Section: New Results
Observability and observer design for nonlinear systems
Participants : JeanPierre Barbot, Wilfrid Perruquetti, Lotfi Belkoura, Thierry Floquet, Gang Zheng.
Observability analysis and observer design are important issues in the field of control theory. Some recent results are listed below:

[32] investigates the observability and observer design for a class of single output switched systems with high frequency switching, where classical observers cannot be applied directly since the high frequency switching signals are not derivable. By assuming that these signals are integrable in the less restrictive way, and defining a new output, this study shows that algebraic observer can be adopted to estimate the states of the studied switched systems. Although the main idea is explained via normal forms, it can be easily extended to treat generic switched systems with high frequency switching.

Observability of a class of switched systems with Zeno phenomenon or high switching frequency is treated in [31] . Particularly, three observability forms are proposed and the observability for each form with knowledge of filtered switching signal is analyzed. Meanwhile, sufficient and necessary conditions for the existence of a diffeomorphism to transform a class of switched systems into one of such forms are presented.

A triangular canonical form for a class of 0flat nonlinear systems is studied in [13] . Necessary and sufficient geometrical conditions are given in order to guarantee the existence of a local diffeomorphism to transform the studied nonlinear systems into the proposed 0flat canonical form, which enables us to compute the flat output as well.

A fault tolerant control for induction motors based on backstepping strategy is designed in [44] . The proposed approach permits to compensate both the rotor resistance variations and the load torque disturbance. Moreover, to avoid the use of speed and flux sensors, a second order sliding mode observer is used to estimate the flux and the speed. The designed observer converges in a finite time and gives a good estimate of flux and speed even in the presence of rotor resistance variations and load torque disturbance.

[42] studies the observability problem of a general class of singular linear systems with unknown inputs. It is shown that, under some assumptions, the problem is equivalent to study the observability of a standard linear system with unknown inputs satisfying algebraic constraints. We obtain necessary and sufficient conditions for observability in terms of the zeros of the system matrix.

[53] is concerned with the study of observability properties of systems without inputs via homogeneous approximations. This approximation is induced by a filtration on the space of observation. A corresponding filtration on a Lie algebra of vector fields is defined and allows to construct the approximation that preserve observability properties. An explicit construction is given in [53] .