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

TL;DR

This paper addresses the challenge of expressing observer dynamics in the same coordinates as the nonlinear system by proposing methods for augmentation and extension of coordinate transformations, ensuring implementation feasibility while preserving convergence.

Contribution

It introduces novel techniques for augmentation and extension of coordinate transformations to unify observer and system coordinates without altering observer dynamics.

Findings

Augmentation can be achieved via a continuous completion of vector families.

Extension can be done through function extension to cover the entire space.

Methods preserve observer convergence while aligning coordinates.

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.