## Section: Scientific Foundations

### Observers

Participants : Ismaïl Gourragui, Abderrahman Iggidr, Mohamed Mabrouk, Gauthier Sallet, Jean-Claude Vivalda.

Consider a real system modeled by the differential system:

where the observation function hrepresents the set of measures made on the physical system. An observer is an auxiliary dynamical system:

which provides at any time tan estimation of the real state x( t) . More specifically, we have:

If every parameter of system ( 1 ) is known with enough precision and if it is possible to design an observer, a differential equation solver can give an estimation of the state of system ( 1 ).

The team investigates the theory of observability and observers for finite dimensional systems. More specifically, a current subject of interest is the design of observers for some mechanical systems (Ph.D. thesis of M. Mabrouk) or for the switched reluctance motor (Ph.D. thesis of I. Gourragui).

Another subject is the design of observers for biological systems, more specifically for systems which model the evolution of fishes populations submitted to a fishing effort. In this case, our aim is to stabilize the size of the population around an equilibrium point but, since it is not
possible to measure all the state variables, it is necessary to design an observer which gets an estimation of the sizes of the different age classes. Before designing an observer, it is necessary to investigate the observability property. One definition of this last concept is as follows:
system (
1 ) is said
*observable* if, given two initial conditions
, there exists an input
usuch that the solutions
x(
t) and
starting from
x_{0} and
satisfy the inequality
for all
tin a set of nonzero measure. If we are concerned with the construction of observers that converge with an arbitrary speed, the observability condition is necessary. From a theoretical point of view, it is then important to know ``how many'' systems are observable,
that's why we study the problem of the genericity of the observability for discrete-time