Desenvolvimento de um controlador para pêndulo invertido baseado no regulador linear quadrático

Abstract

The main objective of this study was to design a Linear Quadratic Regulator (LQR) controller to stabilize an inverted pendulum in its vertical position, minimizing the position error and control energy consumption, while ensuring system stability. The specific objectives included defining the DC motor parameters, verifying the system stability, defining the optimal gain matrix K, determining the rank and controllability matrix, as well as simulations under different initial conditions. The methodology involved obtaining the parameters of a DC motor through experimental tests, such as measurements of armature resistance, current, angular velocity and moment of inertia. The system was modeled in state space based on the Euler-Lagrange equations, followed by linearization via Taylor series expansion and Jacobian matrix. The resulting linear model was implemented in the MATLAB/Simulink environment, where the LQR controller was developed and simulated. The results demonstrated that the originally unstable system presented full controllability (rank of the controllability matrix equal to 4) and was successfully stabilized. The cost function associated with the control was minimized using the Riccati equation, ensuring a balance between performance and control effort. The simulations proved the robustness of the controller against parametric variations and external disturbances, validating the effectiveness of the proposed approach.