{"id":237032,"date":"2025-06-16T18:58:44","date_gmt":"2025-06-16T18:58:44","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=237032"},"modified":"2025-06-16T18:58:46","modified_gmt":"2025-06-16T18:58:46","slug":"find-k1-and-k2-so-that-peak-overshoot-0-25-and-peak-time-is-2-seconds-when-unit-step-input-is-applied","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/16\/find-k1-and-k2-so-that-peak-overshoot-0-25-and-peak-time-is-2-seconds-when-unit-step-input-is-applied\/","title":{"rendered":"Find\u00a0K1\u00a0and\u00a0K2\u00a0so that peak overshoot = 0.25 and peak time is 2 seconds when unit step input is applied.\u00a0"},"content":{"rendered":"\n
Find K1 and K2 so that peak overshoot = 0.25 and peak time is 2 seconds when unit step input is applied. <\/pre>\n\n\n\n
\u00a0<\/p>\n\n\n\n


.\u00a0<\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

<\/p>\n\n\n\n
To solve for K1K_1 and K2K_2, we use the standard second-order system equations.<\/p>\n\n\n\n
Given that peak time TpT_p is 2 seconds and maximum overshoot MpM_p is 0.25, we use the following relations:<\/p>\n\n\n\n
    \n
  1. Peak time<\/strong> equation:<\/li>\n<\/ol>\n\n\n\n
    Tp=\u03c0\u03c9dT_p = \\frac{\\pi}{\\omega_d}<\/p>\n\n\n\n
    where \u03c9d=\u03c9n1\u2212\u03b62\\omega_d = \\omega_n \\sqrt{1 – \\zeta^2}.<\/p>\n\n\n\n
      \n
    1. Maximum overshoot<\/strong> equation:<\/li>\n<\/ol>\n\n\n\n
      Mp=e\u2212\u03c0\u03b61\u2212\u03b62M_p = e^{\\frac{-\\pi \\zeta}{\\sqrt{1 – \\zeta^2}}}<\/p>\n\n\n\n
      Solving for damping ratio \u03b6\\zeta,<\/p>\n\n\n\n
      0.25=e\u2212\u03c0\u03b61\u2212\u03b620.25 = e^{\\frac{-\\pi \\zeta}{\\sqrt{1 – \\zeta^2}}}<\/p>\n\n\n\n
      Taking natural logarithm,<\/p>\n\n\n\n
      ln\u2061(0.25)=\u2212\u03c0\u03b61\u2212\u03b62\\ln(0.25) = \\frac{-\\pi \\zeta}{\\sqrt{1 – \\zeta^2}}<\/p>\n\n\n\n
      Solving for \u03b6\\zeta, we get approximately 0.404<\/strong>.<\/p>\n\n\n\n
        \n
      1. Substituting Tp=2T_p = 2 into the peak time equation,<\/li>\n<\/ol>\n\n\n\n
        2=\u03c0\u03c9n1\u22120.40422 = \\frac{\\pi}{\\omega_n \\sqrt{1 – 0.404^2}}<\/p>\n\n\n\n
        Solving for \u03c9n\\omega_n, we get 2.61 rad\/s<\/strong>.<\/p>\n\n\n\n
        For the second problem, we equate given system parameters to the standard second-order system equation:<\/p>\n\n\n\n
        \u03c9n=5,\u03b6=0.7\\omega_n = 5, \\quad \\zeta = 0.7<\/p>\n\n\n\n
        Solving for the system parameters KK and aa, we match them with standard equations, obtaining K=25K = 25 and a=7a = 7.<\/p>\n\n\n\n
        Explanation<\/h3>\n\n\n\n
        Control systems are analyzed using standard second-order system equations that define transient response characteristics. Peak overshoot indicates how much the system output exceeds the desired steady-state value, while peak time determines how quickly the system reaches this first peak. The damping ratio and natural frequency influence system stability and responsiveness.<\/p>\n\n\n\n
        Using logarithmic transformations, we solve for the damping ratio from the overshoot equation. Substituting into the peak time equation helps determine the system\u2019s natural frequency. These relationships enable fine-tuning of control parameters to achieve desired performance characteristics.<\/p>\n\n\n\n
        For the second problem, damping ratio and natural frequency provide constraints. By relating them to standard transfer functions, we derive system parameters that meet specified criteria, ensuring stable and predictable response.<\/p>\n\n\n\n
        \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
        Find K1 and K2 so that peak overshoot = 0.25 and peak time is 2 seconds when unit step input is applied.  \u00a0 .\u00a0 The Correct Answer and Explanation is: To solve for K1K_1 and K2K_2, we use the standard second-order system equations. Given that peak time TpT_p is 2 seconds and maximum overshoot MpM_p is 0.25, we use […]<\/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-237032","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237032","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=237032"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237032\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=237032"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=237032"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=237032"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}