Contents I. Introduction
Model-based fault detection methods are based on the use of the mathematical models of the monitored system to exploit analytical redundancy. Many model-based fault detection techniques, mostly based on linear models, have been investigated and developed in the literature over the last few years. The use of FDI linear-based methods has been extended to non-linear systems using linearization around an operating point[1]. However, for systems with high nonlinearity and a wide operating range, the linearized approach fails to give satisfactory results. To tackle this problem new fault detection methods based on non-linear models have been developed to address this problem. Methods range from the direct use of non-linear models to the use of neural networks, TS fuzzy systems and neuro-fuzzy systems [1]. Alternatively, LPV models (coming from the gain-scheduling control of non-linear systems) have been recently attracted the attention of the Fault Detection and Isolation (FDI) research community. Such models can be used efficiently to represent some nonlinear systems (see [2], [3]). This has motivated some researchers from the FDI community to develop model-based methods using LPV models (see [4], [5], among others). But even with the use of Linear Parameter Varying (LPV) models, modeling errors and disturbances are inevitable in complex engineering systems. So in order to increase reliability and performance of model-based fault detection the development of robust fault detection algorithms should be addressed. The robustness of a fault detection system means that it must be only sensitive to faults, even in the presence of model-reality differences [1]. One of the approaches to robustness, known as active, is based on generating residuals which are insensitive to uncertainty, while at the same time sensitive to faults. Another approach to solve this problem, known as passive, is based on enhancing the robustness of the fault detection system at the decision-making stage. The aim with the passive approach is usually to determine, given a set of models, if there is any member in the set that can explain the measurements. A common approach to this problem is to propagate the model uncertainty to the alarm limits of the residuals. When the residuals are outside of the alarm limits it is argued that model uncertainty alone can not explain the residual and therefore a fault must have occurred. This approach has the drawback that faults that produce a residual deviation smaller than the residual uncertainty due to parameter uncertainty will not be detected. Another approach to the passive robust fault detection problem is to explicitly calculate the set of states that are consistent with the measurements. When a measurement is found to be inconsistent with this set, a fault is assumed to have occurred. As an exact representation of the set of states consistent with the measurements is hard to calculate, approximating sets that provide outer bounds are often used instead. In the literature several approximating sets to enclose the set of possible states has been proposed. In [6], a state estimator based on enclosing the set of states by the smallest ellipsoid is proposed following the algorithms proposed by [7]. However, in this approach only additive uncertainty is considered, but not the multiplicative uncertainty introduced by modeling uncertainty located in the parameters. In this paper, both types of uncertainties are considered as in [8], but there only system trajectories obtained from the uncertain parameter interval vertices are considered, assuming that the monotonicity property holds.
