A cautious approach to robust design with model parameter uncertainty

Industrial robust design methods rely on empirical process models that relate an output response variable to a set of controllable input variables and a set of uncontrollable noise variables. However, when determining the input settings that minimize output variability, model uncertainty is typically neglected. Using a Bayesian problem formulation similar to what has been termed cautious control in the adaptive feedback control literature, this article develops a cautious robust design approach that takes model parameter uncertainty into account via the posterior (given the experimental data) parameter covariance. A tractable and interpretable expression for the posterior response variance and mean square error is derived that is well suited for numerical optimization and that also provides insight into the impact of parameter uncertainty on the robust design objective. The approach is cautious in the sense that as parameter uncertainty increases, the input settings are often chosen closer to the center of the experimental design region or, more generally, in a manner that mitigates the adverse effects of parameter uncertainty. A brief discussion on an extension of the approach to consider model structure uncertainty is presented.