# Control Systems - Pole Location

Hello there I am trying to solve a question and in the question I have a system with two complex conjugate poles. The question asks me to analyse the Maximum % Overshoot, Settling-Peak-Rise Time and Damping ratio if those poles were moved to the left. Here are my conclusions:

• Natural frequency (wn) increases since the radial distance of the poles are going to be greater. Angle theta is going to decrease.

• Since theta decreases damping ratio (zeta) is going to increase since it is equal to cos(theta).

• Maximum percent overshoot is going to decrease.
• Settling time is going to decrease.

I could not reach to a certain conclusion about the Peak Time and Rise Time. I know that Peak time is equal to pi/wn.(sqrt(1-zeta^2)). And I also know that Rise time cannot be calculated directly.

Can you help me with that?

Here is the question:

I am of course not trying to get the answer. I am just trying to get some feedback.

• Everything you seems to be saying sounds correct but you need to ask a specific question and highlight what in terms of proofs (as in math proofs) you need to undertake in order to fulfill the requirements of the question. Commented May 22, 2020 at 17:12
• I added the question just to let everybody know. Can you inspect what I've done so far one more time? Thank you so much Commented May 22, 2020 at 17:20

• Natural frequency (wn) increases since the radial distance of the poles are going to be greater. Angle theta is going to decrease.

Correct.

• Since theta decreases damping ratio (zeta) is going to increase since it is equal to cos(theta).

Correct.

• Maximum percent overshoot is going to decrease.

Yes, because $$\\zeta\$$ has increased

• Settling time is going to decrease.

Because $$\\zeta\$$ has increased, overshoot has decreased hence settling time to within "some percentage" limits are decreased. But it depends on how you define settling time. Once the step has caused the output response to "hit" the first overshoot, it will settle more quickly to stability. But as $$\\zeta\$$ approaches 1 (the limit for conjugate poles) it does take a long time to hit that first overshoot.

For instance if $$\\zeta\$$ is 0.90 it will hit the first overshoot in about 70% of the time it would take if $$\\zeta\$$ was 0.95. But, once that overshoot is hit, a $$\\zeta\$$ value of 0.95 will settle down more quickly.

So, it depends on how you define settling time.