A dither is a high frequency signal introduced into a non-linear system in order to improve its performance. It is shown that the stability of a dithered system can be predicted rigorously by establishing the stability of its corresponding smoothed system when the dither has a sufficiently large amplitude as well as high–enough frequency, and the transfer function of the ‘linearized’ feedback system has a high-frequency attenuation property. A generally parametrized controller, advanced by Youlaet al., and a dither as its auxiliary are simultaneously introduced to augment the stability of the non–linear feedback system, where the controlled plant are not necessarily stable. The main feature of this study is that the technique of robustness optimization is used to find a lower bound on the dither amplitude for stabilization of the non–linear feedback system. Furthermore, there arc no stable, square, and minimum-phase constraints on the controlled plant.