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Asymptotic Hyperstability and Input–Output Energy Positivity of a Single-Input Single-Output System Which Incorporates a Memoryless Non-Linear Device in the Feed-Forward Loop

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  • Manuel De la Sen

    (Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country (UPV/EHU), 48940 Leioa, Bizkaia, Spain)

Abstract

This paper visualizes the role of hyperstable controllers in the closed-loop asymptotic stability of a single-input single-output system subject to any nonlinear and eventually time-varying controller within the hyperstable class. The feed-forward controlled loop (or controlled plant) contains a strongly strictly positive real transfer function in parallel with a non-linear and memory-free device. The properties of positivity and boundedness of the input–output energy are examined based on the “ad hoc” use of the Rayleigh energy theorem on the truncated relevant signals for finite time intervals. The cases of minimal and non-minimal state-space realizations of the linear part are characterized from a global asymptotic stability (asymptotic hyperstability) point of view. Some related extended results are obtained for the case when the linear part is both positive real and externally positive and for the case of incorporation of other linear components which are stable but not necessarily positive real.

Suggested Citation

  • Manuel De la Sen, 2022. "Asymptotic Hyperstability and Input–Output Energy Positivity of a Single-Input Single-Output System Which Incorporates a Memoryless Non-Linear Device in the Feed-Forward Loop," Mathematics, MDPI, vol. 10(12), pages 1-20, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:2051-:d:837899
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    References listed on IDEAS

    as
    1. Liping Chen & Tingting Li & YangQuan Chen & Ranchao Wu & Suoliang Ge, 2019. "Robust passivity and feedback passification of a class of uncertain fractional-order linear systems," International Journal of Systems Science, Taylor & Francis Journals, vol. 50(6), pages 1149-1162, April.
    2. M. De la Sen & A. Ibeas & S. Alonso-Quesada, 2013. "Asymptotic Hyperstability of a Class of Linear Systems under Impulsive Controls Subject to an Integral Popovian Constraint," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-14, October.
    3. Zhongwei Lin & Jizhen Liu & Yuguang Niu, 2013. "Robust Passivity and Feedback Design for Nonlinear Stochastic Systems with Structural Uncertainty," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-9, March.
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