Figure 4.3. The matrix relationships in PLS [145], T and U show PLS scores matrices on X and Y blocks, respectively, P, X loadings, W and Q represent weight matrices for each block, E and F are residual matrices formed by the variation in the data that were left out of modeling.

At this point, the convergence is checked by comparing ti in Eq. 4.21 with the ti from the preceding iteration. If they are equal within rounding error, one proceeds to Eq. 4.23 to calculate X data block loadings pi and weights W[ are rescaled using the converged ui. Otherwise, Ui from Eq. 4.22 is used.

The regression coefficient b for the inner relation is computed as

Once the scores and loadings have been calculated for the first latent variable, X- and Y-block residuals are computed as

The entire procedure is now repeated for the next latent variable starting with Eq. 4.21. X and Y are replaced with the residuals Ei and Fi, respectively, and all subscripts are incremented by 1. Hence, the variability explained by the earlier latent variables is filtered out from X and Y by replacing them in the next iteration with their residuals that contain unexplained variation.

Several enhancements have been made to the PLS algorithm [118, 198, 343, 363, 664, 660, 668] and software is available for developing PLS models [472, 548].

4.3 Input-Output Modeling of Dynamic Processes

Methods for developing models to describe steady state relationships of processes are presented in Sections 4.1 and 4.2. The description of batch fermentation processes and the general form of their model equations in Chapter 2 (for example Eq. 2.1 or Eq. 2.3) indicate that dynamic input-output models are more appropriate for representing the behavior of these processes. Two types of dynamic models are introduced in this section: time series models (Section 4.3.1) and state space models (Section 4.3.2). State estimators are also presented in conjunction with state space models. The linear model structures are discussed in this section. They can handle mild nonlinearities. They can also result from linearization around an operating point. Their extensions to nonlinear models are discussed in Section 4.7. Use of these modeling paradigms to develop more complex models of batch and semi-batch processes is reported in Section 4.3.4.

Inputs, outputs, disturbances and state variables will be denoted as u, y, d and x, respectively. The models can be in continuous time (differential equations) or discrete time (difference equations). For multivariable processes where ui(t), U2(t), ■ ■■, um(t) are the m inputs, the input vector u(i) at time t is written as a column vector. Similarly, the p outputs, and the n state variables are defined by column vectors:

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