## 5 Evaluation
### Example 1: Interacting a Two Tank Liquid Level System
#### 5.1.1 Example 1: Closed Loop Stability Analysis  
The stability of the closed loop liquid level control and pressure control with the RTD-A controller was checked according to [PERSON] et al. and [PERSON] et al. (2011, 2012). The stability analysis concept in the state variable form was discussed in Section 2.1.1. It is necessary that the roots of the characteristics equation in (17) should lie within the unit circle for closed loop stability. The roots of the liquid level control characteristic equation in the state variable form were calculated using (17) for different tuning methods and listed in Table 9. The location of roots in the \(z\)-plane was plotted and shown in Figure 19. Since all the roots were located inside the unit circle, the system was stable for all tuning methods. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **GSO** & **GA** & **PSO** & **Firefly** & **GWA** \\ \hline Maximum & 125 & 30 & 30 & 50 & 70 \\ Iteration & 100 & 50 & 70 & 60 & 50 \\ Swarm Size & 100 & 50 & 70 & 60 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Iteration and swarm size used in the optimization algorithm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & \multicolumn{2}{c}{**Parameter**} \\ \hline  & \multicolumn{2}{c}{Inertia coefficient = 1} \\ PSO & \multicolumn{2}{c}{Damping coefficient = 0.99} \\  & \multicolumn{2}{c}{Acceleration coefficients c1 = 1.5, c2 = 2} \\ \hline Firefly & \multicolumn{2}{c}{\(\alpha=0.2\), \(\beta=2\), \(\gamma=1\)} \\ \hline GSO & Acceleration coefficients c1 = c2 = c3 = c4 = 2.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Additional parameters for galactic swarm optimization (CSO), PSO and firefly. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & \multicolumn{2}{c}{**\(\theta_{R}\)**} & \multicolumn{1}{c}{\(\theta_{T}\)} & \multicolumn{1}{c}{\(\theta_{D}\)} & \multicolumn{1}{c}{\(\theta_{A}\)} \\ \hline GSO & 0.9597 & 0.4782 & 0.8880 & 0.0356 \\ GA & 0.9581 & 0.00103 & 0.67 & 0.0211 \\ PSO & 0.8931 & 0.2532 & 0.5945 & 0.03 \\ Firefly & 0.9663 & 0.0001 & 0.519 & 0.033 \\ GWA & 0.9470 & 0.0015 & 0.914 & 0.0016 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Example 1: Tuning parameters with an optimization tool. The stability could also be analyzed on the basis of the Jury's stability criterion discussed in 2.1.2. The coefficients \(c_{0}\) and \(c_{n}\) were calculated using (21) and (22). The value of the polynomial \(P(z)\) when \(z\) = 1 and \(z\) = \(-1\) was calculated according to (18). The \(\theta_{\mathit{ref}}\) value also was calculated to check the stability in terms of \(\theta_{R}\) using (25). The calculated values are tabulated in Table 10. The value of \(c_{n}\) = 51,164, \(\theta_{\mathit{ref}}\) = 0.07 and the order of \(P(z)\) was an even number. The necessary conditions in (20), (23), (24) and (25) were checked for stability. It was found that all the conditions were satisfied for closed loop stability for the calculated tuning values. Therefore, the designed RTD-A controller ensured closed loop stability in liquid level control. A similar analysis was done for pressure control in Section 5.2.1. The open loop gain margin (GM) of the liquid level process was 4.1 dB and the closed loop GM with PID was 5.86 dB. Thus the closed loop stability with PID would be ensured by choosing the proportional gain value less than open loop gain margin value.