# Design of PIDD(r) Controller for Robust Performance of Process Plants
## 3 Simulation Study
### Automatic Voltage Regulation System  
The AVR system regulates the terminal voltage of a generator to a predetermined nominal value by using a sensor located at the generator's output, which measures the actual value, calculates the error signal, and sends it back to the controller. The controller then sends a control signal to the amplifier, exciter, and generator, correcting the output voltage [13]. The AVR system can be represented as a transfer function given by the following equation:  
\[G_{p}(s)=\frac{0.1s+10}{0.0004s^{4}+0.045s^{3}+0.555s^{2}+1.51s+11}. \tag{13}\]  
The system response in Figure 16 shows that it is a stable system, although it reaches a steady state at around 10 s and suffers tremendous overshoot values. So, the control action must be able to reduce the overshoot value, rise time, and settling time to achieve its objective. All four controllers control the plant, similar to previous sections where the controller parameters, transient response characteristics, and step response are shown in Table 5 and Figure 17.  
Figure 16: Open-loop response of an AVR system.  
Figure 15: Disturbance rejection of controller structure for magnetic levitation system.  
From the simulation results, the PIDD\({}^{2}\) controller and the proposed PIDD\({}^{\alpha}\) controller exhibit superior performance in all three transient response characteristics. Regarding overshoot performances, PIDD\({}^{2}\) and PIDD\({}^{\alpha}\) trump their competitors with values of 0.2782% and 0.1923% compared to PID and FOPID, with values of 4.7876% and 3.8753%, respectively. The same is true for \(t_{r}\), where PIDD\({}^{2}\) and PIDD\({}^{\alpha}\) achieve 0.0245 s and 0.0232 s compared to 0.2582 s and 0.2196 s for PID and FOPID. For settling time, PIDD\({}^{2}\) and PIDD\({}^{\alpha}\) achieved 0.0373 s and 0.0984 s, respectively, compared to the 1.1458 s and 0.9012 s that PID and FOPID achieved. The proposed controller performs better between PIDD\({}^{2}\) and the proposed PIDD\({}^{\alpha}\) controller with the most negligible overshoot value and \(t_{r}\) of 0.1923% and 0.0232 s but at a higher \(t_{s}\) of 0.0984 s.  
The set-point is initially set to 1, then changes to 2, 3, and 1.5 at 2 s, 4 s, and 6 s, respectively. These timed set-point changes ensure that step responses can reach their respective steady-state values. Each controller structure in Figure 18 can effectively control the output to reach the set-point changes. Regarding disturbance rejection, a disturbance magnitude of \(-\)0.3 is introduced to the controller structure at 2 s to simulate real-world plant behaviour. Figure 19 shows that each controller structure can reject the disturbance by effectively controlling the output to reach the set-point following the step response. The proposed PIDD\({}^{\alpha}\) controller outperforms competitors, delivering the best set-point tracking and disturbance rejection performance.  
Figure 17: Step response of controller structure for an AVR system.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Controller** & \(K_{p}\) & \(K_{t}\) & \(K_{dt}\) & \(K_{dt2}\) & \(\mu\) & \(\beta\) & \(\alpha\) & \(Mp\) (\%) & \(t_{r}\) (\(s\)) & \(t_{s}\) (\(s\)) \\ \hline PID & 4.8938 & 3.2301 & 2.2479 & - & - & - & 4.7876 & 0.2582 & 1.1458 \\ PIDD\({}^{2}\) & 4.8938 & 3.2301 & 2.2479 & 0.2048 & - & - & - & 0.2872 & 0.0245 & 0.0373 \\ FOPID & 4.8938 & 3.2301 & 2.2479 & - & 1.1122 & 1.0624 & - & 3.8758 & 0.2196 & 0.9012 \\ PIDD\({}^{\alpha}\) & 4.8938 & 3.2301 & 2.2479 & 0.2048 & - & - & 2.01 & 0.1923 & 0.0232 & 0.0984 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Controller parameters and transient response characteristics for an AVR system.