## 5 Evaluation
### Example 1: Interacting a Two Tank Liquid Level System
The servo response with the PID and RTD-A for zero uncertainty and zero disturbance is shown in Figure 10. Similarly, the servo response with RTD-A tuned using the optimization algorithm is shown in Figure 11.  
Figure 8: Open-loop step response with reduced models. Figure 9: Level control system with the RTD-A controller. The efficiency of the proposed tool was examined by adding the uncertainty parameter (\(\lambda\)) and disturbance to the plant. The RTD-A and PID controlled servo response with disturbance and uncertainty are shown in Figures 12 and 13. The RTD-A controller requires values for the sampling time, noise and uncertainty parameter as an input. The tool II was provided with the same input parameters for comparing the two different tuning algorithms. Whenever the user chooses the tuning rule, the tool will acclimate to the chosen tuning rule. In Figure 12, the response with the Ogunnaike tuning method was more susceptible to noise and also it exhibited a more aggressive controller response. It had some overshoot as compared with the Kariwala tuning method. The process response with the first method deviated more from the setpoint and this deviation increased further with the increase in the uncertainty parameter (\(\lambda\)). The Kariwala, tuning method was more flexible with respect to the controller aggressiveness and uncertainty. The response with PID had more rise and settling time with less overshoot and control variation than the Ogunnaike method. In Figure 13 the servo response with optimal tuned parameters after adding uncertainty is shown. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Ogunnaike** & **Kariwala** \\ \hline \(\theta_{R}\) & 0.5 & 0.95 \\ \(\theta_{T}\) & 0.9180 & 0.9207 \\ \(\theta_{D}\) & 0.5 & 0.0405 \\ \(\theta_{A}\) & 0.1971 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Example 1: Tuning parameters with two tuning methods [4]. Figure 10: Example 1: Servo response with the RTD-A controller using two tuning methods. Figure 11: Example 1: Servo response with the RTD-A controller using optimization algorithms. Figure 12: Example 1: Servo response with the RTD-A controller with disturbance and uncertainties. The disturbance rejection ability of the RTD-A controller was experimented and the regulatory response without optimally tuned parameters is shown in Figure 14. In Figure 15, the response with disturbance rejection is shown for some optimally tuned values. The disturbance of 20% of setpoint was applied to the process at 50 s. It was found that all the controllers were able to reject the disturbance effect on the process output. Figure 14: Example 1: Regulatory response with the RTD-A controller using tuning rules and proportional integral derivative (PID). Figure 13: Example 1: Servo response with the RTD-A controller with uncertainty using optimization algorithms. The unit step change in setpoint was applied to the closed loop system and time response was obtained. The time domain specifications could be calculated from the step response of the closed loop system using the stepinfo command in MATLAB. The RTD-A controller performance for two tuning rules and optimal tuning was compared in terms of the integral absolute error (IAE) and other time domain specifications with PID performance and it is presented in Table 5. In this example performance with RTD-A was better than PID. Between the first two tuning rules, the Kariwala tuning rule-based process response had a good performance compared to the Ogunnaike tuning method excluding the rise time. However, in the overall comparison the firefly algorithm-based result was slightly better than the other methods. The execution time of the RTD-A and PID controller implementation for the servo response of level control without uncertainty was 0.034279 s and 0.062699 s respectively.