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Adaptive Attenuation of Multiple Sparse Unknown and Time-Varying Narrow-Band Disturbances

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Book cover Adaptive and Robust Active Vibration Control

Part of the book series: Advances in Industrial Control ((AIC))

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Abstract

This chapter considers the possible solutions for adaptive attenuation of multiple sparse unknown and time-varying narrow-band disturbances. One takes also into account the possible presence of low damped complex zeros in the vicinity of the attenuation region. The problem of the design of the underlined linear controller for the known disturbance case is itself a challenging problem and is discussed first. The adaptive schemes proposed are obtained by extending the linear solutions to the case of unknown characteristics of the disturbances. Comparative experimental evaluation of the various solutions on a test bench are given.

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Notes

  1. 1.

    The complex variable \(z^{-1}\) is used to characterize the system’s behaviour in the frequency domain and the delay operator \(q^{-1}\) will be used for the time domain analysis.

  2. 2.

    As indicated earlier, it is assumed that a reliable model identification is achieved and therefore the estimated model is assumed to be equal to the true model.

  3. 3.

    For frequencies bellow \(0.17\,{f_S}\) (\({f_S}\) is the sampling frequency) the design can be done with a very good precision directly in discrete time [5].

  4. 4.

    Its structure in a mirror symmetric form guarantees that the roots are always on the unit circle.

  5. 5.

    The argument \((q^{-1})\) will be dropped in some of the following equations.

  6. 6.

    For the development of the equation for the adaptation error one assumes that the estimated parameters have constant values which allows to use the commutativity property of the various operators.

  7. 7.

    The details of the developments leading to this equation are given in the Appendix C.

  8. 8.

    Neglecting the non-commutativity of the time-varying operators.

  9. 9.

    The disturbance passes through a so called “primary path” which is not represented in Fig. 13.5.

  10. 10.

    The chirp will be considered only for complexity evaluation, for other results concerning chirp disturbance see [33, 34].

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Authors and Affiliations

  1. Department of Automatic Control, GIPSA-LAB (Grenoble INP/CNRS/UJF/Stendhal), 11 rue des Mathématiques, Grenoble Campus, BP46, Saint Martin d’heres , F-38402, France

    Ioan Doré Landau

  2. IMS-lab (University of Bordeaux, Bordeaux INP, CNRS) UMR 5218, Talence , F-33405, France

    Tudor-Bogdan Airimitoaie

  3. Grenoble, France

    Abraham Castellanos-Silva

  4. Saint-Lazare, QC, Canada

    Aurelian Constantinescu

Authors
  1. Ioan Doré Landau
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  2. Tudor-Bogdan Airimitoaie
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  3. Abraham Castellanos-Silva
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  4. Aurelian Constantinescu
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Correspondence to Ioan Doré Landau .

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Landau, I.D., Airimitoaie, TB., Castellanos-Silva, A., Constantinescu, A. (2017). Adaptive Attenuation of Multiple Sparse Unknown and Time-Varying Narrow-Band Disturbances. In: Adaptive and Robust Active Vibration Control. Advances in Industrial Control. Springer, Cham. https://doi.org/10.1007/978-3-319-41450-8_13

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  • DOI: https://doi.org/10.1007/978-3-319-41450-8_13

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  • Online ISBN: 978-3-319-41450-8

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