Systems & Control Letters 39 (2000) 141–151 www.elsevier.com/locate/sysconle A reduced-order observer for non-linear discrete-time systems M. Boutayeb ∗ , M. Darouach CRAN-CNRS, University of Henri Poincar e, Nancy I, 186, rue de Lorraine, 54400 Cosnes et Romain, France Received 14 November 1998; received in revised form 25 October 1999 Abstract In this note, a simple and useful reduced-order observer for a large class of non-linear discrete-time systems is presented. Neither detectability of the linear part nor resolution of the Sylvester constraint is required. From the stability analysis we show that the proposed state observer has strong tracking properties under very general conditions and generalizes results in (M. Boutayeb et al., IEEE. Trans. Automat. Control 42(4) (1997) 581–586). High performances, from the design and implementation point of view, will be shown through three numerical examples of physical processes largely used in industrial applications. c  2000 Elsevier Science B.V. All rights reserved. Keywords: Reduced-order observers; Non-linear discrete-time systems; Stability analysis 1. Introduction If the state estimation of non-linear continuous systems has received large attention in the last two decades, design of reduced-order observers remains, however, a challenge as can be shown from the few works devel- oped in this eld and where the discrete-time case was rarely addressed. Many of the results developed by now consider the class of non-linear systems with linear parts in the state and measurement equations [16,20,9,2,1] that lead among others to a classical algebraic constraint called the Sylvester equation. We note, by the way, that a simple and useful technique to solve the general form of the Sylvester equation was proposed in [7] with an extension to a class of non-linear descriptor systems [2]. A second and recent approach [9] consists rst in using a non-linear change of coordinates to bring the original systems into a canonical form on which a reduced-order Luenberger-like observer is applied. Let us remark, however, that state observers based on non-linear change of coordinates concerns only a class of dynamic systems under strong assumptions (see [11]). Furthermore, design and implementation in practice seems to be fastidious and require high computational capacity [13] especially when the inverse of some matrices is needed [9,5]. In this contribution we propose a useful local reduced-order observer for a large class of multi-input– multi-output (MIMO) non-linear discrete-time systems with general non linear dynamics and not necessarily with the presence of a linear part in the state. First, we provide a simple structure of state observers of order n − p where n and p are the state and output dimensions, respectively. The second part concerns the stability * Corresponding author. E-mail address: boutayeb@iut-longwy.u-nancy.fr (M. Boutayeb) 0167-6911/00/$ - see front matter c  2000 Elsevier Science B.V. All rights reserved. PII: S0167-6911(99)00102-4