## 6 Comparisons with Other Methods
#### Case 4  
The following high-order non-minimum phase process is chosen:  
\[G_{P4}(s)=\frac{0.5(1-0.5s)e^{-0.7s}}{s(1+0.4s)(1+0.1s)(1+0.5s)}. \tag{60}\]  
An experiment using this process was already carried out by [PERSON] and [PERSON] [106] ([PERSON]), and [PERSON] and [PERSON] [103] ([PERSON]).  
The tracking and the disturbance-rejection closed-loop responses between the aforementioned tuning rules are compared in Figures 34 and 35. Some methods are missing because they are not suitable for higher-order process models. The criteria values are presented in Figure 36.  
Figure 33: Criteria values for the process \(G_{P3}\).  
Figure 34: The process \(G_{P4}\) closed-loop responses. The PID controllers were tuned by the PID\({}_{2}\), PID\({}_{\text{fh}}\), Anil–Padma [103], and Medarametta [106] methods.  
Figure 35: The process \(G_{P4}\) closed-loop responses. The PID controllers were tuned by the PID\({}_{2}\), PID\({}_{\text{fh}}\), Astrôm 2, and Astrôm 1.4 [18] methods.  
The lowest IAE and the IAT\({}^{2}\)E criteria for tracking were obtained with the PID\({}_{\text{fh}}\) controller. The Medarametla method was ranked second, and the proposed PID\({}_{\text{f2}}\) controller ranked third. Similarly, the RVy \(\times\) IAE criterion favoured the proposed PID\({}_{\text{fh}}\) controller, the Anil-Padma method was ranked second, and the Medarametla method ranked third.  
Similar results were obtained for the disturbance-rejection performance. The proposed PID\({}_{\text{f2}}\) and PID\({}_{\text{fh}}\) controllers were ranked first, and the Medarametla method was ranked second.  
Concerning sensitivity M\({}_{\text{S}}\), Anil-Padma and Medarametla suggested M\({}_{\text{S}}\) = 2.81 for this process, while the proposed method gave values below 3.2. As expected, the M\({}_{\text{S}}\) values under the Astrom methods were lower since the tuning method is based on the M\({}_{\text{S}}\) values.  
**Case 5**  
The next high-order process is chosen:  
\[G_{P4}(s)=\frac{1}{s(1+0.25s)^{8}}. \tag{61}\]  
The tracking and the disturbance-rejection closed-loop responses between the chosen tuning rules are compared in Figure 37. The criteria values are presented in Figure 38.  
Figure 36: Criteria values for the process \(G_{P4}\).  
Figure 37: The process \(G_{\text{PS}}\) closed-loop responses. The PID controllers were tuned by the PID\({}_{\text{2}}\), PID\({}_{\text{Th}}\), Aström 2, and Aström 1.4 [18] methods.  
Figure 38: Criteria values for the process \(G_{\text{PS}}\).  
Only the proposed and Astrom methods were appropriate for the high-order process. According to the criteria, the proposed controllers (PID\({}_{\text{I2}}\) and PID\({}_{\text{fh}}\)) ranked first in tracking and disturbance-rejection responses. As expected, the M\({}_{\text{S}}\) values under the Astrom methods were lower since the tuning method is based on the M\({}_{\text{S}}\) values.  
All experiment results showed that the proposed PID\({}_{\text{I2}}\) and PID\({}_{\text{fh}}\) controllers, compared to some other tuning methods based on similar tuning procedures, result in very good tracking and disturbance-rejection performance for a wide range of process models. The proposed controllers were ranked mostly first. The closed-loop responses were stable and fast, with only slight overshoots.