Frequency Domain Analysis - the math

[phiby]

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/

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?

 

[reddvoid]

here is another way

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

i put s = - jw to get value of a, if you put s = jw you will get the value of a bar

 

[phiby]

here is another way

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

i put s = - jw to get value of a, if you put s = jw you will get the value of a bar

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?

 

[ajith.mk91]

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.

 

[DeeNos]

The value of a is derived from Equation (8-2), which is the transfer function of a system in the frequency domain. This transfer function is defined as G(s) = Y(s)/X(s), where Y(s) is the output of the system and X(s) is the input. In this case, X(s) is the input signal ωX and Y(s) is the output signal a(s^2 + ω^2).

To find the value of a, we can substitute s = jω into the transfer function G(s). This is because in the frequency domain, s represents the complex frequency jω. By substituting s = jω, we get G(jω) = a(jω^2 + ω^2). This can be simplified to G(jω) = aω^2 + ajω^2.

Now, we can compare this to the given expression for a = G(s) (ωX) /(s^2 + ω^2). By substituting s = jω, we get a = G(jω) (ωX) /(jω^2 + ω^2). Since G(jω) = aω^2 + ajω^2, we can substitute this into the equation and simplify to get a = G(jω) (ωX) /(jω^2 + ω^2) = (aω^2 + ajω^2) (ωX) /(jω^2 + ω^2) = a (ωX) /(jω^2 + ω^2). This is the same as the given expression, so we can conclude that a = G(jω) (ωX) /(jω^2 + ω^2).

To answer your second question, they substituted s = -jω because this is the negative frequency of s = jω. In control theory, it is common to analyze systems in the frequency domain using complex frequencies s = jω and s = -jω. By substituting s = -jω, we can analyze the system's response at negative frequencies and gain a more complete understanding of its behavior.