Research Article Robust Takagi-Sugeno Fuzzy Dynamic Regulator for Trajectory Tracking of a Pendulum-Cart System Miguel A. Llama, 1 Wilfredo De La Torre, 1 Francisco Jurado, 1 and Ramon Garcia-Hernandez 2 1 Tecnol´ ogico Nacional de M´ exico, Instituto Tecnol´ ogico de La Laguna, Boulevard Revoluci´ on y Calzada Cuauht´ emoc, 27000 Torre´ on, COAH, Mexico 2 Universidad Aut´ onoma del Carmen, Facultad de Ingenier´ ıa, Calle 56 No. 4 x Avenida Concordia, 24180 Ciudad del Carmen, CAM, Mexico Correspondence should be addressed to Miguel A. Llama; mllama@itlalaguna.edu.mx Received 17 August 2014; Accepted 16 October 2014 Academic Editor: Luis Rodolfo Garcia Carrillo Copyright © Miguel A. Llama et al. his 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 efectiveness of the proposed schemes, keeping the pendulum in its upright position while the cart follows a reference signal, standing out the RTSFDR. 1. Introduction A great number of nonlinear systems can be represented by Takagi-Sugeno (T-S) fuzzy models. hey are considered universal approximators [1]. In [2–4], the T-S fuzzy control system stability has been veriied considering a common Lya- punov function determined using linear matrix inequalities (LMIs) and optimization algorithms. New relaxed stability conditions and designs based on LMI for fuzzy control systems in continuous and discrete time have been presented in [5] and its utility is demonstrated with a fuzzy regulator and a fuzzy observer design. he pendulum-cart system is a perfect test bed for dem- onstrating the theoretical and practical aspects of the con- trol theory because of its inherently unstable open-loop with highly nonlinear dynamics. Two diferent dynamics of the pendulum and the cart are coupled together. here are several limitations in controlling the system, such as the limited length of the rail, and the restriction on the maximum control action. here are many works about the swing-up and stabiliza- tion of the pendulum-cart system using several methods, for instance, [6–12]. In [6] the energy control method is used to swing the pendulum up from its pendant position to around the upright position, and a linear servo state feedback controller design by coeicient diagram method is used to stabilize the pendulum. In [7], a hybrid fuzzy controller with fuzzy swing-up and parallel distributed pole assignment schemes is adopted to position the pendulum and the cart at the desired states. he T-S fuzzy model proposed is obtained via linearization with respect to diferent operating points; it consists of seven fuzzy rules and friction is considered. Hindawi Publishing Corporation Mathematical Problems in Engineering Article ID 247682