## 6 Comparisons with Other Methods  
As far as we know, no other tuning method for IPs offers tuning based on both the measurement of a simple process open- or closed-loop time response in the time domain and a general process transfer function with an arbitrary order with time-delay. This represents a significant advantage over alternative tuning methods for PID controllers. Nevertheless, in this section, the results of the simulations of five different IP models that often appear in the literature are compared with those of other PID tuning methods for IPs. Notably, the selected tuning methods are not model-free controller tuning methods. Some methods cannot be used on arbitrary IP models. However, all the chosen methods provide analytical tuning formulas for the calculation of the controller parameters. The proposed tuning method for the PID\({}_{\text{f2}}\) and PID\({}_{\text{fh}}\) controllers was compared to five other methods [18; 77; 97; 103; 106]. The first method is [PERSON]'s method [77] (hereafter denoted as Ali-Majhi) for 1-DOF PID controllers. This method achieves optimal load disturbance-rejection for IPs with Integral of the Squared Error (ISE) criterion minimisation with Nyquist curve constraints, e.g., the slope has a specified inclination at the gain crossover frequency. The second method is that of [PERSON] and [PERSON] [97] (hereafter denoted as Taguchi) for 2-DOF PID controllers. This method optimises the settling time while keeping the overshoot under 20%. The third one is [PERSON] and [PERSON]'s method [106] (hereafter denoted as Medarametla), which is based on the loop sensitivity transfer function. This controller consists of a 1-DOF PID controller with a second-order Lead/Lag filter and a fourth-order reference filter that cancels some controller zeros. The fourth method is that of [PERSON] [103] (hereafter denoted as Anil-Padma), which employs a pole-placement strategy, and the tuning parameter is the M\({}_{\text{S}}\). The controller consists of a 2-DOF PID controller with a first-order Lead/Lag filter. The last is [PERSON] and [PERSON]'s method [18]. This method is based on selecting a maximal sensitivity value M\({}_{\text{S}}=1.4\) (hereafter denoted as Astrom 1.4) or M\({}_{\text{S}}=2.0\) (hereafter denoted as Astrom 2). The method employs a 2-DOF PID controller. Note that these methods employ different controller structures. The closed-loop tracking (\(r=1\)) and disturbance-rejection (\(d=1\)) responses were measured on all five process models. In all cases, the a priori chosen controller filter time constant was \(T_{F}=0.01\), with the exception of the Ali-Majhi method, where the controller filter time constant was set to the proposed value of \(T_{F}=T_{D}\times 0.1\). The calculated controller parameters (for all presented tuning methods and the proposed tuning method) for all five IP models are found in the Tables S1--S10 of the Supplementary Materials Section 1. The tracking and disturbance-rejection performance of the tuning methods can be evaluated by the following criteria:  
\[ \begin{array}{l}\text{IAE}=\int_{0}^{\infty}\lvert\text{el}(t)dt,\\ \text{IAT}^{2}\text{E}=\int_{0}^{\infty}t^{2}\cdot\lvert\text{el}(t)dt,\\ \text{RVy}\times\text{IAE},\end{array}  \tag{55}\]  
where RVy is the relative variance of the process output signal:  
\[ \begin{array}{l}\text{RVy (tracking)}=\ \frac{\int_{0}^{\infty}\lvert \frac{x(t)}{dt}\rvert_{dt}}{\lvert y(\infty)-y(0)\rvert},\\ \text{RVy (disturbance)}=\ \frac{\int_{0}^{\infty}\lvert\frac{2x(t)}{dt} \rvert_{dt}}{\lvert y_{1}-y(0)\rvert+\lvert y_{2}-y_{1}\rvert+\lvert y(\infty )-y_{2}\rvert},\end{array}  \tag{56}\]  
and \(y_{1}\) and \(y_{2}\) are the first and the second extrema of the process output signal. The RVy for tracking becomes \(1\) for a monotone process output signal. For any other response (overshoots, undershoots,or oscillations), the value will increase. The RVy for disturbance becomes 1 for the process output signal, which has only one maximum and one minimum. This is an ideal disturbance-rejection response.