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Mathematical Problems in Engineering
Volume 2015, Article ID 187284, 10 pages
Research Article

In-Domain Control of a Heat Equation: An Approach Combining Zero-Dynamics Inverse and Differential Flatness

1Department of Basic Courses, Southwest Jiaotong University, Emeishan, Sichuan 614202, China
2Department of Electrical Engineering, Polytechnique Montreal, P.O. Box 6079, Station Centre-Ville, Montreal, QC, Canada H3T 1J4

Received 11 November 2015; Revised 17 December 2015; Accepted 21 December 2015

Academic Editor: Rafael Morales

Copyright © 2015 Jun Zheng and Guchuan Zhu. 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 addresses the set-point control problem of a one-dimensional heat equation with in-domain actuation. The proposed scheme is based on the framework of zero-dynamics inverse combined with flat system control. Moreover, the set-point control is cast into a motion planning problem of a multiple-input, multiple-output system, which is solved by a Green’s function-based reference trajectory decomposition. The validity of the proposed method is assessed through the analysis of the invertibility of the map generated by Green’s function and the convergence of the regulation error. The performance of the developed control scheme and the viability of the proposed approach are confirmed by numerical simulation of a representative system.