Explicit Formulas for the Equivalence Between Fault Detection Filters and Observers

This article establishes the explicit formulas for the equivalence between fault detection filters and observers. The method relies on solving a Sylvester equation, the Schur decomposition, and algebraic manipulations. The equivalence being demonstrated valid from a generic point of view, it is shown that
$H_\infty,H_-,H_{2},\text{ and }H_{2g}$
robust residual generators and
$H_\infty /\mu$
fault estimators are particular cases. A numerical example and a realistic satellite's example illustrate the equivalence.