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

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.

Dynamic event–triggered security control for stochastic partial differential equation systems via a unified pointwise measurement/control strategy

“…Figure 7. The trajectories of y(0:3, t) and y(0:6, t) in different time domain: (a) the trajectories of y(0:3, t) (1) , (b) the trajectories of y(0:6, t) (1) , (c) the trajectories of y(0:3, t) (2) , and (d) the trajectories of y(0:6, t) (2) . observable, and the sensors and actuators cannot be installed in the same place all the time, the above method will be invalid.…”

“…Nonlinear partial differential equation (PDE) systems are extensively utilized to model the practical physical processes, such as the control of robotic aircraft wings, 1 chemical reaction procedure, 2 the process of thermal diffusion, 3 and so on. Compared with the ordinary differential equation (ODE) systems, PDE systems have both spatial and temporal characteristics, which is more suitable for describing practical systems.…”