Design a model predictive controller for the process
G_{p}(s)=\frac{e^{-6 s}}{10 s+1} \quad G_{v}=G_{m}=1
Select a value of N based on 95 \% completion of the step response and \Delta t=2. Simulate the closed-loop response for a set-point change using the following design parameters:
\begin{array}{lll}\text { (a) } M=1 & P=7 & R = 0 \end{array} \begin{array}{lll}\text { (b) } M=1 & P=5 & R = 0 \end{array} \begin{array}{lll}\text { (c) } M=4 & P=30 & R=0\end{array}The open-loop unit step response of G_{p}(s) is
y(t)= L ^{-1}\left(\frac{e^{-6 s}}{10 s+1} \frac{1}{s}\right)= L ^{-1}\left(e^{-6 s}\left(\frac{1}{s}-\frac{10}{10 s+1}\right)\right)=S(t-6)\left[1-e^{-(t-6) / 10}\right]
By trial and error, y(34)<0.95, y(36)>0.95.
Therefore N \Delta t=36 or N=18.
The coefficients \left\{S_{i}\right\} are obtained from the expression for y(t) and the predictive controller is obtained following the procedure of Example 20.5. The closed-loop responses for a unit set-point change are shown below for the three sets of controller design parameters.