Unlike for linear systems, no systematic method exists for the design of observers for nonlinear systems. However, observer design may be more or less straightforward depending on the coordinates we choose to express the system dynamics. In particular, some specific structures, called normal forms, have been identified for allowing a direct and easier observer construction. It follows that a common way of addressing the problem consists in looking for a reversible change of coordinates transforming the expression of the system dynamics into one of those normal forms, design an observer in those coordinates, and finally deduce an estimate of the system state in the initial coordinates via inversion of the transformation. This talk gives contributions to each of those three steps.

First, we show the interest of a new triangular normal form with continuous (non-Lipschitz) nonlinearities. Indeed, we have noticed that systems which are observable for any input but with an order of differential observability larger than the system dimension, may not be transformable into the standard Lipschitz triangular form, but rather into an "only continuous" triangular form. In this case, the famous high gain observer no longer is sufficient, and we propose to use homogeneous observers instead.

Another canonical form of interest is the Hurwitz linear form which admits a trivial observer. The question of transforming a nonlinear system into such a form has only been addressed for autonomous systems with the so-called Lunberger or Kazantzis-Kravaris observers. This design consists in solving a PDE and we show here how it can be extended to time-varying/controlled systems.

As for the inversion of the transformation, this step is far from trivial in practice, in particular when the domain and image spaces have different dimensions. When no explicit expression for a global inverse is available, numerical inversion usually relies on the resolution of a minimization problem with a heavy computational cost. That is why we have developed a method to avoid the explicit inversion of the transformation by bringing the observer dynamics (expressed in the canonical form coordinates) back into the initial system coordinates. This is done by dynamic extension, i.e. by adding some new coordinates to the system and transforming an injective immersion into a surjective diffeomorphism.

Bio. Pauline Bernard graduated from MINES ParisTech in 2014 with a Master degree in Applied Mathematics and Automatic Control. In 2017, she obtained her Ph.D. in Mathematics and Automatic Control at PSL Reserch University, prepared at the Systems and Control Center, MINES ParisTech under the supervision of Laurent Praly and Vincent Andrieu.

### Séminaire d'Automatique du Plateau de Saclay : Observer design for nonlinear systems

Séminaire le 30 Novembre 2017, 10h00 à CentraleSupelec (Gif-sur-Yvette) Salle du conseil du L2S - B4.40

**Pauline Bernard (PhD, PSL Reserch University, Systems and Control Center, MINES ParisTech)**