Control theory - overshoot max

Question

So, during my last class the teacher asked if we could go from:

MO = e(-πξ / √1-ξ²) to ξ= (-ln(MO)) / (√π²+ln²(MO))

MO = Max overshoot ξ = zeta e = exp

Does anyone understand what he meant by that? Did I just misunderstand? Tried a few times but I just can't make sense of how to go from the MO equation to the zeta one.

Answer 1

It would be good to see what you tried and I'm sure your teacher would too, but does this make it any clearer?

$$ MO = e^{-\pi \frac{\zeta}{\sqrt{1-\zeta^2}}} $$ $$ ln(MO) = -\pi\frac{\zeta}{\sqrt{1-\zeta^2}} $$

$$ ln(MO)^2 = (ln(MO)^2 +\pi^2)\zeta^2 $$

$$ \zeta^2 = \frac{ln(MO)^2}{ln(MO)^2 +\pi^2} $$