A class of non-linear error-detector characteristics is considered where, for large magnitudes of error, the error signal decreases with increased error. Sinusoidal and random-input analysis of a closed-loop system containing such a non-linearity is shown to necessitate the calculation of a family of describing-function curves to cover operation with a d.c. input signal to the non-linear error detector. The peculiarities of such a non-linear system are discussed, and a simple method using a non-linear circuit to correct the adverse behaviour is described.