Ucl

4. Set up search space for ith landmark. This consists of a vector of possible landmarks range for the ith landmark of reference set (£m (1 x iii) and an,x K, matrix of n.L replicates of the test curve to be warped with respect to the possible landmarks' space.

(a) Align rii test curves by landmark registration. If i > 2 use previously estimated and stored landmarks in Li during registration.

(b) Calculate the sum of squared errors (SSE) between each of the nl test curves and the reference trajectory (calculate reference trajectory from previously aligned reference set).

When the minimum SSE is found, select the ijth. landmark as the 2th landmark for the test trajectory.

(e) Store estimated zth landmark in vector L,.

6. Warp test trajectory (and the rest of the trajectories in the new batch) with respect to the m estimated landmarks.

Example. The implementation of time alignment by landmark registration differs in postmortem and online analysis. When there is a database of historical batches of different lengths that alignment is to be performed, all of the multivariate batch data are re-sampled to a common number (e.g. the median length) of data points by interpolation first, then the registration technique is implemented. However, in the case of online registration with landmark detection, unknown batch length (process termination time or campaign run length is not known) of the new batch represents some difficulties in implementation. To represent online landmark detection in real-time, it is assumed that a fixed batch termination point is determined. Although all of the batches are to come to a completion at the same final time point, a temporal variation in important landmark locations is still present. The critical implementation issues of the alignment technique are presented in an orderly fashion in this example.

Determination of the landmark locations in reference batches

The regularization technique for mixed case is implemented to simulated fed-batch penicillin fermentation data of 40 batches sampled at 0.5 h on 16 variables for 500 h resulting in 1000 measurements. The decision should be made using engineering judgment on choosing appropriate variable trajectories that may contain physiological information about the location of the process landmarks. Figure 6.36 shows some of the process variable trajectories as well as two of the manipulated variables (base and acid flow rates) of a reference batch run under nominal operating conditions. Note that, the concentration of hydrogen ion (pH) is associated with biomass growth [61] as explained in Section 2.7.1, hence, becoming a good indicator for tracking physiological activities.

It is inferred that there are three process landmarks separating four distinct phases in fed-batch penicillin fermentations based on expert knowledge about penicillin fermentation. The first phase (lag phase and pre-exponential phase) corresponds to batch operation where a lag exists on the inception of penicillin production while cells are consuming substrates to grow. The landmark for the first phase can be found easily in any of the trajectories as shown in Figure 6.36. Exponential cell growth along with the start of penicillin production is observed in the second phase where glucose feed is initiated. The location of the second landmark is not apparent on each trajectory. Normally it corresponds to the time when biomass concentration begins to level off. In the vicinity of that point, hydrogen ion concentration starts to decrease and consequently the need for base addition (Fba.se) is reduced as shown in Figure 6.36. Hence, base flow rate is an excellent candidate for determining the location of the second landmark that indicates the beginning of the third phase (stationary phase). A

0 100 200 300 400 500 Time, h

"0 100 200 300 400 500 Time, h

Figure 6.36. Physiologically important variable trajectories (F: glucose feed rate, 5: glucose concentration, DO: percent oxygen saturation, X: biomass concentration, P: penicillin concentration, V: culture volume, -fbase: base flow rate and Fac;<j: acid flow rate).

0 100 200 300 400 500 Time, h

0 100 200 300 400 500 Time, h

"0 100 200 300 400 500 Time, h

Figure 6.36. Physiologically important variable trajectories (F: glucose feed rate, 5: glucose concentration, DO: percent oxygen saturation, X: biomass concentration, P: penicillin concentration, V: culture volume, -fbase: base flow rate and Fac;<j: acid flow rate).

similar line of thought can be followed for detecting the temporal location of the third landmark which is the start of the death phase towards harvesting the fermentation. At this phase, biomass concentration begins to decline resulting in the decrease in hydrogen ion concentration level. Note that, a set point gap is defined for acid flow rate controller action to avoid excessive acid addition during the simulations resulting in a small increase at pH right after the third landmark [61]. Therefore, the instant when acid addition takes place after stationary phase can be used to determine the location of the third landmark and the beginning of the death phase.

Once the decision about the choice of the variables that contain landmark information is made, these variables (biomass concentration (X), base (■fbase) and acid (FacKi) flow rates in Figure 6.36) are investigated in each batch of the reference set and landmark locations are stored. In this example, reference landmark locations matrix £m is of size (3 x 40). Note

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