*4.5. The Poorly Damped Process*

A final case study is considered in this section, with the process described by the following transfer function:

$$P(s) = \frac{22.24}{s^2 + 0.6934s + 5.244}e^{-0.8s} \tag{22}$$

The indirect autotuning methods based on an S-shaped response cannot be applied for (21). An FO-PI controller tuned according to [36] is compared with a FO-PID obtained using the method in [40] and a FO-PI controller determined using [49]. First, the relay method is used to estimate the critical gain as *Kcr* = 0.0709 and *Pcr* = 2.8. These critical gain and period of oscillations allow the design of a FO-PID controller according to the first set of tuning rules in [42]. However, the proportional gain obtained in this way is negative and destabilizes the closed-loop system. Thus, the design is not included in this comparison. To tune the FO-PI controller [49], a sine test is firstly applied to the process to determine its phase, magnitude and phase slope. Then, the parameters of the FO-PI controller are determined such that the open-loop system achieves a gain crossover frequency of 0.09 rad/s and a phase margin of 75◦, along with the iso-damping property. The parameters of the fractional-order controllers are given in Table 10. Figure 18 shows the closed-loop results, as well as the required input signals. The performance measures are indicated in Table 11. The simulation results in Figure 18 and Table 11 show that the fastest settling time is achieved by the FO-PID controller [40], with a zero overshoot. However, in this case, the required control effort is the largest. The two FO-PI controllers determined using [36,49] have a similar overshoot, as well as control effort. For the latter, the settling and the disturbance rejection times are larger.


**Table 10.** FO-PID parameters computed for the poorly damped process.

**Figure 18.** (**a**) Output signals for FO-PID control of the poorly damped process. (**b**) Input signals for FO-PID control of poorly damped process. Controllers tuned according to [36,40,49].


**Table 11.** Closed-loop results obtained with the FO-PID controller for the poorly damped process.

For second-order poorly damped processes, most fractional order autotuning methods cannot be applied, except for [47,49]. The direct autotuning method in [47] leads to a FO-PI in series with a FO-PD controller, of the form given in (20), whereas the method in [49] produces a simpler FO-PI controller. Similarly to the results in Table 7, a faster settling time and better disturbance rejection are achieved using the fractional-order controller in [47], due to the FO-PD component.
