**3. Direct Fractional Order Autotuning Methods**

Most of the direct autotuning methods are based on using the relay test to determine the process critical gain and critical period of oscillations, but other methods have been developed [34].

Several generalizations of the Ziegler-Nichols ultimate gain method have been proposed over the years for the tuning of FO-PIDs. A new tuning method for such a controller that combines both the Ziegler-Nichols as well as the Astrom and Hagglund methods has been proposed in [35]. The idea is based on obtaining the process critical frequency and critical gain and then computing the *kp* and *Ti* parameters using the classical Ziegler-Nichols method. For a specified phase margin, the *Td* parameter is computed using the Astrom and Hagglund method. Two equations referring to the controller's real and imaginary parts are obtained. Fine tuning of *Td* is employed to achieve the best numerical solution of the equations, for each specified phase margin. Matlab®'s built in functions, such as fsolve, are used to solve the two equations to obtain numerical values of *λ* and *μ* by considering the new value of *Td* for each specified phase margin. An optimization Simulink model is used to obtain a better step response. The least squares method is used in the optimization model and the optimized FO-PID parameters are obtained. The approach is tedious and involves three controller designs before the final optimized FO-PID is obtained. However, the design allows for a direct specification of the loop phase margin.

In [36], an extension of the modified Ziegler-Nichols tuning rules for fractional-order controllers is presented. The proposed design approach is only suitable for tuning fractional order PI controllers. The tuning rules are derived without any knowledge of the process model, but they require the critical frequency *ωcr*, as well as the corresponding critical gain *kcr*. Based on this process information, the FO-PI autotuning objective is to determine the controller parameters such that the loop frequency response is moved to a point in the Nyquist plane where a performance criterion is minimized, according to a constraint. The performance criterion is mathematically expressed as a measure of the system ability to handle low-frequency load disturbances, subjected to a robustness constraint referring to the maximum sensitivity function of the closed-loop system. The tuning rules are given by:

$$\lambda = \frac{1.11k\_{180} + 0.084}{k\_{180} + 0.07}, \; k\_p = k\_{\varepsilon r} r\_b \cos \beta + k\_{\varepsilon r} r\_b \cot \lambda \sin \beta \text{ and } T\_i = \frac{k\_p}{k\_i} \tag{15}$$

where *k*<sup>180</sup> = <sup>1</sup> *kcrk* , *rb* <sup>=</sup> 0.34*k*180+0.03 *<sup>k</sup>*180+0.52 , *<sup>β</sup>* <sup>=</sup> <sup>−</sup>0.92*k*180−0.012 *<sup>k</sup>*180+0.6 , *ki* <sup>=</sup> <sup>−</sup>*kcrrbω<sup>λ</sup> cr* sin *<sup>γ</sup>* sin *<sup>β</sup>* and *<sup>γ</sup>* <sup>=</sup> *<sup>π</sup>* <sup>2</sup> *λ*, with *k* the process gain as indicated in (7).

The method is compared with several other direct and indirect autotuning methods for integer order PIDs and it provides good performance results. The method is also compared to some similar autotuning approaches developed in [37,38] and the results demonstrate the superiority of the current approach.

A similar idea as the one used in [31] is employed in [39], where an error filter is cascaded with a FO-PI controller. Unlike the autotuning approach taken in [31], the research in [39] is focused on estimating the parameters of an integer order PI controller using the relay method. An estimation of the process critical gain and period of oscillation is firstly determined, which in turn leads to the computation of an integer order PI controller parameters according to the standard Ziegler-Nichols approach. The same error filter is used in [39] as in [31] with the same advantage. Various values for the fractional order integration are used and the results evaluated on a steam temperature process. Experimental results show that the FO-PI controller leads to better performance during the set-point change and load disturbance test in terms of output and control effort. However, poor closed-loop performance is obtained if *λ* is set too low. Even though both the direct [39] and the indirect [31] autotuning methods are simple enough for designing the FO-PI controllers, there is no clear advice on the selection of the fractional order of integration.

A modification of the Ziegler-Nichols closed-loop method is proposed in [40]. The method provides for an improvement of the standard Ziegler-Nichols results. The idea is based on the fact that a fractional order can help shape the "direction" of the loop frequency response in a fixed point in the Nyquist plot and thus keep the loop frequency response further away from the −1 point. The process critical frequency of oscillation, as well as the critical gain are obtained based on the relay test. To simplify the tuning method, the same fractional order of integration and differentiation is used in the FO-PID, similarly to [41]:

$$\mathcal{C}\_{\text{FO}\to\text{PID}}(s) = k\_p \left( 1 + \frac{1}{\left( \tau\_i s \right)^{\lambda}} + \left( \tau\_d s \right)^{\lambda} \right) \tag{16}$$

where *Ti* = *τ<sup>i</sup> <sup>λ</sup>* and *Td* = *τ<sup>d</sup> <sup>λ</sup>*. The ratio *r* = *<sup>τ</sup><sup>i</sup> <sup>τ</sup><sup>d</sup>* between the integral and derivative time constants is considered to be a design parameter. The final tuning rules are exemplified for a ratio *r* = 4, similarly to [6,41]. Unlike the standard Ziegler-Nichols approach, the tuning rules depend not only on the process critical gain *Kcr* and critical period of oscillation *Pcr*, but also on the fractional order. The parameters of the FO-PID controller can thus be easily computed, without any complex optimization procedure [40], as indicated in Table 1.

**Table 1.** FO-PID parameters according to the modified Ziegler-Nichols method and for different values of the fractional order *λ* [40].


The critical process gain and period of oscillation are used in [42] to determine the parameters of a FO-PID controller. Three sets of tuning rules are developed. Processes described as FOPDT systems are used for two of the sets, whereas for the third one, integrative processes are considered. The first set of tuning rules applies when the critical period of oscillations *Pcr* ≤ 8 and *PcrKcr* ≤ 640. For the case when *Pcr* ≤ 2, a second set of tuning rules is developed. Both of these are quite restrictive and do not often work properly for plants with a pole at the origin [42]. The third set of rules is designed specifically for integrative processes (without time delay), but can be used only when 0.2 ≤ *Pcr* ≤ 5 and 1 ≤ *Kcr* ≤ 200. The research in [42] concludes that the closed-loop performance can be poor near the borders of the mentioned range. All of these rules were developed in order to meet certain performance specifications regarding the loop gain crossover frequency, phase margin, iso-damping, rejection of high-frequency noise and output disturbance. All tuning rules are developed similarly to those in [32], by minimizing the magnitude equation in (2) and using the remaining conditions in (3)–(6) as design constraints. The controller parameters are obtained by polynomial fitting using least squares. The coefficients of the polynomials for the three sets of tuning rules are indicated in Figures 6–8.


**Figure 6.** Parameters for the first set of tuning rules for processes with critical gain and period (*P* = *kp*, *I* = *kp/Ti*, *D* = *kpTd*) [42].


**Figure 7.** Parameters for the second set of tuning rules for processes with critical gain and period (*P* = *kp*, *I* = *kp/Ti*, *D* = *kpTd*) [42].


**Figure 8.** Parameters for the third set of tuning rules for processes with critical gain and period (*P* = *kp*, *I* = *kp/Ti*, *D* = *kpTd*) [42].

The relay test is also used in [43], but with a variation that includes also a time delay, as indicated in Figure 9. The process frequency response at any frequency can be identified using this scheme. The main issue is to determine the correct value of the time delay that corresponds to a specific frequency. An iterative method is used [44] and two initial values for the time delay and their corresponding frequencies are needed to start the iteration.

**Figure 9.** Relay autotuning scheme with delay [43].

The autotuning method is based on specifying an iso-damping property, a gain crossover frequency and a phase margin. A fractional order PI controller is designed first, followed by a fractional order PD controller with a filter. The fractional order PI controller will be used to ensure the iso-damping property around the gain crossover frequency *wcg*. The slope of the phase of the plant is computed using the gain crossover frequency and the corresponding phase and a supplementary frequency and its corresponding process phase as resulting from the relay experiment. Once the slope is cancelled using the FO-PI controller, the FO-PD controller is designed to fulfill the design specifications of gain crossover frequency and phase margin. To ensure a maximum robustness to plant gain variations, a robustness criterion based on the flatness of the phase curve of the FO-PD controller is used such that the resulting phase of the open-loop system will be the flattest possible. The procedure is rather lengthy. A mechanical unit consisting mainly of a servo motor is used to experimentally validate the proposed method. The experimental results illustrate the effectiveness of this method.

The same method is described in [45], where experimental results with the FO-PID on a similar servo motor are used to validate the efficiency of the approach. A refinement of the relay feedback test in [43,45] is introduced in [46]. The improvement is based on adding a moving average filter. Simulation results for the control of a position servo with time delay are presented and validate the autotuning algorithm. The same autotuning method for determining a FO-PID controller for the servo system in [46] is presented in [47]. A similar approach is detailed in [48] for the design of FO-PID controllers. Two numerical case studies are provided for a double-integrator process and a fractional order integrative process. The simulation results validate the autotuning method.

Instead of using the relay test to determine the process magnitude, phase and phase slope, a single sine test at the gain crossover frequency is used in [34]. Novel filtering techniques are used to determine the process phase slope, as indicated in Figure 10. To determine the parameters of either a FO-PI or a FO-PD controller, performance specifications regarding the phase margin, gain crossover frequency and iso-damping property are used. The process magnitude, phase and phase slope previously determined are used in the resulting nonlinear equations. Optimization techniques or graphical methods are then employed to determine the controller parameters. Numerical examples are used to validate the approach. A different approach is presented in [49], where a forbidden region circle is defined based on the iso-damping property and phase margin specifications. The same sine test used in [34] is required here as well, in order to estimate the process phase, magnitude and phase slope. Instead of using optimization routines, the parameters of the optimal fractional order PID controller are determined by minimizing the slope difference between the circle border and the loop-frequency response. Numerical results are presented to validate the approach.

**Figure 10.** Experimental scheme used to compute the phase slope of the process at the gain crossover frequency (refer to [34] for details).
