**Title**: Stabilization of non-uniformly observable control systems and infinite dimensional observers

**The jury:**

Prieur, Christophe, CNRS Research Director, GIPSA-lab, Rapporteur

Wirth, Fabian, University Professor, University of Passau, Rapporteur

Bernard, Pauline, Senior Lecturer, MINES ParisTech, Examiner

Coron, Jean-Michel, University Professor, LJLL, Examiner

Jacob, Birgit, University Professor, University of Wuppertal, Examiner

Maschke, Bernhard, University Professor, University of Lyon 1, Examiner

Andrieu, Vincent, CNRS Research Director, LAGEPP, Thesis Director

Serres, Ulysse, Senior Lecturer, University of Lyon 1, Thesis Co-supervisor

Gauthier, Jean-Paul, Professor emeritus, University of Toulon, Thesis co-supervisor

**Abstract**:

This thesis is structured around two different but related themes. In the first part, we are interested in the problem of stabilization by dynamic output looping. When only a part of the state of a dynamic system is known, a stabilising state loop cannot be implemented. Therefore, a possible strategy to stabilise the state at a target point is to design an observer, to asymptotically estimate the state by filtering the output over time, and to use the stabilising control law applied to the observer as a controller. This approach is known to be effective on uniformly observable systems, i.e. observable for any input. However, non-linear systems are generically not uniformly observable when the dimension of the output is less than or equal to that of the input. Thus, in the presence of observability singularities, new techniques remain to be developed.

In a second part, we deal with the observer synthesis problem for infinite dimensional linear time-varying systems. The objective is to design a dynamic system capable of estimating the state of the initial system from a measurement and its dynamics. The notion of observability can be generalised in several ways in infinite dimension. In particular, we distinguish between exact and approximate observability. While an exponential convergence of Luenberger observers can generally be shown under exact observability assumptions, results on approximate observability assumptions, which we are interested in, are rarer. These observers can also be used in the context of reconstructing the initial condition of a system. The procedure, called Back and Forth Nudging (BFN), is then based on successive iterations of observers in positive and retrograde time. These methods can be applied to a batch crystallisation process, where the state to be estimated is the particle size distribution (PSD).

**Date/heure**

Date(s) - 19 May 2021*10 h 00 min - 12 h 00 min*

**Emplacement**

Salle Fontannes, Université de Lyon1

**Catégories**