*4.1. The FOPDT Lag-Dominant Process*

The following FOPDT lag-dominant process taken from [26] is considered:

$$P(s) = \frac{2.4351}{12.5688s + 1}e^{-1.0787s} \tag{17}$$

In this case, *k* = 2.4315, *L* = 1.0787, *T* = 12.5688. Based on the relay test, the critical gain is *Kcr* = 7.78 and *Pcr* = 4.175. The parameters of the fractional-order controllers used for comparative purposes are indicated in Table 2. Indirect [26,27,32] and direct tuning methods [36,40,42,49] are used. The direct autotuning method in [34] produces the same result as in [49] and, therefore, was omitted from the comparison. Only the first set of tuning rules in [32] is used, as the second set of tuning rules cannot be applied. The same is valid for the direct autotuning method in [42], where only the first set of tuning rules is used, since the other two sets of tuning rules cannot be applied to this particular process.

**Table 2.** FO-PID parameters computed for the lag-dominant process.


The FO-PID [42] leads to a highly oscillating closed-loop response, while the FO-PID [32] is an unstable controller, which suggests that the proposed tuning rules work poorly for the lag-dominant system in (17). In fact, in both cases the expected phase margin is 38◦ [32,42], which explains the highly oscillating character. The FO-PI [49] was tuned to meet the iso-damping property, as well as a gain crossover frequency of 0.2 rad/s and a phase margin of 70◦. These performance specifications were selected in order to obtain a small overshoot, as well as the fastest possible settling time. The closed-loop results considering step reference tracking and disturbance rejection are given in Figure 11, while the numerical values of the overshoot, settling time and disturbance rejection time are given in Table 3. The results show that the smallest overshoot is obtained using the FO-PI in [49], at the expense of a large settling time and the time required to reject the load disturbance. Small overshoot is obtained also using the FO-PID of Tepljakov in [27] or using the F-MIGO method [26], while the settling time is slightly larger in the latter case. However, the required control effort for Tepljakov's FO-PID [27] is extremely large compared to the other methods, as indicated in Figure 11b). A larger control effort is also observed in the case of the FO-PI controller in [40], which achieves the fastest settling time and the smallest disturbance rejection time. A decent control effort is necessary when using FO-PI controllers tuned according to [26,36,49]. Among these, the fastest settling time is obtained with the FO-PI controller [26], while the smallest overshoot is achieved by the FO-PI controller [49]. Improved settling time might be possible in this last case, if a FO-PD controller is designed and implemented in series with the FO-PI.

**Figure 11.** (**a**) Output signals for FO-PID control of lag-dominant process (**b**) Input signals for FO-PID control of lag-dominant process. Controllers tuned according to [26,27,36,40,49].

**Table 3.** Closed-loop results obtained with the FO-PID controller for the lag-dominant process.


*4.2. The Higher Order Process*

The following higher order process taken from [36] is considered:

$$P(s) = \frac{1}{\left(s+1\right)^4} \tag{18}$$

In this case, *k* = 1, *L* = 1.42, *T* = 2.92 and the critical gain is *Kcr* = 4 and *Pcr* = 6.28 [36]. The parameters of the fractional-order controllers used for comparative purposes are indicated in Table 4. Indirect [26,32] and direct tuning methods [36,40,42,49] are used. The direct autotuning method in [34] produces the same result as in [49] and therefore was omitted from the comparison. Only the first set of tuning rules in [32] is used, as the second set of tuning rules cannot be applied. The same is valid for the direct autotuning method in [42], where only the first set of tuning rules is used, since the other two sets of tuning rules cannot be applied to this particular processes.

**Table 4.** FO-PID parameters computed for the higher order process.


Figure 12 shows the closed-loop results obtained with the first three controllers in Table 4, while Figure 13 presents the closed-loop simulations obtained with the last three controllers. Note that the FO-PI [49] controller was first tuned for a gain crossover frequency of 0.5 rad/s and a phase margin of 38◦. This is in agreement to the performance specifications used in [32,42] for the first set of tuning rules. The results in Figure 13 show that indeed similar overshoot and settling times are obtained with the fractional-order controllers tuned according to [32,42,49].

**Figure 12.** (**a**) Output signals for FO-PID control of higher order process (**b**) Input signals for FO-PID control of higher order process (controllers tuned according to [26,36,40]).

**Figure 13.** (**a**) Output signals for FO-PID control of higher order process (**b**) Input signals for FO-PID control of higher order process (controllers tuned according to [32,42,49]).

Table 5 contains the performance evaluation of the fractional-order controllers from Figure 12. The remaining three controllers in Figure 13 are not evaluated due to the large overshoot and settling time. The direct autotuning method from [49] can be used to tune a better FO-PI controller. Table 4 shows the resulting parameters of this improved FO-PI controller, which was tuned to meet a gain crossover frequency of 0.2 rad/s and a phase margin of 75◦. The performance of this better FO-PI controller is compared to that of the FO-PI controller in [36], which achieves the best overshoot and settling time. The comparative simulation results are given in Figure 14 and in Table 5.

**Table 5.** Closed-loop results obtained with the FO-PID controller for the higher order process.


**Figure 14.** (**a**) Comparative results for the output signals for FO-PID control of higher order process. (**b**) Input signals for FO-PID control of higher order process (controllers tuned according to [36,49]).

To estimate the quantitative results given in Table 5 for the FO-PI and FO-PID controllers designed according to [26,40], the closed-loop simulation results from Figure 12 were considered. The quantitative results in Table 5 show that the smallest overshoot is possible using the FO-PI controller tuned using the methods in [36,49], which are also suitable to achieve a quick disturbance rejection. Similarly to the lag-dominant case study, in this case as well, the FO-PID controller [40] achieves the smallest settling time and the fastest disturbance rejection time, at the expense of an increased control effort similar to that of the FO-PID [42] and larger compared to the other controllers.
