This study is about fault detection observer design in finite‐frequency domain for Lipschitz non‐linear systems with actuator fault. To make the generated residual sensitive to fault while robust against disturbance, the authors derive design conditions for the fault detection observer to achieve fault sensitivity performance in finite‐frequency domain and disturbance attenuation performance. The authors further transform the design conditions as a set of linear matrix inequalities to facilitate the observer design. Besides, some extra design degrees of freedom parameters are provided in the proposed method to further reduce some conservatism. Numerical simulations of a single‐link flexible joint robot are conducted to verify the effectiveness of the proposed approach.