Zames-Falb multipliers for absolute stability: From O׳Shea׳s contribution to convex searches

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.