The integrals are evaluated numerically by using traditional tools such as the trapezoidal rule.

An alternative computation of w can be made by attaching a penalty term toEq. (4.139):

where 6j controls the roughness of the estimated weight functions. The solution can be found by using the pointwise approach which results in w = [T(t)TT{t) + JftT0]-1rTA(i) (4.148)

where 0 = diag(9o, • • • , 9m-i). Alternatively, the basis function approach can be used to compute w.

The FDA framework can be used to model the trajectories of the process variables of a batch process. This model can be used to generate the estimates of the future portion of the trajectories for implementing SPM during the progress of the batch. The FDA based prediction provides remarkable improvement in trajectory estimation illustrated in Figures 4.7 and 4.8. In this illustration, data are generated using the simulator based on the unstructured nonlinear multivariable model of penicillin fermentation (Section 2.7.1) and data-based models are developed using multiway PCA (MPCA) and FDA frameworks. The trajectories to the end of the batch are estimated based on these models and the "data collected" (Solid curves in Figures 4.7 and 4.8) up to the present time in the current batch.

Two cases are generated for penicillin concentration profile to illustrate and compare estimation methods. Curves labelled 1-3 are based on the estimation methods described in Section 6.5.1, curve 4 is based on the PDA model and curve 5 is the PDA based estimation with EWMA-type local weights on "measured" and "estimated" data. In the first case, data generated under normal operation are used. Estimation performed starting at 150 h onward to the end of the batch run resulted in comparatively close results for all methods (Figure 4.7(a)). Mean-centered profiles are also given in Figure 4.7(b) to provide a magnified look at the predictions. Although the predictions are close in this case, PDA with EWMA-type local weightings produced the best result (Curve 5). The problem can also be cast into a framework of Kalman filter-type correction of the predicted values. The results are identical when the EWMA weight and Kalman filter gain are matched. A drift disturbance in the substrate feed rate from the start of fed-batch operation until the end of the batch is generated as the second case. Estimation is started at 180 h onward to the end of the batch (Figure 4.8). The best estimates are given by PDA with local weighting of data (Curve 5).

The FDA approach provides a framework to develop methods for data pretreatment, adjustment of data length of different batches, detection of landmarks for the beginning and ending of various stages during the batch, PCA, and estimators for final product properties. Landmark detection and data synchronization using FDA are discussed in Section 6.3.3. Furthermore, the differential equations generated can be converted to a state-space

(b) Mean-centered profiles

Figure 4.7. Estimates of the penicillin concentration trajectory under normal operation.

(b) Mean-centered profiles

Figure 4.7. Estimates of the penicillin concentration trajectory under normal operation.

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