Zames-Falb multipliers for absolute stability : from O'Shea's contribution to convex searches

Absolute stability attracted much attention in the 1960s. Several stability conditions for loops with slope-restricted nonlinearities were developed. Results such as the Circle Criterion and the Popov Criterion form part of the core curriculum for students of control. Moreover, the equivalence of results obtained by different techniques, specifically Lyapunov and Popov׳s stability theories, led to one of the most important results in control engineering: the KYP Lemma. For Lurye1 systems this work culminated in the class of multipliers proposed by O׳Shea in 1966 and formalized by Zames and Falb in 1968. The superiority of this class was quickly and widely accepted. Nevertheless the result was ahead of its time as graphical techniques were preferred in the absence of readily available computer optimization. Its first systematic use as a stability criterion came 20 years after the initial proposal of the class. A further 20 years have been required to develop a proper understanding of the different techniques that can be used. In this long gestation some significant knowledge has been overlooked or forgotten. Most significantly, O׳Shea׳s contribution and insight is no longer acknowledged; his papers are barely cited despite his original parameterization of the class. This tutorial paper aims to provide a clear and comprehensive introduction to the topic from a user׳s viewpoint. We review the main results: the stability theory, the properties of the multipliers (including their phase properties, phase-equivalence results and the issues associated with causality), and convex searches. For clarity of exposition we restrict our attention to continuous time multipliers for single-input single-output results. Nevertheless we include several recent significant developments by the authors and others. We illustrate all these topics using an example proposed by O׳Shea himself.