Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 527413, 22 pages
Research Article

Two-Valued Control for a Second-Order Plant with Additive External Disturbance

1Programa de Ingeniería Ambiental, Facultad de Ingeniería y Arquitectura, Universidad Católica de Manizales, Cr 23 no. 60-30, Manizales 170002, Colombia
2Percepción y Control Inteligente, Departamento de Ingeniería Eléctrica, Electrónica y Computación, Facultad de Ingeniería y Arquitectura, Universidad Nacional de Colombia, Bloque Q, Campus La Nubia, Manizales 170003, Colombia

Received 29 October 2011; Revised 11 January 2012; Accepted 21 January 2012

Academic Editor: Oded Gottlieb

Copyright © 2012 Alejandro Rincon 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.


In this work a two-valued state feedback control for a plant of second order with known constant coefficients and an additive bounded disturbance is designed. In this controller the control signal can take only two possible values. The controller design is based on Lyapunov-like function method, achieving the convergence of the tracking error to a user-defined residual set. A boundedness condition for the user-defined reference signal is defined, which is necessary to allow out-put tracking. The developed scheme avoids large commutation rate of the control input. The controller design and stability analysis have important contributions with respect to closely related controllers based on the direct Lyapunov method, namely, (i) conditions to guarantee the expected convergence of the tracking error are established. These conditions are imposed on the reference signal and the extreme values of the control input. The stability analysis is developed by means of the Lyapunov-like function method and the Barbalat's Lemma and includes (ii) the bounded nature of the Lyapunov function, (iii) the monotonic convergence of the Lyapunov function to a residual set, and (iv) the asymptotic convergence of the tracking error to a residual set of user-defined size.