(37c) Application of Observation Techniques in a Model Predictive Control Framework of Fed-Batch Crystallization of Ammonium Sulphate

Batch crystallization is often used for production of high purity, high added-value materials with tight specifications on crystal properties (size, purity, morphology). In order to meet these requirements, an effective control strategy is needed. With the current progress in crystallization modeling, models having predictive capabilities become available. In principle it is possible to use these models in a Model Predictive Control (MPC) framework.

In this study, a simple moment model is used at first. The studied system is ammonium sulphate ? water. The model of a seeded fed-batch crystallizer contains calculation of leading moments of Crystal Size Distribution (CSD), mass balance for solute and crystal mass, and an empirical relation for crystal growth. The model does not have any crystal birth term. It is assumed that seed crystals grow up and result in the product crystals. Due to a high seed loading supersaturation is at a low level at all times during a batch [1]. Under these conditions, secondary nucleation is moderate and therefore, can be neglected.

The advantage of using the moment model is its simplicity. It does not require much computation time and therefore, does not need further model reduction. This makes it suitable for on-line application in the MPC framework.

In order to make sure that the moment model is a good representation of the crystallization system at hand, model validation is performed. A number of seeded fed-batch evaporative crystallization experiments are carried out in a 75-l Draft-Tube (DT) crystallizer. The experiments are conducted with different seed sizes, seed loadings, and operating conditions. The CSD measured by a laser diffraction instrument (Helos/Vario, SympaTech, Germany) and liquid fraction (inferred from the product density signal) are used for estimation of parameters of the empirical crystal-growth expression.

Model predictive control uses the crystallizer model and measurements of important variables to compute optimal control actions corresponding to a certain objective function. It often happens that not all variables can be (reliably) measured. Therefore, those unmeasured variables should be estimated using available measurements and the crystallizer model using state estimators or observers. A mismatch between process and model can occur due to unmeasured process disturbances and model imperfections. A properly designed observer can correct for those discrepancies and provide estimates of a high quality.

There are various techniques in soft-sensing technology. In this study, we focus on techniques applicable to non-linear systems. We investigate the possibility of using an Extended Kalman Filter (EKF) and an extended Luenberger observer.

An EKF is an extension of linear Kalman filter that can be applied to non-linear systems [2]. Since a single linear model cannot approximate the entire process due to intrinsic dynamic nature of batch processes, the process model is linearized at different operating points along a certain (optimal) trajectory. The time interval between two adjacent operating points is two minutes. It corresponds to the sampling frequency of the CSD measurement device. The ensemble of the linear models is subjected to observability analysis. From the observability analysis it follows that supersaturation that is not measured, can be observed from the measured leading moments of CSD.

The second non-linear observer is an extension of the linear Luenberger observer [3]. As it was shown by different authors [4, 5], the extended Luenberger observer is an adequate choice for the non-linear system at hand as it directly accounts for the non-linear process dynamics. Analysis of nonlinear observability matrix shows that the unmeasured state is observable and therefore, can be estimated.

In conclusion, a comparison of two observation techniques is made. Conclusions about usage of the simple moment model for observation in on-line model predictive control are presented.

References

[1] Doki, N.; Kubota, N.; Yokota, M. and Chianese, A. (2002). Determination of Critical Seed Loading Ratio for the Production of Crystals of Uni-Modal Size Distribution in Batch Cooling Crystallization of Potassium Alum. J. Chem. Eng. Japan 35, pp. 670-676.

[2] Lee, J. and N. Ricker (1994). Extended Kalman filter based nonlinear model predictive control. Ind. Eng. Chem. Res. 33, 1530-1541.

[3] Luenberger, D. (1971). An introduction to observers. IEEE Transactions on Automatic Control 16 (6), 596-602.

[4] Ciccarella, G., M. Dalla Mora, and A. Germani (1993). A Luenberger-like observer for nonlinear systems. Int. J. Control 57 (3), 537-556.

[5] Muusze, J. and S. Dijkstra (1998). Optimal control of batch crystallisation processes with the use of observer techniques. Proceedings of the 13th International Congress of Chemical and Process Engineering (CHISA '98), August 23-28, Praha, Czech Republic.