Approximate observability and back and forth observer of a PDE model of crystallisation process

TL;DR

This paper demonstrates the approximate observability of a PDE model for a crystallization process and applies a back-and-forth nudging algorithm to estimate particle size distributions from chord length measurements.

Contribution

It proves approximate observability of a PDE system modeling crystallization and introduces a BFN algorithm for PSD reconstruction from CLD data.

Findings

Proved approximate observability of the PDE system.

Successfully applied BFN algorithm for PSD estimation.

Validated approach with theoretical guarantees.

Abstract

In this paper, we are interested in the estimation of Particle Size Distributions (PSDs) during a batch crystallization process in which particles of two different shapes coexist and evolve simultaneously. The PSDs are estimated thanks to a measurement of an apparent Chord Length Distribution (CLD), a measure that we model for crystals of spheroidal shape. Our main result is to prove the approximate observability of the infinite-dimensional system in any positive time. Under this observability condition, we are able to apply a Back and Forth Nudging (BFN) algorithm to reconstruct the PSD.