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

TL;DR

This paper explores the conditions under which Zames-Falb multipliers are necessary and sufficient for ensuring the robust feedback stability of LTI systems against various classes of nonlinearities, extending classical results.

Contribution

It introduces a generalized S-procedure lossless theorem and clarifies the necessity and sufficiency of Zames-Falb multipliers for different classes of uncertain systems.

Findings

Zames-Falb multipliers are necessary and sufficient for certain classes of nonlinearities.

A generalized S-procedure lossless theorem involving infinite quadratic forms is developed.

The classical stability results are recovered for static monotone nonlinearities.

Abstract

This paper analyzes the robust feedback stability of a single-input-single-output stable linear time-invariant (LTI) system against four 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 implies the existence of an appropriate Zames-Falb multiplier based on the generalised S-procedure lossless theorem. Meanwhile, we show that the existence of such a Zames-Falb multiplier is sufficient for the robust feedback stability over a smaller class of uncertain systems. Thirdly, when restricted to be static (a.k.a. memoryless), the second 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 second class is demonstrated to be a subset of the third, and the existence of a Zames-Falb multiplier is shown to be sufficient but not necessary for the robust feedback stability.