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Bi-homogeneous Differentiators

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Sliding-Mode Control and Variable-Structure Systems

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 490))

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Abstract

In many applications, it is important to be able to estimate online some number of derivatives of a given (differentiable) signal. Some famous algorithms solving the problem comprise linear high-gain observers and Levant’s exact differentiators, that is discontinuous. They are both homogeneous, as are many other ones. A disadvantage of continuous algorithms is that they are able to calculate exactly the derivatives only for a very small class of (polynomial) time signals. The discontinuous Levant’s differentiator, in contrast, can calculate in finite-time and exactly the derivatives of Lipschitz signals, which is a much larger class. However, it has the drawback that its convergence time increases very strongly with the size of the initial conditions. Thus, a combination of both algorithms seems advantageous, and this has been proposed recently by the author in [38]. In this work, some techniques used to design differentiators are reviewed and it is shown how the combination of two different homogeneous algorithms can be realized and that it leads to interesting properties. A novelty is the derivation of a very simple realization of the family of bi-homogeneous differentiators proposed in [38]. The methodological framework is based on the use of smooth Lyapunov functions to carry out their performance and convergence analysis.

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Notes

  1. 1.

    Note that these relations do not imply that \(\varphi _{i}\) and \(\phi _{i}\) are homogeneous of degree one, because this equality is not valid for arbitrary (positive) constants multiplying s but only for the ratio \(\frac{\alpha }{L^{n}}\), which is used in the definition of \(\varphi _{i}\). Functions \(\varphi _{i}\) and \(\phi _{i}\) are bi-homogeneous.

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Acknowledgements

The author would like to thank the financial support from PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica), project IN106323.

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  1. Instituto de Ingeniería, Universidad Nacional Autónoma de México (UNAM), 04510, Coyoacán, Ciudad de México, Mexico

    Jaime A. Moreno

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Correspondence to Jaime A. Moreno .

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  1. Department of Electronics and Telecommunication Engineering, State University of Rio de Janeiro (UERJ), Rio de Janeiro, Brazil

    Tiago Roux Oliveira

  2. Department of Control Engineering and Robotics, National Autonomous University of Mexico (UNAM), Mexico City, Mexico

    Leonid Fridman

  3. Department of Electrical Engineering, Federal University of Rio de Janeiro (COPPE/UFRJ), Rio de Janeiro, Brazil

    Liu Hsu

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Moreno, J.A. (2023). Bi-homogeneous Differentiators. In: Oliveira, T.R., Fridman, L., Hsu, L. (eds) Sliding-Mode Control and Variable-Structure Systems. Studies in Systems, Decision and Control, vol 490. Springer, Cham. https://doi.org/10.1007/978-3-031-37089-2_4

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