**5. Applications and Self-Tuned FO-PIDs**

Autotuning methods have been used to produce fractional-order controllers for different processes. The purpose of this section is to provide some applicative examples of autotuning methods for fractional-order controllers designed mostly according to the methods presented in Sections 2 and 3. The autotuning method in [34] is applied to a multivariable time-delay process to tune the FO-PI controllers for each loop [51]. The method in [49] is applied for designing fractional-order controller for a multivariable refrigeration system using vapor compression [52], a heterogeneous dynamic system [53] and to a highly coupled multivariable system [54]. A robust autotuning method is described and implemented for controlling an aerodynamic system in [55]. An experimental validation of the direct autotuning method in [49] is provided in [55] for controlling an UR10 robot. The autotuning method in [34] is applied to tune a FO-PD controller for vibration suppression in a smart beam [56]. An autotuning method designed for poorly damped systems that shapes the closed-loop system in order to achieve better damping is proposed in [57]. The design is performed in the frequency domain and requires information regarding the process magnitude and phase for five frequencies. Experimental results are given to validate the efficiency of the method.

A "plug and play" solution for a multivariable FO-PI controller is developed in [58] for controlling a multivariable twin-rotor aerodynamical system. A decentralized approach is considered and three performance specifications, as in (2)–(4), are used to compute the parameters of the two FO-PI controllers, one for azimuth and one for pitch angle control. The design is based on a novel, simplified algorithm using vector theory, where the proportional *<sup>z</sup>*<sup>1</sup> = / /*kp* / / and integral *z*<sup>2</sup> = / / / 1 *Ti* (*jω*) −*λ* / / / terms are defined as vectors. The vectorial representation of the FO-PI controller as the sum of *z*<sup>1</sup> and *z*<sup>2</sup> is indicated in Figure 19.

**Figure 19.** Vector form of a FO-PI controller.

Then, using classical trigonometric equations based on Figure 1, the proportional gain and integral time constant of the FO-PI controller are determined as a function of the fractional order *λ*, using the gain crossover equation (2) and the phase margin equation in (3). The procedure is iterative and computes the *kp* and *Ti* parameters for small increments of 0 < *λ* < 1. Then, the iso-damping property in (4) is evaluated and *λ* is selected to be the value that minimizes (4). Finally, *kp* and *Ti* are computed using the selected value of *λ*. The fractional-order controller is implemented in a self-tuning structure as indicated in Figure 20, where the "Controller designer" block includes the iterative procedure. The "System identifier" block is used to estimate the process parameters online which are then used in the iterative procedure to determine the new values for the FO-PI controller. A recursive simple least squares algorithm is implemented in the "System identifier" block.

**Figure 20.** The self-tuning FO-PI controller.

Experimental results are provided to demonstrate the efficiency of the autotuning method. A step reference change of −1 rad for the azimuth angle and a step change of 0.2 rad for the pitch angle is considered, with the experimental results provided in Figure 21, demonstrating that reference tracking can be achieved successfully using the proposed multivariable self-tuned FO-PI control strategy.

**Figure 21.** Experimental results of self-tuned FO-PI controllers for a twin-rotor system.

An autotuning approach for FO-PIDs is used to control the air-conditioning fan coil unit [59]. A basic differential evolution algorithm is modified by varying the mutation factor and crossover rate and used to tune the five parameters of indoor temperature FO- PID controller. Numerical simulations are presented that show that the approach is reliable and the related control performance indexes meet the requirements of comfortable air-conditioning design and control criteria.

Improvements in FO-PID controller design have been considered in order to determine algorithms that perform a better tuning in real time. One solution to this issue is the selftuned FO-PID controller. The purpose of this last part of the manuscript is to present some ideas regarding additional solutions to autotuning methods that could facilitate the industrial acceptance of FO-PID controllers. In what follows, the manuscript covers an important part of adaptive control algorithms, namely, self-tuning methods, applied to fractional-order controllers. Only the most recent findings in this area of research are reviewed.

Fuzzy logic is usually used to achieve the self-tuning property, such a FO-PI selftuned controller is presented in [60], in a differential mobile robot. Three different types of controllers are evaluated and compared to a classical controller, with its parameters being acquired through traditional methods. A similar self-tuned fuzzy FO-PI controller for a steam distillation process is evaluated in [61]. The numerical results show that this

controller leads to better closed-loop performance in comparison to the integer order PI, the FO-PI and self-tuning fuzzy PI. The control of the horizontal motion of a dual-axis photo voltaic sun-tracker is presented in [62]. A new technique for online self-tuning of a FO-PID controller based on both a type-1 fuzzy and a Takaji-Sugeno Fuzzy is developed. Satisfactory results were obtained in numerical simulations. Takagi-Sugeno (TS) fuzzy technique combined with interval type-2 fuzzy sets is used in [63] to design a new adaptive self-tuning FO-PID controller. A modified FO-PID controller is obtained using TS, while the interval type-2 fuzzy sets are used as a tuner to update the gains of the FO-PID. Three types of interval type-2 fuzzy sets tuning methods are used and applied to load-frequency control as a case study of a power system comprising a single area. Comparative studies with type-1 fuzzy sets are carried. The simulation results show that the proposed approach works well considering disturbance changes and parameter uncertainties. A fuzzy FO-PID is used in [64] to control the position of a robotic manipulator. A fuzzy system combined with the particle swarm optimization method is used to determine the parameters of a FO-PID controller. Numerical simulations and comparisons with a fuzzy PID are performed. The simulation results show that the FO-PID is able to reduce the overshoot and the oscillatory dynamics, compared to the fuzzy PID. Three self-tuned fuzzy controllers are implemented in [65], namely, a FO-PD, a FO-PI and a FO-PID. The controllers are then evaluated in a servo-regulatory mechanism. The simulation results show that the self-tuned fuzzy FO-PID leads to the best closed-loop performance. The control of a mover position of a direct drive linear voice coil motor (VCM) is performed in [66] using a self-tuning FO-PID. The five FO-PID control parameters are optimized dynamically and concurrently using an adaptive differential evolution algorithm. Experimental results are provided and demonstrate that the proposed self-tuning FO-PID achieves better performance compared to PID and FOPID controllers, under both nominal and payload conditions.

The control of an inverted pendulum system is described in [67], where two selftuned FO-PD controllers are designed to vertically balance the pendulum and for accurate positioning. The proportional and derivative gains of the two controllers are dynamically adjusted using particle swarm optimization after each sampling interval using piecewise nonlinear functions of their respective state-variations. Hardware-in-the-loop experiments are performed and the proposed approach is compared to fixed gain dual-PD and dual-FO-PD control schemes.

A direct autotuning method for a FO-PI controller is used in [68] to control the speed of a permanent magnet synchronous motor. Only the measured input-output data of the closed-loop servo system is required to tune the FO-PI controller. The FO-PI parameters are determined using a virtual reference feedback tuning with an incorporated Bode ideal transfer function, which allows the properties of the resulting system to be approximated to the desired fractional-order reference model. Optimal performance constraints, such as sensitivity criteria, frequency-domain and time-domain characteristics are considered in the autotuning. Experimental results are provided to illustrate the efficiency of the proposed model-free FO-PI control method for the servo system. The extremum seeking approach is used as a non-model-based method that searches online for the FO-PID parameters that minimizes a cost function related to the performance of the controller [69]. Simulation examples are provided to demonstrate the effectiveness of the proposed algorithm.

A novel self-tuning FO-PID controller using the optimal model reference adaptive control (MRAC) is applied to power system load-frequency control [70]. The requirements for the control systems are embedded in the model reference, mathematically described as a first- or second-order system. A harmony search optimization method is used to determine the parameters of MRAC. Three methods for self-tuning FO-PID control are presented. The first two methods assume some of the FO-PID parameters to be fixed and adjust the remaining ones, while the third method was developed to adjust all five parameters of the FO-PID controller, at the same time. The simulation results show that the latter method achieves better disturbance rejection, as well as improved handling of system uncertainty.

The control of coupled and non-linear 2-link rigid robot is tackled in [71] using a novel non-linear FO-PID that includes a non-linear hyperbolic function cascaded with a FO-PID. The fractional orders allow for greater flexibility in the controller design, while the adaptive feature is incorporated in the non-linear function. The parameters of the FO-PID are determined according to the multi-objective non-dominated sorting genetic algorithm II (NSGA-II) for small variations in control and error signal. Comparisons with a non-linear PID, FO-PID, non-linear hyperbolic function cascaded with an integer order PID or traditional PID are performed. The simulation results demonstrate that the proposed method provides robust and efficient control of the robotic arm.

A fractional fuzzy controller is designed in [72], without using an actual model of the robot and only well-known structural properties of mechanical systems. The entire implementation is model-free and tackles the control of robotic manipulators. To ensure improved disturbance rejection, a fuzzy logic formulation is used with an online adaptation of the outputs to achieve a better closed-loop response. To demonstrate the efficiency of the approach, simulations and experimental results are presented. An innovative design method, suitable for many industrial applications is presented in [73]. A self-tuning fractional-order controller is designed using fractional order pole placement and indirect adaptation profiles. Simulation results are provided for an air-lubricated capstan drive for precision positioning. The results show that, indeed, better closed-loop performance is possible using the proposed method instead of a similar one based on integer order pole placement.

A fractional-order self-tuned fuzzy PID controller is designed for a three-link rigid robotic manipulator system in [74]. The controller is tuned using a cuckoo search algorithm to minimize the weighted sum of the integral of absolute error and the integral of absolute change in controller output. The same tuning procedure is used to tune a fractionalorder fuzzy PID and an integer-order self-tuning fuzzy PID. Comparative simulation results are provided and demonstrate better trajectory tracking, disturbance rejection, noise suppression and robustness to model uncertainty in the case of the proposed fractionalorder self-tuned fuzzy PID controller.

In [75], an online identification of the parameters of a fractional order process is performed based on a particle swarm optimization algorithm. Then, a fractional order self-tuning regulator is designed using differential evolution algorithms. Simulation results show that the proposed method is robust and leads to good closed-loop results.

A self-tuning controller is designed in [76] using fuzzy logic for the control of microgrid systems. A fractional-order controller is developed in combination with a fuzzy logic algorithm for load-frequency control of the off-grid microgrid. An optimal way to estimate the input and output scale coefficients of the fuzzy controller and fractional orders of the fractional-order controller is developed based on a novel meta-heuristic whale algorithm. The case study consists in a microgrid containing a diesel generator, wind turbine, photovoltaic systems and energy storage devices. Simulation results show that the proposed optimized fractional-order self-tuning fuzzy controller manages to outperform the classical PID controller in terms of operation characteristics, settling time and load-disturbance attenuation.

The active suspension system of a quarter car is considered as the case study in [77], where a self-tuned robust fractional-order fuzzy proportional-derivative controller is developed. The design of the controller attempts to minimize the root mean square of vertical vibration acceleration of car body. Tracking force, ratio between tire dynamics and static loads and suspension travel are considered as design constraints. Genetic algorithms are used to optimize the parameters online for a sinusoidal road surface. However, simulations were performed for random road surfaces and bumps. The proposed self-tuned fractionalorder fuzzy proportional-derivative controller achieved better results compared to passive solutions, as well as to its integer order counterpart.

The cuckoo search algorithm is also proposed in [78] in the design of a self-tuned fractional-order fuzzy PID controller. The optimization algorithm is based on the minimization of an objective function defined as the sum of integral of squared error and integral of

the squared deviation of controller output. The final controller consists in a Takagi-Sugeno model-based fuzzy adaptive controller containing non-integer-order differ-integral operators. For comparative purposes, the integer order counterpart of this controller is also designed. Simulation results indicate the increased robustness of the self-tuned fractionalorder fuzzy PID controller when applied to the control of an integrated power system.
