Find the percent overshoot, settling time, rise time, and peak time for T(s) 14.145 (s2+0.824s+2.892)(s+5)<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n To determine the percent overshoot, settling time, rise time, and peak time for the transfer function ( T(s) = \\frac{14.145}{(s^2 + 0.824s + 2.892)(s + 5)} ), we need to analyze the system’s behavior in the time domain. Here\u2019s a step-by-step process to derive these characteristics.<\/p>\n\n\n\n We begin by simplifying the transfer function into a standard second-order system form and finding the poles of the system. The given transfer function has two terms in the denominator:<\/p>\n\n\n\n First, we can solve for the poles of the second-order system by using the quadratic formula on ( s^2 + 0.824s + 2.892 = 0 ):<\/p>\n\n\n\n [ Solving this:<\/p>\n\n\n\n [ This gives complex conjugate poles, indicating underdamped behavior. We can express these poles as ( s = \\sigma \\pm j \\omega_n ), where ( \\sigma ) is the real part, and ( \\omega_n ) is the imaginary part.<\/p>\n\n\n\n From the quadratic equation, we can find the damping ratio ( \\zeta ) and the natural frequency ( \\omega_n ). For a second-order system ( s^2 + 2\\zeta \\omega_n s + \\omega_n^2 ), the values are:<\/p>\n\n\n\n Using these values, we can calculate the time-domain specifications.<\/p>\n\n\n\n The calculated time-domain specifications for the system are:<\/p>\n\n\n\n These parameters provide insights into the transient response of the system.<\/p>\n","protected":false},"excerpt":{"rendered":" Find the percent overshoot, settling time, rise time, and peak time for T(s) 14.145 (s2+0.824s+2.892)(s+5) The Correct Answer and Explanation is : To determine the percent overshoot, settling time, rise time, and peak time for the transfer function ( T(s) = \\frac{14.145}{(s^2 + 0.824s + 2.892)(s + 5)} ), we need to analyze the system’s […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","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":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-189571","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189571","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=189571"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189571\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=189571"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=189571"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=189571"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Identify the system’s poles<\/h3>\n\n\n\n
\n
s = \\frac{-0.824 \\pm \\sqrt{(0.824)^2 – 4(1)(2.892)}}{2(1)}
]<\/p>\n\n\n\n
s = \\frac{-0.824 \\pm \\sqrt{0.6781 – 11.568}}{2} = \\frac{-0.824 \\pm \\sqrt{-10.8899}}{2}
]<\/p>\n\n\n\nStep 2: Determine the damping ratio and natural frequency<\/h3>\n\n\n\n
\n
Step 3: Time domain specifications<\/h3>\n\n\n\n
\n
The formula for percent overshoot is: [
PO = 100e^{\\frac{-\\zeta \\pi}{\\sqrt{1 – \\zeta^2}}}
] Substituting ( \\zeta = 0.423 ): [
PO = 100e^{\\frac{-0.423 \\pi}{\\sqrt{1 – 0.423^2}}} \\approx 16.75\\%
]<\/li>\n\n\n\n
Settling time for a second-order system is approximately: [
Ts = \\frac{4}{\\zeta \\omega_n} = \\frac{4}{0.423 \\times 1.7} \\approx 5.8 \\, \\text{seconds}
]<\/li>\n\n\n\n
The rise time is approximated for a second-order system by: [
Tr \\approx \\frac{\\pi – \\theta}{\\omega_n \\sqrt{1 – \\zeta^2}}
] where ( \\theta = \\arccos(\\zeta) ). Using ( \\zeta = 0.423 ): [
Tr \\approx \\frac{\\pi – \\arccos(0.423)}{1.7} \\approx 3.1 \\, \\text{seconds}
]<\/li>\n\n\n\n
The peak time for a second-order system is: [
Tp = \\frac{\\pi}{\\omega_n \\sqrt{1 – \\zeta^2}} \\approx \\frac{\\pi}{1.7 \\times \\sqrt{1 – 0.423^2}} \\approx 3.1 \\, \\text{seconds}
]<\/li>\n<\/ol>\n\n\n\nConclusion<\/h3>\n\n\n\n
\n