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## Q. 20.9

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:

$\begin{array}{lll}\text { (a) } M=1 & P=7 & R = 0 \end{array}$ $\begin{array}{lll}\text { (b) } M=1 & P=5 & R = 0 \end{array}$ $\begin{array}{lll}\text { (c) } M=4 & P=30 & R=0\end{array}$

## Verified Solution

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.