ar X iv : m at h / 06 12 19 3 v 3 [ m at h . O C ] 5 A pr 2 00 8 1 Symmetry - preserving observers

This paper presents three non-linear observers for three examples of engineering interest: a nonholonomic car, a chemical reactor, 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 of a copy of the system equation and a correction term, and we propose a constructive method (based on the Cartan moving-frame method) to find all the symmetry-preserving correction terms. The construction relies 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 on the use of invariant state-errors, a symmetry-preserving way to define the estimation error. Index Terms Nonlinear observer, invariants, symmetry, moving frame, inertial navigation, chemical reactor.