Product design problems often require finding the raw materials and/or operating conditions (inputs) that are needed to achieve some pre-assigned quality specifications on the product (outputs). The problem can be tackled by first building a latent-variable model (e.g., partial least-squares regression) on historical manufacturing data of products similar to the new one, and then using model inversion to find the input conditions required to obtain the target product. However, in most practical cases, the variables characterizing product quality are correlated, and this may raise singularity issues upon algebraic model inversion. The most popular approach to cope with this relies on removing a priori some of the correlated quality variables from the model output matrix, and on building the latent-variable model in such a way that the inputs be related to the remaining outputs only. However, in this case the inputs obtained upon model inversion may not be able to ensure that the quality variables that were not included in the output matrix will be close enough to their targets. We propose a novel algebraic formulation of the latent-variable model inversion problem, named regularized direct inversion, which can cope with output correlation by design. The proposed formulation enables one to retain in the model output matrix all quality variables, and addresses output correlation by removing a posteriori only the non-systematic information that would cause singularity issues. Therefore, no structural information about the relation between inputs and out-puts is left out of the model by design, which in turn improves the performance of model inversion. The supe-riority of regularized direct inversion over the standard approach, and its ability to cope with model uncertainty, are proved using two simulated batch processes: a generic fermentation process, and a process for the manufacturing of penicillin.
Digital design of new products: accounting for output correlation via a novel algebraic formulation of the latent-variable model inversion problem
Arnese Feffin, E;Facco, P;Bezzo, F;Barolo, M
2022
Abstract
Product design problems often require finding the raw materials and/or operating conditions (inputs) that are needed to achieve some pre-assigned quality specifications on the product (outputs). The problem can be tackled by first building a latent-variable model (e.g., partial least-squares regression) on historical manufacturing data of products similar to the new one, and then using model inversion to find the input conditions required to obtain the target product. However, in most practical cases, the variables characterizing product quality are correlated, and this may raise singularity issues upon algebraic model inversion. The most popular approach to cope with this relies on removing a priori some of the correlated quality variables from the model output matrix, and on building the latent-variable model in such a way that the inputs be related to the remaining outputs only. However, in this case the inputs obtained upon model inversion may not be able to ensure that the quality variables that were not included in the output matrix will be close enough to their targets. We propose a novel algebraic formulation of the latent-variable model inversion problem, named regularized direct inversion, which can cope with output correlation by design. The proposed formulation enables one to retain in the model output matrix all quality variables, and addresses output correlation by removing a posteriori only the non-systematic information that would cause singularity issues. Therefore, no structural information about the relation between inputs and out-puts is left out of the model by design, which in turn improves the performance of model inversion. The supe-riority of regularized direct inversion over the standard approach, and its ability to cope with model uncertainty, are proved using two simulated batch processes: a generic fermentation process, and a process for the manufacturing of penicillin.
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