Abstract
This chapter addresses the question of the extension of an injective immersion into a diffeomorphism. This is done by complementing continuously the full-rank rectangular Jacobian of the injective immersion into an invertible square matrix. Indeed, when this is possible, an explicit formula for the diffeomorphism is given. Several sufficient conditions for a Jacobian complementation are given with either explicit formulas or constructive algorithms and are illustrated in the example of an oscillator with unknown frequency.
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- 1.
Texts of Chap. 9 are reproduced from [3] with permission from SIAM.
- 2.
For a positive real number \(\varepsilon \) and \(z_0\) in \(\mathbb {R}^p\), \(B_\varepsilon (z_0)\) is the open ball centered at \(z_0\) and with radius \(\varepsilon \).
- 3.
\(F:\mathbb {R}^{d_\xi }\rightarrow \mathbb {R}^n\) with \({d_\xi }\ge n\) is a submersion on \(\mathscr {V}\) if \(\frac{\partial F}{\partial \xi }(\xi )\) is full-rank for all \(\xi \) in \(\mathscr {V}\).
References
Andrieu, V., Eytard, J.B., Praly, L.: Dynamic extension without inversion for observers. In: IEEE Conference on Decision and Control, pp. 878–883 (2014)
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
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)
Bott, R., Milnor, J.: On the parallelizability of the spheres. Bull. Am. Math. Soc. 64(3), 87–89 (1958)
Dugundgi, J.: Topology. Allyn and Bacon, Boston (1966)
Eckmann, B.: Mathematical Survey Lectures 1943–2004. Springer, Berlin (2006)
Wazewski, T.: Sur les matrices dont les éléments sont des fonctions continues. Compos. Math. 2, 63–68 (1935)
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Bernard, P. (2019). Around Problem 8.1: Augmenting an Injective Immersion into 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_9
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DOI: https://doi.org/10.1007/978-3-030-11146-5_9
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