**2. Indirect Fractional Order Autotuning Methods**

A popular indirect autotuning method, suitable only for FO-PIs, was developed by extending the *M*<sup>s</sup> constrained integral (MIGO)-based controller design approach [26]. F-MIGO tuning determines the optimum parameters of the FO-PI controller such that the load disturbance rejection is optimized, with a constraint on the maximum or peak sensitivity. The F-MIGO method provides the tuning rules for the FO-PI controller provided that the process step response has an S-shaped form, as indicated in Figure 3, that could be approximated by the following transfer function:

$$P(s) = \frac{k}{Ts+1}e^{-Ls} \tag{7}$$

where *k* is the process gain, *L* is the delay and *T* is the time constant of the process. The relative dead time can be computed as:

$$
\pi = \frac{L}{T+L} \tag{8}
$$

Systems where *L* >> *T* are delay dominant, whereas systems in which *T* >> *L* are lag dominant. Research studies performed in [26] revealed that the FO-PI fractional order is almost independent of *L*, but depends on the relative dead time. For some particular situations, where 0.4 ≤ *τ* < 0.6, an integer order PI controller was determined to be more suitable for controlling the process. A summary of the results is indicated below:

$$
\lambda = \begin{cases}
 & \text{1.1}, & \tau \ge 0.6 \\
 & \text{1}, & 0.4 \le \tau < 0.6 \\
 & 0.9, & 0.1 \le \tau < 0.4 \\
 & 0.7, & \tau < 0.1
\end{cases}
\tag{9}
$$

The proportional and integrative gains of the FO-PI controller were also determined as a function of the relative dead time:

$$k\_p = \frac{1}{k} \frac{0.2978}{\tau + 0.00037} \text{ and } T\_i = T \frac{0.8578}{\tau^2 - 3.402\tau + 2.405} \tag{10}$$

An indirect autotuning method that applies to the S-shaped step response process was developed in [27]. The tuning is unnecessarily complicated as the parameters of (7) are firstly estimated and then used to determine the process critical frequency *ωcr* and critical gain *kcr*, according to:

$$
\omega\_{cr}T = -\tan(\omega\_{cr}L) \text{ and } k\_{cr} = -\frac{1+\omega\_{cr}T^2}{k(\cos(\omega\_{cr}L) - \omega\_{cr}T\sin(\omega\_{cr}L))}\tag{11}
$$

Then, the parameters of an integer order PID are determined using the previously computed process critical frequency and gain, as well as three additional design parameters referring to the ratio of the integral and derivative time constants, loop phase and gain:

$$k\_p = k\_{cr} r\_b \cos \mathcal{Q}\_{b^\*} \ T\_i = -\frac{T\_{cr}}{\pi} \frac{\pi \cos \mathcal{Q}\_b}{\sin \mathcal{Q}\_b + 1} \text{ and } T\_d = \varkappa T\_i \tag{12}$$

where *<sup>α</sup>*, *rb* and ∅*<sup>b</sup>* are design parameters [28]. Once the PID controller parameters are computed, a possible range for the fractional orders in the FO-PID is selected and an optimization routine is performed. The algorithm attempts to minimize the integral time absolute error with the open loop gain and phase margin imposed as design specifications.

Another indirect tuning method is proposed in [29] for processes that produce an S-shaped step response. The method is based on determining first the process dead time *L* and time constant *T*, as well as the value at which the system reaches steady state *k*. The standard Ziegler-Nichols equations are used then to estimate the *kp*, *Ti* and *Td* parameters of an integer order PID. Then, the fractional orders of differentiation and integration are determined by the Nelder-Mead optimization algorithm in order to meet certain phase and gain margins. A second approach based on the standard Cohen-Coon method is also used in [29], for processes that exhibit first order plus dead time dynamics. Based on the process parameters, the integer order PID parameters are first computed. The Nelder-Mead optimization algorithm is used afterwards to estimate the fractional orders of differentiation and integration based on certain phase and gain margin requirements. The Cohen-Coon tuning method is proposed as an alternative to the Ziegler-Nichols approach in order to improve the slow, steady state response of the latter.

An indirect autotuning method for designing only FO-PI controllers using the Ziegler-Nichols open-loop approach is described in [30]. The parameters of the integer order PI controller are firstly determined using the standard Ziegler-Nichols approach. In order to improve the overall closed-loop response, the research suggests that the PI performance can be improved a lot with a fractional order of integration. An error filter as proposed in [31] is used for steady state error compensation:

$$G\_t(s) = \frac{s+n}{s} \tag{13}$$

where *n* is chosen to be small enough so that high frequency specifications are maintained and the system gain will not be altered drastically. The research in [30] proposed a modification of (13) such that the value of *n* is adjustable with respect to the fractional order of integration. The tuning of the fractional order and of the filter is performed by trial and error for a specific type of process. The method is evaluated experimentally on a steam temperature process and compared to the F-MIGO method [26] in terms of robustness for set point changes and disturbance rejection. The proposed controller shows better performance compared to the F-MIGO autotuning method, but it also requires higher control effort.

In [32], two existing analytical methods for tuning the parameters of fractional PIDs are reviewed. Then, for two specific sets of performance criteria similar to (2)–(6), the corresponding sets of tuning rules are developed based on an optimization method applied to the FO-PID control of an *S*-shaped process dynamics similar to (7). The newly developed tuning rules for fractional order PIDs use the time delay *L* value and the estimated process time constant, *T*, much like the standard S-shaped Ziegler-Nichols approach, to produce the controller parameters. The method works provided the step response of the process is *S*-shaped. These two methods were initially presented in [33]. The first set of rules developed works if 0.1 ≤ *T* ≤ 50 and *L* ≤ 2, while the second set of rules can be applied for processes with 0.1 ≤ *T* ≤ 50 and *L* ≤ 0.5. Both sets of rules are determined in a similar manner. For a batch of process described as FOPDT systems, a set of performance specifications is imposed. The set included values for the gain crossover frequency, for the phase margin, a high-frequency value for the improved high-frequency noise cancellation and the corresponding maximum magnitude limit, as well as a low frequency value for improved output disturbance and the corresponding maximum magnitude. Tuning by minimization is then applied using the fmincon Matlab® (Natick, Massachusetts, USA) function, where the magnitude equation in (2) is used as the main function to minimize, whereas the remaining conditions in (3)–(6) are used as constraints. Using least squares fit, polynomials are determined to compute the controller parameters based on the process time constant *<sup>T</sup>* and time delay *<sup>L</sup>*: *<sup>P</sup>* <sup>=</sup> <sup>−</sup>0.0048 <sup>+</sup> 0.2664*<sup>L</sup>* <sup>+</sup> 0.4982*<sup>T</sup>* <sup>+</sup> 0.0232*L*<sup>2</sup> <sup>−</sup> 0.0720*T*<sup>2</sup> <sup>−</sup> 0.0348*LT*. Using Figures 4 and 5, for the first set of rules and for the second one, the FO-PID controller parameters, as indicated in (1), can be finally computed:

$$k\_p = P\_\prime \ T\_i = \frac{k\_p}{I} \text{ and } T\_d = \frac{D}{k\_p} \tag{14}$$


**Figure 4.** Parameters for the first set of tuning rules for S-shaped response processes (*P* = *kp*, *I* = *kp*/*Ti*, *D* = *kpTd*) [32].


**Figure 5.** Parameters for the second set of tuning rules for S-shaped response processes (*P* = *kp*, *I* = *kp/Ti*, *D* = *kpTd*) [32].
