This paper proposes a robust fault detection framework for micro thrusters of drag-free satellites. Due to the small magnitude of fault signals and their susceptibility to interference and noise, the method is designed to have robust residual signals against disturbances and noise, while remaining sensitive to fault signals in order to achieve separation. This results in a framework where the robust fault detection problem is transformed into an observer design problem. By employing factorization methods, a unified solution to the multi-objective optimization problem can be obtained, leading to the optimal observer design and thus the optimal residual generation. The method has the advantages of low computational complexity and providing analytical results, rendering it exceptionally well-suited for on-board fault detection within space applications. The simulation results demonstrate the efficacy of the suggested approach when implemented in a drag-free spacecraft.