Reduction and Observer Design for a Grey-Box Model in Continuous Pharmaceutical Manufacturing

TL;DR

This paper introduces a reduced order model for a pharmaceutical dryer that maintains high accuracy and significantly speeds up computations, enabling effective observer design for real-time state estimation.

Contribution

A novel ROM formulation for a grey-box pharmaceutical dryer model using $ ext{H}_2$-norm projection and bilinear form, integrated with a nonlinear Kalman filter-based observer.

Findings

ROM achieves less than 0.3% mean square error compared to FOM.

Computational time reduced by up to 40 times.

Observer converges reliably regardless of initial conditions.

Abstract

In this contribution, a novel Reduced Order Model (ROM) formulation of the grey-box model proposed in Elkhashap et al. (2020a) for the pharmaceutical continuous vibrated fluid bed dryer (VFBD) is presented. The ROM exploits the $\mathcal{H}_2$-norm projection-based model order reduction method after a special solution formulation of the model's infinite-dimensional part. This is mainly by introducing a vector field mapping between the model parts casting the semi-discretized PDE into a bilinear form. The ROM produced is then integrated into an nonlinear Kalman Filtering-based observer design also handling the estimation of the model's algebraic variables. Evaluations of the FOM, ROM, ROM-based observer variants, and the FOM-based observer are performed using Monte-Carlo simulations as well as simulations based on experimental data of the real system. It is shown that the ROM could reproduce the FOM states accurately with a relative mean square error below $0.3\,\%$ for the experimental data simulation. This is while reaching a computational-time reduction up to a factor of $40$. The ROM-based observer with algebraic states correction is shown (using Monte-Carlo simulations) to be able to converge to the true values for all cases regardless of initialization. Moreover, it is also shown that the performance degradation of the observer due to reduction is practically insignificant. This is while the computational speedup of the observer due to reduction reached a factor of more than third order of magnitude.