On the Necessity and Sufficiency of the Zames-Falb Multipliers for Bounded Operators

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 fourfold. Firstly, we present a generalised S-procedure lossless theorem that involves a countably infinite number of quadratic forms. Secondly, we identify a class of uncertain systems over which the robust feedback stability is equivalent to the existence of an appropriate Zames-Falb multiplier based on the generalised S-procedure lossless theorem. Thirdly, 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. Lastly, 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.

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