Abstract
The previous two parts have shown that the observer design usually consists in transforming the system into a favorable form for which a global observer is known. It follows that the plant’s and observer’s dynamics are not expressed in the same coordinates and may not have the same dimensions: in order to obtain an estimate of the plant’s state, it is therefore necessary to inverse the transformation. When no explicit expression for a global inverse is available, numerical inversion usually relies on the resolution of a minimization problem with a heavy computational cost. Other methods rely on gradient/Newton algorithms which provide only local convergence. In Part III, a recently developed method is presented, whose goal is to bring the observer dynamics (expressed in the normal form coordinates) back into the initial plant’s coordinates. In the case where the transformation is a stationary injective immersion, a first sufficient condition is given, namely that the transformation can be extended into a surjective diffeomorphism.
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Notes
- 1.
The saturation function is defined by \(\mathsf {sat}_M(s)\;=\; \min \left\{ M,\max \left\{ s,-M\right\} \right\} \).
- 2.
For instance, we can take \( {\mathscr {T}}(\xi ) =\left( \xi _1\, ,\, \xi _2\, ,\, -\frac{\xi _1\xi _3+\xi _4\xi _2}{\max \{\xi _1^2+\xi _2^2,\frac{1}{r^2}\}} \right) . \)
- 3.
Indeed, consider any x in \(\mathscr {S}\) and \(\mathscr {V}\) an open neighborhood of x such that \(\mathtt {cl}(\mathscr {V})\) is contained in \(\mathscr {S}\). According to the Lipschitz injectivity of \({T}\) on \(\mathtt {cl}(\mathscr {V})\), there exists \(\mathfrak a\) such that for all v in \(\mathbb {R}^3\) and for all h in \(\mathbb {R}\) such that \(x+hv\) is in \(\mathscr {V}\), \( |v|\le \mathfrak a\frac{|{T}(x+hv)-{T}(x)|}{|h|} \) and thus by taking h to zero, \( |v|\le \mathfrak a\left| \frac{\partial {T}}{\partial x}(x)v\right| \) which means that \(\displaystyle \frac{\partial {T}}{\partial x}(x)\) is full-rank.
- 4.
We will also refer to the x-coordinates as the “given coordinates” because they are chosen by the user to describe the model dynamics.
- 5.
- 6.
See Definition 1.1.
- 7.
We abusively denote in same way \(t\mapsto x(t)\) and \(s\mapsto x(\tau ^{-1}(s))\), \(t\mapsto y(t)\) and \(s\mapsto y(\tau ^{-1}(s))\), \(\dot{x}\) and \(\frac{d x}{ds}\) although those are in different time frames.
- 8.
For instance, choosing a center C inside the parabola, define the function \(\alpha \) such that for any \(\xi \in \mathbb {R}^2\) outside \(\mathscr {P}\), \(\alpha (\xi )\) is the unique positive number such that \(C+\alpha (\xi )(\xi -C)\in \mathscr {P}\). Then, take \(\overline{T}_{23}(x_2) = C+\alpha (T_{23}(x_2))(T_{23}(x_2) -C)\).
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Bernard, P. (2019). Motivation and Problem Statement. 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_8
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