When ζ is negative, Mr ceases to have any meaning? (Book Benjamin Kuo)

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In the 7th edition of "Automatic Control Systems" by Benjamin C. Kuo, it is stated that when the damping ratio (ζ) is negative, the maximum value of the transfer function |H(jω)|, denoted as Mr, ceases to have meaning due to system instability. However, the discussion reveals that Mr remains valid regardless of the sign of ζ, as the amplitude response to sinusoidal inputs remains unchanged. The analysis emphasizes that even with negative damping, the frequency domain analysis retains significance, contradicting Kuo's assertion.

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In the book Automatic Control Systems, Benjamin C.Kuo, 7th edition, on page 548, he says:
https://imagizer.imageshack.us/v2/720x154q90/540/ML8zmu.jpg
He is doing an analysis of the following transfer function:
http://imagizer.imageshack.us/a/img911/4839/XPKxoO.gif
Mr is the maximum value that | H (jw) | can reach with w ranging from 0 to infinity.

He says that if ζ is negative, the system is unstable and the value of Mr ceases to have any meaning.
I disagree with that. Whereas if ζ is negative, it is clear that this puts the two complex poles to the right side of the real axis, but the function | H (jw) | is exactly the same in the cases of ζ be positive or negative and it is not by fact a transfer function having its poles right that Mr ceases to have meaning.
Mr will be:
http://imageshack.com/a/img538/9372/tFTOaZ.gif (as discussed on page 546)
independent of ζ to be negative or positive.
Ie, it is not because the system is unstable to a step in the time that their analysis in the frequency domain loses meaning.
I would like to be corrected if my view is wrong. Thank you.
 
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If a system could exist with a negative damping ratio, what would it do when disturbed? Would there be a resonant peak for this system?
 
billy_joule said:
If a system could exist with a negative damping ratio, what would it do when disturbed? Would there be a resonant peak for this system?
For a sinusoidal input, the amplitude is the same as it would if ζ were positive, so the maximum is the same. And the angle is the same, but negative.
I think that's it.
 
What is its response to a small impulse, e.g., thermal noise?
 
xorg said:
For a sinusoidal input, the amplitude is the same as it would if ζ were positive, so the maximum is the same. And the angle is the same, but negative.
I think that's it.

Nope..
Damping, like friction, is an energy loss.

If a system with negative friction or negative damping could exist it would have very unusual characteristics...
It would defy energy conservation and would make a fine perpetual motion machine.
 
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