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Around Problem 8.2: Image Extension of a Diffeomorphism

  • Pauline BernardEmail author
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 479)

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.

References

  1. 1.
    Astolfi, D., Praly, L.: Output feedback stabilization for SISO nonlinear systems with an observer in the original coordinate. In: IEEE Conference on Decision and Control, pp. 5927–5932 (2013)Google Scholar
  2. 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. 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)MathSciNetCrossRefGoogle Scholar
  4. 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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hirsch, M.: Differential Topology. Springer, Berlin (1976)CrossRefGoogle Scholar
  6. 6.
    Maggiore, M., Passino, K.: A separation principle for a class of non uniformly completely observable systems. IEEE Trans. Autom. Control 48 (2003)Google Scholar
  7. 7.
    Milnor, J.: Lectures on the \(h\)-Cobordism Theorem. Notes by L. Siebenmann and J. Sondow. Princeton University Press, Princeton (1965)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

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