This paper proposes a robust fault detection observer (RFDO) for a continuous non-linear system with polynomial non-linearities up to any finite degree. The observer error and residual are robust with respect to unknown inputs. Sufficient conditions are given, in an efficient algebraic compact form, for a RFDO( observer and residual) to exist. Useable results on detectablility conditions are proved. Numerical prodedures for the design of a RFDO and a sub-optimal RFDO (SORFDO) are given. The theory is illustrated with an application to a ( benchmark) three-tank system. One linear and three non-linear models are considered with corresponding RDFOs ( including a sub-optimal RFDO), and the performances of the corresponding residuals are compared under two initial conditions for a scenario consisting of a leak ( fault) and a disturbance.