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

Abstract

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.