This paper presents a robust FDI scheme using a sliding mode observer based on an LPV system, with fault reconstruction capability. Both actuator and sensor fault reconstruction schemes are considered which possess robustness against a certain class of uncertainty and corrupted measurements. For actuator fault reconstruction, the input distribution matrix (associated with the actuators being monitored) is factorized into fixed and varying components. LMIs are used to design the key observer parameters in order to minimize the effect of uncertainty and measurement corruption on the fault reconstruction signal. The faults are reconstructed using the output error injection signal associated with the nonlinear term of the sliding mode observer. For sensor fault reconstruction, the idea is to re-formulate the problem into an actuator fault reconstruction scenario so that the same design procedure can be applied. This is achieved by augmenting the original system with the filtered sensors being monitored. Simulations using a full nonlinear model of a large transport aircraft are presented, and show good fault reconstruction performance.