Oscillations in feedback systems subjected to periodic interference, and stability of a self-optimising control system

A theory of stability for nonlinear feedback systems interfered with by an external periodic signal is developed. It makes use of a conventional dual-input describing function which may be available in an analytic or graphical form. The interference may take a variety of forms: the external signal may be injected into the loop, modulate the loop gain or vary any other parameter of the system, or it may interfere in a combination of the above ways. The nonlinearity may be analytic or discontinuous but it must be independent of frequency. A simple servomechanism with backlash and an injected sinusoidal signal is studied, and oscillations at frequencies subharmonic to that of the signal are examined. The stability of a self-optimising control system with sinusoidal perturbation of one parameter is examined. Symmetrical and asymmetrical criterion curves are considered. Second- and third-subharmonic oscillations are found possible: in some circumstances they are stable and self-starting; in others they are unstable, in which case external excitation may cause their onset. The systems were simulated, and experimental results were found to agree well with the theory.