This paper presents a design of a sliding mode observer (SMO) combined with a Luenberger observer so that the fault-detection residual remains sensitive to incipient faults while being robust to disturbances. In contrast with the conventional design of the SMO, the design presented in this paper is based on transformation of the system into two subsystems. One of these subsystems is free from disturbances and/or modeling errors: for this subsystem, a Luenberger observer is constructed; the other subsystem is subject to the disturbances: an SMO is designed for this subsystem so that the effect of the disturbances is eliminated. Together, the entire system achieves both robustness to disturbances and sensitivity to incipient faults. Stability and convergence of the proposed SMO and Lueneberger observer are proved, and an aircraft example is employed to illustrate the effectiveness of the proposed design approach.