Structural equivalence between direct residuals based on parity space and indirect residuals based on unknown input observers

In order to solve the fault detection and isolation problem, a diagnostic procedure is used. This procedure is composed of two steps: residuals generation followed by their evaluation within decision functions. Many ways have been developed to generate residuals. Among them, we quote the well known parity space approach and the observer based approach. These methods are known to produce structural equivalent residuals. As a consequence, one wonders which method to use in order to design a more robust on-line detection system. Besides, unknown input observers are known to be helpful, within Generalised Observer Schemes, to improve robustness with respect to systems uncertainties. Then, it is important to know if this equivalence is still true for residuals based on unknown input observers. So the goal of this paper is to give theorems so that answers to the previous interrogations could be given and discussed.