Consider a process with transfer function:
\[
G_p = \frac{2e^{-s}}{(5s + 1)^2}
\]
A first-order plus dead time (FOPDT) model is to be fitted to the unit step process reaction curve (PRC) by applying the maximum slope method. Let \( \tau_m \) and \( \theta_m \) denote the time constant and dead time, respectively, of the fitted FOPDT model. The value of \( \frac{\tau_m}{\theta_m} \) is ________ (rounded off to 2 decimal places).
% Given Data
Given:
For \( G = \frac{1}{(\tau s + 1)^2} \), the unit step output response is:
\[
y(t) = 1 - \left(1 + \frac{t}{\tau}\right)e^{-t/\tau}
\]
The first and second derivatives of \( y(t) \) are:
\[
\frac{dy(t)}{dt} = \frac{t}{\tau^2} e^{-t/\tau}
\]
\[
\frac{d^2y(t)}{dt^2} = \frac{1}{\tau^2} \left(1 - \frac{t}{\tau}\right) e^{-t/\tau}
\]