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<title>Describing function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Describing function</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Control_systems" class="mw-redirect" title="Control systems">control systems theory</a>, the <b>describing function</b> (DF) method, developed by <a href="Nikolay_Mitrofanovich_Krylov" class="mw-redirect" title="Nikolay Mitrofanovich Krylov">Nikolay Mitrofanovich Krylov</a> and <a href="Nikolay_Bogoliubov" class="mw-redirect" title="Nikolay Bogoliubov">Nikolay Bogoliubov</a> in the 1930s,<sup id="cite_ref-Krylov_1-0" class="reference"><a href="#cite_note-Krylov-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Blaquiere_2-0" class="reference"><a href="#cite_note-Blaquiere-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and extended by Ralph Kochenburger<sup id="cite_ref-Kochenburger_3-0" class="reference"><a href="#cite_note-Kochenburger-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> is an approximate procedure for analyzing certain <a href="Nonlinear_control" title="Nonlinear control">nonlinear control</a> problems. It is based on <a href="Quasi-linearization" class="mw-redirect" title="Quasi-linearization">quasi-linearization</a>, which is the approximation of the non-linear system under investigation by a <a href="LTI_system" class="mw-redirect" title="LTI system">linear time-invariant</a> (LTI) <a href="Transfer_function" title="Transfer function">transfer function</a> that depends on the <a href="Amplitude" title="Amplitude">amplitude</a> 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 <a href="Linear_system" title="Linear system">linear</a>. 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 <a href="Limit_cycle" title="Limit cycle">limit cycles</a> in <a href="Closed-loop_controller" title="Closed-loop controller">closed-loop controllers</a>, such as industrial process controls, servomechanisms, and <a href="Electronic_oscillator" title="Electronic oscillator">electronic oscillators</a>.

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<div class="mw-heading mw-heading2"><h2 id="The_method">The method</h2></div>

<p>Consider feedback around a discontinuous (but piecewise continuous) nonlinearity (e.g., an amplifier with saturation, or an element with <a href="Deadband" title="Deadband">deadband</a> 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 <a href="Oscillation" title="Oscillation">oscillations</a>. 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 <a href="Low-pass" class="mw-redirect" title="Low-pass">low-pass</a> or <a href="Bandpass" class="mw-redirect" title="Bandpass">bandpass</a> 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 <a href="Sliding_mode_control" title="Sliding mode control">sliding mode</a> of the feedback system.

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<p>Using this low-pass assumption, the system response can be described by one of a family of <a href="Sine_wave" title="Sine wave">sinusoidal waveforms</a>; in this case the system would be characterized by a sine input describing function (SIDF) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(A,\,j\omega )}">

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<annotation encoding="application/x-tex">{\displaystyle H(A,\,j\omega )}</annotation>

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</math></span><img src="./545de4c8473f14f4752e267b73f6ed3f8686785f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.441ex; height:2.843ex;" alt="{\displaystyle H(A,\,j\omega )}" loading="lazy"></span> giving the system response to an input consisting of a sine wave of amplitude A and frequency <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span>. This SIDF is a modification of the <a href="Transfer_function" title="Transfer function">transfer function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(j\omega )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle H(j\omega )}</annotation>

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</math></span><img src="./54b9cf918573394f5d6d888c7c8519bfc73eb7a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.277ex; height:2.843ex;" alt="{\displaystyle H(j\omega )}" loading="lazy"></span> 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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(A,\,j\omega )}">

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<annotation encoding="application/x-tex">{\displaystyle H(A,\,j\omega )}</annotation>

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</math></span><img src="./545de4c8473f14f4752e267b73f6ed3f8686785f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.441ex; height:2.843ex;" alt="{\displaystyle H(A,\,j\omega )}" loading="lazy"></span>. 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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> as the input. This is because such systems may possess intrinsic <a href="Low-pass" class="mw-redirect" title="Low-pass">low-pass</a> or <a href="Bandpass" class="mw-redirect" title="Bandpass">bandpass</a> characteristics such that harmonics are naturally attenuated, or because external <a href="Filter_(signal_processing)" title="Filter (signal processing)">filters</a> are added for this purpose. An important application of the SIDF technique is to estimate the oscillation amplitude in sinusoidal <a href="Electronic_oscillator" title="Electronic oscillator">electronic oscillators</a>.

</p><p>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 <a href="Higher_order_sinusoidal_input_describing_function" class="mw-redirect" title="Higher order sinusoidal input describing function">higher order sinusoidal input describing functions</a> (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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<div class="mw-heading mw-heading2"><h2 id="Caveats">Caveats</h2></div>

<p>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 <a href="Bang%E2%80%93bang_control" title="Bang–bang control">bang–bang</a> systems.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A fairly similar example is a closed-loop oscillator consisting of a non-inverting <a href="Schmitt_trigger" title="Schmitt trigger">Schmitt trigger</a> followed by an <i>inverting</i> <a href="Integrator" title="Integrator">integrator</a> that feeds back its output to the Schmitt trigger's input. The output of the Schmitt trigger is going to be a <a href="Square_wave_(waveform)" title="Square wave (waveform)">square waveform</a>, while that of the integrator (following it) is going to have a <a href="Triangle_wave" title="Triangle wave">triangle waveform</a> with peaks coinciding with the transitions in the square wave. Each of these two oscillator stages lags the signal exactly by 90 degrees (relative to its input). If one were to perform DF analysis on this circuit, the triangle wave at the Schmitt trigger's input would be replaced by its fundamental (sine wave), which passing through the trigger would cause a phase shift of less than 90 degrees (because the sine wave would trigger it sooner than the triangle wave does) so the system would appear not to oscillate in the same (simple) way.<sup id="cite_ref-LurieEnright2000_5-0" class="reference"><a href="#cite_note-LurieEnright2000-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>Also, in the case where the conditions for <a href="Aizerman's_conjecture" title="Aizerman's conjecture">Aizerman's</a> or <a href="Kalman's_conjecture" title="Kalman's conjecture">Kalman conjectures</a> are fulfilled, there are no periodic solutions by describing function method,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> but counterexamples with <a href="Hidden_attractor" title="Hidden attractor">hidden periodic attractors</a> are known. Counterexamples to the describing function method can be constructed for discontinuous dynamical systems when a rest segment destroys predicted limit cycles.<sup id="cite_ref-2018-AIP-Keldysh-problem_8-0" class="reference"><a href="#cite_note-2018-AIP-Keldysh-problem-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Therefore, the application of the describing function method requires additional justification.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2011-IJBC-Hidden-attractors_10-0" class="reference"><a href="#cite_note-2011-IJBC-Hidden-attractors-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFKrylovN._Bogoliubov1943" class="citation book cs1">Krylov, N. M.; N. Bogoliubov (1943). <a rel="nofollow" class="external text" href="https://archive.today/20130620041142/http://libra.msra.cn/Publication/3271320/introduction-to-nonlinear-mechanics"><i>Introduction to Nonlinear Mechanics</i></a>. Princeton, US: Princeton Univ. Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0691079854</bdi>. Archived from <a rel="nofollow" class="external text" href="http://libra.msra.cn/Publication/3271320/introduction-to-nonlinear-mechanics">the original</a> on 2013-06-20.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></span>

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<li id="cite_note-Blaquiere-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Blaquiere_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBlaquiere2012" class="citation book cs1">Blaquiere, Austin (2012-12-02). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LC2_lK9HZQgC&q=krylov+bogoliubov&pg=PA177"><i>Nonlinear System Analysis</i></a>. Elsevier Science. p. 177. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0323151665</bdi>.</cite></span>

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<li id="cite_note-Kochenburger-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kochenburger_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFKochenburger1950" class="citation journal cs1">Kochenburger, Ralph J. (January 1950). "A Frequency Response Method for Analyzing and Synthesizing Contactor Servomechanisms". <i>Trans. AIEE</i>. <b>69</b> (1). American Institute of Electrical Engineers: <span class="nowrap">270–</span>284. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ft-aiee.1950.5060149">10.1109/t-aiee.1950.5060149</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51646567">51646567</a>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFTsypkin1984" class="citation book cs1">Tsypkin, Yakov Z. (1984). <i>Relay Control Systems</i>. Cambridge: Univ Press.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: publisher location (link)</span></span>

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<li id="cite_note-LurieEnright2000-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-LurieEnright2000_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBoris_LuriePaul_Enright2000" class="citation book cs1">Boris Lurie; Paul Enright (2000). <i>Classical Feedback Control: With MATLAB</i>. CRC Press. pp. <span class="nowrap">298–</span>299. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8247-0370-7</bdi>.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFLeonov_G.A.Kuznetsov_N.V.2011" class="citation journal cs1">Leonov G.A.; Kuznetsov N.V. (2011). <a rel="nofollow" class="external text" href="http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf">"Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems"</a> <span class="cs1-format">(PDF)</span>. <i>Doklady Mathematics</i>. <b>84</b> (1): <span class="nowrap">475–</span>481. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2FS1064562411040120">10.1134/S1064562411040120</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120692391">120692391</a>.</cite>,</span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.spbu.ru/user/nk/PDF/Harmonic_balance_Absolute_stability.pdf">"Aizerman's and Kalman's conjectures and describing function method"</a> <span class="cs1-format">(PDF)</span>.</cite></span>

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<li id="cite_note-2018-AIP-Keldysh-problem-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-2018-AIP-Keldysh-problem_8-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFLeonov_G.A.Kuznetsov_N.V.2018" class="citation journal cs1">Leonov G.A.; Kuznetsov N.V. (2018). "On the Keldysh problem of flutter suppression". <i>AIP Conference Proceedings</i>. <b>1959</b> (1): art. num. 020002. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1803.06920">1803.06920</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018AIPC.1959b0002L">2018AIPC.1959b0002L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.5034578">10.1063/1.5034578</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:55340847">55340847</a>.</cite></span>

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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFBragin_V.O.Vagaitsev_V.I.Kuznetsov_N.V.Leonov_G.A.2011" class="citation journal cs1">Bragin V.O.; Vagaitsev V.I.; Kuznetsov N.V.; Leonov G.A. (2011). <a rel="nofollow" class="external text" href="http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf">"Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Computer and Systems Sciences International</i>. <b>50</b> (4): <span class="nowrap">511–</span>543. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2FS106423071104006X">10.1134/S106423071104006X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:21657305">21657305</a>.</cite></span>

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<li id="cite_note-2011-IJBC-Hidden-attractors-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-2011-IJBC-Hidden-attractors_10-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFLeonov_G.A.Kuznetsov_N.V.2013" class="citation journal cs1">Leonov G.A.; Kuznetsov N.V. (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0218127413300024">"Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits"</a>. <i>International Journal of Bifurcation and Chaos</i>. <b>23</b> (1): <span class="nowrap">1330002–</span>219. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013IJBC...2330002L">2013IJBC...2330002L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0218127413300024">10.1142/S0218127413300024</a></span>.</cite></span>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li>N. Krylov and N. Bogolyubov: <i>Introduction to Nonlinear Mechanics</i>, Princeton University Press, 1947</li>

<li>A. Gelb and W. E. Vander Velde: <a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-30-estimation-and-control-of-aerospace-systems-spring-2004/readings/#Downloadable"><i>Multiple-Input Describing Functions and Nonlinear System Design</i></a>, McGraw Hill, 1968.</li>

<li>James K. Roberge, <i>Operational Amplifiers: Theory and Practice,</i> <a rel="nofollow" class="external text" href="http://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/textbook/MITRES_6-010S13_chap06.pdf">chapter 6: Non-Linear Systems</a>, 1975; free copy courtesy of <a href="MIT_OpenCourseWare" title="MIT OpenCourseWare">MIT OpenCourseWare</a> 6.010 (2013); see also (1985) video recording of Roberge's lecture on <a rel="nofollow" class="external text" href="http://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/course-videos/lecture-15-describing-functions/">describing functions</a></li>

<li>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)</li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.ee.unb.ca/jtaylor/Publications/EEncyc_final.pdf">Electrical Engineering Encyclopedia: Describing Functions</a></li></ul></div><div><div class="zim-footer">

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