## 6 Comparisons with Other Methods
Since the RVy measure favours monotonic responses, while IAE favours fast responses, the combined measure (RVy \(\times\) IAE) offers the best compromise between speed and monotonicity. The RVy measures are based on the TV0y and TV2y measures introduced in References [48,128]. Note that the TV0y and TV2y measures are 0 for monotone responses (or responses with one maximum and one minimum). Since we multiply the RVy with IAE, the RVy is modified to give values of 1 instead of 0 under ideal circumstances. Like RVy \(\times\) IAE, the IAT\({}^{2}\)E measure favours fast and non-oscillatory responses. Note that the measurements of the criteria were taken after the process output completely settled, meaning that the simulations were longer than those shown for the closed-loop responses below. **Case 1**  
The next pure integrator plus time-delay is chosen as  
\[G_{\rm P1}(s)=\frac{e^{-s}}{s}. \tag{57}\]  
The tracking and the disturbance-rejection closed-loop responses between the aforementioned tuning rules are compared in Figures 26 and 27. The  
Figure 27: The process \(G_{\text{P1}}\) closed-loop responses. The PID controllers were tuned by the \(\text{PID}_{\text{E2}}\), \(\text{PID}_{\text{fh}}\), Ali–Majhi [77], \(\hat{\text{Astr\'{o}m}}\) 2, and \(\hat{\text{Astr\'{o}m}}\) 1.4 [18] methods. Figure 28: Criteria values for the process \(G_{\text{P1}}\). According to the IAE criterion, the proposed controllers ranked first, and according to the IAT\({}^{2}\)E criterion, they ranked first and third in tracking performance. The proposed PID\({}_{\text{fh}}\) controller also ranked first for the RVy \(\times\) IAE criterion. The Ali-Majhi method showed a large overshoot, and the Astrom 2 method exhibited a somewhat oscillatory response. The Taguchi method obtained the lowest criteria for disturbance-rejection. The proposed tuning method was ranked second. However, the Taguchi method exhibited a slightly oscillatory response. Concerning sensitivity M\({}_{\text{S}}\), [PERSON] and [PERSON] suggested M\({}_{\text{S}}\) = 2 for processes without zero, while the proposed method gave values below 3.3. **Case 2**  
The following slightly delayed first-order process is chosen:  
\[G_{P2}(s)=\frac{e^{-0.35s}}{s(1+1.35s)}. \tag{58}\]  
Figures 29 and 30 compare the closed-loop tracking and disturbance-rejection responses for all mentioned tuning rules. The criteria values of the aforementioned tuning rules are presented in Figure 31.  
Figure 31: Criteria values for the process \(G_{P2}\). Figure 30: The process \(G_{P2}\) closed-loop responses. The PID controllers were tuned by the \(\text{PID}_{\text{E}}\), \(\text{PID}_{\text{fh}}\), Ali–Majhi [77], \(\hat{\text{Astrôm 2}}\), and \(\hat{\text{Astrôm 1.4}}\)[18] methods. The lowest criteria for tracking was obtained by the proposed PID\({}_{\text{fh}}\) controller. The Medarametla method ranked second, and the PID\({}_{\text{f2}}\) ranked third. The Taguchi and the Ali-Majhi method exhibited a somewhat oscillatory response. For the disturbance-rejection performance, the criteria favoured the Medarametla method. However, the proposed method was ranked second, close to Medarametla. Like tracking, the Taguchi and Ali-Majhi methods exhibited oscillatory responses. Concerning sensitivity M\({}_{\text{S}}\), [PERSON] and Medarametla suggested M\({}_{\text{S}}\) = 2 for processes without zero, while the proposed method gave values below 3. **Case 3**  
The next second-order process with time-delay is chosen:  
\[G_{\text{P3}}(s)=\frac{e^{-s}}{s{(1+1.5s)}^{2}}. \tag{59}\]  
The tracking and disturbance-rejection closed-loop responses between the tuning rules are compared in Figure 32. Some methods are missing because they are not suitable for second-order process models. The criteria values of the tested tuning methods are presented in Figure 33.  
Figure 27:0 The process \(G_{P3}\) closed-loop responses. The PID controllers were tuned by the PID\({}_{\text{f2}}\), PID\({}_{\text{fh}}\), Taguchi [97], Astrôm 2, and Astrôm 1.4 [18] methods.