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On the necessity and sufficiency of the Zames–Falb multipliers for bounded operators

journal contribution
posted on 03.09.2021, 11:52 by Sei Zhen Khong, Lanlan Su
This paper analyzes the robust feedback stability of a single-input-single-output stable linear time-invariant (LTI) system against three different classes of nonlinear systems using the Zames–Falb multipliers. The contribution is threefold. Firstly, we identify a class of uncertain systems over which the robust feedback stability is equivalent to the existence of an appropriate Zames–Falb multiplier. Secondly, when restricted to be static (a.k.a. memoryless), such a class of systems coincides with the class of sloped-restricted monotone nonlinearities, and the classical result of using the Zames–Falb multipliers to ensure feedback stability is recovered. Thirdly, when restricted to be LTI, the first class is demonstrated to be a subset of the second, and the existence of a Zames–Falb multiplier is shown to be sufficient but not necessary for the robust feedback stability.

History

Citation

Automatica, Volume 131, September 2021, 109787

Author affiliation

School of Engineering

Version

AM (Accepted Manuscript)

Published in

Automatica

Volume

131

Pagination

109787

Publisher

Elsevier BV

issn

0005-1098

Acceptance date

12/05/2021

Copyright date

2021

Available date

25/06/2022

Language

en