## 1 Introduction
### Literature Review  
The use of benchmark problems in the research and development of control systems is fundamental for analyzing a proposed system's quality. Besides, this allows its comparison with other existing systems. Perhaps one of the most famous benchmarks in control development is the inverted pendulum system. Recently, a book on its use in control and robotics was published [40]. This is because, despite its structural simplicity, it is challenging to obtain a proper control law of the inverted pendulum system for many standard control techniques, in particular given its high nonlinearity, non-minimum phase, and underactuated nature. Research work on the inverted pendulum system focuses on developing controllers capable of swinging the pendulum while exhibiting robustness in the presence of external disturbances. Multiple model-based control approaches are found in the literature considering the inverted pendulum benchmark. Nonlinear Model Predictive Control (NMPC) was employed in [41] for the swing-up of the parallel double inverted pendulum. The authors considered the Real-Time Iteration (RTI) scheme to allow the control to be implemented in hardware. In [42], the Integral Sliding Mode Control (ISMC) and the State-Dependent Riccati Equation (SDRE) were combined to perform the swing-up of the rotary inverted pendulum. The resulting robust continuous-time nonlinear optimal controller was experimentally tested. The work in [43] presented a Switched Robust Integral Sliding Mode (SRISM) control to address unmatched uncertainties. A Backstepping Sliding Mode Control (BSMC) scheme was proposed in [44] by transforming the dynamic system model into the so-called regular form. In this way, they took advantage of the convenient features that characterize the backstepping and sliding mode control schemes. A similar approach is that found in [45]. Many works have recently been reported in the literature for multiple types of nonlinear controllers featuring automatic tuning methodologies. In particular, metaheuristic algorithms to perform online tuning of the control parameters have been reported. For instance, [PERSON] presented a sliding mode control for the cart-pole system swing-up featuring the culture bees algorithm for online parameter tuning [46]. [PERSON] et al. complemented the hybrid sliding mode control by self-tuning the control parameters with the fireworks algorithm [47]. In addition, [PERSON] and [PERSON] considered the backstepping control scheme with the genetic algorithm for a similar scope [48]. Other authors considered the energy method to address the cart-pole swing-up control task. For instance, [PERSON] and [PERSON] considered this approach for the swing-up and switches to a Linear Quadratic Regulator Controller (LQRC) to stabilize in the vertical position [49]. In contrast to the mentioned model-based controllers, a model-free approach is that found in [50]. There, the energy method was employed in an online setup with no information of the system's physical parameters. The algorithm self-learns a weight that is used in normalizing the energy injected to the pendulum. An approach considering neural networks to model the controller was presented in [51]. There, the swing-up problem of the double inverted pendulum was solved in a data-driven framework employing a neural network controller with parameters optimized via the hybrid uniform design multiobjective genetic algorithm. In [52], the Advantage Actor-Critic (A2C) was used to stabilize the cart-pole system [53]. This RL algorithm is considered in two different setups for the observations. One is with the state of the system, while the other employs a convolutional neural network to extract features from the animation produced by the simulation. Both cases consider a continuous observation space and a discrete control space. Recently, there has been interest in combining model-free and model-based approaches for controller development. Some works with this perspective are [54; 55; 56].