## 1 Introduction
In [5], an adaptive fuzzy algorithm is designed for the tracking of the cart and stabilization of the pendulum; nevertheless, a PD action had to be incorporated to deal with the states of the cart (position and velocity). The controller was successfully applied in experimentation but the stability of the overall closed-loop system is not studied and it is an open problem. More recent efforts have been made to take into account the states of the cart in this kind of scheme. In [6] an adaptive PD-like fuzzy controller is used with a good performance but only simulation results are presented and the stability of the algorithm is not proven. In [7], the authors present the study of dynamic response of the inverted pendulum in terms of classical control theory. Theoretical and experimental results using LQR design are presented. In [8] motion equations of a linear inverted pendulum system, and classical and artificial intelligence adaptive control algorithms are designed and implemented for real-time control. Classic PID controller and PID fuzzy logic controller methods are used to control the system. However, the authors in [7; 8] do not use evolutionary techniques, nor is a stability analysis is presented. In [9], the authors present the design of a fuzzy-evidential controller in order to realize the stability control of the planar inverted pendulum system. The authors do not present a stability analysis. In this paper we propose an adaptive fuzzy controller with a full-state feedback, i.e., a feedback of the four states of the inverted pendulum system (positions and velocities of both, pendulum and cart). This algorithm is meant to solve the trajectory tracking of the cart and the stabilization of the pendulum at the unstable equilibrium (vertical position). The full-state feedback action is inspired in the work presented by [3], where the nominal dynamic model of the plant is used in the control equation and only the disturbances are monitored through an adaptive fuzzy system. However, the stability of the system in closed-loop with the total control law is not studied. Those are the main differences regarding this work since the algorithm proposed here does not use any knowledge of the dynamic model; the adaptive fuzzy systems estimate the plant dynamics. The stability of the total control law and the adaptive fuzzy action together with the full-state feedback is proven through a classic adaptive system analysis [10] based on the Lyapunov theory. The latter guarantees that the solutions (positions and velocities of pendulum and cart) are bounded. Another part of this paper deals with the application of evolutionary algorithms (EA), which are a subset of evolutionary computing and they can be considered as generic optimization metaheuristic algorithms. Most EA are based upon swarm intelligence or bio-inspired computation and they have been gaining wide interest and attention in the community of optimization since they have some advantages in comparison to classic (deterministic) algorithms. Some of those advantages are that EA can address a problem even in the presence of discontinuities in the objective function and they can find optimal or near optimal solutions in big multi-dimensional search spaces with no dependence of the initial guess [11]. Applications of EA have been done mainly in engineering problems. Particularly in control engineering, there exists a vast literature mainly used for tuning parameters in control equations. In [12] the authors propose the use of the differential evolution (DE) algorithm with fuzzy logic for parameter adaptation in the optimal design of fuzzy controllers for nonlinear plants. The DE algorithm is enhanced using Type-1 and Interval Type-2 fuzzy systems for achieving dynamic adaptation of the mutation parameter. Four control optimization problems in which the DE algorithm optimizes the membership functions of the fuzzy controllers are presented.