The issue of state estimation of an aggregation process through (1) using model reduction to obtain a tractable approximation of the governing dynamics and (2) designing a fast moving-horizon estimator for the reduced-order model is addressed. The method of moments is first used to reduce the governing integro-differential equation down to a nonlinear ordinary differential equation. This reduced-order model is then simulated for both batch and continuous processes and the results are shown to agree with constant Number Monte Carlo simulation results of the original model. Next, the states of the reduced order model are estimated in a moving horizon estimation approach. For this purpose, Carleman linearization is first employed and the nonlinear system is represented in a bilinear form. This representation lessens the computation burden of the estimation problem by allowing for analytical solution of the state variables as well as sensitivities with respect to decision variables. V C 2016 American Institute of Chemical Engineers AIChE J, 62: 1557-1567, 2016 Keywords: mathematical modeling, simulation, moving horizon estimation, model reduction, coagulation process
IntroductionIn chemical engineering, material science, and biology, there are a multitude of processes characterized by dispersed phenomena. Systems involving such processes merit a particle population study rather than the traditional mass balance performed for continuous media. There are numerous examples of these processes ranging from industrial applications to biological reactions. Crystallization, polymerization, viral infections, and collocalization of enzymes in cells are just a few instances of these processes that occur in our everyday life and motivate the work in this article. The common characteristic of all these processes are the individual members of the population or the particles. These particles are distinguished by their type, size and/or composition. Mostly, a population balance governs the dynamic behavior of particulate systems. This results in complex mathematical models, specifically for systems in which the particles are characterized by two or more properties. The simplest system, in this area, is a twocomponent aggregation system with no chemical reactions. This system is mostly used in pharmaceutical applications where through use of a solvent called excipient the particles in a drug powder adhere together and form granules. In an ideal granulation process, the components, solvent, and solute, are well mixed and the distribution of the solute mass is uniform. The deviation from the mean of the solute mass in aggregates quantifies the granulation quality which is referred to as blending degree. However, this output property is not easily measurable during evolution of the process. This article focuses on these bicomponent granulation processes and estimation of the distribution of components in the product.The population balance equation for granulation systems determines the dynamics of a bivariate distribution function in fo...