Abstract
When the sufficient conditions presented in the previous chapter are not satisfied, it is shown here how the problem may still be solved by changing or choosing in a more cunning way the original transformation itself. Some applicative examples such as an image-based aircraft landing are also considered to see how this whole methodology extends to time-varying transformations.
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Notes
- 1.
- 2.
Actually, \(I\) can be replaced by any \(C^1\) function B the values of which are \({d_\xi }\times {d_\xi }\) matrices with positive definite symmetric part.
- 3.
In [6], it is shown that for any \(r>0\), there exists \(L_r>0\) such that for all \((x_a,x_b)\) in \(\mathbb {R}^2\times (0,r)\), \(|x_{1,a}-x_{1,b}|+|x_{2,a}-x_{2,b}|+\frac{x_{1,a}+x_{1,b}+x_{2,a}+x_{2,b}}{2}|x_{3,a}-x_{3,b}|\le L_r |{T}_{14}(x_a)-{T}_{14}(x_b)|\) where \({T}_{14}\) denotes the first four components of \({T}\). Therefore, \({T}_{14}(x_a)={T}_{14}(x_b)\) implies that \(x_{1,a}=x_{1,b}\) and \(x_{2,a}=x_{2,b}\): Either one of them is nonzero, and in that case, the inequality says that we have also \(x_{3,a}=x_{3,b}\), or they are all zero but then \({T}_{5}(x_a)={T}_{5}(x_b)\) implies that \(x_{3,a}=x_{3,b}\). We conclude that \({T}\) is injective on \(\tilde{\mathscr {S}}\). Now, applying the inequality between x and \(x+hv\) and making h go to zero, we get that \(\frac{\partial {T}_{14}}{\partial x}(x)v=0\) implies that \(v_1=v_2=0\) and \(v_3=0\) if either \(x_1\) or \(x_2\) is nonzero. If they are both zero, \(\frac{\partial {T}_{5}}{\partial x}(x)v=0\) with \(v_1=v_2=0\) gives \(v_3=0\). Thus, \(\frac{\partial {T}}{\partial x}(x)\) is full-rank.
- 4.
Note that whatever the number of chosen lines and points in the image, the model can always be written in this form, only the dimensions of \(x_m\) and the input change.
- 5.
This comes back to choosing one particular input law, but the reader may check that the same design works for any input such that the observability assumption and the saturation by \({\overline{\varPhi }}\) in (11.18) are valid.
- 6.
Take for instance \(\mathscr {T}_0(\xi ,t) = \left( \xi _m,\frac{\varPi (\xi _m)^T(\xi _d-\varSigma (\xi _m,t))}{\max \{\delta (\xi _m,t),\varepsilon \}} \right) \).
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Bernard, P. (2019). Generalizations and Examples. 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_11
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DOI: https://doi.org/10.1007/978-3-030-11146-5_11
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