State and Unknown Inputs Estimation for a Class of Discrete-time Takagi-Sugeno Descriptor Models Karim Bouassem 1,2 , Jalal Soulami 1,2 , Abdellatif El Assoudi 1,2 , El Hassane El Yaagoubi 1,2 1 Laboratory of High Energy Physics and Condensed Matter, Faculty of Science Hassan II University of Casablanca, B.P 5366, Maarif, Casablanca, Morocco 2 ECPI, Department of Electrical Engineering, ENSEM Hassan II University of Casablanca, B.P 8118, Oasis, Casablanca Morocco. Email: jalal.soulami@gmail.com Abstract—In this paper, the design problem of simultaneous estimation of unmeasurable states and unknown inputs (UIs) is investigated for a class of discrete-time Takagi-Sugeno descriptor models (DTSDMs) with measurable premise variables. The UIs affect both state and output of the system. The approach is based on the separation between dynamic and static relations in the considered DTSDM. First, the method permitting to separate dynamic equations from static equations is exposed. Next, an augmented fuzzy explicit model which contains the dynamic equations and the UIs is constructed. Then a fuzzy unknown inputs observer (FUIO) design in explicit structure is developed. The exponential convergence of the state estimation error is studied by using the Lyapunov theory and the stability conditions are given in terms of linear matrix inequalities (LMIs). Finally, an illustrative example is given to show the good performances of the proposed method. Keywords: Discrete-time Takagi-Sugeno descriptor model, unknown inputs, fuzzy unknown inputs observer, LMI. In this paper, some notations used are fair standard. For example, X> 0 means the matrix X is symmetric and positive definite. X T denotes the transpose of X. The symbol I (or 0) represents the identity matrix (or zero matrix) with appropriate dimension. q  i,j=1 μ i μ j = q  i=1 q  j=1 μ i μ j ,  X ∗ Z Y  =  X Z T Z Y  . I. I NTRODUCTION AND PROBLEM STATEMENT Descriptor dynamic models, known as a generalization of standard dynamic models, constitute a powerful modeling tool allowing to describe the dynamic behavior of processes governed by both dynamic and static equations. They represent physical phenomenas that can not be described by standard models, see [1], [2], [3] for some real applications of descriptor models. Moreover, the ordinary T-S fuzzy model [4], [5] has been successfully developed to study nonlinear control systems, see e.g. [6], [7] and the references therein. In [8], [9], a fuzzy descriptor system is defined by extending the T- S fuzzy model [4]. Notice that, UIs can result either from uncertainty in the model or from the presence of unknown external excitation. Thus, due to the increasing demand for reliability and maintenability of the automatic control process, unknown inputs observer design is widely used in the area of fault detection and design of fault tolerant control strategy. This is one of the most attractive research areas in both theoretical and practical fields during these last two decades, see e.g. [10], [11], [12] for works using different approaches. In this paper, the following class of DTSDMs subject to UIs which affect both state and output of the system is considered:            MZ k+1 = q  i=1 μ i (η k )(A i Z k + B i u k + C i d k ) y k = q  i=1 μ i (η k )(D i Z k + E i u k + F i d k ) (1) where Z T k =[Z 1 k T Z 2 k T ] ∈ R n is the state vector with Z 1 k ∈ R n1 is the vector of difference variables, Z 2 k ∈ R n2 is the vector of algebraic variables with n 1 +n 2 = n, u k ∈ R m is the control input, d k ∈ R r is the unknown control input, y k ∈ R p is the measured output. A i ∈ R n×n , B i ∈ R n×m , C i ∈ R n×r , D i ∈ R p×n , E i ∈ R p×m , F i ∈ R p×r , M ∈ R n×n such that rank(M )= n 1 are real known constant matrices with: M =  I 0 0 0  ; A i =  A 11i A 12i A 21i A 22i  (2) B i =  B 1i B 2i  ; C i =  C 1i C 2i  ; D i = ( D 1i D 2i ) (3) where constant matrices A 22i are supposed invertible. q is the number of sub-models. η k is the premise variable which is supposed here to be real-time accessible and the μ i (η k ) (i =1, ..., q) are the weighting functions that ensure the transition between the contribution of each sub model:  MZ k+1 = A i Z k + B i u k + C i d k y k = D i Z k + E i u k + F i d k (4) They verify the so-called convex sum properties:      q  i=1 μ i (η k )=1 0 ≤ μ i (η k ) ≤ 1 i =1,...,q (5) 5th International Conference on Control Engineering&Information Technology (CEIT-2017) Proceeding of Engineering and Technology –PET Vol.32 pp.16-20 Copyright IPCO-2017 ISSN 2356-5608