of the batch, the contribution plot for SPE signals an unusual situation for variable 3 (Figure 6.80c). Variable 3 and 11 are found to be the most affected variables because of the fault according to T2 contribution plot. Deviation from average batch behavior plot is ineffective in indicating the most affected variable(s) in this case (Figure 6.83a).
Quality prediction ability of the integrated MSPM framework is also tested via two cases. A normal batch is investigated first. As expected, SPE plot produced no out-of-control signal and final product quality on all five variables (shown as a solid star) is successfully predicted (Figure 6.84). The prediction capability is somewhat poor in the beginning because of limited data, but it gets better as more data become available. In the second case, where a drift of magnitude -0.05% ft-1 is introduced into substrate feed rate at the beginning of the fed-batch phase until the end of operation, SPE plot signaled out-of-control right after the sixth quality prediction point (80% completion of phase 2). Because MPLSB model is not valid beyond this point no further confidence limit is plotted (Figure 6.85). Although the predictions of MPLSB model might not be accurate for the seventh (and final) value, the framework generated fairly close predictions of the inferior quality. Predicting the values of end-of-batch quality during the progress of the batch provided a useful insight to anticipate the effects of excursions from normal operation on final quality. □
6.5.4 Kalman filters for Estimation of Final Product Quality
Information on final product quality complements the information obtained from process variable trajectories. A model of the batch process and process variable measurements can be used to estimate final product properties before the completion of the batch. Consider the differential-algebraic nonlinear equation system that describes the batch process and its final state:
where x are the state variables, u the manipulated inputs, v and w the state and output disturbances, y the measured outputs, and q the final product quality at the end of the batch(i = tf). If a fundamental model
of the process were available, the final product quality can be estimated by using an Extended Kalman filter. When a fundamental dynamic model is not available, an empirical model could be developed by using historical data records of successful batches. The problem may be cast as a regression problem where the measurements y upto the current time tc, and inputs u upto the end of the batch are used at any time tc to estimate q. Note that the inputs at t = tc, • • • . t f have not been implemented yet and have to be assumed. A linear predictor for final product quality has been proposed by using a least squares estimator obtained through biased regression (by using PCA or PLS) and extended to recursive least squares prediction through a Kalman filter .
Batch process monitoring techniques presented in previous sections focus on detecting abnormalities in the current batch run by comparing it with performance criteria developed using a historical database. In some batch processes, disturbances may evolve over several batches, causing a gradual drift in product quality and eventually leading to significant quality deviation. MSPM methods that track chanties in "between-batch" correlation
structure, mean or drift are needed to track systematic variations in successive batch runs. Subspace identification (Section 4.3.2) where the time index is replaced by a batch index can be used to develop a framework for monitoring batch to batch (between-batch) variations .
Consider measurement data generated in a batch run arranged as I batches, J variables, and K sampling instants. Let yhk denote the vector of mean centered and scaled J process measurements of batch i at sampling time k. Collecting all process measurement vectors for k — 1, • • • , K, the J x K data are unfolded to vectors of length JK for each k (called as lifting in subspace identification literature):
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