Closed-loop frequency response of nonlinear systems

WHEN A NONLINEAR feedback control system includes saturation, the output of the system may oscillate with either the input frequency or the l/n order subharmonic of the input frequency, where n is a positive integer greater than l. When the frequency of the output is l/n times that of the input, the output oscillation is called the l/n order pure subharmonic oscillation. The possibility of the occurrence of pure subharmonic oscillations as the response of a nonlinear feedback control system may be predicted from an accurate closed-loop attenuation-frequency curve for a given input amplitude. That is, if the maximum ratio of the system output amplitude to the system input amplitude is large (about 3 db or more), there is a possibility that the output of the system may oscillate with a subharmonic of the input frequency. However, if the maximum ratio of the output amplitude to the input amplitude is small (about 2 db or less), the possibility of the occurrence of pure subharmonic oscillations is very small, if any. Because of the fact that the wave form of the output of a nonlinear element is often extremely nonsinusoidal and it is difficult, in general, to handle nonlinear differential equations, graphical techniques are usually employed for finding the closed-loop frequency response of nonlinear feedback control systems.