IFAC PapersOnLine 50-1 (2017) 2756–2761 ScienceDirect Available online at www.sciencedirect.com 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2017.08.583 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: H ∞ dynamic observer, discrete-time systems, LMI, estimation and filtering, fault detection and diagnosis. 1. INTRODUCTION AND PROBLEM FORMULATION A considerable amount of research has been devoted to observer design for linear discrete-time systems, ever since Luenberger presented the results on the observer Luen- berger (1971). In Leondes and Novak (1974), a reduced- order observer was designed for linear discrete-time sys- tems. A class of general structured discrete-time deter- ministic observers were developed in Cheok et al. (1982). It is important to design observer for the system with disturbances, since the disturbance is very common in practice and has negative influence. One method to deal with the disturbance is disturbance observer, which can estimate the state and disturbance simultaneously. The authors of Kim and Rew (2013) presented an output-based disturbance observer for linear discrete-time systems. In Kang (2014), the disturbance observer design problem was solved, based on the solution of linear matrix inequality (LMI). Another method named H ∞ observer, which can decrease the influence of disturbance directly, has been widely used recently. The applications of H ∞ observer can be found in Pe˜ narrocha Al´ os et al. (2009), Abbaszadeh and Marquez (2009) and references therein. Other method such as Kalman filter can be found in Zorzi (2016) and references therein. The observers introduced previously are proportional ob- servers (PO). In order to deal with the static error in the ⋆ This work is supported by the Fundamental Research Funds for the Central Universities (under Grant 310822171005 ), in Chang’an University, China. PO estimation procedure, the proportional integral ob- server (PIO) was introduced (see Orjuela et al. (2007) and references therein). Recently, a new structure of observer named dynamic observer (DO) has been developed as an extension of the classical static observer. In Darouach et al. (2013), an H ∞ DO (HDO), which generalized the existing results on the PO and PIO was proposed. More recently, a more generalized form of DO was proposed in Gao et al. (2014b) for uncertain systems and in Gao et al. (2014a) for systems in the presence of disturbances and unknown inputs. The contribution of this paper is to extend the result of Gao et al. (2014a) for continuous-time systems to discrete- time systems in the presence of disturbances. The observer is derived from the solution of LMIs, based on the algebraic constraints obtained through the analysis of the estimation error. The rest of the paper is organized as follows. Section 1 presents the problem. The parameterizations of algebraic constraints are presented in section 2. The observer design problem is solved in section 3. Section 4 provides a numerical example to illustrate our observer design procedure and its performance. Finally, in section 5, some conclusions are given. Notation : R n is the n dimensional Euclidean space; R n×m is the set of all n × m real matrices; A T denotes the transpose of matrix A; matrix A is symmetric positive definite if and only if A T = A and A> 0; A + denotes any generalized inverse of matrix A which satisfies AA + A = A; ‖.‖ ∞ is the H ∞ norm; I i denotes the i × i identity matrix and 0 denotes the zero matrix of appropriate dimension; A ⊥ denotes the left orthogonal complement of matrix A, * School of Automobile, Chang’an University, Middle-section of Nan’er Huan Road, Xi’an, ShaanXi Province, 710064, China (e-mail: gaonan@chd.edu.cn). ** CRAN-CNRS UMR7039, University de Lorraine, IUT de Longwy, 186, Rue de Lorraine, Cosnes et Romain 54400, France (e-mail: Mohamed.Darouach@univ-lorraine.fr) *** CRAN-CNRS UMR7039, University de Lorraine, IUT de Longwy, 186, Rue de Lorraine, Cosnes et Romain 54400, France (e-mail: marouane.alma@univ-lorraine.fr) Abstract: The objective of this paper is to propose a new form of H ∞ dynamic observer for linear discrete-time systems. The widely used proportional observer (PO) and proportional integral observer (PIO) can be considered as particular cases of the proposed observer. The observer design is derived from the solution of linear matrix inequalities (LMIs), based on the solutions of algebraic constraints obtained from the unbiasedness conditions of estimation error. A numerical example is provided to show the performance of the proposed observer, compared with PO and PIO. Nao GAO * Mohamed DAROUACH ** Marouane ALMA *** H ∞ dynamic observer design for linear discrete-time systems ⋆