  When designing observers for nonlinear systems, the dynamics of the given
system and of the designed observer are usually not expressed in the same
coordinates or even have states evolving in different spaces. In general, the
function, denoted $\tau$ (or its inverse, denoted $\tau^*$) giving one state in
terms of the other is not explicitly known and this creates implementation
issues. We propose to round this problem by expressing the observer dynamics in
the the same coordinates as the given system. But this may impose to add extra
coordinates, problem that we call augmentation. This may also impose to modify
the domain or the range of the augmented" $\tau$ or $\tau^*$, problem that we
call extension. We show that the augmentation problem can be solved partly by a
continuous completion of a free family of vectors and that the extension
problem can be solved by a function extension making the image of the extended
function the whole space. We also show how augmentation and extension can be
done without modifying the observer dynamics and therefore with maintaining
convergence.Several examples illustrate our results.
