The injection of high-frequency signals, commonly called dithers : into a nonlinear large-scale system may improve its performance. Stability of the dithered large-scale system is related to that of its corresponding model-the smoothed large-scale system. The dithers' amplitudes, but not their frequencies, affect the sectors of nonlinearities. The importance of dithers' frequencies is found in their effect on the deviation of the smoothed large-scale system from the dithered large-scale system, and the deviation can be improved as the frequencies of dithers increase. Dithers of sufficiently high frequencies may result in outputs of the smoothed large-scale system and of the dithered large-scale system as close as desired. This fact enables a rigorous prediction of stability of the dithered large-scale system by establishing that of its corresponding smoothed large-scale system. The main characteristic of this work is that an algorithm is proposed to find the lower bound of each dither's amplitude for stabilizing the nonlinear large-scale system.