This paper presents an analysis of a class of two-relay feedback systems, motivated by the idea of estimating a relay-type nonlinearity on the feedback path of a plant by adding another relay. The time response of the resulting system is first analyzed, and stability conditions for possible limit cycles are then proposed using Poincare＇s method. On this basis, a simple procedure for estimating the unknown value (e.g. Coulomb friction) of the relay on a plant feedback, by a closed-loop experiment, is finally presented and illustrated on both simulated and real-time examples.