A process control system contains the following transfer functions:
\begin{aligned}G_{p}(s) &=\frac{2 e^{-1.5 s}}{(60 s+1)(5 s+1)} \\G_{v}(s) &=\frac{0.5 e^{-0.3 s}}{3 s+1} \\G_{m}(s) &=\frac{3 e^{-0.2 s}}{2 s+1} \\G_{c}(s) &=K_{c}\end{aligned}
(a) Show how G_{O L}(s) can be approximated by a FOPTD model;
G_{p} G_{v} G_{m} G_{c}=G_{O L}(s) \approx \frac{K e^{-\theta s}}{\tau s+1}
Find K, \tau, and \theta for the open-loop process transfer function.
(b) Use direct substitution and your FOPTD model to find the range of K_{c} values that will yield a stable closed-loop system. Check your value of K_{c m} for the full-order model using simulation.
