PID gains from the transfer function of a plant
Your Plant transfer function is a simple pole (1st order).

From a theoretical perspective your controller requirement is first order including an integrator for non-zero error. This is also known as a PI controller.

My advice is to proceed as follows:


Consider the controller to be k(s+a)/s i.e. an integrator, a zero and a gain; a.k.a a PI controller
Select a so that it is 2x the pole frequency of the plant (i.e. 2* 0.497)
Convert pole zero values to P and I coefficients.


This design will yield approximately 70 degrees of Phase Margin. Use SISOTOOL in Matlab to tune for overshoot.

Alternatively, you could easily calculate overshoot based on ZETA from closed loop equations or from open loop Phase Margin... a so called "paper design". The following graph is useful for finding out how Damping Ratio and Phase Margin Equate to Overshoot requirements.



The conversion from ZPK (zero-pole-gain) model to PI follows from simple algebra by setting the PI transfer function equal to the zpk version and identifying the parameter equivalence by inspection. For example:

Kp + Ki/s = k(s+b)/s

(sKp + Ki)/s = k(s + b)/s

Kp(s + Ki/Kp) = k(s + b)

so Kp = k
and Ki = Kp.b