EXACT DIFFERENTIATION VIA SLIDING MODE OBSERVER FOR SWITCHED SYSTEMS H. Saadaoui ∗ , M. Djema¨ ı ∗ , N. Manamanni ∗∗ , T. Floquet ∗∗∗ , J-P. Barbot ∗ ∗ Equipe Commande des Syst` emes (ECS), ENSEA, 6 Av. du Ponceau, 95014 Cergy-Pontoise Cedex, France. {djemai,saadaoui,barbot}@ensea.fr ∗∗ CReSTIC, University of Reims, Moulin de la Housse BP 1039, 51687 Reims cedex 2 - France, noureddine.manamanni@univ-reims.fr ∗∗∗ LAGIS, UMR CNRS 8146, Ecole Centrale de Lille, Cit´ e Scientifique, BP 48, 59651 Villeneuve d’Ascq Cedex, France, floquet@ec-lille.fr Abstract: The main topic of this paper is the problem of observer synthesis for switched systems, which includes, as a specific case, the design of observers based on high order sliding mode technique. High order sliding mode is used to overcome the occurring chattering phenomena which induces some irrelevant decision of switching between the subsystems when the trajectory is in the neighborhood of the switching manifold. Moreover, in this paper, after presenting the general structure of the step by step differentiator, well show the step by step finite time convergence of the estimation error and the discrete state estimation. Two simulation examples illustrate the efficiency of the proposed approach. Copyright c  IFAC 2006. Keywords: Hybrid system, Switched systems, Non linear observer, Sliding mode, Exact differentiator, Finite time convergence. 1. INTRODUCTION Switched systems are a class of Hybrid Sys- tems (HS) which consist of several subsystems that switch according to a given switching law (Antsaklis, 2000). A rich and thorough bibliography deals with sta- bility problems of switched systems, see (Branicky, 1998), (Liberzon, 2000), (Michel and Sun, 2003) and references therein. More recently, various re- searchers have studied observability and observer design for such systems. Some sufficient geomet- rical conditions to analyze the observability of hybrid dynamical systems were given in (Boutat et al., 2004). These conditions are refined for the particular class of piecewise linear and nonlinear systems. The so-called extended joint observabil- ity matrix was proposed in (Vidal et al., 2003), to analyze the observability of jump linear sys- tems. In (Sontag, 1979), Sontag introduced a set of observability related definitions and examined the implications among the various concepts of observability. In the same way, other works deal with the hybrid observer design. Indeed, in (Balluchi et al., 2002) a methodol- ogy was presented for the design of dynamical observers of hybrid systems that reconstruct the discrete state and the continuous state from the knowledge of the continuous and discrete outputs. The design of linear observers for a class of linear hybrid systems was addressed in (De laSen and Preprints of the 2nd IFAC Conf. on Analysis and Design of Hybrid Systems (Alghero, Italy), 7-9 June 2006 124