Stabilize a system, so there is an overshoot of 10%

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Fredje

Joined Jun 1, 2022
15
I have a TF: \( H(s)\;=\;\cfrac{(s+\alpha)}{(s+\beta)* (3s^2 - 5s + 25)} \)

Question: Choose α and β so that this is possible with a simple proportional controller, but do not make them equal. Choose Kc so that the overshoot is 10%. If this is not possible, find Kc so that the overshoot is as small as possible.

I calculated the Closed Loop TF: \( H(s)\;=\;\cfrac{(Kc(s+\alpha))}{(s+\beta)(3s^2 + (-5+Kc)s + (25+Kc*\alpha))} \)
PO (percentual overshoot) has to be 10 so I calculated that ζ has to be 0.59
I would think if I assume second order dominance I could calculate Kc by saying:
Wn² = 25 + Kc * \alpha
2 * 0.59 * Wn = -5 + Kc

But I don't really understand the first half of the question. "Choose α and β so that this is possible with a simple proportional controller". isn't it always possible with a P-controller? or what do I need to take in account?

How can I get good values for alpha and beta?
 
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