This article presents the design of a fuzzy controller to stability analysis base on the convergence of the linguistic trajectory in the state of space, for an inverted car-pendulum system, the fuzzy controller is of the Mamdani type and, it consists of 25 rules, 3 input variables and each one is composed of five memberships functions. The inverted car-pendulum system is represented by a non-linear model, which is obtained from a linearized equivalent model under the consideration of small oscillations. Results are validated against a PID control base on the trajectory on the phase plane to evaluate the efficiency and effectiveness of the fuzzy controller. The dynamic behavior of the system of both controllers is obtained with a unit impulse input, the simulation of the control system is developed on the MATLAB / SIMULINK software using the FUZZY LOGIC TOOLBOX, which allows to perform test and simulations, and also it shows results of graphic form.

This article presents the design of an LQR controller for a Ball-Beam system, as well as its stability analysis, the control of a Ball-Beam system is one of the most interesting for control engineering since it is a highly non-dynamic system linear. The objectives of this document focus on the performance of the system using an LQR control for different disturbances as well as obtaining the phase plans. The work starts with the modeling of the ballgirder system, which consists of two mechanical arms, a gear box and a DC servomotor, later the LQR control is designed, this allows simulation and obtaining the response of the controller under different conditions. In the system the input torque is generated from the DC servo motor to control the position of the ball on the beam, where the ball rolls freely on the beam. Performance analysis is performed using robust LQR and the performance characteristics of the system are presented. Finally, the stability analysis is carried out by plotting the phase planes.