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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 314964, 18 pages
Research Article

𝐻 Neural-Network-Based Discrete-Time Fuzzy Control of Continuous-Time Nonlinear Systems with Dither

1Department of Computer Science and Information Engineering, Asia University, No. 500 Lioufeng Road, Wufeng Distinguish, Taichung City 41354, Taiwan
2Department of Electronic Engineering, Jinwen University of Science and Technology, No. 99 Anchung Road, Xindian Distinguish, New Taipei City 23154, Taiwan

Received 23 April 2011; Revised 26 June 2011; Accepted 6 July 2011

Academic Editor: Zidong Wang

Copyright © 2012 Zhi-Ren Tsai and Jiing-Dong Hwang. 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 study presents an effective approach to stabilizing a continuous-time (CT) nonlinear system using dithers and a discrete-time (DT) fuzzy controller. A CT nonlinear system is first discretized to a DT nonlinear system. Then, a Neural-Network (NN) system is established to approximate a DT nonlinear system. Next, a Linear Difference Inclusion state-space representation is established for the dynamics of the NN system. Subsequently, a Takagi-Sugeno DT fuzzy controller is designed to stabilize this NN system. If the DT fuzzy controller cannot stabilize the NN system, a dither, as an auxiliary of the controller, is simultaneously introduced to stabilize the closed-loop CT nonlinear system by using the Simplex optimization and the linear matrix inequality method. This dither can be injected into the original CT nonlinear system by the proposed injecting procedure, and this NN system is established to approximate this dithered system. When the discretized frequency or sampling frequency of the CT system is sufficiently high, the DT system can maintain the dynamic of the CT system. We can design the sampling frequency, so the trajectory of the DT system and the relaxed CT system can be made as close as desired.