On invariant asymptotic observers

For dynamics x˙=f(x) with output y=h(x) invariant with respect to a transformation group G, we define invariant asymptotic observer of the form xˆ˙=fˆ(xˆ,y) where y=h(x) is the measured output and xˆan estimation of the unmeasured state x. Such a definition is motivated by a class of chemical reactors and treated in details, when the transformation group corresponds to unit changes and the output V to ratio of concentrations. We propose a constructive method that guarantees automatically the observer invariance xˆ˙=fˆ(xˆ,y): it is based on invariant vector fields and scalar functions, called invariant estimation errors, that can be computed via the Cartan moving frame method. 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 ofˆ/∂xˆis definite negative (contraction).