 Describing function

The Describing function (DF) method of Nikolay Mitrofanovich Krylov and Nikolay Bogolyubov is an approximate procedure for analyzing certain nonlinear control problems. It is based on quasilinearization, which is the approximation of the nonlinear system under investigation by an LTI system 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 nonlinear system behavior.
For example, consider when feedback is placed around a discontinuous (but piecewise continuous) nonlinearity (e.g., saturation^{[disambiguation needed ]} or deadband effects) cascaded with a slow stable linear system. Depending on the amplitude of the output of the linear system, the feedback presented to the nonlinearity will be in a different continuous region. As the output of the linear system 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 lowpass 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.
Using this lowpass assumption, the system response can be described by one of a family of sinusoidal waveforms; in this case the system would be characterized by a sine input describing function (SIDF) giving the system response to an input consisting of a sine wave of amplitude A and frequency ω. This SIDF is a modification of the transfer function H(jω) used to characterize linear systems. In a quasilinear system, when the input is a sine wave, the output will be a sine wave of the same frequency but with a scaled amplitude and shifted phase as given by . Many systems are approximately quasilinear 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 ω as the input. This is because such systems may possess intrinsic lowpass or bandpass 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. Although 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. In addition the higher order sinusoidal input describing functions (HOSIDF), describe the response of a class of nonlinear systems at harmonics of the input frequency of a sinusoidal input. The HOSIDFs are an extension of the SIDF for systems where the nonlinearities are significant in the response.
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Caveats
Although the DF method may find oscillations for wide class of systems, it is well known that DF method can lead to incorrect results. Such examples have been presented by Tzypkin in a bang–bang systems.^{[1]} Also, in the case when conditions of Aizerman's or Kalman conjectures are fulfilled, there is no periodic solutions by DF method,^{[2]}^{[3]} but counterexamples with periodic solutions are well known. Therefore, the application of the DF method requires additional justifications.^{[4]}
References
 ^ Tsypkin, Yakov Z. (1984). Relay Control Systems. Cambridge: Univ Press.
 ^ Leonov G.A., Kuznetsov N.V. (2011). "Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems". Doklady Mathematics 84 (1): 475–481. doi:10.1134/S1064562411040120.,
 ^ "Aizerman's and Kalman's conjectures and DF method". http://www.math.spbu.ru/user/nk/PDF/Harmonic_balance_Absolute_stability.pdf.
 ^ Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A. (2011). "Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits". Journal of Computer and Systems Sciences International 50 (4): 511–543. doi:10.1134/S106423071104006X.
Further reading
 Krylov N., and N. Bogolyubov: Introduction to Nonlinear Mechanics, Princeton University Press, 1947
 Gelb, A., and W. E. Vander Velde: MultipleInput Describing Functions and Nonlinear System Design, McGraw Hill, 1968.
 P.W.J.M. Nuij, O.H. Bosgra, M. Steinbuch, Higher Order Sinusoidal Input Describing Functions for the Analysis of Nonlinear Systems with Harmonic Responses, Mechanical Systems and Signal Processing, 20(8), 1883–1904, (2006)
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