# Describing function

﻿
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 quasi-linearization, which is the approximation of the non-linear 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 non-linear 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 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.

Nonlinear system in the state of harmonic balance

Using this low-pass 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) $H(A,\,j\omega)$ 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 quasi-linear 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 $H(A,\,j\omega)$. 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 ω as the input. This is because such systems may possess intrinsic low-pass 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.

## 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

1. ^ Tsypkin, Yakov Z. (1984). Relay Control Systems. Cambridge: Univ Press.
2. ^ 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. ,
3. ^
4. ^ 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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• describing function method — harmoninio balanso metodas statusas T sritis automatika atitikmenys: angl. describing function method vok. Methode der harmonischen Bilanz, f rus. метод гармонического баланса, m pranc. méthode de balance harmonique, f …   Automatikos terminų žodynas

• Function (biology) — A function is part of an answer to a question about why some object or process occurred in a system that evolved through a process of selection. Thus, function refers forward from the object or process, along some chain of causation to the goal… …   Wikipedia

• Function model — A function model or functional model is a structured representation of the functions, activities or processes within the modeled system or subject area. [http://www.itl.nist.gov/fipspubs/idef02.doc FIPS Publication 183] released of IDEFØ December …   Wikipedia

• Wave function — Not to be confused with the related concept of the Wave equation Some trajectories of a harmonic oscillator (a ball attached to a spring) in classical mechanics (A B) and quantum mechanics (C H). In quantum mechanics (C H), the ball has a wave… …   Wikipedia

• Smooth function — A bump function is a smooth function with compact support. In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to …   Wikipedia

• Characteristic function (probability theory) — The characteristic function of a uniform U(–1,1) random variable. This function is real valued because it corresponds to a random variable that is symmetric around the origin; however in general case characteristic functions may be complex valued …   Wikipedia

• Implicit function — In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable has not been given explicitly in terms of the independent variable. To give a function f explicitly is to provide a… …   Wikipedia

• Gaussian function — In mathematics, a Gaussian function (named after Carl Friedrich Gauss) is a function of the form::f(x) = a e^{ { (x b)^2 over 2 c^2 } }for some real constants a > 0, b , c > 0, and e ≈ 2.718281828 (Euler s number).The graph of a Gaussian is a… …   Wikipedia

• Generalized function — In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognised theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and (going …   Wikipedia

• Size function — Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half plane to the natural numbers, counting certain connected components of a topological space. They are used in pattern recognition and… …   Wikipedia