Describing-Function Analysis
The describing function approximates a nonlinearity by its gain to a sinusoid, extending frequency-domain analysis to predict limit cycles.
A quasi-linear approximation
Frequency-response methods assume linearity, but many loops contain a single static nonlinearity, saturation, a relay, a dead zone, backlash. The describing function replaces the nonlinearity with an equivalent complex gain: feed it a sinusoid, keep only the fundamental harmonic of the output, and take the ratio of that fundamental to the input as an amplitude-dependent gain. Unlike a linear gain, the describing function depends on the input amplitude.
Predicting limit cycles
With the nonlinearity replaced by its describing function N(A), depending on amplitude A, the loop looks linear and the Nyquist-like harmonic-balance condition applies. A sustained oscillation, a limit cycle, is predicted where the loop transfer function G(jw) equals minus one over N(A). Graphically, one plots the negative reciprocal of the describing function and the Nyquist plot of G; intersections predict the amplitude and frequency of possible limit cycles.
Accuracy and limits
The method assumes the linear part is a good low-pass filter, so higher harmonics generated by the nonlinearity are attenuated and the fundamental dominates, the filtering hypothesis. When this holds, predictions of limit-cycle amplitude and frequency are often remarkably accurate. When the linear part passes harmonics, or when there are multiple nonlinearities, the approximation degrades and predictions can be wrong. Stability of a predicted limit cycle is judged by how the intersection point moves with amplitude.
- Nonlinearity replaced by an amplitude-dependent complex gain
- Limit cycle where G(jw) equals minus one over N(A)
- Graphical test intersects Nyquist plot with minus one over N
- Relies on the linear part filtering higher harmonics
Describing functions exist in closed form for common nonlinearities: a saturation's gain falls as amplitude grows past the linear region, a relay's gain falls inversely with amplitude. Dual-input describing functions extend the idea to a bias plus sinusoid.
For a design-stage loop with actuator saturation or relay elements, describing-function analysis quickly predicts whether self-sustained oscillation can occur and at what amplitude, guiding compensator design before detailed simulation.