## 4 Examples and Comparisons
### Comparisons with Other Methods  
To the best of our knowledge, no other tuning method for IP offers both tuning based on a general process transfer function with arbitrary order with time-delay, and on the measurement of a simple process open- or closed-loop time response in the time-domain. In practical applications, this represents a substantial advantage over other tuning methods. However, the proposed tuning method was compared to some other existing tuning methods for PI controller. Selected methods are suitable for IP and employed a relatively simple experiment on the process to calculate the controller parameters. We compared four different IP models that are frequently encountered in the literature. The reference-tracking (\(r=1\)) and disturbance-rejection responses (\(d=0.1\)) were measured on all processes. Tracking and disturbance-rejection performances were evaluated by the following criterion:  
\[ITSE=\int_{0}^{\infty}t\cdot e^{2}(t)dt, \tag{31}\]  
where \(e(t)\) is control error (\(r\)-\(y\)). The integral-time-square-error (ITSE) criterion favours fast and non-oscillatory responses. Tracking response was additionally evaluated by measuring 5% settling time. In the proposed tuning method, controller parameter \(b=0.5\) was used in all four cases. This represents a trade-off between tracking and disturbance-rejection performance, and the reference-tracking or disturbance-rejection performance of the proposed method can be additionally improved/adjusted by changing parameter \(b\). **Case 1**  
The first comparison was performed on the first-order process with a relatively small delay:  
\[G_{\mathrm{PI}}(s)=\frac{e^{-0.05s}}{s(1+s)}. \tag{32}\]  
Figure 12: Closed-loop amplitude responses for all four tested processes. On the basis of the given process model, areas were determined from Equation (9):  
\[A_{0}=1,\ A_{1}=1.05,\ A_{2}=1.0513. \tag{33}\]  
Then, the controller was designed according to Equations (12) and (14):  
\[K_{P}=0.48,\ K_{l}=0.087,\ b=0.5. \tag{34}\]  
The obtained controller was compared to three other methods [1; 37; 65]. The first one was [PERSON]'s method [1]. This method is based on choosing maximal sensitivity value Ms = 1.4 (_Ms_14) or Ms = 2.0 (_Ms_20). The second method was that of [PERSON] and Araki (TA) [65]. This method calculates the controller parameters in order to optimize settling time while keeping overshoot below 20%. The third method was that of [PERSON] (ALI) [37]. This method optimises the integral-square-error (ISE) criterion and the shape of the Nyquist curve. The calculated PI controller parameters for the presented methods are given in Table 3.  
Figure 13 compares closed-loop responses for tracking and for disturbance-rejection between the mentioned tuning rules applied to 2-DOF PI controllers. The values of the ITSE criterion for all tested methods are given in Table 4 (for reference tracking) and Table 5 (for disturbance-rejection). Tracking settling times within \(\pm 5\%\) of the final value were comparable for the first four methods (see Table 6), where MOMI ranked second, just after Ms14. The Ms20 method showed a somewhat large overshoot, while the TA and ALI methods exhibited an oscillatory response. According to the ITSE criterion, reference-tracking performance (Table 4) was the best with the TA method. The lowest ITSE criterion for disturbance-rejection was obtained with the ALI method (measurement of disturbance-rejection was taken after the process output completely settled). However, due to a relatively high degree of oscillations, neither the TA nor the ALI method could be considered as the most appropriate for the \(\mathrm{G_{P1}}\) process. The proposed method response was between methods Ms20 and Ms14, faster than Ms20 but slower than Ms14.