Error models for nonlinear observers

We revisit the concept of observation error for nonlinear observers of nonlinear systems. In order to obtain a coordinate free notion of such an error we define it using fiber bundles over the system manifold. The new notion ties in nicely with Brockett’s and Willems’ classical description of nonlinear systems as bundles as well as with the related description of observers as bundles due to van der Schaft. It identifies the nonlinear observer design problem as the problem of choosing a nonlinear connection form (horizontal distribution plus affine offset) in this bundle with certain desirable properties. We demonstrate how a particular solution to this connection design problem is given by the invariant observers of Bonnabel, Martin and Rouchon. In the case of systems on Lie groups we recover the recent results by Lageman and two of the authors.