The research community has deeply studied the classical PID controller. These controllers have been applied to industrial processes, electromechanical systems, electric and electronic systems, and aeronautical and aerospace systems. The PID controller’s success is focused on its simplicity and robustness. One important characteristic of this controller is that the integral term offers robustness against disturbances and parameter uncertainty. Several tuning methods are applied to these controllers, e.g., root locus, frequency, or graphical methods, which are relatively easy to implement. These facts make the controller attractive for implementing the regulation of almost any system or process. A weak characteristic of PID controllers is related to the derivative term. This term is sensitive to the noise originating that the resulting control signal is noisy. This fact negatively affects the performance of the closed-loop system. Different proposals of PID structures have been developed in place of classical PIDs, such as fuzzy PID controllers [
1,
2], adaptive PID controllers [
3,
4], sliding mode PID controllers [
5,
6], intelligent PID controllers [
7,
8], and generalized PI and PID controllers [
9,
10,
11]. All of these developments try to improve the performance of classical PIDs and give better characteristics that classical PIDs lack.
Another variation of the classical PID controller is the fractional-order PID controller (FOPID). This variation is based on fractional calculus. During the last decades, fractional calculus has been the focus of automatic control researchers. Fractional calculus has been applied to modeling systems that have fractional-order differential equations. At first, models were developed based on fractional order equations for mechanical [
12,
13] and electrical systems [
14,
15]. Later, the first continuous FOPID (cFOPID) controller was introduced by Podlubny in 1999 [
16]. This controller has two additional parameters compared with classical PIDs. These parameters are the constants of the fractional-order integral and fractional-order derivative. This new characteristic added to the classical PID generates a more flexible controller and provides the opportunity to better adjust the dynamical properties of the closed-loop system [
16]. After this first development, the use of the FOPID controller to regulate different types of systems increased. Examples of the applications in engineering systems that are regulated by FOPIDs are: smart reactors [
17], electronic power converters [
18,
19,
20], rehabilitation systems [
21]; automatic voltage regulators [
22,
23], industrial process models [
24,
25], robotic systems [
26], power systems using synchronous generators [
27], and wind turbines [
28].
The gains-tuning process is essential in the implementation of FOPIDs for regulating dynamic systems, which is a challenging aspect. This process can be manually facilitated, but the performance of the closed-loop system could be better. Different methods for tuning FOPID controllers have been developed to solve this problem, e.g., rule-based, analytical, and numerical methods [
29]. We focus on numerical methods, specifically evolutionary algorithms. An evolutionary algorithm is an optimization algorithm that is inspired by a natural, biological, or physical phenomenon. Among the evolutionary algorithms, genetic algorithms (GAs) and their improvements have been an attractive solution used to tune fractional-order PID controllers. In order to start the optimization process, it is necessary to establish a suitable objective function. Many popular objective functions have been defined and used for tracking error minimizations, such as the integral squared error (ISE), integral of the time-weighted squared error (ITSE), integral absolute error (IAE), and integral time-weighted absolute error (ITAE) indices [
27], with the aim of obtaining the minimum value of these functions. In [
30], the authors used a GA to tune the fractional factors of a cFOPID while considering the ISE index as the objective function and setting the fractional order bound as ranging from 0 to 100. A third-order continuous transfer function was utilized as a controlled system. The results showed that the cFOPID controller can regulate the system, even if the fractional factor values exceed the general bound for these two parameters [
29]. In [
31], the authors implemented a simulation of a cFOPID controller with a three-DOF robot system that is driven by DC motors. In this work, the objective function considered the overshoot, settling time, and IAE index. The optimization was performed to find all FOPID parameters, and the simulation results show that GAs can find optimal values for all FOPID parameters. In [
32], the authors discussed the control of a CD motor fed by a buck–boost converter using a cFOPID, and its performance was compared with the classical PID controller; in this work, the controller was tuned by applying a GA, and the tuning process was conducted while considering three different cost functions: the ISE, ITSE, and mean square error (MSE). The results obtained show that, for this study case, the ISE function achieved the best performance for the cFOPID. In [
33], the authors addressed the tuning of a cFOPID device using GAs and particle swarm optimization for a two-DOF robot trajectory control. The authors used three different cost functions during the tuning process: the root mean squared error (MRSE), mean absolute error (MAE), and mean minimum fuel and absolute error (MMFAE). Each cost function is used individually to obtain the optimal values of both controllers. The results show that the MRSE cost function provides the best controller parameters with the lowest fitness values. In [
34], the authors addressed the application of a multi-objective genetic algorithm (MOGA) fractional-order PID controller for semi-active magnetorheological damped seat suspension. The authors selected the gain crossover frequency and phase margin for the optimization problem of the FOPID controller as objective functions. The simulations performed show that cFOPID offers superior performance over the integer controllers. An improvement of GAs called multi-objective evolutionary non-dominated sorting genetic algorithm II (NSGAII), which is used to tune a cFOPID, was applied to a hydraulic turbine and addressed in [
35]. The performance of the cFOPID was compared with the performance of a PID. The authors considered two objective functions for optimizing the controller parameters in the system: the ISE as the cost function one and ITSE as cost function two. The results show that the NSGAII algorithm is an efficient algorithm for tuning a cFOPID. In [
36], the authors implemented a novel optimization algorithm called cloud-model-based quantum genetic algorithm (CQGA) to tune a cFOPID to control the motion of an autonomous underwater vehicle (AUV) in real time, and the authors considered the linear transfer functions of the AUV dynamics. CQGA is a fusion of cloud-model theory and the quantum genetic algorithm (QGA). The QGA algorithm is based on the principles and concepts of quantum computing. The authors adopted the ITAE as the objective function. The results show that the cFOPID performs the best for the heading control, diving control, and path-following systems compared with a classical PID. The authors established the limits of the fractional order factors as
, but their optimization result shows values of
and
out of this interval. The authors in [
37] constructed the design of a cFOPID device tuned using GAs for a conical tank and compared its performance with a PID device. From the nonlinear model of the tank, the authors analyzed the dynamical response in an open-loop system at different operating points. After this process, a linear transfer function at the height of 40 cm was selected to perform the tuning process of the cFOPID. The ITAE was chosen as the objective function. The cFOPID achieved the best transient response and performance at the operation point of 40 cm compared with the results obtained by applying the PID. In [
38], the authors presented the application of a robust queen bee assisted genetic algorithm (QBGA) for tuning a cFOPID to control a boost converter with a non-minimum phase behavior. The authors selected the ISE as the objective function to minimize. The tests performed show that the cFOPID performs better than two PID controllers that were tuned based on the transient response and ISE using the QBGA. The robustness of the controllers of the closed-loop system was tested by applying parameter variations to the model. The results show that the cFOPID has better robustness than the PID controllers.
From a general point of view, the FOPID tuning process using GAs employs objective cost functions depending on the tracking error (e.g., ISE, ITAE, and ITSE) and simplified linear models of the system under control. This motivated our study to explore the behavior of GAs by using a cost function that incorporates the weighted quadratic error and control effort, as well as the nonlinear system model, to perform the optimization process. Considering the control input in the cost function allows us to penalize large control inputs, which improves the dynamic performance [
39]. In this research work, we present the study of the performance of cFOPID and dFOPID controllers in real time. The performance of the FOPID controllers was compared with the performance of continuous and discrete classical PID controllers, which were also tuned using GAs. The work contribution is summarized as follows: