{"id":3924,"date":"2021-01-26T09:00:57","date_gmt":"2021-01-26T17:00:57","guid":{"rendered":"https:\/\/www.linquip.com\/blog\/?p=3924"},"modified":"2025-08-28T01:18:22","modified_gmt":"2025-08-28T09:18:22","slug":"final-value-theorem","status":"publish","type":"post","link":"https:\/\/www.linquip.com\/blog\/final-value-theorem\/","title":{"rendered":"All You Need to Know about Final Value Theorem"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.linquip.com\/blog\/final-value-theorem\/#Deducing_lim_tto_infty_ft\" >Deducing \\lim _{ t\\to \\infty } f(t)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.linquip.com\/blog\/final-value-theorem\/#Deducing_lim_s_to_0_sFs\" >Deducing { \\lim _{ s\\, \\to \\, 0 }{ sF(s) } }<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.linquip.com\/blog\/final-value-theorem\/#Examples\" >Examples<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.linquip.com\/blog\/final-value-theorem\/#Download_Final_Value_Theorem_PDF\" >Download Final Value Theorem PDF<\/a><\/li><\/ul><\/nav><\/div>\n<p>The final value theorem (FVT) is one theorem utilized to relate <a href=\"https:\/\/en.wikipedia.org\/wiki\/Frequency_domain\" target=\"_blank\" rel=\"noopener\">frequency domain<\/a> expression to the time domain behavior as time approaches infinity.<\/p>\n<div class=\"mceTemp\"><\/div>\n<p><span style=\"font-weight: 400;\">The Linquip website provides information about a variety of scientific topics. Besides researchers and those who are interested in general or specific knowledge about these topics, users and entrepreneurs who are eager to expanding their knowledge of the working principles and concepts behind industrial equipment and tools can also benefit from them. The final value theorem is a fundamental mathematical and calculating concept that is used by some specific industrial devices. You may find it helpful to visit Linquip&#8217;s page entitled &#8220;<\/span><a href=\"https:\/\/www.linquip.com\/industrial-directories\/400\/electrical-power-transmission\"><b>What Is Electrical Power Transmission<\/b><\/a><span style=\"font-weight: 400;\">.&#8221;<\/span><\/p>\n<p><span style=\"font-weight: 400;\">To get the most out of the Linquip platform and fully utilize its features, you must become a<\/span><a href=\"https:\/\/www.linquip.com\/experts\/leaderboard\"> <b>Linquip Expert<\/b><\/a><span style=\"font-weight: 400;\">. A Linquip expert account lets you demonstrate your expertise in the field of industrial equipment in a way that is specifically tailored to meet the needs of your industry. Would you like to be a guest writer on the Linquip website and contribute to the content of the site? It is possible to publish your content directly on Linquip&#8217;s website through the<\/span><a href=\"https:\/\/www.linquip.com\/blog\/user-guest-post\"> <b>Guest Posting<\/b><\/a><span style=\"font-weight: 400;\"> feature.<\/span><\/p>\n<p>Mathematically, if\u00a0 f(t) in continuous time has Laplace transform F(s) then a final value theorem establishes situations under which<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">{ \\lim _{ t\\to \\infty } f(t)=\\lim _{ s\\, \\to \\, 0 }{ sF(s) } }<\/span>\n<p>Similarly, if f[k] in discrete time has Z-transform F(z) then a final value theorem establishes conditions under which<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\lim _{ k\\to \\infty } f[k]=\\lim _{ z\\to 1 }{ (z-1)F(z) }<\/span>\n<p>The Abelian final value theorem assumes the time domain of\u00a0 f(t)\u00a0(or\u00a0 f[k]) to calculate<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">{ \\lim _{ s\\, \\to \\, 0 }{ sF(s) } }<\/span>\n<p>On the other hand, a Tauberian final value theorem makes assumptions about the frequency-domain of\u00a0 F(s) to calculate<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\lim _{ t\\to \\infty } f(t)(or{ \\lim _{ k\\to \\infty } f[k] })<\/span>\n<h2><span class=\"ez-toc-section\" id=\"Deducing_lim_tto_infty_ft\"><\/span>Deducing <span class=\"katex-eq\" data-katex-display=\"false\">\\lim _{ t\\to \\infty } f(t)<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Final value theorems for obtaining\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\lim _{t\\to \\infty }f(t)<\/span> have usage in establishing the long-term stability of a specific system<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\lim _{t\\to \\infty }f(t)<\/span>\n<h3>Standard Final Value Theorem<\/h3>\n<p>Suppose that every pole of\u00a0 F(s) is at the origin or in the open left half plane, and that\u00a0 F(s) has at most one pole at the origin. Then<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">sF(s)\\to L\\in { R\\quad }<\/span>\n<p>as<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">{ s\\to 0 }<\/span>\n<p>and<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">{ \\lim _{ t\\to \\infty } f(t)=L }.<\/span>\n<p>&nbsp;<\/p>\n<h3>Final Value Theorem in Laplace Transform of the Derivative<\/h3>\n<p>If f(t) and\u00a0 f'(t) both have Laplace transforms that exist for all\u00a0 s&gt;0 , and<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\lim _{ t\\to \\infty } f(t)<\/span>\n<p>and<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">{ \\lim _{ s\\, \\to \\, 0 }{ sF(s) } }<\/span>\n<p>exists, then<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">{ \\lim _{ t\\to \\infty } f(t)=\\lim _{ s\\, \\to \\, 0 }{ sF(s) } }.<\/span>\n<p>Note:<br \/>\nBoth limits must exist in order that the theorem holds. For instance, if<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">f(t)=\\sin (t)<\/span>\n<p>then<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\lim _{ t\\to \\infty } f(t)<\/span>\n<p>does not exist. However,<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\lim _{ s\\, \\to \\, 0 }{ sF(s) } =\\lim _{ s\\, \\to \\, 0 }{ \\frac { s }{ s^{ 2 }+1 } } =0<\/span>\n<p>&nbsp;<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Deducing_lim_s_to_0_sFs\"><\/span>Deducing <span class=\"katex-eq\" data-katex-display=\"false\">{ \\lim _{ s\\, \\to \\, 0 }{ sF(s) } }<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Another application of final value theorems for obtaining<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\u00a0\\lim _{s\\,\\to \\,0}{sF(s)}<\/span>\n<p>In probability and statistics is to find the moments of a random variable.<\/p>\n<h3>Final Value Theorem in Laplace Transform of the Derivative<\/h3>\n<p>Suppose that all of the conditions below are satisfied:<br \/>\n1. <span class=\"katex-eq\" data-katex-display=\"false\">f:(0,\\infty )\\to { C }<\/span> is constantly differentiable and both f and\u00a0 f&#8217; have a Laplace Transform<br \/>\n2.\u00a0 f&#8217; is completly integrable, that is <span class=\"katex-eq\" data-katex-display=\"false\">{ \\int _{ 0 }^{ \\infty } |f&#039;(\\tau )|\\, d\\tau } <\/span> is finite<br \/>\n3.<span class=\"katex-eq\" data-katex-display=\"false\">{ \\lim _{ t\\to \\infty } f(t) }<\/span>\u00a0 is finite<br \/>\nThen<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">{ \\lim _{ s\\to 0^{ + } } sF(s)=\\lim _{ t\\to \\infty } f(t) }.<\/span>\n<h3>Final Value Theorem for the Mean of a Function<\/h3>\n<p>Assume that <span class=\"katex-eq\" data-katex-display=\"false\">{ f:(0,\\infty )\\to { C } }<\/span> be a continuous and bounded function such that the following limit exists<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">{ \\lim _{ T\\to \\infty }{ \\frac { 1 }{ T } } \\int _{ 0 }^{ T } f(t)\\, dt=\\alpha \\in { C } }<\/span>\n<p>Then<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">{ \\lim _{ s\\, \\to \\, 0,\\, s&gt;0 }{ sF(s) } =\\alpha }.<\/span>\n<h2><span class=\"ez-toc-section\" id=\"Examples\"><\/span>Examples<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3>An Example FVT Is Applicable<\/h3>\n<p>For instance, for a system described by transfer function<br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">G(s)=\\frac { 3 }{ s+4 } ,<\/span>\nand so the impulse response converges to<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\lim _{ t\\to \\infty } g(t)=\\lim _{ s\\to 0 } \\frac { 3s }{ s+4 } =0.<\/span>\n<p>The system comes to zero after being disturbed by a short impulse. Nevertheless, the Laplace transform of the unit step response is<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">H(s)=\\frac { 1 }{ s } \\frac { 3 }{ s+4 } <\/span>\n<p>Thus the step response converges to<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\lim _{ t\\to \\infty } h(t)=\\lim _{ s\\to 0 } \\frac { s }{ s } \\frac { 3 }{ s+4 } =\\frac { 3 }{ 4 } =0.75<\/span>\n<p>Thus, a zero-state system will follow an exponential rise to a final value of 0.75.<\/p>\n<h3>An Example FVT Is Not Applicable<\/h3>\n<p>For a system determined by the transfer function<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">H(s)=\\frac { 16 }{ s^{ 2 }+16 } ,<\/span>\n<p>the final value theorem seems to predict the final value of the step response to be one and the final value of the impulse response to be zero. Though, the time-domain limit does not exist, and so the final value theorem forecasts are not valid.<\/p>\n<p>Both the step response and impulse response oscillate, and (in this special case) the final value theorem determines the average values where the responses oscillate.<br \/>\nThere are two analyses performed in Control theory that confirm valid results for the Final Value Theorem:<br \/>\nAll non-zero roots in the denominator of H(s) must contain negative real parts.<br \/>\nH(s) must not possess more than one pole at the origin.<br \/>\nRule 1 was not satisfied in this case, in that the roots of the denominator are\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">0+j4 <\/span>and\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">0-j4.<\/span>\n<h2><span class=\"ez-toc-section\" id=\"Download_Final_Value_Theorem_PDF\"><\/span><strong>Download Final Value Theorem PDF<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">If you would like to save a PDF version of this article to your computer or laptop for future reference, please click the link provided below.<\/span><\/p>\n<div class=\"su-button-center\"><a href=\"https:\/\/www.linquip.com\/blog\/wp-content\/uploads\/2021\/01\/linquip.com-All-You-Need-to-Know-about-Final-Value-Theorem.pdf\" class=\"su-button su-button-style-default su-button-wide\" style=\"color:#FFFFFF;background-color:#2D89EF;border-color:#246ec0;border-radius:12px\" target=\"_blank\" rel=\"noopener noreferrer\"><span style=\"color:#FFFFFF;padding:0px 30px;font-size:22px;line-height:44px;border-color:#6cadf4;border-radius:12px;text-shadow:none\"> Download PDF<\/span><\/a><\/div>\n<p>&nbsp;<\/p>\n<p><em><strong>Read More on Linquip<\/strong><\/em><\/p>\n<ul>\n<li><span style=\"text-decoration: underline;\"><strong><span style=\"font-size: 10pt; font-family: verdana, geneva, sans-serif;\"><a href=\"https:\/\/www.linquip.com\/blog\/convenient-preventive-maintenance-checklist\/\" target=\"_blank\" rel=\"noopener\">Your Convenient Preventive Maintenance Checklist<\/a><\/span><\/strong><\/span><\/li>\n<li><span style=\"text-decoration: underline;\"><strong><span style=\"font-size: 10pt; font-family: verdana, geneva, sans-serif;\"><a href=\"https:\/\/www.linquip.com\/blog\/the-best-civil-engineering-colleges-in-the-united-states\/\" target=\"_blank\" rel=\"noopener\">The Best Civil Engineering Colleges in the United States<\/a><\/span><\/strong><\/span><\/li>\n<li><span style=\"text-decoration: underline;\"><strong><span style=\"font-size: 10pt; font-family: verdana, geneva, sans-serif;\"><a href=\"https:\/\/www.linquip.com\/blog\/role-of-universities-in-teaching-and-research-in-agriculture\/\" target=\"_blank\" rel=\"noopener\">What is The Role of Universities in Teaching and Research in Agriculture?<\/a><\/span><\/strong><\/span><\/li>\n<li><span style=\"text-decoration: underline;\"><strong><span style=\"font-size: 10pt; font-family: verdana, geneva, sans-serif;\"><a href=\"https:\/\/www.linquip.com\/blog\/what-is-a-septic-tank-and-how-does-it-work\/\" target=\"_blank\" rel=\"noopener\">What is Septic Tank and How Does It Work? 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The Linquip website provides information about a variety of scientific topics. Besides researchers and those who are interested in general or specific knowledge about these topics, users and entrepreneurs who are eager &#8230;<\/p>\n","protected":false},"author":12,"featured_media":3927,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","footnotes":""},"categories":[24],"tags":[],"class_list":["post-3924","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-science"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/posts\/3924","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/users\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/comments?post=3924"}],"version-history":[{"count":6,"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/posts\/3924\/revisions"}],"predecessor-version":[{"id":36542,"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/posts\/3924\/revisions\/36542"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/media\/3927"}],"wp:attachment":[{"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/media?parent=3924"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/categories?post=3924"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.linquip.com\/blog\/wp-json\/wp\/v2\/tags?post=3924"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}