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Describing function

The Describing function method of Krylov and Bogolyubov is an approximate procedure for analyzing nonlinear control problems. It is based on quasi-linearization, that is the approximation of the non-linear system under investigation by a system that is linear except for a dependence on the amplitude of the input waveform. This quasi-linearization must be carried out for a specific family of input waveforms.

One choice might be to describe the system response to the family of sine wave inputs; in this case the system would be characterized by an SIDF or sine input describing function $H(A,,jomega)$ giving the system response to an input consisting of a sine wave of amplitude A and frequency $omega$ . This SIDF is a generalization of the transfer function $H(jomega)$ used to characterize linear systems. In a quasi-linear system when the input is a sine wave, the output will be a sine wave of the same frequency but with different amplitude and phase as given by $H(A,,jomega)$ . Many systems are approximately quasi-linear in the sense that although the response to a sine wave is not a pure sine wave, most of the energy in the output is indeed at the same frequency $omega$ as the input. This is because such systems may possess intrinsic low-pass or band-pass characteristics such that harmonics are naturally attenuated, or because external filters are added for this purpose. An important application of the SIDF technique is to estimate the oscillation amplitude in sinusoidal electronic oscillators.

Other types of describing functions that have been used are DFs for level inputs and for Gaussian noise inputs. While not a complete description of the system, the DFs often suffice to answer specific questions about control and stability. DF methods are best for analyzing systems with relatively weak nonlinearities, such as saturation or deadband effects.

References

Krylov N., and N. Bogolyubov: Introduction to Nonlinear Mechanics, Princeton University Press, 1947
Gelb, A., and W. E. Vander Velde: Multiple-Input Describing Functions and Nonlinear System Design, McGraw Hill, 1968.
http://ocw.mit.edu/OcwWeb/Aeronautics-and-Astronautics/16-30Spring2004/Readings/index.htm#Downloadable (downloadable version of above book).