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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-01199791
Contributor : Pauline Bernard <>
Submitted on : Monday, February 19, 2018 - 6:17:53 PM
Last modification on : Wednesday, October 14, 2020 - 4:10:22 AM
Long-term archiving on: : Sunday, May 6, 2018 - 4:38:07 AM

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### Citation

Pauline Bernard, Vincent Andrieu, Laurent Praly. Expressing an observer in preferred coordinates by transforming an injective immersion into a surjective diffeomorphism. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2018, 56 (3), pp.2327-2352. ⟨10.1137/15M1037755⟩. ⟨hal-01199791v6⟩

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