Luenberger Observers for Infinite-Dimensional Systems, Back and Forth Nudging, and Application to a Crystallization Process

Abstract

This paper deals with the observer design problem for time-varying linear infinitedimensional systems. We address both the problem of online estimation of the state of the system from the output via an asymptotic observer, and the problem of offline estimation of the initial state via a Back and Forth Nudging (BFN) algorithm. In both contexts, we show under a weak detectability assumption that a Luenberger-like observer may reconstruct the so-called observable subspace of the system. However, since no exact observability hypothesis is required, only a weak convergence of the observer holds in general. Additional conditions on the system are required to show the strong convergence. We provide an application of our results to a batch crystallization process modeled by a one-dimensional transport equation with periodic boundary conditions, in which we try to estimate the Crystal Size Distribution from the Chord Length Distribution.

New perspectives on output feedback stabilization at an unobservable target

“…This infinite-dimensional Luenberger observer has been investigated in [11] (see also [8]), in which it is proved that ε(t) w ⇀ 0 as t goes to infinity if u is a regularly persistent input. Our goal is to embed the original system (1) into a unitary system, and to use this observer design in the context of dynamic output feedback stabilization.…”

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

“…Clearly, exact observability implies approximate observability, and they are equivalent on finite dimensional systems. Unfortunately, the function k i ( , r) being bounded, the system is not exactly observable according to [6,Proposition 6.3]. Therefore, we focus on approximate observability, as in [10], [13], [28], and more recently in [6].…”

“…Unfortunately, the function k i ( , r) being bounded, the system is not exactly observable according to [6,Proposition 6.3]. Therefore, we focus on approximate observability, as in [10], [13], [28], and more recently in [6]. Let…”

“…As shown in [13] (which extended the results of [14], [23] which focused on exactly observable systems), the type of convergence depends on the observability properties of the system. These results have been extended to the nonautonomous context (which is the case here since G i is timevarying) in [6] and applied to a crystallization process in [8].…”

New inversion methods for the single/multi-shape CLD-to-PSD problem with spheroid particles

“…Observers have proved to be very useful in the context of batch crystallization [25,26,28,31,34,35]. Here we apply the Back and Forth Nudging (BFN) algorithm [2,3,4,5,9,13,14,15,18,33], which is an inverse problem technique based on dynamical observers. We prove the convergence of this method when crystals are split into two clusters: spheres and elongated spheroids, which happens, for example, in [11].…”