Let's say I have following situation. I have recorded a step response of a dynamic system in following form
The data have been gathered with the sampling period $100\,\mu s$. My goal is to find the analytical expression of the step response. Based on the dynamic model of the system I expected that the system should have following transfer function
$$\frac{A}{\tau\cdot s + 1}\cdot e^{-s\cdot T_d}.$$
Based on that I supposed the Laplace transform of the step response in following form
$$A\cdot\frac{\frac{1}{\tau}}{s\cdot(s + \frac{1}{\tau})}\cdot e^{-s\cdot T_d}.$$
i.e. I supposed that I have been looking for a function in following form
$$A\cdot\left[1 - e^{\left(\frac{-(t-T_d)}{\tau}\right)}\right].$$
Then I have used the least square method for finding the unknown parameters $A, T_d, \tau$. Unfortunately this approach resulted in following poor conformity between the measured data and fitted curve.
It seems that the dynamic of the system is more complex than I expected. Can anybody give me an advice how to find the analytical expression for the measured step response?