European Journal of Control (1998) 4: 2- 16 © 1998 EUCA European Journal of Control Fault Detection and Isolation for State Affine Systems H. Hammouri l , M. Kinnaert 2 and E. H. El Yaagoubi 3 1 LAGEP, Universite Claude Bernard Lyon I, Villeurbanne Cedex, France; 2 Laboratoire d'Automatique, Universite Libre de Bruxelles, Brussels, Belgium; J ENSEM, Casablanca, Morocco The problem of residual generation for fault detection and isolation (FDI) in state affine systems is studied. Sufficient conditions for the existence of a solution are presented, and the connection with the geometric theory of FDI for linear systems is emphasised. Kalman-like time-varying observers are usedfor the synthesis of the residual generators. The theoretical results are illu- strated by simulations of diagnosis systems for a chemical reactor and a distillation column. Keywords: Fault detection and isolation; Non-linear observer; Residual generator; State affine system 1. Introduction A typical model-based diagnosis system consists of three parts: a residual generator, a residual evaluation module and a decision system. The first one takes the actuator commands and the measurements as inputs, and it generates signals called residuals. These are nominally equal to zero in the absence of faults, and some of them become distinguishably different from zero upon occurrence of a fault. The residual evaluation module determines whether each residual is significantly different from zero. Finally, by analys- ing the pattern of zero and non-zero residuals, the decision system isolates the fault(s), i.e. it localises the faulty component(s). In this paper we only con- sider the design of residual generators. Several approaches exist for the synthesis of resid- ual generators for linear systems. Among them, the observer-based methods have been extensively Correspondence and offprint requests to: H. Hammouri, Universite Claude Bernard Lyon I, Bat ESCPE-L YON-308G, 43 Bd du 11 novembre 1918 , 69622 Villeurbanne Cedex, France. E-mail: ham- mouri @lagep.cpe.fr studied. Initially, the residual generators were designed so as to include an observer for the whole state of the plant [1 , 10,13,17,22]. Later it was realised that only a specific part of the plant state needs to be estimated to obtain a residual [4,14,26]. In this last framework, necessary and sufficient conditions for the existence of a solution to a basic problem of fault detection and isolation called the fundamental problem of residual generation (FPRG) can be expressed very clearly both in geometric terms and in the frequency domain [14]. In this paper, we attempt to generalise the latter approach to the class of state affine systems up to output injection. To this end, we give a careful for- mulation of the FPRG in order to assure that the dependence of the residual with respect to the failure it has to detect is expressed in a tractable way. The geometric approach to system theory is then used to present sufficient conditions for the existence of a solution to the problem. Of course, the obtained results apply to bilinear and linear systems, as they belong to subclasses of the considered model class. The connection with the geometric approach to FDI for linear systems is emphasised. Several algebraic approaches to the problem of FPRG for bilinear systems which are linear or bilinear in the failure modes have been developed [11,12,15,27]. In the first two, the observers or filters that are used are linear time-invariant (up to an out- put injection). On the contrary, in [11,12], Kalman- like time-varying observers are considered. The latter are also utilised here. Indeed, even if the attention is restricted to bilinear systems only, those observers allow one to handle the synthesis of residual genera- tors for a larger class of such systems. By particular- Received 2 October 1996; Accepted in revised form I August 1997. Recommended by A. S. Morse and P. M. Frank.