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Frequency Domain Analysis - the math

  1. Nov 12, 2011 #1
    I am studying Control Theory from Ogata. My math is a little rusty, so this is a math question about Frequency Domain Analysis.

    Check this page - http://www.flickr.com/photos/66943862@N06/6337116280/sizes/l/in/photostream/ [Broken]

    I get everything upto Equation 8-4

    However I don't get the line after that.

    The line "where the constant a can be evaluated from Equation (8-2) as follows"

    And they write the value of a.

    How is
    a = G(s) (ωX) /(s^2 + ω^2).

    How do they arrive at this value of a?

    Can someone help?

    And why did they substitute s = -jω after that?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Nov 12, 2011 #2
    here is another way

    http://img41.imageshack.us/content_round.php?page=done&l=img41/4643/12112011130.jpg [Broken]

    i put s = - jw to get value of a, if you put s = jw you will get the value of a bar
     
    Last edited by a moderator: May 5, 2017
  4. Nov 12, 2011 #3
    Thank you. That's a little clearer.

    You get values of a & abar by evaluating the limit as s = -jω & s = jω

    However, I still don't get why you evaluate these limits?
     
    Last edited by a moderator: May 5, 2017
  5. Nov 12, 2011 #4
    Its simply a mathematical way of obtaining the coefficients. You can simply factorize the denominator into complex factors and proceed to obtain the inverse laplace transform.
     
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