## 5 Evaluation
### Example 1: Interacting a Two Tank Liquid Level System
#### 5.2.1 Example 2: Closed Loop Stability Analysis  
The roots of the characteristics equation of pressure control process were calculated for different tuning methods using (17) and listed in Table 13. The location of roots in the \(z\)-plane is shown in Figure 22. It was observed that all the roots were located inside the unit circle. As mentioned in Section 5.1.1, the jury's stability test was performed on the pressure control process. The value of \(P(z)\) for \(z=1\) and \(z=-1\), \(\theta_{ref}\), coefficients \(c_{0}\) and \(c_{n}\) were calculated using (18), (21), (22) and (25). The values are tabulated in Table 14. The calculated values were checked for stability necessary conditions in (20) and (23)-(25). The value of \(c_{n}=1.5929\)e + 07, \(\theta_{Ref}\) = 0.3295 and the order of \(P(z)\) was an odd number. It was found that the RTD-A tuning parameter satisfied all the necessary conditions for the closed loop stability. It is guaranteed that pressure control will be stable with the RTD-A controller. The gain margin of the open loop pressure system was 35.3 dB and the closed loop gain margin with PID was 24.1 dB. The closed loop stability with PID is guaranteed if the proportional gain is chosen less than the maximum open loop gain margin value. The setpoint 53 psi was considered for the pressure control process. The RTD-A and PID controller tuned values were used to control the pressure process. The servo response with RTD-A for a different tuning method and PID is shown in Figure 23. The pressure response with RTD-A using optimization algorithms is shown in Figure 24. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Tuning Method** & **Ogunnaike** & **Kariwala** & **GSO** & **GA** & **PSO** & **FF** & **GWA** \\ \hline \multirow{4}{*}{Closed Loop Roots} & 0.6032 & 0.7360 & 0.2247 & 0.3681 & 0.1985 & 0.5930 & 0.4940 \\  & 0.3840 & 0.2621 & 0.6452 & 0.7198 & 0.4132 & 0.2589 & 0.7538 \\ \cline{1-1}  & 0.0108 & 0.0164 & 0.2302 & 0.2725 & 0.0837 & 0.1515 & 0.2980 \\ \cline{1-1}  & 0.0056 & 0.0056 & 0.0056 & 0.0056 & 0.0056 & 0.0056 & 0.0056 \\ \cline{1-1}  & 0.0054 & 0.0054 & 0.0054 & 0.0054 & 0.0054 & 0.0054 & 0.0054 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Example 2: Closed loop roots with different tuning methods. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & \(\theta_{R}\) & \(\theta_{T}\) & \(\theta_{D}\) & \(\theta_{A}\) \\ \hline GSO & 0.5121 & 0.0036 & 0.176 & 0.0352 \\ \hline GA & 0.4620 & 0.00101 & 0.2450 & 0.0255 \\ \hline PSO & 0.4097 & 0.8845 & 0.8346 & 0.0196 \\ \hline Firefly & 0.7398 & 0.7424 & 0.3032 & 0.0044 \\ \hline GWA & 0.4356 & 0.0083 & 0.3270 & 0.0163 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Example 2: Tuning parameters with an optimization tool. The execution time of RTD-A and PID controller implementation for the servo response of pressure control was 0.047388 s and 1.643077 s respectively when the simulation time was 100 s and sampling time was 0.1 s.  
Figure 23: Comparison of the servo response for the pressure control station with two tuning rules and PID. Figure 22: Example 2: \(z\)-plane plot for the roots of the characteristic equation. The regulatory response of the pressure control process without optimally tuned values and with optimally tuned values is shown in Figures 25 and 26 respectively. The setpoint 50 psi was considered in the investigation of disturbance rejection ability. The 10% of setpoint was applied as the disturbance to the process at 50 s in Figure 25 and at 25 s in Figure 26. The disturbance rejection ability of the optimally tuned RTD-A controller was better than the other method. Figure 24: Comparison of the servo response for the pressure control station with optimization algorithms. Figure 25: Example 2: Regulatory response for the pressure control station with tuning rules and PID. The overall performance comparison for example 2 is done in Table 15. There was no noticeable difference in the performance of pressure control among the optimally tuned responses excluding GSO.