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| In [[Control systems|control systems theory]], the '''describing function''' (DF) method, developed by [[Nikolay Mitrofanovich Krylov]] and [[Nikolay Bogoliubov]] in the 1930s,<ref name="Krylov">{{cite book
| | Hi there. Let me begin by introducing the author, her title is Sophia. Office supervising is where my main earnings comes from but I've always wanted my personal company. She is truly fond of caving but she doesn't have the time recently. Her family members life in Ohio but her husband desires them to transfer.<br><br>My website :: spirit messages ([http://test.jeka-nn.ru/node/129 http://test.jeka-nn.ru/node/129]) |
| | last = Krylov
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| | first = N. M.
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| | coauthors = N. Bogoliubov
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| | title = Introduction to Nonlinear Mechanics
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| | publisher = Princeton Univ. Press
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| | year = 1943
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| | location = Princeton, US
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| | pages =
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| | url = http://libra.msra.cn/Publication/3271320/introduction-to-nonlinear-mechanics
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| | doi =
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| | id =
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| | isbn = 0691079854}}</ref><ref name="Blaquiere">{{cite book
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| | last = Blaquiere
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| | first = Austin
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| | title = Nonlinear System Analysis
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| | publisher = Elsevier Science
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| | date =
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| | location =
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| | pages = 177
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| | url = http://books.google.com/books?id=LC2_lK9HZQgC&pg=PA177&dq=krylov+bogoliubov
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| | isbn = 0323151663}}</ref> and extended by Ralph Kochenburger<ref name="Kochenburger">{{cite journal
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| | last = Kochenburger
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| | first = Ralph J.
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| | title = A Frequency Response Method for Analyzing and Synthesizing Contactor Servomechanisms
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| | journal = Trans. of the AIEE
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| | volume = 69
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| | issue = 1
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| | pages = 270–284
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| | publisher = American Institute of Electrical Engineers
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| | location =
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| |date=January 1950
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| | url = http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=5060149&contentType=Journals+%26+Magazines
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| | accessdate = June 18, 2013}}</ref> 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 [[LTI system|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 system|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 cycle]]s in [[closed-loop controller]]s, such as industrial process controls, servomechanisms, and [[electronic oscillator]]s.
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| ==The method==
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| 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. 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 [[oscillation]]s. 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 control|sliding mode]] of the feedback system.
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| [[File:Function-block-harmonic-balance.png|thumb|center|600px|Nonlinear system in the state of harmonic balance]]
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| Using this low-pass assumption, the system response can be described by one of a family of [[sine wave|sinusoidal waveform]]s; in this case the system would be characterized by a sine input describing function (SIDF) <math>H(A,\,j\omega)</math> giving the system response to an input consisting of a sine wave of amplitude A and frequency <math>\omega</math>. This SIDF is a modification of the [[transfer function]] <math>H(j\omega)</math> 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 <math>H(A,\,j\omega)</math>. 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 <math>\omega</math> 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 [[filter (signal processing)|filter]]s are added for this purpose. An important application of the SIDF technique is to estimate the oscillation amplitude in sinusoidal [[electronic oscillator]]s.
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| 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 function]]s (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==
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| Although the describing function method can produce reasonably accurate results for a wide class of systems, it can fail badly for others. For example, the method can fail if the system emphasizes higher harmonics of the nonlinearity. Such examples have been presented by Tzypkin for [[bang–bang control|bang–bang]] systems.<ref>{{cite book|last=Tsypkin|first=Yakov Z.|title=Relay Control Systems|publisher=Cambridge: Univ Press|year=1984}}</ref> Also, in the case where the conditions for [[Aizerman's conjecture| Aizerman's]] or [[Kalman's conjecture| Kalman conjectures]] are fulfilled, there are no periodic solutions by describing function method,<ref>{{cite journal
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| | author = Leonov G.A., Kuznetsov N.V.
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| | year = 2011
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| | title = Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems
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| | journal = Doklady Mathematics
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| | volume = 84
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| | number = 1
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| | url = http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf
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| | pages = 475–481
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| | doi = 10.1134/S1064562411040120}},
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| </ref><ref>{{cite web|url=http://www.math.spbu.ru/user/nk/PDF/Harmonic_balance_Absolute_stability.pdf |title=Aizerman's and Kalman's conjectures and describing function method}}</ref> but counterexamples with periodic solutions ([[hidden oscillation]]) are well known. Therefore, the application of the describing function method requires additional justification.<ref>{{cite journal
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| | author = Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A.
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| | year = 2011
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| | title = Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits
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| | journal = Journal of Computer and Systems Sciences International
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| | volume = 50
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| | number = 4
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| | pages = 511–543
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| | url = http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf
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| | doi = 10.1134/S106423071104006X}}
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| </ref><ref name=2011-IJBC-Hidden-attractors>{{cite journal |
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| author = Leonov G.A., Kuznetsov N.V. |
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| year = 2013 |
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| title = Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits |
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| journal = International Journal of Bifurcation and Chaos |
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| volume = 23 |
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| issue = 1 |
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| pages = art. no. 1330002|
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| url = http://www.worldscientific.com/doi/pdf/10.1142/S0218127413300024|
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| doi = 10.1142/S0218127413300024}}
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| </ref>
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| ==References==
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| {{Reflist|2}}
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| ==Further reading==
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| {{refbegin}}
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| * Krylov N., and N. Bogolyubov: ''Introduction to Nonlinear Mechanics'', Princeton University Press, 1947
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| * [http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-30-estimation-and-control-of-aerospace-systems-spring-2004/readings/#Downloadable Gelb, A., and W. E. Vander Velde: ''Multiple-Input Describing Functions and Nonlinear System Design'', McGraw Hill, 1968.]
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| * 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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| {{refend}}
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| == External links ==
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| * [http://www.facstaff.bucknell.edu/mastascu/eControlHTML/Nonlinear/NonLinear2DescFcn.htm The Describing Function: A Tool for Predicting Nonlinear System Oscillation]
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| *[http://www.ee.unb.ca/jtaylor/Publications/EEncyc_final.pdf Electrical Engineering Encyclopedia: Describing Functions]
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| *[http://user.chol.com/~limdj/nonlinear/chap5.pdf Dong-Jin Lim: Nonlinear systems control,Ch.5 - The Describing Function]
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| *[http://www.atp.ruhr-uni-bochum.de/rt1/nonlin/node12.html D. P. Atherton: The Describing Function (teaching module)]
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| {{DEFAULTSORT:Describing Function}}
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| [[Category:Nonlinear control]]
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Hi there. Let me begin by introducing the author, her title is Sophia. Office supervising is where my main earnings comes from but I've always wanted my personal company. She is truly fond of caving but she doesn't have the time recently. Her family members life in Ohio but her husband desires them to transfer.
My website :: spirit messages (http://test.jeka-nn.ru/node/129)