Zames-Falb Multipliers: don't panic

Zames-Falb multipliers are mathematical constructs which can be used to prove stability of so-called Lur'e systems: systems that consist of a feedback interconnection of a linear element and a static nonlinear element. The main advantage of Zames-Falb multipliers is that they enable "passivity"-like results to be obtained but with a level of conservatism much lower than \emph{pure} passivity results. However, some of the papers describing the development of the Zames-Falb multiplier machinery are somewhat abstruse and not entirely clear. This article attempts to provide a relatively simple construction of Zames and Falb's main results which will hopefully be understandable to most graduate-level control engineers.