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Concept
Describing function
Summary
In control systems theory, the describing function (DF) method, developed by Nikolay Mitrofanovich Krylov and Nikolay Bogoliubov in the 1930s, and extended by Ralph Kochenburger is an approximate procedure for analyzing certain nonlinear control problems. It is based on quasi-linearization, which is the approximation of the non-linear system under investigation by a linear time-invariant (LTI) transfer function that depends on the amplitude of the input waveform. By definition, a transfer function of a true LTI system cannot depend on the amplitude of the input function because an LTI system is linear. Thus, this dependence on amplitude generates a family of linear systems that are combined in an attempt to capture salient features of the non-linear system behavior. The describing function is one of the few widely applicable methods for designing nonlinear systems, and is very widely used as a standard mathematical tool for analyzing limit cycles in closed-loop controllers, such as industrial process controls, servomechanisms, and electronic oscillators.
Consider feedback around a discontinuous (but piecewise continuous) nonlinearity (e.g., an amplifier with saturation, or an element with deadband effects) cascaded with a slow stable linear system. The continuous region in which the feedback is presented to the nonlinearity depends on the amplitude of the output of the linear system. As the linear system's output amplitude decays, the nonlinearity may move into a different continuous region. This switching from one continuous region to another can generate periodic oscillations. The describing function method attempts to predict characteristics of those oscillations (e.g., their fundamental frequency) by assuming that the slow system acts like a low-pass or bandpass filter that concentrates all energy around a single frequency. Even if the output waveform has several modes, the method can still provide intuition about properties like frequency and possibly amplitude; in this case, the describing function method can be thought of as describing the sliding mode of the feedback system.
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