Variable No.

Figure 6.72. On-line monitoring of a faulty batch using AHPCA. The subscript "140-180" in figures (b) and (d) indicate that contributions are averaged between 140th and 180th measurements.

6.5.3 Online MSPM and Quality Prediction by Preserving Variable Direction

A different online MSPM framework can be established by unfolding the three-way data array by preserving variable direction [203, 232, 663]. In this MSPM framework, it is not necessary to estimate the future portions of variable trajectories. MPCA or MPLS models can be developed and used for online monitoring. A new methodology has been proposed based on developing an MPLS model between process variable matrix that is unfolded in the variable direction and local time stamp to use in the alignment of

Figure 6.73. Three-way array formation and unfolding, (a) by preserving batch direction, (b) by preserving variable direction.

Figure 6.73. Three-way array formation and unfolding, (a) by preserving batch direction, (b) by preserving variable direction.

trajectories [663].

Process measurements array X can be unfolded to X (IK x J) by preserving the variable direction [232, 552, 663]. In this case, X can be thought of as a combination of slices of matrices of size (KxJ) for each batch (Figure 6.73(a)). X is formed after rearrangement of these slices. This type of unfolding suggests a different multivariate modeling approach [232, 606, 663]. Batch evolution can be monitored by developing an MPLS model between X (IK x J) and a time stamp vector z (IK x 1) (Figure 6.75(b)). In this case, MPLS decomposes X and z into a combination of scores matrix T (IK x R), loadings matrix P (J x R) and vector q (R x 1) and weight matrix W (J x R) with different sizes compared to conventional MPLS decomposition discussed in Section 4.5.2

where E and f are the residuals matrix and vector, respectively.

Figure 6.74. Comparison of different scalings applied to X matrix. Biomass (left figures) and penicillin concentration profiles (right figures). Raw profiles (a)-(b), mean-centered (by subtracting mean trajectories) and unit variance scaled profiles (c)-(d), mean-centered (by subtracting variable means) and unit variance scaled profiles (e)-(f).

During the progress of a new batch, a vector xnew of size 1 x J becomes available at each time interval k. After applying the same scaling to new observations vector as reference sets, scores can be predicted for time instant k by using the MPLS model parameters t = xnewW(PTW)-1. (6.130)

Since the size of the resulting matrix from the operation W(PTW)-1 is J x R, online monitoring of the new batch can be performed without any future value estimation.

In the pre-processing step, X is mean-centered by subtracting variable means and usually scaled to unit variance. This pre-processing differs from the conventional approach (Figure 6.74(c)-(d)) in that the dynamic nonlinear behavior of trajectories in X is retained (Figure 6.74(e)-(f)). This technique can also be combined with conventional MPLS for predicting product quality after the completion of the batch run [606, 663].

KJ t

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