  This paper presents three non-linear observers on three examples of
engineering interest: a chemical reactor, a non-holonomic car, and an inertial
navigation system. For each example, the design is based on physical
symmetries. This motivates the theoretical development of invariant observers,
i.e, symmetry-preserving observers. We consider an observer to consist in a
copy of the system equation and a correction term, and we give a constructive
method (based on the Cartan moving-frame method) to find all the
symmetry-preserving correction terms. They rely on an invariant frame (a
classical notion) and on an invariant output-error, a less standard notion
precisely defined here. For each example, the convergence analysis relies also
on symmetries consideration with a key use of invariant state-errors. For the
non-holonomic car and the inertial navigation system, the invariant
state-errors are shown to obey an autonomous differential equation independent
of the system trajectory. This allows us to prove convergence, with almost
global stability for the non-holonomic car and with semi-global stability for
the inertial navigation system. Simulations including noise and bias show the
practical interest of such invariant asymptotic observers for the inertial
navigation system.
