## 3 Results
### Summary of Results
#### 3.5.1 Process 1  
Generally, MPC gives better performance for underdamped systems and systems with larger time delays. On the other hand, the PID controllers give better performance when the time delays are smaller and the process is overdamped, when both TV and IAE are considered. For disturbance rejection, MPC moved relatively less across all values of \(r_{w}\). Hence, both aggressive and conservative MPC controllers moved less than the corresponding PID controllers, which is suitable for actuators and controller robustness. The IAE values for disturbance rejection show that the MPC had better performance when \(\frac{\theta}{\tau}\) is at least 3. However, there is a noticeable drop in performance from left to right of the  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & & \multicolumn{4}{c}{**Damping Ratio (\(\mathbf{\xi}\))**} \\ \(r_{w}\) & **0** & **0.2** & **0.4** & **0.6** & **0.8** & **1** & **2** & **3** & **4** & **5** \\ \hline \multirow{8}{*}{**10**} & **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 \\  & **0.1** & 1.84 & 1.05 & 1.00 & 1.00 & 1.36 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **3** & 3.56 & 2.50 & 1.76 & 1.28 & 3.69 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **5** & 4.45 & 3.49 & 2.59 & 1.95 & 4.84 & 1.47 & 1.00 & 1.00 & 1.00 \\  & **7** & 4.97 & 4.16 & 3.23 & 2.48 & 5.51 & 1.94 & 1.03 & 1.00 & 1.00 \\  & **10** & 5.48 & 4.91 & 3.96 & 3.16 & 6.10 & 2.55 & 1.42 & 1.00 & 1.00 \\ \hline \hline \end{tabular}
\end{table}
Table 13: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & \multicolumn{4}{c}{**Damping Ratio (\(\mathbf{\xi}\))**} \\ \(r_{w}\) & **0** & **0.2** & **0.4** & **0.6** & **0.8** & **1** & **2** & **3** & **4** & **5** \\ \hline \multirow{8}{*}{**0.01**} & **0.01** & 1.00 & 1.21 & 1.29 & 1.37 & 1.00 & 1.00 & 2.33 & 4.83 & 5.95 \\  & **0.1** & 1.01 & 1.26 & 1.44 & 1.54 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **1** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **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 \multirow{8}{*}{**0.1**} & **0.01** & 2.73 & 3.04 & 2.83 & 2.61 & 1.00 & 1.00 & 2.76 & 5.80 & 7.48 \\  & **0.1** & 2.68 & 3.03 & 3.03 & 2.86 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **1** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & **3** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 table for \(\frac{\theta}{\tau}<3\), i.e., with decreasing damping ration \(\zeta\). For the setpoint response, the TV values when \(0.01\leq r_{w}\leq 0.1\) indicate that the MPC controller moved more when \(\frac{\theta}{\tau}>3\) for all values of \(\zeta\) or when \(\frac{\theta}{\tau}>1\) and \(\zeta<1\). The corresponding IAE values show that the MPC controller had lower IAE values for \(\frac{\theta}{\tau}\geq 1\). For \(\frac{\theta}{\tau}\leq 1\), there is a noticeable increase in IAE values from left to right of the table, i.e., with increasing damping. The boundary of the performance improvement in terms of IAE is seen to gradually move from \(\frac{\theta}{\tau}\geq 1\) for \(r_{w}=0.01\) to \(\frac{\theta}{\tau}\geq 5\) for \(r_{w}=10\).  
Hence, MPC tends to give better performance than PID when \(\frac{\theta}{\tau}>1\), irrespective of the damping ratio. But when \(\frac{\theta}{\tau}\leq 1\), the PID controller tends to give a better performance. It is also observed that the relative performance of MPC increases as damping is decreased. Hence, even for lower values of \(\frac{\theta}{\tau}\), MPC is likely to perform better if the system is underdamped.