Interval analysis based learning for fault model identification. Application to control surfaces oscillatory failures

Interval models may be seen as a trade-off between numerical and qualitative models. They have been often referred as semi-qualitative models. The interval algebra is indeed a specific qualitative algebra with advantageous algebraic properties. This paper presents the application of an interval based parameter estimation method, which is used for learning fault models supporting the detection of Oscillatory Failure Cases (OFC) in Electrical Flight Control System (EFCS) of civil airplanes. The interval estimation method results are guaranteed and computations are performed in finite time. Failures are identified using the fault models which are checked against system input and output measurements. Introduction Model based reasoning relies on the soundness of the models supporting the reasoning. This is particularly true for model based fault detection and diagnosis. Nevertheless building models turns out to be an awkward task. At some stage of the process, one may face two kinds of uncertainties. On one side, unstructured uncertainties mean that deriving a complete equational model from the physical phenomena is impossible. On the other side, when the structure of the equations is known but some of the parameters are not, uncertainties are said to be structured. In addition to these uncertainties, it is not always possible to get informations about disturbances and noises acting on the system. In such cases, assuming bounded uncertainties may be a solution. Considering structured uncertainties, an interesting way to go is then to use guaranteed estimation methods, which learn the state and/or parameters of the models from data. These methods rely on interval analysis that first appeared in (Moore 1966). They are now subject of a growing interest in various communities and are applied for many tasks (Alamo, Bravo, & Camacho 2005; Armengol et al. 2001; Guerra, Puig, & Ingimundarson 2006; Jaulin et al. 2001; Kieffer & Walter 1998; Kieffer, Jaulin, & Walter 2002; Lesecq, Barraud, & Dinh 2003; Ribot 2006; Ribot, Jauberthie, & Travé-Massuyès 2007). Copyright © 2008, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. This paper presents a fault detection method using interval parameter estimation. Parameters of the model are estimated from the input and output measurements of the system. The consistency of this estimation is then checked against parameters computed from a theoretical (possibly faulty) model of the system. Computations use the set inversion algorithm SIVIA (Jaulin & Walter 1993; Jaulin et al. 2001). The results are approximated but are bounded in a guaranteed way. The method is applied to detect Oscillatory Failure Cases (OFC) in Electrical Flight Control System (EFCS) of civil airplanes. The article is organised as follows. Next section positions interval models with respect to qualitative models. Then second section provides an overview of interval analysis, its original purpose and its use for fault detection. The error bounded context is then presented more precisely with parametric estimation using intervals in the fourth section. In fifth section, the case study is presented: we describe what are OFC, and their consequences on the aircraft control surfaces, why such failures must be detected in time and one of the methods currently used on Airbus aircrafts for OFC detection. In sixth section the application and the obtained results are analyzed. Finally some conclusions are outlined in last section. Qualitative versus interval models Providing models representing physical systems is a common concern spread over all scientific and engineering communities. Modelling depends on the available knowledge about the physical system. This is why pure numerical models are sometimes disregarded to the benefit of qualitative models which naturally cope with uncertain and inaccurate knowledge. Within the qualitative framework, numerical values are replaced by qualitative values that can be seen as (absolute) orders of magnitude1. Absolute orders of magnitude are based on partitioning the real line R into a finite set of basic qualitative valRelative orders of magnitude refer to different formalisms based on binary relations used to compare quantities (Dague 1993a; 1993b; Travé-Massuyès et al. 2005). ues. Considering the order relation given by set inclusion, it allows one to build the whole set of qualitative values, organised along to a high semi-lattice (Travé-Massuyès & Piera 1989; Travé-Massuyès, Ironi, & Dague 2003). As an example, (De Kleer & Brown 1984; Forbus 1984; Kuipers 1984) introduced sign algebra for which a parameter or a variable x takes values in {−, 0,+, ?} depending on whether it is negative, zero, positive, or undetermined. Unfortunately, many operations, e.g. (+) − (+), lead to an undetermined result. Absolute order of magnitude algebras were proposed to hinder this problem (Travé-Massuyès, Ironi, & Dague 2003). The real line partitioning defines the quantity space of a variable thanks to landmark values (Kuipers 1994). It captures the intuition that there are only a few qualitative important values associated to different qualitative behaviors. Whatever partitioning is chosen, an algebra and arithmetical operations can be defined. The interval algebra can be seen as an extreme case in which the partition elements are provided by every real number and intervals are closed and connected subsets of R. Interval analysis may then be interpreted as a specific case of order of magnitude reasoning.