*4.3. The Integrating Time-Delay Process*

An integrating time-delay process [46] is considered here. The transfer function is:

$$P(\text{s}) = \frac{0.55}{s(0.6\text{s} + 1)} e^{-0.05\text{s}} \tag{19}$$

The classical Ziegler-Nichols autotuning method has a major disadvantage: poor results are obtained regarding setpoint tracking, especially when used with integrating systems [1]. Several extensions and improvements have been developed over the years to deal with such systems. The autotuning methods based on an *S*-shaped response of the process cannot be used in this particular situation.

In [47], an iterative experiment of a relay with delay is applied to the process in order to determine the process magnitude, phase and phase slope at a specific gain crossover frequency 2.3 rad/s. Then, a FO-PI in series with a FO-PD controller are designed to meet the iso-damping property, a gain crossover frequency of 2.3 rad/s for the open-loop system and a phase margin of 72◦. The resulting fractional-order controller is given by [47]:

$$\mathcal{L}\_{MONIE}(s) = \left(\frac{0.4348s + 1}{s}\right)^{1.1803} \left(\frac{3.7282s + 1}{0.0037s + 1}\right)^{1.1580} \tag{20}$$

Four other direct autotuning methods are used for comparison purposes. First, based on the relay test, the critical gain is *Kcr* = 36.88 and *Pcr* = 1.1043. The parameters of the fractional-order controllers used for comparative purposes are indicated in Table 6, where the fractional-order controllers have been determined using [40,42,49]. The first and the third set of tuning rules in [42] are used to estimate the FO-PID controller parameters, as the second set cannot be applied for the process in (19). The tuning rules in [42] were developed for integrative processes without time delays (third set) and for FOPDT processes (first and second set). For the process in (19), the third set of tuning rules [42] leads to an unstable controller. Figure 15 shows the simulation results. The quantitative performance results are indicated in Table 7. Similar overshoot is obtained for the fractional-order controller in (20) designed using [47] and for the FO-PI controller [49], despite the latter having a large settling time. The fastest FO-PID controller is yet again the one designed according to [40]. The poorest overshoot along with a significant settling time is obtained with the FO-PID [42]. A comparison of the required control effort based on Figure 15b,c shows the increased amplitudes of the input signals are necessary for the FO-PIDs designed based on [40,42,47]. The smallest control effort is required by the FO-PI controller tuned according to [49], which also exhibits the largest settling time and a significant disturbance rejection time. However, this controller is also the simplest one, without any derivative effect.

**Table 6.** FO-PID parameters computed for the integrative time-delay process.


**Figure 15.** (**a**) Output signals for FO-PID control of integrative time-delay process. (**b**) Input signals for FO-PID control of integrative time-delay process required for setpoint tracking. (**c**) Input signals for FO-PID control of integrative time-delay process required for disturbance rejection. Controllers tuned according to [40,42,47,49].

**Table 7.** Closed-loop results obtained with the FO-PID controller for the integrative time-delay process.

