## 1 Introduction  
The common application areas of pneumatic systems are mostly in automatic controllers and the automation industry [1]. Pneumatic actuators (PAs) are considered safe and reliable; they are smaller than most other actuators and exhibit high power-to-weight ratios [2]. They have a fast time response and can be used in high-temperature or radioactive situations without causing harm. Pneumatic systems have advantages since gases are not limited by temperature [3; 4; 5]. The challenges in controlling PAs are mostly due to the presence of nonlinearities caused by the pneumatic actuator's high friction force, air compressibility, and the valve dead zone [6]. Hence, it is hard to precisely control the PAs. From the literature [7], there are numerous methods available focusing on improving the performance of pneumatic positioning systems. Many types of linear controllers (such as PID and pole placement) and intelligent controllers (such as neural and fuzzy networks) were researched in the early stages of developing a controller [8]. For example, [PERSON] et al. [9] implemented a PI controller on their designed experimental setup while [PERSON] et al. [10] developed a proportional-integral-derivative fuzzy logic (fuzzy-PID) controller. Based on the results of the simulation and the experiments [10],the fuzzy-PID controller yielded the best performance. A pole-placement feedback control mechanism was presented by [PERSON] et al. [11] to control the positioning system of the IPA system. In addition, IPA system position control was achieved by utilizing a generalized predictive controller based on the bat algorithm (GPC-BA), which was developed by [PERSON] et al. [12]. The use of a predictive functional controller (PFC) for IPA systems' control was proposed by [PERSON] et al. [13] where a real-time system was developed to validate the results. However, recently, [PERSON] et al. [14] reported that the control of a pneumatic positioning system using the predictive functional controller (PFC) with a novel observer yielded promising results. In previous decades, due to its increased flexibility in meeting the requirements of specific control applications, numerous research projects have been carried out in fractional-order control (FOC) [15; 16]. The FOPID controller, on the other hand, has been applied in diverse areas, such as motor control [17], robotics [18], power systems [19], magnetic levitation systems [20], and time delay systems [21; 22]; it also exhibits superior time response attributes when compared to systems using a traditional PID controller. Other research works on FOC include [PERSON] et al. [23], where a fractional-order PID controller for the IPA positioning system and its performance were compared with a conventional PID controller. In addition, [PERSON] et al. [24] used fractional-order PID for frequency regulation of a dual-area power system and yielded significant results. This work is mainly motivated by the need to design and develop a model of the pneumatic system controller. In this work, to obtain the pneumatic system model, the system identification (SI) approach was used. The CFOPID design was used as the new control mechanism for the pneumatic system. To increase the overall system performance, the proposed controller was comprised of an FOPI and was coupled in cascade with an FOPD. Meanwhile, to obtain the best possible design of the CFOPID controller, the PSO method was used by minimizing the time response parameters. The performance of the controller was evaluated using MATLAB simulation, and the results were compared to those obtained using fractional-order proportional-integral-derivative (FOPID). To evaluate the performance of the proposed controllers, a simulated ball and beam system was used. The ball and beam system is considered a model control engineering concept whose fundamental notion can serve in stabilizing a wide range of systems.