How do I get the transfer function from this step response?

I have the following step response of a closed loop system. Using the calculations below, I tried to calculate the transfer function. I am trying to get the same response in Matlab. However, I seem not get it right. Can someone please tell me what I am doing wrong? My result in Matlab as well is given below.   • Envelope calcs: $\alpha=\frac12\cdot 1.993=0.9965$, $\omega_{_0}=\sqrt{28.41}\approx 5.33\:\frac{\text{rad}}{\text{s}}$, $\zeta=\frac{\alpha}{\omega_{_0}}\approx 0.187$. So, I get a very low damping factor; which would appear to explain your matlab results. The value of $\omega_{_0}$ also appears to match, at least by eye.
– jonk
Mar 21 '21 at 21:40
• It's because the system is nonlinear, so you can't estimate it with 2nd order linear system. You can see the difference: yours (linear) VS. scope (non-linear). Mar 21 '21 at 22:29
• Further it has a dead time, a closed loop would have also a numerator part with laplace operators. Could be a kind of controller with limiters that introduce the non-linearity. Mar 21 '21 at 22:34