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Observer Design for Nonlinear Systems pp 53–73Cite as

Transformations into State-Affine Normal Forms

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

Abstract

In this chapter, sufficient conditions on a nonlinear system are given to ensure the existence of a transformation into one of the state-affine normal forms. This includes the linearization by output injection and the nonlinear Luenberger design. The former consists in transforming the system into linear dynamics (possibly depending on the input/output), and such that the output is a linear function of the new state. On the other hand, the latter aims at transforming the system into linear dynamics with a stationary Hurwitz linear part but where the output can be any nonlinear function of the new state. In particular, it is shown that under a rather weak backward distinguishability property, any nonlinear system can be transformed into a Hurwitz linear form in an injective way, but through a time-varying transformation.

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Notes

  1. 1.

    \(T:\mathbb {R}^{d_x}\rightarrow \mathbb {R}^{d_\xi }\) is an immersion if the rank of \(\frac{\partial T}{\partial x}\) is \({d_x}\). Contrary to a diffeomorphism, this allows to take \({d_\xi }\ge {d_x}\).

  2. 2.

    In [19], the transformation is called an immersion, but as defined in [19, Definition 2.2], and not in the usual differential geometry sense as defined in this book (see previous footnote).

  3. 3.

    See Definition 1.2.

  4. 4.

    Dynamics with u equal to a constant.

  5. 5.

    \(\xi =T \, x\) follows the dynamics (6.3) if and only if (6.4).

  6. 6.

    Any solution exiting \(\mathscr {X}\) in finite time must cross the boundary of \(\mathscr {X}\). See [2, Definition 1].

  7. 7.

    Separating the real/imaginary parts, the observer is thus of dimension \(2({d_x}+1){d_y}\) on \(\mathbb {R}\). See Remark 6.1.

  8. 8.

    This notion is similar to the distinguishability defined in Definition 1.2 but in negative time and with the constraint that \(\overline{t}\) occurs when both solutions are still in \(\mathscr {S}\).

  9. 9.

    See Definition 5.2 in the autonomous case.

  10. 10.

    Some texts of Sect. 6.2.2 are reproduced from [5] with permission from IEEE.

  11. 11.

    We could have considered a more general Hurwitz form \(\dot{\xi } = A \, \xi + B(y)\) with B any nonlinear function, but taking B linear is sufficient to obtain satisfactory results.

  12. 12.

    The function T depends on u in \(\mathscr {U}\), and we should write \(T_u\) as in Theorem 1.1. But we drop this too heavy notation in this chapter to ease the comprehension. What is important is that the target Hurwitz form, namely \({d_\xi }\), A, and B, is the same for all u in \(\mathscr {U}\).

  13. 13.

    This property is named completeness within \(\mathscr {X}'\) in [2].

  14. 14.

    \(H(\cdot ,\overline{\nu }_m)\) is injective on \(\mathscr {S}\), and \(\frac{\partial H}{\partial x}(x,\overline{\nu }_m)\) is full-rank for any x in \(\mathscr {S}\).

  15. 15.

    This set depends on u, and unfortunately, there is no guarantee that \(\displaystyle \bigcup _{u\in \mathscr {U}}\mathscr {R}_u\) is also of zero Lebesgue measure.

  16. 16.

    Uniformly instantaneously observable, see Definition 1.2.

  17. 17.

    The conditions given in [1] correspond to those obtained from Theorem 6.4. A weaker condition is given by Theorem 6.5.

  18. 18.

    In the case where \(A(\theta )\) is in companion form, an explicit inversion algorithm is available in [1].

  19. 19.

    \(\eta = \varPhi \omega \left( \begin{array}{c} -\sin \theta \\ \cos \theta \end{array}\right) \), with \(\theta \) the electrical angle, and \(\omega =\dot{\theta }\).

  20. 20.

    P is symmetric so it is sufficient to compute \(\frac{{d_\xi }({d_\xi }+1)}{2}\) components.

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

    Pauline Bernard

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Bernard, P. (2019). Transformations into State-Affine Normal Forms. 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_6

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