposed control strategy. In Fig. 6a, it can be seen that the target PSD is achieved in a few iterations. Fig. 6b shows the 2-norm error percentages (Enorm) reduction at each iteration while Fig. 7 shows the evolution of the flow-rate of ammonium oxalate in order to reject the persistent disturbance. The 2-norm error percentage is calculated as: Enorm ¼

∥NPSD;target −N PSD;control ∥  100 ∥N PSD;target ∥

ð16Þ

where NPSD, target and NPSD, control are the end-point PSD of the target and the controlled batch respectively. 4.3.2. Scenario 2: batch-to-batch control to achieve a new product quality target In this case, the objective is to design a new product when the process is affected by a persistent batch-to-batch disturbance in the initial concentration of cobalt chloride (with 60 (g/L) instead of 65 (g/L)). As shown in Fig. 8 the new target PSD which referred to as target 2 is quite different from that used in Scenario 1. The performance of the proposed approach is shown in Figs. 9 and 10, where it can be seen that the desired PSD is gradually achieved by using the hybrid model based batch-to-batch control strategy. 5. Conclusions A batch-to-batch control strategy using hybrid model to regulate the endpoint particle distribution in cobalt oxalate synthesis process is presented in the paper. In the proposed approach, a hybrid model comprising of a simplified principal model for cobalt oxalate synthesis process

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