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Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 1241545, 25 pages
Research Article

MIMO PI Controllers for LTI Systems with Multiple Time Delays Based on ILMIs and Sensitivity Functions

National Institute of Applied Sciences and Technology (INSAT), Centre Urbain Nord, BP 676, 1080 Tunis Cedex, Tunisia

Correspondence should be addressed to Olfa Boubaker; nt.unr.tasni@rekabuob.aflo

Received 12 August 2016; Revised 9 November 2016; Accepted 5 December 2016; Published 20 February 2017

Academic Editor: Tamas Kalmar-Nagy

Copyright © 2017 Wajdi Belhaj and Olfa Boubaker. 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 paper, a MIMO PI design procedure is proposed for linear time invariant (LTI) systems with multiple time delays. The controller tuning is established in two stages and guarantees performances for set-point changes, disturbance variations, and parametric uncertainties. In the first stage, an iterative linear matrix inequality (ILMI) approach is extended to design PI controllers for systems with multiple time delays without performance guarantee, a priori. The second stage is devoted to improve the closed-loop performances by minimizing sensitivity functions. Simulations results carried out on the unstable distillation column, the stable industrial scale polymerization (ISP) reactor, and the non-minimum phase 4-tank benchmark prove the efficiency of the proposed approach. A comparative analysis with the conventional internal model control (IMC) approach, a multiloop IMC-PI approach, and a previous ILMI PID approach proves the superiority of the proposed approach compared to the related ones.