The rotational inverted pendulum (Furuta pendulum) is a non-linear dynamic testbed to demonstrate and develop advanced control strategies either linear or non-linear on the unstable underactuated dynamic plant. This particular Furuta pendulum was designed and built in the laboratory of the school of mechanical engineering of the Universidad Industrial de Santander as a learning challenge of implementing these control laws strategies for graduate students. This research work developed a full dynamics model using Lagrangian formulation. The math dynamic model includes real physic parameters, size and length of vertical and horizontal links, the masses of each component, inertia terms, dimensions, viscous frictions, and DC high torque motor dynamics. Through using The MATLAB tool and the SIMULINK toolbox allowed modeling the equations with the Simulink blocks tool, with this, it’s possible to get to simulate the real behavior of the plant, and later it allowed to evaluate the behavior of the pendulum.Three control strategies were implemented: Optimal Control (LQR), LQR plus Kalman filter as linear quadratic state estimator (LQG), and a PID control just for performance comparison as the reference condition. For the optimal control strategies as well as the PID approach, a linearized state space model of the system dynamics was developed around a specific operating point. The optimal control gain for the linear quadratic regulator (LQR) controller and the linear quadratic Gaussian (LQG) regulator design were computed and the dynamic and disturbance response was evaluated for each control strategy. Different performance indexes were evaluated and compared to determine the best performance: a lower percentage of Peak Overshoot (PO), Steady-State Error (), lower settling time (Settling Time ()), and control (u) cost.Based on the dynamic response, the linear quadratic regulator (LQR) and LQR plus Kalman filter as linear quadratic state estimator or linear quadratic Gaussian (LQG) controllers produced the best response, the LQG controller for its robustness presented better performance against disturbances, while the control strategy (LQR) achieved good performance in settling time, error in steady state, overshoot considering the cost-performance ratio, the PID controller presented the lowest performance in the control response to disturbances, unlike LQR, this does not take into account the cost-performance ratio, the controllers LQG plus Kalman filter and LQR modeled in the state space produce a feedback gain matrix which allows the Furuta pendulum to have regulated behavior making the pendulum quickly find the equilibrium point of the vertical link faster than the PID control strategy.

On the Dynamics and Optimal Control of the Rotational Inverted Pendulum

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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