Info

Cooling water flow rate, L/h

example is chosen from pharmaceutical granules production case (Section 6.4.5).

Example. A set of data is produced using the simulator of fed-batch penicillin production (based on the unstructured model discussed in Section 2.7.1) under normal operating conditions. The values of the initial conditions and set points of input variables are slightly varied for each batch, resulting in unequal and unsynchronized batch trajectories that are typical in most experimental cases. Batch lengths varied between 375 h and 390 h. One of the batches that has a batch length of 382 h, close to the median batch length is chosen as a reference batch. Data of the other batches are equalized based on multivariate DTW algorithm discussed in Section 6.3.2. Type II symmetric local continuity constraint (Figure 6.24(b)) with smoothed path weightings, Itakura global constraint, and Sakoe and Chiba band constraint (with Qmax = 2 and Kq = 50) are used for data synchronization. Multivariate DTW synchronization procedure was applied for a maximum of five iterations (Figure 6.43).

The reference set is comprised of 42 batches containing 14 variables (sampled at 0.5 h). A three-way array of size 41 x 14 x 764 is formed based on this initial analysis. The variables are listed in Table 6.6. Although on-line real-time measurement availability of some of the product related variables such as biomass and penicillin concentrations is somewhat limited in reality, it is assumed that these can be measured along with frequently measurable variables such as feed rates and temperature. If the sampling rates are different, an estimator such as Kalman filter can be used to estimate these variables from measured values of frequently measured variables. A number of product quality variables are also recorded at the end of the batch (Table 6.7 and Figure 6.41).

Three additional batches were simulated to illustrate detection, diagnosis and prediction capabilities of the MPCA and MPLS models used for both end-of-batch and on-line SPM. Fault scenarios are chosen such that they resemble the ones generally encountered in industry. First fault is a 10% step decrease in agitator power input about its set point during early in the second phase of the fermentation between 70 and 90 hrs (between the 140th and 180th samples). The second fault is a small drift in the glucose feed rate right after start of feeding in fed-batch operation. In the latter case, the abnormal operation develops slowly and none of the individual measurements reveal it clearly when their univariate charts are examined. The third fault is the same as the second fault, only the slope of the drift is higher. The first faulty batch is of length 375 h (750 samples), the second is 380 h (760 samples) and the third batch of length 382 h (764 samples). Figure 6.44 shows the trajectories of these faulty batches along with a normal batch trajectory set.

6.4.2 Multivariate Charts for SPM

The following multivariate charts are used as visual aids for interpreting multivariate statistics calculated based on empirical models. Each chart can be constructed to monitor batches or performance of one batch during its evolution.

Score biplots or 3D plots are used to detect any departure from the in-control region defined by the confidence limits calculated from the ref-

Table 6.7. Quality variables measured after the completion of batches

No.

Quality Variables

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