ABOUT THE OBSERVABILITY OF NONLINEAR SWITCHING SYSTEMS Rudy KAJDAN * Didier AUBRY ** Fr´ ed´ eric KRATZ * * Laboratoire de Vision et Robotique, ENSI de Bourges, 88 Boulevard Lahitolle, 18020 Bourges, Cedex, France firstname.lastname@ensi-bourges.fr ** Laboratoire de Vision et Robotique, IUT de Bourges, Universit´ e d’Orl´ eans, 63 avenue De Lattre de Tassigny, 18020 Bourges, Cedex, France Didier.Aubry@bourges.univ-orleans.fr Abstract: The purpose of this paper is to give sufficient conditions to ensure observ- ability of linear and non-linear switching systems with discrete time representation. These conditions may be useful for the design of observers for such systems. An example on a two tank system will be given in order to illustrate our method. Copyright c  2007 IFAC Keywords: Observability, Hybrid systems, Nonlinear systems. 1. INTRODUCTION Hybrid dynamical systems are systems whose be- havior involves continuous and discrete variables. The evolution of such systems can be described by a succession of modes. Each mode corresponds to a modality of the discrete variables. For a given mode, the hybrid system’s evolution in this mode is described by a continuous system. Modes transitions obey conditions on the discrete inputs and/or on the states. Hybrid models possess a great number of applications: e.g. batch processes (Both and Hanisch 2002), air traffic manage- ment (Tomlin et al. 1998), bioprocesses (Vald´ es- Gonz´ alez and Flaus 2001) etc. The study of hybrid systems has reached a large interest nowadays. The stability of these systems was studied for instance in (Johansson and Rantzer 1998) or in (Branicky 1994). In (Bemporad and Morari 1999), the authors propose an approach for the control of a class of hybrid systems called mixed logical dynamical (MLD) systems. Observability of switching systems has already been studied in several papers. In (Vidal et al. 2002), the authors deal with the observabil- ity of linear discrete-time switching systems. In their contribution, the system is supposed to re- main in each mode during a period which is at least equal to twice his joint-observability index. The work exposed in (Vidal et al. 2003) is the continuous-time counterpart of the previous arti- cle. In (Babaali and Egerstedt 2004), the authors study the joint observability for autonomous and non autonomous linear systems which may switch at each sample time. In this paper, the pathwise observability and the forward joint observabil- ity are shown to be decidable for autonomous discrete-time switching systems. The observability of linear system with MLD form was studied in (Bemporad et al. 2000). To our knowledge, it seems that few authors deal with the problem of observability of switching systems in the nonlinear case. In (Boutat et al. 2004), sufficient geometrical conditions are given to analyse the observability