Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 247682, 11 pages

http://dx.doi.org/10.1155/2015/247682

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

^{1}Tecnológico Nacional de México, Instituto Tecnológico de La Laguna, Boulevard Revolución y Calzada Cuauhtémoc, 27000 Torreón, COAH, Mexico^{2}Universidad 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.

#### Abstract

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.

#### 1. Introduction

A great number of nonlinear systems can be represented by Takagi-Sugeno (T-S) fuzzy models. They are considered universal approximators [1]. In [2–4], the T-S fuzzy control system stability has been verified considering a common Lyapunov 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.

The pendulum-cart system is a perfect test bed for demonstrating the theoretical and practical aspects of the control theory because of its inherently unstable open-loop with highly nonlinear dynamics. Two different dynamics of the pendulum and the cart are coupled together. There are several limitations in controlling the system, such as the limited length of the rail, and the restriction on the maximum control action.

There are many works about the swing-up and stabilization 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 coefficient 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. The T-S fuzzy model proposed is obtained via linearization with respect to different operating points; it consists of seven fuzzy rules and friction is considered. The effectiveness of the proposed controller is validated via numerical simulations. In [8] a hybrid fuzzy controller is proposed to swing and stabilize the pendulum-cart system. The controller is designed to have a robust performance using the LMIs technique for T-S fuzzy systems. The T-S fuzzy model proposed consists of three fuzzy rules obtained through linearization via Taylor’s series where friction has not been considered. The effectiveness of this method is validated via simulation and real-time experiment. In [9] a swing-up and tracking controller design for a pendulum-cart system using hybrid fuzzy control has been proposed. A fuzzy tracking controller is designed based on a synthesis of the tracking control theory of linear multivariable control and the T-S fuzzy model. A stabilizing compensator based on observer is chosen. The Takagi-Sugeno fuzzy model is obtained via Taylor’s series linearization and consists of three fuzzy rules where friction has not been considered and both controller and observer gains are obtained via poles placement method. In [10] the robust fuzzy control problem for uncertain continuous-time nonlinear systems is considered. The T-S fuzzy model with norm-bounded parameter uncertainties is adopted. Parallel distributed compensation (PDC) scheme is employed to design, independently, the robust fuzzy controller and the robust fuzzy observer from the T-S fuzzy models. The number of rules is only two. Simulation on an inverted pendulum system demonstrates the effectiveness and the applicability of the proposed approach. On the other hand, in [11] robust controller design methodologies for T-S descriptors are considered. Two different approaches, based on LMIs, are proposed. The first one involves classical closed-loop dynamics formulation and the second one redundancy closed-loop dynamics approach. The provided conditions are obtained through a fuzzy Lyapunov function candidate and a non-PDC control law. Both the classical and redundancy approaches are compared. It is shown that the latter leads to less conservative stability conditions. To show the applicability of the proposed approaches, the benchmark stabilization of an inverted pendulum on a cart is considered. Finally, in [12] a T-S fuzzy dynamic regulator for a pendulum-cart system is proposed using local approximation in fuzzy partition spaces to derive the T-S fuzzy model of the nonlinear system. Both a fuzzy controller and a fuzzy observer are designed via PDC scheme for which feedback gains are obtained via LMIs technique. Real-time experiments validate the effectiveness of this approach for the regulation case only.

In this paper, unlike [12], the focus is placed on the trajectory tracking problem, that is, stabilizing the pendulum in its upright position while the cart follows a reference signal. Starting from a nonlinear model for a pendulum-cart system, on which viscous friction is considered, a Takagi-Sugeno 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: a T-S fuzzy servo controller (TSFSC) designed from the TSFAM; a robust TSFSC (RTSFSC) designed from the TSFAMwU; and a robust T-S fuzzy dynamic regulator (RTSFDR) designed from the RTSFSC with the addition of a T-S fuzzy observer, designed also via PDC using the* separation principle*, which estimates cart and pendulum velocities. Both TSFAM and TSFAMwU are comprised of only two fuzzy rules and designed via local approximation in fuzzy partition spaces technique. The three T-S fuzzy controllers are designed via PDC scheme for which the state feedback gains of the local linear controllers are obtained via LMIs technique for Takagi-Sugeno fuzzy systems. A nonfuzzy swing-up controller is developed to swing the pendulum up from its pendant position to its upright position, where any of the three T-S fuzzy controllers takes action. Real-time experiments validate the effectiveness of the three proposed schemes, keeping the pendulum in its upright position while the cart follows a reference signal. The performance of the three proposed controllers is evaluated using the norm of the stable state errors of the cart and pendulum, based on the norm , standing out between the three controllers the RTSFDR, which presents the smaller errors.

This paper is organized as follows. Section 2 describes the state-space model of the pendulum-cart system. In Section 3 the framework of the T-S fuzzy modeling is described and also shows how the servo compensator model is introduced into a Takagi-Sugeno fuzzy model. The design of the three proposed fuzzy controllers is developed in Section 4. Real-time experimentation results are shown in Section 5. Finally, in Section 6 the conclusions are given.

#### 2. State-Space Model of the Pendulum-Cart System

The state-space representation of the pendulum-cart system is given as in [12] (see Figure 1):
where denotes the position of the cart from the center of the rail [m], denotes the angle of the pendulum from the upright position [rad], is the velocity of the cart [m/s], is the angular velocity of the pendulum [rad/s], is the gravity constant [], is the mass of the pendulum [kg], is the mass of the cart [kg], is the distance from the axis of rotation to the center of mass of the pendulum-cart system [m], is the moment of inertia of the pendulum-cart system with respect to the center of mass [kg·m^{2}], is the force applied to the cart [N], and and represent the viscous friction of the cart and the pendulum [N·m·s/rad], respectively; , , , , and .