Describing-Function Analysis

The describing function approximates a nonlinearity by its gain to a sinusoid, letting frequency methods predict limit cycles.

Frequency Analysis of Nonlinearities

Linear frequency-domain tools cannot be applied directly to nonlinear elements such as saturation, dead zones, or relays. The describing function method extends them approximately by representing a nonlinearity with an equivalent gain, computed as the ratio of the fundamental harmonic of its output to a sinusoidal input. This equivalent gain generally depends on the input amplitude, unlike a linear gain.

The core idea

Feed a nonlinearity a sinusoid of amplitude A. Its output is periodic but not sinusoidal, containing harmonics. Keep only the fundamental (the harmonic at the input frequency) and express it as a complex gain N(A). If the rest of the loop is low-pass enough to filter out the higher harmonics, the nonlinearity behaves approximately like this amplitude-dependent gain.

Predicting limit cycles

• Model the loop as a linear part L(jw) in series with a nonlinearity described by N(A).
• A sustained oscillation (limit cycle) requires L(jw) = -1/N(A) for some amplitude and frequency.
• Plot the linear frequency response and the negative inverse describing function on the same axes; intersections predict limit cycles.
• The amplitude and frequency of the intersection estimate the oscillation's size and rate.

Assessing stability of the limit cycle

Beyond predicting whether a limit cycle exists, the geometry of the crossing indicates whether it is stable, meaning the system settles into it, or unstable, meaning small perturbations grow or decay away from it. This tells the designer whether a sustained oscillation is a persistent operating mode or a transient boundary.

Scope and limits

The describing function is approximate: it assumes the higher harmonics are filtered out by the linear part, so it works best when that part is genuinely low-pass. It can miss oscillations dominated by harmonics and does not give exact amplitudes. Still, for the common nonlinearities it handles, it is a fast and insightful design tool that connects nonlinear behavior to familiar Nyquist-style plots.

It complements phase-plane analysis, which is exact but limited to second-order systems, by extending approximate limit-cycle prediction to higher-order loops.