This paper presents a fault detection and isolation (FDI) method applied to a wind turbine system. The approach utilizes a nonlinear sliding mode observer (SMO) to effectively reconstruct faults in both the hydraulic pitch actuator and generator torque actuator of the wind turbine. A Linear Matrix Inequality (LMI) optimization approach is employed for the design. The blade pitch angle and generator torque in the wind turbine have significantly different orders of magnitudes, rendering them vulnerable to faults of different magnitudes. This discrepancy poses a challenge for the simultaneous reconstruction of faults. To resolve this challenge, a modification is made to the observer. To examine the effectiveness of the modified SMO, two fault scenarios were considered for the hydraulic pitch actuator and generator torque actuator. In the first case, faults are introduced separately, while in the second case, faults occur simultaneously. Simulation results demonstrate accurate detection, isolation, and reconstruction of these faults, whether in the case of separate or simultaneous fault occurrences.