Consider a first-order-plus-time-delay model:
\frac{Y(s) }{U(s)}=\frac{Ke^{-θs} }{\tau s+1} (20-7)
(a) Derive the equivalent step-response model by considering the analytical solution to a unit step change in the input.
(b) Calculate the step-response coefficients, { S_{i} }, for the following parameter values: K = 5, \tau= 15 min, θ = 3 min, and a sampling period of Δt = 1 min . Also, calculate and plot the response y(k) for 0 ≤ k ≤ 80 after a step change in u from 0 to 3 occurs at t = 2 min .