Use the IMC design method to design two controllers for the FOPTD model in Equation
\widetilde{G}(s) = \frac{Ke^{-\theta s}}{\tau s+1} (1)
Assume that f is specified by Equation
f = \frac{1}{(\tau _{c}s+1)^{r}} (2)
with r=1, and consider two approximations for the time-delay term:
(a) 1/1 Padé approximation:
e^{-\theta s}\cong \frac{1-\frac{\theta }{2}s }{1+\frac{\theta }{2}s} (3.1)
(b) First-order Taylor series approximation:
e^{-\theta s}\cong 1-\theta s (3.2)