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Stability and Control of Fractional Periodic Time-Delayed Systems

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Time Delay Systems

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 7))

Abstract

In this chapter, two new methods are proposed to study the stability of linear fractional periodic time-delayed (FPTD) systems. First, the explicit harmonic balance (EHB) method is proposed to find necessary and sufficient conditions for fold, flip, and secondary Hopf transition curves in linear FPTD systems, from which the stability boundaries are obtained as a subset. Transition curves of the fractional damped delayed Mathieu equation are obtained by using the EHB method. Next, an approximated monodromy operator in a Banach space is defined for FPTD systems, which gives the linear map between two solutions. The fractional Chebyshev collocation (FCC) method is proposed to approximate this monodromy operator. The FCC method is outlined and illustrated with three practical problems including obtaining the parametric stability charts of the fractional Hayes equation and the fractional second-order system with delay, and designing an optimal linear periodic gain fractional delayed state feedback control for the damped delayed Mathieu equation.

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Authors and Affiliations

  1. Aerospace and Mechanical Engineering Department, University of Arizona, Tucson, AZ, 85721, USA

    Eric A. Butcher, Arman Dabiri & Morad Nazari

Authors
  1. Eric A. Butcher
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  2. Arman Dabiri
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  3. Morad Nazari
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Corresponding author

Correspondence to Eric A. Butcher .

Editor information

Editors and Affiliations

  1. Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary

    Tamás Insperger

  2. Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, USA

    Tulga Ersal

  3. Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, USA

    Gábor Orosz

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Butcher, E.A., Dabiri, A., Nazari, M. (2017). Stability and Control of Fractional Periodic Time-Delayed Systems. In: Insperger, T., Ersal, T., Orosz, G. (eds) Time Delay Systems. Advances in Delays and Dynamics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-53426-8_8

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  • DOI: https://doi.org/10.1007/978-3-319-53426-8_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53425-1

  • Online ISBN: 978-3-319-53426-8

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