*4.4. The FOPDT Delay-Dominant Process*

A FOPDT delay-dominant process is considered in the comparison, with the transfer function given by:

$$P(s) = \frac{1}{0.2s + 1}e^{-0.4s} \tag{21}$$

In this case, *k* = 1, *L* = 0.4, *T* = 0.2 and the critical gain is *Kcr* = 1.5202 and *Pcr* = 1.0985. Three indirect autotuning methods are used in the comparisons with the FO-PID controllers computed according to [32] using the first and the second set of rules for S-shaped process response. The third method is the F-MIGO method described in [26]. The resulting controller parameters are indicated in Table 8. Five direct autotuning methods are also used for the comparison, namely: FO-PID tuned according to the first and second set of rules in [42], FO-PID computed based on the method in [40] and two FO-PI controllers determined using [36,49]. The controller parameters for these cases are also given in Table 8. The FO-PI [49] is tuned to meet the iso-damping property, a gain crossover frequency 1.2 rad/s and a 70◦ phase margin. The closed-loop results are indicated in Figures 16 and 17, while the performance is evaluated using quantitative measures as indicated in Table 9. The FO-PID controller obtained using the second set of tuning rules in [42] is not included in the comparison, due to its highly oscillating nature.

**Table 8.** FO-PID parameters computed for the integrative time-delay process.


**Figure 16.** (**a**) Output signals for FO-PID control of delay-dominant process. (**b**) Input signals for FO-PID control of the delay-dominant process (controllers tuned according to [26,32,36] second set and [49]).

A small overshoot is obtained with the FO-PI controllers [26,36,49], combined with small settling times and fast disturbance rejection. The control effort in all these cases is similar, according to Figure 16b. FO-PID tuned using [40] manages to achieve a small settling time for this case study, as well. Good results are also obtained for disturbance rejection, at the expense of a larger control effort, compared to the other controllers (Figure 17b). FO-PIDs determined according to [32,42] lead to larger overshoots and increased settling times, as well as a poorer disturbance rejection, as indicated in Figure 17a. The required control effort for these controllers is small (Figure 17b), comparable to the input amplitudes given in Figure 16b.

**Figure 17.** (**a**) Output signals for FO-PID control of delay-dominant process. (**b**) Input signals for FO-PID control of the delay-dominant process (controllers tuned according to [40,42] first set and [32] first set).

**Table 9.** Closed-loop results obtained with the FO-PID controller for the delay-dominant process.

