Fault Tolerant Estimation for Multisensor Nonlinear Systems A.M Lienhardt1, A. Aïtouche1, B. Ould Bouamama2 LAGIS UMR CNRS 8146 1ERASM-HEI, 13 rue de Toul. 59046 Lille Cedex Phone: +333328384858, Fax: +33328384804, E-mail: abdel.aitouche@hei.fr 2Polytech’Lille, 59655 Villeneuve d’Ascq, France Phone: +3332337139, Fax: +33320337189, E-mail: belkacem.bouamama@univ-lille1.fr Abstract - This paper deals with the analysis of fault tolerance properties of multisensor nonlinear systems. These properties do not depend on the kind and size of the faults but only of the subset of the faulty sensors. For linear systems, fault tolerance estimation has been studied. In this paper, we have provided in the case of nonlinear system, the computation of observability indices and minimal/redundant sensors. Multisensor systems can accept sensor failures as long their objective (estimate some functional of state) can still be achieved. The fault tolerance evaluation with respect to sensor failures is based on redundancy degrees for which observability properties remain satisfied. An analysis of the graph built using redundant and minimal sensor sets allows computing the redundancy degrees with respect to sensor losses. An example of a nonlinear three-tank system demonstrates the effectiveness of the approach. I. INTRODUCTION Fault tolerance is highly required for modeling complex control systems. Sensors, actuators or process failures may drastically change the system behavior ranging from performance degradation to instability. Fault- Tolerant Control (FTC) is needed in order to preserve the ability of the system to achieve the objectives that have been assigned and to reach new objectives so as to avoid unexpected behaviors. A survey of different approaches has been proposed by [Patton, 1997]. The design of fault-tolerant systems needs the system to remain observable in the presence of sensor failures. The observability is a property which characterizes the system and that has been introduced by [Kalman and Bucy, 1961]. The notions of observability are of great importance in state estimation design for dynamical systems. Observability is a necessary and sufficient condition for the existence of an estimator. For linear systems, the notions of observability are wellunderstood and do not depend on the region within which the system operates, the properties of the system are global properties. There are several notions of observability in LTI systems that include complete and partial observability and observability of an eigenvalue. It has since then motivated many works in linear systems: necessary and sufficient observability condition [Kailath, 1991], structural observability [Izadi-Zamanabadi and Staroswiecki, 2000], observability gramian proposed to estimate the system recoverability [Frei et al, 199], [Wu et al, 2000)]. However, for nonlinear systems, the notions of observability are very complex and do depend on the region within the system operates, the properties of the system are local properties and difficult to assess in the general case. Then, for general nonlinear systems global or complete observabilty cannot usually be expected, and therefore local [Gauthier and Kupka, 1994] or generic observability [Deleon et al, 2000] could be used. Observabilty analysis is the main step to identify the system monitorable part and provides the computation for the estimation algorithms and their reconfiguration. Fault tolerance estimation of linear systems has been studied by [Staroswiecki et al, 1999] based on the use of individual observability indices. In this paper, we intend to expand this approach in the case of nonlinear systems thus proposing a computation of these indices. Our analysis of observability leant on the definition of minimal and redundant sensor sets. This paper is organized as follows: some notions related to observability of nonlinear systems with reference to the observability problem of a functional of the system state are presented in section 2. In section 3, recoverability that is the possibility to accommodate the faults or to reconfigure the system when faults occur and minimal/redundant sensor sets are introduced. In the section 4, the fault tolerance effectiveness of multi sensor systems is evaluated through structural criteria: strong and weak redundancy degrees. Finally, an analysis of fault tolerant estimation of the laboratory three-tank benchmark system is presented. II. PRELIMINARIES First some notations are introduced, which are used throughout this paper. .....