{"id":22306,"date":"2025-06-16T16:04:03","date_gmt":"2025-06-16T16:04:03","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=22306"},"modified":"2025-06-16T16:04:04","modified_gmt":"2025-06-16T16:04:04","slug":"for-the-unity-feedback-system-with-feedforward-transfer-function-gs","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/for-the-unity-feedback-system-with-feedforward-transfer-function-gs\/","title":{"rendered":"For the unity feedback system with feedforward transfer function G(s)"},"content":{"rendered":"\n<p>For the unity feedback system with feedforward transfer function G(s) a. find the system type b. What steady state error can be expected for a step input of 12?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p>Sure! To answer your question about the unity feedback system with feedforward transfer function G(s)G(s):<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Given:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Unity feedback system<\/li>\n\n\n\n<li>Forward transfer function G(s)G(s)<\/li>\n\n\n\n<li>Input: Step input of magnitude 12<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">a. Find the system type<\/h2>\n\n\n\n<p><strong>System type<\/strong> depends on the number of poles of G(s)G(s) at the origin (i.e., number of integrators).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Type 0 system:<\/strong> No poles at s=0s=0<\/li>\n\n\n\n<li><strong>Type 1 system:<\/strong> One pole at s=0s=0<\/li>\n\n\n\n<li><strong>Type 2 system:<\/strong> Two poles at s=0s=0, and so on&#8230;<\/li>\n<\/ul>\n\n\n\n<p>The system type indicates the system\u2019s ability to track certain inputs with zero steady-state error.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">b. What steady-state error can be expected for a step input of magnitude 12?<\/h2>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Steady-state error formula for unity feedback system<\/h3>\n\n\n\n<p>For a unity feedback system with forward transfer function G(s)G(s), the error transfer function is: E(s)=11+G(s)\u22c5R(s)E(s) = \\frac{1}{1 + G(s)} \\cdot R(s)<\/p>\n\n\n\n<p>where R(s)R(s) is the Laplace transform of the input.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Use Final Value Theorem<\/h3>\n\n\n\n<p>The steady-state error is: ess=lim\u2061t\u2192\u221ee(t)=lim\u2061s\u21920sE(s)=lim\u2061s\u21920s\u22c511+G(s)\u22c5R(s)e_{ss} = \\lim_{t \\to \\infty} e(t) = \\lim_{s \\to 0} s E(s) = \\lim_{s \\to 0} s \\cdot \\frac{1}{1 + G(s)} \\cdot R(s)<\/p>\n\n\n\n<p>For a step input of magnitude 12: r(t)=12\u22c5u(t)\u21d2R(s)=12sr(t) = 12 \\cdot u(t) \\Rightarrow R(s) = \\frac{12}{s}<\/p>\n\n\n\n<p>Plug into steady-state error: ess=lim\u2061s\u21920s\u22c511+G(s)\u22c512s=lim\u2061s\u21920121+G(s)=121+Kpe_{ss} = \\lim_{s \\to 0} s \\cdot \\frac{1}{1 + G(s)} \\cdot \\frac{12}{s} = \\lim_{s \\to 0} \\frac{12}{1 + G(s)} = \\frac{12}{1 + K_p}<\/p>\n\n\n\n<p>where Kp=lim\u2061s\u21920G(s)K_p = \\lim_{s \\to 0} G(s)<\/p>\n\n\n\n<p>KpK_p is the <strong>position error constant<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Interpreting the error based on system type<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Type 0:<\/strong> KpK_p is finite. The steady-state error to step input is finite:<\/li>\n<\/ul>\n\n\n\n<p>ess=121+Kpe_{ss} = \\frac{12}{1 + K_p}<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Type 1 or higher:<\/strong> Since G(s)G(s) has at least one pole at zero, G(0)\u2192\u221eG(0) \\to \\infty, so<\/li>\n<\/ul>\n\n\n\n<p>ess=121+\u221e=0e_{ss} = \\frac{12}{1 + \\infty} = 0<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\"><strong>Summary<\/strong><\/h1>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>System Type<\/th><th>Number of Poles at 0<\/th><th>KpK_p<\/th><th>Step Input Steady-State Error<\/th><\/tr><\/thead><tbody><tr><td>0<\/td><td>0<\/td><td>Finite<\/td><td>121+Kp\\frac{12}{1 + K_p} (Non-zero)<\/td><\/tr><tr><td>1<\/td><td>1<\/td><td>Infinite<\/td><td>0<\/td><\/tr><tr><td>2 or more<\/td><td>\u2265 2<\/td><td>Infinite<\/td><td>0<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\"><strong>Example<\/strong><\/h1>\n\n\n\n<p>If G(s)=10s+2G(s) = \\frac{10}{s+2}<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>No pole at zero \u2192 Type 0 system<\/li>\n\n\n\n<li>Kp=lim\u2061s\u21920G(s)=102=5K_p = \\lim_{s \\to 0} G(s) = \\frac{10}{2} = 5<\/li>\n\n\n\n<li>Steady-state error for step 12:<\/li>\n<\/ul>\n\n\n\n<p>ess=121+5=126=2e_{ss} = \\frac{12}{1 + 5} = \\frac{12}{6} = 2<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\"><strong>Explanation<\/strong><\/h1>\n\n\n\n<p>In control systems, <strong>system type<\/strong> is crucial for predicting steady-state error for standard inputs like step, ramp, or parabolic signals.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>system type<\/strong> is determined by the number of pure integrators in the forward path. Integrators help eliminate steady-state error for certain inputs.<\/li>\n\n\n\n<li>For a step input, a <strong>Type 0 system<\/strong> can only reduce the error to a finite nonzero value depending on the gain KpK_p.<\/li>\n\n\n\n<li>A <strong>Type 1 or higher system<\/strong> has at least one integrator, making the position error constant infinite, resulting in zero steady-state error for step inputs.<\/li>\n\n\n\n<li>This means increasing system type improves tracking of lower-order polynomial inputs (step, ramp, parabolic, etc.).<\/li>\n\n\n\n<li>If the exact G(s)G(s) is known, calculating KpK_p is straightforward and gives the exact steady-state error.<\/li>\n\n\n\n<li>The magnitude of the step input scales the steady-state error linearly, as seen in the formula.<\/li>\n<\/ul>\n\n\n\n<p>Understanding system type and error constants allows control engineers to design feedback systems that meet accuracy requirements for specific inputs.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>For the unity feedback system with feedforward transfer function G(s) a. find the system type b. What steady state error can be expected for a step input of 12? The correct answer and explanation is: Sure! To answer your question about the unity feedback system with feedforward transfer function G(s)G(s): Given: a. Find the system [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-22306","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22306","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=22306"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22306\/revisions"}],"predecessor-version":[{"id":22307,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22306\/revisions\/22307"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=22306"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=22306"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=22306"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}