Structure-Preserving Uncertainty Quantification and Control of Population Balance Models

This article describes a systematic method for propagating and controlling the effects of parametric uncertainties in population balance models (PBMs) using polynomial chaos expansions. PBMs are an important element in the model-based control of many particulate manufacturing processes. Direct stochastic Galerkin projections of PBMs with uncertain parameters allow for structure-preserving uncertainty quantification of the population distribution across the full intrinsic variable domain. With the proposed method, more accurate real-time risk analysis to ensure safe and controlled particulate manufacturing under distributed parameter system model constraints becomes accessible on compute-limited hardware. The method in particular shows a considerable computational speedup and memory usage decrease relative to conventional sampling-based methods, enabling fast on-line model predictive control implementations. The method also retains full population distribution information in contrast to less-accurate, structure-reducing techniques such as the method of moments. Throughout, we discuss approaches to mitigate common numerical accuracy losses appearing in the discretization of PBMs.