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Mathematical Problems in Engineering
Volume 2015, Article ID 160542, 10 pages
Research Article

Design of a Backstepping Tracking Controller for a Class of Linear Systems with Actuator Delay

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received 29 April 2015; Revised 2 October 2015; Accepted 15 October 2015

Academic Editor: Alfonso Banos

Copyright © 2015 Yonglong Liao and Fucheng Liao. 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 presents a method for designing a backstepping tracking controller for a class of continuous-time linear systems with actuator delay subject to a reference signal. The actuator delay can be modeled by a first-order hyperbolic PDE, and then a PDE-ODE coupled system is obtained. By applying the backstepping transformation to the coupled system, a feedback controller that includes the state of the system, the integral of the input control, and the integral of the tracking error is derived. We show that the closed-loop system is asymptotically stable at the equilibrium point and achieves complete regulation under the stabilizability assumption. The designs in this paper are illustrated with numerical simulations.