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Robust State Estimation for Linear Time-Varying Systems Using High-Order Sliding-Modes Observers

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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

This chapter presents two algorithms for state estimation of linear time-varying systems affected by unknown inputs. The chapter is divided into two parts. The first part presents an observer for the class of strongly observable linear time-varying systems with unknown inputs. The proposed observer uses a cascade structure to guarantee the correct state reconstruction despite bounded unknown inputs and system instability. The second part of this chapter presents a finite-time observer that exploits the structural properties of the system through a linear operator. The particular design of this observer allows for avoiding the use of a cascade structure, providing with reduced complexity an exact estimate of the states after a finite transient time, even in the presence of possible instability of the system and the effects of bounded unknown inputs.

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©2022 John Wiley and Sons. Reprinted with permission from Galván-Guerra, R., Fridman, L., and Dávila, J. (2017) High-order sliding-mode observer for linear time-varying systems with unknown inputs. Int. J. Robust. Nonlinear Control, 27: 2338–2356. https://doi.org/10.1002/rnc.3698.

©2022 IEEE. Reprinted with permission from J. Dávila, M. Tranninger and L. Fridman, “Finite-Time State Observer for a Class of Linear Time-Varying Systems With Unknown Inputs,” in IEEE Transactions on Automatic Control, vol. 67, no. 6, pp. 3149–3156, June 2022, https://doi.org/10.1109/TAC.2021.3096863.

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Notes

  1. 1.

    Here we can use any left generalized inverse \(\left[ K(t)\mathcal {\tilde{O}}_{(\tilde{{A}},C),l_{o}}\right] ^{+}\), such that

    $$ \left[ K(t)\mathcal {\tilde{O}}_{(\tilde{{A}},C),l_{o}}\right] ^{+}K(t)\mathcal{{\tilde{O}}}_{(\tilde{{A}},C),l_{o}}=I; $$

    and it can be computed using any methodology (see, e.g., [26], for more details about generalized inverses and its computation).

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Author information

Authors and Affiliations

  1. Instituto Politécnico Nacional, ESIME-UPT, Section of Graduate Studies and Research, Mexico City, 07340, Mexico

    Jorge Dávila

  2. Facultad de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, 04510, Mexico

    Leonid Fridman

  3. School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, 6997801, Israel

    Arie Levant

Authors
  1. Jorge Dávila
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  2. Leonid Fridman
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  3. Arie Levant
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Correspondence to Jorge Dávila .

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Editors and Affiliations

  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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Dávila, J., Fridman, L., Levant, A. (2023). Robust State Estimation for Linear Time-Varying Systems Using High-Order Sliding-Modes Observers. 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_6

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