Abstract
In this chapter, triangular forms are considered, namely systems made of a cascade of integrators, and of nonlinearities depending in a triangular way on the state. In order to observe those systems, it is standard to use a high gain in the correction terms, which is able to compensate for the nonlinearities if taken sufficiently large. The structure–homogeneity–of the correction terms must be adapted to the regularity of the nonlinearities–Lipschitz, Hölder, continuous.
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Notes
- 1.
See Definition 1.2.
- 2.
This property can be relaxed: See Sect. 4.1.5.
- 3.
See Sect. 4.1.5.
- 4.
This may be relaxed: See Sect. 4.1.5.
- 5.
If \(\xi _1\) is a block of dimension \({d_y}\), those notations apply component-wise, namely \(\left\lfloor a\right\rceil ^{b}=(\left\lfloor a_1\right\rceil ^{b},\ldots ,\left\lfloor a_m\right\rceil ^{b})^\top \).
- 6.
See Remark 1.1.
- 7.
See Remark 1.1.
- 8.
The saturation function is defined on \(\mathbb {R}\) by \( \mathsf {sat}_a(x)=\max \{\min \{x, a\},-a\} \ . \)
- 9.
- 10.
By strictly contained, we mean that \({\tilde{\mathscr {M}}}\) is contained in the interior of \(\mathscr {M}\).
- 11.
Let \(\varDelta \varPhi _3(\xi _1,\xi _3,e_1,e_3)= |\xi _3+e_3+\xi _1+e_1|^{\frac{4}{5}}\left\lfloor \xi _1+e_1\right\rceil ^{\frac{1}{5}}- |\xi _3+\xi _1|^{\frac{4}{5}} \left\lfloor \xi _1\right\rceil ^{\frac{1}{5}}=|\xi _3+\xi _1|^{\frac{4}{5}}\left( \left\lfloor \xi _1+e_1\right\rceil ^{\frac{1}{5}}-\left\lfloor \xi _1\right\rceil ^{\frac{1}{5}}\right) + \left\lfloor \xi _1+e_1\right\rceil ^{\frac{1}{5}}\left( |\xi _3+\xi _1+e_3+e_1|^{\frac{4}{5}}-|\xi _3+\xi _1|^{\frac{4}{5}}\right) \). By Lemma A.5, we have \(\left| \left\lfloor \xi _1+e_1\right\rceil ^{\frac{1}{5}}-\left\lfloor \xi _1\right\rceil ^{\frac{1}{5}}\right| \le 2^{\frac{4}{5}}|e_1|^{\frac{1}{5}}\) and \(\left| |\xi _3+\xi _1+e_3+e_1|^{\frac{4}{5}}-|\xi _3+\xi _1|^{\frac{4}{5}}\right| \le 2^{\frac{1}{5}}(|e_3|+|e_1|)^{\frac{4}{5}}\le 2^{\frac{1}{5}}(|e_3|^{\frac{4}{5}}+|e_1|^{\frac{4}{5}})\). Besides, \(|\xi _1+e_1|^{\frac{1}{5}}\le |\xi _1|^{\frac{1}{5}}+|e_1|^{\frac{1}{5}}\), so that for \(\xi _1\) and \(\xi _3\) in compact sets, \(|\varDelta \varPhi _3(\xi _1,\xi _3,e_1,e_3)|\le c_1|e_1|^{\frac{1}{5}}+c_2|e_3|^{\frac{4}{5}}+c_3|e_1|^{\frac{4}{5}}+c_4|e_1|^{\frac{1}{5}}|e_3|^{\frac{4}{5}}+c_5|e_1|\). By Young’s inequality, \(|e_1|^{\frac{1}{5}}|e_3|^{\frac{4}{5}}\le \frac{1}{5}|e_1|+\frac{4}{5}|e_3|\), and finally, for \(e_1\) and \(e_3\) in compact sets, \(|\varDelta \varPhi _3(\xi _1,\xi _3,e_1,e_3)|\le {\tilde{c}}_1|e_1|^{\frac{1}{5}} +{\tilde{c}}_3|e_3|^{\frac{4}{5}}\).
- 12.
See Definition 2.1.
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Bernard, P. (2019). Triangular 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_4
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