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Mathematical Problems in Engineering
Volume 2015, Article ID 247682, 11 pages
Research Article

Robust Takagi-Sugeno Fuzzy Dynamic Regulator for Trajectory Tracking of a Pendulum-Cart System

1Tecnológico Nacional de México, Instituto Tecnológico de La Laguna, Boulevard Revolución y Calzada Cuauhtémoc, 27000 Torreón, COAH, Mexico
2Universidad Autónoma del Carmen, Facultad de Ingeniería, Calle 56 No. 4 x Avenida Concordia, 24180 Ciudad del Carmen, CAM, Mexico

Received 17 August 2014; Accepted 16 October 2014

Academic Editor: Luis Rodolfo Garcia Carrillo

Copyright © 2015 Miguel A. Llama et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Starting from a nonlinear model for a pendulum-cart system, on which viscous friction is considered, a Takagi-Sugeno (T-S) fuzzy augmented model (TSFAM) as well as a TSFAM with uncertainty (TSFAMwU) is proposed. Since the design of a T-S fuzzy controller is based on the T-S fuzzy model of the nonlinear system, then, to address the trajectory tracking problem of the pendulum-cart system, three T-S fuzzy controllers are proposed via parallel distributed compensation: (1) a T-S fuzzy servo controller (TSFSC) designed from the TSFAM; (2) a robust TSFSC (RTSFSC) designed from the TSFAMwU; and (3) a robust T-S fuzzy dynamic regulator (RTSFDR) designed from the RTSFSC with the addition of a T-S fuzzy observer, which estimates cart and pendulum velocities. Both TSFAM and TSFAMwU are comprised of two fuzzy rules and designed via local approximation in fuzzy partition spaces technique. Feedback gains for the three fuzzy controllers are obtained via linear matrix inequalities approach. A swing-up controller is developed to swing the pendulum up from its pendant position to its upright position. Real-time experiments validate the effectiveness of the proposed schemes, keeping the pendulum in its upright position while the cart follows a reference signal, standing out the RTSFDR.