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If I have a closed loop second order transfer function such as:

$$T(s) = \frac{10-s}{0.3s^2+3.1s+(1+24K_{c})}$$

Can I still use this formula for overshoot (when a step input is applied) ?: $$\frac{A}{B}=e^{\frac{-\pi \zeta}{\sqrt{1-\zeta^2}}}$$ Where $B$ is the step input size

I don't think you can but I'm not sure, can someone confirm?

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    $\begingroup$ I guess the formula you are referring to is derived for 2nd order transfer functions with no s in the numerator. $\endgroup$
    – Karlo
    Apr 18 '16 at 11:17
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I think you should still be able to find the inverse laplace transforms of the closed loop TF, at set values of Kc. Such that the overshoot can be determined in the time domain.

What is the step input in the s-domain, $\frac{B}{s}$?

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Yes you can, but only for underdamped response. The formula is derived from a second order system response, the one in your question looks to be a first order system but under a PI controller. That usually isn't relevant for the formula, given that the parameters are so arranged that the response is underdamped.

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