Ultimate gain and ultimate period

A few questions regarding your transfer function.

Firstly, is this the transfer function of your plant (what you are trying to control)? Or does it also include your controller? I assume it represents only what you are trying to control.

Secondly, please clarify your transfer function. Do you mean:

a) T(s) = 2e^-0.5s / (2s+1)

or

b) T(s) = 2^-0.5s / (2s+1)

2e^-0.5s / (2s+1) is the Laplace transform of the function f(t)=e^-(0.5t-0.25) where f(t) = 0 for t<0.5.

I do not recognize 2^-0.5s / (2s+1) as a typical transfer function. If this is what you meant, what does it represent (real world)?