Justification of the Describing Function Method

Explicit conditions are given under which the use of the describing function method to investigate the nature of oscillations in autonomous nonlinear feedback systems is justified. When these conditions are satisfied, bounds are given for the frequency, fundamental magnitude, and higher harmonics of the oscillation based on the describing function approximation. The feedback systems considered are those which can be decomposed into a linear time-invariant subsystem, not necessarily causal, stable, or finite-dimensional, and a nonlinear frequency independent subsystem, possibly containing hysteresis. The approach taken is to consider the describing function equation as an approximation to a determining equation for periodic solutions of the autonomous system’s operator equation, and to use local degree theory to guarantee the existence of a solution to the determining equation.