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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 970786, 14 pages
Research Article

A Multivariable Adaptive Control Approach for Stabilization of a Cart-Type Double Inverted Pendulum

1Control Engineering Department, Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 5166616471, Iran
2Electrical Engineering Department, Tehran University, Tehran 5157747343, Iran
3Electrical Engineering Department, Tarbiat Modares University, Tehran 5156913547, Iran

Received 9 August 2010; Revised 5 December 2010; Accepted 7 March 2011

Academic Editor: Maria do Rosário de Pinho

Copyright © 2011 I. Hassanzadeh 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.


This paper considers the design and practical implementation of linear-based controllers for a cart-type double inverted pendulum (DIPC). A constitution of two linked pendulums placed on a sliding cart, presenting a three Degrees of Freedom and single controlling input structure. The controller objective is to keep both pendulums in an up-up unstable equilibrium point. Modeling is based on the Euler-Lagrange equations, and the resulted nonlinear model is linearized around up-up position. First, the LQR method is used to stabilize DIPC by a feedback gain matrix in order to minimize a quadratic cost function. Without using an observer to estimate the unmeasured states, in the next step we make use of LQG controller which combines the Kalman-Bucy filter estimation and LQR feedback control to obtain a better steady-state performance, but poor robustness. Eventually, to overcome the unknown nonlinear model parameters, an adaptive controller is designed. This controller is based on Model Reference Adaptive System (MRAS) method, which uses the Lyapunov function to eliminate the defined state error. This controller improves both the steady-state and disturbance responses.