## 1 Introduction  
An inverted pendulum system is known as an underactuated mechanism, the main objective of which is to maintain in a vertical position a rod that rolls freely. Particularly, the inverted pendulum on a cart is composed by a rail on which the cart is mounted and has horizontal linear movement whereas the rod is attached to it. This is one of the most known inverted pendulum configurations and it is the plant under study of this work. Such a system is highly nonlinear, underactuated and it has an unstable equilibrium point, which makes it a pretty challenging exercise; for those reasons, it has become an important, well established and extensively studied problem in control theory. Regarding inverted pendulums, several examples of applications can be found in sectors such as aerospace, biomechanics and transport. For example, in aerospace, active control of a rocket is required to keep it in an upside-down vertical position during takeoff. Here, the angle of inclination of the rocket is controlled by varying the angle of application of the thrust force, placed at the base of said rocket. In biomechanics the inverted pendulum is frequently used to model walking bipeds, such as the humanoid robots. In biped robots, the support leg in contact with the ground is often modeled as an inverted pendulum, while the moving leg behaves like a pendulum that oscillates freely, suspended from the hip of the humanoid. Finally, it can be seen that a Segway human transporter is an inverted pendulum, the control of which is based on sensory inputs of gyroscopes mounted at the base of the Segway and a computer control system that maintains the balance while people walk on the vehicle. As well as the examples shown above, there are many others associated with the applications of inverted pendulums, among which are the stabilization of cranes, the stabilization of the balancing of ships and cars, the positioning of a satellite, etc. Therefore, a study of inverted pendulum systems is an excellent starting point for understanding problems in dynamics and nonlinear control. Fuzzy adaptive techniques have been successfully applied to several control problems like in [1], where a fuzzy adaptive scheme is proposed to achieve the motion control objective of robot manipulators and it is implemented in a two degree of freedom robotic arm. Fundamentally, fuzzy adaptive control is a nonlinear scheme that uses fuzzy systems with adjustable parameters as function approximators, which compute the control law without necessarily having to incorporate the dynamic model of the plant [2]. This kind of controllers may be robust enough and can deal with high nonlinear effects or with dynamics of the plant that changes during a task execution. That means, fuzzy adaptive controllers can compensate diverse phenomena, such as uncertainties, unmodeled dynamics or unknown variation in the parameters of the plant. That is one of the motivations to tackle the control of the inverted pendulum system through a fuzzy adaptive algorithm. There exists a wide frame of previous works for the control of the inverted pendulum problem, and several of them with fuzzy adaptive approaches. However, most of them address the problem as a second-order system based on feedback linearization, dealing only with the rod dynamics and leading to the presence of unstable zero dynamics (the cart dynamics). Consequently, this might conduct to undesired behavior of the cart or hindering the experimental implementation of the controllers (as stated in [3]). In [4] a fuzzy adaptive controller in combination with a sliding mode action is proposed for operation of an inverted pendulum. The fuzzy system compensates for the plant nonlinearities and forces the inverted pendulum to track a prescribed reference model. When matching with the model occurs, the pendulum is stabilized at an upright position and the cart should return to its zero position.