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

Expressing an observer in preferred coordinates by transforming an injective immersion into a surjective diffeomorphism