## 1 Introduction  
The proportional-integral-derivative (PID) controller was initially conceived in the 1920s and has since gained widespread use. Although numerous other control strategies have been put out over the past 100 years, PID remains the most widely used process control technology in the industrial domain. This is due to the fact that an integer order is used to model most systems. The PID controller offers reliable performance and an easy-to-implement structure at the same time. Additionally, numerous potential methods for adjusting the PID controller's parameters have been documented in the literature. Nonetheless, in certain scenarios, the system can and ought to be represented as a non-integer-order system; representing the system as an integer order is merely a rough approach [1]. A system that is modeled as a fractional order will be a better approach. Fractional-order control will undoubtedly work better for fractional-order systems. A sophisticated control method that has gained substantial popularity recently is the fractional-order proportional-integral-derivative (FOPID) controller [2, 3]. The FOPID controllers have a higher degree of freedom and greater flexibility due to their additional parameters, which are fractional-order in the integral and derivative terms. The FOPID controller has many benefits when compared with a PID controller. This includes enhanced set-point tracking, strong disturbance rejection, and greater processing ability to withstand model uncertainties [4]. The FOPID controller is appropriate for a variety of applications because it can more precisely capture the dynamic behavior of complex systems when fractional order is used in integrals and derivatives [5]. The FOPID controller has benefits like higher tracking performance, increased stability margins, and robustness to uncertainty. The FOPID controller offers a promising foundation for improving the performance of control systems and overcoming the difficulties posed by complex, nonlinear systems. The foundations of the FOPID controller and its applications are examined in this paper. Along with the most recent advancements and research trends in the area, several techniques for adjusting the FOPID controller parameters are also explored. Multi-input, multi-output (MIMO) systems are extensively used in the process control sector. Designing an effective controller is challenging due to the interactions between the loops. For the conventional single-input, single-output (SISO) proportional-integral-derivative (PID) controller, a number of tuning techniques are offered [6]. The design approaches that should be taken into account when designing fractional controllers are also outlined. To achieve the required performance and stability, FOPID controller parameters must be properly selected, which is a significant cause for concern. Model-based and model-free tuning techniques are the two categories into which FOPID controller tuning approaches are categorized in the literature. Several research works have been conducted to provide effective model-based tuning rules and techniques for FOPID controllers [7]. Trajectory tracking of a rotating flexible-joint system was assessed using a state feedback-based fractional integral control approach [8]. For a rotary inverted pendulum, a two-degree-of-freedom FOPID controller was used [9]. In [10], the topic of tuning FO controllers for industrial use was examined. However, for complicated nonlinear systems, these approaches require a precise dynamic model, which is not accessible [11; 12]. Conversely, no model or process identification is present in model-free tuning techniques [11]. As a result, ref. [13] looked into the model-free tuning technique for FOPID controlling. In [14], when the characteristics of the system varied over time, a model-free adaptive FOPID tuning technique was applied.