Phase Properties of the Generalised Zames- Falb Multipliers

Abstract

The Zames- Falb multipliers are a classical tool in the analysis of Lure systems. They are the widest known class of multipliers (up to phase equivalence) that preserve the positivity of memoryless monotone and bounded nonlinearities. They can be used to prove that the Kalman Conjecture is true for third order systems. This paper brings together two separate and recent developments in the area. On the one hand it is possible to derive generalised multipliers applicable to nonlinearities that need be neither memoryless nor monotone, but that can be bounded by input-output maps with such properties. On the other, it is possible to derive analytic constraints on the phase properties of Zames- Falb multipliers and to interpret these in terms of the Kalman Conjecture. We derive and discuss such analytic phase restrictions for the generalised Zames- Falb multipliers. We discuss the implications for nonlinearities with partial symmetry and for Lure systems with persistent disturbances.