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Around Problem 8.2: Image Extension of a Diffeomorphism

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Observer Design for Nonlinear Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 479))

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Abstract

In this chapter, we study how a diffeomorphism can be extended to make its image cover the whole space, namely to make it surjective. In some cases, the construction of the extension is explicit and is illustrated on examples. In particular, this extension enables to guarantee the completeness of solutions to a high-gain observer written in the initial coordinates for a bioreactor.

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Notes

  1. 1.

    Texts of Chap. 10 are reproduced from [3] with permission from SIAM.

  2. 2.

    If not replace \(\chi \) by \(\frac{\chi }{\sqrt{1+|\chi |^2}}\).

  3. 3.

    It is omitted in this book and can be found in [3].

  4. 4.

    See Definition 5.3.

  5. 5.

    See Definition 5.2.

References

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  2. Bernard, P., Praly, L., Andrieu, V.: Expressing an observer in given coordinates by augmenting and extending an injective immersion to a surjective diffeomorphism (2018). https://hal.archives-ouvertes.fr/hal-01199791v6

  3. Bernard, P., Praly, L., Andrieu, V.: Expressing an observer in preferred coordinates by transforming an injective immersion into a surjective diffeomorphism. SIAM J. Control Optim. 56(3), 2327–2352 (2018)

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  4. Gauthier, J.P., Hammouri, H., Othman, S.: A simple observer for nonlinear systems application to bioreactors. IEEE Trans. Autom. Control 37(6), 875–880 (1992)

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Authors and Affiliations

  1. Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi”, University of Bologna, Bologna, Italy

    Pauline Bernard

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  1. Pauline Bernard
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Correspondence to Pauline Bernard .

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Bernard, P. (2019). Around Problem 8.2: Image Extension of a Diffeomorphism. In: Observer Design for Nonlinear Systems. Lecture Notes in Control and Information Sciences, vol 479. Springer, Cham. https://doi.org/10.1007/978-3-030-11146-5_10

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