Controle e estabilização do Duplo Pêndulo Invertido sobre um carro

Abstract

The sub-actuated mechanical system of the double inverted pendulum on a cart is a complex subject in the field of control theory. The main reason researchers have focus their attention on this system is its characteristic difficulty on balancing two rods in vertical only using the movement of the cart. Although the evolution of the control theory throughout the years, the development of a robust controller is still an open question. In order to provide a study on more efficient control techniques, considering aspects of efficiency and robustness, we performed a comparative study from which we chose the most appropriate to make part of the model presented in this work. Since there is no a common standard in the different control laws applied to the system, it is modeled, linearized and its dynamic movements are studied. Three control techniques are designed to this modeled system: linear quadratic regulator (LQR), fuzzy logic control and sliding mode control. To the last technique we applied an improvement in the switching function in order to decrease the effects of the chattering problem, using the saturation function. The asymptotic stability of the sliding mode surfaces in each one of these subsystems is proved theoretically using the Barbalat s lemma and the Lyapunov stability. The results of the simulations to the LQR and fuzzy control applied to the linearized system proved its efficiency in stabilizing the system, and the fuzzy controller as a better approximation of the state variables of the equilibrium point, however, their response is slow. The results with the sliding mode controller also present better results when compared to the LQR, with a faster response. The results with the hierarchical sliding mode controller proved its efficiency in face of the uncertainty of the system and the function used to decrease the effects of the chattering problem showed itself useful.