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<title>Closed-loop transfer function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Closed-loop transfer function</span></span>

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<p>In <a href="Control_theory" title="Control theory">control theory</a>, a <b>closed-loop transfer function</b> is a <a href="Mathematical_function" class="mw-redirect" title="Mathematical function">mathematical function</a> describing the net result of the effects of a <a href="Feedback_control_loop" class="mw-redirect" title="Feedback control loop">feedback control loop</a> on the input <a href="Signal_(information_theory)" class="mw-redirect" title="Signal (information theory)">signal</a> to the <a href="Plant_(control_theory)" title="Plant (control theory)">plant</a> under control.

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

<p>The closed-loop <a href="Transfer_function" title="Transfer function">transfer function</a> is measured at the output. The output signal can be calculated from the closed-loop transfer function and the input signal. Signals may be <a href="Waveform" title="Waveform">waveforms</a>, <a href="Image" title="Image">images</a>, or other types of <a href="Data_stream" title="Data stream">data streams</a>.

</p><p>An example of a closed-loop block diagram, from which a transfer function may be computed, is shown below:

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</p><p>The summing node and the <i>G</i>(<i>s</i>) and <i>H</i>(<i>s</i>) blocks can all be combined into one block, which would have the following transfer function:

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<annotation encoding="application/x-tex">{\displaystyle {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}}</annotation>

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</math></span><img src="./27b1c92941fdb75fc156530553db8548692e5005.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.343ex; height:6.509ex;" alt="{\displaystyle {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle G(s)}</annotation>

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</math></span><img src="./cb2e6025c8f4c9d44fb1dc2da68407e4eb56f9db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.726ex; height:2.843ex;" alt="{\displaystyle G(s)}" loading="lazy"></span> is called the <a href="Feed_forward_(control)" title="Feed forward (control)">feed forward</a> transfer function, <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(s)}">

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

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</math></span><img src="./f1b91390324fc9c33ec00fe57e3924ad7118cc1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}" loading="lazy"></span> is called the <a href="Feedback#Control_theory" title="Feedback">feedback</a> transfer function, and their product <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 G(s)H(s)}">

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

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</math></span><img src="./0e227a74f1c944c40c62587c58c226dcb792f035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.69ex; height:2.843ex;" alt="{\displaystyle G(s)H(s)}" loading="lazy"></span> is called the <b>open-loop transfer function</b>.

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

<p>We define an intermediate signal Z (also known as <a href="Error_signal" class="mw-redirect" title="Error signal">error signal</a>) shown as follows:

</p><p>Using this figure we write:

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<annotation encoding="application/x-tex">{\displaystyle Y(s)=G(s)Z(s)}</annotation>

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</math></span><img src="./1b47428583190ec3033199a2c986d2be6b763aee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.078ex; height:2.843ex;" alt="{\displaystyle Y(s)=G(s)Z(s)}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle Z(s)=X(s)-H(s)Y(s)}</annotation>

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</math></span><img src="./91c75dc929609feae3823eb2f874f94f444648c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.035ex; height:2.843ex;" alt="{\displaystyle Z(s)=X(s)-H(s)Y(s)}" loading="lazy"></span></dd></dl>

<p>Now, plug the second equation into the first to eliminate Z(s):

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<annotation encoding="application/x-tex">{\displaystyle Y(s)=G(s)[X(s)-H(s)Y(s)]}</annotation>

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</math></span><img src="./28ca32117ca5a1061eb68777e15e264aae796d46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.148ex; height:2.843ex;" alt="{\displaystyle Y(s)=G(s)[X(s)-H(s)Y(s)]}" loading="lazy"></span></dd></dl>

<p>Move all the terms with Y(s) to the left hand side, and keep the term with X(s) on the right hand side:

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<annotation encoding="application/x-tex">{\displaystyle Y(s)+G(s)H(s)Y(s)=G(s)X(s)}</annotation>

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</math></span><img src="./64b4430ec5a68abf54791727697338471ee6cbfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.581ex; height:2.843ex;" alt="{\displaystyle Y(s)+G(s)H(s)Y(s)=G(s)X(s)}" loading="lazy"></span></dd></dl>

<p>Therefore,

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<annotation encoding="application/x-tex">{\displaystyle Y(s)(1+G(s)H(s))=G(s)X(s)}</annotation>

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</math></span><img src="./dcb68ced3c4569f59075f7f529bd37b8a9a69a18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.88ex; height:2.843ex;" alt="{\displaystyle Y(s)(1+G(s)H(s))=G(s)X(s)}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}}</annotation>

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</math></span><img src="./03adda2962df8216bb815f64f23bb85cf12b3a9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.312ex; height:6.509ex;" alt="{\displaystyle \Rightarrow {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}}" loading="lazy"></span></dd></dl>

<ul><li><a href="Federal_Standard_1037C" title="Federal Standard 1037C">Federal Standard 1037C</a></li>

<li><a href="Open-loop_controller" title="Open-loop controller">Open-loop controller</a></li>

<li><a href="Control_theory#Open-loop_and_closed-loop_(feedback)_control" title="Control theory">Control theory § Open-loop and closed-loop (feedback) control</a></li></ul>

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

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</style><span class="citation 1037C"><span class="noviewer" typeof="mw:File"><span></span></span> This article incorporates <a href="Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220122224547/https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm"><i>Federal Standard 1037C</i></a>. <a href="General_Services_Administration" title="General Services Administration">General Services Administration</a>. Archived from <a rel="nofollow" class="external text" href="https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm">the original</a> on 2022-01-22.</cite></span></li></ul></div><div><div class="zim-footer">

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