On invariant asymptotic observers

For dynamics ẋ = f(x) with output y = h(x) invariant with respect to a transformation group G, we define invariant asymptotic observer of the form ̇̂ x = f̂(x̂, y) where y = h(x) is the measured output and x̂ an estimation of the unmeasured state x. Such a definition is motivated by a class of chemical reactors, treated in details, when the group of transformations corresponds to unit changes and the output y to ratio of concentrations. We propose a constructive method that guaranties automatically the observer invariance ̇̂ x = f̂(x̂, y): it is based on invariant vector fields and scalar functions, called invariant estimation errors, that can be computed via Darboux-Cartan moving frame methods. The observer convergence remains, in the general case, an open problem. But for the class of chemical reactors considered here, the invariant observer convergence is proved by showing that, in a Killing metric associated to the action of G, the symmetric part of the Jacobian matrix ∂f̂/∂x̂ is definite negative (contraction).