## 3 Results
### Process 2: \(K_{p}=5\) and \(\tau=1\)  
The simulation was carried out with the gain and time constant of the SOPDT model set to 5 and 1, respectively, to obtain the following transfer function:  
\[\frac{5e^{-\theta s}}{s^{2}+2^{2}_{0}s+1} \tag{20}\]  
Simulations were then carried out using the protocol defined in Section 2.3. The results obtained for setpoint response and disturbance rejection are tabulated in Tables 5-7. For disturbance rejection, MPC had a \(TV_{N}=1\) for all the tuned controllers. As such, the results are not presented in a table.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & & & & & & \\ \(r_{w}\) & \(\theta\) & **0.2** & **0.4** & **0.6** & **0.8** & **1** & **2** & **3** & **4** & **5** \\ \hline \multirow{6}{*}{**10**} & **0.01** & 11.48 & 9.03 & 8.60 & 14.30 & 9.84 & 119.28 & 146.69 & 159.52 & 168.03 \\  & **0.1** & 9.40 & 8.91 & 9.25 & 9.99 & 10.66 & 13.65 & 15.26 & 16.33 & 17.12 \\  & **1** & 1.24 & 1.27 & 1.68 & 1.89 & 2.21 & 2.89 & 3.14 & 3.31 & 3.43 \\  & **3** & 1.24 & 1.02 & 1.03 & 1.06 & 1.12 & 1.25 & 1.55 & 1.64 & 1.68 \\  & **5** & 1.29 & 1.04 & 1.00 & 1.00 & 1.00 & 1.02 & 1.06 & 1.17 & 1.30 \\  & **7** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **10** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\ \hline \hline \end{tabular}
\end{table}
Table 4: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(r_{w}\)} & **0.01** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.02 & 1.95 & 3.14 & 4.12 \\  & **0.1** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.24 & 1.43 & 1.53 & 1.60 \\  & **1** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
**0.01** & **3** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **5** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **7** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **10** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Normalized IAE for MPC (Process 2, setpoint response).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & & \multicolumn{3}{c}{**Damping Ratio (\(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\bm}}}}}}}}}}}}}}}}\) & & \\ \(r_{w}\) & \(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ }}}}}}}}}}}}}}}\) & **0** & **0.2** & **0.4** & **0.6** & **0.8** & **1** & **2** & **3** & **4** & **5** \\ \hline \multirow{9}{*}{**0.01**} & **0.01** & 4.60 & 2.46 & 1.66 & 1.27 & 1.01 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **0.1** & 5.29 & 2.82 & 1.93 & 1.46 & 1.16 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **1** & 25.81 & 13.35 & 9.00 & 6.71 & 11.16 & 2.64 & 1.54 & 1.17 & 1.00 \\  & **3** & 49.55 & 31.55 & 23.23 & 18.36 & 29.43 & 7.96 & 4.67 & 3.59 & 2.88 \\  & **5** & 61.52 & 43.36 & 33.66 & 27.36 & 41.17 & 12.87 & 7.73 & 6.06 & 4.82 \\  & **7** & 69.15 & 51.81 & 41.58 & 34.72 & 49.66 & 17.32 & 10.65 & 8.42 & 6.72 \\  & **10** & 77.69 & 61.53 & 51.09 & 43.72 & 59.61 & 23.36 & 14.81 & 11.77 & 9.60 \\ \hline \multirow{9}{*}{**0.1**} & **0.01** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **0.1** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **1** & 3.99 & 1.85 & 1.22 & 1.00 & 1.64 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **3** & 7.67 & 4.36 & 3.14 & 2.55 & 4.33 & 1.66 & 1.26 & 1.01 & 1.00 \\  & **5** & 9.52 & 5.99 & 4.55 & 3.80 & 6.06 & 2.68 & 2.08 & 1.71 & 1.50 \\  & **7** & 10.70 & 7.16 & 5.62 & 4.82 & 7.31 & 3.60 & 2.86 & 2.37 & 2.09 \\  & **10** & 12.02 & 8.50 & 6.90 & 6.07 & 8.78 & 4.86 & 3.98 & 3.