Product optimization and control in the latent variable space of nonlinear PLS models

Process optimization is usually performed on causal models built from fundamentals or from designed experiments that relate the independent effects of all the adjustable process variables to all the response variables of interest. However, in many complex processes, there exist a very large number of adjustable (manipulated) variables, and these variables are often highly correlated through process or operational constraints (e.g. temperature profile along the barrel of an injection molding machine). In such cases, completely causal models may not be easily obtained. However, a significant amount of data is often available from process operations over a wide range of conditions. These data may allow one to build restricted causal models that show how groups of process and raw material variables affect the product quality in a subspace of the original variables. Linear or nonlinear latent variable models built using PLS are ideal for such a purpose.In this paper, a methodology is proposed for process development and control based on optimization in the subspaces defined by the latent variable models built from such data. Nonlinear latent variable models are developed and optimization strategies using sequential quadratic programming are implemented to solve several important product development and process control objectives. The optimization problem is formulated such that the solutions are constrained to lie in the latent variable space of the model defined by the training data set, and to satisfy additional operational constraints within this space. The methodology is applied to an industrial over-molding injection process, and is shown to be very effective in finding process operating conditions that achieve desired quality objectives, that reduce the variability of the final product quality and that compensate for variations in both raw material and environmental factors.