On Invariant Observers

A definition of invariant observer and compatible output function is proposed and motivated. For systems admitting a Lie symmetry-group G of dimension less or equal to the state dimension and with a G-compatible output, an explicit procedure based on the moving frame method is proposed to construct such invariant observers. It relies on an invariant frame and a complete set of invariant estimation errors. Two examples of engineering interest are considered: an exothermic chemical reactor and an inertial navigation problem. For both examples we show how invariance and the proposed construction can be a useful guide to design non-linear convergent observers, although the part of the design procedure which achieves asymptotic stability is not systematic and must take into account the specific nonlinearities of the case under study.