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Journal of Control Science and Engineering
Volume 2015, Article ID 636909, 10 pages
Research Article

Observer Design for a Class of Nonlinear Descriptor Systems: A Takagi-Sugeno Approach with Unmeasurable Premise Variables

1ECPI, Department of Electrical Engineering, ENSEM, University Hassan II of Casablanca, BP 8118, Oasis, 20103 Casablanca, Morocco
2ESPCM, Department of Physics, Faculty of Science, University Ibn Tofail of Kenitra, BP 133, 14 000 Kenitra, Morocco

Received 18 June 2014; Revised 20 January 2015; Accepted 8 March 2015

Academic Editor: Ai-Guo Wu

Copyright © 2015 J. Soulami 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.


The Takagi-Sugeno (T-S) fuzzy observer for dynamical systems described by ordinary differential equations is widely discussed in the literature. The aim of this paper is to extend this observer design to a class of T-S descriptor systems with unmeasurable premise variables. In practice, the computation of solutions of differential-algebraic equations requires the combination of an ordinary differential equations (ODE) routine together with an optimization algorithm. Therefore, a natural way permitting to estimate the state of such a system is to design a procedure based on a similar numerical algorithm. Beside some numerical difficulties, the drawback of such a method lies in the fact that it is not easy to establish a rigorous proof of the convergence of the observer. The main result of this paper consists in showing that the state estimation problem for a class of T-S descriptor systems can be achieved by using a fuzzy observer having only an ODE structure. The convergence of the state estimation error is studied using the Lyapunov theory and the stability conditions are given in terms of linear matrix inequalities (LMIs). Finally, an application to a model of a heat exchanger pilot process is given to illustrate the performance of the proposed observer.